Datasets:
AlgorithmicResearchGroup
/
arxiv_s2orc_parsed

Dataset card Data Studio Files Community
1
title
listlengths
0
18
author
listlengths
0
4.41k
authoraffiliation
listlengths
0
6.45k
venue
listlengths
0
9
abstract
stringlengths
1
37.6k
doi
stringlengths
10
114
⌀
pdfurls
listlengths
1
3
⌀
corpusid
int64
158
259M
arxivid
stringlengths
9
16
pdfsha
stringlengths
40
40
text
stringlengths
66
715k
github_urls
listlengths
0
36
[ "A Quantum Internet Architecture", "A Quantum Internet Architecture", "A Quantum Internet Architecture", "A Quantum Internet Architecture" ]
[ "Rodney Van Meter \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Ryosuke Satoh \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Naphan Benchasattabuse \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Takaaki Matsuo \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Michal Hajdušek \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Takahiko Satoh \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Shota Nagayama \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Shigeya Suzuki \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Rodney Van Meter \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Ryosuke Satoh \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Naphan Benchasattabuse \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Takaaki Matsuo \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Michal Hajdušek \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Takahiko Satoh \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Shota Nagayama \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n", "Shigeya Suzuki \nWIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n\n" ]
[ "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n", "WIDE Project\nInc\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\nKeio University\n" ]
[]
Entangled quantum communication is advancing rapidly, with laboratory and metropolitan testbeds under development, but to date there is no unifying Quantum Internet architecture. We propose a Quantum Internet architecture centered around the Quantum Recursive Network Architecture (QRNA), using RuleSet-based connections established using a two-pass connection setup. Scalability and internetworking (for both technological and administrative boundaries) are achieved using recursion in naming and connection control. In the near term, this architecture will support end-to-end, two-party entanglement on minimal hardware, and it will extend smoothly to multi-party entanglement and the use of quantum error correction on advanced hardware in the future. For a network internal gateway protocol, we recommend (but do not require) qDijkstra with seconds per Bell pair as link cost for routing; the external gateway protocol is designed to build recursively. The strength of our architecture is shown by assessing extensibility and demonstrating how robust protocol operation can be confirmed using the RuleSet paradigm. arXiv:2112.07092v1 [quant-ph]
10.1109/qce53715.2022.00055
[ "https://export.arxiv.org/pdf/2112.07092v1.pdf" ]
245,130,959
2112.07092
3937195e904d9bf2e75ab0630cf524368045d56d
A Quantum Internet Architecture Rodney Van Meter WIDE Project Inc Keio University Keio University Keio University Keio University Keio University Keio University Ryosuke Satoh WIDE Project Inc Keio University Keio University Keio University Keio University Keio University Keio University Naphan Benchasattabuse WIDE Project Inc Keio University Keio University Keio University Keio University Keio University Keio University Takaaki Matsuo WIDE Project Inc Keio University Keio University Keio University Keio University Keio University Keio University Michal Hajdušek WIDE Project Inc Keio University Keio University Keio University Keio University Keio University Keio University Takahiko Satoh WIDE Project Inc Keio University Keio University Keio University Keio University Keio University Keio University Shota Nagayama WIDE Project Inc Keio University Keio University Keio University Keio University Keio University Keio University Shigeya Suzuki WIDE Project Inc Keio University Keio University Keio University Keio University Keio University Keio University A Quantum Internet Architecture Entangled quantum communication is advancing rapidly, with laboratory and metropolitan testbeds under development, but to date there is no unifying Quantum Internet architecture. We propose a Quantum Internet architecture centered around the Quantum Recursive Network Architecture (QRNA), using RuleSet-based connections established using a two-pass connection setup. Scalability and internetworking (for both technological and administrative boundaries) are achieved using recursion in naming and connection control. In the near term, this architecture will support end-to-end, two-party entanglement on minimal hardware, and it will extend smoothly to multi-party entanglement and the use of quantum error correction on advanced hardware in the future. For a network internal gateway protocol, we recommend (but do not require) qDijkstra with seconds per Bell pair as link cost for routing; the external gateway protocol is designed to build recursively. The strength of our architecture is shown by assessing extensibility and demonstrating how robust protocol operation can be confirmed using the RuleSet paradigm. arXiv:2112.07092v1 [quant-ph] Introduction The coming Quantum Internet will provide new encryption services, enhance the sensitivity of sensor networks, and couple distant quantum computers to enhance secure computation, share quantum data and increase the size of problems that can be attacked [50,53,81,89]. Hardware components are in rapid development [5]. Numerous architecture and protocol factors have also been investigated, but not yet brought together into a coherent architecture [4,22,52,65,85,86]. And yet, our decades of experience with the classical Internet clearly show that architecture and hardware must develop in tandem, and that of the two architecture matures more slowly. Thus, it is imperative to begin laying the foundation for an architecture, driving development of hardware and learning from proposed applications as we go. It is important to recognize that there will be an internetwork, a network of networks [80]. Without a doubt there will be more than one network architecture; but to build a true Quantum Internet there will ultimately have to be only a single internetwork architecture. Quantum Communication is Different We can summarize quantum communication as follows: nonlocality is the goal, teleportation is the heart, decoherence is the reality, and the speed of light is still the constraint. Quantum entanglement arises from quantum nonlocality, a phenomenon in which distant systems obeying quantum mechanics share a state, allowing them to demonstrate correlations as if they are in direct, seemingly instantaneous communication. Entangled states can be either bipartite or multipartite. Teleportation is currently the heart of quantum networking [11], as it is the primary method of transferring quantum information encoded in physical quantum states. In quantum teleportation, the state of a quantum variable is destroyed in one place and reconstructed in another. Teleportation from network node A to node B consumes a special entangled state spanning A and B, known as a Bell pair; hence, the task of a quantum network is to continually produce enough end-toend entanglement to satisfy applications. Moreover, a form of teleportation known as entanglement swapping is used to stretch link-level entanglement into end-to-end entanglement. Other types of quantum networking, e.g., involving superposition but not teleportation, appear to be limited to single-hop configurations and are thus not considered further here. Unfortunately, quantum data is exceedingly fragile. Photons get lost, so generally speaking we must use acknowledged link layers (though there are exceptions), dramatically affecting throughput. Errors in quantum states caused by noise, imperfect control of memories, etc. are collectively called decoherence. One measure of decoherence suffered is fidelity, an estimate of the closeness between the actuallyachieved and desired quantum states. Finally, although entanglement shows nonlocal correlations, it cannot be used to communicate faster than the speed of light. Essentially, all quantum communications require supporting classical communication, which is naturally limited to c. Measurement outcomes on entangled qubits are (anti)correlated and at a first glance may appear to violate special relativity. However, the measurement collapse is random and cannot be controlled, making faster-than-light communication impossible. All quantum communication relies on a classical communication infrastructure to enable control and coordination. This classical infrastructure is a distinct communication system that operates at the application layer, similar to how some routing protocols run as an application to manage router forwarding tables. This classical network need not share paths or topology with the quantum network it manages, but necessarily interconnects every controllable quantum network component, whether quantum (e.g., teleportation repeater) or classical (e.g., optical switch). To read this paper, readers need only the notions above, along with the general idea that we are working with qubits, quantum binary digits that can be entangled with each other and follow a few simple rules [25]. Qubits can be encoded into photons (using a variety of encoding methods) or stored in stationary memories (implementable in many different physical systems). For a brief summary of quantum information concepts and both popular and technical references, see Appendix A. Quantum Communication is Desirable The unusual characteristics just described would be little more than a curiosity (or a physics experiment) without compelling reasons to integrate quantum communications into our existing IT ecosystem to provide new or better services. We can divide applications into three main, overlapping areas: cryptographic services, sensor networks, and distributed quantum computation [21,71,81,88]. The best-known quantum cryptographic service is quantum key distribution (QKD), in which quantum characteristics are used to assess the probability of the presence of an eavesdropper as a stream of shared, random bits is created 1 . These random, shared, believed-to-be-secret [29,69,91] bits can be used in key cryptographic protocols [2,30,60]. However, this is not the only cryptographic service that is possible; secret sharing [20,41,48,55], secure election protocols [78], and byzantine agreement protocols [9,77] are all known. The second category, sensors, encompasses a range of uses. Arguably, QKD itself is a sensor application, as it physically detects the presence or absence of an eavesdropper. Other 1 Roughly speaking, QKD can be done using single photons [10,67,91] or E2E entangled states [12,28]. Single-photon demonstration networks have existed since the early 2000s [30], but without the ability to store and manipulate states mid-path, they are single-purpose networks and do not provide E2E security; instead, they depend on classical relays with only hop-by-hop security. Here, we focus on more general, entanglement-based systems. uses include enhanced interferometry for telescopes [35,49] and higher-precision clock synchronization [44,51], both of which can be viewed as using entanglement as a form of reference frame for time and space [8,45,56,66,73]. Challenges include determining whether the required precision for classical control of the quantum elements exceeds the gains from the use of entanglement in practice, and the extremely high data rates (entanglement generation rates) required. The final area is distributed quantum computation [15,21,72,88], where individual quantum processors are networked together, communicating and sharing their resources to carry out quantum information processing tasks in a coordinated way. Extension of the paradigm of delegated quantum computation leads to applications such as blind quantum computation [14,32], where a client is able to delegate her computation to a quantum server without revealing information about its input, the computation itself or its output. Quantum Repeaters Quantum repeaters are very different from classical signal repeaters; quantum states cannot be amplified or simply regenerated 2 , and as a general rule cannot be faithfully copied. Instead, the work of the network is to perform a distributed computation that builds the end-to-end entanglement that applications consume. Repeaters and routers serve as waypoints in that E2E problem, and perform four main tasks: 1. Creating base entanglement: Typically using single photons (though there are exceptions to this rule [23]), neighboring repeaters entangle stationary memory qubits. The most common outcome of this process is a Bell pair. A number of different link architectures can be used to achieve this task [47]. 2. Entanglement extension: Achieved via entanglement swapping [43] shown in Fig. 1(a), two entangled Bell pairs, A ↔ B and B ↔ C can be spliced to form a single A ↔ C Bell pair. Classical communication is required. 3. Error management: Loss of photons is handled using acknowledged link layers, but state errors and operation (gate) errors must be addressed as well; purification is a form of error detection, shown in Fig. 1 Figure 1: Quantum repeaters build end-to-end distributed entanglement for use by applications at end nodes. In the basic form shown in (a), that process is a distributed computation, depending on entanglement swapping (ES) to lengthen entanglement to span multiple hops and a form of error detection, shown in (b), known as purification, where multiple low quality Bell pairs can be winnowed down to a single pair of higher quality through a testing protocol that consumes some pairs. of the quantum network, both of which operate over a classical network that interconnects quantum devices at the classical application layer. This is the focus of this paper. The most commonly discussed architecture uses purification and entanglement swapping; unless otherwise stated, in this paper we are discussing these first generation, or 1G, networks. Purification requires bilateral confirmation of a qubit measurement result; on even parity, the entangled state is kept and proceeds, while on odd parity the state must be discarded. Entanglement swapping transfers entanglement from one node to another, which requires communicating with two nodes, one of which may be required to adjust its state using information known as a Pauli frame correction. Coordination of these operations in a robust but maximally asynchronous fashion is one of the primary tasks of the network protocol. Architecture Decision Points In developing a Quantum Internet architecture, our goals are similar to those of the classical Internet: we want a system that is robust in operation; easy to implement; and meets requirements such as scalability, security, manageability, and autonomy. Good definitions of interfaces will allow subsystems and hardware implementations to evolve independently and systems will continue to interoperate over time spans of (human) generations. Because we are designing an internetwork, our goal is to create a homogeneous service over heterogeneous subpaths, however, this must be balanced against the fact that early hardware generations will have substantial differences in capabilities. A number of key design decisions must be made: Figure 2: Present-day quantum repeaters [68] represent the absolute minimal form of hardware: a single transceiver qubit (e − ), a single buffer memory qubit (atom symbol ), a twoport optical switch in front, and the ability to initialize, store, manipulate and measure the qubits. This repeater can only attempt to build entanglement to either the left or the right in a given cycle; e.g., after succeeding in making entanglement to the left (Step 1), then the transceiver qubit's state is transferred to to the buffer qubit (Step 2), and entanglement to the right is attempted (Step 3). Once entanglement to the right is achieved, entanglement swapping is performed via a Bell state measurement (joint measurement) of the two qubits (Step 4). This is followed by classical communication with the neighbors (Step 5, not shown). Figure 3: A full quantum router with hardware architecture similar to today's commercial Internet routers will have QNICs (line cards) coupled via a backplane consisting of optical ports, an optical switch, and Bell State Analyzer measurement devices. Using the BSAs, the qubits in the backplane buffers at the top of the line cards are entangled while the transceiver qubits in the lower portion attempt to create entanglement with neighboring nodes. Once both backplane and neighbor entangled states are made, entanglement swapping is used within each line card to splice the long-distance entanglement. A number of steps in hardware complexity (and cost) will exist between the minimal configuration of Fig. 2 The above list is by no means exhaustive but covers the critical points. For a more complete list, the interested reader can turn to the QIRG Internet Draft [53]. After proposing answers to these questions in the next several sections, we provide some evidence for the correctness of our choices (Sec. 6) before concluding (Sec. 7). Quantum Network Services Semantics: Bell Pairs and On-Path Distributed Computation Entanglement is the resource that will fuel quantum applications such as QKD, teleportation, quantum sensing, or delegated quantum computation. Continuous, reliable and efficient replenishment of this resource is one of the primary tasks of a quantum network. However, entangled states come in many shapes and sizes [36,42]. Bell pairs are the most basic bipartite states and form the fundamental building blocks of entangled quantum networks. They can be generated by a network link equipped with stationary quantum memories at each end that are entangled via flying photons. Due to the Bell pairs' importance to virtually all quantum communication protocols it is generally agreed that they will be part of the fundamental network service. End-to-end multipartite states such as GHZ, W and graph states [39] are resources for a variety of multiparty protocols, and therefore are likely to be extremely valuable. Bell pairs alone would be sufficient; multipartite states can be built using them, but because the efficiency will matter, it is an open question whether multipartite states are part of the fundamental service or should be created and managed entirely by applications running at end nodes. For this reason, we focus on distribution of Bell pairs in this manuscript. This distribution can be achieved in a number of ways and depending on the nature of the connection networks are classified into three generations [63]. 1G quantum networks build E2E entangled pairs using physical Bell pairs, spliced and error tested using entanglement swapping and purification. Such connections are the most basic way of establishing E2E entanglement and therefore the first implementations of quantum networks are expected to be 1G. 2G and 3G networks are designed to reach and maintain higher fidelities (especially useful for distributed computation) by utilizing quantum error correction, placing very demanding requirements on the hardware. These error management schemes require work at each repeater, but with the goal of E2E error management [64]. All of the above generations of quantum networks rely on quantum memories to store the qubits while the networks entangle them using flying photons. It is prudent to mention that all-photonic quantum repeaters have been proposed that do not require quantum memories [6,40]. In Sec. 1.3, we noted that this distribution of Bell pairs is, in fact, a limited form of distributed computation all along the path. The semantics of the network service must be defined to take this into account. Finally, it is important to note that time is part of the service [53]. For sensor applications in particular, very high precision timestamps of some events are necessary information, and must be provided to applications. Application Access: Quantum Sockets Once entangled qubits are ready, an application consumes them. As mentioned earlier, there are three types of currently envisioned applications. It is possible to categorize the three into two types: those that use qubits in larger quantum applications and those that measure the qubits to produce classical information right away. Applications that consume qubits directly will measure the result immediately after the execution of the application; thus, eventually, both cases have a classical result. We are designing the Quantum Socket API (API from now on) in an analogy with the classical socket API widely used in the classical Internet. The API is very similar to the classical socket API. Like the classical socket API, the API has functionalities such as: creating a socket, connecting to the socket endpoint, reading, writing, setting options, and destroying the socket. The API is node-type agnostic; i.e., it can handle three end-node types (MEAS, COMP, SNSR described in Sec. 4.1) corresponding to three different application classes. Operations on nodes of the type that return classical information (MEAS, SNSR) as described earlier are synchronous since the result will eventually become classical reads (read system calls, etc.). For these applications, the stochastic arrival time of completed Bell pairs is not a problem. In contrast, COMP nodes involve substantial coordination with other work at a quantum computer. How to build distributed quantum programs that deal robustly with stochastic entanglement delivery, e.g. via asynchronous callbacks, is an open problem. Both applications or controlling programs can configure specific parameters for each physical interface, such as the RuleSet specific to the QNIC, via the ioctl-like interface. In other words, classical components communicate with physical quantum components via the socket API. Expressing Connection Semantics: RuleSets Having just established that the core service of a quantum internet is building E2E entanglement, now we need an internetwork protocol capable of communicating the actions necessary to span different connection architectures (1G, 2G). Here, we describe a mechanism efficient enough for use as the basis of a network protocol, and rich and abstract enough for use as an internetwork protocol (Sec. 5). 1G networks need a mechanism for conveying requests such as, "Bob, once you get a Bell pair with Alice and a Bell pair with Charlie, execute entanglement swapping, then send the Pauli frame correction to Charlie and a notice-ofentanglement-transfer to Alice," and "If you have two Bell pairs with Alice, both with fidelity less than 0.9, then purify." 2G networks will work on logical qubits encoded using quantum error correction, making for complex operations for entanglement swapping and error correction while presenting high-fidelity logical qubits to applications. Our approach is to define Rules that have a condition clause and an action clause, very analogous to the Open-Flow extensions of classical software defined networking (SDN) [59]. For a connection, each node is given a Rule-Set that should comprehensively define what to do as local events occur (entanglement success, timeout, etc.) and as messages arrive [57,58]. This RuleSet-based operation is the heart of our work, and allows for explicit reasoning about how to achieve the maximum asynchrony and autonomy in the network (rather than waiting for explicit instructions at every operation or attempting to make everything proceed in lockstep). Our version of RuleSets includes local state, which cannot be expressed in SDN OpenFlow. There is one RuleSet for each connection. Once a resource (e.g., link level Bell pair) is assigned to a RuleSet, that assignment does not change. How that assignment is done is the responsibility of the multiplexing scheme (Sec. 4.3). RuleSets and any qubits at the nodes that are currently assigned to a particular connection are connection state that must be held at each repeater/router. The scalability of this needs to be assessed, and it affects AAA (Sec. 4.5), but we currently see no approach to quantum networking that allows mid-path routers and repeaters to be fully stateless. We have adapted the RuleSet approach from [57,58] and incorporated an additional construct called a Stage. A RuleSet can be thought of as a program that oversees the processing of the states. Entangled states are allocated to a given Stage. In each Stage, there can be multiple Rules. Each Stage can also have its own variables which are shared by Rules in the same Stage. After one of the Rules in the Stage fires, the entangled state is either promoted to the next Stage or declared defunct and the physical resources are returned to the pool available for reuse. This ensures that the flow of qubits is unidirectional and terminates with being either delivered to an application or service, or discarded (either consumed as part of the protocol operation or determined to likely be in error). Naming States (Qubits) Managing the qubits and agreeing on the consumption of entangled states among shared nodes are two of the most critical tasks for RuleSets. In the IP architecture, network addresses are associated with a network interface; here, we assume the same. Thus, a physical qubit can be uniquely identified by its network address and the index of the qubit within the QNIC, using the tuple <QNICAddress,QubitIndex>. QNIC firmware applies quantum operations based on that index, and the tuple is unique within the scope of the network address. However, rather than this physical address, we are usually interested in the state (e.g., half of a Bell pair) that is held in the qubit, which is dynamic and has a finite lifetime; the distinction is philosophically similar to a register versus a temporary variable. Therefore, when nodes that share entangled resources want to communicate changes to other parties, they use another (external) name which is only known by the shared parties. This name needs to be unique. Initially, the name is determined by one of the nodes involved in the creation of the link-level Bell pair (e.g., the BSA node described in the next section). The name might be, for example, the tuple <NodeAddress,Timestamp>, where the timestamp is of high enough precision that at most one Bell pair may have been created. The mapping of that external name to internal qubit address is maintained independently and privately by each node. When entanglement swapping is completed, a new name for the Bell pair is created by the node performing the swapping. That name is communicated to the two end points as part of the notification of the entanglement being transferred to new partners. Messages Tab. 1 lists the primary messages included in our protocol. (Naturally, each message transmission is initiated by an Action clause, and its reception matches a Condition clause; in the interest of space these are not included in Tabs. 2 and 3.) Purification involves testing the parity of two qubits at each end and exchanging the results using a measurement outcome (MEAS) message. Each end compares the parity it calculates to the parity it receives, and either discards both Bell pairs (on mismatch) or raises the software's estimated fidelity of one and discards the other (on match). Entanglement swapping requires that both ends be notified of the transfer of entanglement to new partners, and one end must also receive a Pauli frame update. Condition Clauses Tab. 2 shows the Condition Clauses that can be defined in Rules. The Condition Clauses can be thought of as defining the trigger for moving from one state to another in a state machine, while the Action Clause for the Rule defines the side effects. Sometimes, a Condition Clause needs to match only one entangled state, for example when matching a Bell pair and deciding to deliver it to an application (in which case it passes out of our ken). More often, it needs to match two: either with Action Clauses Tab. 3 shows the Action Clauses that can be defined in Rules. The Action Clauses can be thought of as defining sequences of local quantum operations and messages to be sent. The actions are chosen from a restricted set of options and do not include loop primitives; despite the existence of QCIRC which applies a quantum circuit, this is not a Turing complete computation platform. Conditional execution is done by creating separate rules with distinct Condition clauses. These restrictions make it easier to reason about distributed protocol actions in terms of termination, robustness, deadlock, security, and other issues. As noted above, message generation is not included in this table but is a natural consequence of QCIRC, MEAS and some of the local software actions. Two-Pass Connection Setup Our approach to connection setup uses two passes, as proposed by Van Meter and Matsuo [85]. On the outbound leg (starting at the Initiator), information about links and available resources is collected. The connection request eventually reaches the Responder, which takes that information and builds RuleSets for every node along the path. Those Rule-Sets are distributed in a return pass, then the operation for the connection begins. Setup within a single network is illustrated in Fig. 4. As in the classical Internet, we expect that the majority of connections will be initiated by a client node reaching out to a server. This architecture places the server in charge of RuleSet creation, allowing service providers a single point of innovation; if they create better RuleSets than their competitors, then connections will be faster or more robust, providing a competitive advantage. Matches Bell pairs. Used commonly for purification and entanglement swapping. Used to check fidelity of Bell pairs, this also serves as the primary "meets application requirements" clause for delivering to apps at EndNodes. Networking The previous section discussed individual connections in the abstract; here we show how to operate in complex topologies with complex traffic patterns and actors whose interests aren't always perfectly aligned. Quantum Network Components Quantum networks are distinct from their classical counterparts because they cannot exist in isolation; quantum networks incorporate and rely on classical networks to interconnect their components to enable classical control. So despite the name, a quantum network is really a hybrid of a quantum and a classical network. Just as today's classical Internet consists of Ethernet switches, IP routers of varying capabilities, home routers, WLAN access points, and terminals of various types, nodes comprising the Quantum Internet will come in a variety of flavors. All of the node types below can be implemented in numerous technologies (NV diamond, ion traps, superconducting, quantum dot) [54], using a variety of optical qubit representations (polarization, time bin, spatial path, energy/wavelength, etc.). We divide these into three categories: end nodes, repeater nodes, and support nodes. End nodes represent hosts that wish to execute a quantum application such as quantum key distribution, secret sharing and blind quantum computation. The technological maturity required of an end node heavily depends on the desired application. There are three major kinds of end nodes: MEAS A node that can only measure received photons (in at least two different ways) and does not store qubits is actually surprisingly useful. A pair of such nodes can conduct quantum key distribution, or a single node of this type can serve as a terminal connecting to a full COMP node in order to execute one form of secure blind quantum computation [62]. However, its error management capabilities are very limited. COMP Computational end node capable of measuring quantum states as well as storing them in a quantum memory. This greatly enhances the nodes functionality and leads to advanced applications such as blind quantum computation [14,38]. This node may vary in its processing abilities. Simple clients may be only able to generate, store and manipulate single-qubit states while advanced quantum servers may be able to create large multi-qubit entangled states and hence be capable of universal fault-tolerant quantum computation. SNSR A sensor node uses the entangled states in a cyber-physical operation, e.g. as a reference frame for interferometry or clock synchronization. For these nodes in particular, recall that time is part of the service. Quantum repeaters are responsible for distribution and management of entanglement across the quantum network. We have three kinds of repeater nodes: REP1 A 1G repeater. Always has two interfaces; a recent experiment ( Fig. 2 and [68]) allows only one to be active at a time, but the generalized form allows both to be active simultaneously. Its primary task is to perform entanglement swapping and error management in the form of purification on physical qubits. REP2 A 2G repeater. Has the same primary task of entanglement swapping as REP1 but operates at the level of encoded logical qubits composed of multiple physical qubits. Error management is achieved via error correction, signified by the check mark in the REP2 icon. REP2 must be equipped with hardware capable of handling a large number of physical qubits, which necessitates more advanced computational capabilities. Fig. 3, a router likely consists of multiple line cards and a backplane, but for network architectural purposes, the important fact is that a router runs a full suite of protocols governing network operations. Typically, an RTR will have three or more network interfaces, and is capable of governing a network border, where it may be called upon to speak both 1G and 2G protocols and to rewrite RuleSets, behaving as a Responder for connection requests (outbound or transit). RTR A router. As in Finally, support nodes are tasked with aiding end and repeater nodes in entanglement distribution. There are five kinds of support nodes: EPPS An entangled photon pair source, implemented using e.g. symmetric parametric down conversion (SPDC). An EPPS simply produces pairs of entangled photons, which must be captured or measured at link end points. An EPPS can be used in terrestrial links [47] or on a satellite, with the photons captured by telescopes on the ground [92]. BSA Bell State Analyzer, which projects two photons into one of the Bell states; usually used to swap memory-photon and photon-memory entanglement to memorymemory entanglement. The theoretical efficiency limit with linear optics implementation is 50%. The hardware complexity of the BSA depends on the particular qubit encoding. RGSS Repeater Graph State Source generates entangled multipartite photonic states used in memoryless repeater networks. It sends one half of the generated repeater graph state to its neighboring nodes where the photons are measured. ABSA Advanced Bell State Analyzer. The basic BSA always performs the same operation, but all-optical repeaters based on repeater graph states require two-photon and single-photon measurements. The measurement basis (type of measurement) is selected dynamically based on prior measurement outcomes as well as the logical encoding and structure of the underlying repeater graph state. This makes the hardware, software and protocol implementations much more complex than a BSA. OSW Optical switches (nanomechanical or otherwise) can be incorporated into the above node types, but they can also stand alone in the network, switching photons from link to link without measuring them. This list is by no means exhaustive but covers the main components of a quantum network. The division into end, repeater and support nodes is not mutually exclusive, as there may be some overlap in functionality. For example, the ABSA may be viewed as a type of repeater node as well, as it realizes the task of entanglement swapping. The ABSA requires sophisticated RuleSets and is visible in the connect planning process; the simpler BSA, on the other hand, is tasked only with notifying two nodes about the success of entanglement creation, and need not be visible to nodes farther away in the path. Routing Routing is the process of determining the path of communication between a given set of end nodes. In quantum networks, there are two distinct routes used: one that consists of quan-tum nodes, and a separate set of classical routes between the control mechanisms of each of those quantum devices. Picking a route can be achieved with qDijkstra (quantum Dijkstra's algorithm) [86]. The link cost in this case is defined as "seconds per Bell pair of some index fidelity F". Fidelity is not an easy metric to obtain in practice, and requires constant link monitoring. An expensive but accurate measure is via tomography of the link; lower-cost means of characterizing quantum states is an active area of research [27]. By including fidelity in the link metric, route calculation automatically takes into account the tradeoff between links with high data rate but poor fidelity versus those with low data rate and high fidelity. This approach has yielded good agreement between calculated path cost and throughput obtained via simulation of various paths with heterogeneous links [86]. One of the big open questions that we are investigating is how to combine paths with multiplexing and resource reservation (and starvation), which we take up next. Multiplexing and Resource Reservation Circuit switching, time-division multiplexing, statistical multiplexing (like Internet best-effort forwarding) and buffer space multiplexing are all possible approaches. In buffer space multiplexing, each qubit at each router or repeater node is assigned to one of the specific connections passing through the node, akin to network slicing [7]. Aparicio studied aggregate throughput and fairness for these approaches, and found that statistical multiplexing works pretty well [3,4]. Statmux allows separate regions of the network to work productively at the same time while sharing the bottleneck link, surpassing circuit switching in terms of aggregate throughput. However, those simulations were for small-scale networks. We believe this topic needs to be studied in much more detail to assess robustness in the face of complex, varying traffic patterns. In particular, we fear that something akin to congestion collapse is possible, or that short-distance connections can starve long-distance connections. Multiplexing has to coordinate with routing and with AAA, below. Naturally, we want to avoid a fully blocking multiplexing protocol if possible. Any multiplexing scheme that results in extended occupation of resources requires us to determine how those resources are to be allocated, and such a policy will involve identity and likely some form of payment or at minimum debit against some system credit. Authentication, Authorization and Accounting As just noted, it seems likely that performance well below demand will force early implementations to adopt fixed allocation of resources to individual connections. This, in turn, implies that authentication, authorization and accounting (AAA) will become important elements of the architecture [31]. Layer k-1 Host Router Virtual node Layer k E 1 E 2 R 1 R 2 Figure 5: QRNA uses a fully recursive architecture that can virtualize a network as a node. Note that QRNA can work down to the link layer, or networks can be internally different [52] as long as they participate at the network border. Economics may come to define who has access to the early networks, unless an AAA architecture that explicitly focuses on fairness or some metric other than direct bids for access is put into place. Security Quantum mechanics promises unprecedented levels of confidentiality between communicating parties, which is why quantum key distribution has attracted attention of the theoretical physics and computer science community. However, the focus on QKD also painted a skewed and incomplete picture of security in quantum networks as a whole. This has been slowly changing lately and it has been recognized that while in principle quantum mechanics offers new methods of detecting malicious players in a network, it also enables new vectors of attack [75]. All of the protocols discussed above need authentication and tamper resistance; whether privacy is also required or useful is an open question. Given the previous Internet (and, to a lesser extent, telephone network) experiences with lack of security in routing, accounting, etc., and the likely high cost of quantum connections, it is imperative to have a solid framework in place very early in the Quantum Internet, ideally well before a truly operational network is implemented. This ties into the multiplexing and AAA decisions as outlined above. Internetworking and Scalability: Recursion An idealization of today's Internet is that it is a two-level system. External gateway protocols such BGP are used for routing between networks while internal gateway protocols such as OSPF and IS-IS are responsible for routing within the networks. The reality, however, is not so elegant. Tunneling, switched Ethernets requiring spanning tree protocol underneath even though they are nominally "link-layer", and recent emphasis on virtualization of networks and services [7] has shaped the Internet into a multi-tier system with ad hoc interactions at each level. Given the opportunity to create the system from scratch, and knowing the evolution path that the Internet has taken, we would probably design the Internet in a unified way that naturally takes into account interactions across multiple layers. One such unified approach, known as the Recursive Network Architecture, was proposed by Touch et al. [79]. RNA presents an attractive blueprint for the design of the Quantum Internet, which Van Meter et al. named the Quantum Recursive Network Architecture (QRNA) [87]. This approach is intended to provide scalability to global proportions, including connecting physically and logically heterogeneous networks and providing autonomy, security and privacy. Recursion naturally affects naming (Sec. 3.1) and routing (Sec. 4.2). Recursion describes the hierarchy of names; the relationship among names can be described as a directed acyclic graph. This approach provides scalability in naming and routing, and enhances autonomy, security and privacy. Traditionally, connections may be of two types; boundaryto-boundary for transit and boundary-to-end node for termination. In QRNA, both of these connections are treated as the same thing but at different levels of the network. In Fig. 5, a host node E 1 wishes to establish a end-to-end connection with another host node E 2 at Layer k. From the perspective of Layer k the path to E 2 is straightforward and leads through routers R 1 and R 2 . When the connection request reaches the first router it is embedded and passed to Layer k − 1 by the border router. The border router is then responsible for requesting an end-to-end connection across Layer k − 1 to an appropriate border router that then passes the original connection request up to Layer k. The recursive nature of the architecture allows the connection requests to be embedded into as many levels as is required. Connection Setup: Two-Pass with Rewrite Recursion must work with the two-pass connection setup described in Sec. 3.5. We accomplish this via RuleSet rewriting where crossing recursion layers, such as at network boundaries. Setup in an internetwork is shown in Fig. 6 (compare to Fig. 4). In order to maintain network autonomy and privacy and improve scalability, the border router rewrites the existing set of link information into a single hop, much like a single hop in BGP routing hides network internal topological information for the same purposes. The border router acts as a Responder to the original Initiator, and its estimate of the performance of the path from the Initiator to its location is used to derive the performance characteristics it reports when describing the virtual link at a higher layer of recursion. The arrows indicate how the initial connection request gets passed up/down the Layers and between the networks. Each Layer has its own Initiator (I) and Responder (R), due to the recursive encapsulation of the connection request Layer k − 1 has two pairs I-R. In the Quantum Internet (at least through the first two generations), a connection is a form of distributed computation, with all nodes on the path participating in purification and entanglement swapping. Connections will have to be established in advance along the path, and will be stateful. During connection setup, at every layer, the node is given a Responder (destination) address and can determine the nexthop based on local selection policy. To advance the setup process, the node sends the request to the neighbor (if we have reached the physical link level) or recurses. At layer k, we recurse to layer k − 1 by translating our k address and the k layer nexthop to layer k − 1 addresses, then passing to layer k − 1 with the latter as the new Responder address. Each network constructs provisional RuleSets upon the connection request reaching the corresponding Layer k − 1 Responders (3.99 and 4.1 in Fig. 6). These RuleSets are distributed backwards along the network path (not shown in Fig. 6). Upon acceptance of the connection request by the Server, a reply is sent backwards along the same path confirming the RuleSets. Performing recursion at administrative boundaries has several benefits: a) it limits the amount of information each node has to have on hand about the entire internetwork, enhancing E2E scalability and network autonomy; b) it allows Responders to innovate (within the bounds of the RuleSet architecture); c) it allows us to reason about connections effectively; d) it serves as a convenient point for 1G-2G inter-operation as new technologies are deployed [65]; and e) it facilitates interoperation with different network architectures [52]. Routing, Multiplexing and AAA The core routing problem of selecting a path in Internet-scale systems is solved, as noted above, using two-level or threelevel systems, with the Internet's top level being the global BGP. However, issues of policy, economics and especially of security still exist at the top level [26]. In QRNA, this approach is generalized and extended using full recursion; a routing protocol is required at each layer. At the lowest layer, we follow the qDijkstra link cost metric of seconds per Bell pair at a particular fidelity. Using QRNA's recursion, at the next layer up, the intra-network path will appear as a link. This link will, in turn, have a reportable performance metric of Bell pair creation rate. However, as the intra-network RuleSet can be tuned with different numbers of rounds of purification, that rate can be traded off for higher fidelity. Prior work has shown that performing more purification closer to the link level results in higher end-to-end throughput [84], so we expect the policy to be set such that each network presents a slower but higher quality link. A bigger problem is multiplexing, which as noted requires AAA. Any network will have many connections originating, terminating or transiting. The Internet community unfortunately provides less guidance here; inter-domain QoS mechanisms have been under development since the 1990s but are not widely deployed. Thus, we consider this to be one of the most important open research issues. Evidence In an ideal world, the long-term proof of an architecture would be widespread adoption. In reality, of course, plenty of splendid architectures (processors, OSes, communication systems) have fallen by the wayside for reasons unrelated to technology. Moreover, such a retrospective view does not help us assess a prospective architecture. Here, we discuss how the RuleSet approach leads to robust protocols, and how we are validating our architecture via simulation, documenting the protocols, and working toward real-world implementation. Reasoning with RuleSets A key purpose of the RuleSet architecture is to make it possible to reason rigorously about distributed behavior. At any point in time, we can enumerate the set of possible events at all nodes and ask if execution of specific Action Clauses will result in unwanted operation, such as leapfrogging. In leapfrogging, in a chain of nodes A-B-C-D, if B and C are each tasked with performing entanglement swapping, uncoordinated selection of resource states can result in A-C and B-D entanglement, rather than the desired A-D entanglement. In another example, if A-B entanglement has been achieved and B is waiting on B-C entanglement to perform swapping, a race condition can occur in which A decides to discard the Bell pair (due to memory decoherence incurred during a long wait) just as B receives notification of B-C entanglement and performs the swapping operation. The message from B to A informing A of the swapping event arrives too late, and A has already reinitialized its qubit for reuse. This is especially problematic if C chooses, upon receipt of notification from B, to use the ersatz A-C pair to teleport C's important data to A. Using RuleSet logic, we can detect this potential race condition and define Rules such that A will not discard the Bell pair until after B does, by giving B a discard timer that is more than the one-way messaging latency with A. Simulation, Specification and Implementation To validate our designs, we are implementing a highly scalable simulator called QuISP (Quantum Internet Simulation Package) [74]. 1G networks, entanglement swapping and purification governed by RuleSets, and connection setup are complete (but continue to evolve); rudimentary routing and circuit switched multiplexing are all functional and pass included tests, but remain in active development. All-optical paths are in active development. RuleSets are currently being designed for 2G and multi-party states. The full QRNA protocol set is in design, and the simulator's performance has been measured to scale adequately for hundreds of nodes on a laptop, enough to demonstrate complex, multi-level, recursive internetworking. Any network system, especially one intended to be open, must be supported by specifications for protocols and behavior. The difficulty of writing such documents can be viewed as one piece of evidence about the elegance and simplicity of an architecture. Our simulator work began with a set of design documents, and we have specifications for some of the core protocols now in development. Moreover, we are working closely with the Quantum Internet Task Force (QITF), a quantum Internet testbed initiative that expects to build not only a single network but to actually focus on scalability in network and internetwork architecture. We expect some aspects of the architecture presented here to be adopted directly, while others doubtless will undergo significant evolution as a result of the collaboration. Conclusion Ultimately, our proposed quantum internetwork architecture builds on three critical points: a recursive architecture for internetworking and scalability, RuleSet-based connection operation providing the right vocabulary across disparate hardware, and a two-pass connection setup routine (outbound info collection, inbound RuleSet distribution). This structure will allow for continuing evolution of the internetwork, providing a platform for distributed, independent advances in physical technology and in protocols. Our work is maturing rapidly, with design, specification, and simulation well advanced and real-world implementation in the serious planning stages. In particular, with different groups now involved in detailed discussions, the RuleSet design will be challenged to work in heterogeneous environments, which we expect to further validate the general approach even as it is likely that details will change. Although there is solid work on routing and multiplexing, designing a system that will be robust at scale and that will serve us well for decades is perhaps the area of most concern. With this structure in place, we feel that architecture and protocols are on pace to mature to usable levels alongside hardware, though as noted in the introduction experience shows that architecture matures more slowly. However, we expect to take full advantage of knowledge gained over the last half-century of data networking research and development. This should carry us through evolutionary stages to a full Quantum Internet supporting cryptographic, sensor, and distributed computation applications. A Quantum Concepts There are many good introductions to quantum computing, on the web [34] and in print [76], but for convenience the following is a brief summary of the key aspects of quantum communication and computation that impact network and system architecture. The primary difference between quantum mechanics and classical probability is that quantum mechanics uses probability amplitudes, rather than straight probabilities [1]. Probability amplitudes can be complex numbers; if the amplitude of a given state is α, then the probability of finding that state is |α| 2 . Most of the concepts below derive fairly directly from this fact and the general wave nature of quantum systems. Quantum information is most often discussed in terms of qubits. A qubit, like a classical bit, is something with two possible values that we can label zero and one. Unlike a classical bit, a qubit can occupy both values simultaneously, known as superposition. To understand quantum computation, we need seven basic concepts: Superposition. A qubit can represent multiple values in different proportions at the same time, e.g., two-thirds of a "one" and one-third of a "zero". This superposition determines the relative probability of finding each value when we measure the state. Entanglement (and Bell pairs). Groups of qubits can exhibit strong correlation between the qubits that cannot be explained by independent probabilities for individual qubits. Instead, the group must be considered as a whole, with interdependent probabilities. This phenomenon is known as quantum entanglement. A special entangled state known as a Bell pair or EPR pair, consisting of two quantum bits, figures prominently in quantum communication. Each qubit in the pair has a 50% probability of having a value of 1 and a 50% probability of having a value of 0 when we measure it. Although we cannot predict which will be found, when we measure one member of the pair, the value of the other is immediately determined. This happens independent of the distance between the two members of the Bell pair. Interference. Quantum algorithms use some building blocks derived from classical concepts, such as adder designs, but the overall thrust of a quantum algorithm is very different from that of a classical algorithm. Rather than attempting to solve a problem and checking for the answer, a quantum algorithm's goal is to create interference between the elements of a superposition quantum state. Constructive interference reinforces desirable states, increasing the probability of finding a desirable outcome on measurement, while destructive interference reduces the probability. Unitary, or reversible, gates. Manipulating those probability amplitudes, including creating entanglement and making the interference patterns, involves the use of logical operations known as gates. These gates are similar to Boolean logic, but must be reversible, which in mathematical terms means they are represented by a unitary transformation matrix. Measurement. As described above, when we measure a qubit, we get only a single classical bit of information (the "one" or "zero"), and the superposition collapses. The probability of finding a zero or a one depends on the probability amplitudes. Decoherence. Unfortunately, any physical operation (including simply storing a qubit) gradually degrades the state. Decoherence is the single most important technological fact driving quantum computer and quantum network implementations. We can counter this by using a form of error correction or detection. No cloning. As mentioned above, a key restriction of quantum systems is that we cannot make independent copies of an unknown state [90]. This makes error correction difficult. A few additional concepts will augment understanding quantum networks. Fidelity. The quality of a quantum state is described by its fidelity, which is, roughly, the probability that we correctly understand the state -if we ran the same experiment many times and measured the results, how close to our desired statistics would we be? This is one simple measure of the amount of decoherence. Purification. The form of error detection historically favored in quantum repeater networks is purification, which uses minimal resources [13]. It sacrifices some quantum states to test the fidelity of others. There are various purification mechanisms, with different purification algorithms and different methods for determining which states are sacrificed, each with particular tradeoffs. Quantum error correction (QEC). QEC may be based on classical codes or purely quantum concepts. The primary difficulties are extraction of errors without damaging quantum state, avoiding error propagation, and the increased resources required. (See references contained in [80], [46] and [33].) Teleportation. Teleportation destroys the state of a qubit at the sender and recreates that state at the destination, teleporting information rather than matter [11]. The process uses a Bell pair's long-distance correlation, followed by transmission of a pair of classical bits. Teleportation consumes a Bell pair. Entanglement swapping. Splicing two long-distance Bell pairs together to make one longer Bell pair is known as entanglement swapping. With these basic concepts, we can begin to construct networks. For those interested in a more research-oriented, indepth survey of quantum computing systems, we recommend the following short list of papers: [19,24,25,37,54,61,70,82,83]. For communication, we recommend: [5,13,50,53,88,89]. Figure 4 : 4Two-pass connection setup (CS) within a single quantum network. RuleSets are created by the Responder, offering a distributed innovation point. the same end points, for purification, or with different end points, for entanglement swapping. Figure 6 : 6Two-pass connection setup in an internetwork. Figure 7 : 7Our open source simulator, QuISP, focuses on protocol and scaling issues in order to further network and internetwork architecture research. Here, 100 COMP, 110 REP1, and 10 RTR nodes, collectively having 44,000 memory qubits, are connected via 220 links. 100 of the links use BSA nodes; the rest are direct connections. and this one.connection? Is this centralized or distributed? (Secs. 3.5, 4.3 and 5) 6. Node types. The state of technology determines the types of nodes we can build; the above items determine the types of nodes required to build a quantum network. (Sec. 4.1) 7. Routing. How do we pick a path or route through the network? (Sec. 4.2) 8. Multiplexing discipline for resources. Options for multi- plexing the use of quantum resources may include circuit switching, time division muxing, statistical muxing or buffer space muxing. Naturally, stateful connections and many of the muxing candidates require authentication, authorization and accounting. (Secs. 4.3, 4.4) 9. Security. Quantum networks allow numerous new attack vectors which have to be considered [75]. These attacks sometimes coincide with the defining property of the service provided by the quantum network, e.g., as in QKD; in other cases, such as for distributed computation, they represent challenges to be overcome. (Sec. 4.5) 10. Making an internetwork. How should the networks come together to create an internetwork and what is the nature of their interactions? (Sec. 5) Table 1 : 1Protocol MessagesName Descriptive Name Arguments Comments Remote Events (Message Transmission) FREE Release a state Partner addr., resource IDs Release a state back to the free pool. Used after purification. UPDATE State change notification Partner addr., resource IDs, Pauli frame correction Used to indicate a Pauli frame correction to a state. Most commonly used with TRANSFER to complete entanglement swapping. MEAS Measurement outcome Partner addr., resource IDs, result Exchange purification results. Each partner sends this message, and a separate rule will recognize whether purification results agree and proceed appropriately. Numerous types are possible. TRANSFER Entanglement transfer notification Partner addr., resource IDs Distribute the result of a swapping circuit. Generalizes to a notice of entanglement transfer from one location or partner to another. Carries a new resource ID to used for the resulting state. Table 2 : 2Condition ClausesName Descriptive Name Arguments Comments Local Software Events CMP Check whether a variable is equal, less than, or greater than some values variable ID, comparison operator, value Used to track number of operations done (e.g. purification count, measurement count, or number of notification message received) TIMER Timer expiration Timer ID Must be used with caution when dealing with distributed states; race conditions can occur. Quantum State Events (Local Hardware Notifications, Message Reception) RES Enough Resources Partner address (or wildcard) and fidelity Table 3 : 3Action ClausesName Descriptive Name Arguments Comments Local Software Actions (Classical) SETTIMER Set timer Timer ID Use with caution; distributed race conditions can occur. PROMOTE Promotion of qubits Qubit IDs, Rule ID, Stage Used to transfer the control/ownership of Qubits from current Rule (Stage) to Another Rule (Stage) FREE Free qubits Qubit IDs Release qubits to the pool of unallocated resources. SET change value of a Rule/RuleSet variable variable identifier Can be used to track how many measurements have occurred for tomography. Local Hardware Actions (Quantum) MEAS Measure qubits Qubit IDs, meas. basis Measure one or more qubits in specified basis or a randomly chosen one. QCIRC Apply quantum circuit Qubit identifiers, Qcircuit Apply a general unitary quantum operation on one or more qubits, without measuring. Bell state measurement, purification, and entanglement swapping execute QCIRC first, then MEAS. Encoding into logical qubits also uses. Quantum amplifiers[16,18] are an existing quantum technology capable of boosting certain quantum signals, however quantum states where this is possible have limited use in the context of quantum communication[17]. https://github.com/sfc-aqua/quisp#quisp Acknowledgments and AvailabilityThis material is based upon work supported by the Air Force Office of Scientific Research under award number FA2386-19-1-4038.The authors thank Joe Touch for clarification of past contributions.Our open source simulator 3 and in-preparation RFC-like specifications are or will be made available on the web. Quantum computing since Democritus. Scott Aaronson, Cambridge University PressScott Aaronson. Quantum computing since Democritus. Cambridge University Press, 2013. Using quantum key distribution for cryptographic purposes: A survey. R Alléaume, C Branciard, J Bouda, T Debuisschert, M Dianati, N Gisin, M Godfrey, P Grangier, T Länger, N Lütkenhaus, C Monyk, P Painchault, M Peev, A Poppe, T Pornin, J Rarity, R Renner, G Ribordy, M Riguidel, L Salvail, A Shields, H Weinfurter, A Zeilinger, Theoretical Aspects of Quantum Cryptography. 560celebrating 30 years of BB84R. Alléaume, C. Branciard, J. Bouda, T. Debuisschert, M. Dianati, N. Gisin, M. Godfrey, P. Grangier, T. Länger, N. Lütkenhaus, C. Monyk, P. Painchault, M. Peev, A. Poppe, T. Pornin, J. Rarity, R. Renner, G. Ribordy, M. Riguidel, L. Salvail, A. Shields, H. Weinfurter, and A. Zeilinger. Using quantum key distribution for cryp- tographic purposes: A survey. Theoretical Computer Science, 560, Part 1:62 -81, 2014. Theoretical As- pects of Quantum Cryptography, celebrating 30 years of BB84. Design and evaluation of communication protocols for quantum repeater networks. Luciano Aparicio, University of TokyoMaster's thesisLuciano Aparicio. Design and evaluation of communica- tion protocols for quantum repeater networks. Master's thesis, University of Tokyo, 2011. Multiplexing schemes for quantum repeater networks. Luciano Aparicio, Rodney Van Meter, Proc. SPIE. SPIE8163816308Luciano Aparicio and Rodney Van Meter. Multiplexing schemes for quantum repeater networks. In Proc. SPIE, volume 8163, page 816308, August 2011. Development of quantum interconnects (quics) for next-generation information technologies. David Awschalom, Karl K Berggren, Hannes Bernien, Sunil Bhave, Lincoln D Carr, Paul Davids, Sophia E Economou, Dirk Englund, Andrei Faraon, Martin Fejer, Saikat Guha, Martin V Gustafsson, Evelyn Hu, Liang Jiang, Jungsang Kim, Boris Korzh, Prem Kumar, Paul G Kwiat, Marko Lončar, Mikhail D Lukin, A B David, Christopher Miller, Sae Monroe, Prineha Woo Nam, Jason S Narang, Michael G Orcutt, Raymer, H Amir, Maria Safavi-Naeini, Kartik Spiropulu, Shuo Srinivasan, Jelena Sun, Edo Vučković, Ronald Waks, Andrew M Walsworth, Zheshen Weiner, Zhang, PRX Quantum. 217002David Awschalom, Karl K. Berggren, Hannes Bernien, Sunil Bhave, Lincoln D. Carr, Paul Davids, Sophia E. Economou, Dirk Englund, Andrei Faraon, Martin Fejer, Saikat Guha, Martin V. Gustafsson, Evelyn Hu, Liang Jiang, Jungsang Kim, Boris Korzh, Prem Kumar, Paul G. Kwiat, Marko Lončar, Mikhail D. Lukin, David A.B. Miller, Christopher Monroe, Sae Woo Nam, Prineha Narang, Jason S. Orcutt, Michael G. Raymer, Amir H. Safavi-Naeini, Maria Spiropulu, Kartik Srinivasan, Shuo Sun, Jelena Vučković, Edo Waks, Ronald Walsworth, Andrew M. Weiner, and Zheshen Zhang. Development of quantum interconnects (quics) for next-generation in- formation technologies. PRX Quantum, 2:017002, Feb 2021. Allphotonic quantum repeaters. Koji Azuma, Kiyoshi Tamaki, Hoi-Kwong Lo, Nature Communications. 61Koji Azuma, Kiyoshi Tamaki, and Hoi-Kwong Lo. All- photonic quantum repeaters. Nature Communications, 6(1):1-7, 2015. 5G network slicing using SDN and NFV: A survey of taxonomy, architectures and future challenges. Alex Alcardo, Arslan Barakabitze, Rashid Ahmad, Andrew Mijumbi, Hines, Computer Networks. 167106984Alcardo Alex Barakabitze, Arslan Ahmad, Rashid Mi- jumbi, and Andrew Hines. 5G network slicing using SDN and NFV: A survey of taxonomy, architectures and future challenges. Computer Networks, 167:106984, 2020. Reference frames, superselection rules, and quantum information. Stephen D Bartlett, Terry Rudolph, Robert W Spekkens, Rev. Mod. Phys. 79Stephen D. Bartlett, Terry Rudolph, and Robert W. Spekkens. Reference frames, superselection rules, and quantum information. Rev. Mod. Phys., 79:555-609, Apr 2007. Fast quantum Byzantine agreement. M Ben-Or, A Hassidim, Proceedings of the thirty-seventh annual ACM symposium on Theory of computing. the thirty-seventh annual ACM symposium on Theory of computingACMM. Ben-Or and A. Hassidim. Fast quantum Byzantine agreement. In Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pages 481- 485. ACM, 2005. Quantum cryptography: Public key distribution and coin tossing. C H Bennett, G Brassard, Proc. IEEE International Conference on Computers, Systems, and Signal Processing. IEEE International Conference on Computers, Systems, and Signal essingIEEEC. H. Bennett and G. Brassard. Quantum cryptography: Public key distribution and coin tossing. In Proc. IEEE International Conference on Computers, Systems, and Signal Processing, pages 175-179. IEEE, December 1984. Teleporting an unknown quantum state via dual classical and EPR channels. C H Bennett, G Brassard, C Crépeau, R Josza, A Peres, W Wootters, Physical Review Letters. 70C. H. Bennett, G. Brassard, C. Crépeau, R. Josza, A. Peres, and W. Wootters. Teleporting an unknown quantum state via dual classical and EPR channels. Phys- ical Review Letters, 70:1895-1899, 1993. Quantum cryptography without Bell's theorem. Charles H Bennett, Gilles Brassard, N David Mermin, Phys. Rev. Lett. 68Charles H. Bennett, Gilles Brassard, and N. David Mer- min. Quantum cryptography without Bell's theorem. Phys. Rev. Lett., 68:557-559, Feb 1992. Quantum repeaters: the role of imperfect local operations in quantum communication. H.-J Briegel, W Dür, J I Cirac, P Zoller, Physical Review Letters. 81H.-J. Briegel, W. Dür, J.I. Cirac, and P. Zoller. Quan- tum repeaters: the role of imperfect local operations in quantum communication. Physical Review Letters, 81:5932-5935, 1998. Universal blind quantum computation. Anne Broadbent, Joseph Fitzsimons, Elham Kashefi, 50Anne Broadbent, Joseph Fitzsimons, and Elham Kashefi. Universal blind quantum computation. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science. Annual IEEE Symposium on Foundations of Computer Science, pages 517-526, 2009. Mathematical Foundations of Computer Science. Harry Buhrman, Hein Röhrig, chapter Distributed Quantum Computing. Springer-VerlagHarry Buhrman and Hein Röhrig. Mathematical Foun- dations of Computer Science 2003, chapter Distributed Quantum Computing, pages 1-20. Springer-Verlag, 2003. Quantum limits on phase-preserving linear amplifiers. Carlton M Caves, Joshua Combes, Zhang Jiang, Shashank Pandey, Phys. Rev. A. 8663802Carlton M. Caves, Joshua Combes, Zhang Jiang, and Shashank Pandey. Quantum limits on phase-preserving linear amplifiers. Phys. Rev. A, 86:063802, Dec 2012. Phase-preserving linear amplifiers not simulable by the parametric amplifier. A Chia, M Hajdušek, R Nair, R Fazio, L C Kwek, V Vedral, Phys. Rev. Lett. 125163603A. Chia, M. Hajdušek, R. Nair, R. Fazio, L. C. Kwek, and V. Vedral. Phase-preserving linear amplifiers not simulable by the parametric amplifier. Phys. Rev. Lett., 125:163603, Oct 2020. Phase diffusion and the small-noise approximation in linear amplifiers: Limitations and beyond. Andy Chia, Michal Hajdušek, Rosario Fazio, Leong-Chuan, Vlatko Kwek, Vedral, 200Andy Chia, Michal Hajdušek, Rosario Fazio, Leong- Chuan Kwek, and Vlatko Vedral. Phase diffusion and the small-noise approximation in linear amplifiers: Lim- itations and beyond. Quantum, 3:200, November 2019. Programming languages and compiler design for realistic quantum hardware. T Frederic, Diana Chong, Margaret Franklin, Martonosi, Nature. 5497671180Frederic T Chong, Diana Franklin, and Margaret Martonosi. Programming languages and compiler design for realistic quantum hardware. Nature, 549(7671):180, 2017. Secure multi-party quantum computation. Claude Creṕeau, Daniel Gottesman, Adam Smith, Proc. Symposium on Theory of Computing. Symposium on Theory of ComputingACMClaude Creṕeau, Daniel Gottesman, and Adam Smith. Secure multi-party quantum computation. In Proc. Sym- posium on Theory of Computing. ACM, 2002. Towards a distributed quantum computing ecosystem. Daniele Cuomo, Marcello Caleffi, Angela Sara Cacciapuoti, IET Quantum Communication. 11Daniele Cuomo, Marcello Caleffi, and Angela Sara Cac- ciapuoti. Towards a distributed quantum computing ecosystem. IET Quantum Communication, 1(1):3-8, 2020. A link layer protocol for quantum networks. Axel Dahlberg, Matthew Skrzypczyk, Tim Coopmans, Leon Wubben, Filip Rozpędek, Matteo Pompili, Arian Stolk, Przemysław Pawełczak, Robert Knegjens, Julio De Oliveira, Ronald Filho, Stephanie Hanson, Wehner, Proceedings of the ACM Special Interest Group on Data Communication, SIGCOMM '19. the ACM Special Interest Group on Data Communication, SIGCOMM '19New York, NY, USAAssociation for Computing MachineryAxel Dahlberg, Matthew Skrzypczyk, Tim Coopmans, Leon Wubben, Filip Rozpędek, Matteo Pompili, Ar- ian Stolk, Przemysław Pawełczak, Robert Knegjens, Julio de Oliveira Filho, Ronald Hanson, and Stephanie Wehner. A link layer protocol for quantum networks. In Proceedings of the ACM Special Interest Group on Data Communication, SIGCOMM '19, page 159-173, New York, NY, USA, 2019. Association for Computing Machinery. High-speed quantum networking by ship. J Simon, Andrew D Devitt, Ashley M Greentree, Rodney Stephens, Van Meter, Scientific Reports. 636163Simon J Devitt, Andrew D Greentree, Ashley M Stephens, and Rodney Van Meter. High-speed quan- tum networking by ship. Scientific Reports, 6:36163, 2016. Quantum error correction for beginners. J Simon, Devitt, J William, Kae Munro, Nemoto, Reports on Progress in Physics. 76776001Simon J Devitt, William J Munro, and Kae Nemoto. Quantum error correction for beginners. Reports on Progress in Physics, 76(7):076001, 2013. The physical implementation of quantum computation. D P Divincenzo, Fortschritte der Physik. 489D.P. DiVincenzo. The physical implementation of quan- tum computation. Fortschritte der Physik, 48(9-11):771- 783, 2000. . Jerome Durand, Ivan Pepelnjak, Gert Doering, BGP Operations and Security. RFC. 7454Jerome Durand, Ivan Pepelnjak, and Gert Doering. BGP Operations and Security. RFC 7454, February 2015. Quantum certification and benchmarking. Jens Eisert, Dominik Hangleiter, Nathan Walk, Ingo Roth, Damian Markham, Rhea Parekh, Ulysse Chabaud, Elham Kashefi, Nature Reviews Physics. 27Jens Eisert, Dominik Hangleiter, Nathan Walk, Ingo Roth, Damian Markham, Rhea Parekh, Ulysse Chabaud, and Elham Kashefi. Quantum certification and bench- marking. Nature Reviews Physics, 2(7):382-390, 2020. Quantum cryptography based on Bell's theorem. A K Ekert, Physical Review Letters. 676A.K. Ekert. Quantum cryptography based on Bell's theorem. Physical Review Letters, 67(6):661-663, 1991. The ultimate physical limits of privacy. Artur Ekert, Renato Renner, Nature. 5077493Artur Ekert and Renato Renner. The ultimate physical limits of privacy. Nature, 507(7493):443-447, 2014. Quantum cryptography in practice. Chip Elliott, David Pearson, Gregory Troxel, Proc. SIGCOMM 2003. ACM, ACM. SIGCOMM 2003. ACM, ACMChip Elliott, David Pearson, and Gregory Troxel. Quan- tum cryptography in practice. In Proc. SIGCOMM 2003. ACM, ACM, August 2003. Diameter Base Protocol. RFC 6733. Victor Fajardo, Jari Arkko, John A Loughney, Glen Zorn, Victor Fajardo, Jari Arkko, John A. Loughney, and Glen Zorn. Diameter Base Protocol. RFC 6733, October 2012. Private quantum computation: an introduction to blind quantum computing and related protocols. F Joseph, Fitzsimons, npj Quantum Information. 3Joseph F Fitzsimons. Private quantum computation: an introduction to blind quantum computing and related protocols. npj Quantum Information, 3(1):1-11, 2017. Hollenberg. Surface code quantum communication. Austin G Fowler, David S Wang, Charles D Hill, Thaddeus D Ladd, Rodney Van Meter, Lloyd C L , Phys. Rev. Lett. 10418180503Austin G. Fowler, David S. Wang, Charles D. Hill, Thad- deus D. Ladd, Rodney Van Meter, and Lloyd C. L. Hol- lenberg. Surface code quantum communication. Phys. Rev. Lett., 104(18):180503, May 2010. Understanding quantum computers. Future LearnFuture Learn. Understanding quantum comput- ers. https://www.futurelearn.com/courses/ intro-to-quantum-computing. Longer-baseline telescopes using quantum repeaters. Daniel Gottesman, Thomas Jennewein, Sarah Croke, Phys. Rev. Lett. 10970503Daniel Gottesman, Thomas Jennewein, and Sarah Croke. Longer-baseline telescopes using quantum repeaters. Phys. Rev. Lett., 109:070503, Aug 2012. Entanglement in pure and thermal cluster states. Michal Hajdušek, Vlatko Vedral, New Journal of Physics. 12553015Michal Hajdušek and Vlatko Vedral. Entanglement in pure and thermal cluster states. New Journal of Physics, 12(5):053015, 2010. Quantum computational supremacy. W Aram, Ashley Harrow, Montanaro, Nature. 5497671203Aram W Harrow and Ashley Montanaro. Quantum com- putational supremacy. Nature, 549(7671):203, 2017. Selfguaranteed measurement-based quantum computation. Masahito Hayashi, Michal Hajdušek, Phys. Rev. A. 9752308Masahito Hayashi and Michal Hajdušek. Self- guaranteed measurement-based quantum computation. Phys. Rev. A, 97:052308, May 2018. Multiparty entanglement in graph states. M Hein, J Eisert, H J Briegel, Phys. Rev. A. 6962311M. Hein, J. Eisert, and H. J. Briegel. Multiparty entan- glement in graph states. Phys. Rev. A, 69:062311, Jun 2004. Resource requirements for efficient quantum communication using all-photonic graph states generated from a few matter qubits. Paul Hilaire, Edwin Barnes, Sophia E Economou, 397Paul Hilaire, Edwin Barnes, and Sophia E. Economou. Resource requirements for efficient quantum communi- cation using all-photonic graph states generated from a few matter qubits. Quantum, 5:397, February 2021. Quantum secret sharing. M Hillery, V Bužek, A Berthiaume, Physical Review A. 593M. Hillery, V. Bužek, and A. Berthiaume. Quantum secret sharing. Physical Review A, 59(3):1829-1834, 1999. Quantum entanglement. Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, Karol Horodecki, Rev. Mod. Phys. 81Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum en- tanglement. Rev. Mod. Phys., 81:865-942, Jun 2009. Event-ready-detectors" Bell experiment via entanglement swapping. M Żukowski, A Zeilinger, M A Horne, A K Ekert, Phys. Rev. Lett. 71M.Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert. "Event-ready-detectors" Bell experiment via en- tanglement swapping. Phys. Rev. Lett., 71:4287-4290, Dec 1993. Remote quantum clock synchronization without synchronized clocks. Louis Ebubechukwu O Ilo-Okeke, Jonathan P Tessler, Tim Dowling, Byrnes, npj Quantum Information. 4Ebubechukwu O Ilo-Okeke, Louis Tessler, Jonathan P Dowling, and Tim Byrnes. Remote quantum clock syn- chronization without synchronized clocks. npj Quantum Information, 4(1):1-5, 2018. Spatial reference frame agreement in quantum networks. Tanvirul Islam, Loïck Magnin, Brandon Sorg, Stephanie Wehner, New Journal of Physics. 16663040Tanvirul Islam, Loïck Magnin, Brandon Sorg, and Stephanie Wehner. Spatial reference frame agree- ment in quantum networks. New Journal of Physics, 16(6):063040, 2014. Quantum repeater with encoding. J M Liang Jiang, Kae Taylor, W J Nemoto, Rodney Munro, M D Van Meter, Lukin, Phys. Rev. A. 79332325Liang Jiang, J. M. Taylor, Kae Nemoto, W. J. Munro, Rodney Van Meter, and M. D. Lukin. Quantum repeater with encoding. Phys. Rev. A, 79(3):032325, Mar 2009. Design and analysis of communication protocols for quantum repeater networks. Cody Jones, Danny Kim, T Matthew, Rakher, G Paul, Thaddeus D Kwiat, Ladd, New Journal of Physics. 18883015Cody Jones, Danny Kim, Matthew T Rakher, Paul G Kwiat, and Thaddeus D Ladd. Design and analysis of communication protocols for quantum repeater net- works. New Journal of Physics, 18(8):083015, 2016. Quantum entanglement for secret sharing and secret splitting. Anders Karlsson, Masato Koashi, Nobuyuki Imoto, Phys. Rev. A. 59Anders Karlsson, Masato Koashi, and Nobuyuki Imoto. Quantum entanglement for secret sharing and secret splitting. Phys. Rev. A, 59:162-168, Jan 1999. Quantum-assisted telescope arrays. E T Khabiboulline, J Borregaard, K De Greve, M D Lukin, Phys. Rev. A. 10022316E. T. Khabiboulline, J. Borregaard, K. De Greve, and M. D. Lukin. Quantum-assisted telescope arrays. Phys. Rev. A, 100:022316, Aug 2019. The quantum Internet. H J Kimble, Nature. 453H. J. Kimble. The quantum Internet. Nature, 453:1023- 1030, June 2008. A quantum network of clocks. P Kómár, E M Kessler, M Bishof, L Jiang, A S Sorensen, M D Lukin, Nature Physics. P. Kómár, E.M. Kessler, M. Bishof, L. Jiang, A. S. Sorensen, and M. D. Lukin. A quantum network of clocks. Nature Physics, June 2014. Designing a quantum network protocol. Wojciech Kozlowski, Axel Dahlberg, Stephanie Wehner, Proceedings of the 16th International Conference on Emerging Networking EXperiments and Technologies, CoNEXT '20. the 16th International Conference on Emerging Networking EXperiments and Technologies, CoNEXT '20New York, NY, USAAssociation for Computing MachineryWojciech Kozlowski, Axel Dahlberg, and Stephanie Wehner. Designing a quantum network protocol. In Proceedings of the 16th International Conference on Emerging Networking EXperiments and Technologies, CoNEXT '20, page 1-16, New York, NY, USA, 2020. Association for Computing Machinery. Angela Sara Cacciapuoti, Marcello Caleffi, and Shota Nagayama. Architectural principles for a quantum internet. Wojciech Kozlowski, Stephanie Wehner, Rodney Van Meter, Bruno Rijsman, IETF Secretariat. Internet-Draft draft-irtfqirg-principles-06Wojciech Kozlowski, Stephanie Wehner, Rodney Van Meter, Bruno Rijsman, Angela Sara Cacciapuoti, Mar- cello Caleffi, and Shota Nagayama. Architectural prin- ciples for a quantum internet. Internet-Draft draft-irtf- qirg-principles-06, IETF Secretariat, February 2021. Quantum computers. T D Ladd, F Jelezko, R Laflamme, Y Nakamura, C Monroe, J L O&apos;brien, Nature. 464T.D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J.L. O'Brien. Quantum computers. Nature, 464:45-53, March 2010. Graph states for quantum secret sharing. D Markham, B C Sanders, Physical Review A. 78442309D. Markham and B.C. Sanders. Graph states for quan- tum secret sharing. Physical Review A, 78(4):42309, 2008. Optimal extraction of information from finite quantum ensembles. S Massar, S Popescu, Phys. Rev. Lett. 74S. Massar and S. Popescu. Optimal extraction of infor- mation from finite quantum ensembles. Phys. Rev. Lett., 74:1259-1263, Feb 1995. Simulation of a dynamic, ruleset-based quantum network. Takaaki Matsuo, arXiv:1908.10758arXiv preprintTakaaki Matsuo. Simulation of a dynamic, ruleset-based quantum network. arXiv preprint arXiv:1908.10758, 2019. Quantum link bootstrapping using a ruleset-based communication protocol. Takaaki Matsuo, Clément Durand, Rodney Van Meter, Phys. Rev. A. 10052320Takaaki Matsuo, Clément Durand, and Rodney Van Me- ter. Quantum link bootstrapping using a ruleset-based communication protocol. Phys. Rev. A, 100:052320, Nov 2019. Openflow: enabling innovation in campus networks. Nick Mckeown, Tom Anderson, Hari Balakrishnan, Guru Parulkar, Larry Peterson, Jennifer Rexford, Scott Shenker, Jonathan Turner, ACM SIGCOMM computer communication review. 382Nick McKeown, Tom Anderson, Hari Balakrishnan, Guru Parulkar, Larry Peterson, Jennifer Rexford, Scott Shenker, and Jonathan Turner. Openflow: enabling inno- vation in campus networks. ACM SIGCOMM computer communication review, 38(2):69-74, 2008. Quantum key distribution (QKD) and commodity security protocols: Introduction and integration. Alan Mink, Sheila Frankel, Ray Perlner, International Journal of Network Security & Its Applications (IJNSA). 12Alan Mink, Sheila Frankel, and Ray Perlner. Quantum key distribution (QKD) and commodity security proto- cols: Introduction and integration. International Journal of Network Security & Its Applications (IJNSA), 1(2), July 2009. Ashley Montanaro, Quantum algorithms: an overview. npj Quantum Information. 215023Ashley Montanaro. Quantum algorithms: an overview. npj Quantum Information, 2:15023, 2016. Blind quantum computation protocol in which alice only makes measurements. Tomoyuki Morimae, Keisuke Fujii, Phys. Rev. A. 8750301Tomoyuki Morimae and Keisuke Fujii. Blind quantum computation protocol in which alice only makes mea- surements. Phys. Rev. A, 87:050301, May 2013. Optimal architectures for long distance quantum communication. Linshu Sreraman Muralidharan, Jungsang Li, Norbert Kim, Lütkenhaus, Liang Mikhail D Lukin, Jiang, Scientific Reports. 620463Sreraman Muralidharan, Linshu Li, Jungsang Kim, Nor- bert Lütkenhaus, Mikhail D Lukin, and Liang Jiang. Op- timal architectures for long distance quantum communi- cation. Scientific Reports, 6:20463, 2016. Towards end-to-end error management for a quantum internet. Shota Nagayama, Released simultaneously to arXiv.Shota Nagayama. Towards end-to-end error manage- ment for a quantum internet. Released simultaneously to arXiv., December 2021. Interoperability in encoded quantum repeater networks. Shota Nagayama, Byung-Soo Choi, Simon Devitt, Shigeya Suzuki, Rodney Van Meter, Phys. Rev. A. 9342338Shota Nagayama, Byung-Soo Choi, Simon Devitt, Shigeya Suzuki, and Rodney Van Meter. Interoperabil- ity in encoded quantum repeater networks. Phys. Rev. A, 93:042338, Apr 2016. Transmission of a cartesian frame by a quantum system. Asher Peres, Petra F Scudo, Phys. Rev. Lett. 87167901Asher Peres and Petra F. Scudo. Transmission of a cartesian frame by a quantum system. Phys. Rev. Lett., 87:167901, Sep 2001. S Pirandola, U L Andersen, L Banchi, M Berta, D Bunandar, R Colbeck, D Englund, T Gehring, C Lupo, C Ottaviani, J L Pereira, M Razavi, J Shaari, M Tomamichel, V C Usenko, G Vallone, P Villoresi, P Wallden, Advances in quantum cryptography. 12S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Sham- sul Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden. Advances in quantum cryp- tography. Adv. Opt. Photon., 12(4):1012-1236, Dec 2020. Realization of a multinode quantum network of remote solid-state qubits. Matteo Pompili, L N Sophie, Simon Hermans, Baier, K C Hans, Beukers, C Peter, Raymond N Humphreys, Schouten, F L Raymond, Vermeulen, J Marijn, Laura Tiggelman, Santos Martins, Bas Dirkse, Science. 3726539Matteo Pompili, Sophie LN Hermans, Simon Baier, Hans KC Beukers, Peter C Humphreys, Raymond N Schouten, Raymond FL Vermeulen, Marijn J Tiggelman, Laura dos Santos Martins, Bas Dirkse, et al. Realization of a multinode quantum network of remote solid-state qubits. Science, 372(6539):259-264, 2021. Security ain Quantum Cryptography. Christopher Portmann, Renato Renner, arXiv:2102.00021v12021Christopher Portmann and Renato Renner. Security ain Quantum Cryptography, 2021. arXiv:2102.00021v1. John Preskill, Quantum computing in the NISQ era and beyond. Quantum. 279John Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2:79, 2018. . Quantum Protocol Zoo. Protocol library. Quantum Protocol Zoo. Protocol library. Exponential separation of quantum and classical communication complexity. R Raz, Proceedings of the thirtyfirst annual ACM symposium on Theory of computing. the thirtyfirst annual ACM symposium on Theory of computingR. Raz. Exponential separation of quantum and classical communication complexity. Proceedings of the thirty- first annual ACM symposium on Theory of computing, pages 358-367, 1999. Quantum communication complexity of establishing a shared reference frame. Terry Rudolph, Lov Grover, Phys. Rev. Lett. 91217905Terry Rudolph and Lov Grover. Quantum communica- tion complexity of establishing a shared reference frame. Phys. Rev. Lett., 91:217905, Nov 2003. QuISP: a quantum internet simulation package. Ryosuke Satoh, Michal Hajdušek, Naphan Benchasattabuse, Shota Nagayama, Kentaro Teramoto, Takaaki Matsuo, Sara Ayman Metwalli, Takahiko Satoh, Shigeya Suzuki, Rodney Van Meter, Released simultaneously to arXiv.Ryosuke Satoh, Michal Hajdušek, Naphan Benchasat- tabuse, Shota Nagayama, Kentaro Teramoto, Takaaki Matsuo, Sara Ayman Metwalli, Takahiko Satoh, Shigeya Suzuki, and Rodney Van Meter. QuISP: a quantum in- ternet simulation package. Released simultaneously to arXiv., December 2021. Attacking the quantum internet. Takahiko Satoh, Shota Nagayama, Shigeya Suzuki, Takaaki Matsuo, Michal Hajdušek, Rodney Van Meter, IEEE Transactions on Quantum Engineering. 2Takahiko Satoh, Shota Nagayama, Shigeya Suzuki, Takaaki Matsuo, Michal Hajdušek, and Rodney Van Me- ter. Attacking the quantum internet. IEEE Transactions on Quantum Engineering, 2:1-17, 2021. Dancing with Qubits. Robert S Sutor, Packt PublishingRobert S. Sutor. Dancing with Qubits. Packt Publishing, 2019. Resource-aware system architecture model for implementation of quantum aided byzantine agreement on quantum repeater networks. Keivan Mohammand Amin Taherkhani, Rodney Navi, Van Meter, Quantum Science and Technology. 3114011Mohammand Amin Taherkhani, Keivan Navi, and Rod- ney Van Meter. Resource-aware system architecture model for implementation of quantum aided byzantine agreement on quantum repeater networks. Quantum Science and Technology, 3(1):014011, 2018. Exact quantum algorithms for the leader election problem. Seiichiro Tani, Hirotada Kobayashi, Keiji Matsumoto, 1:1- 1:24ACM Trans. Comput. Theory. 41Seiichiro Tani, Hirotada Kobayashi, and Keiji Mat- sumoto. Exact quantum algorithms for the leader elec- tion problem. ACM Trans. Comput. Theory, 4(1):1:1- 1:24, March 2012. A recursive network architecture. ISI. Joe Touch, Yu-Shun Wang, Venkata Pingali, 626Tech. RepJoe Touch, Yu-Shun Wang, and Venkata Pingali. A re- cursive network architecture. ISI, Tech. Rep, 626, 2006. Quantum networking and internetworking. Rodney Van Meter, IEEE Network. 264Rodney Van Meter. Quantum networking and inter- networking. IEEE Network, 26(4):59-64, July/August 2012. Rodney Van Meter, Quantum Networking. Wiley-ISTE. Rodney Van Meter. Quantum Networking. Wiley-ISTE, April 2014. A #quantumcomputerarchitecture tweetstorm. Rodney Van Meter, Rodney Van Meter. A #quantumcomputerarchitecture tweetstorm. September 2019. A blueprint for building a quantum computer. Rodney Van Meter, C Horsman, Communications of the ACM. 5310Rodney Van Meter and C. Horsman. A blueprint for building a quantum computer. Communications of the ACM, 53(10):84-93, October 2013. System design for a long-line quantum repeater. Rodney Van Meter, Thaddeus D Ladd, W J Munro, Kae Nemoto, IEEE/ACM Transactions on Networking. 173Rodney Van Meter, Thaddeus D. Ladd, W. J. Munro, and Kae Nemoto. System design for a long-line quan- tum repeater. IEEE/ACM Transactions on Networking, 17(3):1002-1013, June 2009. Internet-Draft draft-vanmeter-qirg-quantum-connection-setup-01. Rodney Van Meter, Takaaki Matsuo, IETF Secretariat. Connection setup in a quantum networkRodney Van Meter and Takaaki Matsuo. Connection setup in a quantum network. Internet-Draft draft-van- meter-qirg-quantum-connection-setup-01, IETF Secre- tariat, September 2019. Path selection for quantum repeater networks. Rodney Van Meter, Takahiko Satoh, Thaddeus D Ladd, William J Munro, Kae Nemoto, Networking Science. 31Rodney Van Meter, Takahiko Satoh, Thaddeus D. Ladd, William J. Munro, and Kae Nemoto. Path selection for quantum repeater networks. Networking Science, 3(1):82-95, 2013. Rodney Van Meter, Joe Touch, C Horsman, arXiv:1105.1238Recursive quantum repeater networks. arXiv preprintRodney Van Meter, Joe Touch, and C. Horsman. Re- cursive quantum repeater networks. arXiv preprint arXiv:1105.1238, 2011. Applications and use cases for the quantum internet. Internet-Draft draft-irtf-qirg-quantuminternet-use-cases-06, IETF Secretariat. Chonggang Wang, Akbar Rahman, Ruidong Li, Melchior Aelmans, Chonggang Wang, Akbar Rahman, Ruidong Li, and Mel- chior Aelmans. Applications and use cases for the quantum internet. Internet-Draft draft-irtf-qirg-quantum- internet-use-cases-06, IETF Secretariat, May 2021. Quantum internet: A vision for the road ahead. Stephanie Wehner, David Elkouss, Ronald Hanson, Science. 3626412Stephanie Wehner, David Elkouss, and Ronald Hanson. Quantum internet: A vision for the road ahead. Science, 362(6412), 2018. A single quantum cannot be cloned. W K Wootters, W H Zurek, Nature. 299W. K. Wootters and W. H. Zurek. A single quantum cannot be cloned. Nature, 299:802-803, October 1982. Secure quantum key distribution with realistic devices. Feihu Xu, Xiongfeng Ma, Qiang Zhang, Hoi-Kwong Lo, Jian-Wei Pan, Rev. Mod. Phys. 9225002Feihu Xu, Xiongfeng Ma, Qiang Zhang, Hoi-Kwong Lo, and Jian-Wei Pan. Secure quantum key distribution with realistic devices. Rev. Mod. Phys., 92:025002, May 2020. Satellite-based entanglement distribution over 1200 kilometers. Juan Yin, Yuan Cao, Yu-Huai Li, Sheng-Kai Liao, Liang Zhang, Ji-Gang Ren, Wen-Qi Cai, Wei-Yue Liu, Bo Li, Hui Dai, Guang-Bing Li, Qi-Ming Lu, Yun-Hong Gong, Yu Xu, Shuang-Lin Li, Science. Feng-Zhi Li, Ya-Yun Yin, Zi-Qing Jiang, Ming Li, Jian-Jun Jia, Ge Ren, Dong He, Yi-Lin Zhou, Xiao-Xiang Zhang, Na Wang, Xiang Chang, Zhen-Cai Zhu, Nai-Le Liu, Yu-Ao Chen, Chao-Yang Lu, Rong Shu, Cheng-Zhi Peng, Jian-Yu Wang, and Jian-Wei Pan3566343Juan Yin, Yuan Cao, Yu-Huai Li, Sheng-Kai Liao, Liang Zhang, Ji-Gang Ren, Wen-Qi Cai, Wei-Yue Liu, Bo Li, Hui Dai, Guang-Bing Li, Qi-Ming Lu, Yun-Hong Gong, Yu Xu, Shuang-Lin Li, Feng-Zhi Li, Ya-Yun Yin, Zi- Qing Jiang, Ming Li, Jian-Jun Jia, Ge Ren, Dong He, Yi- Lin Zhou, Xiao-Xiang Zhang, Na Wang, Xiang Chang, Zhen-Cai Zhu, Nai-Le Liu, Yu-Ao Chen, Chao-Yang Lu, Rong Shu, Cheng-Zhi Peng, Jian-Yu Wang, and Jian- Wei Pan. Satellite-based entanglement distribution over 1200 kilometers. Science, 356(6343):1140-1144, 2017.
[ "https://github.com/sfc-aqua/quisp#quisp" ]
[]
[ "Zhi-Gang Wang 1e-mail:[email protected]. \nDepartment of Physics\nNorth China Electric Power University\n071003BaodingP. R. China\n" ]
[ "Department of Physics\nNorth China Electric Power University\n071003BaodingP. R. China" ]
[]
In this article, we take the Y (4260/4220) as the vector tetraquark state with J P C = 1 −− , and construct the Cγ5⊗ ↔ ∂ µ ⊗γ5C type diquark-antidiquark current to study its mass and pole residue with the QCD sum rules in details by taking into account the vacuum condensates up to dimension 10 in a consistent way. The predicted mass MY = 4.24 ± 0.10 GeV is in excellent agreement with experimental data and supports assigning the Y (4260/4220) to be the Cγ5⊗ ↔ ∂ µ ⊗γ5C type vector tetraquark state, and disfavors assigning the Zc(4100) to be the Cγ5⊗ ↔ ∂ µ ⊗γ5C type vector tetraquark state. It is the first time that the QCD sum rules have reproduced the mass of the Y (4260/4220) as a vector tetraquark state.
10.1140/epjc/s10052-018-6417-5
[ "https://arxiv.org/pdf/1809.10299v2.pdf" ]
119,374,452
1809.10299
fe6e2708130295b7eaaf82a89cec2a9cb138cd79
29 Oct 2018 Zhi-Gang Wang 1e-mail:[email protected]. Department of Physics North China Electric Power University 071003BaodingP. R. China 29 Oct 2018Lowest vector tetraquark states: Y (4260/4220) or Z c (4100)number: 1239Mk1238Lg Key words: Tetraquark stateQCD sum rules In this article, we take the Y (4260/4220) as the vector tetraquark state with J P C = 1 −− , and construct the Cγ5⊗ ↔ ∂ µ ⊗γ5C type diquark-antidiquark current to study its mass and pole residue with the QCD sum rules in details by taking into account the vacuum condensates up to dimension 10 in a consistent way. The predicted mass MY = 4.24 ± 0.10 GeV is in excellent agreement with experimental data and supports assigning the Y (4260/4220) to be the Cγ5⊗ ↔ ∂ µ ⊗γ5C type vector tetraquark state, and disfavors assigning the Zc(4100) to be the Cγ5⊗ ↔ ∂ µ ⊗γ5C type vector tetraquark state. It is the first time that the QCD sum rules have reproduced the mass of the Y (4260/4220) as a vector tetraquark state. Introduction In 2005, the BaBar collaboration observed the Y (4260) in the π + π − J/ψ mass spectrum in the initial-state radiation process e + e − → γ ISR π + π − J/ψ [1]. Then the Y (4260) was confirmed by the Belle and CLEO collaborations [2,3]. There have been several possible assignments for the Y (4260) since its observation, such as the tetraquark state [4,5,6,7,8,9,10,11], hybrid states [12,13,14], hadro-charmonium state [15], molecular state [16,17], kinematical effect [18,19], baryonium state [20], etc. In 2014, the BES collaboration observed a resonance in the ωχ c0 cross section in the processes e + e − → ωχ c0/c1/c2 , the measured mass and width are 4230 ± 8 ± 6 MeV and 38 ± 12 ± 2 MeV, respectively [21]. In 2016, the BES collaboration observed the Y (4220) and Y (4390) in the process e + e − → π + π − h c , the measured masses and widths are M Y (4220) = 4218.4 ± 4.0 ± 0.9 MeV, M Y (4390) = 4391.6 ± 6.3 ± 1.0 MeV, Γ Y (4220) = 66.0 ± 9.0 ± 0.4 MeV and Γ Y (4390) = 139.5 ± 16.1 ± 0.6 MeV, respectively [22]. Also in 2016, the BES collaboration observed the Y (4220) and Y (4320) by precisely measuring the cross section of the process e + e − → π + π − J/ψ, the measured masses and widths are M Y (4220) = 4222.0 ± 3.1 ± 1.4 MeV, M Y (4320) = 4320.0 ± 10.4 ± 7.0 MeV, Γ Y (4220) = 44.1 ± 4.3 ± 2.0 MeV and Γ Y (4320) = 101.4 +25. 3 −19.7 ± 10.2 MeV, respectively [23]. The Y (4260) and Y (4220) may be the same particle, while the Y (4360) and Y (4320) may be the same particle according to the analogous masses and widths. In Ref. [4], L. Maiani et al assign the Y (4260) to be the diquark-antidiquark type tetraquark state with the angular momentum L = 1 based on the effective Hamiltonian with the spin-spin and spinorbit interactions. In the type-II diquark-antidiquark model [5], where the spin-spin interactions between the quarks and antiquarks are neglected, L. Maiani et al interpret the Y (4008), Y (4260), Y (4290/4220) and Y (4630) as the four ground states with L = 1. By incorporating the dominant spin-spin, spin-orbit and tensor interactions, A. Ali et al observe that the preferred assignments of the ground state tetraquark states with L = 1 are the Y (4220), Y (4330), Y (4390), Y (4660) rather than the Y (4008), Y (4260), Y (4360), Y (4660) [6]. The QCD sum rules can reproduce the experimental values of the masses of the Y (4360) and Y (4660) in the scenario of the tetraquark states [8,9,10,11,24,25,26,27]. The diquarks ε ijk q T j CΓq ′ k have five structures in Dirac spinor space, where CΓ = Cγ 5 , C, Cγ µ γ 5 , Cγ µ and Cσ µν for the scalar, pseudoscalar, vector, axialvector and tensor diquarks, respectively, the i, j, k are color indexes. The attractive interactions of one-gluon exchange favor formation of the diquarks in color antitriplet, flavor antitriplet and spin singlet [28], while the favored configurations are the scalar (Cγ 5 ) and axialvector (Cγ µ ) diquark states based on the QCD sum rules [29,30,31,32]. We can take the Cγ 5 and Cγ µ diquark states as basic constituents to construct the scalar and axialvector tetraquark states [33,34]. In the non-relativistic quark models, we have to introduce additional P-waves explicitly to study the vector tetraquark states, while in the quantum field theory, we can also take other diquark states (C, Cγ µ γ 5 and Cσ µν ) as basic constituents without introducing the explicit P-waves to study the vector tetraquark states [8,9,10,11,24,25,35,36]. However, up to now, the QCD sum rules cannot reproduce the experimental value of the mass of the Y (4260/4220) in the scenario of the tetraquark state [8,9,10,11,24,25,26,27]. We often obtain much larger mass than the M Y (4260/4220) . The net effects of the relative P-waves between the heavy (anti)quarks and light (anti)quarks in the heavy (anti)diquarks are embodied in the underlined γ 5 in the Cγ 5 γ 5 ⊗ γ µ C type and Cγ 5 ⊗ γ 5 γ µ C type currents or in the underlined γ α in the Cγ α γ α ⊗ γ µ C type currents [27]. If we introduce the relative P-waves between the heavy (anti)quarks and light (anti)quarks in obtain the hadronic representation [40,41]. After isolating the ground state contribution of the vector tetraquark state Y (4260/4220), we get the result, Π µν (p) = λ 2 Y M 2 Y − p 2 −g µν + p µ p ν p 2 + · · · , = Π(p 2 ) −g µν + p µ p ν p 2 + Π 0 (p 2 ) p µ p ν p 2 ,(4) where the pole residue λ Y is defined by 0|J µ (0)|Y (p) = λ Y ε µ , the ε µ is the polarization vector of the vector tetraquark state Y (4260/4220). The vector and scalar tetraquark states contribute to the components Π(p 2 ) and Π 0 (p 2 ), respectively. In this article, we choose the tensor structure −g µν + pµpν p 2 for analysis, the scalar tetraquark states have no contaminations. Now we briefly outline the operator product expansion for the correlation function Π µν (p) in perturbative QCD. We contract the u, d and c quark fields in the correlation function Π µν (p) with Wick theorem, obtain the result: Π µν (p) = − iε ijk ε imn ε i ′ j ′ k ′ ε i ′ m ′ n ′ 2 d 4 xe ip·x Tr γ 5 C kk ′ (x)γ 5 CS jj ′ T (x)C ∂ µ ∂ ν Tr γ 5 C n ′ n (−x)γ 5 CS m ′ mT (−x)C −∂ µ Tr γ 5 C kk ′ (x)γ 5 CS jj ′ T (x)C ∂ ν Tr γ 5 C n ′ n (−x)γ 5 CS m ′ mT (−x)C −∂ ν Tr γ 5 C kk ′ (x)γ 5 CS jj ′ T (x)C ∂ µ Tr γ 5 C n ′ n (−x)γ 5 CS m ′ mT (−x)C +∂ µ ∂ ν Tr γ 5 C kk ′ (x)γ 5 CS jj ′ T (x)C Tr γ 5 C n ′ n (−x)γ 5 CS m ′ mT (−x)C ,(5) where the S ij (x) and C ij (x) are the full u/d and c quark propagators respectively, S ij (x) = iδ ij x 2π 2 x 4 − δ ij qq 12 − δ ij x 2 sg s σGs 192 − ig s G a αβ t a ij ( xσ αβ + σ αβ x) 32π 2 x 2 − δ ij x 4 qq g 2 s GG 27648 − 1 8 q j σ µν q i σ µν + · · · ,(6)C ij (x) = i (2π) 4 d 4 ke −ik·x δ ij k − m c − g s G n αβ t n ij 4 σ αβ ( k + m c ) + ( k + m c )σ αβ (k 2 − m 2 c ) 2 − g 2 s (t a t b ) ij G a αβ G b µν (f αβµν + f αµβν + f αµνβ ) 4(k 2 − m 2 c ) 5 + · · · , f λαβ = ( k + m c )γ λ ( k + m c )γ α ( k + m c )γ β ( k + m c ) , f αβµν = ( k + m c )γ α ( k + m c )γ β ( k + m c )γ µ ( k + m c )γ ν ( k + m c ) ,(7) and t n = λ n 2 , the λ n is the Gell-Mann matrix [41,42]. In Eq. (6), we retain the term q j σ µν q i originate from the Fierz re-arrangement of the q iqj to absorb the gluons emitted from other quark lines to extract the mixed condensate qg s σGq [24,33]. It is very difficult (or cumbersome) to carry out the integrals both in the coordinate and momentum spaces directly due to appearance of the partial derives ∂ µ and ∂ ν . We perform integral by parts to exclude the terms proportional to the tensor structure pµpν p 2 , which only contributes to the scalar tetraquark states, and simplify the correlation function Π µν (p) greatly, Π µν (p) = 2iε ijk ε imn ε i ′ j ′ k ′ ε i ′ m ′ n ′ d 4 xe ip·x ∂ µ Tr γ 5 C kk ′ (x)γ 5 CS jj ′ T (x)C ∂ ν Tr γ 5 C n ′ n (−x)γ 5 CS m ′ mT (−x)C .(8) Then we compute the integrals both in the coordinate and momentum spaces, and obtain the correlation function Π(p 2 ) therefore the spectral density at the level of quark-gluon degrees of freedom. Once analytical expressions of the QCD spectral density are obtained, we can take the quarkhadron duality below the continuum threshold s 0 and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the QCD sum rules: λ 2 Y exp − M 2 Y T 2 = s0 4m 2 c ds ρ(s) exp − s T 2 ,(9) where ρ(s) = ρ 0 (s) + ρ 3 (s) + ρ 4 (s) + ρ 5 (s) + ρ 6 (s) + ρ 7 (s) + ρ 8 (s) + ρ 10 (s) ,(10)ρ 0 (s) = 1 61440π 6 dydz yz (1 − y − z) 4 1 − y s − m 2 c 4 s − m 2 c − 2y 8s − 3m 2 c − 1 12288π 6 dydz yz 2 (1 − y − z) 3 1 − y s − m 2 c 4 3s − m 2 c + 1 3840π 6 dydz y 2 z 2 (1 − y − z) 3 s − m 2 c 3 18s 2 − 16sm 2 c + 3m 4 c + 1 20480π 6 dydz yz (1 − y − z) 3 s − m 2 c 4 s − m 2 c + 4y 8s − 3m 2 c ,(11)ρ 3 (s) = − m c qq 48π 4 dydz yz (1 − y − z) s − m 2 c 2 s − m 2 c + 3y 2s − m 2 c ,(12)ρ 4 (s) = − m 2 c 9216π 4 α s GG π dydz z (1 − y − z) 4 y 2 (1 − y) s − m 2 c s − m 2 c − 2y 5s − 3m 2 c + m 2 c 9216π 4 α s GG π dydz z 2 (1 − y − z) 3 y 2 (1 − y) s − m 2 c 9s − 5m 2 c − m 2 c 1152π 4 α s GG π dydz z 2 (1 − y − z) 3 y 15s 2 − 20sm 2 c + 6m 4 c − m 2 c 1024π 4 α s GG π dydz z (1 − y − z) 3 y 2 s − m 2 c 2 + m 2 c 3072π 4 α s GG π dydz z y 2 + y z 2 (1 − y − z) 3 s − m 2 c s − m 2 c − 2y 5s − 3m 2 c + 1 6144π 4 α s GG π dydz z (1 − y − z) 3 1 − y s − m 2 c 2 s − m 2 c − 6y 2s − m 2 c − 1 6144π 4 α s GG π dydz z 2 (1 − y − z) 2 1 − y s − m 2 c 2 11s − 5m 2 c + 1 256π 4 α s GG π dydz yz 2 (1 − y − z) 2 s − m 2 c 7s 2 − 8sm 2 c + 2m 4 c − 1 2048π 4 α s GG π dydz z (1 − y − z) 2 s − m 2 c 2 s − m 2 c − 6y 2s − m 2 c ,(13)ρ 5 (s) = m c qg s σGq 64π 4 dydz yz s − m 2 c s − m 2 c + y 5s − 3m 2 c + m c qg s σGq 128π 4 dydz y (1 − y − z) (1 − y) s − m 2 c 2 − m c qg s σGq 128π 4 dydz y 2 (1 − y − z) s − m 2 c 9s − 5m 2 c − 3m c qg s σGq 128π 4 dydz (y + z) (1 − y − z) s − m 2 c 2 + m c qg s σGq 128π 4 dydz y (1 − y − z) s − m 2 c s − m 2 c − 2y 5s − 3m 2 c ,(14)ρ 6 (s) = m 2 c qq 2 12π 2 dy y (1 − y) s − m 2 c ,(15)ρ 7 (s) = m 3 c qq 144π 2 α s GG π dydz z y 2 + y z 2 (1 − y − z) 1 + 3y + ys δ s − m 2 c − m c qq 48π 2 α s GG π dydz y (1 − y − z) z s − m 2 c + y 4s − 3m 2 c + m c qq 192π 2 α s GG π dydz z 3 s − m 2 c − 2y 4s − 3m 2 c − m c qq 288π 2 α s GG π dydz y (1 − y) s − m 2 c + m c qq 288π 2 α s GG π dydz y 2 7s − 5m 2 c − m c qq 288π 2 α s GG π dy y (1 − y) s − m 2 c + y 4s − 3 m 2 c ,(16)ρ 8 (s) = − m 2 c qq qg s σGq 24π 2 dy y (1 − y) 3 + s δ s − m 2 c + m 2 c qq qg s σGq 24π 2 dy ,(17)ρ 10 (s) = 203m 2 c qg s σGq 2 9216π 2 dy δ s − m 2 c + m 2 c qg s σGq 2 32π 2 dy y (1 − y) 1 + 2s 3T 2 + s 2 6T 4 δ s − m 2 c − m 2 c qg s σGq 2 48π 2 dy 1 + s 2T 2 δ s − m 2 c − m 4 c qq 2 108T 2 α s GG π dy 1 − y y 2 δ s − m 2 c + m 2 c qq 2 36 α s GG π dy 1 − y y δ s − m 2 c − m 2 c qq 2 108 α s GG π dy 1 + s 2T 2 δ s − m 2 c + m 2 c qq 2 36 α s GG π dy y (1 − y) 1 + 2s 3T 2 + s 2 6T 4 δ s − m 2 c ,(18) where dydz = s ), respectively, and are discarded [24,33]. We derive Eq.(9) with respect to τ = 1 T 2 , then eliminate the pole residue λ Y , and obtain the QCD sum rules for the mass of the vector tetraquark state Y (4260/4220), y f yi dy 1−y zi dz, y f = 1+ √ 1−4m 2 c /s 2 , y i = 1− √ 1−4m 2 c /s 2 , z i =M 2 Y = − s0 4m 2 c ds d dτ ρ(s) exp (−τ s) s0 4m 2 c dsρ(s) exp (−τ s) .(19) Numerical results and discussions We take the standard values of the vacuum condensates qq = −(0.24 ± 0.01 GeV) 3 , qg s σGq = m 2 0 qq , m 2 0 = (0.8 ± 0.1) GeV 2 , αsGG π = (0.33 GeV) 4 at the energy scale µ = 1 GeV [40,41,43], and choose the M S mass m c (m c ) = (1.275 ± 0.025) GeV from the Particle Data Group [44], and set m u = m d = 0. Moreover, we take into account the energy-scale dependence of the input parameters on the QCD side, qq (µ) = qq (Q) α s (Q) α s (µ) 12 25 , qg s σGq (µ) = qg s σGq (Q) α s (Q) α s (µ) 2 25 , m c (µ) = m c (m c ) α s (µ) α s (m c ) 12 25 , α s (µ) = 1 b 0 t 1 − b 1 b 2 0 log t t + b 2 1 (log 2 t − log t − 1) + b 0 b 2 b 4 0 t 2 ,(20) where t = log µ 2 Λ 2 , b 0 = 33−2n f 12π , b 1 = 153−19n f 24π 2 , b 2 = 2857− 5033 9 n f + 325 27 n 2 f 128π 3 , Λ = 210 MeV, 292 MeV and 332 MeV for the flavors n f = 5, 4 and 3, respectively [44,45], and evolve all the input parameters to the optimal energy scale µ to extract the mass of the vector tetraquark state Y (4260/4220). In this article, we search for the ideal Borel parameter T 2 and continuum threshold parameter s 0 to satisfy the following four criteria: 1. Pole dominance at the phenomenological side; 2. Convergence of the operator product expansion; 3. Appearance of the Borel platforms; 4. Satisfying the energy scale formula, using try and error. In the four-quark system qq ′ QQ, the Q-quark serves as a static well potential and combines with the light quark q to form a heavy diquark D in color antitriplet or combines with the light antiquarkq ′ to form a heavy meson-like state or correlation (not a physical meson) in color singlet, while theQ-quark serves as another static well potential and combines with the light antiquarkq ′ to form a heavy antidiquarkD in color triplet or combines with the light quark state q to form another heavy meson-like state or correlation (not a physical meson) in color singlet [24,34,38]. Then the D andD combine with together to form a compact tetraquark state, the two meson-like states (not two physical mesons) combine together to form a physical molecular state [24,34,38], the two heavy quarks Q andQ stabilize the tetraquark state [7]. The tetraquark states DD are characterized by the effective heavy quark masses M Q and the virtuality [24,34,38]. We cannot obtain energy scale independent QCD sum rules, but we have an energy scale formula to determine the energy scales consistently, which works well even for the hidden-charm pentaquark states [46], the updated value M c = 1.82 GeV [11]. V = M 2 X/Y /Z − (2M Q ) 2 . It is natural to take the energy scale µ = V = M 2 X/Y /Z − (2M Q ) 2 In Refs. [24,33,34], we study the hidden-charm or hidden-bottom tetraquark states, the heavy diquarks and heavy antidiquarks are in relative S-wave, if there exist relative P-waves, the Pwaves lie in between the heavy (anti)quark and light (anti)quark in the heavy (anti)diquark. In the present work, we study the vector tetraquark state which has a relative P-wave between the charmed diquark and charmed antidiquark. If a relative P-wave costs about 0.5 GeV, then the energy scale formula is modified to be µ = M 2 Y − (2M c + 0.5 GeV) 2 = M 2 Y − (4.1 GeV) 2 .(21) In calculations, we observe that if we take the continuum threshold parameter √ s 0 = 4.8±0.1 GeV, Borel parameter T 2 = (2.2 − 2.8) GeV 2 , energy scale µ = 1.1 GeV, the pole contribution of the ground state vector tetraquark state Y (4260/4220) is about (49 − 81)%, the predicted mass is about M Y = 4.24 GeV, the modified energy scale formula is well satisfied. In Fig.1, we plot the pole contribution with variation of the Borel parameter, from the figure, we can see that the pole contribution decreases monotonously with increase of the Borel parameter, the pole contribution reaches about 50% at the point T 2 = 2.8 GeV 2 and √ s 0 = 4.7 GeV, we can obtain the upper bound T 2 max = 2.8 GeV 2 . In Fig.2, we plot the contributions of the vacuum condensates of dimension n in the operator product expansion, which are defined by D(n) = s0 4m 2 c ds ρ n (s) exp − s T 2 s0 4m 2 c ds ρ(s) exp − s T 2 .(22) From the figure, we can see that the contributions of the vacuum condensates of dimensions 3, 5, 6 and 8 are very large, and change quickly with variation of the Borel parameter T 2 at the region T 2 < 2.2 GeV 2 , the operator product expansion is not convergent, we can obtain the lower bound [44], which supports assigning the Y (4260/4220) to be the Cγ 5 ⊗ ↔ ∂ µ ⊗γ 5 C type vector tetraquark state. The average value of the width of the Y (4260) is 55 ± 19 MeV, the relative P-wave between the diquark and antidiquark disfavors rearrangement of the quarks to form meson pairs, which can account for the small width. From Fig.3, we can see that the mass M Zc(4100) lies below the lower bound of the predicted mass of the Cγ 5 ⊗ ↔ ∂ µ ⊗γ 5 C type vector tetraquark state ccqq, which disfavors assigning the Z c (4100) to be the Cγ 5 ⊗ ↔ ∂ µ ⊗γ 5 C type vector tetraquark state ccqq. In Refs. [47,48], we study the Cγ µ ⊗ γ µ C-type, Cγ µ γ 5 ⊗ γ 5 γ µ C-type, Cγ 5 ⊗ γ 5 C-type, C ⊗ Ctype cscs scalar tetraquark states with the QCD sum rules in a systematic way, and obtain the predictions M Cγµ⊗γ µ C = 3.92 +0. 19 −0.18 GeV and M Cγ5⊗γ5C = 3.89±0.05 GeV, which support assigning the X(3915) to be the Cγ µ ⊗ γ µ C-type or Cγ 5 ⊗ γ 5 C-type cscs scalar tetraquark state. In fact, the SU (3) breaking effects of the masses of the cscs and cqcq tetraquark states from the QCD sum rules are rather small, if the scalar tetraquark state cqcq has the mass M Cγµ⊗γ µ C = 3.92 +0. 19 −0.18 GeV, which is compatible with the LHCb data M Zc = 4096 ± 20 +18 −22 MeV and Γ Zc = 152 ± 58 +60 −35 MeV considering the uncertainties [39], and favors assigning the Z c (4100) to be the Cγ µ ⊗ γ µ C-type scalar tetraquark state. In Ref. [27], we choose the C⊗γ µ C type and Cγ 5 ⊗γ 5 γ µ C type vector currents to study the vector tetraquark states, the net effects of the relative P-waves are embodied in the underlined γ 5 in the Cγ 5 γ 5 ⊗ γ µ C type and Cγ 5 ⊗ γ 5 γ µ C type currents or in the underlined γ α in the Cγ α γ α ⊗ γ µ C type currents, and obtain the masses M C⊗γµC = 4.59 ± 0.08 GeV and M Cγ5⊗γ5γµC = 4.34 ± 0.08 GeV. The C ⊗ γ µ C type tetraquark states have larger masses than the corresponding Cγ 5 ⊗ γ 5 γ µ C type tetraquark states, as C⊗γ µ C = Cγ 5 γ 5 ⊗ γ µ C ⊕ Cγ α γ α ⊗ γ µ C and Cγ 5 ⊗γ 5 γ µ C = Cγ 5 ⊗γ 5 γ µ C, the Cγ µ diquark states have slightly larger masses than the corresponding Cγ 5 diquark states from the QCD sum rules [29,30]. The vector tetraquark masses M C⊗γµC and M Cγ5⊗γ5γµC differ from the vector tetraquark mass M Cγ5⊗ ↔ ∂ µ ⊗γ5C greatly. For the conventional ground state cq mesons, the energy gaps between the S-wave and P-wave states are about 0.5 GeV, if the relative P-waves between the q-quark and c-quark in the diquark states cq cost about 0.5 GeV [44], the masses of the ↔ ∂ µ Cγ 5 ⊗ γ 5 C type, Cγ 5 ⊗ ↔ ∂ µ γ 5 C type, ↔ ∂ µ Cγ α ⊗ γ α C type and Cγ α ⊗ ↔ ∂ µ γ α C vector tetraquark states are estimated to be 4.4 GeV according the Cγ µ ⊗ γ µ C-type and Cγ 5 ⊗ γ 5 Ctype scalar tetraquark masses [47,48], which differs from the present prediction M Cγ5⊗ ↔ ∂ µ ⊗γ5C = 4.24 ± 0.10 GeV greatly. Before draw a definite conclusion, we should study the masses of the ↔ ∂ µ Cγ 5 ⊗γ 5 C type, Cγ 5 ⊗ ↔ ∂ µ γ 5 C type, ↔ ∂ µ Cγ α ⊗γ α C type and Cγ α ⊗ ↔ ∂ µ γ α C vector tetraquark states with the QCD sum rules directly, this is our next work. Conclusion In this article, we take the Y (4260/4220) as the vector tetraquark state with J P C = 1 −− , and construct the Cγ 5 ⊗ ↔ ∂ µ ⊗γ 5 C type current to study its mass and pole residue with the QCD sum rules in details by taking into account the vacuum condensates up to dimension 10 in a consistent way in the operator product expansion, and use the modified energy scale formula µ = M 2 X/Y /Z − (2M c + 0.5GeV) 2 with the effective c-quark mass M c to determine the optimal energy scale of the QCD spectral density. The predicted mass M Y = 4.24 ± 0.10 GeV is in excellent agreement with the experimental value M Y (4220) = 4222.0 ± 3.1 ± 1.4 MeV from the BESIII collaboration or the experimental value M Y (4260) = 4230.0 ± 8.0 MeV from Particle Data Group, and supports assigning the Y (4260/4220) to be the Cγ 5 ⊗ ↔ ∂ µ ⊗γ 5 C type vector tetraquark state, and disfavors assigning the Z c (4100) to be the Cγ 5 ⊗ ↔ ∂ µ ⊗γ 5 C type vector tetraquark state. It is the first time that the QCD sum rules have reproduced the mass of the Y (4260/4220) as a vector tetraquark state. when the δ functions δ s − m 2 c and δ s − m 2 c 2appear.In this article, we carry out the operator product expansion up to the vacuum condensates of dimension-10, and take into account the vacuum condensates which are vacuum expectations of the operators of the orders O(α k s ) with k ≤ 1 consistently. The condensates g 3 s GGG , s σGq have the dimensions 6, 8, 9, respectively, but they are the vacuum expectations of the operators of the order O(α3/2 s ), O(α 2 s ), O(α 3/2 T 2 2min = 2.2 GeV 2 . At the region T 2 ≥ 2.2 GeV 2 , the contribution of the vacuum condensate of dimension n = 3 is large, but the contributions of the vacuum condensates of dimensions 3, 5, 6, 8 have the hierarchy D(3) ≫ |D(5)| ∼ D(6) ≫ |D(8)|, the contributions of the vacuum condensates of the dimensions 4, 7, 10 are tiny, the operator product expansion is convergent. The Borel window is T 2 = (2.2 − 2.8) GeV 2 , where operator product expansion is well convergent. We take into account all uncertainties of the input parameters, and obtain the values of the mass and pole residue of the vector tetraquark state Y (4260/4220), which are shown explicitly in Figs.3-4, M Y = 4.24 ± 0.10 GeV , λ Y = (2.31 ± 0.45) × 10 −2 GeV 6 . (23) From Figs.3-4, we can see that there appear platforms in the Borel window. Now the four criteria of the QCD sum rules are all satisfied, and we expect to make reliable predictions. The predicted mass M Y = 4.24 ± 0.10 GeV is in excellent agreement with the experimental value M Y (4220) = 4222.0 ± 3.1 ± 1.4 MeV from the BESIII collaboration [23], or the experimental value M Y (4260) = 4230.0 ± 8.0 MeV from Particle Data Group Figure 1 : 1The pole contributions with variation of the Borel parameter T 2 , where the A, B and C denote the threshold parameters √ s 0 = 4.7 GeV, 4.8 GeV and 4.9 GeV, respectively. Figure 2 : 2The contributions of the vacuum condensates of dimension n with variation of the Borel parameter T 2 for the threshold parameter √ s 0 = 4.8 GeV. Figure 3 : 3The mass of the Y (4260/4220) as vector tetraquark state with variation of the Borel parameter T 2 . Figure 4 : 4The pole residue of the Y (4260/4220) as vector tetraquark state with variation of the Borel parameter T 2 . AcknowledgementsThis work is supported by National Natural Science Foundation, Grant Number 11775079. . B Aubert, Phys. Rev. Lett. 95142001B. Aubert et al, Phys. Rev. Lett. 95 (2005) 142001. . C Z Yuan, Phys. Rev. Lett. 99182004C. Z. Yuan et al, Phys. Rev. Lett. 99 (2007) 182004. . Q He, Phys. Rev. 7491104Q. He et al, Phys. Rev. D74 (2006) 091104. . L Maiani, V Riquer, F Piccinini, A D Polosa, Phys. Rev. 7231502L. Maiani, V. Riquer, F. Piccinini and A. D. Polosa, Phys. Rev. D72 (2005) 031502. . L Maiani, F Piccinini, A D Polosa, V Riquer, Phys. Rev. 89114010L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. D89 (2014) 114010. . A Ali, L Maiani, A V Borisov, I Ahmed, M Aslam, A Y Parkhomenko, A D Polosa, A Rehma, Eur. Phys. J. C78. 29A. Ali, L. Maiani, A. V. Borisov, I. Ahmed, M. Jamil Aslam, A. Y. Parkhomenko, A. D. Polosa and A. Rehma, Eur. Phys. J. C78 (2018) 29. . S J Brodsky, D S Hwang, R F Lebed, Phys. Rev. Lett. 113112001S. J. Brodsky, D. S. Hwang and R. F. Lebed, Phys. Rev. Lett. 113 (2014) 112001. . J R Zhang, M Q Huang, Phys. Rev. 8336005J. R. Zhang and M. Q. Huang, Phys. Rev. D83 (2011) 036005. . J R Zhang, M Q Huang, JHEP. 101157J. R. Zhang and M. Q. Huang, JHEP 1011 (2010) 057. . R M Albuquerque, M Nielsen, Nucl. Phys. 815532009R. M. Albuquerque and M. Nielsen, Nucl. Phys. A815 (2009) 532009; . Erratum-Ibid, 85748Erratum-ibid. A857 (2011) 48. . Z G Wang, Eur. Phys. J. 76387Z. G. Wang, Eur. Phys. J. C76 (2016) 387. . S L Zhu, Phys. Lett. 625212S. L. Zhu, Phys. Lett. B625 (2005) 212; . F E Close, P R Page, Phys. Lett. 628215F. E. Close and P. R. Page, Phys. Lett. B628 (2005) 215. . L Liu, JHEP. 07126L. Liu et al, JHEP 07 (2012) 126. . E Braaten, C Langmack, D. Hudson Smith, Phys. Rev. 9014044E. Braaten, C. Langmack and D. Hudson Smith, Phys. Rev. D90 (2014) 014044. . X Li, M B Voloshin, Mod. Phys. Lett. 291450060X. Li and M. B. Voloshin, Mod. Phys. Lett. A29 (2014) 1450060. . M Cleven, Q Wang, F K Guo, C Hanhart, U G Meissner, Q Zhao, Phys. Rev. 9074039M. Cleven, Q. Wang, F. K. Guo, C. Hanhart, U. G. Meissner and Q. Zhao, Phys. Rev. D90 (2014) 074039. . Z G Wang, Chin. Phys. 4183103Z. G. Wang, Chin. Phys. C41 (2017) 083103. . D Y Chen, J He, X Liu, Phys. Rev. 8354021D. Y. Chen, J. He and X. Liu, Phys. Rev. D83 (2011) 054021; . D Y Chen, X Liu, X Q Li, H W Ke, Phys. Rev. 9314011D. Y. Chen, X. Liu, X. Q. Li and H. W. Ke, Phys. Rev. D93 (2016) 014011. . A Torres, K P Khemchandani, D Gamermann, E Oset, Phys. Rev. 8094012A. Martinez Torres, K. P. Khemchandani, D. Gamermann and E. Oset, Phys. Rev. D80 (2009) 094012; . X H Liu, G Li, Phys. Rev. 8814013X. H. Liu and G. Li, Phys. Rev. D88 (2013) 014013. . C F Qiao, Phys. Lett. 639263C. F. Qiao, Phys. Lett. B639 (2006) 263. . M Ablikim, Phys. Rev. Lett. 11492003M. Ablikim et al, Phys. Rev. Lett. 114 (2015) 092003. . M Ablikim, Phys. Rev. Lett. 11892002M. Ablikim et al, Phys. Rev. Lett. 118 (2017) 092002. . M Ablikim, Phys. Rev. Lett. 11892001M. Ablikim et al, Phys. Rev. Lett. 118 (2017) 092001. . Z G Wang, Eur. Phys. J. 742874Z. G. Wang, Eur. Phys. J. C74 (2014) 2874. . W Chen, S L Zhu, Phys. Rev. 8334010W. Chen and S. L. Zhu, Phys. Rev. D83 (2011) 034010. . J M Dias, R M Albuquerque, M Nielsen, C M Zanetti, Phys. Rev. 86116012J. M. Dias, R. M. Albuquerque, M. Nielsen and C. M. Zanetti, Phys. Rev. D86 (2012) 116012. . Z G Wang, Eur. Phys. J. C78. 518Z. G. Wang, Eur. Phys. J. C78 (2018) 518. . A De Rujula, H Georgi, S L Glashow, Phys. Rev. 12147A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev. D12 (1975) 147; . T Degrand, R L Jaffe, K Johnson, J E Kiskis, Phys. Rev. 122060T. DeGrand, R. L. Jaffe, K. Johnson and J. E. Kiskis, Phys. Rev. D12 (1975) 2060. . Z G Wang, Eur. Phys. J. 711524Z. G. Wang, Eur. Phys. J. C71 (2011) 1524; . R T Kleiv, T G Steele, A Zhang, Phys. Rev. 87125018R. T. Kleiv, T. G. Steele and A. Zhang, Phys. Rev. D87 (2013) 125018. . Z G Wang, Commun. Theor. Phys. 59451Z. G. Wang, Commun. Theor. Phys. 59 (2013) 451. . L Tang, X Q Li, Chin. Phys. 36578L. Tang and X. Q. Li, Chin. Phys. C36 (2012) 578. . H G Dosch, M Jamin, B Stech, Z. Phys. 42167H. G. Dosch, M. Jamin and B. Stech, Z. Phys. C42 (1989) 167; . M Jamin, M Neubert, Phys. Lett. 238387M. Jamin and M. Neubert, Phys. Lett. B238 (1990) 387. . Z G Wang, T Huang, Phys. Rev. 8954019Z. G. Wang and T. Huang, Phys. Rev. D89 (2014) 054019. . Z G Wang, T Huang, Nucl. Phys. 93063Z. G. Wang and T. Huang, Nucl. Phys. A930 (2014) 63; . Z G Wang, Y F Tian, Int. J. Mod. Phys. 301550004Z. G. Wang and Y. F. Tian, Int. J. Mod. Phys. A30 (2015) 1550004; . Z G Wang, Commun. Theor. Phys. 63325Z. G. Wang, Commun. Theor. Phys. 63 (2015) 325; . Z G Wang, Commun. Theor. Phys. 66335Z. G. Wang, Commun. Theor. Phys. 66 (2016) 335. . Z G Wang, Eur. Phys. J. 59675Z. G. Wang, Eur. Phys. J. C59 (2009) 675; . Z G Wang, J. Phys. 3685002Z. G. Wang, J. Phys. G36 (2009) 085002. . H Sundu, S S Agaev, K Azizi, arXiv:1805.04705H. Sundu, S. S. Agaev and K. Azizi, arXiv:1805.04705. . Z G Wang, S L Wan, Chin. Phys. Lett. 233208Z. G. Wang anf S. L. Wan, Chin. Phys. Lett. 23 (2006) 3208. . Z G Wang, T Huang, Eur. Phys. J. 742891Z. G. Wang and T. Huang, Eur. Phys. J. C74 (2014) 2891; . Z G Wang, Eur. Phys. J. 742963Z. G. Wang, Eur. Phys. J. C74 (2014) 2963. . R Aaij, arXiv:1809.07416R. Aaij et al, arXiv:1809.07416. . M A Shifman, A I Vainshtein, V I Zakharov, Nucl. Phys. 147448Nucl. Phys.M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B147 (1979) 385; Nucl. Phys. B147 (1979) 448. . L J Reinders, H Rubinstein, S Yazaki, Phys. Rept. 1271L. J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rept. 127 (1985) 1. QCD: Renormalization for the practitioner. P Pascual, R Tarrach, SpringerBerlin HeidelbergP. Pascual and R. Tarrach, "QCD: Renormalization for the practitioner", Springer Berlin Heidelberg (1984). . P Colangelo, A Khodjamirian, hep-ph/0010175P. Colangelo and A. Khodjamirian, hep-ph/0010175. . M Tanabashi, Phys. Rev. D98. 30001M. Tanabashi et al, Phys. Rev. D98 (2018) 030001. . S Narison, R Tarrach, Phys. Lett. 125 B. 217S. Narison and R. Tarrach, Phys. Lett. 125 B (1983) 217; QCD as a theory of hadrons from partons to confinement. S Narison, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 171S. Narison, "QCD as a theory of hadrons from partons to confinement", Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 17 (2007) 1. . Z G Wang, Eur. Phys. J. 7670Z. G. Wang, Eur. Phys. J. C76 (2016) 70; . Z G Wang, T Huang, Eur. Phys. J. 7643Z. G. Wang and T. Huang, Eur. Phys. J. C76 (2016) 43. . Z G Wang, Eur. Phys. J. 7778Z. G. Wang, Eur. Phys. J. C77 (2017) 78. . Z G Wang, Eur. Phys. J. A53. 19Z. G. Wang, Eur. Phys. J. A53 (2017) 19.
[]
[ "LARGE SCALE STRUCTURE AND COSMIC RAYS REVISITED", "LARGE SCALE STRUCTURE AND COSMIC RAYS REVISITED" ]
[ "R Ugoccioni \nCentro Multidisciplinar de Astrofísica\nInstituto Superior Técnico\nAv. Rovisco Pais 11049-001LisboaPortugal\n", "L Teodoro [email protected] \nCentro Multidisciplinar de Astrofísica\nInstituto Superior Técnico\nAv. Rovisco Pais 11049-001LisboaPortugal\n", "U Wichoski [email protected] \nCentro Multidisciplinar de Astrofísica\nInstituto Superior Técnico\nAv. Rovisco Pais 11049-001LisboaPortugal\n" ]
[ "Centro Multidisciplinar de Astrofísica\nInstituto Superior Técnico\nAv. Rovisco Pais 11049-001LisboaPortugal", "Centro Multidisciplinar de Astrofísica\nInstituto Superior Técnico\nAv. Rovisco Pais 11049-001LisboaPortugal", "Centro Multidisciplinar de Astrofísica\nInstituto Superior Técnico\nAv. Rovisco Pais 11049-001LisboaPortugal" ]
[]
We investigate the possibility that ultra high energy cosmic rays (E > 10 19 eV) are related to the distribution of matter on large scales. The large scale structure (LSS) data stems from the recent IRAS PSCz redshift survey. We present preliminary predictions drawn from an anisotropic distribution of sources which follows the galaxy distribution.
10.1142/9789812811035_0017
[ "https://export.arxiv.org/pdf/astro-ph/0011171v1.pdf" ]
119,342,569
astro-ph/0011171
d699cadb32da10595b014bfc76b9b3a5235dd606
LARGE SCALE STRUCTURE AND COSMIC RAYS REVISITED 8 Nov 2000 R Ugoccioni Centro Multidisciplinar de Astrofísica Instituto Superior Técnico Av. Rovisco Pais 11049-001LisboaPortugal L Teodoro [email protected] Centro Multidisciplinar de Astrofísica Instituto Superior Técnico Av. Rovisco Pais 11049-001LisboaPortugal U Wichoski [email protected] Centro Multidisciplinar de Astrofísica Instituto Superior Técnico Av. Rovisco Pais 11049-001LisboaPortugal LARGE SCALE STRUCTURE AND COSMIC RAYS REVISITED 8 Nov 2000We investigate the possibility that ultra high energy cosmic rays (E > 10 19 eV) are related to the distribution of matter on large scales. The large scale structure (LSS) data stems from the recent IRAS PSCz redshift survey. We present preliminary predictions drawn from an anisotropic distribution of sources which follows the galaxy distribution. We investigate the possibility that ultra high energy cosmic rays (E > 10 19 eV) are related to the distribution of matter on large scales. The large scale structure (LSS) data stems from the recent IRAS PSCz redshift survey. We present preliminary predictions drawn from an anisotropic distribution of sources which follows the galaxy distribution. Introduction FISIST/12-2000/CENTRA Ultra high energy cosmic rays (UHECR) are particles with kinetic energies above ∼ 10 18 eV. 1 The nature of these energetic particles is presently unknown. The reason is twofold: i) These particles interact on the top of the Earth's atmosphere producing extensive air showers (EAS) that can be observed from the ground; in this case, the primary particle can not be observed directly; and ii) at these ultra high energies (UHE) the fluxes are extremely low (less than 1 particle per square kilometer per year). This makes it impracticable to observe the UHE particles directly using balloons, satellites or spacecrafts due to their small acceptance. The observation methods are indirect and rely on the observation of the secondary particles produced in the EAS. The hadronic particles, as well as muons and electrons created by the interactions of the primary particle in the atmosphere, are detected on the ground. The EAS also produces detectable fluorescent light photons due to the excitation of nitrogen molecules in the air by the charged secondary particles. Yet, the secondary charged particles that travel with velocities higher than the velocity of the light in the air generate Cherenkov radiation that can also be detected. Despite the fact that the EAS can be observed by the detection of different kinds of secondaries using various techniques, the determination of the nature of the primary particle is very difficult and model dependent. As the fluxes are low it is necessary to use large ground arrays, and/or many fluorescent light and Cherenkov radiation detectors. Until the present moment, the ground arrays and the fluorescent light detectors have gathered only a handful of events in the UHE range. The number of events has not been enough to tell us whether the sources are extragalactic or are located in the Galaxy. If the primary particle is a γ-ray or a neutrino the arrival direction would point back directly to the source. This would also be the case for charged particles if the Galactic magnetic field is 10 −6 G and the extragalactic magnetic fields are 10 −9 G in the case of extragalactic sources. The distance to the source is also constrained for most kinds of primaries: If the primary particle is a nucleus or a proton (antiproton) the distance to the source is limited to less than ∼ 100 Mpc for particles with arrival energies above ∼ 6 × 10 19 eV (GZK cutoff, see Fig.(1)). Due to interactions with the cosmic microwave background (CMB) photons these particles rapidly lose energy. Sources of γ-rays must be even closer because of the short mean absorption length for the UHE photons traveling in the CMB. The mechanism that provides particles with UHE is also not known. The ignorance about the sources makes it harder to determine the mechanism at work. For the UHE events no source candidate in the vicinity of the region to where the arrival direction points back has been found yet. On the other hand, some analysis of the showers profile have been favoring protons as the primary particle. 2 As it was mentioned above, the number of UHE events is still too small to allow us, based on the statistics, to answer questions about their isotropy and composition. In this work, we assume that the UHECR primary particles above 10 19 eV are predominantly extragalactic protons and that the sources are related to the distribution of matter on large scales. It means that without specifying the sources themselves or the acceleration mechanism, we would expect an excess of events coming from regions with mass overdensities and less events coming from regions with mass underdensities. In the § 2 we briefly describe the formalism used; the propagation code is described in the § 3; and the smoothing procedure of the density field is described in § 4. Our results are presented in the § 5. Formalism In this contribution we apply a generalization of the formalism described in Waxman, Fisher and Piran. 3 We model the population of UHECR sources S within a "box" ∆V centered at (z,Ω) as drawn from a Poisson distribution prob S (z,Ω) =S (z,Ω) S S! exp −S(z,Ω) , ugoccioni: submitted to World Scientific on March 19, 2022 whose mean value isS(z,Ω) =s(z)B δρ(z,Ω) ∆V . Heres(z) denotes the average comoving number of UHECR sources at redshift z and B is some bias functional of the local galaxy distribution δρ(z,Ω). The generating function (g.f.) of such a distribution is f S (u; z,Ω) = exp S (z,Ω)(u − 1) ,(1) where u is a dummy variable. The detected number N of UHECR produced by a source within ∆V with observed energy larger than E is also modeled by a Poisson distribution prob N (≥ E) =N (E, z) N N ! exp −N (E, z) , with mean valueN (E, z) = A Tṅ 0 [E inj (E, z)] s 0 (1 + z) 4πd L (z) 2 , wheres 0 =s(z = 0), d 2 L (z) = 4c 2 H −2 0 (2 + z − 2 √ 1 + z) for an Ω = 1 Universe; A and T denote the detector area and observation time, respectively; E inj is the energy with which a UHECR observed with energy E was produced at redshift z; andṅ 0 is the number of UHE protons emitted by a source per unit time and is assumed to be proportional to dN/dE inj . We have assumed that the source differential spectrum is a power law in energy dN/dE inj ∝ E −(γ+1) inj . The g.f. of the last probability distribution is given by g N (u; z, E) = exp N (E, z)(u − 1) .(2) Hence, it is straightforward to show from equations (1) and (2) that the g.f. for the probability of observing a total of N events from ∆V , with an energy larger than E, is expressed by F (u; z,Ω, E) = exp S (z,Ω) exp N (E, z)(u − 1) − 1 . The overall UHECR distribution coming from a collection of independent volume elements ∆V i has the following g.f.: F (u; i ∆V i , E) = i F u; z i ,Ω i , E , which for a given line of sight (l.o.s.) can be expressed as an integral over z F (u;Ω, E) = exp zmax 0S (z,Ω) exp N (E, z)(u − 1) − 1 dV ,(3) ugoccioni: submitted to World Scientific on March 19, 2022 where dV = c |dt/dz| d 2 L (z)(1 + z) −1 dz. In defining λ(Ω) as λ(Ω) ≡ zmax 0S (z,Ω)dV, one can characterize the distribution of UHECR produced by sources along the l.o.s. with energy larger than E by the g.f.. G(u;Ω, E) ≡ 1 λ(Ω) zmax 0S (z,Ω) exp N (E, z)(u − 1) dV. From equation (3) is then straightforward to show that F (u;Ω, E) = exp λ(Ω) G(u;Ω, E) − 1 , which still is a compound Poisson distribution, although G is not Poissonian. Propagation code The propagation equation takes into account energy losses of the UHE protons due to: i) Adiabatic expansion of the Universe; ii) e + e − pair production; and iii) pion production due to interactions with CMB photons. A proton observed at present (z = 0) with energy E must have been produced at an epoch z with energy E inj = E inj (E, z). We assume that the influence of the magnetic fields on particles with energies E > 10 19 eV is negligible. Figure 1 shows the decrease of energy as a function of the distance from the source for UHE protons. We note that for a proton to be observed with energies above ∼ 6 × 10 19 eV the source must be within ∼ 100 Mpc from the observer irrespective to E inj . Smoothed density field The galaxy distribution is estimated from the IRAS PSCz redshift survey. 4 We have computed the smoothed density field on a spherical grid up to 200 h −1 Mpc. The Gaussian-smoothed density field at a grid point n is given by 1 + δ g (c z n ) = 1 (2π) 3/2 σ 3 sm,n i 1 φ(c z i ) exp − (c z n − c z i ) 2 2σ 2 sm,n .(4) We have divided the sphere in 72 bins of approximately equal area. Radially, the bin size increases in proportion to the IRAS PSCz inter-particle spacing [nφ(cz)] −1/3 . This smoothing scheme is tailored to keep the shot-noise uncertainties in the density field roughly constant through out the sampled volume. A more detailed analysis of the IRAS PSCz density field can be found in Branchini et al. 5 Results We have found that the final results are independent of the cosmological parameters (Ω, Λ). Thus, we have used Ω = 1 throughout our calculations for the sake of simplicity. For the bias functional B[δ( x)] we have considered B[δ( x)] = 1+δ( x). Figure 2 presents maps of fluctuations in the mean Cosmic Ray intensity, δ CR (E,Ω) = 4πN (E,Ω) dΩN (E,Ω) − 1,(5) for E = (6, 10) × 10 19 eV. In the maps we clearly see the regions from where an excess and a deficit of UHECR events is expected following the LSS. The specific predictions for future experiments as the Auger project and HiRes will be presented elsewhere. 6 Figure 1 . 1Propagation of UHE protons in the CMB. The lines represent various proton injection energies, E inj = 10 7 , 10 6 , 10 5 , 10 4 , 10 3 EeV (from top to bottom respectively). Figure 2 . 2Aitoff projection of δ CR , for E inj ≈ 60 and 100 EeV (γ = 1.0). The heavy contour denotes the zero contour. Dark (light) grey contours denote positive (negative) fluctuations equally-spaced at 0.20. The long-dashed line represents the Super-Galactic plane. also been supported by FCT under the project PRAXIS/C/FIS/13196/98. Acknowledgements: This work has been supported by "Fundação para a Ciência e a Tecnologia" (FCT) under the program "PRAXIS XXI". L.T. has ugoccioni: submitted to World Scientific on March 19, 2022 . M Nagano, A A Watson, Rev. Mod. Phys. 72689see also A.A. Watson in these ProceedingsM. Nagano and A.A. Watson, Rev. Mod. Phys. 72, 689 (2000); see also A.A. Watson in these Proceedings. . M Ave, Phys. Rev. Lett. 852244M. Ave et al., Phys. Rev. Lett. 85, 2244 (2000). . E Waxman, K B Fisher, T Piran, Astrophys. J. 4831E. Waxman, K. B. Fisher, and T. Piran, Astrophys. J. 483, 1 (1997). . W Saunders, PSCz collaboration MNRAS. 55317W Saunders et al. PSCz collaboration MNRAS 55, 317 (2000). . E Branchini, MNRAS. 3081E Branchini et al. MNRAS 308, 1 (1999). L Teodoro, R Ugoccioni, U Wichoski, preparation ugoccioni: submitted to World Scientific on March. 192022L. Teodoro, R. Ugoccioni, and U. Wichoski, in preparation ugoccioni: submitted to World Scientific on March 19, 2022
[]
[ "A Novel Multi-Secret Sharing Approach for Secure Data Warehousing and On-Line Analysis Processing in the Cloud", "A Novel Multi-Secret Sharing Approach for Secure Data Warehousing and On-Line Analysis Processing in the Cloud" ]
[ "Varunya Attasena \nUniversité de Lyon (Laboratoire ERIC)\nFrance\n", "Nouria Harbi \nUniversité de Lyon (Laboratoire ERIC)\nFrance\n", "Jérôme Darmont \nUniversité de Lyon (Laboratoire ERIC)\nFrance\n" ]
[ "Université de Lyon (Laboratoire ERIC)\nFrance", "Université de Lyon (Laboratoire ERIC)\nFrance", "Université de Lyon (Laboratoire ERIC)\nFrance" ]
[]
Cloud computing helps reduce costs, increase business agility and deploy solutions with a high return on investment for many types of applications, including data warehouses and online analytical processing. However, storing and transferring sensitive data into the cloud raises legitimate security concerns. In this paper, we propose a new multi-secret sharing approach for deploying data warehouses in the cloud and allowing on-line analysis processing, while enforcing data privacy, integrity and availability. We first validate the relevance of our approach theoretically and then experimentally with both a simple random dataset and the Star Schema Benchmark. We also demonstrate its superiority to related methods.
10.4018/ijdwm.2015040102
[ "https://arxiv.org/pdf/1701.05449v1.pdf" ]
1,897,642
1701.05449
4b3aeb7113166f31342e6f1ea1ac28c0352dd06b
A Novel Multi-Secret Sharing Approach for Secure Data Warehousing and On-Line Analysis Processing in the Cloud Varunya Attasena Université de Lyon (Laboratoire ERIC) France Nouria Harbi Université de Lyon (Laboratoire ERIC) France Jérôme Darmont Université de Lyon (Laboratoire ERIC) France A Novel Multi-Secret Sharing Approach for Secure Data Warehousing and On-Line Analysis Processing in the Cloud Data warehousesOLAPCloud computingSecret sharingData privacyData availabilityData integrity Cloud computing helps reduce costs, increase business agility and deploy solutions with a high return on investment for many types of applications, including data warehouses and online analytical processing. However, storing and transferring sensitive data into the cloud raises legitimate security concerns. In this paper, we propose a new multi-secret sharing approach for deploying data warehouses in the cloud and allowing on-line analysis processing, while enforcing data privacy, integrity and availability. We first validate the relevance of our approach theoretically and then experimentally with both a simple random dataset and the Star Schema Benchmark. We also demonstrate its superiority to related methods. INTRODUCTION Business intelligence (BI) has been an ever-growing trend for more than twenty years, but the recent advent of cloud computing now allows deploying data analytics even more easily. While building a traditional BI system typically necessitates an important initial investment, with the cloud pay-as-you-go model, users can punctually devote small amounts of resources in return for a one-time advantage. This trend is currently supported by numerous "BI as a service" offerings, with high economic stakes. Although cloud computing is currently booming, data security remains a top concern for cloud users and would-be users. Some security issues are inherited from classical distributed architectures, e.g., authentication, network attacks and vulnerability exploitation, but some directly relate to the new framework of the cloud, e.g., cloud service provider or subcontractor espionage, cost-effective defense of availability and uncontrolled mashups (Chow et al., 2009). In the context of cloud BI, privacy is of critical importance. Security issues are currently handled by cloud service providers (CSPs). But with the multiplication of CSPs and subcontractors in many countries, intricate legal issues arise, as well as another fundamental issue: trust. Telling whether trust should be placed in CSPs falls back onto end-users, with the implied costs. Critical security concerns in (especially public) cloud storage are depicted in Figure 1. User data might be deleted, lost or damaged. First, some CSPs have the policy of taking the highest profit. Therefore, unmodified or unaccessed data may be deleted to serve other customers. Second, data loss may also be caused by accidental, e.g., electrical or network failure, or intentional plans, e.g., maintenance or system backup. Moreover, virtual cloud architectures might not be sufficiently safeguarded from inside attacks. Finally, all CSPs cannot guarantee 100% data availability, although some cloud businesses must run on a 7/24 basis. Thus, data privacy, availability and integrity are major issues in cloud data security. Encrypting and replicating data can solve most of these issues, but existing solutions are greedy in resources such as data storage, memory, CPU and bandwidth. Moreover, cloud data warehouses (DWs) must be both highly protected and effectively refreshed and analyzed through on-line analysis processing (OLAP). Thence, while CSPs must optimize service quality and profit, users seek to reduce storage and access costs within the pay-as-you-go paradigm. Thus, in cloud DWs, the tradeoff between data security and large-scale OLAP analysis poses a great challenge (Chow et al., 2009;Sion, 2007). To address this challenge, we propose a global approach that relies on a new multi-secret sharing scheme, a family of encryption methods that enforce privacy and availability by design. Moreover, we incorporate in our approach features for data integrity verification and computation on shared data (or shares). Eventually, we minimize shared data volume. This paper expands (Attasena et al., 2013) along three axes. First, we complement the state of the art and deepen our analysis of related works. Second, we detail the section related to sharing a DW and specify the way OLAP queries run on shares. Finally, we complement our validation effort with new experiments, especially with the Star Schema Benchmark. The remainder of this paper is organized as follows. We first introduce and discuss previous research related to our proposal. Based on this diagnosis, we further motivate and position our work. Then, we detail our secret sharing-based approach, before providing a security analysis and performance evaluation that highlight the relevance of our proposal and demonstrates the enhancements it brings over existing methods. We finally conclude this paper and hint at future research perspectives. Encryption turns original data into unreadable cipher-text. Modern encryption schemes, such as homomorphic (HE -Melchor et al., 2008;Gentry, 2009) and incremental encryption (Bellare et al., 1994), help perform computations and updates on cipher-texts without decrypting them first. Partially HE allows only one operation, e.g., addition or multiplication, whereas fully HE supports several, but still does not allow mixed-operators. Unfortunately, HE is currently too computationally expensive for practical use. An older, well-known encryption strategy is secret sharing (Asmuth & Bloom, 1983;Blakley, 1979;Shamir, 1979), which distributes individually meaningless shares of data to n participants to enforce privacy. A subset of ≤ participants is required to reconstruct the secret. Moreover, up to − participants may disappear without compromising data availability. The drawback of this solution is the multiplication of the initial data volume by the number of participants. Modern secret sharing schemes, such as multi-secret sharing (Liuet al., 2012;Waseda & Soshi, 2012), verifiable secret sharing (Bu & Zhou, 2009), and verifiable multi-secret sharing (Bu & Yang, 2013;Eslami & Ahmadabadi, 2010;Hu et al., 2012), help reduce the volume of shares, verify the honesty of each participant, and both, respectively. Data anonymization (Cormode & Srivastava, 2009;Kenneally & Claffy, 2010;Machanavajjhala et al., 2007;Sweeney, 2002) is also used to enforce data privacy. In a database, only keys or sensitive information are protected (Sedeyao, 2012). Thus, data anonymization straightforwardly allows data querying. There are several models (e.g., -anonymized, -diversity) and techniques (hashing, hiding, permutation, shift…) to protect keys and sensitive information, respectively. For example, the -anonymized model transforms distinguishable records into indistinguishable records (Sweeney, 2002). The -diversity model creates different sensitive values from only one value in each key identification combination (Machanavajjhala et al., 2007). While cheap when accessing data, anonymization is not strong enough to protect against attacks such as homogeneity and background knowledge attacks (Sedeyao, 2012), and is not designed to address data availability and integrity issues. Data replication (Padmanabhan et al., 2008) is the process of copying some or all data from one location to one or several others. Its main purposes are to improve availability, faulttolerance and/or accessibility. A well-known data replication scheme is Reed Solomon (RS) code (Thomas & Schwarz, 2002), which is quite similar to secret sharing. RS code indeed distributes data amongst a group of participants and can reconstruct data even if some participants disappear, thus enforcing availability. RS code and secret sharing mostly differ in their driving goals, i.e., availability and privacy, respectively. Data verification (Bowers et al., 2009;Juels & Kaliski, 2007;Shacham & Waters, 2008;Wang et al., 2009) is the process of checking data integrity, by verifying data corruption caused by either accident or intruder attack, with the help of signatures (digital signature, message authentication, fingerprint…). However, since signature creation typically involves random or hash functions, they cannot guarantee 100% data correctness. Moreover, so-called outer code verifying methods (Juels & Kaliski, 2007) allow checking encrypted data without decrypting them first. Eventually, some security solutions directly relate to ours. Most apply Shamir's (1979) classical secret sharing to relational databases or data warehouses (Emekci et al., 2006;Hadavi & Jalili, 2010), thus, enforcing data privacy, availability and updating. In addition, Thompson et al. (2009), Wang et al. (2011 and Hadavi et al. (2012) also support data verification through HE, a hash function, and checksums and a hash function, respectively. Most of these methods allow computing at least one query type (aggregation, range and match queries) on shares. As in the three last cited approaches (here after denoted TWH for brevity), our strategy is to extend one security scheme presenting interesting characteristics, namely multi-secret sharing, by integrating the missing features needed in cloud DWs. However, in our approach, shared data volume is better controlled than in TWH's, i.e., it is significantly lower than times that of the original data volume. Moreover, we also incorporate both inner and outer code data verification in our solution, whereas TWH only feature inner code data verification. Finally, we also include capabilities from homomorphic and incremental encryption that allow updating and computing basic operations on shares. Thus, to the best of our knowledge, our multisecret sharing-based approach is the first attempt at securing data warehousing and OLAP while minimizing data volume. MULTI-SECRET SHARING OF CLOUD DATA WAREHOUSES The solution we propose is based on trusting neither CSPs nor network data transfers. It is subdivided into two schemes. Scheme-I is a new multi secret sharing scheme that transforms data into blocks (to optimize computing and storage costs), and shares data blocks at several CSPs'. Each CSP only stores part of the shares, which are not exploitable, neither by the CSP nor any intruder, because they have been transformed by a mathematical function. Though performing computations on shares is possible, i.e., data need not be decrypted, it yields meaningless results. It is only when all results are mathematically transformed back at the user's that they can be reconstructed into global, meaningful information. Individual shares and computed results being encrypted, network transfers to and from CSPs are thus safe. Hence, privacy is achieved at any point outside of the user's (network, providers). Finally, to verify the honesty of CSPs and the correctness of shares, we incorporate into Scheme-I two types of hash-based signatures. Signatures help verify data correctness in case some CSPs are not honest, and incorrect or erroneous data before decryption. However, updating and querying data are still difficult and expensive in Scheme-I, because data pieces are dependent on the others in the same block. Thus, Scheme-II builds upon Scheme-I to actually allow sharing and querying a DW in the cloud. Assuming a DW stored in a relational database, each attribute value in each record is shared independently. We first transform each attribute value to at least one block, depending on data type and size (e.g., one block for integers, reals or characters; and blocks for strings of length ), and encrypt each data block with Scheme-I. Then, we allow analyzing data over shares with ad-hoc queries and Relational OLAP (ROLAP) operations, without decrypting all data first whenever possible. All basic OLAP operations (roll-up, drill-down, some slice and dice, pivot and drill-across) can apply directly on shares at the CSPs', with results being reconstructed at the user's. However, other complex queries must be transformed or split first, depending on operations and functions used. Scheme-I: , , Multi-secret sharing with data verification Scheme-I is an , , multi-secret sharing scheme: data pieces are encrypted and shared among CSPs. out of shares can reconstruct the original data. The total volume of shares is only about − 1 . Data are organized into blocks that are encrypted and decrypted all at once. The priorities of blocks and data in the blocks are important because they directly affect the results of data access in Scheme-II. All data pieces in a block are encrypted at once by distinct random -variable linear equations, where variables are data and their signatures and coefficients are pseudorandom. Eventually, we introduce two types of signatures. The first, inner signature is created from all data pieces in one block. It matches with data in the reconstruction process if CSPs return correct shares. The second, outer signature is created from each share. At each CSP's, it verifies shares before transferring them back to the user for reconstruction. Parameters of Scheme-I are listed in Table 2. =1.. are randomly selected from distinct integers and are stored at the user's. is split into blocks with = −1 . If is not a multiple of − 1, the last block is padded with integer values -1 ( Figure 3). Data sharing process Each data block is encrypted independently (Figure 4). Data pieces in block are encrypted as follows. 1. Compute signature _ from data block with homomorphic function 1 : . Figure 4.Data sharing process. _ = 1 . 2. Create distinct random − 1 linear equations (Equation 1). = 1 , ⋯ , = ℎ + 2 × ,ℎ −1 ℎ=1 + × ,(1) Data reconstruction process A dataset D is reconstructed from shares and signatures , , _ , =1… stored at ∈ , where is any group of t CSPs ( Figure 5). There are two phases to reconstruct original data: the initialization phase and the actual reconstruction phase. Figure 5.Data reconstruction process. Initialization phase: In this phase, share correctness is verified and a matrix that is used in the reconstruction phase is created as follows. Verify information at all ∈ . At each CSP's, only shares to be decrypted are verified for correctness. Share , is correct if _ , = 2 , . In case of error at , then another CSP is selected and correctness is verified again. 2. At the user's, matrix is created from of ∈ such that = , × , where , is the ℎ coefficient of . Then, is computed such that = −1 . Let , be an entry in the ℎ row and the ℎ column of matrix . Reconstruction phase: To decrypt data block , share , of ∈ is transferred to the user and decrypted as follows. 1. Compute data block (Equation 2) and its signature _ (Equation 3). −1 −1 + = ,ℎ × ,ℎ ℎ=1 − 2; ∀ 1, − 1 (2) _ = ,ℎ × ,ℎ ℎ=1 (3) 2. If _ = 1 , then data in block are correct. In case of errors, the user can reconstruct data from shares from a new . Scheme-II: Sharing a data warehouse in the cloud In this section, we exploit Scheme-I to share a DW among CSPs. Databases attribute values, except NULL values and primary or foreign keys, are shared in relational databases at CSPs'. Keys help match records in the data reconstruction process and perform JOIN and GROUP BY operations. Any sensitive primary key, such as a social security number, is replaced by an unencrypted sequential integer key. Each attribute in the original tables is transformed into two attributes in encrypted tables, i.e., share and signature attributes. Figure 6 shows the example of a PRODUCT table that is shared among three CSPs. To handle data from a shared DW, we propose solutions to encrypt data of various types, to share the customary DW logical models in the cloud, to perform loading, backup and recovery processes, and to analyze shared data through ROLAP operations. Data types To handle the usual data types featured in databases, we encrypt and handle each data piece independently. Data pieces of any type are first transformed into integers, then split into one or several data blocks (depending on type and value), and finally encrypted with Scheme-I. For sharing an integer, date or timestamp , is split into − 1 pieces =1… −1 such that = −1 mod , where is a prime number and > −1 bits, where is the size of the maximum integer value in bits. Then, =1… −1 is encrypted to shares with Scheme-I. For sharing a real , is transformed into an integer by multiplication. For example, let be stored in numeric format ( , ), where is a precision value and a scale value. Then, is transformed into = × 10 . can then be encrypted as any integer. For sharing a character , is transformed into an integer through its ASCII code. For example, let be 'A'. is transformed into = 65. can then be encrypted as any integer. For sharing a string , is transformed into a set of integers =1… where is the length of , using the ASCII code of each character in . For example, let be 'ABC'. Then, is transformed into =1…3 = 65, 66, 67 . After transformation, each character is encrypted independently as any integer. For sharing a binary string , is transformed into a set of integers =1… where is the length of and is the size of the maximum integer value in bits. For example, let = 1000000000000000000000000000000011 and = 32 bits. Then is split into two smaller binaries: 10 and 00000000000000000000000000000011 sizing less than , which are then transformed into =1,2 = 2, 3 . After transformation, each is encrypted independently as any integer. An example of sharing an integer follows. 1. Sharing parameters are assigned as follows: = 4, = 3 and = 13. 2. Homomorphic and hash functions are 1 = mod ∈ and 2 , = , mod 13. 3. Let = 75, i.e., the shirt's unit price in Figure 6(a). 4. We compute =1,2 as follows: 1 = 75 13 1−1 mod 13 = 10 and 2 = 75 13 2−1 mod 13 = 5. 5. We compute _ 1 = 1 1 = 10 + 5 mod 13 = 2. 6. Let four random 3-variable linear equations be: a. = 1 1 , 2 , 3 = 1 × 1 + 2 + 0 × 2 + 2 + 2 × 3 , b. = 2 1 , 2 , 3 = 3 × 1 + 2 + 1 × 2 + 2 + 0 × 3 , c. = 3 1 , 2 , 3 = 2 × 1 + 2 + 1 × 2 + 2 + 1 × 3 , d. = 4 1 , 2 , 3 = 0 × 1 + 2 + 2 × 2 + 2 + 1 × 3 . 7. We compute 1, =1…4 such that 1,1 = 1 10,5,2 = 1 × 10 + 2 + 0 × 5 + 2 + 2 × 2=16. Similarly, 1,2 = 43, 1,3 = 33 and 1,4 = 14. 8. We compute _ 1, =1…4 such that _ 1,1 = 2 16 = 16 mod 7 = 2. Similarly, _ 1,2 = 1, _ 1,3 = 5 and _ 1,4 = 0. 9. We distribute each couple 1, , _ 1, to . Then, is reconstructed as follows. 1. Suppose 1 , 2 and 3 are selected into . 2. We verify _ 1, =1,2,3 such that _ 1,1 ′ = 2 16 = 16 mod 7 = 2 = _ 1,1 . Then 1,1 is correct. After verification, all three shares 1,1 , 1,2 , 1,3 are found correct. 3. We create matrix from =1,2,3 : = 1 0 2 3 1 0 2 1 1 . 4. We compute matrix = −1 mod = 1 2 −2 −3 −3 6 1 −1 1 3 5. We compute =1,2 as follows. a. 1 = 16 × 1 + 43 × 2 + 33 × −2 3 − 2 = 10. b. 2 = 16 × −3 + 43 × −3 + 33 × 6 3 − 2 = 5. 6. We compute _ 1 = 16 × 1 + 43 × −1 + 33 × 1 3 = 2. 7. We verify the original data. The result is correct since _ 1 ′ = 1 1 = 10 + 5 mod 13 = 2 = _ 1 . Data warehouse sharing Since each table of a shared DW is stored in a relational database at a given CSP's and each attribute value in each record is encrypted independently, Scheme-II straightforwardly helps implement any DW logical model, i.e., star, snowflake or constellation schema. Figures 7(a) and 7(b) show an example of snowflake-modeled DW that is shared among three CSPs. Each shared DW bears the same schema as the original DW's, but type and size of each attribute in each shared table differ from the original tables. All attribute types, except Booleans that are not encrypted to save computation and data storage costs, are indeed transformed into integers. Moreover, Scheme-II supports the storage of data cubes that optimize response time and bandwidth when performing ROLAP operations. Cubes are physically stored into tables that are shared among CSPs, retaining the same structure. For example, Figure 7(c) features a shared cube named cube-I that totalizes total prices and numbers of sales by time period and by product. Shared cubes include signatures for shared aggregate measures and customarily use NULL values to encode superaggregates. Finally, indices can also be shared to improve query performance. However, they must be created from the original data before the sharing process. We envisage lazy index creation on shares in future research, though. Loading, backup and recovery processes For loading data into a shared DW, each data piece is encrypted and loaded independently. New data can be loaded without decrypting previous data first, because each attribute value in each record is encrypted independently. For instance, in Figure However, when updating cubes, some shared aggregates may have to be recomputed. Within Scheme-II, we currently cannot apply all aggregation operations on shares. Thus, such aggregations still require to be computed on the original data. For example, maximum and minimum cannot be computed on shares because original data order is lost in the sharing process. Averaging data must be performed by summing and counting. Hence, to optimize costs, aggregates are first computed on new data, and then aggregated to relevant existing shares, which are decrypted on-the-fly. Finally, a backup process is unnecessary in our scheme, because each share , is actually a backup share of all other shares , , where ∈ 1, … , − 1, + 1, … , . In case a share is erroneous, it can be recovered from other shares. Data analysis over shares Since DWs and cubes can be shared in the cloud, Scheme-II directly supports all basic OLAP operations at the CSPs' through SQL operators and aggregation functions, and helps reconstruct the result on the user's side by performing queries on shared tables. For example, query "select YearID, YearName, TotalPrice from cube-I, year where cube-I.YearID=year.YearID and MonthID=null and DateID=null and CategoryID=null and ProdNo=null" can be run at t CSPs to compute the total price of products per year. However, although some queries apply directly onto shares, others require some or all data to be decrypted. Simple SELECT/FROM queries directly apply onto shares. All join operators, when operating on unencrypted keys, also apply directly. However, when expressing conditions in a WHERE or HAVING clause, the following routine must be followed: Similarly, aggregation functions SUM, AVG and COUNT can directly apply on shares, whereas other aggregation functions, such as MAX and MIN, require all original data to be reconstructed prior to computation. Finally, grouping queries using the GROUP BY or GROUP BY CUBE clauses can directly apply if and only if they target unencrypted key attributes. Again, grouping by other attribute(s) requires all data to be reconstructed at the user's before aggregation by an external program. Consequently, executing a complex query may require either transforming or splitting the query, depending on its clauses and operators, following the above guidelines. Figure 9 shows an example of complex query execution. Figure 9. Example of complex query execution over shares. SECURITY ANALYSIS AND PERFORMANCE EVALUATION In this section, we illustrate the relevance of our approach along two axes. First, we mainly theoretically study the security features of our schemes, which are our primary focus. Second, since our approach applies in the cloud, we both theoretically and experimentally study the factors that influence cost in the pay-as-you-go paradigm, i.e., computing, storage and data transfer costs, with respect to the TWH secret sharing schemes. Security analysis Privacy We focus here on data pilfering. Neither a CSP nor any intruder can decrypt original data from only one share, and data transferred between the user and CSPs are all encrypted. In case an intruder can steal shares from CSPs with ≤ , the probability of discovering (the original data in the ℎ block) remains low, i.e., 1 2 − −1 and 1 2 − −1 in Scheme-I and Scheme-II, respectively ( Figure 10). The probability of discovering depends on the following. 1. The size of control parameters in Scheme-I and in Scheme-II. In Scheme-I, the probability of breaking the secret is low because is a big prime number. In Scheme-II, the probability of breaking the secret ranges between 10 −22 and 10 −10 in Figure 10's example, because depends on . If p = , the probabilities of breaking the secret are equal in both schemes, but storage cost in Scheme-II is not controlled, which falls back to Shamir's (1979) case. 2. The user-defined value of . The higher , the lower the probability of breaking the secret. 3. The number of pilfered shares . The probability of breaking the secret obviously increases with x. However, both our schemes are secure enough since it is difficult to retrieve shares from at least CSPs by attacking them simultaneously. (a) Scheme-I (b) Scheme-II Figure 10.Probability of decrypting a data block from its shares. Within Scheme-II, although some data can be decrypted, if an intruder steals all data from one CSP, s/he must discover the pattern of data blocks and generate all −1 combinations of data pieces stored at the − 1 other CSPs' by brute force. The complexity of Scheme-II's reconstructing process is 2 , since the × matrix must be computed for data pieces. Thus, with = 99,991, = 3 and = 100 (11 KB of data), breaking the secret with a standard desktop computer would take more than 13 years. Thence, even with a botnet available, even partially decrypting a giga or terabyte-scale DW cannot be achieved in reasonable time. Reliability Reliability includes data availability and recovery, which are achieved by design with secret sharing, and data integrity and correctness. Our schemes can verify both the honesty of CSPs and the correctness of shares. Verification performance depends on the user-defined hash functions that define inner and outer signatures. To test the reliability of our signatures, we generate random 32-bits unsigned integers and share them. Then, we generate errors in all shares with respect to a given pattern. Finally, we account the number of incorrect data pieces that are not detected as such. Figure 11 plots the ratio of false positives achieved with inner signature _ = mod ∈ and outer signature _ , = , mod 2 , where 2 is a prime. If only the inner signature is used to verify data, i.e., only the honesty of CSPs is verified, the ratio ranges between 7.7E-6% and 3.2E-2%, inversely depending on . However, all incorrect data pieces can be detected if data are verified by both inner and outer signatures (i.e., share correctness is also verified) and 2 > 61. Figure 11.Rate of incorrect data not being detected. Finally, note that sharing data on one node is a surjective function, i.e., two different initial values may have the same share value. However, since the reconstruction process is achieved by intersection from all nodes, sharing data is overall a bijective function. Thus, querying shares always results in a 100% hit rate. Cost analysis In this section, we provide a cost analysis of the main factors inducing costs when storing a DW in the cloud. For this sake, in addition to theoretical considerations, we run two series of experiments. In the first series of experiments, we run 100 1 GB test cases made of 32-bits unsigned integers and vary parameters , and . In the second series of experiments, we use the Star Schema Benchmark (SSB -O'Neil et al., 2009) and vary parameter with = 3 and = 4. Experiments are conducted with Bloodshed Dev-C++ 5.5.3 and MySQL 5.0.51a on a PC with an Intel(R) Core(TM) i5 2.76 GHz processor with 3 GB of RAM running Microsoft Windows 7. Computing cost The time complexity of Scheme-I's data sharing process is , since -variable linear equations must be computed for data blocks. Moreover, since = ( − 1), complexity can also be expressed as . The time complexity of Scheme-II's data sharing process is also , or since = here. The time complexity of Scheme-I's data reconstruction process is or 2 , since the × matrix must be computed for data blocks and = −1 . Scheme-II's is 2 or 2 , since must be computed for data blocks and = . For example, the execution time of sharing and reconstructing 32-bits unsigned integers with Scheme-II is plotted in Figure 12 with respect to and . The execution time of both processes increase with when = . The execution time of the sharing process increases with when is fixed, whereas does not affect reconstruction. For instance, the execution time of the data sharing and reconstruction processes are about 15 seconds (throughput is 68 MB/s) and 7 seconds (throughput is 144 MB/s) when = = 3 and = 99991. To evaluate the performance of data analysis on shares more accurately, we measure the execution time of SSB's OLAP query workload (Table 3). Quite unexpectedly, query execution is faster on shares than on the original DW. But this is because we run plain SSB queries on the original DW and MySQL does not optimize joins natively. When executing the same query on shares, we have to split the original queries and process subqueries before joining, thus implicitly optimizing join dimensionality. Query response on shares appears reasonable, though it could be further enhanced through shared indices and materialized views. Table 3.OLAP performance with SSB. SSB query Storage cost One advantage of our schemes is that the volume of shares nears that of original data when = , is big, and in Scheme-I and in Scheme-II are small. Shared data volume is only in Scheme-I and in Scheme-II, where and are sizes of and , re-spectively. For example, with Scheme-II, let us consider a set of ten 32-bits unsigned integers that is shared among six CSPs, with five CSPs being sufficient to reconstruct . The volume of is 10 × 32 = 320 bits. Let = 9 bits. Then, the volume of all shares is lower than 10 × 6 × 9 = 540 bits (1.69 × ). The volume of each share is about 10 × 9 = 90 bits (0.17 × ). The volume of our shared 32-bit unsigned integer dataset using Scheme-II is plotted in Figure 13. The volume of each share varies with respect to , and . The volume of all shares ranges between −1 and times the volume of . For example, it is 1.89 GB ( Finally, Table 4 shows the volume of SSB's DW, once shared with Scheme-II. The volume of each share is still smaller than that of the original DW (about 46% smaller). Therefore, this guarantees any shared DW can be stored and queried if the original DW can. If data volume is very large, a higher-performance DBMS can be envisaged, e.g., a parallel DBMS or low-level distributed storage. Although, the volume of all shares is greater than that of original data (about 185% greater), it is smaller than twice that of original data. Shared volume may be reduced to about 139% of original data if the data availability constraint is relaxed, though. Data transfer cost In our context, data transfer cost only relates to the size of query results, since uploads are generally free of charge at major CSPs'. SSB query result size is shown in Table 3. Since many queries do not need to decrypt data, only some parts of the shared DW are transferred when executing SSB's workload. Thus, the transferred data volume is greater than the volume of each query result, but lower than that of all shares. For example, query Q.1.3 ran on shares outputs 32 KB, which is greater the actual result size (11 bytes) but much lower than the volume of all shares (1.5 GB), thus incurring reasonable transfer costs. Comparison of Scheme-II to existing related approaches In this section, we compare Scheme-II to the TWH approaches presented in our state of the art, with respect to security and cost. Table 5. Comparison of database sharing approaches Security All approaches handle data security, availability and integrity issues, but only ours verifies both the correctness of shares and the honesty of CSPs by outer and inner code verification, respectively. Computing cost The time complexity of Scheme-II's sharing process is equal to TH's and better than W's because is normally much bigger than and . Thence, log > > . However, in the context of DWs, where updates are performed off-line, update performance is not as critical as in transactional databases. The time complexity of Scheme-II's reconstructing process is again equal to TH's and better than W's. Since data decryption is part of query response time, it is critical in a DW context. However, shared data access is also part of query response time. In this regard, our approach is faster than TWH, because we can directly query shared tables, whereas TWH must perform ad-hoc queries, aggregate and reconstruct data to achieve the same result. For instance, W cannot perform any aggregation operation on shares. Thence, many shares are transferred back to the user for aggregation. Storage cost In addition to the storage estimations provided in Table 5, let us illustrate the storage gain achieved with our approach through an example. Let = 4, = 3 and be the original data volume. Let us also assume that each share is not bigger than the original data it encrypts. For simplicity, let us finally disregard the volume of signatures that depends on user-defined parameters in all approaches. The result is shown in Table 6, with column Improvement displaying the storage gain achieved by Scheme-II over TWH. Approach Shared data volume Scope Improvement Table 6. Comparison shared data volume in Scheme-II and TWH. Data transfer cost Data transfer cost directly relates to the size of shares when loading data, and to the size of query results when accessing the shared database. Since all approaches allow different operations and vary in shared data volume, it is difficult to compare data transfer costs by proof. However, data transfer cost in our approach is cheaper in the sharing phase because the size of each encrypted data piece is 1/( − 1) smaller than that of TWH. Moreover, by creating shared data cubes, we allow straight computations on shares, and thus only target results are transferred to the user, i.e., with no additional data to decrypt at the user's. CONCLUSION In this paper, we propose an original approach to share a DW in the cloud that simultaneously supports data privacy, availability, integrity and OLAP querying. Our approach is constituted of two schemes. Scheme-I exploits block cryptography and secret sharing to protect data and guarantee data privacy and availability. Moreover, Scheme-I ensures data correctness by utilizing homomorphic and hash functions as signatures. Scheme-II builds upon Scheme-I to allow sharing and querying cloud DWs. It allows analyzing data over shares without decrypting all data first. Our security and performance analysis shows that our schemes are more robust and cheaper than similar existing techniques when storing and querying data. Future research shall run along two axes. First, we plan to further assess the cost of our solution in the cloud pay-as-you-go paradigm. Sharing data indeed implies increasing the initial data volume, and thus storage cost, as well as duplicating computing costs over CSPs. However, it also guarantees data availability. Hence, we plan to run monetary cost evaluations against classical data replication schemes. It would also be very interesting to balance the cost of our solution against the cost of risking data loss or theft. Moreover, parameter assignment affects the security of our schemes. Notably, to enforce security, big values should be assigned to primes , p and number of CSPs needed to decrypt data t. In contrast, small values should be assigned to , , and to reduce execution time and data volume. Thus, a suitable tradeoff must be investigated. Figure 1 . 1Cloud data security issues. Figure 2 . 2Existing data security solutions. Figure 3 . 3Organization of data in blocks. Figure 6 . 6Example of original data and shares at three CSPs'. Figure 7 . 7Example of shared data warehouse and cube. Figure 8 . 88, data from Figure 6 are already shared and the last record (#126) is new. Example of sharing new data. Figure 12 . 12Execution time of Scheme-II. when the volume of original data is 1 GB, = = 3 and = 99991. Figure 13 . 13Shared data volume. Table 1 1summarizes the features of the above security approaches, with respect to data privacy, availability, integrity and full access. No existing approach simultaneously satisfies all criteria.Approaches Data Privacy Data availability Data Integrity Data access Encryption -Homomorphic encryption √ On encrypted data without decryption. -Incremental encryption √ On encrypted data without decryption. -Secret sharing √ √ Summing and averaging shares -Multi-secret sharing √ √ -Verifiable secret sharing √ √ √ -Verifiable multi-secret sharing √ √ √ Data anonymization √ On non-anonymized data Data replication(RS code) √ Data verification √ Table 1. Comparison of data security solutions. Table 2 . 2Scheme-I parameters.Parameters Definitions Number of CSPs CSP number Number of data pieces Number of data blocks Number of shares necessary for reconstructing original data A big prime number Original data such that = 1 , … , and = 1 , … , The ℎ piece of in integer format such that − 2 > ≥ 0 The ℎ block of such that = −1 −1 , … , −1 Identifier number of such that > 0 , Share of stored at _ Signature of original data in such that > _ ≥ 0 _ , Signature of share of stored at where is positive variable, ℎ is the ℎ th positive pseudorandom coefficient seeded at , > ,ℎ ≥ 0 and 1 ≠ 2 if 1 ≠ 2. These functions are used for all blocks. 3. Compute the set of shares , =1...from data block such that , = , _ , and distribute each share , to . 4. Compute signatures _ , =1...with hash function 2 such that _ , = 2 , , and distribute each signature _ , to along with , . Thus, data and their signatures are shared among CSPs. stores pairs of shares and signatures , , _ , =1… Table Original OriginalPercentage of original data volume.SSB 1st share 2nd share 3th share 4th share All shares Volume (KB) Volume (KB) % (1) Volume (KB) % (1) Volume (KB) % (1) Volume (KB) % (1) Volume (KB) % (1) Customer 3,167 2,550 80.52 2,550 80.52 2,550 80.52 2,550 80.52 10,200 322.07 Date 218 205 94.04 205 94.04 205 94.04 205 94.04 820 376.15 Part 18,798 14,940 79.48 14,940 79.48 14,940 79.48 14,940 79.48 59,760 317.91 Supplier 965 768 79.59 758 78.55 758 78.55 758 78.55 3,042 315.23 Lineorder 822,929 373,610 45.40 373,610 45.40 373,610 45.40 373,610 45.40 1,494,440 181.60 All tables 846,077 392,073 46.34 392,063 46.34 392,063 46.34 392,063 46.34 1,568,262 185.36 (1) Table 4 . 4SSB shared data warehouse volume achieved with Scheme-II. Table 5 5synthesizes the features and costs of all approaches, which we discuss below.-Data storagew.r.t. original data volume ≥ 2 + 1 (hash tree)Features and costs Thompson et al. (2009) Wang et al. (2011) Hadavi et al. (2012) Scheme-II Privacy Yes Yes Yes Yes Data availability Yes Yes Yes Yes Data integrity -Inner code verifying Yes Yes Yes Yes -Outer code verifying No No No Yes Target Databases Databases Databases Data warehouses Data types Positive integers Positive integers Positive integers Integers, Reals, Characters, Strings, Dates, Booleans Shared data access -Data updates Yes Yes Yes Yes -Exact match queries No Yes Yes Yes -Range queries No Yes Yes No -Aggregation functions Yes No Yes Yes -OLAP queries No No No Yes Costs ≥ / + / (B++ tree index) + signatures ≥ + 1 (B++ tree index) ≥ / − 1 + signatures -Sharing process execution time max log , -Reconstructing process execution time 2 2 2 Second, although we provide in this paper a raw framework for OLAPing shared data, more research is required to implement all operations needed in OLAP analyses, as well as incremental updates. We notably plan to reuse the strategies ofWang et al. (2011)andHadavi et al. (2012)to achieve range and match queries, e.g., by implementing shared B+ tree indices. A modular Approach to Key Safeguarding. C Asmuth, J Bloom, IEEE Transactions on Information Theory. 292Asmuth, C., & Bloom, J. (1983). A modular Approach to Key Safeguarding. IEEE Transac- tions on Information Theory.29 (2), 208-210. Sharing-based Privacy and Availability of Cloud Data Warehouses. 9 th French-speaking days on Data Warehouse and On-line. V Attasena, N Harbi, J Darmont, Analysis. Attasena, V., Harbi, N., & Darmont, J. (2013) Sharing-based Privacy and Availability of Cloud Data Warehouses. 9 th French-speaking days on Data Warehouse and On-line Analysis (EDA 2013), Blois, France: RNTI, B-9.17-32. Incremental Cryptography: The Case of Hashing and Signing. M Bellare, O Goldreich, S Goldwasser, Advances in Cryptology (CRYPTO'94). 839Bellare, M., Goldreich, O., & Goldwasser, S. (1994). Incremental Cryptography: The Case of Hashing and Signing. Advances in Cryptology (CRYPTO'94): LNCS, 839. 216-233. Safeguarding Cryptographic Keys. G R Blakley, AFIPS National Computer Conference. Monval, NJ, USABlakley, G. R. (1979). Safeguarding Cryptographic Keys.1979 AFIPS National Computer Conference, Monval, NJ, USA. 313-317. K D Bowers, A Juels, A Oprea, Proofs of Retrievability: Theory and Implementation. 1 st ACM Workshop on Cloud Computing Security (CCSW'09). Chicago, IL, USABowers, K.D., Juels, A., & Oprea, A. (2009). Proofs of Retrievability: Theory and Implemen- tation. 1 st ACM Workshop on Cloud Computing Security (CCSW'09), Chicago, IL, USA. 43- 54. S Bu, R Yang, Novel and Effective Multi-Secret Sharing Scheme. International Conference on Information Engineering and Applications (IEA'12): LNEE, 219. Bu, S., & Yang, R. (2013).Novel and Effective Multi-Secret Sharing Scheme. International Conference on Information Engineering and Applications (IEA'12): LNEE, 219. 461-467. A Secret Sharing Scheme Based on NTRU Algorithm. S Bu, H Zhou, International Conference on Wireless Communications, Networking and Mobile Computing (Wi-Com'09). Beijing, ChinaBu, S., & Zhou, H. (2009). A Secret Sharing Scheme Based on NTRU Algorithm. Interna- tional Conference on Wireless Communications, Networking and Mobile Computing (Wi- Com'09), Beijing, China. 1-4. Anonymized Data: Generation, Models, Usage. G Cormode, D Srivastava, ACM SIGMOD International Conference on Management of Data (SIGMOD'09). Providence, Rhode Island, USACormode, G. & Srivastava, D. (2009). Anonymized Data: Generation, Models, Usage. ACM SIGMOD International Conference on Management of Data (SIGMOD'09), Providence, Rhode Island, USA. 1015-1018. F Emekci, D Agrawal, A El Abbadi, A Gulbeden, Privacy Preserving Query Processing Using Third Parties. 22 nd International Conference on Data Engineering (ICDE'06). Atlanta, Georgia, USAEmekci, F., Agrawal, D., El Abbadi, A., & Gulbeden, A. (2006).Privacy Preserving Query Processing Using Third Parties. 22 nd International Conference on Data Engineering (ICDE'06), Atlanta, Georgia, USA. 27-37. A Verifiable Multi-Sharing Scheme Based on Cellular Automata. Z Eslami, J Z Ahmadabadi, Information Sciences. 15Eslami, Z., & Ahmadabadi, J.Z. (2010). A Verifiable Multi-Sharing Scheme Based on Cellu- lar Automata. Information Sciences, 180(15). 2889-2894. Fully Homomorphic Encryption Using Ideal Lattices. C Gentry, 41 st Annual ACM Symposium on Theory of Computing (STOC'09). Bethesda, MD, USAGentry, C. (2009). Fully Homomorphic Encryption Using Ideal Lattices. 41 st Annual ACM Symposium on Theory of Computing (STOC'09), Bethesda, MD, USA. 169-178. Database as a service: Towards a unified solution for security requirements. M A Hadavi, M Noferesti, R Jalili, E Damiani, th Computer Software and Applications Conference Workshops (COMPSACW'12). Izmir, Turkey36Hadavi, M.A., Noferesti, M., Jalili, R., & Damiani, E. (2012). Database as a service: Towards a unified solution for security requirements. 36 th Computer Software and Applications Confe- rence Workshops (COMPSACW'12), Izmir, Turkey. 415-420. Secure Data Outsourcing Based on Threshold Secret Sharing; Towards a More Practical Solution. M A Hadavi, R Jalili, VLDB PhD Workshop. Hadavi, M.A., & Jalili, R. (2010).Secure Data Outsourcing Based on Threshold Secret Shar- ing; Towards a More Practical Solution. VLDB PhD Workshop, Singapore. 54-59. Verifiable Multi-Secret Sharing Based on LFSR Sequences. C Hu, X Liao, X Cheng, Theoretical Computer Science. 445Hu, C., Liao, X., & Cheng, X. (2012). Verifiable Multi-Secret Sharing Based on LFSR Se- quences. Theoretical Computer Science, 445. 52-62. PORs: Proofs of Retrievability for Large Files. 14 th ACM conference on Computer and communications security. A Juels, B Kaliski, CCS'07). Alexandria, VA, USAJuels, A., & Kaliski, B. (2007). PORs: Proofs of Retrievability for Large Files. 14 th ACM con- ference on Computer and communications security (CCS'07), Alexandria, VA, USA. 584-597. Dialing Privacy and Utility: a Proposed Data-sharing Framework to Advance Internet Research. E Kenneally, K Claffy, IEEE Security and Privacy. 84Kenneally, E. & Claffy, K. (2010).Dialing Privacy and Utility: a Proposed Data-sharing Framework to Advance Internet Research. IEEE Security and Privacy, 8(4), 31-39. Efficient ( , ) secret sharing schemes. Y X Liu, L Harn, C N Yang, Y Q Zhang, Journal of Systems and Software. 6Liu, Y.X., Harn, L., Yang, C.N., & Zhang, Y.Q. (2012). Efficient ( , , ) secret sharing schemes. Journal of Systems and Software, 85(6). 1325-1332. A Machanavajjhala, D Kifer, J Gehrke, M Venkitasubramaniam, diversity: Privacy beyond -anonymity. 1Machanavajjhala, A., Kifer, D., Gehrke, J., & Venkitasubramaniam, M. (2007). -diversity: Privacy beyond -anonymity. ACM Transactions on Knowledge Discovery from Data, 1(1). Lattice-based Homomorphic Encryption of Vector Spaces. C A Melchor, G Castagnos, P Gaborit, IEEE International Symposium on Information Theory (ISIT'08). Toronto, ON, CanadaMelchor, C.A., Castagnos, G., & Gaborit, P. (2008). Lattice-based Homomorphic Encryption of Vector Spaces. IEEE International Symposium on Information Theory (ISIT'08), Toronto, ON, Canada. 1858-1862. P O&apos;neil, E O&apos;neil, X Chen, S Revilak, The Star Schema Benchmark and Augmented Fact Table Indexing. 1 st Technology Conference on Performance Evaluation and Benchmarking (TPCTC'09). Lyon, France5895O'Neil, P., O'Neil, E., Chen, X., & Revilak, S. (2009).The Star Schema Benchmark and Augmented Fact Table Indexing. 1 st Technology Conference on Performance Evaluation and Benchmarking (TPCTC'09), Lyon, France: LNCS, 5895. 237-252. A Survey of Data Replication Techniques for Mobile Adhoc Network Databases. P Padmanabhan, L Gruenwald, A Vallur, M Atiquzzaman, VLDB Journal. 5Padmanabhan P., Gruenwald, L., Vallur, A., & Atiquzzaman, M. (2008). A Survey of Data Replication Techniques for Mobile Adhoc Network Databases. VLDB Journal, 17(5). 1143- 1164. Enhancing Cloud Security using Data Anonymization. J Sedeyao, Sedeyao, J. (2012). Enhancing Cloud Security using Data Anonymization. Compact Proofs of Retrievability. 14 th International Conference on the Theory and Application of Cryptology and Information Security. H Shacham, B Waters, Advances in Cryptology (ASIACRYPT'08). Melbourne, AustraliaShacham, H., & Waters, B. (2008). Compact Proofs of Retrievability. 14 th International Con- ference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology (ASIACRYPT'08), Melbourne, Australia. 90-107. How to Share a Secret. A Shamir, Communications of the ACM. 2211Shamir, A. (1979). How to Share a Secret. Communications of the ACM, 22(11). 612-613. Secure Data Outsourcing. R Sion, rd International Conference on Very Large Data Bases (VLDB'07). Vienna, Austria33Sion, R. (2007). Secure Data Outsourcing. 33 rd International Conference on Very Large Data Bases (VLDB'07), Vienna, Austria. 1431-1432. -anonymity: a Model for Protecting Privacy. L Sweeney, International Journal on Uncertainty, Fuzziness and Knowledge-based Systems. 5Sweeney, L. (2002). -anonymity: a Model for Protecting Privacy. International Journal on Uncertainty, Fuzziness and Knowledge-based Systems, 10(5). 557-570. Generalized Reed Solomon Codes for Erasure Correction in SDDS. J Thomas, E Schwarz, Workshop on Distributed Data and Structures (WDAS'02). Paris, FranceThomas J., & Schwarz, E. (2002). Generalized Reed Solomon Codes for Erasure Correction in SDDS. Workshop on Distributed Data and Structures (WDAS'02), Paris, France. 75-86. Privacy-Preserving Computation and Verification of Aggregate Queries on Outsourced Databases. B Thompson, S Haber, W G Horne, T Sander, D Yao, Privacy Enhancing Technologies. 5672Thompson, B., Haber, S., Horne, W. G., Sander, T., & Yao, D. (2009). Privacy-Preserving Computation and Verification of Aggregate Queries on Outsourced Databases. Privacy En- hancing Technologies: LNCS, 5672. 185-201. Enabling Public Verifiability and Data Dynamics for Storage Security in Cloud Computing. Q Wang, C Wang, J Li, K Ren, W Lou, 14 th European Symposium on Research in Computer Security (ESORICS'09). Saint-Malo, FranceWang, Q., Wang, C., Li, J., Ren, K., & Lou, W. (2009). Enabling Public Verifiability and Data Dynamics for Storage Security in Cloud Computing. 14 th European Symposium on Re- search in Computer Security (ESORICS'09), Saint-Malo, France. 355-370. A Comprehensive Framework for Secure Query Processing on Relational Data in the Cloud. S Wang, D Agrawal, A E Abbadi, Secure Data Management. 6933Wang, S., Agrawal, D., & Abbadi, A.E. (2011). A Comprehensive Framework for Secure Query Processing on Relational Data in the Cloud. Secure Data Management: LNCS, 6933. 52-69. Consideration for Multi-Threshold Multi-Secret Sharing Schemes. A Waseda, M Soshi, International Symposium on Information Theory and its Applications (ISITA'12). Honolulu, HI, USAWaseda, A., & Soshi, M. (2012).Consideration for Multi-Threshold Multi-Secret Sharing Schemes. International Symposium on Information Theory and its Applications (ISITA'12), Honolulu, HI, USA. 265-269.
[]
[ "COLLABORATIVE LEARNING MODEL WITH VIRTUAL TEAM IN UBIQUITOUS LEARNING ENVIRONMENT USING CREATIVE PROBLEM- SOLVING PROCESS", "COLLABORATIVE LEARNING MODEL WITH VIRTUAL TEAM IN UBIQUITOUS LEARNING ENVIRONMENT USING CREATIVE PROBLEM- SOLVING PROCESS" ]
[ "Sitthichai Laisema \nPh.D student, Information and Communication Technology for Education Division\nFaculty of Technical Education\nKing Mongkutt's University of Technology North Bangkok\nThailand\n", "Panita Wannapiroon \nFaculty of Technical Education\nAssistant Professor, Information and Communication Technology for Education Division\nKing Mongkutt's University of Technology North\nBangkokThailand\n" ]
[ "Ph.D student, Information and Communication Technology for Education Division\nFaculty of Technical Education\nKing Mongkutt's University of Technology North Bangkok\nThailand", "Faculty of Technical Education\nAssistant Professor, Information and Communication Technology for Education Division\nKing Mongkutt's University of Technology North\nBangkokThailand" ]
[ "International Journal on Integrating Technology in Education (IJITE)" ]
The purposes of this research study were: 1) to develop a Collaborative Learning Model with Virtual Team in u-Learning Environment using Creative Problem-solving Process(U-CCPS Model); 2) to evaluate a U-CCPS Model. The research procedures were divided into two phases. The first phase was to develop U-CCPS Model, and the second phase was to evaluate U-CCPS Model. The sample group in this study consisted of five experts using purposive sampling. Data were analyzed by arithmetic mean and standard deviation. The research findings were as follows: The U-CCPS learning Model consisted of five components as follows: 1) Input factors, 2) Process, 3) Control, 4) Output and 5) Feedback. The input factors consisted of four components as followed: 1) Objectives of U-CCPS Model, 2) Roles of Instructors, 3) Roles of learners and 4) Design of learning media. The process consisted of two components as followed: 1) Preparation before learning, and 2) Instructional management process. The experts agree that a U-CCPS Model was highest suitability.
10.5121/ijite
[ "https://arxiv.org/pdf/1401.2232v1.pdf" ]
1,912,712
1401.2232
ac9cbfdb93a167ef226e8e46f526b41ba74ad6ab
COLLABORATIVE LEARNING MODEL WITH VIRTUAL TEAM IN UBIQUITOUS LEARNING ENVIRONMENT USING CREATIVE PROBLEM- SOLVING PROCESS December 2013 Sitthichai Laisema Ph.D student, Information and Communication Technology for Education Division Faculty of Technical Education King Mongkutt's University of Technology North Bangkok Thailand Panita Wannapiroon Faculty of Technical Education Assistant Professor, Information and Communication Technology for Education Division King Mongkutt's University of Technology North BangkokThailand COLLABORATIVE LEARNING MODEL WITH VIRTUAL TEAM IN UBIQUITOUS LEARNING ENVIRONMENT USING CREATIVE PROBLEM- SOLVING PROCESS International Journal on Integrating Technology in Education (IJITE) 24December 201310.5121/ijite.2013.24011Collaborative Learningu-LearningCreative Problem-solving ProcessVirtual Team The purposes of this research study were: 1) to develop a Collaborative Learning Model with Virtual Team in u-Learning Environment using Creative Problem-solving Process(U-CCPS Model); 2) to evaluate a U-CCPS Model. The research procedures were divided into two phases. The first phase was to develop U-CCPS Model, and the second phase was to evaluate U-CCPS Model. The sample group in this study consisted of five experts using purposive sampling. Data were analyzed by arithmetic mean and standard deviation. The research findings were as follows: The U-CCPS learning Model consisted of five components as follows: 1) Input factors, 2) Process, 3) Control, 4) Output and 5) Feedback. The input factors consisted of four components as followed: 1) Objectives of U-CCPS Model, 2) Roles of Instructors, 3) Roles of learners and 4) Design of learning media. The process consisted of two components as followed: 1) Preparation before learning, and 2) Instructional management process. The experts agree that a U-CCPS Model was highest suitability. INTRODUCTION Partnership for 21 st Century Skills has developed some visions for the success of learners in economic systems of the new world, and for the practitioners to integrate skills with the teaching of academic contents. Accordingly, the 21st Century Student Outcomes and Support Systems were established by the combination of knowledge, specialized skills, expertise, and omniscience, all of which contribute to the success in both career and life [1]. The skills that can insure the learner's preparedness to such a complicated working life as seen in this modern era include Creativity and Innovation, Critical Thinking and Problem Solving, and Communication and Collaboration [1]. As to the dynamic changes of the world in the 21 st century, those who can survive and blend themselves with this modern society must practice their creative thinking. Creative thinking already exists in everybody, but it will be more acute, more active and more sustainable with proper learning and practice. Those equipped with high creative thinking skills will always receive better jobs, live more prosperous life, and do anything more helpful to the world [2]. In addition to creative thinking, collaborative skill is also indispensable for the learners of 21 st century. This skill refers to the ability to work efficiently with others in a team, while having flexibility in their own roles and dedication towards the team's tasks in order to finally achieve the mutual goals. Thereby, in order to work well with others, the learners must have responsibility in collaboration and recognize other team members' performance. The instructional technique based on Creative problem-solving process is so popular that most educators from different institutes have applied it to both adult and youth education [3]. This is because this technique can be easily used in daily life, and easy to learn and understand in all age groups, all situations and all cultures. It is also practical when applied to solve common problems, and above all, it is a technique designed especially to develop problem-solving skill and creative thinking skill [4]. Several scholars have studied and developed the creative problem-solving process. Isaksen, Dorval and Trafinger developed the model of creative problem-solving process, in which they made languages and process thereof more flexible so that it could better solve the problems in different contexts [5]. Once having been tested and used by Maraviglia and Kvashny, the new developed creative problem-solving model was found to have the highest influences on the development of creative thinking [6]. According to National ICT Policy Framework 2011-2020: (ICT2020), Strategy 6, the objective thereof is to reduce the economic and social inequality by creating the equal access to public resources and services, particularly the fundamental services necessary for life and for the wellbeing of citizens [7]. The said strategy will also promote the production and application of all innovative digital learning media, as well as the publication of all electronic media or lessons. Meanwhile, the said policy framework also places an emphasis on the use of information and communication technology in education management. U-learning is an instructional management in which information and communication technology is used in education management. Since this kind of learning is based on Ubiquitous technology, which can create learning in different environments as to the contexts of learners [8], the learners are able to learn anywhere anytime they want through their mobile device with no need of access to the computer. Consequently, there are flexibility in learning and quick access to desired information. It is also a kind of learner-centered instructional management which focuses on the works of learners. So, this kind of learning will encourage learners to create and acquire knowledge by themselves [9]. U-learning can be applied with Constructivism [10]. The use of learning theory to design education can well link knowledge information of learners to environments (Jacobs, 1999). The researcher is interested in collaborative learning, which enables learners of different ability to cooperate in teamwork. The team members will study something of common interest by creating a project. Then, the team will present the knowledge they have got from the said collaborative learning. Also, during the process of managing knowledge information, creating a project together, the learners have to exchange their opinions. This is the highlight of collaborative working, which depends mostly on collective action and mutual understanding [11]. Furthermore, collaborative learning helps learners do their job with high efficiency by combining parts of their existing knowledge and synthesizing them to create the new knowledge [12], [13]. Therefore, collaborative learning is an instructional management that promote the learners to have collaboration skills [14]. Collaborative learning encourages the learners to work in a team in order to create a work piece as required by the instructors. Thereby, this collaborative learning enables the learners to create a virtual team so that the members can contact one another through the media with no face-to-face meeting. Although the members do not stay in the same place [15], [16], [17], the virtual team can still help them work together all 24 hours, providing the team with more various skills and different perspectives. To summarize, collaborative learning in the virtual team, in spite of different places and time, helps the learners work or learn together by means of information and communication technology [17], [18]. It is therefore necessary to develop a learning model with the application of information technology in order that the learners could have both creative thinking skill and collaboration skill. Thus, the researcher is interested in the study of collaborative learning with virtual team via electronic media based on Creative problem-solving process in Ubiquitous learning environment so that the learners could develop creative thinking skill and collaboration skills. PURPOSE OF THE STUDY The purposes of this study were; To design a develop a Collaborative Learning Model with Virtual Team in Ubiquitous Learning Environment using Creative Problem-solving Process to enhance Creative Thinking and Collaboration Skills. To evaluate a Collaborative Learning Model with Virtual Team in Ubiquitous Learning Environment using Creative Problem-solving Process to enhance Creative Thinking and Collaboration Skills. SCOPE OF THE STUDY Population The study population was experts in instructional design, ubiquitous Learning, information technology, creative thinking and collaboration skills. Sample groups The sample groups of study were five experts in instructional design, ubiquitous Learning, information technology, creative thinking and collaboration skills by purposive sampling. Variables of the study An independent variable was Collaborative Learning Model with Virtual Team in u-Learning Environment using Creative Problem-solving Process. A dependent variable was evaluation of the proposed learning model. CONCEPTUAL FRAMEWORK The conceptual framework of this study is shown in Figure 1. five components were used for creating a U-CCPS model which theoretically affected the Collaborative Learning with Virtual team, Creative Problem-Solving Process, Ubiquitous Learning environment, Creative thinking and Collaboration Skills of the learner. Collaborative learning with virtual team Collaborative learning with virtual team refers to the method that enables the learners to cooperate in the team working in order to study something of common interest by creating a project and then presenting the knowledge they have got from the said collaborative learning. The learners in the team study and create knowledge together; thereby, with the collaborative learning with virtual team, they can still work or study together, despite different places and time, by the aid of information and communication technology [13], [15], [16], [17], [18], [19]. Collaborative Learning Model with Virtual Team in Ubiquitous Learning Creative problem-solving process Creative Problem-Solving Process is a method to combine creative thinking with experiences and information research in order to find out the solutions. There are 4 main steps in solving the problems [3], [4], [5]. Understanding the problem In creative problem-solving process, once we completely understand the problem or really know its context, it is easy to find out the solution thereof. Generating ideas To generate any ideas to find out the solution or answer to the questions of the previous step, the extraordinarily new and different ideas are needed. Planning for action T In this step, there are solution finding and acceptance finding. The first one is about analysis, definition, and adaptation of the ideas to be more concrete based on elaborate consideration and examination. The other is about the finding of support and objection in order to bring about the solution. Appraising tasks Creative problem solving process is effective and flexible, and it can be adjusted to suit any individuals, problems or situations. Ubiquitous learning environment (ULE) Ubiquitous Learning is a kind of learning in a form of digital media, in which the learners can study anything anytime and anywhere without a computer. As a result, there are flexibility in learning and fast access to the information; whereby the learning will be in accordance with different environments and contexts of the learners. The characteristics of ULE include [10], [20], [21], [22], [23], [24], [25], [26]: Computer tablets Computer tablets with processing unit and memory are equipped with a system that can check the status of learners before sending them the contents through the said tablets. Wireless Communication Wireless Communication, e.g. Bluetooth, 3G or Wifi, is suitable for the fast data transfer. This research employs Wifi network or 3G, in which the learners can study anything wherever there is Wifi or 3G available. Ubiquitous Learning Mangement System (U-LMS) U-LMS has a host computer for the management of learning, and storage of education resources, media, and education units. The host computer can also provide the learners with understanding and assistance by analyzing and answering the learners' questions through their tablets. Context Awareness Context Awareness will detect the movement and environment according to the learners' context so as to recognize their status. Creative Thinking Creative Thinking refers to the advanced cognitive process that employs several thinking processes to create new things or solve existing problems. Creative thinking exists only when the creators have freedom of thinking, or Divergence Thinking, and ability to adapt or combine the existing thinking, which will lead to the discovery of novelties. Divergence Thinking is a kind of creative thinking. It means the ability of an individual to solve the problems, the ways of thinking that generate different and new things, and the ability that can be applied in different kinds of jobs. Divergence Thinking consists of [27]: Fluency Ability to think and respond to stimuli at the best, or to find out the right but different solutions in the same issue. Flexibility Ability to adjust the thinking and make it comply with different circumstances, focusing on the wide ranges of fluency by means of classification and criteria. Originality Ability to think differently and uniquely, or to modify existing knowledge so as to create new things. Elaboration Ability to see through the details that are invisible to others, and to link different things together in a meaningful manner. Collaboration skills Collaboration skills consists of [1]: 4.5.1. Demonstrate ability to work effectively and respectfully with diverse teams. 4.5.2. Exercise flexibility and willingness to be helpful in making necessary compromises to accomplish a common goal. 4.5.3. Assume shared responsibility for collaborative work, and value the individual contributions made by each team member. METHODOLOGY The first phase Collaborative Learning Model with Virtual Team in Ubiquitous Learning Environment using Creative Problem-solving Process to enhance Creative Thinking and Collaboration Skills with the following method: 5.1.1 To study, analyze and synthesize documents and former research relevant to the elements of Ubiquitous learning, Collaborative Learning with virtual team and Creative Problem-Solving Process. Then, the results thereof were used to set up a conceptual framework in order to develop a model of Collaborative Learning with virtual team. 5.1.2 To study information about learning management by interviewing the instructors in order to synthesize the data of learning model and by interviewing the students about their ability to use information technology and communication for learning, their learning style, and their cognitive style. 5.1.3 The development of the model of U-CCPS model in this phase was derived by analyzing the principles of Collaborative Learning with virtual team and Creative Problem-Solving Process. Then, the results of the study were used to identify U-CCPS model based on the following components: 1) Input factors, 2) U-CCPS process, 3) Control, 4) Output and 5) Feedback. 5.1.4 To present the U-CCPS model to the advisors for consideration and revision. 5.1.5 To present the U-CCPS model model to the experts for consideration by means of in-depth interview. 5.1.6 To create the tools for evaluating the suitability of the model of U-CCPS model. The second phase The second phase of the project was an evaluation of Collaborative Learning Model with Virtual Team in Ubiquitous Learning Environment using Creative Problem-solving Process to enhance Creative Thinking and Collaboration Skills, with a method as follows: 5.2.1 To present the developed activity to the five experts from the fields of instructional design, ubiquitous Learning, information technology, creative thinking and collaboration skills, for suitability evaluation. 5.2.2 To improve the model of Collaborative Learning model according to the suggestions of the experts. 5.2.3 To present the model of Collaborative Learning model in the form of a diagram with report. 5.2.4 To analyze the results of evaluation of the model by mean ( x ) and standard deviation (S.D.) consisting of five criteria for evaluation, according to the ideas of Likert; that is highest, hight, moderate, low and lowest. COLLABORATIVE LEARNING MODEL WITH VIRTUAL TEAM Input Factors The input factors consisted of four components as followed: 1) Objectives of U-CCPS Model, 2) Roles of Instructors, 3) Roles of learners and 4) Design of learning media. Objectives of U-CCPS Model The objectives of this model are to develop creative thinking in terms of fluency, flexibility, elaboration, and collaboration skill. Ubiquitous Learning Environment(ULE) Ubiquitous learning is a seamless learning whenever it is in information space or in physics space, through ubiquitous computing information space and physics space are converged. In ULE Learning, learning demands and learning resources are everywhere; study, life and work are connected each other. When learners meet any practice problem ubiquitous computing help them to resolve it at anytime, anywhere. The learners can easily perception and obtaining learning objects detailed information and content through situational perception of mobile devices. Using dialogue, living community, cooperation studies, social process of internalization, participate in joint activity to realize social learning. Happen of effective ubiquitous learning depends on founding of learning environment [28]. Roles of Instructors The instructors control the learning, maintain it in the learning process, provide advice and assistance, and design learning activities in order to enhance learners' creative thinking and collaboration skills. The instructors also have to prepare instructional media, contents, and management system thereof. Moreover, the instructors have an important role in Ubiquitous learning environment, e.g. speaking with learners, holding seminar, questioning and answering problems with learners, following up the learners, giving feedback information to the learners, and checking the works of learners. Roles of learners The learners do learning activities that are designed by the instructors. The said learning activities encourage the learners to work in groups; whereby the learners have to work with the other members. However, each of them does not have to meet because they can work through their mobile devices. In the learning process, the learners are asked to produce a piece of work together as to the topic provided by the instructors Design of learning media The learning media herein are in the form of online learning media that can be modified, changed, and publicized very quickly. It can also be accessed with no limitation of duration, places, or devices. The learners can choose any contents to learn or any activity to do based on their own interest. The learning media can be displayed on any kind of mobile device. Process The process consisted of two components as followed: 1) Preparation before learning, and 2) Instructional management process. Preparation before learning 6.2.1.1 Orientation This activity was held to provide knowledge and understanding of learning activities, evaluation, tests, and collaborative learning with virtual team in u-learning environment using creative problem-solving process. Register Register through Ubiquitous Learning Management System -All learners had to register in U-LMS to participate in learning activity of the system. Group of the learner The learners were divided into groups based on their interests, and they could use any tools to communicate, to work together, and to analyze any problems. Test of creative thinking and collaboration skills Test of creative thinking and collaboration skills to measurement of the scores before taking part in the developed learning model. U-CCPS process The researcher developed U-CCPS Model with an instructional management process, in which creative thinking was combined with experiences and researches in order to figure out different solutions. The instructional management process using Ubiquitous learning environment includes 4 main steps and 7 sub-steps. Understanding the problem In CPS Process, the true understanding of problems or the contexts thereof will facilitate the finding out of solutions. In this process, there are 3 sub-steps as follow: Constructing opportunities In this step, the learners will find out the problems by themselves. The group members mutually set up general target that leads to problem solving. The instructors provide the topic for the learners to search all relevant problems. The members of the group help one another to search for the problems on the basis of their experiences, roles, and status. Thereby, everybody sees problem solving as a way to construct opportunities. U-LMS system proposes a topic or an issue in order for the learners to find out problems. Then the learners of each group try to search for the problems as to their own knowledge and experiences. The tool used here is Group Discussion via web board and online chat. The learners can work together even though they are in different places and different contexts. When the group members post an issue on the web board, the other members who have similar opinions will leave comments in the said issue so that there would be as many issues as possible from the members. If the other members have different issues to post, they can do so immediately. Whenever the group members leave their comments in any issues, the system will notify the other members about this automatically. In this way, the learners can still work together through their mobile device although they are in different places and different contexts. Exploring data This step focuses on the collection of facts, opinions, satisfaction, confirmation, conflicts, and consideration of overall contexts. The questions are to be posted to get the clarity of issue contexts. Information, understandings, and feelings are also collected in this step to better understand the issues. In other words, those who want to solve the problems have to search for information and classify it (convergence) so as to obtain the information most effective to the problem solving. Each member of the team, in this step, is collecting data relevant to the issues posted on the web board and online chat of U-LMS system. This is to finalize the most important information that can define the said issues. Framing problem This step is designed to specially help the problem solvers and to make the issues more obvious. Thereby, the group members consider and choose the most significant issues, based on the concept provided by instructors, out of all that have been posted. In this step, the members have to cooperate via the web board and group discussion. The tool used here is Google Document on U-LMS System. Generating ideas This step is to find out the ways to solve the problems or answer the questions in the previous steps. The questions in this step have no fixed answers, so there must be divergent thinking in terms of fluency, flexibility, and elaboration. Then, the ideas are collected and the one with highest possibility will be chosen. That is why this step is called "generating ideas". The highlight in this step is the opportunity to extend thinking, both in and out of the box. The learners help one another to look for as many solutions as possible by posting them on their group web board. Besides, the learners search for information, brain storm, and share the data with the other members by means of Google Document on their mobile devices. This is to create online documents derived from their collaboration. Planning for action This step includes: Solution finding This step is to analyze, define, and clarify the ideas so that they are more concrete. It also requires meticulous consideration and investigation. The main point of this step is to scrutinize and select the likely practical option that is manageable and most efficient to the tasks. The highlight of this step is that all group members choose the tools or methods that are really practical by picking out the best idea in order to attain the best solution. The tools here include the group chat where the members can share their ideas, and the web board to post their opinions and choose the optimal solution. Acceptance finding This step is to find acceptance and opposition so as to derive solutions. The selection of solutions is based on individuals, places, equipment, or time that can promote the operation plan to be successful. The management must be proceed as planned; e.g. the members must strictly follow the plan, attend the meeting, and provide acceptance for the solutions. In this step, the group members work together to set up an operation plan and find out the solutions with the aid of Google Document on their mobile devices. Appraising tasks Creative problem-solving process is so effective and flexible that it is chosen as a tool to find out solutions. After the members have produced a work piece as planned, their tasks and their work will be appraised by instructors, friends, and experts. Whereby, the said appraisal is done through U-LMS System on mobile device. Control Learning activities based on U-CCPS Model encourages the learners to have creative thinking and collaboration skills, and to search for knowledge by themselves via U-LMS System provided by the instructors. The learners have to work together, solve the problems, and share their knowledge under the control of the system that is provided by instructors. The control is divided into: Control and Follow up of learners U-LMS will examine and follow the learners in every aspect such as interest in study and activity, enthusiasm in problem solving and finding solutions, determination and responsibility for work. The system also provides feedback information. Set up timeline to examine the tasks U-LMS will set up a timeline to check out the tasks and use it as a tool to control and encourage the learners to study and do all activities in due time. Output Measurement and assessment in each unit are all authentic. The learning outcomes, Torrance creative thinking, and collaboration skills will be measured after collaborative learning with virtual team activities using creative problem-solving process in ubiquitous learning environment. Feedback This refers to the analysis of information from different steps of U-CCPS Model, opinions of the experts, and opinions of the learner. This is to optimize all steps of the instructional model and to achieve the goals as expected. THE EVALUATION RESULTS OF THE U-CCPS MODEL The evaluation is carried out by submitting the developed model to the five experts for a certification on the suitability of its components, U-CCPS process, and for a test [29] who found that the instructional design of ADDIE could be applied in Ubiquitous instructional design. In addition, this research used the concepts and principles of collaborative learning in Ubiquitous environment. This is in compliance with the research of Tseng et al., [30] who found that collaborative learning in Ubiquitous environment encouraged cooperation of learning among the learners everywhere and every time, and it also complied with the learning contexts of learners. 8.2. The results of assessment on the instructional model at the preparation step before learning show that the model is highest suitable. This is because Ubiquitous instructional design is quite new and it requires orientation to get the learners well prepared for learning. The results are in line with the research of Olaham et al., [31] who found that there should be orientation before any learning so that the learners could understand different aspects and be able to study in Ubiquitous environment as efficiently as possible.. 8.3. The results of assessment on the instructional process at the management step show that the process is highest suitable. The preparation step included the steps of understanding the problem, generating ideas, planning for action, appraising tasks, and summarizing the principles and concepts. This corresponds to the research of Treffinger, Isaksen and Dorval [5] , who found that the instructional activities using creative problem-solving process had to include the steps of understanding the problem, generating ideas, planning for action, and appraising tasks. 8.4. The results of assessment on the application of instructional model by the experts show that the model is highest suitable. This is because the instructional model as well as its steps and activities of collaborative learning were all suitable to the creative problem-solving skill. It complies to the research of Maraviglia and Kvashny [6], who applied the instructional model using creative problem-solving process. They found that the newly developed creative problemsolving process had highest influence on the deveopment of creative thinking and creative problem-solving skills. This also corresponds to the research of Suangsuda Pansakul [11], who studied and presented the learning model of creative and collaborative problem-solving process in the internet-based organizations. She found that the sample group, after participating in the learning process, had higher level of creative problem-solving skills than they did before. Moreover, the research of Lau et al. [14] found that collaborative learning is the process that provides the learners with collaboration skills. Figure 1 . 1conceptual framework of U-CCPS Model IN UBIQUITOUS LEARNING ENVIRONMENT USING CREATIVE PROBLEM-SOLVING PROCESS TO ENHANCE CREATIVE THINKING AND COLLABORATION SKILLS (U-CCPS MODEL) U-CCPS Model was developed on the basis of System Approach, consisting of 5 main elements and 10 sub-elements. Figure 2 . 2Collaborative Learning Model with Virtual Team in Ubiquitous Learning Environment using Creative Problem-solving Process to enhance Creative Thinking and Collaboration Skills (U-CCPS Model) . The evaluation result by the expert has shows that the components have highest suitability ( x = 4.52, S.D. = 0.58), the U-CCPS process have the highest suitability ( x = 4.58, S.D. = 0.65), and the overall appropriateness for a test have the high suitability ( x = 4.47, S.D. = 0.55). Table 1 . 1The evaluation results of U-CCPS Model.The table 1. Shows that the experts agree that a U-CCPS components was highest suitability.( x = 4.52, S.D. = 0.58)Table 2. The evaluation results of Instructional Process The table 2. shows that the experts agree that a U-CCPS Process was highest suitability. ( x = 4.58, S.D. = 0.65) Table 3. The evaluation results of U-CCPS model for a test The table 3. shows that the experts agree that a U-CCPS Model was appropriateness for test in the high ( x = 4.47, S.D. = 0.55). The results of assessment on the model's elements show that the 5 main elements of the model; i.e. Input factors, Process, Output, Control and Feedback, are highest suitable. This is because the process of instructional design employed the principle of analysis, design, development, implement and evaluate (ADDIE Model). The results are in accordance with the research of Tekinarslan et al.,Evaluation Lists Results Level of suitability x S.D. 1. Input factors 4.80 0.45 Highest 2. Process 4.60 0.55 Highest 3. Control 4.40 0.55 High 4. Output 4.20 0.45 High 5. Feedback 4.60 0.89 Highest Summary 4.52 0.58 Highest Evaluation Lists Results Level of suitability x S.D. 1. Preparation before learning 4.70 0.59 Highest 1.1 Orientation 4.60 0.55 Highest 1.2 Register 4.80 0.45 Highest 1.3 Group of learners 4.80 0.45 Highest 1.4 Test of creative thinking and collaboration skills 4.60 0.89 Highest 2. U-CCPS Process 4.45 0.71 High 2.1 Understanding the problem 4.60 0.89 Highest 2.2 Generating ideas 4.20 0.84 High 2.3 Planning for action 4.60 0.55 Highest 2.4 Appraising tasks 4.40 0.55 High Summary 4.58 0.65 Highest International Journal on Integrating Technology in Education (IJITE) Vol.2, No.4, December 2013 The Partnership for 21st Century Skills. P21 Framework Definitions. The Partnership for 21st Century Skills. (2013, October 9). P21 Framework Definitions[Online]. Approach to student learning in the 21st century. V Panich, Tathata PublicationV. Panich, (2012) Approach to student learning in the 21st century, Tathata Publication. Creative problem solving : Overview of educational implications. D J Treffinger, Educational Psychology Review. 73D. J. Treffinger, (1995). "Creative problem solving : Overview of educational implications", Educational Psychology Review, Vol. 7, No. 3, pp. 301-312. Creative Approaches to Problem Solving. S G Isaksan, K B Dorval, D J Treffinger, Kendall-HuntDubuque, IAS. G. Isaksan, K. B. Dorval and D. J. Treffinger, (1994). Creative Approaches to Problem Solving. Dubuque, IA: Kendall-Hunt. Creative problem solving (CPSVersion 6.1) A contemporary framework for managing change. D J Treffinger, S G Isaksen, K B Dorval, D. J. Treffinger, S. G. Isaksen and K. B. Dorval, (2003). Creative problem solving (CPSVersion 6.1) A contemporary framework for managing change [Brochure]. Managing Virtual Changes-A Guide to Creative Problem-solving in the Design Professions. R L Maraviglia, A Kvashny, Author HouseBloomington, IndianaR. L. Maraviglia and A. Kvashny, (2006). Managing Virtual Changes-A Guide to Creative Problem-solving in the Design Professions. Bloomington, Indiana: Author House. National ICT Policy Framework. Ministry of Information and Communication Technology. Ministry of Information and Communication Technology, (2011). National ICT Policy Framework 2011-2020. Issues and Challenges in Ubiquitous Computing. K Liyytinen, Y Yoo, Communications of the ACM. 4512K. Liyytinen and Y. Yoo, (2002). Issues and Challenges in Ubiquitous Computing. Communications of the ACM, vol.45, no.12, pp.62-65. Study of Instructional design in Ubiquitous Learning. W Junqi, L Yumei, L Zhibin, Second International Workshop on Education Technology and Computer Science. W. Junqi, L. Yumei and L. Zhibin, (2010). "Study of Instructional design in Ubiquitous Learning" in Second International Workshop on Education Technology and Computer Science, pp. 518-523. Ubiquitous Learning Environment: An Adaptive Teaching System Using Ubiquitous Technology. V Jones, J H Jo, Conference of the 21 st ASCILITE. V. Jones and J.H. Jo, (2004). "Ubiquitous Learning Environment: An Adaptive Teaching System Using Ubiquitous Technology" in Conference of the 21 st ASCILITE, pp. 468-474. Collaborative learning. Pansakul, Educational Journal. 152Pansakul. S, (2000). "Collaborative learning", Educational Journal, Vol. 15, No. 2, pp. 1-8. Is this Collaboration. J M Garlach, Collaboration Learning : Underlying Process and Effective Techniques. Jossey-Bass PublishersJ. M. Garlach, (1994). "Is this Collaboration", Collaboration Learning : Underlying Process and Effective Techniques, Jossey-Bass Publishers, pp. 5-13. CSCL & Model of Instruction collaborative learning. Koschman, Koschman. (2012, December 31). CSCL & Model of Instruction collaborative learning[Online]. Available: http://www.uib.no/People/sinia/CSCL/ web_struktur-975.html Developing Collaborative and Self-Directed Learners through the Affordances of a Virtual Learning Environment. F Y Lau, Celebrating Learning through Active Research. SingaporeF. Y. Lau et al. (2010). "Developing Collaborative and Self-Directed Learners through the Affordances of a Virtual Learning Environment" in Celebrating Learning through Active Research,, Ministry of Education Singapore, pp.79-84. Investigating Cultural Differences in Virtual Software Teams. G Dafoulas, L Macaulay, The Electronic Journal of Information Systems in Developing Countries (EJISDC). G. Dafoulas and L. Macaulay, (2002), "Investigating Cultural Differences in Virtual Software Teams". The Electronic Journal of Information Systems in Developing Countries (EJISDC), no.7, pp. 1-14. Conflict Resolution in Virtual Teams. Y Shin, Organizational Dynamics. Y. Shin, (2005). Conflict Resolution in Virtual Teams . Organizational Dynamics , pp. 331-345. Global virtual teams for value creation and project success: A case study. L Lee-Kelley, T Sankey, International Journal of Project Management. 26L. Lee-kelley and T. Sankey, (2008), "Global virtual teams for value creation and project success: A case study". International Journal of Project Management, No.26, pp. 51-62. Implementing virtual teamworking. Part 1: a literature review of best practice. J Bal, P K Teo, Logistics Information Management. 13J. Bal and P.K. Teo, (2001), "Implementing virtual teamworking. Part 1: a literature review of best practice", Logistics Information Management, No.13, pp. 346 -352. Intelligent virtual team in collaborative design. W Qianping, H Hai, W Xiaoyi, 7th International Conference on Computer-Aided Industrial Design and Conceptual Design. W. Qianping, H. Hai and W. Xiaoyi, (2006). "Intelligent virtual team in collaborative design" in 7th International Conference on Computer-Aided Industrial Design and Conceptual Design, pp. 1-6. Designing and Supporting Cooperative and Ubiquitous Learning Systems for People with Special Needs. L A Fernández, M J Rodríguez, G M Noguera, Proceedings of the Confederated International Workshops and Posters on the Move to Meaningful Internet Systems (OTM). the Confederated International Workshops and Posters on the Move to Meaningful Internet Systems (OTM)L. A. Fernández, M. J. Rodríguez and G. M. Noguera, (2009). "Designing and Supporting Cooperative and Ubiquitous Learning Systems for People with Special Needs" in Proceedings of the Confederated International Workshops and Posters on the Move to Meaningful Internet Systems (OTM), pp. 423-432. Study of Instructional design in Ubiquitous Learning. W Junqi, L Yumei, L Zhibin, Second International Workshop on Education Technology and Computer Science. W. Junqi, L. Yumei and L. Zhibin. (2010). "Study of Instructional design in Ubiquitous Learning" in Second International Workshop on Education Technology and Computer Science, pp. 518-523. Context-Aware Support for Computer-Supported Ubiquitous Learning. H Ogata, Y Yano, Proceedings of the 2nd IEEE International Workshop on Wireless and Mobile Technologies in Education. the 2nd IEEE International Workshop on Wireless and Mobile Technologies in EducationH. Ogata and Y. Yano, (2004). "Context-Aware Support for Computer-Supported Ubiquitous Learning" in Proceedings of the 2nd IEEE International Workshop on Wireless and Mobile Technologies in Education, pp.27-34. Context-Aware Support for Learning Japanese Polite Expressions. H Ogata, C Yin, Y , Yano , Proceedings of the IEEE International Workshop on Wireless and Mobile Technologies in Education, WMTE'04. the IEEE International Workshop on Wireless and Mobile Technologies in Education, WMTE'04H. Ogata, C. Yin and Y, Yano, (2004). "Context-Aware Support for Learning Japanese Polite Expressions" in Proceedings of the IEEE International Workshop on Wireless and Mobile Technologies in Education, WMTE'04. Computer Supported Ubiquitous Learning Environment for Vocabulary Learning Using RFID Tags. H Ogata, R Akamatsu, Y Yano, Proceedings of TEL2004. TEL2004H. Ogata, R. Akamatsu and Y. Yano, (2004). "Computer Supported Ubiquitous Learning Environment for Vocabulary Learning Using RFID Tags" in Proceedings of TEL2004. Ubiquitous Computing Technologies for Ubiquitous Learning. K Sakamura, N Koshizuka, Proceedings of the 2005 IEEE International Workshop on Wireless and MobileTechnologies in Education (WMTE '05). the 2005 IEEE International Workshop on Wireless and MobileTechnologies in Education (WMTE '05)K. Sakamura and N. Koshizuka (2005). "Ubiquitous Computing Technologies for Ubiquitous Learning" in Proceedings of the 2005 IEEE International Workshop on Wireless and MobileTechnologies in Education (WMTE '05), pp.11-20. An Instructional Design Model for Ubiquitous Learning Environments. E Tekinarslan, M D Gurer, R K Agca, Proceedings of International Educational Technology Conference. International Educational Technology ConferenceE. Tekinarslan, M.D. Gurer and R.K. Agca, (2008). "An Instructional Design Model for Ubiquitous Learning Environments" in Proceedings of International Educational Technology Conference, pp. 333-336. The Torrance Tests of Creative Thinking (Norma-technical Manual). E P Torrance, Scholastic Testing Service, IncBensenville, ILE.P. Torrance, (1974), The Torrance Tests of Creative Thinking (Norma-technical Manual).Bensenville, IL: Scholastic Testing Service, Inc. Using the IOT to Construct Ubiquitous Learning Environment. R Xue, L Wang, J Chen, Second International Conference on Mechanic Automation and Control Engineering. R. Xue, L. Wang and J. Chen, (2011). "Using the IOT to Construct Ubiquitous Learning Environment" in Second International Conference on Mechanic Automation and Control Engineering, pp. 7878-7880. A instructional design model for ubiquitous learning environments. E Tekinarslan, M D Gurer, R K Agca, E. Tekinarslan, M. D. Gurer and R. K. Agca, (2013, May 12). A instructional design model for ubiquitous learning environments[Online]. A Collaborative Ubiquitous Learning Approach for Conducting Personal Computer-Assembling Activities. G Tseng, C H Wu, G Hwang, International Conference on Advanced Learning Technologies. G. Tseng, C.H. Wu and G. Hwang, (2010). "A Collaborative Ubiquitous Learning Approach for Conducting Personal Computer-Assembling Activities" in International Conference on Advanced Learning Technologies, pp. 726-727. Ubiquitous devices a preparation for studentteachers' use of technology for teaching and learning. E Oldham, A Fitzgibbon, K Johnston, Society for Information Technology and Teacher Education International Conference. E.Oldham, A. FitzGibbon and K. Johnston, (2005). "Ubiquitous devices a preparation for student- teachers' use of technology for teaching and learning" in Society for Information Technology and Teacher Education International Conference.
[]
[ "Robust Tracking Control for Nonlinear Systems: Performance optimization via extremum seeking", "Robust Tracking Control for Nonlinear Systems: Performance optimization via extremum seeking" ]
[ "Jiapeng Xu ", "Ying Tan ", "Xiang Chen " ]
[]
[]
This paper presents a controller design and optimization framework for nonlinear dynamic systems to track a given reference signal in the presence of disturbances when the task is repeated over a finite-time interval. This novel framework mainly consists of two steps. The first step is to design a robust linear quadratic tracking controller based on the existing control structure with a Youla-type filterQ. Secondly, an extra degree of freedom: a parameterization in terms ofQ, is added to this design framework. This extra design parameter is tuned iteratively from measured tracking cost function with the given disturbances and modeling uncertainties to achieve the best transient performance. The proposed method is validated with simulation placed on a Furuta inverted pendulum, showing significant tracking performance improvement.
10.48550/arxiv.2304.00138
[ "https://export.arxiv.org/pdf/2304.00138v1.pdf" ]
257,912,604
2304.00138
1effbce80aa132a10984310096062775dbdb0a9b
Robust Tracking Control for Nonlinear Systems: Performance optimization via extremum seeking Jiapeng Xu Ying Tan Xiang Chen Robust Tracking Control for Nonlinear Systems: Performance optimization via extremum seeking This paper presents a controller design and optimization framework for nonlinear dynamic systems to track a given reference signal in the presence of disturbances when the task is repeated over a finite-time interval. This novel framework mainly consists of two steps. The first step is to design a robust linear quadratic tracking controller based on the existing control structure with a Youla-type filterQ. Secondly, an extra degree of freedom: a parameterization in terms ofQ, is added to this design framework. This extra design parameter is tuned iteratively from measured tracking cost function with the given disturbances and modeling uncertainties to achieve the best transient performance. The proposed method is validated with simulation placed on a Furuta inverted pendulum, showing significant tracking performance improvement. I. INTRODUCTION Robust tracking control of nonlinear systems has been extensively studied in the literature using various robust techniques, such as H ∞ control [1], [2] and sliding mode control [3]. These methods are in general the worst-case design, which would ensure the stability under the worstcase disturbances. On the other hand, optimal performance such as a linear quadratic form has been the focus of the design in industry applications. However, it is usually hard to analyze the performance for nonlinear dynamics. One of the key reasons to contribute this difficulty comes from the the analysis tool used in stability analysis. Lyapunov direct method [4] has been used to provide sufficient conditions to guarantee the stability of nonlinear dynamics. The optimal control for nonlinear dynamics requires to solve the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear partial differential equation with respect to a given cost. Solving this HJB equation is computational costly. In contrast, both robust and optimal control designs have been extensively investigated for linear time-invariant (LTI) dynamic systems [5]- [8]. In particular, a robust controller design with a Youla-type filterQ [8], [9] has been proposed recently, which is motivated by the generalized internal model control (GIMC) proposed in [6]. The robust controller withQ provides automatic robustness recovery in the linear quadratic Gaussian (LQG)/H 2 control [8]. Its key idea is to use theQ filter to balance the optimal performance without consideration of the disturbance and robust performance using the techniques such as H ∞ control. Since the filter Q is driven by the residual signal indicating the mismatch 1 This work proposes to utilizes the robust controller with Q in [8] to systematically design the feedback control for a nonlinear dynamic system via its linearization. The proposed framework is used to generate optimal tracking performance for a class of nonlinear dynamic systems to track a given reference trajectory. More specifically, the proposed framework first presents a robust linear quadratic tracking (LQT) controller design based on the filterQ. Then by introducing an extra gain factor in terms ofQ, which can be treated as the balance between the LQT performance and the robustness with respect to disturbances and uncertainties coming from linearizations and other external signals, an updating law is generated to tune this gain factor to minimize the tracking cost in the presence of modeling uncertainties and disturbances. The choice of the gain factor does not affect the local stability properties of the closed-loop nonlinear system, while it improves the tracking performance. In this work, the data-driven extremum seeking (ES) approach [13]- [16], which is a model-free optimization method, is adapted to find this optimal gain factor. Alternatively, other model-free optimization techniques such as reinforcement learning can also be considered [2], [17]. The effectiveness of the proposed framework is validated with simulation placed on a Furuta inverted pendulum. It has been shown that this optimal parameter is dependent on the nonlinear dynamics, the type of the reference trajectories, as well as the type of disturbances. The obtained optimal gain can achieve much better transient tracking performance compared with the standard LQT controller and the robust controllers such as H ∞ . The remainder of this paper is organized as follows. Section II formulates the tracking problem for nonlinear systems of interest. Section III presents a design procedure of LQT controller withQ. Section IV further presents a controller design for nonlinear systems where performance is further optimized via ES. Section V provides simulation results on an inverted pendulum. Finally, Section VI concludes this work. II. PROBLEM FORMULATION Nonlinear systems of the following form are considered: x = f (x, u, w), x(0) ∈ R n y = g(x, w),(1) where x ∈ R n is the state of system, y ∈ R p is the measurement output, u ∈ R m is the control input, and w ∈ R nw is a disturbance signal representing external disturbances and/or modeling uncertainties. The nonlinear mappings f : D X × D U × D W → R n and g : R n × D W → R p are continuously differentiable functions. Here D X , D U , and D W are compact sets in R n , R m , and R nw respectively. Moreover, it is assumed that f (0, 0, 0) = 0, indicating the origin is in the set Ω = D X × D U × D W . The linearization of system (1) around (x = 0, u = 0, w = 0) becomes the following LTI system: x = Ax + B 1 w + B 2 u, (2) y = C 2 x + D 21 w,(3) where A = ∂f ∂x x,u,w=0 , B 1 = ∂f ∂w x,u,w=0 , B 2 = ∂f ∂u x,u,w=0 , C 2 = ∂g ∂x x,w=0 , D 21 = ∂g ∂w x,w=0 . The following assumption is standard to stabilize the linearized system (2) by output feedback using the output signal in (3). Assumption 1: (i) (A, B 2 ) is stabilizable, and (ii) (C 2 , A) is detectable. The control objective of this work is to design an appropriate controller such that the output y = Ex, E ∈ R p1×n ,(4) can track a desired (reference) trajectory r that is known a priori. Specifically, we seek to find an optimal control input to minimize the following finite-time quadratic cost function: J(u) = 1 T T 0 [(ỹ − r) Q(ỹ − r) + u Ru]dt,(5) where Q is positive semi-definite and R is positive definite. In this paper, we use " " to represent the transpose of a matrix. We propose to solve the formulated tracking problem for nonlinear systems (1) by two steps. The first step is to design a robust linear quadratic tracking (LQT) controller based on a Youla-type filterQ proposed in [8] , which will be detailed in the next section. The second step is to introduce an extra gain factor to the filterQ and use the data-driven optimization algorithm such as extremum seeking (ES) [13]- [16] to tune this factor. III. ROBUST LINEAR QUADRATIC TRACKING CONTROL WITHQ This section presents a robust LQT controller design based on the filterQ in [8], which balances the robustness with respect to disturbances and the LQ optimal tracking control, for the LTI systems taking the same form as the linearized system (2) and (3). A. Preliminaries of the controller structure withQ The structure of the controller withQ is presented in Fig. 1 (see more discussions in [8]). It contains some performance related variable z defined as z = C 1 x + D 12 u,(6) an observer, and the filterQ(·) satisfyingQ(0) = 0. The input of the filterQ is the residual signal f =ŷ − y, which is the deviation of estimated and actual sensor outputs. It reflects the mismatch between the nominal model and the true system. As shown in [8], the output of the filterQ(·) is designed for robustness recovery, reducing the effect of disturbances. For example, the H ∞ design method can be used. The control input u consists of two parts : u l and u f . The first part u l is a nominal control, which is to design the stabilizing controller with the help of the observer without consideration of w(t). The matrices (F, L) are the state feedback and observer gains, respectively, such that A+B 2 F and A + LC 2 are Hurwitz by Assumption 1. The second part u f is the output of the filter, i.e., u f =Q(f ). If w(t) = 0, ∀t ≥ 0, the residual signal f is zero and produces u f =Q(f ) = 0. Hence the role of this filter is to enhance the robustness. One admissible filterQ is of the following form [8]:Q :ẋ q = A q x q + B q f, u f = F q x q .(7) Matrices A q , B q and F q can be obtained by the standard H ∞ techniques [5], [18] for the augmented system with state (x, e), with e = x −x, and A q is Hurwitz. Following this controller structure, next will introduce the LQT controller design, followed by the robust controller usingQ. B. Linear Quadratic Tracking (LQT) Consider the nominal case of the LTI system (2) and (3), i.e., w = 0 and the initial state x(0) is known: x = Ax + B 2 u, x(0) given.(8) This reduces to a state feedback control problem. The following proposition provides the optimal solution to minimizing (5) by using the Hamilton-Jacobi-Bellman equation [7,Chapter 2]; see also [7,Chapter 4]. Proposition 1: Considering the system consisting of (8) and (4), for a given reference trajectory r and the given performance index (5), the optimal control u * is given by −ḃ = (A + B 2 F ) b + E Qr, b(T ) = 0, (9) u * = F x + R −1 B 2 b,(10) where F = −R −1 B 2 P , and P is the solution of the following differential Riccati equation: −Ṗ = P A + A P −P B 2 R −1 B 2 P + E QE.(11) In practice, to simplify the design of LQT controller, the algebraic Riccati equation instead of differential Riccati equation (11) can be considered, such that a time-invariant stabilizing control gain F is obtained. For convenience, the feed-forward control law in (10) is denoted as u r := R −1 B 2 b. The diagram of control system with LQ optimal tracking is shown in Fig. 2. C. Design of LQT withQ Now we design the LQT controller withQ to robustify the feedback system. To do this, we redefine the performance variable z in (6) as z = C 1 (ỹ − r) + D 12 u,(12) such that it coincides with the tracking cost in (5). We can let C 1 C 1 = Q, D 12 D 12 = R and C 1 D 12 = 0 to reduce the weighing matrix parameters. To design the filterQ, set the reference signal r to be zero such that u r = 0 and introduce an observer gain L such that A + LC 2 is Hurwitz. Then the design procedure ofQ can be the same as in [8], i.e., (7). More specifically, the filterQ is designed according to the following H ∞ performance criterion: T zw (s) ∞ < γ,(13) where γ > 0 is a prescribed value and T zw (s) is the closedloop transfer function from w to z. See [8] for more details. By combing the LQT controller, the observer gain L and the filterQ designed using H ∞ technique, the diagram of the robust LQT withQ is shown in Fig. 3. Remark 1: The transfer functions from u r to u ,y, andỹ in the closed-loop system are independent of the filterQ. These transfer function are the same as those in the LQT design. Therefore, the proposed tracking design scheme allows us to selectQ to attenuate disturbances without affecting the tracking capability in the LQT design. IV. PERFORMANCE OPTIMIZATION FOR NONLINEAR SYSTEMS VIA EXTREMUM SEEKING We propose to solve the formulated optimal tracking problem based on the linearized model (2) and (3). However, the performance of the robust LQT controller with Q designed in the previous section cannot be guaranteed for nonlinear systems due to the existence of linearization errors. Moreover, this linear controller cannot give an optimal control input minimizing the performance index (5) for our tracking problem for nonlinear systems (1). In the sequel, we will design an optimal control input by introducing an extra gain factor α ∈ R to the filterQ and then optimize this factor by the data-driven ES algorithm. A. An extra gain factor forQ To deal with linearization errors and unmodeled disturbances, a gain factor α ∈ R for the output of the filterQ is introduced as an extra degree of freedom, such that a new Q denoted byQ α is obtained: Q α :ẋ q = A q x q + B q f, u f = αF q x q .(14) When α = 0, it leads to an optimal tracking controller as indicated in Proposition 1 while α = 1 leads to the LQT controller withQ shown in Fig. 3. Intuitively, the choice of this α reflects some balance between the LQT and the robustness. This leads to a closed-loop nonlinear system with the LQT controller withQ α shown in Fig. 4. Now the tracking performance index J(u) in (5) can be written as a function of α, i.e., J(α) = 1 T T 0 [(ỹ − r) Q(ỹ − r) + u Ru]dt,(15) and we seek to find an optimal α parameter to minimize the above cost function. Before solving the α optimization problem, we present the following result about the local stability properties of such a closed-loop nonlinear system. Theorem 1: Let the origin (x = 0, u = 0, w = 0) be an equilibrium point for the nonlinear system (1) and let system (2) and (3) be the linearization of (1) about the origin. Then for any α ∈ R, the origin of the closed-loop nonlinear system in Fig. 4 is locally exponentially stable. Proof: Let x c := x qx . Then it follows from the design of the LQT controller withQ shown in Fig. 3 and the expression ofQ α in (14) that we have the following linear tracking controller: x c = A c x c + B c y + B r u r , u = F c x c + u r ,(16) where A c = A q B q C 2 αB 2 F q A + B 2 F + LC 2 , B c = −B q −L , B r = 0 B 2 , F c = αF q F . Hence, by a linear transformation, it can be shown that the closed-loop matrix for the linearized system (2) and (3) has the following triangular form A =Ā(α) =   A + B 2 F αB 2 F q B 2 F 0 A q B q C 2 0 0 A + LC 2   .(17) Since A + B 2 F , A + LC 2 and A q are all Hurwitz,Ā is Hurwitz. It can be verified that the origin (x = 0, x c = 0, w = 0) is an equilibrium point of the closed-loop nonlinear system. Thus, we can conclude the local exponential stability of the closed-loop nonlinear system in Fig. 4 at the origin for any α ∈ R [4, Section 12.2]. Theorem 1 shows that the choice of α does not affect the local stability properties of the closed-loop nonlinear system presented in Fig. 4 while it affects the performance, in particular, the transient performance. This work tries to find an optimal α to tracking control for nonlinear systems. In the sequel, the data-driven extremum seeking (ES) [13]- [16] will be used tune the parameter α for a given disturbance w. B. Performance optimization via extremum seeking It is highlighted that in the linearized model consisting of (2) and (3), the disturbances coming from two parts: one is unmodelled uncertainties coming from the linearization residue and unmodelled dynamics. Such uncertainties are related to the size of compact sets D X , D U , and D W . The other is other types of deterministic and random noises. In this paper, our focus is the unmodelled uncertainties and deterministic noises that are repeatable when the nonlinear dynamics run over a fixed time interval [0, T ]. Thus, the disturbance can be re-written as w = w(t, x, u). Assumption 2: It is assumed that for a deterministic and repeatable disturbance w(t, x, u), t ∈ [0, T ], for any x ∈ D X , and u ∈ D U , there exists a unique optimal α * ∈ R such that the cost function defined (15) can reach a minimum. Under this assumption, the ES algorithm presented in [13] is used to tune the parameter α for a given disturbance w(t, x, u) by repeatedly running the closed-loop nonlinear system consisting of (1) with the control law (16) over the finite time [0, T ]. ES is a model-free optimization method which uses only input-output data to see an optimal input with respect to a given cost [13]. The ES algorithm adopted here works for the iteration domain, which is the same as the one used in [14]: ζ(k + 1) = −hζ(k) + J(α(k)), α(k + 1) =α(k) − δβ cos(ωk)[J(α(k)) − (1 + h)ζ(k)], α(k + 1) =α(k + 1) + β cos(ω(k + 1)),(18) where k is the iteration number, 0 < h < 1, ζ(k) is a scalar, δ is the step size, and β is the perturbation amplitude. Stability and convergence are mainly influenced by the values of δ and β. The modulation frequency ω is chosen such that ω = aπ, where a satisfies 0 < a < 1. The overall ES α tuning scheme is summarized in Fig. 5. The local convergence analysis of such ES algorithm when Assumption 2 holds locally can be found in [19]. Using the similar analysis techniques as in [15] for the continuous-time systems, non-local convergence can be achieved when Assumption 2 holds. The design procedure of the proposed robust optimal tracking control scheme can be summarized as the following three steps. • Step 1: Obtain the linearized model around the origin (x = 0, u = 0, w = 0) from nonlinear system (1); • Step 2: Design the LQT controller withQ α as in Fig. 3 in whichQ in (7) is replaced byQ α in (14); • Step 3: Tune α via the ES algorithm (18) for a given disturbance w(t, x, u), shown in Fig. 5. Remark 2: It is noted that the focus of this work is to ensure that the transient behaviour of the closed-loop system presented in Fig. 5 over the finite time interval [0, T ] can be improved over iteration when the disturbance w is repeatable over iteration. If Assumption 2 holds, the ES algorithm in (18) can ensure that the cost function (15) decreases over iterations for appropriately tuned parameters (a, δ, β, h) in (18). The decrease of the cost function has been verified in simulations. V. SIMULATION RESULTS In order to verify the effectiveness of the proposed robust tracking control in Fig. 5, a simulation example on a Furula inverted pendulum (the Quanser QUBE Servo 2 rotary pendulum [20]) is presented. A. Linearization and design parameters The Furuta inverted pendulum is an under-actuated (unstable) nonlinear system with the state variable x = θ 1 θ 2θ1θ2 , consisting of the base arm angle θ 1 and the vertical arm (pendulum) angle θ 2 , as well as the corresponding angular velocities. When the pendulum is in the upright position, x = 0 0 0 0 . The nonlinear equations of motion for the pendulum system in disturbancefree case are [20]: Mθ 1 =m p lrθ 2 cos θ 2 − J pθ1θ2 sin 2θ 2 − m p lrθ 2 2 sin θ 2 − b rθ1 + τ, J pθ2 =m p lrθ 1 cos θ 2 + 0.5J pθ 2 1 sin 2θ 2 + m p gl sin θ 2 − b pθ2 ,(19) where M = J r + J p sin 2 θ 2 and τ = k m /R m (u − k mθ1 ) is the applied torque at the base of the rotary arm with u the control input/voltage. Following the 3-step design procedure in the previous section, we need to linearize the nonlinear system first. The origin (x = 0, u = 0) is an equilibrium point for pendulum system (19), leading to the following linearization model with assumed disturbance/noise w where J t = J p J r −m 2 p l 2 r 2 . The values of system parameters of the rotary pendulum are presented in Table I. Since w is considered as linearization errors and deterministic unmodelled disturbances, the disturbance matrices B 1 and D 21 above are chosen by trial and error. Let y = 1 0 y,(20) such that E = 1 0 0 0 , meaning that the base arm angle θ 1 needs to track a reference signal and the vertical pendulum needs to be balanced in the upright position. The weighting matrices for the performance variable z in (6) and for the tracking performance (15) are selected as C 1 = 15 0 , D 12 = 0 √ 2 , R = C 1 C 1 and Q = D 12 D 12 . Then according to Step 2, we need to design the LQT controller withQ α . The LQT controller in Proposition 1 is first designed for the linearized pendulum system, in which the closed-loop poles are located at (−16.55 ± j12.80, −21.20 ± j1.76). The observer gain L is chosen such that the eigenvalues of A + LC 2 are (−59.40 ± j80.54, −61.04 ± j76.24). The filterQ is then designed to satisfy the H ∞ performance (13) [8], where γ = 0.21. Following Step 3, we shall tune α via the ES algorithm (18) in simulation under different disturbance signals, shown in the next subsection. B. Simulation results We consider that the disturbance signal w is injected into the control voltage u and the measurement output y is perturbed by white noise. In particular, two different disturbance signals are considered: a square wave with the amplitude of 2 and frequency of 0.5Hz, denoted by w 1 , and a kind of vanishing signal w 2 (t) = 5e −0.1t . Two different references are considered: Case A tracks a square wave with amplitude of 20 degrees and frequency of 0.05Hz while Case B tracks a sinusoidal signal r = π 3 sin πt. In the simulation, the initial value of ES is selected as α(0) = 1, indicating the local H ∞ performance at the first iteration. For all simulations, the quadratic cost function J(α) in (15) is used with T = 20s and the parameters a, h and β in the ES algorithm (18) are set to a = 0.8, h = 0.1, β = 0.015. The choice of the step size δ in the ES algorithm depends on a specific disturbance signal and reference signal. The optimal α tuned via ES is denoted as α * . Now we tune α for disturbances w 1 and w 2 . Table II summarizes the α * values in which minimum costs are reached for Case A and Case B in the presence of w 1 and w 2 . Figs. 6 and 7 show that ES minimizes the cost function (15) with convergence to the parameter α that produces a minimum for Case A and Case B in the presence of w 1 respectively. It can be seen that the controller with α * obtained from ES tuning yields a much better closed-loop performance in terms of transient behaviours compared with the cases when α = 0 (LQT performance) and α = 1 (H ∞ performance). Figs. 6 and 7 show the effectiveness of the proposed ES algorithm when dealing with deterministic uncertainties and linearization errors. Similar results are observed when dealing with w 2 . VI. CONCLUSION In this paper, we develop a novel tracking control scheme for a classe of nonlinear systems in the presence of disturbances based on the robust controller with a Youla-type filterQ and the data-driven ES technique. A key point is that a gain factor α is introduced to the filterQ, which allows an extra degree of freedom to be optimized for tracking performance. It has been shown from the simulation results that the proposed tracking controller with a tuned α via ES can achieve an optimal tracking performance for nonlinear systems with disturbances. Fig. 1 . 1The diagram of the controller withQ. Fig. 2 . 2The diagram of LQ optimal tracking control. Fig. 3 . 3The diagram of robust LQT with parameterizedQ. Fig. 4 . 4The diagram of robust LQT control with parameterizedQα for nonlinear systems. Fig. 5 . 5The overall ES α tuning scheme. The ES algorithm updates the parameter α(k) in the filterQα to minimize the cost J(α). Fig. 6 .Fig. 7 . 67Case A: ES α tuning for w 1 illustrated by (a) the evolution of the cost function and (b) the parameter α during ES tuning of the closed-loop system with w 1 . The lower plots present (c) the output signal θ 1 and (d) the control input signal u. Case B: ES α tuning for w 1 illustrated by (a) the evolution of the cost function and (b) the parameter α during ES tuning of the closed-loop system with w 1 . The lower plots present (c) the output signal θ 1 and (d) the control input signal u. Jiapeng Xu and Xiang Chen are with the Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON N9B 3P4, Canada. [email protected], [email protected] Ying Tan is with the Department of Mechanical Engineering, The University of Melbourne, Melbourne, VIC 3010, [email protected] between the nominal model and the true system,Q is only activated when there exists unmodelled dynamics or external disturbances, such that this kind of controller design can lead to a high performance in the presence of disturbances and uncertainties. This technique is quite different from the traditional mixed H 2 /H ∞ control, a trade-off design [10]-[12].2 TABLE I ISYSTEM PARAMETERS OF THE ROTARY PENDULUM.Parameter Description Value Unit Rm Terminal resistance 8.4 Ω km Motor back-emf constant 0.042 V·s/rad r Length of the base arm 0.085 m 2l Length of the pendulum 0.129 m mp Mass of the pendulum 0.024 Kg g Gravity constant 9.81 N/Kg br Damping on the rotary arm 0.0005 N·m· s/rad bp Damping on the pendulum 0.0001 N·m· s/rad Jr Inertia of the rotary arm 2.3060 × 10 −4 N·m · s 2 /rad Jp Inertia of the pendulum 1.3313 × 10 −4 N·m · s 2 /rad TABLE II α II* FOR DISTURBANCES w 1 AND w 2 .w1 w2 Case A α * = 1.20 α * = 1.21 Case B α * = 1.38 α * = 1.47 (a) (b) (c) (d) Nonlinear H∞-control, Hamiltonian systems and Hamilton-Jacobi equations. M Aliyu, CRC PressBoca Raton, FLM. Aliyu, Nonlinear H∞-control, Hamiltonian systems and Hamilton- Jacobi equations. Boca Raton, FL: CRC Press, 2011. H∞ tracking control of completely unknown continuous-time systems via off-policy reinforcement learning. H Modares, F L Lewis, Z.-P Jiang, IEEE Transactions on Neural Networks and Learning systems. 2610H. Modares, F. L. Lewis, and Z.-P. Jiang, "H∞ tracking control of completely unknown continuous-time systems via off-policy reinforce- ment learning," IEEE Transactions on Neural Networks and Learning systems, vol. 26, no. 10, pp. 2550-2562, 2015. Robust output tracking control of nonlinear mimo systems via sliding mode technique. H Elmali, N Olgac, Automatica. 281H. Elmali and N. Olgac, "Robust output tracking control of nonlinear mimo systems via sliding mode technique," Automatica, vol. 28, no. 1, pp. 145-151, 1992. H K Khalil, Nonlinear systems. Upper Saddle River, NJPrentice-Hall3rd edH. K. Khalil, Nonlinear systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002. Robust and optimal control. K Zhou, J Doyle, K Glover, NJ. Prentice HallK. Zhou, J. Doyle, and K. Glover, Robust and optimal control. NJ: Prentice Hall, 1996. A new controller architecture for high performance, robust, and fault-tolerant control. K Zhou, Z Ren, IEEE Transactions on Automatic Control. 4610K. Zhou and Z. Ren, "A new controller architecture for high per- formance, robust, and fault-tolerant control," IEEE Transactions on Automatic Control, vol. 46, no. 10, pp. 1613-1618, 2001. Optimal control: Linear quadratic methods. B D Anderson, J B Moore, Prentice HallEnglewood Cliffs, NJB. D. Anderson and J. B. Moore, Optimal control: Linear quadratic methods. Englewood Cliffs, NJ: Prentice Hall, 1989. Revisit of LQG control-A new paradigm with recovered robustness. X Chen, K Zhou, Y Tan, Porceedings of the IEEE 58th Conference on Decision and Control (CDC). X. Chen, K. Zhou, and Y. Tan, "Revisit of LQG control-A new paradigm with recovered robustness," in Porceedings of the IEEE 58th Conference on Decision and Control (CDC), 2019, pp. 5819-5825. A dual-loop robust control scheme with performance separation: Theory and experimental validation. T He, X Chen, G G Zhu, IEEE Transactions on Industrial Electronics. 6912T. He, X. Chen, and G. G. Zhu, "A dual-loop robust control scheme with performance separation: Theory and experimental validation," IEEE Transactions on Industrial Electronics, vol. 69, no. 12, pp. 13 483-13 493, 2022. A Nash game approach to mixed H 2 /H∞ control. D J Limebeer, B D Anderson, B Hendel, IEEE Transactions on Automatic Control. 391D. J. Limebeer, B. D. Anderson, and B. Hendel, "A Nash game approach to mixed H 2 /H∞ control," IEEE Transactions on Automatic Control, vol. 39, no. 1, pp. 69-82, 1994. Mixed H 2 and H∞ performance objectives II: Optimal control. J Doyle, K Zhou, K Glover, B Bodenheimer, IEEE Transactions on Automatic Control. 398J. Doyle, K. Zhou, K. Glover, and B. Bodenheimer, "Mixed H 2 and H∞ performance objectives II: Optimal control," IEEE Transactions on Automatic Control, vol. 39, no. 8, pp. 1575-1587, 1994. Multiobjective H∞/H 2 control design. X Chen, K Zhou, SIAM Journal on Control and Optimization. 402X. Chen and K. Zhou, "Multiobjective H∞/H 2 control design," SIAM Journal on Control and Optimization, vol. 40, no. 2, pp. 628-660, 2001. Real-time optimization by extremumseeking control. K B Ariyur, M Krstic, NJ. John Wiley & SonsK. B. Ariyur and M. Krstic, Real-time optimization by extremum- seeking control. NJ: John Wiley & Sons, 2003. PID tuning using extremum seeking: Online, model-free performance optimization. N J Killingsworth, M Krstic, IEEE Control Systems Magazine. 261N. J. Killingsworth and M. Krstic, "PID tuning using extremum seeking: Online, model-free performance optimization," IEEE Control Systems Magazine, vol. 26, no. 1, pp. 70-79, 2006. Extremum seeking from 1922 to 2010. Y Tan, W H Moase, C Manzie, D Nešić, I M Mareels, Proceedings of the 29th Chinese Control Conference (CCC). the 29th Chinese Control Conference (CCC)Y. Tan, W. H. Moase, C. Manzie, D. Nešić, and I. M. Mareels, "Extremum seeking from 1922 to 2010," in Proceedings of the 29th Chinese Control Conference (CCC), 2010, pp. 14-26. Model-guided extremum seeking for diesel engine fuel injection optimization. Q Tan, P Divekar, Y Tan, X Chen, M Zheng, IEEE/ASME Transactions on Mechatronics. 232Q. Tan, P. Divekar, Y. Tan, X. Chen, and M. Zheng, "Model-guided extremum seeking for diesel engine fuel injection optimization," IEEE/ASME Transactions on Mechatronics, vol. 23, no. 2, pp. 936- 946, 2018. Towards a theoretical foundation of policy optimization for learning control policies. B Hu, K Zhang, N Li, M Mesbahi, M Fazel, T Başar, arXiv:2210.04810arXiv preprintB. Hu, K. Zhang, N. Li, M. Mesbahi, M. Fazel, and T. Başar, "Towards a theoretical foundation of policy optimization for learning control policies," arXiv preprint arXiv:2210.04810, 2022. State-space solutions to standard H 2 and H∞ control problems. J Doyle, K Glover, P Khargonekar, B Francis, IEEE Transactions on Automatic Control. 348J. Doyle, K. Glover, P. Khargonekar, and B. Francis, "State-space so- lutions to standard H 2 and H∞ control problems," IEEE Transactions on Automatic Control, vol. 34, no. 8, pp. 831-847, 1989. Extremum seeking control for discrete-time systems. J.-Y Choi, M Krstic, K B Ariyur, J S Lee, IEEE Transactions on Automatic Control. 472J.-Y. Choi, M. Krstic, K. B. Ariyur, and J. S. Lee, "Extremum seeking control for discrete-time systems," IEEE Transactions on Automatic Control, vol. 47, no. 2, pp. 318-323, 2002. QUBE-Servo 2 Workbook (Student): Experiment for Matlab/Simulink Users. J Apkarian, M Levis, Quanser, IncJ. Apkarian and M. Levis, QUBE-Servo 2 Workbook (Student): Ex- periment for Matlab/Simulink Users, Quanser, Inc., 2021.
[]
[ "On the Fundamental Feedback-vs-Performance Tradeoff over the MISO-BC with Imperfect and Delayed CSIT", "On the Fundamental Feedback-vs-Performance Tradeoff over the MISO-BC with Imperfect and Delayed CSIT" ]
[ "Jinyuan Chen ", "Sheng Yang ", "Petros Elia " ]
[]
[]
This work considers the multiuser multiple-input single-output (MISO) broadcast channel (BC), where a transmitter with M antennas transmits information to K single-antenna users, and where -as expected -the quality and timeliness of channel state information at the transmitter (CSIT) is imperfect. Motivated by the fundamental question of how much feedback is necessary to achieve a certain performance, this work seeks to establish bounds on the tradeoff between degrees-of-freedom (DoF) performance and CSIT feedback quality. Specifically, this work provides a novel DoF region outer bound for the general K-user M × 1 MISO BC with partial current CSIT, which naturally bridges the gap between the case of having no current CSIT (only delayed CSIT, or no CSIT) and the case with full CSIT. The work then characterizes the minimum CSIT feedback that is necessary for any point of the sum DoF, which is optimal for the case with M ≥ K, and the case with M = 2, K = 3. R i log P , i = 1, 2, · · · , K. The corresponding DoF region D is then the set of all achievable DoF tuples (d1, d2, · · · , dK).
10.1109/isit.2013.6620376
[ "https://arxiv.org/pdf/1302.0806v1.pdf" ]
11,915,882
1302.0806
7b71a31f904ae918f4eea449100cdff7cefb77b1
On the Fundamental Feedback-vs-Performance Tradeoff over the MISO-BC with Imperfect and Delayed CSIT 4 Feb 2013 Jinyuan Chen Sheng Yang Petros Elia On the Fundamental Feedback-vs-Performance Tradeoff over the MISO-BC with Imperfect and Delayed CSIT 4 Feb 2013 This work considers the multiuser multiple-input single-output (MISO) broadcast channel (BC), where a transmitter with M antennas transmits information to K single-antenna users, and where -as expected -the quality and timeliness of channel state information at the transmitter (CSIT) is imperfect. Motivated by the fundamental question of how much feedback is necessary to achieve a certain performance, this work seeks to establish bounds on the tradeoff between degrees-of-freedom (DoF) performance and CSIT feedback quality. Specifically, this work provides a novel DoF region outer bound for the general K-user M × 1 MISO BC with partial current CSIT, which naturally bridges the gap between the case of having no current CSIT (only delayed CSIT, or no CSIT) and the case with full CSIT. The work then characterizes the minimum CSIT feedback that is necessary for any point of the sum DoF, which is optimal for the case with M ≥ K, and the case with M = 2, K = 3. R i log P , i = 1, 2, · · · , K. The corresponding DoF region D is then the set of all achievable DoF tuples (d1, d2, · · · , dK). I. INTRODUCTION We consider the multiuser multiple-input single-output (MISO) broadcast channel (BC), where a transmitter with M antennas, transmits information to K single-antenna users. In this setting, the received signal at time t, is of the form y k,t = h T k,t x t + z k,t , k = 1, · · · , K where h k,t denotes the M × 1 channel vector for user k, z k,t denotes the unit power AWGN noise, and where x t denotes the transmitted signal vector adhering to a power constraint E[||x t || 2 ] ≤ P , for P taking the role of the signal-to-noise ratio (snr). We here consider that the fading coefficients h k,t , k = 1, · · · , K, are independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance, and are i.i.d. over time. It is well known that the performance of the BC is greatly affected by the timeliness and quality of feedback; having full CSIT allows for the optimal min{M, K} sum degrees-of-freedom (DoF) (cf. [2]) 1 , while the absence of any CSIT reduces this to just 1 sum DoF (cf. [3], [4]). This gap has spurred a plethora of works that seek to analyze and optimize BC communications in the presence of delayed and imperfect feedback. One of the works that stands out is the work by Maddah-Ali and Tse [5] which recently revealed the benefits of employing delayed CSIT over the BC, even if this CSIT is completely obsolete. Several interesting generalizations followed, including the work in [6] which showed that in the BC setting with K = M + 1, combining delayed CSIT with perfect (current) CSIT (over the last An initial version of this paper has been reported as Research Report No. RR-12-275 at EURECOM, December 7, 2012 (see in [1]). This paper was submitted in part to the ISIT 2013. J. Chen and P. Elia are with the Mobile Communications Department, EURECOM, Sophia Antipolis, France (email: {chenji, elia}@eurecom.fr). S. Yang is with the Telecommunications department of SUPELEC, 3 rue Joliot-Curie, 91190 Gif-sur-Yvette, France (e-mail: [email protected]). The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 257616 (CONECT), from the FP7 CELTIC SPECTRA project, and from Agence Nationale de la Recherche project ANR-IMAGENET. 1 We remind the reader that for an achievable rate tuple (R1, R2, · · · , RK), where Ri is for user i, the corresponding DoF tuple (d1, d2, · · · , dK ) is given by di = limP →∞ K−1 K fraction of communication period) allows for the optimal sum DoF M corresponding to full CSIT. A similar approach was exploited in [7] which revealed that, to achieve the maximum sum DoF min{M, K}, each user has to symmetrically feed back perfect CSIT over a min{M,K} K fraction of the communication time, and that this fraction is optimal. Other interesting works in the context of utilizing delayed and current CSIT, can be found in [8]- [11] which explored the setting of combining perfect delayed CSIT with immediately available imperfect CSIT, the work in [12], [13] which additionally considered the effects of the quality of delayed CSIT, the work in [14] which considered alternating CSIT feedback, the work in [15] which considered delayed and progressively evolving (progressively improving) current CSIT, and the works in [16]- [22] and many other publications. Our work here generalizes many of the above settings, and seeks to establish fundamental tradeoff between DoF performance and CSIT feedback quality, over the general K-user M × 1 MISO BC. A. CSIT quantification and feedback model We proceed to describe the quality and timeliness measure of CSIT feedback, and how this measure relates to existing work. We here useĥ k,t to denote the current channel estimate (for channel h k,t ) at the transmitter at timeslot t, and useh k,t = h k,t −ĥ k,t to denote the estimate error assumed to be mutually independent ofĥ k,t and assumed to have i.i.d. Gaussian entries with power E h k,t 2 . = P −α k,t , for some CSI quality exponent α k,t ∈ [0, 1] describing the quality of this estimate. We note that α k,t = 0 implies very little current CSIT knowledge, and that α k,t = 1 implies perfect CSIT in terms of the DoF performance 2 . The approach extends over non-alternating CSIT settings in [5] and [8]- [11], as well as over an alternating CSIT setting (cf. [7], [14]) where CSIT knowledge alternates between perfect CSIT (α k,t = 1), and delayed or no CSIT (α k,t = 0). In a setting where communication takes place over n such coherence periods (t = 1, 2, · · · , n), this approach offers a natural measure of a per-user average feedback cost, in the form of α k 1 n n t=1 α k,t , k = 1, 2, · · · , K, as well as a measure of current CSIT feedback cost C C K k=1ᾱ k ,(2) accumulated over all users. 1) Alternating CSIT setting: In a setting where delayed CSIT is always available, the above model captures the alternating CSIT setting where the exponents are binary (α k,t = 0, 1), in which casē α k = δ P,k simply describes the fraction of time during which user k feeds back perfect CSIT, with C C = C P K k=1 δ P,k describing the total perfect CSIT feedback cost. 2) Symmetric and asymmetric CSIT feedback: Motivated by the fact that different users might have different feedback capabilities due to the feedback channels with different capacities and different reliabilities, symmetric CSIT feedback (ᾱ 1 = · · · =ᾱ K ) and asymmetric CSIT feedback (ᾱ k =ᾱ k ′ ∀ k = k ′ ) are considered in this work. B. Structure of the paper and Summary of Contributions Section II provides the main results of this work: • In Theorem 1 we first provide a novel outer bound on the DoF region, for the K-user M ×1 MISO BC with partial current CSIT quantized with {α k,t } k,t , which bridges the case with no current CSIT (only delayed CSIT, or no CSIT) and the case with full CSIT. This result manages to generalize the results by Maddah-Ali and Tse (α k,t = 0, ∀t, k), Yang et al. and Gou and Jafar (K = 2, α k,t = α, ∀t, k), Maleki et al. (K = 2, α 1,t = 1, α 2,t = 0, ∀t), Chen and Elia (K = 2, α 1,t = α 2,t , ∀t), Lee and Heath (M = K + 1, α k,t ∈ {0, 1}, ∀t, k), and Tandon et al. (α k,t ∈ {0, 1}, ∀t, k). • From Theorem 1, we then provide the upper bound on the sum DoF, which is tight for the case with M ≥ K (cf. Theorem 2) and the case with M = 2, K = 3 (cf. Theorem 3, Corollary 3a). • Furthermore, Theorem 4 characterizes the minimum total current CSIT feedback cost C ⋆ P to achieve the maximum sum DoF, where the total feedback cost C ⋆ P can be distributed among all the users with any (asymmetric and symmetric) combinations {δ P,k } k . • In addition, the work considers some other general settings of BC and provides the DoF inner bound as a function of the CSIT feedback cost. The main converse proof, that is for Theorem 1, is shown in the Section III and appendix. Most of the achievability proofs are shown in the Section IV. Finally Section V concludes the paper. C. Notation and conventions Throughout this paper, we will consider communication over n coherence periods where, for clarity of notation, we will focus on the case where we employ a single channel use per such coherence period (unit coherence period). Furthermore, unless stated otherwise, we assume perfect delayed CSIT, as well as adhere to the common convention (see [5], [7], [9], [10], [14], [24]), and assume perfect and global knowledge of channel state information at the receivers. In terms of notation, (•) T , (•) H , tr(•) and || • || F denote the transpose, conjugate transpose, trace and Frobenius norm of a matrix respectively, while diag(•) denotes a diagonal matrix, || • || denotes the Euclidean norm, and | • | denotes either the magnitude of a scalar or the cardinality of a set. o(•) and O(•) come from the standard Landau notation, where f (x) = o(g(x)) implies lim x→∞ f (x)/g(x) = 0. with f (x) = O(g(x)) implying that lim sup x→∞ |f (x)/g(x)| < ∞. We also use . = to denote exponential equality, i.e., we write f (P II. MAIN RESULTS A. Outer bounds We first present the DoF region outer bound for the general K-user M × 1 MISO BC. Theorem 1 (DoF region outer bound): The DoF region of the K-user M × 1 MISO BC, is outer bounded as K k=1 d π(k) min{k, M} ≤ 1+ K−1 k=1 1 min{k, M} − 1 min{K, M} ᾱ π(k) (3) d k ≤ 1, k = 1, 2, · · · , K(4) where π denotes a permutation of the ordered set {1, 2, · · · , K}, and π(k) denotes the k th element of set π. Proof: The proof is shown in Section III. Remark 1: It is noted that the bound captures the results in [5] (α k,t = 0, ∀t, k), in [9], [10] (K = 2, α k,t = α, ∀t, k), in [24] (M = K = 2, α 1,t = 1, α 2,t = 0, ∀ t), in [11] (K = 2, α 1,t = α 2,t , ∀ t), in [7], [14] (α k,t ∈ {0, 1}, ∀t, k), as well as in [25] (α (k) t = α, ∀t, k). Summing up the K different bounds from the above, we directly have the following upper bound on the sum DoF d Σ K k=1 d k , which is presented using the following notation d MAT K 1 + 1 min{2,M } + 1 min{3,M } + · · · + 1 min{K,M } (5) Γ M K−M i=1 1 i ( M −1 M ) i−1 + ( M −1 M ) K−M ( K i=K−M +1 1 i ) .(6) Corollary 1a (Sum DoF outer bound): For the K-user M ×1 MISO BC, the sum DoF is outer bounded as d Σ ≤ d MAT + 1 − d MAT min{K, M} K k=1ᾱ k .(7) The above then readily translates onto a lower bound on the minimum possible total current CSIT feedback cost C C = K k=1ᾱ k needed to achieve the maximum sum DoF 3 d Σ = min{K, M}. Corollary 1b (Bound on CSIT cost for maximum DoF): The minimum C C required to achieve the maximum sum DoF min{K, M} of the K-user M × 1 MISO BC, is lower bounded as C ⋆ C ≥ min{K, M}.(8) Transitioning to the alternating CSIT setting where α k,t ∈ {0, 1}, we have the following sum-DoF outer bound as a function of the perfect-CSIT durationᾱ k = δ P,k = δ P , ∀ k. We note that the bound holds irrespective of whether, in the remaining fraction of the time 1 − δ P , the CSIT is delayed or non existent. Corollary 1c (Outer bound, alternating CSIT): For the K-user M ×1 MISO BC, the sum DoF is outer bounded as d Σ ≤ d MAT + K − Kd MAT min{K, M} min δ P , min{K, M} K .(9) 3 Naturally the result is limited to the case where min{K, M } > 1. d ∑ 1 K 0 MAT P MAT d d K d + - = å d ) ( MAT d P d Fig. 2. Optimal sum DoF dΣ vs. δP for the MISO BC with M ≥ K . B. Optimal cases of DoF characterizations We now provide the optimal cases of DoF characterizations. The case with M ≥ K is first considered in the following. Theorem 2 (Optimal case, M ≥ K): For the K-user M × 1 MISO BC with M ≥ K, the optimal sum DoF is characterized as d Σ = (K − d MAT ) min{δ P , 1} + d MAT .(10) Proof: The converse and achievability proofs are derived from Corollary 1c and Proposition 2 (shown in the next subsection), respectively. Remark 2: It is noted that, for the special case with M = K = 2, the above characterization captures the result in [14]. Moving to the case where M < K, we have the following optimal sum DoF characterizations for the case with M = 2, K = 3. The first interest is placed on the minimum C ⋆ P (d Σ ) to achieve a sum DoF d Σ , recalling that C ⋆ P = K k=1 δ P,k describes the total perfect CSIT feedback cost. Theorem 3 (Optimal case, M = 2, K = 3): For the three-user 2 × 1 MISO BC, the minimum total perfect CSIT feedback cost is characterized as C ⋆ P (d Σ ) = (4d Σ − 6) + , ∀ d Σ ∈ [0, 2](11) where the total feedback cost C ⋆ P (d Σ ) can be distributed among all the users with some combinations {δ P,k } k such that δ P,k ≤ C ⋆ P (d Σ )/2 for any k. Proof: The converse proof is directly from Corollary 1a, while the achievability proof is shown in Section IV-B. Theorem 3 reveals the fundamental tradeoff between sum DoF and total perfect CSIT feedback cost (see Fig 3). The following examples are provided to offer some insights corresponding to Theorem 3. Example 1: For the target sum DoF d Σ = 3/2, 7/4, 2, the minimum total perfect CSIT feedback cost is C ⋆ P = 0, 1, 2, respectively. Example 2: The target d Σ = 7/4 is achievable with asymmetric feedback δ P = [1/6 1/3 1/2], and symmetric feedback δ P = [1/3 1/3 1/3], and some other feedback such that C ⋆ P (7/4) = 1. Example 3: The target d Σ = 2 is achievable with asymmetric feedback δ P = [1/3 2/3 1], and symmetric feedback δ P = [2/3 2/3 2/3], and some other feedback such that C ⋆ P (2) = 2. Transitioning to the symmetric setting where δ P,k = δ P ∀ k, from Theorem 3 we have the fundamental tradeoff between optimal sum DoF and CSIT feedback cost δ P . Corollary 3a (Optimal case, M = 2, K = 3, δ P ): For the three-user 2×1 MISO BC with symmetrically alternating CSIT feedback, the optimal sum DoF is characterized as d Σ = min 3(2 + δ P ) 4 , 2 .(12) Now we address the questions of what is the minimum C ⋆ P to achieve the maximum sum DoF min{M, K} for the general BC, and how to distribute C ⋆ P among all the users, recalling again that C ⋆ P is the total perfect CSIT feedback cost. Theorem 4 (Minimum cost for maximum DoF): For the K-user M × 1 MISO BC, the minimum total perfect CSIT feedback cost to achieve the maximum DoF is characterized as C ⋆ P (min{M, K}) = 0, if min{M, K} = 1 min{M, K}, if min{M, K} > 1 where the total feedback cost C ⋆ P can be distributed among all the users with any combinations {δ P,k } k . Proof: For the case with min{M, K} = 1, simple TDMA is optimal in terms of the DoF performance. For the case with min{M, K} > 1, the converse proof is directly derived from Corollary 1b, while the achievability proof is shown in Section IV-A. It is noted that Theorem 4 is a generalization of the result in [7] where only symmetric feedback was considered. The following examples are provided to offer some insights corresponding to Theorem 4. Example 4: For the case where M = 2, K = 4, the optimal 2 sum DoF performance is achievable, with asymmetric feedback δ P = [1/5 2/5 3/5 4/5], and symmetric feedback δ P = [1/2 1/2 1/2 1/2], and any other feedback such that C ⋆ P = 2. Example 5: For the case where M = 3, K = 5, the optimal 3 sum DoF performance is achievable, with asymmetric feedback δ P = [1/5 2/5 3/5 4/5 1], and symmetric feedback δ P = [3/5 3/5 3/5 3/5 3/5], and any other feedback such that C ⋆ P = 3. The following corollary is derived from Theorem 4, where the case with min{M, K} > 1 is considered. C. Inner bounds In this subsection, we provide the following inner bounds on the sum DoF as a function of the CSIT cost, which are tight for many cases as stated. d ∑ 2/K 2 3/2 0 P K d d 4 2 3 + = å P d Fig. 4. Achievable sum DoF dΣ vs. δP for the K(≥ 3)-user 2 × 1 MISO BC. d ∑ M/K M 0 G + G - = å P M K K d d ) ( G P dd Σ ≥ 3 2 + K 4 min{δ P , 2 K }.(13) Proof: The proof is shown in Section IV-C. Proposition 2 (Inner bound, M ≥ K and M < K): For the K-user M × 1 MISO BC, the sum DoF for the case with M ≥ K is bounded as d Σ ≥ (K − d MAT ) min{δ P , 1} + d MAT ,(14) while for the case with M < K, the sum DoF is bounded as d Σ ≥ (K − KΓ M ) min{δ P , M K } + Γ.(15) Proof: The proof is shown in Section IV-D. Finally, we consider a case of BC with delayed CSIT feedback only, where δ P = 0. In this case, we use δ D,k to denote the fraction of time during which CSIT fed back from user k is delayed, and focus on the case with δ D,k = δ D , ∀k. d ∑ 9/(8K) 2/(3K) 3/2 1 4/3 0 11 / ) 3 ( 4 + = å D K d d 2 / 1 D K d d + = å D d Fig. 6. Achievable sum DoF dΣ vs. δD for the MISO BC with K ≥ 3, M = 2, where δP = 0. Proposition 3 (Inner bound on DoF with delayed CSIT): For the K(≥ 3)-user 2 × 1 MISO BC, and for the case of δ P = 0, the sum DoF is bounded as d Σ ≥ min 1 + K 2 δ D , 12 11 + 4K 11 δ D , 3 2 .(16) Proof: The proof is shown in Section IV-E. Remark 3: For the K-user MISO BC with current and delayed CSIT feedback, by increasing the number of users, the same DoF performance can be achievable with decreasing feedback cost per user. For example, for the K-user MISO BC with M = 2, by increasing K we can achieve any fixed DoF within the range of (1, 2], with decreasing δ P ≤ 2 K , and δ D ≤ 9 8K , both of which approach to 0 as K is large. III. CONVERSE PROOF OF THEOREM 1 We first provide the Proposition 4 to be used, where we drop the time index for simplicity. Proposition 4: Let y k = h T k x + z k , y k [y 1 y 2 · · · y k ] T z k [z 1 z 2 · · · z k ] T H k [h 1 h 2 · · · h k ] T H [h 1 h 2 · · · h K ] T H =Ĥ +H whereh i ∈ C M ×1 has i.i.d. N C (0, σ 2 i ) entries. Then, for any U such that p X|UĤH = p X|UĤ and K ≥ m ≥ l, we have l ′ h(y m |U,Ĥ,H) − m ′ h(y l |U,Ĥ,H) ≤ −(m ′ − l ′ ) l i=1 log σ 2 i + o(log snr)(17) where we define l ′ min {l, M} and m ′ min {m, M}. Proof: The proof is shown in the Section VI. Now giving the observations and messages of users 1, . . . , k − 1 to user k, we establish the following genie-aided upper bounds on the achievable rates nR 1 ≤ I(W 1 ; y n 1 | Ω n ) + nǫ (18) nR 2 ≤ I(W 2 ; y n 1 , y n 2 | W 1 , Ω n ) + nǫ (19) . . . nR K ≤ I(W K ; y n 1 , y n 2 , . . . , y n K | W 1 , . . . , W K−1 , Ω n ) + nǫ(20) where we apply Fano's inequality and some basic chain rules of mutual information using the fact that messages from different users are independent, where we define S t h 1,t · · · h K,t T S t ĥ 1,t · · ·ĥ K,t T Ω n {S t ,Ŝ t } n t=1 y n k {y k,t } n t=1 . Alternatively, we have nR 1 ≤ h(y n 1 | Ω n ) − h(y n 1 | W 1 , Ω n ) + nǫ (21) nR 2 ≤ h(y n 1 , y n 2 | W 1 , Ω n ) − h(y n 1 , y n 2 | W 1 , W 2 , Ω n ) + nǫ (22) . . . nR K ≤ h(y n 1 , . . . , y n K | W 1 , . . . , W K−1 , Ω n ) − h(y n 1 , . . . , y n K | W 1 , . . . , W K , Ω n ) + nǫ.(23) Therefore, it follows that K k=1 n k ′ (R k − ǫ) ≤ K−1 k=1 1 (k + 1) ′ h(y n 1 , . . . , y n k+1 | W 1 , . . . , W k , Ω n ) − 1 k ′ h(y n 1 , . . . , y n k | W 1 , . . . , W k , Ω n ) + h(y n 1 | Ω n ) − 1 K ′ h(y n 1 , . . . , y n K | W 1 , . . . , W K , Ω n )(24)≤ K−1 k=1 n t=1 1 (k + 1) ′ h(y 1,t , . . . , y k+1,t | y t−1 1 , . . . , y t−1 k , W 1 , . . . , W k , Ω n ) − 1 k ′ h(y 1,t , . . . , y k,t | y t−1 1 , . . . , y t−1 k , W 1 , . . . , W k , Ω n ) + n log P + n o(log P )(25)≤ log P K−1 k=1 n t=1 (k + 1) ′ − k ′ k ′ (k + 1) ′ k i=1 α i,t + n log P + n o(log P )(26) = n log P K−1 k=1 (k + 1) ′ − k ′ k ′ (k + 1) ′ k i=1ᾱ i + n log P + n o(log P )(27) = n log P K−1 k=1 1 k ′ − 1 K ′ ᾱ k + n log P + n o(log P )(28) where we define k ′ min {k, M} ;(29) the inequality (25) is due to 1) the chain rule of differential entropy, 2) the fact that removing condition does not decrease differential entropy, 3) h(y 1,t | Ω n ) ≤ log P + o(log P ), i.e., Gaussian distribution maximizes differential entropy under covariance constraint, and 4) h(y n 1 , . . . , y n K | W 1 , . . . , W K , Ω n ) = h(z 1,1 , z 1,2 , . . . , z K,n ) > 0; (26) is from Proposition 4 by setting U = {y t−1 1 , . . . , y t−1 k , W 1 , . . . , W k , Ω n } \ {S t ,Ŝ t }, H = S t , andĤ =Ŝ t ; the last equality is obtained after putting the summation over k inside the summation over i and some basic manipulations. Similarly, we can interchange the roles of the users and obtain the same genie-aided bounds. Finally, the single antenna constraint gives that d i ≤ 1, i = 1, · · · , K. With this, we complete the proof. IV. DETAILS OF ACHIEVABILITY PROOFS In this section, we provide the details of the achievability proofs. Specifically, the achievability proof of Theorem 4 is first described in Section IV-A, which can be applied in parts for the achievability proof of Theorem IV-B shown in Section IV-B, with the proposition proofs shown in the rest of this section. A. Achievability proof of Theorem 4 We will prove that, the optimal sum DoF d Σ = min{M, K} is achievable with any CSIT feedback cost δ P [δ P,1 δ P,2 · · · δ P,K ] ∈ R K such that C P = K k=1 δ P,k = min{M, K}. First of all, we note that there exists a minimum number n such that δ ′ P [δ ′ P,1 δ ′ P,2 · · · δ ′ P,K ] nδ P = [nδ P,1 nδ P,2 · · · nδ P,K ] ∈ Z K is an integer vector. The explicit communication with n channel uses is given as follows: • Step 1: Initially set time index t = 1. • Step 2: Permute user indices orderly into a set U such that δ ′ P,U (1) ≤ δ ′ P,U (2) ≤ · · · ≤ δ ′ P,U (K) , where U(k) denotes the k th element of set U, and where U(k) ∈ {1, 2, · · · , K}. • Step 3: Select min{M, K} users to communicate: users U(K−min{M, K}+1), · · · , U(K−1), U(K). • Step 4: Let selected users feed back perfect CSIT at time t, keeping the rest users silent. • Step 5: The transmitter sends min{M, K} independent symbols to those selected users respectively, which can be done with simple zero-forcing. • Step 6: Set δ ′ P,U (k) = δ ′ P,U (k) − 1, k = K − min{M, K} + 1, · · · , K − 1, K. • Step 7: Set t = t + 1. If renewed t > n then terminate, else go back to step 2. In the above communication with n channel uses, the algorithm guarantees that user i is selected by δ ′ P,k = nδ P,k times totally, and that min{M, K} different users are selected in each channel use. As a result, the optimal sum DoF d Σ = min{M, K} is achievable. Now we consider an example with M = 2, K = 3, and δ P = [1/3 2/3 1], and show that the optimal sum DoF d Σ = 2 is achievable with the following communication: • Let n = 3. Initially δ ′ P,1 = nδ P,1 = 1, δ ′ P,2 = nδ P,2 = 2, δ ′ P,3 = nδ P,3 = 3. In the above communication with three channel uses, the transmitter sends two symbols in each channel use, which allows for the optimal sum DoF d Σ = 2 (see Table I). B. Achievability proof of Theorem 3 We proceed to show that, any sum DoF d Σ ∈ [3/2, 2] is achievable with the feedback δ P,k ≤ C P 2 , k = 1, 2, 3, such that C P = 3 k=1 δ P,k = 4d Σ − 6. First of all, we note that there exists a minimum number n such that [2nδ P,1 /C P 2nδ P,2 /C P n2δ P,3 /C P ] ∈ Z 3 , and 2n/C P ∈ Z. The scheme has two blocks, with the first block consisting of n channel uses, and the second block consisting of n ′ = 2n/C P − n channel uses. In the first block, we use the algorithm shown in the Section IV-A to achieve the full sum DoF in those n channel uses, during which user k feeds back perfect CSIT in 2nδ P,3 /C P channel uses, for k = 1, 2, 3. In the second block, we use the Maddah-Ali and Tse scheme in [5] to achieve 3/2 sum DoF in those n ′ channel uses, during which each user feeds back delayed CSIT only. The communication with n channel uses for the first block is given as follows: • Step 1: Let δ ′ P,k = 2nδ P,k /C P for all k. Initially, set t = 1. • The steps 2, 3, 4, 5, 6 are the same as those in the algorithm shown in Section IV-A, for M = 2, K = 3. • Step 7: Set t = t + 1. If renewed t > n then terminate, else go back to step 2. In the above communication with n channel uses, the algorithm guarantees that user k, k = 1, 2, 3, is selected by δ ′ P,k = 2nδ P,k /C P times. We note that δ ′ P,k ≤ n under the constraint δ P,k ≤ C P /2 for any k, and that K k=1 δ ′ P,k = 2n, to suggest that in each timeslot two different users are selected, which allows for the optimal 2 sum DoF in this block. As stated, in the second block, we use the MAT scheme to achieve the 3/2 sum DoF in those n ′ channel uses, during which each user feeds back delayed CSIT only. As a result, in the total n + n ′ channel uses communication, user k = 1, 2, 3 feeds back perfect CSIT in 2nδ P,k /(C P (n + n ′ )) = δ P,k fraction of communication period, with achievable sum DoF given as d Σ = 2n (n + n ′ ) + 3n ′ 2(n + n ′ ) = 3 2 + 1 4 C P . We note that the achievability scheme applies to the case of having some δ P,1 , δ P,2 , δ P,3 ≤ C P /2 such that C P = 4d Σ − 6, and allows to achieve any sum DoF d Σ ∈ [3/2, 2]. Apparently, C P = 0 allows for any sum DoF d Σ ∈ [0, 3/2], which completes the proof. C. Proof of Proposition 1 The achievability scheme is based on time sharing between two strategies of CSIT feedback, i.e., delayed CSIT feedback with δ ′ P = 0 and alternating CSIT feedback with δ ′′ P = 2 K , where the first strategy achieves d ′ Σ = 3/2 by applying Maddah-Ali and Tse (MAT) scheme (see in [5]), with the second strategy achieving d ′′ Σ = 2 by using alternating CSIT feedback manner (see in [7]). Let ∆ ∈ [0, 1] (res. 1 − ∆) be the fraction of time during which the first (res. second) CSIT feedback strategy is used in the communication. As a result, the final feedback cost (per user) is given as δ P = δ ′ P ∆ + δ ′′ P (1 − ∆),(30) implying that ∆ = δ ′′ P − δ P δ ′′ P − δ ′ P ,(31) with final sum DoF given as d Σ = d ′ Σ ∆ + d ′′ Σ (1 − ∆) = d ′′ Σ + ∆(d ′ Σ − d ′′ Σ ) = d ′′ Σ + (d ′ Σ − d ′′ Σ ) δ ′′ P − δ P δ ′′ P − δ ′ P = 3 2 + K 4 δ P(32) which completes the proof. D. Proof of Proposition 2 For the case with M ≥ K, the proposed scheme is based on time sharing between delayed CSIT feedback with δ ′ P = 0 and full CSIT feedback with δ ′′ P = 1, where the first feedback strategy achieves d ′ = d MAT by applying MAT scheme, with the second one achieving d ′′ = K. As a result, following the steps in (30), (31), (32), the final sum DoF is calculated as d Σ = d ′′ Σ + (d ′ Σ − d ′′ Σ ) δ ′′ P − δ P δ ′′ P − δ ′ P = (K − d MAT )δ P + d MAT where δ P ∈ [0, 1] is the final feedback cost (per user) for this case. Similar approach is exploited for the case with M < K. In this case, we apply time sharing between delayed CSIT feedback with δ ′ P = 0 and alternating CSIT feedback with δ ′′ P = M/K. In this case, the first feedback strategy achieves d ′ Σ = Γ by applying MAT scheme, with the second strategy achieving d ′′ Σ = M by using alternating CSIT feedback manner. As a result, for δ P ∈ [0, M K ] being the final feedback cost for this case, the final sum DoF is calculated as d Σ = d ′′ Σ + (d ′ Σ − d ′′ Σ ) δ ′′ P − δ P δ ′′ P − δ ′ P = (K − KΓ M )δ P + Γ which completes the proof. block index 1 2 3 · · · K No. of channel uses 3 3 3 · · · 3 Active users user 1, 2 user 2, 3 user 3, 4 · · · user K, 1 Delayed CSIT feedback user 1: 1/3 user 2: 1/3 user 3: 1/3 · · · user K: 1/3 fraction in a block user 2: 1/3 user 3: 1/3 user 4: 1/3 user 1: 1/3 the rest: 0 the rest: 0 the rest: 0 the rest: 0 Sum DoF 4/3 4/3 4/3 · · · 4/3 in a block E. Proof of Proposition 3 As shown in the Fig 6, the sum DoF performance has three regions: d Σ =    1 + K 2 δ D , δ D ∈ [0, 2 3K ] 12 11 + 4K 11 δ D , δ D ∈ [ 2 3K , 9 8K ] 3/2, δ D ∈ [ 9 8K , 1]. In the following, we will prove that the sum DoF d Σ = 1, 4 3 , 3 2 are achievable with δ D = 0, 2 3K , 9 8K , respectively. At the end, the whole DoF performance declared can be achievable by time sharing between those performance points. The proposed scheme achieving d Σ = 4 3 with δ D = 2 3K , is a modified version of the MAT scheme in [5]. The new scheme has K blocks, with each block consisting of three channel uses. In each block, four independent symbols are sent to two orderly selected users, which can be done with MAT scheme with each of two chosen user feeding back delayed CSIT in one channel use. As a result, d Σ = 4 3 is achievable with δ D = 2 3K , using the fact that each of K users needs to feed back delayed CSIT twice only in the whole communication (see Table II). Similarly, the proposed scheme achieving d Σ = 3 2 with δ D = 9 8K has K blocks, with each block consisting of 8 channel uses. In each block, 3 out of K users are selected to communicate. In this case, 12 independent symbols are sent to the chosen users during each block, which can be done with another MAT scheme with each of chosen users feeding back delayed CSIT in 3 channel uses. As a result, d Σ = 3 2 is achievable with δ D = 9 8K , using the fact that each of K users needs to feed back delayed CSIT 9 times only in the whole communication (see Table III). Finally, d Σ = 1 is achievable without any CSIT. By now, we complete the proof. V. CONCLUSIONS This work considered the general multiuser MISO BC, and established inner and outer bounds on the tradeoff between DoF performance and CSIT feedback quality, which are optimal for many cases. Those bounds, as well as some analysis, were provided with the aim of giving insights on how much CSIT feedback to achieve a certain DoF performance. VI. APPENDIX -PROOF DETAILS OF PROPOSITION 4 In the following, we will prove Proposition 4 used for the converse proof, as well as three lemmas to be used here. Lemma 1: 4 Let G =Ĝ +G ∈ C m×m whereG has i.i.d. N c (0, 1) entries, andG is independent of G. Then, we have EG log det (G H G) = τ i=1 log λ i (Ĝ HĜ ) + o(log snr)(33) where λ i (Ĝ HĜ ) denotes the i th largest eigenvalue ofĜ HĜ ; τ is the number of eigenvalues ofĜ HĜ that do not vanish with snr, i.e., λ i (Ĝ HĜ ) = o(1) when snr is large, ∀ i > τ . Lemma 2: For P ∈ C m×m a permutation matrix and A ∈ C m×m , let AP = QR be the QR decomposition of the column permuted version of A. Then, there exist at least one permutation matrix P such that closed-form term in the last inequality is due to [28] with ψ(·) being Euler's digamma function. In the following, we show that E log det I + D −1 1 M 11 2 ≥ O(1) as well. To that end, we use the fact that the distribution of M 11 is invariant to rotation, and so for D −1 1 M 11 . Specifically, introducing θ ∼ Unif(0, 2π] that is independent of the rest of the random variables, we have r 2 ii ≥ 1 m − i + 1 λ i (A H A), i = 1, . . . , m(34)E M 11 log det I + D −1 1 M 11 2 = E M 11 ,θ log det I + D −1 1 M 11 e jθ 2 (41) = E M 11 ,θ log det e −jθ I + D −1 1 M 11 2 (42) = τ ′ i=1 E J E θ log|e −jθ + λ i (D −1 M 11 ) J i | 2 (43) = τ ′ i=1 E J E θ [log(1 + |J i | 2 + 2|J i | cos(θ + φ(J i )))] (44) = τ ′ i=1 E J E θ [log(1 + |J i | 2 + 2|J i | cos(θ))] (45) ≥ τ ′ i=1 E J log(1 + |J i | 2 ) − 1 (46) ≥ −τ ′(47) where the first equality is from the fact that M 11 is equivalent to M 11 e jθ as long as θ is independent of M 11 and that M 11 has independent circularly symmetric Gaussian entries; (43) is due to the characteristic polynomial of the matrix −D −1 M 11 ; in (44) we define φ(J i ) the argument of J i that is independent of θ; (45) is from the fact that mod(θ + φ) 2π ∼ Unif(0, 2π] and is independent of φ, as long as θ ∼ Unif(0, 2π] and is independent of φ, also known as the Crypto Lemma [29]; (46) is from the identity 1 0 log(a + b cos(2πt)) dt = log a+ (40) and (47), we have the lower bound √ a 2 −b 2 2 ≥ log(a) − 1, ∀ a ≥ b > 0. CombiningEG log det (G H G) ≥ log det (D 1 ) 2 + O(1)(48)= log|det (D 1 )| 2 + o(log snr)(51) Putting the lower and upper bounds together, we have E log det (G H G) = log|det (D 1 )| 2 + o(log snr). Finally, note that, since λ i (Ĝ HĜ ) . = snr 0 , i = τ ′ + 1, . . . , τ , we have log det (D 1 ) 2 = τ ′ i=1 log λ i (Ĝ HĜ ) (53) = τ i=1 log λ i (Ĝ HĜ ) − τ i=τ ′ +1 log λ i (Ĝ HĜ ) (54) = τ i=1 log λ i (Ĝ HĜ ) + o(log snr)(55) from which the proof is complete. B. Proof of Lemma 2 The existence is proved by construction. Let a j , j = 1, . . . , m, be the j th column of A. We define j * 1 as the index of the column that has the largest Euclidean norm, i.e., j * 1 = arg max j=1,...,m a j .(56) Swapping the j * 1 and the first column, and denoting A 1 = A, we have B 1 A 1 T 1,j * 1(57) where T ij ∈ C m×m denotes the permutation matrix that swaps the i th and j th columns. Now, let U 1 ∈ C m×m be any unitary matrix such that the first column is aligned with the first column of B 1 , i.e., equal to a j * 1 a j * 1 . Then, we can construct a block-upper-triangular matrix R 1 = U H 1 B 1 = U H 1 A 1 T 1,j * 1 with the following form R 1 = r 11 * 0 (m−1)×1 A 2(58) where it is readily shown that r 2 11 = a j * 1 2 (59) ≥ 1 m ||A 1 || 2 F (60) ≥ 1 m λ 1 (A H 1 A 1 ).(61) Repeating the same procedure on A 2 , we will have R 2 = U H 2 B 2 = U H 2 A 2 T 2,j * 2 where all the involved matrices are similarly defined as above except for the reduced dimension (m − 1) × (m − 1) and R 2 = r 22 * 0 (m−2)×1 A 3(62) where it is readily shown that r 2 22 ≥ 1 m − 1 λ 1 (A H 2 A 2 )(63)≥ 1 m − 1 λ 2 (A H 1 A 1 ).(64) Here, the last inequality is from the fact that, for any matrix C and a submatrix C k by removing k rows or columns, we have [30, Corollary 3.1.3] λ i (C H k C k ) ≥ λ i+k (C H C)(65) where we recall that λ i is the i th largest eigenvalue. Let us continue the procedure on A 3 and so on. At the end, we will have all the {U i } and T i,j * i such that I m−1 U H m · · · I 2 U H 3 1 U H 2 U H 1 Q H A T 1,j * 1 1 T 2,j * 2 I 2 T 3,j * 3 · · · I m−1 T m,j * m P =     r 11 * * * r 22 * * . . . . . . r mm     R(66) where it is obvious that P is a permutation matrix and Q is unitary. The proof is thus completed. C. Proof of Lemma 3 LetĀ AP = QR with P a permutation matrix such that (34) where the first inequality results from the Cauchy-Binet formula, and the last inequality is due to Lemma 2. D. Proof of Proposition 4 The inequality (17) In the following, we focus on the term inside the expection overĤ in (72), i.e., for a given realization ofĤ. Since y l is a degraded version of y m , we can apply the results in [31,Corollary 4] and obtain the optimality of Gaussian input, i.e., for any µ ≥ 1. The next step is to upper bound the right hand side (RHS) of (73). Next, let Ψ Ψ Ψ = V Λ Λ ΛV H be the eigenvalue decomposition of the covariance matrix Ψ Ψ Ψ where Λ Λ Λ is a diagonal matrix and V is unitary. Note that it is without loss of generality to assume that all eigenvalues of Ψ Ψ Ψ are strictly positive, i.e., λ i (Ψ Ψ Ψ) ≥ c > 0, ∀i, in the sense that log det (I + HΨ Ψ ΨH H ) ≤ log det (I + H(cI + Ψ Ψ Ψ)H H ) ≤ log det (I + HΨ Ψ ΨH H ) + log det (I + cHH H ) . (74) In other words, a constant lift of the eigenvalues of Ψ Ψ Ψ does not have any impact on the high snr behavior. This regularization will however simplify the analysis. The following is an upper bound for the first term in the RHS of (73). Fig. 1 . 1System model of K-user MISO BC with CSIT feedback. Fig. 3 . 3Optimal sum DoF (dΣ) vs. total perfect CSIT feedback cost (CP) for three-user 2 × 1 MISO BC. Corollary 4a ( 4aMinimum cost for maximum DoF): For the K-user M × 1 MISO BC, where J users instantaneously feed back perfect (current) CSIT, with the other users feeding back delayed CSIT, then the minimum number J is min{M, K}, in order to achieve the maximum sum DoF min{M, K}. Fig. 5 . 5Achievable sum DoF dΣ vs. δP for the MISO BC with M < K. Proposition 1 (Inner bound, M = 2, K ≥ 3): For the K(≥ 3)-user 2 × 1 MISO BC, the sum DoF is bounded as • For t = 1, we have U = {1, 2, 3}, and δ ′ P,U (1) = 1, δ ′ P,U (2) = 2, δ ′ P,U (3) = 3. Users 3 and 2 are selected to communicate. • For t = 2, we update the parameters as U = {1, 2, 3}, and δ ′ P,U (1) = 1, δ ′ P,U (2) = 1, δ ′ P,U (3) = 2. At this time, again user 3 and user 2 are selected to communicate. • For t = 3, we update the parameters as U = {2, 1, 3}, and δ ′ P,U (1) = 0, δ ′ P,U (2) = 1, δ ′ P,U (3) = 1. At this time, user 3 and user 1 are selected to communicate. After that the communication terminates. = where as stated λ i (A H A) is the i th largest eigenvalue of A H A; r ii is the i th diagonal elements of R.Lemma 3: For any matrix A ∈ C m×m , there exists a column permuted versionĀ, such thatdet(Ā H IĀ I ) ≥ m −|I| i∈I λ i (A H A), ∀ I ⊆ {1, . . . , m}(35)whereĀ I = [A ji : j ∈ {1, . . . , m} , i ∈ I] ∈ C m×|I| is the submatrix of A formed by the columns with indices in I.A. Proof of Lemma 1 Let us perform a singular value decomposition (SVD) on the matrixĜ, i.e.,Ĝ = U D 1 D 2 V H where U , V ∈ C m×m are unitary matrices and D 1 and D 2 are τ ′ × τ ′ and (m − τ ′ ) × (m − τ ′ ) diagonal matrices of the singular values ofĜ. Without loss of generality, we assume that the i th singluar value, i = 1, . . . , m, scales with snr as snr b i , when snr is large. Moreover, the singular values in D 1 are such that b i > 0 and those in D 2 verify b i ≤ 0. First, we have the following lower bound EG log det (G H G) E M log det (D 1 + M 11 ) det M 22 − M 21 (D 1 + M 11 ) det (D 1 ) 2 + E M 11 log det I + D −1 1 M 11 2 + E B EM log det(M H (I + BB H )M ) (39) ≥ log det (D 1 ) 2 + E M 11 log det I + D −1 1 M 11 2 + EM log det(M HM ) (ln 2) −1 m−τ ′ −1 l=0 ψ(m−τ ′ −l)=O(1) (40) where we define M U HG V = M 11 M 12 M 21 M 22 with M 11 ∈ C τ ′ ×τ ′ , and remind that the entries of M , thus of M ij , i, j = 1, 2, are also i.i.d. N c (0, 1); (37) is from the fact that expectation of the log determinant of a non-central Wishart matrix is non-decreasing with in the "line-of-sight" component [27]; (38) is due to the identity det N 11 N 12 N 21 N 22 = det(N 11 ) det(N 22 − N 21 N −1 11 N 12 ) whenever N 11 is square and invertible; in (39), we notice that, given the matrix B M 21 (D 1 + M 11 ) −1 , the columns of M 22 − BM 12 are i.i.d. N c (0, I + BB H ), from which |det(M 22 − BM 12 )| 2 is equivalent in distribution to det(M H (I + BB H )M ) whereM ∈ C (m−τ ′ )×(m−τ ′ ) has i.i.d. N c (0, 1) entries; the last inequality is fromM H (I + BB H )M M HM and therefore det(M H (I + BB H )M ) ≥ det(M HM ), ∀ B; the + E[M H M ] (50) = log|det (D 1 )| 2 + log det I + mD −2 1 o(1) + log det mI + D 2 2 o(log snr) is trivial when m ≥ l ≥ M, i.e., l ′ = m ′ = M. From the chain rule h(y m | U,Ĥ,H) = h(y l | U,Ĥ,H) + h(y l+1 , . . . , y m | y l ,Ĥ,H) = h(y l | U,Ĥ,H) + o(log snr), since with l ≥ M, the observations y l+1 , . . . , y m can be represented as a linear combination of y l , up to the noise error. In the following, we focus on the case l ≤ M. First of all, let us write h(y m |U,Ĥ,H) − µ h(y l |U,Ĥ,H) = EĤ EH[h(H m x + z m | U,Ĥ =Ĥ,H =H)] − µ EH[h(H l x + z l | U,Ĥ =Ĥ,H =H)] (72) E [tr(XX H )]≤snr EH h(y m |U,Ĥ =Ĥ,H =H) − µ EH h(y l |U,Ĥ =Ĥ,H =H) = max Ψ Ψ Ψ 0:tr(Ψ Ψ Ψ)≤snr EH log det (I + H m Ψ Ψ ΨH H m ) − µ EH log det (I + H l Ψ Ψ ΨH H l ) TABLE I SUMMARY IOF THE SCHEME FOR ACHIEVING d * = 2 WITH C ⋆P = 2, WHERE M = 2, K = 3, δP,1 = 1/3, δP,2 = 2/3, δP,3 = 1. time t 1 2 3 U {1, 2, 3} {1, 2, 3} {2, 1, 3} {δ ′ P,U (1) , δ ′ P,U (2) , δ ′ P,U (3) } {1, 2, 3} {1, 1, 2} {0, 1, 1} Active users user 2, 3 user 2, 3 user 1, 3 Perfect CSIT feedback user 3: yes user 3: yes user 3: yes user 2: yes user 2: yes user 2: no user 1: no user 1: no user 1: yes No. of transmitted symbols 2 2 2 TABLE II IISUMMARY OF THE ACHIEVABILITY SCHEME FOR ACHIEVING dΣ = 4 3 WITH δD = 2 3K . TABLE III SUMMARY IIIOF THE ACHIEVABILITY SCHEME FOR ACHIEVING dΣ = 3 2 WITH δD = 9 8K . block index 1 2 3 · · · K No. of channel uses 8 8 8 · · · 8 Active users user 1, 2, 3 user 2, 3, 4 user 3, 4, 5 · · · user K, 1, 2 Delayed CSIT feedback user 1: 3/8 user 2: 3/8 user 3: 3/8 · · · user K: 3/8 fraction in a block user 2: 3/8 user 3: 3/8 user 4: 3/8 user 1: 3/8 user 3: 3/8 user 4: 3/8 user 5: 3/8 user 2: 3/8 the rest: 0 the rest: 0 the rest: 0 the rest: 0 Sum DoF 3/2 3/2 3/2 · · · 3/2 in a block when snr is large. In fact, it has been shown that the O(1) term here, sum of the O(1) term in (40) and −τ ′ in (47), does not depend on snr at all. The next step is to derive an upper bound on E log det (G H G) . Following Jensen's inequality, we haveEG log det(G H G) ≤ log det EG[G H G](49)= log det holds. Then, we havedet(Ā H IĀ I ) = det(R H I Q H QR I ) (67) = det(R H I R I ) (68) ≥ det(R H II R II ) (69) = i∈I r 2 ii (70) ≥ m −|I| i∈I λ i (A H A) EH log det I + H m Ψ Ψ ΨH H m = EH log det I M + Ψ Ψ Ψ ≤ EH log det I M + Ψ Ψ Ψ = EH log det I m ′ + H m i=1 log λ i (Ψ Ψ Ψ) + log det (c −1 + m + Ĥ m 2 = Ψ Ψ Ψ; (76) is due to fact that H H m H m U H Hm 21 2 H H m H m Ψ Ψ Ψ 1 2 (75) 1 2 U H Hm 2 F I m ′ 0 UΨ Ψ Ψ 1 2 (76) 2 FΨ Ψ Ψ (77) = EH log det Ψ Ψ Ψ + EH log det Ψ Ψ Ψ −1 + H m 2 F I (78) ≤ m ′ 2 F )I o(log snr) (79) ≤ log det(Λ Λ Λ) + o(log snr) (80) where Ψ Ψ Ψ 1 2 is such that Ψ Ψ Ψ 1 2 This can be readily derived, using for example the work in[23]. We note that Lemma 1 is a slightly more general version of the result in [26,Lemma 6]. mI +Ĥ H mĤ m ; the last inequality is from the assumption that every eigenvalue of Ψ Ψ Ψ is lower-bounded by some constant c > 0 independent of snr. Now, we need to lower bound the second expectation in the RHS of (73). To this end, let us writewhere(82)is an application of the identity det(I + AB) = det(I + BA); in (83), we definein (84), we define Φ Φ Φ I [Φ ji : j = 1, . . . , l, i ∈ I] ∈ C l×|I| as the submatrix of Φ Φ Φ with columns indexed in I and Λ Λ Λ II = [Λ ji : i, j ∈ I] ∈ C |I|×|I| , with I denoting a nonempty set; the equality (84) is an application of the identity[32]det(for any A ∈ C M ×M ; in (85), we define I 1 , . . . , I M as the so-called sliding window of indicesi.e.,with mod(x) M being the modulo operator; (86) is from the fact that arithmetic mean is not smaller than geometric mean; in (87), we use the fact that M j=1 det(Λ Λ Λ I j I j ) = det(Λ Λ Λ) l . Without loss of generality, we assume that the M columns of H l V are ordered in such a way that 1) the first l columns are linearly independent, i.e.,Φ Φ Φ I 1 has full rank, and 2) A =Φ Φ Φ I 1 satisfies Lemma 3. Note that the former condition can almost always be satisfied since rank(Φ Φ Φ) = l almost surely. Hence, we havewhere (90) is from Lemma 1 by noticing thatis defined as the i th largest eigenvalue of A H A; and the last inequality is due to Lemma 3. Summing over all j, we have≥ l log, ∀ i = 1, . . . , l; the last equality is from the fact that Φ Φ Φ I 1 = Σ Σ Σ −1Ĥ l V I 1 and thatĤ l V I 1 has full rank by construction. From(87)and(99), we obtainand finally(101) When m < M, the above bound (101) is not tight. However, we can show that, in this case, (101) still holds when we replace M with m. To see this, let us define Λ Λ Λ ′ diag(λ 1 , . . . , λ m ). First, note that when m < M, (80) holds if we replace Λ Λ Λ with Λ Λ Λ ′ on the RHS. Then, the RHS of (81) becomes a lower bound if we replace Λ Λ Λ with Λ Λ Λ ′ and V with V ′ ∈ C M ×m , the first m columns of V . From then on, every step holds with M replaced by m. (101) thus follows with M replaced by m. By taking the expectation on both sides of (101) overĤ and plugging it into (72), we complete the proof of(17). How much CSIT feedback is necessary for the multiuser MISO broadcast channels?. J Chen, S Yang, P Elia, No. RR-12-275EURECOM reportJ. Chen, S. Yang, and P. Elia, "How much CSIT feedback is necessary for the multiuser MISO broadcast channels?" December 7, 2012, EURECOM report No. RR-12-275, available on: www.eurecom.fr/en/publication/3893/copyright?popup=1. On the achievable throughput of a multiantenna Gaussian broadcast channel. G Caire, S Shamai, IEEE Trans. Inf. Theory. 497G. Caire and S. Shamai, "On the achievable throughput of a multiantenna Gaussian broadcast channel," IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1691 -1706, Jul. 2003. Isotropic fading vector broadcast channels: The scalar upper bound and loss in degrees of freedom. S Jafar, A Goldsmith, IEEE Trans. Inf. Theory. 513S. Jafar and A. Goldsmith, "Isotropic fading vector broadcast channels: The scalar upper bound and loss in degrees of freedom," IEEE Trans. Inf. Theory, vol. 51, no. 3, pp. 848 -857, Mar. 2005. On degrees of freedom region of MIMO networks without channel state information at transmitters. C Huang, S A Jafar, S Shamai, S Vishwanath, IEEE Trans. Inf. Theory. 582C. Huang, S. A. Jafar, S. Shamai, and S. Vishwanath, "On degrees of freedom region of MIMO networks without channel state information at transmitters," IEEE Trans. Inf. Theory, vol. 58, no. 2, pp. 849-857, Feb. 2012. Completely stale transmitter channel state information is still very useful. M A Maddah-Ali, D N C Tse, IEEE Trans. Inf. Theory. 587M. A. Maddah-Ali and D. N. C. Tse, "Completely stale transmitter channel state information is still very useful," IEEE Trans. Inf. Theory, vol. 58, no. 7, pp. 4418 -4431, Jul. 2012. Not too delayed CSIT achieves the optimal degrees of freedom. N Lee, R W HeathJr, arXiv:1207.2211Proc. Allerton Conf. Communication, Control and Computing. Allerton Conf. Communication, Control and ComputingN. Lee and R. W. Heath Jr., "Not too delayed CSIT achieves the optimal degrees of freedom," in Proc. Allerton Conf. Communication, Control and Computing, Oct. 2012, available on arXiv:1207.2211. Minimum CSIT to achieve maximum degrees of freedom for the MISO BC. R Tandon, S A Jafar, S Shamai, arXiv:1211.4254v2R. Tandon, S. A. Jafar, and S. Shamai, "Minimum CSIT to achieve maximum degrees of freedom for the MISO BC," Nov. 2012, available on arXiv:1211.4254v2. On the degrees of freedom of time correlated MISO broadcast channel with delayed CSIT. M Kobayashi, S Yang, D Gesbert, X Yi, Proc. IEEE Int. Symp. Information Theory (ISIT). IEEE Int. Symp. Information Theory (ISIT)M. Kobayashi, S. Yang, D. Gesbert, and X. Yi, "On the degrees of freedom of time correlated MISO broadcast channel with delayed CSIT," in Proc. IEEE Int. Symp. Information Theory (ISIT), Jul. 2012. Degrees of freedom of time correlated MISO broadcast channel with delayed CSIT. S Yang, M Kobayashi, D Gesbert, X Yi, IEEE Trans. Inf. Theory. 591S. Yang, M. Kobayashi, D. Gesbert, and X. Yi, "Degrees of freedom of time correlated MISO broadcast channel with delayed CSIT," IEEE Trans. Inf. Theory, vol. 59, no. 1, pp. 315-328, Jan. 2013. Optimal use of current and outdated channel state information: Degrees of freedom of the MISO BC with mixed CSIT. T Gou, S Jafar, IEEE Communications Letters. 167T. Gou and S. Jafar, "Optimal use of current and outdated channel state information: Degrees of freedom of the MISO BC with mixed CSIT," IEEE Communications Letters, vol. 16, no. 7, pp. 1084 -1087, Jul. 2012. Degrees-of-freedom region of the MISO broadcast channel with general mixed-CSIT. J Chen, P Elia, arXiv:1205.3474v1J. Chen and P. Elia, "Degrees-of-freedom region of the MISO broadcast channel with general mixed-CSIT," May 2012, available on arXiv:1205.3474v1. Can imperfect delayed CSIT be as useful as perfect delayed CSIT? DoF analysis and constructions for the BC. Proc. Allerton Conf. Communication, Control and Computing. Allerton Conf. Communication, Control and Computing--, "Can imperfect delayed CSIT be as useful as perfect delayed CSIT? DoF analysis and constructions for the BC," in Proc. Allerton Conf. Communication, Control and Computing, Oct. 2012. Imperfect delayed CSIT can be as useful as perfect delayed CSIT: DoF and precoding schemes for BC. IEEE Trans. Inform. Theory. --, "Imperfect delayed CSIT can be as useful as perfect delayed CSIT: DoF and precoding schemes for BC," Oct. 2012, submitted to IEEE Trans. Inform. Theory. On the synergistic benefits of alternating CSIT for the MISO BC. R Tandon, S A Jafar, S Shamai, H V Poor, arXiv:1208.5071IEEE Trans. Inform. Theory. R. Tandon, S. A. Jafar, S. Shamai, and H. V. Poor, "On the synergistic benefits of alternating CSIT for the MISO BC," Aug. 2012, submitted to IEEE Trans. Inform. Theory, available on arXiv:1208.5071. MISO broadcast channel with delayed and evolving CSIT. J Chen, P Elia, arXiv:1211.1622IEEE Trans. Inform. Theory. J. Chen and P. Elia, "MISO broadcast channel with delayed and evolving CSIT," Nov. 2012, submitted to IEEE Trans. Inform. Theory, available on arXiv:1211.1622. The degrees of freedom region of two-user and certain three-user MIMO broadcast channel with delayed CSI. C S Vaze, M K Varanasi, arXiv:1101.0306IEEE Trans. Inf. Theory. C. S. Vaze and M. K. Varanasi, "The degrees of freedom region of two-user and certain three-user MIMO broadcast channel with delayed CSI," Dec. 2011, submitted to IEEE Trans. Inf. Theory, available on arXiv:1101.0306. On the degrees of freedom of X channel with delayed CSIT. A Ghasemi, A S Motahari, A K Khandani, Proc. IEEE Int. Symp. Information Theory (ISIT). IEEE Int. Symp. Information Theory (ISIT)A. Ghasemi, A. S. Motahari, and A. K. Khandani, "On the degrees of freedom of X channel with delayed CSIT," in Proc. IEEE Int. Symp. Information Theory (ISIT), Jul. 2011. On the degrees of freedom of three-user MIMO broadcast channel with delayed CSIT. M J Abdoli, A Ghasemi, A K Khandani, Proc. IEEE Int. Symp. Information Theory (ISIT). IEEE Int. Symp. Information Theory (ISIT)M. J. Abdoli, A. Ghasemi, and A. K. Khandani, "On the degrees of freedom of three-user MIMO broadcast channel with delayed CSIT," in Proc. IEEE Int. Symp. Information Theory (ISIT), Jul. 2011. Interference alignment for the MIMO interference channel with delayed local CSIT. A Ghasemi, A S Motahari, A K Khandani, arXiv:1102.5673v1A. Ghasemi, A. S. Motahari, and A. K. Khandani, "Interference alignment for the MIMO interference channel with delayed local CSIT," Feb. 2011, available on arXiv:1102.5673v1. Broadcast channels with delayed finite-rate feedback: Predict or observe?. J Xu, J G Andrews, S A Jafar, IEEE Trans. Wireless Commun. 114J. Xu, J. G. Andrews, and S. A. Jafar, "Broadcast channels with delayed finite-rate feedback: Predict or observe?" IEEE Trans. Wireless Commun., vol. 11, no. 4, pp. 1456 -1467, Apr. 2012. Degrees of freedom in the MISO BC with delayed-CSIT and finite coherence time: A simple optimal scheme. Y Lejosne, D Slock, Y Yuan-Wu, Proc. IEEE Int. Conf. on Signal Processing. IEEE Int. Conf. on Signal essingY. Lejosne, D. Slock, and Y. Yuan-Wu, "Degrees of freedom in the MISO BC with delayed-CSIT and finite coherence time: A simple optimal scheme," in Proc. IEEE Int. Conf. on Signal Processing, Communications and Control (ICSPCC), Aug. 2012. On fading broadcast channels with partial channel state information at the transmitter. R Tandon, M A Maddah-Ali, A Tulino, H V Poor, S Shamai, Proc. Int. Symp. on Wireless Communication Systems (ISWCS). Int. Symp. on Wireless Communication Systems (ISWCS)R. Tandon, M. A. Maddah-Ali, A. Tulino, H. V. Poor, and S. Shamai, "On fading broadcast channels with partial channel state information at the transmitter," in Proc. Int. Symp. on Wireless Communication Systems (ISWCS), Aug. 2012. Multiuser MIMO achievable rates with downlink training and channel state feedback. G Caire, N Jindal, M Kobayashi, N Ravindran, IEEE Trans. Inf. Theory. 566G. Caire, N. Jindal, M. Kobayashi, and N. Ravindran, "Multiuser MIMO achievable rates with downlink training and channel state feedback," IEEE Trans. Inf. Theory, vol. 56, no. 6, pp. 2845 -2866, Jun. 2010. Retrospective interference alignment over interference networks. H Maleki, S Jafar, S Shamai, IEEE Journal of Selected Topics in Signal Processing. 63H. Maleki, S. Jafar, and S. Shamai, "Retrospective interference alignment over interference networks," IEEE Journal of Selected Topics in Signal Processing, vol. 6, no. 3, pp. 228 -240, Mar. 2012. On the degrees of freedom of the K-user time correlated broadcast channel with delayed CSIT. P De Kerret, X Yi, D Gesbert, arXiv:1301.2138P. de Kerret, X. Yi, and D. Gesbert, "On the degrees of freedom of the K-user time correlated broadcast channel with delayed CSIT," Jan. 2013, available on arXiv:1301.2138. The degrees of freedom region of temporally-correlated MIMO networks with delayed CSIT. X Yi, S Yang, D Gesbert, M Kobayashi, arXiv:1211.3322IEEE Trans. Inform. Theory. X. Yi, S. Yang, D. Gesbert, and M. Kobayashi, "The degrees of freedom region of temporally-correlated MIMO networks with delayed CSIT," Nov. 2012, submitted to IEEE Trans. Inform. Theory, available on arXiv:1211.3322. On the log determinant of noncentral wishart matrices. Y H Kim, A Lapidoth, Proc. IEEE Int. Symp. Information Theory (ISIT). IEEE Int. Symp. Information Theory (ISIT)Y. H. Kim and A. Lapidoth, "On the log determinant of noncentral wishart matrices," in Proc. IEEE Int. Symp. Information Theory (ISIT), Jul. 2003. R J Muirhead, Aspects of Multivariate Statistical Theory. New YorkWileyR. J. Muirhead, Aspects of Multivariate Statistical Theory. New York: Wiley, 1982. On the role of MMSE estimation in approaching the information-theoretic limits of linear Gaussian channels: Shannon meets Wiener. G D ForneyJr, Proc. Allerton Conf. Communication, Control and Computing. Allerton Conf. Communication, Control and ComputingG. D. Forney Jr., "On the role of MMSE estimation in approaching the information-theoretic limits of linear Gaussian channels: Shannon meets Wiener," in Proc. Allerton Conf. Communication, Control and Computing, Oct. 2003. R A Horn, C R Johnson, Topics in Matrix Analysis. Cambridge University PressR. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge University Press, 1991. The capacity region of the degraded multiple-input multiple-output compound broadcast channel. H Weingarten, T Liu, S Shamai, Y Steinberg, P Viswanath, IEEE Trans. Inf. Theory. 5511H. Weingarten, T. Liu, S. Shamai, Y. Steinberg, and P. Viswanath, "The capacity region of the degraded multiple-input multiple-output compound broadcast channel," IEEE Trans. Inf. Theory, vol. 55, no. 11, pp. 5011 -5023, Nov. 2009. A C Aitken, Determinants and Matrices. Oliver and Boyd8A. C. Aitken, Determinants and Matrices, 8th ed. Edinburgh: Oliver and Boyd, 1954.
[]
[ "Large Extra Dimension Effects on the Spin Configuration of the Top Quark Pair at e + e − Colliders", "Large Extra Dimension Effects on the Spin Configuration of the Top Quark Pair at e + e − Colliders" ]
[ ") Kang ", "Young Lee \nDepartment of Physics\nSeoul National University\n151-742SeoulKorea\n", "H S Song ", "Jeonghyeon Song \nDepartment of Physics\nSeoul National University\n151-742SeoulKorea\n", ") Chaehyun ", "Yu ", "\nCenter for Theoretical Physics\nSeoul National University\n151-742SeoulKorea (\n" ]
[ "Department of Physics\nSeoul National University\n151-742SeoulKorea", "Department of Physics\nSeoul National University\n151-742SeoulKorea", "Center for Theoretical Physics\nSeoul National University\n151-742SeoulKorea (" ]
[]
Large extra dimension effects on the spin configuration of the top quark pair at the e + e − → tt process are studied. It is shown that the TeV scale quantum gravity effects cause significant deviations from the Standard Model predictions for the spin configuration in the off-diagonal basis: they lead to substantial cross sections of the like-spin states of the top quark pair, which vanish in the SM; they weaken the pure dominance of the processes, the Up-Down (Down-Up) spin states for the left-handed (right-handed) beam. In addition it is shown that the angular cut −0.5 < cos θ < 0 is very effective to determine the sign of the quantum gravity corrections.
10.1103/physrevd.60.093002
[ "https://export.arxiv.org/pdf/hep-ph/9905227v2.pdf" ]
15,426,025
hep-ph/9905227
9a79140e2bc3a9153526726809b2d24a4294e81b
Large Extra Dimension Effects on the Spin Configuration of the Top Quark Pair at e + e − Colliders May 1999 ) Kang Young Lee Department of Physics Seoul National University 151-742SeoulKorea H S Song Jeonghyeon Song Department of Physics Seoul National University 151-742SeoulKorea ) Chaehyun Yu Center for Theoretical Physics Seoul National University 151-742SeoulKorea ( Large Extra Dimension Effects on the Spin Configuration of the Top Quark Pair at e + e − Colliders May 1999arXiv:hep-ph/9905227v2 7 SNUTP 99 − 022 Large extra dimension effects on the spin configuration of the top quark pair at the e + e − → tt process are studied. It is shown that the TeV scale quantum gravity effects cause significant deviations from the Standard Model predictions for the spin configuration in the off-diagonal basis: they lead to substantial cross sections of the like-spin states of the top quark pair, which vanish in the SM; they weaken the pure dominance of the processes, the Up-Down (Down-Up) spin states for the left-handed (right-handed) beam. In addition it is shown that the angular cut −0.5 < cos θ < 0 is very effective to determine the sign of the quantum gravity corrections. There has been an increasing interest in the low scale quantum gravity as an extension of the Kaluza-Klein (KK) scenario. Recently Arkani-Hamed, Dimopoulos, and Dvali (ADD) [1] have suggested that the size of the extra dimensions could be large enough to be detectable if we confine the matter fields to the 4-dimensional world where we live. According to their idea, the weakness of the gravity in our world is caused by the suppression factor from the large extra dimensions since gravitons are the only fields freely propagating in the whole (4 + N)-dimensional spacetime. Considering the macroscopic Gauss' Law for the Newtonian gravity, ADD have derived the relation between the Planck scale and the size of the extra dimensions as Indeed interesting is that this idea is testable by the collider phenomenologies in the near future. When the graviton momentum does not exceed the scale of M S , the spacetime where collisions take place can be approximately described by the linear expansion around the flat metric. Within the framework of the linearized gravity, the effective action in the 4-dimensional spacetime is derived after compactifying the extra dimensions, which leads to corresponding Feynman rules [3,4]. The KK reduction from the whole (4 + N)-dimension to our 4-dimension yields towers of massive KK states in the 4-dimensional effective theory, of which the massive spectrum is cutoff at the scale M S . Each graviton in such KK towers interacts with the ordinary matter fields with the couplings suppressed by the Planck scale. M 2 P l ∼ M N +2 S R N ,(1) The production of a single graviton is enhanced by the kinematic factor and has been studied as a source of the missing energy in the e + e − → γE / or pp → γE / process [4,5]. The indirect effects of massive graviton exchange may be enhanced by the sum of the tower of the KK states and provide various signals in the collider phenomenologies [6][7][8][9][10][11][12]. In particular the spin two nature of the gravitons will result in some characteristic effects on the polarization observables [12]. In the present work, we consider the top quark pair production in the Next e + e − Linear Collider (NLC) [13]. It is a promising testing ground for new physics effects because of Since the remarkable successes of the SM in explaining all the high energy experiments can be retained by assuming the SM fields confined on our world, i.e. T µj = T ij = 0 (µ = 1, · · · 4; i, j = 5, · · · (4 + N)), only the spin two gravitons and the dilaton modes of the spin zero gravitons interact with the ordinary matter. In addition the coupling of the dilaton modes to the electron is to be neglected at high energy collisions, since it is proportional to the fermion mass. Thus the process e + e − → tt receives the effects of TeV scale quantum gravity by the s-channel Feynman diagram mediated by the spin two Kaluza-Klein gravitons. Furthermore, the produced top quark is known to be in the unique spin configuration at the polarized e + e − collider [14] and the information of the top spin is not lost through hadronization since its lifetime is too short for the top quark to constitute hadrons. We can read out the information on the top polarization through the angular distribution of the decay products [15]. The neutral current nature of the graviton interactions leaves the electroweak decay of top quark pair intact, implying that the SM prediction of the angular correlations between the decay products and the spin orientation of each top quark remains valid [15]. Therefore the spin configuration of the top quark pair can be a probe of the effects of KK gravitons. For the process e − (k 1 ) + e + (k 2 ) → t(p 1 ) + t(p 2 ) ,(2) the scattering amplitude of the s-channel Feynman diagram mediated by the spin two gravitons summed over the Kaluza-Klein tower can be written by M G = λ M 4 S (k 1 − k 2 ) · (p 1 − p 2 )v(k 2 )γ µ u(k 1 )u(p 1 )γ µ v(p 2 ) + v(k 2 )(p 1 / − p 2 / )u(k 1 )u(p 1 )(k 1 / − k 2 / )v(p 2 ) ,(3) where the order one parameter λ depends on the number of extra dimensions and the compactification models. Since the exact value of the λ including sign is not determined unless the full quantum gravity theory is provided, two representative cases of λ = ± are considered in the following discussion [6]. It is also to be noted that the amplitude in Eq. (3) as well as the SM amplitudes at the tree level are CP invariant. In order to analyze the spin configuration of the top quark pair, let us briefly review a generic spin basis discussed in Ref. [14]. We define the spin states of the top quark and top anti-quark in their own rest-frame by decomposing their spins along the reference axesη and η, respectively. The CP invariance, which is valid at the tree level even with the large extra dimension effects, does not allow the T odd quantity, i.e. σ t · ( k 1 | (t−rest) × p 2 | (t−rest) ) where the σ t is the spin of the top quark, and k 1 | t−rest and p 2 | t−rest are the momenta of the electron and the top anti-quark in the rest frame of the top [14,16]. Thus the top and anti-top spins are to lie in the production plane. The spin four-vectors of the top quark pair are chosen to be back-to-back in the zero momentum frame. Theη is expressed by an angle ξ, the angle betweenη and the top anti-quark momentum in the rest frame of the top quark. The usual helicity basis is obtained by taking ξ = π. In this general spin basis the differential cross sections of the e + e − → tt process with the large extra dimension effects are dσ d cos θ (e − L e + R → t ↑t↑ or t ↓t↓ ) = N c πα 2 β 2s |Ã L cos ξ −B L sin ξ| 2 , dσ d cos θ (e − L e + R → t ↑t↓ or t ↓t↑ ) = N c πα 2 β 2s |Ã L sin ξ +B L cos ξ ±D L | 2 , dσ d cos θ (e − R e + L → t ↑t↑ or t ↓t↓ ) = N c πα 2 β 2s |Ã R cos ξ −B R sin ξ| 2 , dσ d cos θ (e − R e + L → t ↑t↓ or t ↓t↑ ) = N c πα 2 β 2s |Ã R sin ξ +B R cos ξ ∓D R | 2 ,(4) where t ↑ (t ↓ ) denotes the top spin along (against) theη, N c is the number of color, α is the fine structure constant, β = 1 − 4m 2 t /s and A L = 1 2 (f LL + f LR ) sin θ 1 − β 2 − f G sin 2θ 1 − β 2 , B L = 1 2 f LL (cos θ + β) + f LR (cos θ − β) − f G cos 2θ, D L = 1 2 f LL (1 + β cos θ) + f LR (1 − β cos θ) − f G cos θ, A R = 1 2 (f RR + f RL ) sin θ 1 − β 2 − f G sin 2θ 1 − β 2 , B R = 1 2 f RR (cos θ + β) + f RL (cos θ − β) − f G cos 2θ, D R = 1 2 f RR (1 + β cos θ) + f RL (1 − β cos θ) − f G cos θ.(5) The large extra dimension effects are altogether included in the quantity f G defined by f G = βs 2 4α λ M 4 S .(6) Here f IJ 's (I, J = L or R) are f IJ = Q γ (e)Q γ (t) + Q I Z (e)Q J Z (t) 1 sin 2 θ W s s − M 2 Z ,(7) and θ is the scattering angle of the top quark with respect to the electron beam. The electric charges and couplings to the Z boson of the electron and the top quark are given by Q γ (e) = −1 , Q γ (t) = 2 3 , Q L Z (e) = 2 sin 2 θ W − 1 2 cos θ W , Q R Z (e) = sin 2 θ W cos θ W , Q L Z (t) = 3 − 4 sin 2 θ W 6 cos θ W , Q R Z (t) = − 2 sin 2 θ W 3 cos θ W .(8) indicating f LL > f LR and f RR > f RL . There exist the angles ξ L and ξ R such that the differential cross sections for the t ↑t↑ and t ↓t↓ , i.e., like-spin states vanish for the left-and right-handed electron beam, respectively. It is called the "off-diagonal basis", of which the name originated in this feature [14]. This is general in any 2 → 2 process if the CP is conserved and the spin four-vectors of the outgoing particles are back-to-back in the zero momentum frame. In the SM, the angles ξ L,R are taken to be cos ξ I = − B I A 2 I + B 2 I , sin ξ I = − A I A 2 I + B 2 I ,(10) where I = L, R, A I =Ã I | f G =0 , and B I =B I | f G =0 . The differential cross sections in Eq. (4) are reduced to, in this SM off-diagonal basis, dσ d cos θ (e − L e + R → t ↑t↑ or t ↓t↓ ) = N c πα 2 β 2s f 2 G sin ξ L cos 2θ − cos ξ L sin 2θ 1 − β 2 2 , dσ d cos θ (e − L e + R → t ↑t↓ or t ↓t↑ ) = N c πα 2 β 2s A 2 L + B 2 L ∓ D L +f G (cos ξ L cos 2θ + sin ξ L sin 2θ 1 − β 2 ± cos θ) 2 , dσ d cos θ (e − R e + L → t ↑t↑ or t ↓t↓ ) = N c πα 2 β 2s f 2 G sin ξ R cos 2θ − cos ξ R sin 2θ 1 − β 2 2 , dσ d cos θ (e − R e + L → t ↑t↓ or t ↓t↑ ) = N c πα 2 β 2s A 2 R + B 2 R ± D R +f G (cos ξ R cos 2θ + sin ξ R sin 2θ 1 − β 2 ∓ cos θ) 2 ,(11) where D I =D I | f G =0 . There while those to the total cross section is about ∼ 30% at √ s ∼ 500 GeV [17,18]. The TeV scale quantum gravity modifies these two features. First the differential cross sections of the like-spin states acquire the quantum gravity effects. Secondly the presence of f G in the differential cross sections of the dominant spin configurations corresponding to the incident beam polarizations pollutes their pure dominance. In Fig. 1 and 2 We present the change of the degree of dominance corresponding to the beam polarization due to the large extra dimensions. Table 1 shows the ratios of the cross section for the dominant process to the total cross section at √ s = 500 GeV and √ s = 1 TeV for λ = ±1, which are expected to be stable by the one-loop QCD corrections. As the beam energy increases enough, the λ = −1 case gives rise to more deviations from the SM prediction. And it can be easily seen that the use of the high energy right-handed electron beam is more efficient in detecting the corrections. interference occurs when λ = +1 (λ = −1). In the next dominant processes (the Down-Up spin state with e − L and the Up-Down one with e − R ), enormous quantum corrections with λ = +1 to the backward direction are reduced and dispersed. Therefore we suggest that angular cut, e.g., −0.5 < cos θ < 0 be very effective to probe the TeV scale quantum gravity corrections. The usefulness of the angular cut is demonstrated in Table 2 ∆σ(e − L e + R → t ↓t↑ )/∆σ(e − L e + R → t ↑t↓ ) 0.14 0.035 2.15 (λ = +1) (λ = +1) (λ = −1) σ(e − L e + R → t ↑t↓ )/σ(e − L e + R → ∆σ(e − R e + L → t ↑t↓ )/∆σ(e − R e + L → t ↓t↑ ) 0.083 0.034 26.92 Table 2. The ratios of the cross sections for the next-dominant processes to those for the dominant ones with the angular cut −0.5 < cos θ < 0 at √ s = 1 TeV. In conclusion, we have studied the spin configuration of the top quark pair at the process 1 2 t ↑ t ↑ t ↑ t ↓ t ↓ t ↓ t ↓ t ↑ M S =2.5 TeV5 t ↑ t ↑ t ↑ t ↓ t ↓ t ↓ t ↓ t ↑ M S =2.5 TeV √s=0.5 TeV (×10 −3 ) (×10 −3 ) (×10 −3 ) − − − − cosθ1 2 t ↑ t ↑ t ↑ t ↓ t ↓ t ↓ t ↓ t ↑0.1 t ↑ t ↑ t ↑ t ↓ t ↓ t ↓ t ↓ t ↑ where M S is the only fundamental scale of nature, of which the value is comparable with the electroweak scale. The N = 1 case is excluded by this simple relation because the corresponding size of the extra dimension is order of 10 13 cm if the M S is at a few TeV order which is phenomenologically interesting. The N = 2 case which implies mm scale extra dimensions is not excluded by the current macroscopic measurement of gravitational force [2]. For the cases of N > 2 there exist no other serious constraints up to now. The hierarchy of the Planck scale M P l and the electroweak scale M W is reduced to the revelation of the effects of the large extra dimensions. the heavy mass of top quark. The presence of large extra dimensions affects the top quark pair production at e + e − collisions through the interactions between the fermions and the Kaluza-Klein tower of gravitons. In the view point of our 4-dimensional world, the massless gravitons freely propagating in the (4 + N)-dimensional bulk are massive spin two, spin one and spin zero gravitons under the compactification of the extra dimensional manifold. A comment on the behaviors of the f IJ 's are in order here for they play a central role in the SM background. Numerical results show that the f IJ 's are negative quantity and their dependence on the beam energy is weak at high energy. For instance, their numerical values at √ s =500 GeV are f LL = −1.21, f LR = −0.43, f RL = −0.20, f RR = −0.87, are two important characteristic features of the SM background. First, the differential cross sections for the like-spin states of the top quark pair vanish. Or we have chosen the spin configuration in that way. Secondly, the process for t ↑t↓ (t ↓t↑ ) is dominant when the left-handed (right-handed) electron beam is used. This can be easily seen in Eq.(11) since the negative quantities A's, B's and D's are the same order of magnitude. At high energy, the degree of this dominance is close to 100%[17]. This pure dominance of the Up-Down state for the left-handed electron beam and the Down-Up state for the right-handed one is fairly stable by the one-loop QCD corrections where the soft gluon emissions are dominant so that the QCD corrections are factored out. For instance, the O(α s ) corrections to the ratio of the cross section for the dominant process to the total cross section is just ∼ 0.01% we plot the differential cross sections with respect to the top quark scattering angle, broken down to the spin configuration of the top quark pair at √ s = 500 GeV with the left-and right-handed beams, respectively. The solid lines denote the SM background, and the dotted (dashed) lines include the quantum gravity effects with λ = +1 (λ = −1). The M S is set to be 2.5 TeV. Figures 3 and 4 illustrate the same observables at √ s = 1 TeV. At √ s = 500 GeV, the corrections are small over all the spin configurations, and become a little larger in the case of the right-handed electron beam. Figures 3 and 4 show that the quantum gravity corrections increase with the beam energy. In particular, the like-spin states, which are zero in the tree level SM, gain sizable cross sections. Finally let us observe that the angular distribution of the cross sections have provided valuable information on the nature of the interactions between gravitons and fermions. According to the sign of the λ, the quantum gravity corrections act in a different way. In the dominant processes (the Up-Down spin state with the left-handed electron beam and the Down-Up spin state with the right-handed one), the virtual graviton exchanges cause destructive (constructive) interference with the SM diagrams to the backward direction when λ = +1 (λ = −1). To the forward direction, on the contrary, constructive (destructive) through the ratios between the cross sections for the next-dominant and dominant processes at√ s = 1 TeV,where ∆σ ≡ 0 −0.5 d cos θ(dσ/d cos θ). This observable clearly discriminates the λ = −1 case from the λ = +1 case, as its value is different by a few orders of magnitude. e + e − → tt with large extra dimensions. The TeV scale quantum gravity affects the process through the s-channel Feynman diagram mediated by the Kaluza-Klein tower of spin two gravitons. The framework of the off-diagonal basis, in which the cross sections of the like-spin states of the top quark pair vanish, leads to the SM result that the left-handed (right-handed) electron beam almost completely prefer the spin configuration of the top and anti-top spins as Up-Down (Down-Up). The presence of large extra dimensions modifies these features significantly at high energies. It yields non-zero cross sections for the like-spin states and relieves the partiality of the beam polarization for a specific spin configuration of the top quark pair. In addition, it is shown that the angular cut −0.5 < cos θ < 0 is very effective to determine the sign of the quantum gravity corrections. FIG. 1 . 1The differential cross section with respect to the scattering angle of the top quark at √ s = 500 GeV with the left-handed electron beam, broken down to the spin configuration of the top quark pair. The dotted (dashed) line includes the large extra dimension effects when λ = +1 (λ = −1) and M S = 2.5 TeV. The solid line denotes the SM background. FIG. 2 . 2The differential cross section with respect to the scattering angle of the top quark at √ s = 500 GeV with the right-handed electron beam, broken down to the spin configuration of the top quark pair. The dotted (dashed) line includes the large extra dimension effects when λ = +1 (λ = −1) and M S = 2.5 TeV. The solid line denotes the SM background. FIG. 3 . 3The differential cross section with respect to the scattering angle of the top quark at √ s = 1 TeV with the left-handed electron beam, broken down to the spin configuration of the top quark pair. The dotted (dashed) line includes the large extra dimension effects when λ = +1 (λ = −1) and M S = 2.5 TeV. The solid line denotes the SM background. FIG. 4 . 4The differential cross section with respect to the scattering angle of the top quark at √ s = 1 TeV with the right-handed electron beam, broken down to the spin configuration of the top quark pair. The dotted (dashed) line includes the large extra dimension effects when λ = +1 (λ = −1) and M S = 2.5 TeV. The solid line denotes the SM background. ACKNOWLEDGMENTSWe would like to appreciate valuable discussions with S.Y. Choi. This work is supported by the Korean Science and Engineering Foundation (KOSEF) through the SRC program of the Center for Theoretical Physics (CTP) at Seoul National University. . N Arkani-Hamed, S Dimopoulos, G Dvali, Phys. Lett. 429263N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B429 (1998) 263. . E G J C See, H W Long, J C Chan, Price, Nucl. Phys. 53923See, e.g. J.C. Long, H.W. Chan, and J.C. Price, Nucl. Phys. B539 (1999) 23. . T Han, J D Lykken, R Zhang, Phys. Rev. 59105006T. Han, J.D. Lykken and R. Zhang, Phys. Rev. D59 (1999) 105006. . G F Giudice, R Rattazzi, J D Wells, Nucl. Phys. 5443G.F. Giudice, R. Rattazzi, and J.D. Wells, Nucl. Phys. B544 (1999) 3. . E A Mirabelli, M Perelstein, M E Peskin, Phys. Rev. Lett. 822236E.A. Mirabelli, M. Perelstein, and M.E. Peskin, Phys. Rev. Lett. 82 (1999) 2236. . J L Hewett, hep-ph/9811356J. L. Hewett, hep-ph/9811356. hep-ph/9904380; hep-ph/9903475; hep-ph/9902273. T G Rizzo, hep-ph/9901209T.G. Rizzo, hep-ph/9904380; hep-ph/9903475; hep-ph/9902273; hep-ph/9901209. . P Mathews, S Raychaudhuri, K Sridhar, Phys. Lett. 450343P. Mathews, S. Raychaudhuri, and K. Sridhar, Phys. Lett. B450 (1999) 343. . K Agashe, N G Deshpande, hep-ph/9902263K. Agashe and N.G. Deshpande, hep-ph/9902263. . C Balazs, H.-J He, W W Repko, C.-P Yuan, hep-ph/9904220C. Balazs, H.-J. He, W. W. Repko, and C.-P. Yuan, hep-ph/9904220. . K Cheung, hep-ph/9903294K. Cheung, hep-ph/9904266; hep-ph/9903294. . K Y Lee, H S Song, J Song, hep-ph/9904355K.Y. Lee, H.S. Song, and J. Song, hep-ph/9904355. . E Accomando, No. DESY 97-100ReportE. Accomando et. al. Report No. DESY 97-100. . S Parke, Y Shadmi, Phys. Lett. 387199S. Parke and Y. Shadmi, Phys. Lett. B387 (1996) 199. . T Arens, L M Sehgal, Nucl. Phys. 3934372Phys. Rev.T. Arens and L.M. Sehgal, Nucl. Phys. B393 (1993) 46; Phys. Rev. D50 (1994) 4372; . G Mahlon, S Parke, Phys. Rev. 534886G. Mahlon and S. Parke, Phys. Rev. D53 (1996) 4886. . G L Kane, G A Ladinsky, C.-P Yuan, Phys. Rev. 45124G.L. Kane, G.A. Ladinsky, and C.-P. Yuan, Phys. Rev. D45 (1992) 124. . S Parke, hep-ph/9807573S. Parke, hep-ph/9807573. . J Kodaira, T Nasuno, S Parke, Phys. Rev. 5914023J. Kodaira, T. Nasuno, and S. Parke, Phys. Rev. D59 (1999) 014023.
[]
[ "A Quantitative Simulation-based Modeling Approach for College Counseling Centers", "A Quantitative Simulation-based Modeling Approach for College Counseling Centers", "A Quantitative Simulation-based Modeling Approach for College Counseling Centers", "A Quantitative Simulation-based Modeling Approach for College Counseling Centers" ]
[ "Sohom Chaterjee \nDepartment of Industrial and Systems Engineering\n\n", "Youssef Hebaish \nDepartment of Industrial and Systems Engineering\n\n", "Lewis Ntaimo \nDepartment of Industrial and Systems Engineering\n\n", "James Deegear \nCounseling & Psychological Services Texas A&M University\n77843College StationTXUnited States\n", "Miles Rucker \nCounseling & Psychological Services Texas A&M University\n77843College StationTXUnited States\n", "Hrayer Aprahamian \nDepartment of Industrial and Systems Engineering\n\n", "Sohom Chaterjee \nDepartment of Industrial and Systems Engineering\n\n", "Youssef Hebaish \nDepartment of Industrial and Systems Engineering\n\n", "Lewis Ntaimo \nDepartment of Industrial and Systems Engineering\n\n", "James Deegear \nCounseling & Psychological Services Texas A&M University\n77843College StationTXUnited States\n", "Miles Rucker \nCounseling & Psychological Services Texas A&M University\n77843College StationTXUnited States\n", "Hrayer Aprahamian \nDepartment of Industrial and Systems Engineering\n\n" ]
[ "Department of Industrial and Systems Engineering\n", "Department of Industrial and Systems Engineering\n", "Department of Industrial and Systems Engineering\n", "Counseling & Psychological Services Texas A&M University\n77843College StationTXUnited States", "Counseling & Psychological Services Texas A&M University\n77843College StationTXUnited States", "Department of Industrial and Systems Engineering\n", "Department of Industrial and Systems Engineering\n", "Department of Industrial and Systems Engineering\n", "Department of Industrial and Systems Engineering\n", "Counseling & Psychological Services Texas A&M University\n77843College StationTXUnited States", "Counseling & Psychological Services Texas A&M University\n77843College StationTXUnited States", "Department of Industrial and Systems Engineering\n" ]
[]
College counseling centers in various universities have been tasked with the important responsibility of attending to the mental health needs of their students. Owing to the unprecedented recent surge of demand for such services, college counseling centers are facing several crippling resource-level challenges. This is leading to longer wait times which limits access to critical mental health services. To address these challenges, we construct a discrete-event simulation model that captures several intricate details of their operations and provides a data-driven framework to quantify the effect of different policy changes. In contrast to existing work on this matter, which are primarily based on qualitative assessments, the considered quantitative approach has the potential to lead to key observations that can assist counseling directors in constructing a system with desirable performance. To demonstrate the benefit of the considered simulation model, we use data specific to Texas A&M's Counseling & Psychological Services to run a series of numerical experiments. Our results demonstrate the predictive power of the simulation model, highlight a number of key observations, and identify policy changes that result in desirable system performance.
10.1177/00375497231159675
[ "https://export.arxiv.org/pdf/2206.01790v1.pdf" ]
249,394,820
2206.01790
af0bbb9670bb5aa911e9fa4d2a0d62c9fe006046
A Quantitative Simulation-based Modeling Approach for College Counseling Centers June 7, 2022 Sohom Chaterjee Department of Industrial and Systems Engineering Youssef Hebaish Department of Industrial and Systems Engineering Lewis Ntaimo Department of Industrial and Systems Engineering James Deegear Counseling & Psychological Services Texas A&M University 77843College StationTXUnited States Miles Rucker Counseling & Psychological Services Texas A&M University 77843College StationTXUnited States Hrayer Aprahamian Department of Industrial and Systems Engineering A Quantitative Simulation-based Modeling Approach for College Counseling Centers June 7, 2022College counselingmental healthmodelingdiscrete-event simulationsystem performanceaccess time College counseling centers in various universities have been tasked with the important responsibility of attending to the mental health needs of their students. Owing to the unprecedented recent surge of demand for such services, college counseling centers are facing several crippling resource-level challenges. This is leading to longer wait times which limits access to critical mental health services. To address these challenges, we construct a discrete-event simulation model that captures several intricate details of their operations and provides a data-driven framework to quantify the effect of different policy changes. In contrast to existing work on this matter, which are primarily based on qualitative assessments, the considered quantitative approach has the potential to lead to key observations that can assist counseling directors in constructing a system with desirable performance. To demonstrate the benefit of the considered simulation model, we use data specific to Texas A&M's Counseling & Psychological Services to run a series of numerical experiments. Our results demonstrate the predictive power of the simulation model, highlight a number of key observations, and identify policy changes that result in desirable system performance. Introduction and Motivation Since the advent of the COVID-19 pandemic and its impact on our everyday lives, mental health related problems have seen a sharp increase across the population [1,2], with a notable negative impact on students [3,4,5]. Today, student mental health in higher education is considered one of the primary hurdles in the path to academic success. Studies have shown that students experience their first onset of mental health problems, as well as an increase in pre-existing symptoms, during their college years [6]. Unfortunately, over the past few decades, there has been an alarming increase in psychological issues that college students exhibit. For example, according to the 2006 National Survey of Counseling Center Directors, almost half of college-aged individuals were reported to have some kind of psychiatric disorder [7]. More recent reports from the 2019 annual survey by the Association for University and College Counseling Center Directors (AUCCCD) demonstrate that college students suffer from a multitude of mental disorders, with anxiety, depression, and stress being the most prevalent psychological issues reported by counseling centers [8]. To combat this worrying trend, higher education institutions have set up university Counseling and Psychological Service (CAPS) centers to provide preventive and remedial counseling to help students identify and attain personal, academic, and career goals [9]. However, the increasing ethnic, racial, and social diversity within the student population, as well as the changing trends in students' needs, has altered the traditional mission of these centers, with student mental health support being one of the primary services [10]. These challenges have been exacerbated by the pandemic [11]. Campus closures and the suspension of in-person classes result in adverse psychological effects on students -such as loneliness, isolation, stress, anxiety, and depression -leading to a drastic increase in psychological needs [12]. This surge in demand has imposed additional strains on counseling centers, and the current model is unable to meet the growing mental health needs of students and is resulting in staff burnout [13,14,15], thus calling for a major transformation in the delivery of CAPS at university campuses. In this paper, we propose a quantitative framework to provide CAPS with data-driven insights and recommendations to better manage their resources and to streamline and optimize their operations. Counseling centers in universities are different from counseling centers that cater to the general population in two ways: First, the demand for CAPS services on college campuses follows a cyclical nature related to the student academic calendar. For instance, at the beginning of the semester, or right before major holidays (e.g., Thanksgiving, Spring Break), demand is usually sparse due to low academic pressure, but the demand starts increasing as the semester progresses. This trend is mainly due to the stress that students develop as they approach their exams and the accumulation of academic workload. This semester-based cyclic nature of demand often imposes strains on counseling centers' resources. Second, the set of mental health disorders, and their prevalence, experienced by students is different from that of the general population. As such, CAPS facilities need to tailor their services and resources to cater to this need. Given these unique characteristics of CAPS, we focus the analysis on college counseling centers which narrows down the scope to a more manageable target population with a clear set of psychological disorders. Moreover, one of the objectives of this work is to take advantage of the recurring semester-based demand trends to help CAPS better understand their system and its performance. College counseling centers face a myriad of challenges, which can be broadly categorized into two classes: Resource-level and patient-level challenges. Patient-level challenges pertain to the relationship between patient improvement and the treatment plan they undertake, while resourcelevel challenges involve resource planning and allocation. Patients have unique needs and hence are assigned tailored treatment plans which are often a combination of various treatment options (e.g., one-on-one counseling sessions, psychiatric treatment, etc.). The main challenge lies in identifying a treatment plan that yields the best possible improvement. Studies have shown that a patient's improvement tends to increase with the number of attended sessions [16]. However, the study also reveals that there are no clear pathways to construct treatment plans for patients based on their specific case. Hence, there is no guarantee that the treatment plan they receive maximizes improvement. In addition, although there is a direct relation between the number of sessions and patient improvement, it may be infeasible-on a resource level-to provide every patient with as many sessions as they might need. Therefore, counseling centers also face challenges with regard to resource planning. As previously pointed out, there is a natural growth in demand due to the increasing college student population as well as their psychological issues [17,18]. This increase in demand has negatively affected the waiting time for patients to receive care. According to the 2019 survey by AUCCCD that included 562 counseling centers, the average wait time for a first triage appointment (which we refer to as access time) was 6.1 days, while the average wait time for the following clinical appointment was 8.7 days. These numbers are expected to be much higher during the pandemic. The most straightforward solution to this issue is to hire more counselors as the most frequently reported barrier to meeting demand, as reported by counseling centers directors, is understaffing issues [19]. However, limited funding availability, which is the root cause of understaffing [20], prevents the adoption of such solutions. The study in this paper focuses on addressing the resource-level challenges facing CAPS with the aim of providing recommendations that do not impact the current staffing structure and strategy for deciding on treatment plans. To combat the aforementioned challenges, a number of attempts have been made by counseling centers to meet the surge in demand. One example is external referral of patients where students are referred to off-campus mental health providers whenever they require a higher level of specialization that cannot be provided on-campus [21]. This is done so that counseling centers can meet their students' needs [22]. For example, Iarussi and Shaw [23] propose a "Collaborative Process Model" for referral, consisting of four phases: (i) informed consent, (ii) assessment of patient's case, (iii) collaborative decision making, and (iv) execution. The purpose of the first phase is to provide patients with all the necessary information about the scope of services provided by the counseling center and the possibility of a referral to an off-campus provider [23]. The second phase is assessing the patient's case during the first appointment to gather information about the patient and their needs. The third phase personalizes referral options according to the patient's psychological needs and financial circumstances. The fourth and final phase is to initiate the referral process by connecting the patient to the new provider and following up with them [23]. Owen et al. [24] suggest a similar referral process that entails meeting the patient and then following up with them. Their study showed that 58% of patients at a university counseling center were successful in connecting with an off-campus provider. Although such models might result in more successful referral cases due to the personalization nature of the process, they have some issues concerning resource allocation and functionality. First, the patient's involvement in the process increases the likelihood of successful referral (even if it is not optimal to do so); however, this claim is not substantiated by empirical evidence that supports its validity. Second, such models do not tackle one of the significant challenges that counseling centers face, access time [8]. Because the initiation of the process begins with the first phase during the first appointment with the patient, such models do not affect the expected access time. In addition, such studies do not provide any information regarding the impact external referrals have on the overall system performance. Another example of an attempt made by counseling centers is to set an upper bound on the number of sessions that a student can utilize. This decision was driven by the fact that a considerable number of students end up utilizing a large number of treatment sessions which prevents other students from accessing mental health services. While such a policy has the potential to reduce the load on CAPS and increase access to care, it is still not clear how such a change will impact the overall performance. Moreover, no framework currently exists to help CAPS facilities determine an appropriate maximum number of sessions. Due to the ever-growing complexities of healthcare systems, quantitative analytical methods that attempt to enhance services and reduce costs have been receiving significant attention [25]. An example is optimization which is one of the most commonly used operations research tools in healthcare settings [26]. While the field is rich with a vast amount of literature, studies that focus on counseling centers in universities are scarce. For instance, Began and Queyranne [27] discuss an efficient method to solve a single-server appointment scheduling problem. Their objective is to design an optimal schedule for a given sequence of jobs with the assumption that job durations are integer random values. Denton and Gupta [28] also consider a single-server scheduling problem using a two-stage stochastic linear program, solving it using a modified L-shaped algorithm, with the objective of minimizing waiting time and service overtime. Although these optimization models are robust and provide a mathematical basis for scheduling policies, they exhibit difficulty capturing the complexities of most healthcare settings. For instance, analytical models are often limited to Exponential or Erlang service times [29]. In addition, as seen in [27,28], such optimization models are often formulated for a single server for simplicity. While multi-server approaches have been considered (e.g., [30,31,32,33]), existing studies often impose the unrealistic assumption of identical servers since the incorporation of nonidentical servers leads to analytical intractability. Another commonly utilized analytical approach in healthcare settings is discrete-event simulation (DES). DES is a technique used to study discrete-event dynamic systems by generating "sample paths" that mimic a system's behavior [34]. A major advantage that DES models offer is the ability to capture real-life complexities that healthcare systems exhibit (e.g., patient cancellations and no-shows, dynamic schedule adjustments, etc.). Such complicating factors, which are often neglected in optimization models for tractability reasons, can play a crucial role in governing the performance of the system and hence should not be ignored. In addition, a DES model is a valuable tool that allows conducting various what-if experiments, especially when real-life implementation are expensive or not possible to conduct. Thus, DES models are a powerful tool to study scheduling problems in complex, highly stochastic, systems such as our setting. Although numerous DES models have been developed for various healthcare settings [34,35,36,37], simulations of counseling centers are not abundantly found in the literature, leaving a critical gap. In this paper, given the aforementioned distinct advantages of simulation models, we build a DES model to study and analyze counseling centers' performance. The main objective is to provide a quantitative framework to help answer fundamental questions about CAPS's operations so as to propose data-driven solutions and recommendations that improve system performance. By utilizing this DES model, this paper aims to address three key research questions: (i) what is the effect of external referrals on the performance of the system and what proportion of external referrals is needed to attain desirable performance? (ii) what is the impact of a maximum session policy on the system performance and what upper bound provides good performance? (iii) how does the schedule topology impact system performance and what topologies lead to favorable performance? Addressing these research questions is of importance as doing so can result in significant performance improvement. This, in turn, will positively impact students by increasing access to critical mental health care. To measure system performance, the existing literature on simulation models that tackle healthcare related scheduling operations often use waiting time as the Key Performance Indicator (KPI), e.g., [38,39,40,41]. In this study, we follow suit but use two KPIs which are the average wait time for first appointment (access time) and the average wait time of crisis patients (referred to as crisis time). These KPIs were chosen based on lengthy discussions with our collaborator Texas A&M (TAMU) Counseling & Psychological Services. In fact, our close collaboration with CAPS provides domain expertise and real-life data of patients' arrival and service times, attended sessions, and information on cancellation and no-show. This is of great value as it enabled us to construct and validate a simulation model that is realistic [42]. In summary, the contributions of this paper are two-fold: First, to the best of our knowledge, this paper is the first to present a DES model that is specifically tailored to CAPS. The model serves as a quantitative platform to help CAPS facilities identify ways to improve system performance. The simulation model will be publicly available for other counseling centers to benefit from it. Second, we conduct a case study using data specific to TAMU CAPS. Our simulation results provide data-driven insights on the impact of external referrals, maximum session policy, and scheduling topology on system performance. The remainder of this paper is organized as follows: Section 2 provides an overview of the scheduling system implemented at CAPS centers. Then, Section 3 presents a high-level description of the operational flow at CAPS centers. Section 4 details the different elements of the discreteevent simulation model and their implementation. Section 5 discusses findings and results from a series of numerical experiments conducted on TAMU CAPS. Lastly, Section 6 concludes the paper and provides future research directions. Overview of CAPS Scheduling System College counseling centers are equipped with a wide range of employees, including clinical staff, trainees, and supporting staff. The clinical staff (hereafter collectively referred to as counselors) includes licensed psychologists and psychiatrists who provide counseling, assessment, and treatment for behavioral, emotional, and mental disorders. While all counselors are trained as generalists, Figure 1: Example of a schedule topology for two counselors across two days. and last service type is referred to as "other" which includes a host of activities (e.g., organizing and conducting workshops, service activities) that counselors are responsible for. The allocation of the counselors' time across these four service types plays a key factor in characterizing the overall performance of the system. In an effort to have a well-planned semester cycle, it is typical for CAPS directors to construct and commit to a master schedule plan at the beginning of each semester. This plan outlines, for each counselor, a distribution of time commitment for each of the aforementioned four service types that spans the entire semester. This distribution is referred to as a schedule topology. Figure 1 provides an example of a schedule topology for two counselors across two days. In this example, a day is made up of eight slots each of which is allocated to one of the four service types. For example, counselor 1 has a higher emphasis for serving ongoing patients with slots 1, 4, 7, and 8 of day 1 and slots 3 and 7 of day 2 allocated to serving ongoing patients. Counselor 2, on the other hand, has a higher commitment to the "other" service type with slots 1, 3, 5, and 6 of day 1 and slots 2, 4, 6, and 8 of day 2 all allocated to "other". Establishing such a schedule topology at the beginning of each semester cycle is of great importance to CAPS directors because it provides three distinct advantages: First, doing so leads to a structured, and transparent, mechanism that greatly facilitates the effective management of a large number of counselors. Second, it allows directors to craft schedules that take advantage of specific strengths of counselors. For example, if a specific counselor has been trained to effectively handle crisis patients, then the director may lean towards a schedule topology that allocates more "crisis" service types to that counselor. Third, a schedule topology provides directors a high-level picture of all operations which allows them to devise schedules with sought after specifications. For example, a schedule topology can be used to ensure a fair and equitable distribution of workload among counselors. Alternatively, a schedule topology can be used to make sure certain thresholds (with respect to time committed to each service type) are met. For instance, a director may want to ensure that at least 30% of a counselor's time is committed to serving first-time patients. Once a topology is committed to, arriving patients throughout the semester are scheduled in a manner that follows the selected topology. For example, if a first-time patient enters the system, then they can only be scheduled to "first-time" slots (i.e., time slots that have been allocated to handle first-time patients). Clearly, the schedule topology will play a major role in determining the overall performance of the system (measured through the two KPIs discussed in Section 1). For example, a schedule topology that does not commit enough time to vulnerable crisis patients will inevitably lead to high average wait times for crisis patients. This is undesirable as it can potentially lead to detrimental consequences such as the patient being unsafe to themselves or to people around them. Similarly, if the schedule topology does not dedicate enough first-time slots, then this will increase the average access time (i.e., the wait time for a first triage appointment) of patients. Again, this is undesirable as it might deter students from accessing critical mental health services. As such, determining an appropriate schedule topology that provides favorable system performance is of utmost importance. However, identifying such a topology is extremely challenging as the decision needs to be made under high levels of uncertainty. This is the case because the schedule topology is set up at the beginning of the semester prior to realizing any of the patient arrivals. Part of the objective of this paper is to provide a simulation-based mechanism to quantify the performance of a given topology, especially ones that take advantage of certain trends that are specific to CAPS. For example, owing to the cyclical nature of demand, a good performing topology is expected to contain a high number of slots dedicated to first-time patients right before final exams. Similarly, during major holidays, a good performing topology might dedicate more slots to the service type "other". Such a tool is extremely valuable to CAPS directors as it can assist them in identifying good performing topologies that are specifically tailored to their needs. Overview of CAPS Operational Flow This section aims to provide a high-level overview of the operational flow at CAPS and the sequence of processes (referred to as paths) that patients go through when entering the system. A patient's path is heavily governed by its type (i.e., first-time, ongoing, crisis). In general, there are two ways by which patients can enter the system: First, if the arrival is a first-time patient, then students must book a first-time appointment by either using an online portal or, if they need additional assistance, by visiting CAPS in-person. While there are some operational differences between these two types of booking approaches (e.g., an in-person first-time appointment booking requires students to complete a survey-based assessment report), the differences do not involve counselors' time and do not impact the two considered KPIs. Consequently, in this study we do not differentiate between these two first-time appointment booking methods. Note that the time between a patient's request for a first-time appointment and the actual appointment time is the access time (which is one of our two KPIs). Second, if the arrival is a crisis patient, then the process is simpler as no appointments are required and the patient can simply walk in during service hours and request a crisis meeting. Given the urgency of the situation, it is key for CAPS to provide the service as soon as possible. Of course, to be able to achieve such a service level, the schedule topology discussed in Section 2 must allocate sufficient slots for crisis patients. This, however, comes at a cost of having fewer first-time and ongoing slots which negatively impacts access time. Such tradeoffs between the two KPIs further highlight the difficulty in identifying good performing topologies. For both system entry types, counselors are selected in a probabilistic manner based on their availability; however, preferences are given to counselors that are available earlier. Upon completing the first session (whether as a first-time or crisis patient), counselors determine the best course of action for their patients. Consequently, the subsequent path that students end up going through heavily depends on their specific characteristics. In fact, the set of all possible patient paths is quite diverse. For example, one of the most common paths is based on a traditional one-on-one treatment cycle. In such a path, the patient is assigned to a specific counselor that they see on a periodic basis (e.g., weekly). The frequency of the meetings depends on several factors including, but not limited to, patient needs, counselor availability, and scheduling restrictions. The student continues attending one-on-one sessions until the counselor determines that the patient no longer needs treatment. The number of sessions that students end up attending can thus vary greatly; however, with the use of historical record data, it is possible to identify an appropriate distribution that replicates the current behavior. Replicating the current treatment behavior brings us back to the fact that this paper focuses on the resource-level challenges facing CAPS. That is, the aim is to analyze the system in a manner that does not impact the current strategy for deciding on treatment plans. While the structure of treatment plans certainly plays a major role in characterizing the performance of the system, such patient-level challenges are beyond the scope of the present paper. Another example of a student path involves psychiatric treatment in which a student is assigned (in addition to a counselor) to a psychiatrist. In such a path, students end up attending two independent streams of sessions (one with a counselor and another with a psychiatrist). While such paths are much rarer than the aforementioned conventional one-on-one path, it is still important to model them as the number of available psychiatrists at CAPS is often severely limited (owing to the high costs associated with hiring licensed psychiatrists). In both of these paths, the type of the patient will transition from "first-time" to "ongoing". This transition is not always guaranteed to occur for all paths (e.g., patient paths that involve referring the student to off-campus mental health providers). While it is not possible to concisely go over all possible patient paths, numerous lengthy discussions with TAMU CAPS enabled us to comprehensively identify and factor in all possible paths that students can take. from the left, patients arrive and enter the system. If the arriving patient type is "first-time", then an appointment for the first-time visit is booked and, when the appointment time is attained, the patient proceeds to CAPS. As shown in the figure, the difference between the request and appointment times is the access time. If, on the other hand, the arriving patient type is "crisis", then the patient directly proceeds to CAPS and the time needed to serve the patient is the crisis time. Once the patient arrives at CAPS they will be seen by a counselor and at the end of session it will be decided whether the treatment for the patient needs to be continued or not. If yes, the patient is sent to the "scheduler" where new appointments are booked based on the patient type "ongoing". The patient will then wait until the selected appointment time is attained, after which they will head back to CAPS. Alternatively, if it is deemed by the counselor at the end of the session that no further treatment is required (or if the student is referred to off-campus mental health providers), the patient proceeds to exiting the system. The operational flow at CAPS is subject to high levels of uncertainties and variations that arise from numerous sources. For instance, as previously discussed, the paths that students end up going through varies substantially by a number of random and uncontrollable factors. These include the patient type, the specific characteristics and needs of the patient, and the counselors' decision for determining an appropriate treatment plan. In addition to the variations in the patients' paths, several other sources of randomness exist. For example, students might cancel their appointments. In such a case, the time slot that was previously booked needs to be freed up. It is also possible for students not to show up to an appointment (referred to as no-shows). In this case it is difficult to make the time slot available for immediate booking. Therefore, in case of no-shows, time slots are often reallocated to serve crisis patients. Other examples of uncertainty include time-varying arrival rates (recall the cyclical demand pattern), counselor downtime, and random services times. These uncertainties, when compounded, result in a highly stochastic system that is challenging to analyze and understand. However, DES models are particularly suited to handle such systems and this was one of the primary motivations for us to consider such models in the first place. 1 1 0 3 0 2 0 0 2 0 3 2 0 3 0 4 1 0 2 0 5 0 0 3 0 6 2 0 3 0 7 1 0 2 0 8 1 0 1 0 2 9 0 0 2 0 10 2 0 3 0 11 1 0 2 0 12 0 0 3 0 13 2 0 1 0 14 0 0 3 0 15 1 0 2 0 16 2 0 3 0 † "First-time"=0, " Ongoing"=1, "Crisis"=2, and "Other"=3. ‡ "No"=0 and "Yes"=1. The Simulation Model The discrete-event simulation model was implemented in Simio ® which is a popular commercial simulation software that has a set of expansive features that allow for modeling complex simulation processes [43]. Our software choice is motivated by three factors: First, Simio provides several layers of customization which facilitates the incorporation of all the necessary factors into the simulation. In particular, the "Add-on Processes" feature available in Simio provides users the flexibility to add custom logic specific to the application at hand. Second, Simio has a built-in feature that allows it to read from, and dynamically modify, external files while the simulation is running. As will become clear, this feature is crucial for the successful implementation of the schedule topology discussed in Section 2. Lastly, Simio provides a relatively simple drag-and-drop user-friendly interface. This ease of use not only facilitates the process of constructing the simulation, but it also allows nondomain experts (e.g., CAPS directors) to more easily visualize, interpret, and use the simulation model. The latter is of particular importance because it provides CAPS with a long-term and sustainable tool to effectively manage their operations. Recall that part of the objective of the simulation is to quantify the performance of the system for a given schedule topology. Given this, we first discuss how a schedule topology is defined and embedded within the Simio framework. We characterize a schedule topology by an external excel file that stores all the necessary information in an easily readable tabular format. Table 1 provides an example of this tabular format for the schedule topology presented in Figure 1. Each row of the table represents a time slot. For example, the first 8 rows of the table represent the 8 time slots of day 1 while the next 8 rows represent the 8 time slots of day 2. The first two columns of the table provide information regarding the day number and slot number. The remaining columns are partitioned based on the counselors with each counselor having two columns: "Service type" and "Booked?". The "Service type" column provides information about the specific type of service allocated to that particular counselor and time slot. The values 0, 1, 2, and 3 respectively correspond to the service types "first-time", "ongoing", "crisis", and "other". For example, looking at row 1 of the table (i.e., the first time slot of day 1), counselor 1 has a value of 1 (i.e., "ongoing") while counselor 2 has a value of 3 (i.e., "other"). The column "Booked?", on the other hand, provides information regarding the current availability of that particular counselor and time slot. A value of 0 represents a situation in which the slot is not booked (i.e., it is available for booking) while a value of 1 indicates that this slot has already been booked and hence is not available for future patients. Notice that the example provided in Table 1 has all of the "Booked?" columns set to zero for all counselors. This particular instance, which represents a situation in which all counselors are available, is typically used as the initial table in the simulation. This is the case because at the beginning of a semester cycle (prior to realizing any of the patient arrivals) counselors have not yet been booked and hence are available. The considered approach, however, is flexible enough to consider other initial tables where certain counselor/slot pairs are already booked. Such situations may arise, for instance, when treatment cycles of patients span across multiple semesters. The external excel file discussed in the previous paragraph plays a pivotal role in governing the overall performance of the simulation. In addition to the schedule topology, the file also embeds critical information regarding the current scheduling status of the system. As the simulation progresses, this external file needs to be dynamically updated to reflect recent bookings and modifications. For example, if a crisis patient arrives at time 0, then (based on the schedule provided in Table 1) the nearest available crisis slot is given by slot 2 of counselor 2. In such a case, a reservation will be made for the student to attend that session and the value of the corresponding "Booked?" column must be updated from 0 to 1. Doing such a change on-the-fly is critical as it prevents future patients from accessing the already reserved slot. Another example is when a student cancels an appointment. Here, the value in the column "Booked?" must be updated from 1 to 0 to make the slot available for future patients. Since topology information is embedded within the tabular form, all resulting scheduling processes are guaranteed to adhere to the selected schedule topology. It is worth highlighting that in certain circumstances the column "Service type" may be dynamically updated as well. For example, if a no-show occurs, then the service type of the corresponding slot is modified to 2 to make the slot available to crisis patients. While this inherently modifies the selected topology, these modifications may be set into place by CAPS directors. Such specific complexities are extremely difficult to consider using other modeling approaches (e.g., optimization), which further motivates the use of a simulation-based model. Having discussed how a schedule topology is defined and embedded within the Simio framework, we now turn our attention to simulating the operational flow of CAPS (discussed in Section 3) for a given schedule topology. Primarily, the simulation model has two main elements: Counselors and patients. Counselors are modeled as servers while patients are represented as discrete entities that enter the system seeking service. The general sequence of events is as follows: Patients enter the system according to a non-stationary process that is obtained from historical record data. This arrival process will embed critical cyclical demand patterns that arise in practice. Arriving patients are then categorized to two types: Regular and crisis. Regular patients are patients that do not encounter any crisis attacks throughout their treatment cycle. In contrast, crisis patients are patients that will encounter at least one crisis attack during their course at CAPS. The crisis attacks can either occur at the beginning of their treatment cycle (i.e., they enter the system via a crisis session) or they can occur in between one-on-one sessions. Depending on availability, whenever a patient experiences a crisis attack, the counselor assigned may be different from any previous crisis counselor they have visited, or their regular counselor with which they have periodic one-on-one sessions. All of these crisis attacks will contribute to the overall average waiting time of crisis patients. The patients' type will heavily impact the path that they will end up going through (as observed from historical record data), which is why these two categories of patients were considered in the first place. The proportion of regular and crisis patients is determined from historical data. Upon arrival, each patient entity is assigned certain key features depending on their type. For example, some features for regular patients include the total number of sessions, and no-show and cancellation probabilities. Crisis patients, on the other hand, are assigned additional features such as crisis attack probabilities. Differentiating patients allows us to customize the features based on their type. For instance, an analysis of the data reveals that crisis patients end up requiring more sessions than regular patients. Also, the no-show and cancellation probabilities between the two types of patients are different. These features, which are not necessarily deterministic, will be obtained by conducting an input analysis of the data. For instance, for the total number of sessions of regular patients, an input analysis on a subset of the data that only includes regular patients will generate a probability mass function. This distribution can then be used generate random realizations for the total number of sessions which are then assigned to newly arriving regular patients. Note that this procedure is essentially replicating the current treatment behavior at CAPS. Again, this brings us back to the fact that this paper focuses on the resource-level challenges facing CAPS with the aim of analyzing the system in a manner that does not impact the current strategy for deciding on treatment plans. Collectively, the features that are assigned to a patient will heavily govern the path that the patient ends up going through. Once the initial set of features has been assigned for each patient, the patient enters a scheduler station in which they will be assigned the time and counselor for their first-time session. This assignment process heavily depends on the schedule topology and the current availability of the counselors, which are recorded in the external file discussed at the beginning of this section. To achieve the assignment, we first note that the service type of a triage session can either be "firsttime" or "crisis", and the scheduling process is different between the two. In particular, since a crisis is an emergency which needs urgent attention, patients who require a crisis session (which are patients that show up to CAPS without any prior reservation) are allotted the earliest available crisis time slot among all the counselors. In contrast, patients seeking non-crisis sessions (which must occur through a reservation), might be inclined to meet a specific counselor for their triage session, who might not necessarily have the earliest available slot among all counselors. To incorporate such patient-specific preferences into the simulation, we construct a custom probability distribution for the patient to choose an appropriate counselor based on the earliest availability of all counselors at the time of scheduling. The structure of the distribution is based on the fact that patients will have a higher preference for counselors with earlier availability, and thus the custom distribution assigns a higher probability to such counselors. Once the first-time session is booked, the external excel file is updated to reflect this change and the patient waits until the assigned session time and then visits their assigned counselor. At the end of the session, a number of checks are performed which are based on the features that have been assigned to the patient. For example, if (according to the features) additional sessions are not required, then the patient will exit the system. Similarly, if it is deemed that the patient will be referred to an off-campus mental health provider, then the patient will exit the system. This referral decision is probabilistic and depends on the patient type. Alternatively, if additional sessions are required, then the patient is sent back to the scheduler to determine an appropriate time-slot for their next session. The appointment time of the next session depends on several factors such as counselor preferences, patient needs, and schedule availability. For example, based on an analysis of the data, the vast majority of counselors prefer to see patients on a weekly or bi-weekly basis; however, the data also reveals significant deviations from this behavior. To accurately emulate this random behavior we consider a two-step approach: First, a random number is generated that represents when (in weeks) the patient should ideally be seen. For example, a counselor might decide to want to see the patient one week from now. This probability distribution is determined from historical data (see, for example, Figure 5 in Section 5.2). Generally, a decreasing trend is often observed, that is, more counselors opt to see patients sooner rather than later. Owing to schedule availability, however, a counselor's ideal choice may not be feasible. For instance, it might not be possible to schedule an appointment exactly one week from now. Consequently, in the second step, we run a procedure that searches the schedule for available time slots. This search procedure starts on the counselor's ideal day (which was defined in the first step), and if an available time slot (with an appropriate service type) is found then a reservation is made. If, on the other hand, an available time slot is not found on the counselor's ideal day, the search algorithm attempts to identify slots "around" the ideal day. This is achieved through a sequential procedure in which the algorithm searches for available time slots within one day, two days, etc. This is repeated until a time slot is identified. Note that a consequence of this procedure is that the final reserved time slot might substantially deviate from a counselor's ideal time. This behavior is consistent with the data (e.g., in Figure 5 the bars around the weekly peaks represent such deviations). Once the next session is scheduled, the external excel file is updated to reflect this change and the process repeats (i.e., the patient waits until the assigned session time and then visits their assigned counselor). Case Study: TAMU CAPS In this section, we use the established DES model to conduct a case study on the specific operations of Texas A&M's counseling Center (TAMU CAPS). In Fall 2021, TAMU reported a total student enrollment of 72,982 (92% of which enrolled in the main campus in College Station) making it one of the largest universities (by enrollment) in the entire nation [44]. Owing to the large student population, TAMU CAPS has been particularly affected by the unprecedented surge in demand for counseling services. This, in turn, makes TAMU CAPS an ideal candidate for such an analysis as it is in need of identifying strategies that increase access to mental health services. The main objective of this study is to utilize the DES model to quantify the impact of certain key parameters on the overall operations of CAPS. In particular, we perform a series of experiments to investigate the impact, on the KPIs defined in Section 1, of: (i) varying the proportion of external referrals, (ii) imposing a maximum session limit, and (iii) the structure of the schedule topology. In what follows, we first provide a detailed description of the data sources used in this study (Section 5.1). Next, we discuss the main input parameters used in the DES model and describe the methods used to obtain these parameters from the data sources (Section 5.2). We then validate our DES model by running an experiment with the current operating parameters and comparing its performance to the reported performance of TAMU CAPS (Section 5.3). Finally, we run the aforementioned series of simulation experiments with varying operating parameters and perform sensitivity analysis for each of our KPIs (Section 5.4). on the overall performance of the system. In terms of data, the analysis in this section utilizes two main datasets both of which provided by TAMU CAPS. The first dataset provides detailed patient-level record data while the second provides information regarding counselor scheduling. We provide a detailed description of the structure of each of these datasets below. Time Horizon and Data Description The patient-level record dataset provided by TAMU CAPS contains anonymized information about each session that has been scheduled at CAPS across the entire semester. Each row corresponds to a unique session for a patient and contains information such as the patient ID (a unique identifier for each patient), the session type (e.g., crisis, ongoing, or other), whether a cancellation or no-show occurred, the name of the assigned counselor, and the session date and time. This dataset is an integral component of our study since it provides detailed session-level information, and most of the input parameters for the simulation is obtained by analyzing this dataset. Before performing any analysis, however, the data was carefully reviewed in order to remove any potential anomalies such as technical issues related to data collection. For instance, there were several duplicate rows in the dataset which needed to be removed. There were also some instances where a patient had two ongoing sessions on the same day at different points of time. After lengthy discussions with TAMU CAPS, it was confirmed that these instances were due to technical errors, and were therefore removed from the dataset. Once the data was cleaned, we augmented it with some extra features to facilitate the analysis. For instance, we added a Boolean feature that identifies whether the session for each row corresponds to a crisis or a regular patient. Similarly, for each session/patient pair, we added a categorical feature that provides the session number that the patient is attending (i.e., first session, second session, etc.). The resulting dataset can then be analyzed to obtain the various input parameters of our DES model. The details of obtaining these parameters and their usage in the model are discussed in Section 5.2. The counselor schedule dataset contains information about the number of sessions allocated to each service type by each of CAPS' counselors. This dataset contains a total of 36 rows, one for each unique counselor at TAMU CAPS. Each column of this dataset provides information on the total time per week that the counselors are expected to allot for each unique service. Recall that other than direct counseling of patients via first-time, ongoing, or crisis sessions, counselors also need to allocate time for other services such as organizing workshops, intern supervision, student documentation, etc. For the purpose of this study, these time slots are allotted to the "other" service type as discussed in Section 2. It is important to note that the information provided in this dataset does not characterize a unique schedule topology. However, it provides an idea of the proportion of time that each counselor spends for each service category across the entire semester. In order to get a realistic representation of the schedule topology based on this dataset, we use the proportions of each service category in the schedule to generate random realizations of schedule topologies. In our experiment in Sections 5.3 and 5.4, we perform several replications of the simulation, each with a different random topology realization, and calculate the average estimates of the KPIs across the replications for a specific proportion distribution. Such an analysis is desirable because implementing a specific topology may be challenging. Instead, we aim to provide recommendations on the proportion of time that each counselor should ideally dedicate for each service type. This can be used as a master plan by CAPS and can potentially act as a guideline for the counselors to plan their schedules for the semester. Input Analysis Input analysis is an important component of discrete-event simulation models as the simulation output heavily depends on the input. If a model's input parameters (e.g., arrival rates, no-show and cancellations probabilities, etc.) do not accurately represent real-life behaviors, then the output of a simulation could be unreliable even if the simulation logic is accurate. Therefore, the main purpose of input analysis is to theorize probability distributions that correctly describe the real-life system parameters. In this section, we describe the process of obtaining these input parameters from the previously mentioned patient-level record data. Figure 3 which plots the number of new patient arrivals as a function of time. The data clearly reveals an interesting pattern: As the semester starts, a rush of requests is observed. This demand gradually reduces with a sudden drop right before Thanksgiving break (shaded blue region). The demand then picks up right before final exams (shaded green region). This pattern is consistently observed across semesters (e.g, for the spring semester a similar trend is observed however the drop in demand is observed right before Spring Break). Recall that part of the objective of this analysis is to take advantage of such predictable cyclical demand patterns. To obtain the per hour arrival rates from this data, we divide the total number of arrivals for each week with the total number of work hours per week (set to 40). As common in the literature, we assume that arrivals follow a non-stationary poison process with arrival rates obtained from Figure 3. It is important to note that the above arrival rate takes into account arrivals of both regular and crisis patients. Consequently, determining whether the arriving patient is a crisis or a regular patient is a core element in modeling the arrival process and the simulation. This is because upon arrival, the subsequent processes that patients undergo are dependent on whether they are a regular or a crisis patient. In order to model the differences in arrival rates for regular and crisis patients, we adopt a Poisson split. This is a commonly used technique that allows for splitting a Poisson process into two separate Poisson processes, each with a different arrival rate. This split is based on the proportions of different entities that are expected to arrive in the simulation. For this case study, we use the proportion of crisis and regular patients as the proportions for the Poisson split. Using the patient-level record data, the ratio of crisis patients, obtained by dividing the number of patients who received at least one crisis session by the total number of patients, is equal to 9.6%. As a result, the ratio of regular patients is equal to 90.4%. Thus, whenever an arrival occurs, there is a 9.6% (90.4%) chance that the patient is a crisis (regular) patient. Upon arrival, several important initial features are allotted to each patient. One of the most Another important initial feature that is specific to crisis patients is the crisis attack probability. Recall that crisis patients may arrive into the CAPS system during an emergency crisis attack. However, the patient-level record data also reveals that crisis attacks can occur at any point in between sessions. This aspect is modeled into the simulation using the crisis attack probability. At the end of a session, and once the next appointment has been decided upon by the counselor and the patient, crisis patients have a probability (characterized by the crisis attack probability) of experiencing a crisis attack before the next scheduled session. This probability is estimated from the patient-level record dataset filtered on only crisis patients. Then, for each crisis patient, a crisis attack probability is obtained by dividing the number of crisis sessions that occurred between two sessions by the total number of gaps between every two sessions. The overall crisis attack probability is then calculated by taking the average of all individual probabilities, and is found to be 14.6%. That is, after the end of a session a crisis patient has a 14.6% probability of experiencing a crisis attack before their next scheduled session. Table 2. Observe that the no-show and cancellation probabilities are lower for crisis patients in comparison to regular patients, which highlights that crisis patients have a higher inclination to attend their counseling sessions. It is also observed that the exit probabilities are notably lower for crisis patients, which alludes to the fact that crisis patients have a higher tendency to complete their full course of sessions at CAPS in comparison to regular patients. At the end of each session, the counselor and the patient need to decide on a date and time for their next meeting (this excludes situations in which the patient is done with their total number of sessions). The time between sessions is typically a function of the counselors expert opinion on session frequency (which might be case specific), as well as schedule availability. In fact, an analysis of the number of days between sessions performed on the patient-level record dataset reveals some interesting features (see Figure 5 which provides the empirical distribution of the time between sessions) that can be explained on the basis of these two factors. Observe that the figure clearly depicts three distinct spikes corresponding to 7, 14, and 21 days, which highlights the fact that counselors conventionally prefer meeting on a weekly, bi-weekly, or tri-weekly basis. We suspect that this is based the counselor's expert opinion on when then next session should ideally occur. Another key observation is that the proportion of cases of 7 days is much more common than cases for 14 or 21 days. This indicates that counselors often see the need to schedule weekly appointments. Finally, note that the lower proportions around the peaks can be explained by the second factor, i.e., the inability (due to slot availability) to find a time on exactly 7, 14, or 21 days after their session. As a result, based on schedule availability, the two parties might agree on a day in and around the initially estimated preferred date. This is an important feature of the operations at CAPS that needs to be intricately modeled into the simulation. To achieve this, while determining the date for the next session, we first pick an initial preferred date based on the probabilities of the three peaks. If that exact date is unavailable (i.e., no ongoing sessions slots are available), we start searching for available slots around that day until one is found. Doing so incorporates both of the aforementioned factors when deciding the next patient appointment time. Model Validation An important step before conducting the simulation experiments is to validate the model. That is, to determine whether the simulation is accurately representing the real-world performance. To achieve this, we run the simulation model using the Fall 2019 datasets and compare the system performance measured from the output of the simulation model to the reported Fall 2019 performance of the system. We mainly focus on the access time as it is the main performance metric reported by CAPS. To run this experiment, we first need to define the current schedule topology being information about the proportion of time that each counselor spends for each service type across the entire semester. In this study, in an effort to capture the wide spectrum of topologies that adhere to these proportions, we utilize the information provide by the counselor schedule dataset to generate representative random topology realizations. For each of these realizations, the simulation is executed and the expected access time is stored. To obtain the current proportions from the counselor schedule dataset, we sum the total number of hours dedicated to each service type for all of the counselors across the semester. Then, assuming a 40 hour work week and a time span of 18 weeks, we divide the sum of time allocated for each service type by the total work time across the semester. Doing so reveals that, on average, counselors spend 8% of their time on first-time sessions, 4% on crisis sessions, 25% on ongoing sessions, and the remaining 63% of the time is dedicated for "other" services. To generate a topology realization that adheres to these proportions, we execute a pre-processing procedure at the very beginning of the simulation on the external excel file discussed in Section 4. This procedure enumerates over all rows of the file and assigns, for each counselor, a random service type ID that is generated based on the distribution obtained from the counselor schedule dataset. Specifically, each time slot has an 8% probability of being assigned a first-time service type, 25% probability of being assigned an ongoing service type, 4% probability of being assigned a crisis slot, and a 63% probability of being assigned a "other" service type. The resulting schedule topology ensures that, on average, the proportion of time committed to each service type by each counselor matches that of the counselor schedule dataset. Once a schedule topology is realized, the simulation is executed and the expected access time is recorded. This entire procedure is repeated a number of times to generate a distribution of expected access times. Note that while the overall proportions across iterations are preserved, the specific schedule topology can vary. Such an experiment can provide valuable information about the variability of the system performance for a fixed proportion distribution. We run the aforementioned procedure for a total of 32 replications. Note that each replication represents a simulation run across the entire semester for a randomly generated schedule topology. the simulation is accurately describing the real-world performance of the system. Notice, however, that the reported 7.8 days does not coincide with the simulation confidence interval (represented by the shaded blue region). That is, the simulation is slightly overestimating the expected access time. This behavior is somewhat expected as the simulation experiment was executed for random schedule topologies. In practice, however, CAPS directors put effort in designing good performing topologies which are expected to outperform randomly generated topologies. It is interesting to note that TAMU CAPS is doing a good job in identifying good performing schedule topologies with only 31% of the randomly generated topologies outperforming the current implementation. However, based on our experiments, it is possible to further improve the performance of the current implementation by up to 8%. Simulation Experiments and Sensitivity Analysis Having ensured that the simulation model is accurately depicting the real-world system perform, in this section we perform a series of numerical experiments. As discussed at the beginning of Section 5 we conduct three experiments to investigate the impact of: (i) varying the proportion of external referrals, (ii) imposing a maximum session limit, and (iii) the structure of the schedule topology. These three experiments are outlined below. In the first experiment, we vary the proportion of external referrals from CAPS and observe its referrals without modifying the schedule topology will not have the desired impact on the overall performance of the system. For the next set of experiments we perform a sensitivity analysis on the maximum number of sessions allowed per patient, which is a concept that was introduced in Section 1. Such a policy is introduced in various CAPS systems to increase access to newer patients. In our model, this policy is implemented in the following manner: We generate a random number of total sessions from the distribution described in Section 5.2, and then select the minimum value among the generated number and the maximum session limit as the number of sessions to be allotted to the patient. Next, for our experimental setup, we consider a range of values for the maximum session limit Table 4, which reports the mean values of our two KPIs as well as the half-width (HW) of a 95% confidence interval in parenthesis. Observe that changing the maximum limit does not lead to any statistically significant changes in the expected access time. This is more clearly demonstrated in Figure 8a. Although this can seem anomalous, some insight into the operating scheme of TAMU CAPS makes the observation quite intuitive. This is because the access time is the time taken from the patient's request for a first-time session to the appointment time. The primary factor that impacts this time is the proportion of first-time slots available in the schedule topology. Since the schedule topology remains the same across the cases on average, merely changing the maximum session limit does not influence the access time for patients. More interestingly, the crisis time is increasing with the maximum limit. Again, this is more clearly demonstrated in Figure 8b. This can be explained by the fact that as patients tend to attend a higher number of sessions, it increases the probability of having crisis attacks. Clients attending more sessions (which is a consequence of a higher maximum limit) will inevitably have a higher probability of experiencing a crisis attack (see Section 5.2). Since the proportion distribution is preserved in this experiment, then the total number of crisis slots in the schedule remains the same. This increase in demand for fixed number of crisis slots ultimately leads to higher crisis The first two experiments revealed that, for a fixed proportion distribution, the expected access time is not impacted when varying the proportion of external referrals and the maximum session limit. To observe the impact of the proportion distribution on the two KPIs, we run a two-dimensional sensitivity analysis on the proportions of first-time slots (p f ) and crisis slots (p c ). We vary these values while fixing the proportion of slots dedicated to the "other" category. Consequently, the change in p f and p c will ultimately impact the proportion of ongoing slots (to preserve a total proportion of 1). Following the approach discussed in Section 5.3, we generate a random topology for a given pair (p f , p c ) and run the simulation for the entire semester. This process is repeated for a total of 25 replications. We conduct this analysis for a range of p f values in {5%, 7%, 9%, 11%, 13%, 15%, 17%} and p c values in {1%, 3%, 5%, 7%, 9%}. The results are summarized in Table 5 which reports the expected access and crisis times (and the half width in parentheses) for all scenarios. The results reveal several interesting behaviors, discussed below. Looking at the expected access time, the results show that it is heavily dependent on p f with higher proportions leading to lower expected access times. Note that even small variations can lead to large differences. For example, reducing p f from 8% (current implementation) to 7% increases Shifting the focus to the expected crisis time, Table 5 reveals similar trends. In particular, the expected crisis time is decreasing with p c with the greatest reduction observed for lower p c values. This diminishing returns feature is more clearly depicted in Figure 9b. For example, reducing p c from 4% (current implementation) to 3% increase the expected crises time by an average of 130% while increasing p c to 5% improves the expected crisis time by an average of 50%. Also, the results reveal that the expected crisis time does not seem to be correlated with p f . However, when p c is substantially low, a slight positive correlation is observed with p f . For example, when p c = 1% a p f of 5% leads to an expected crisis time of 61.24 hours while a p f of 17% leads to an expected crisis time of 71.06 hours. We suspect that this positive correlation is attributed to the fact that a higher p f value results in lower proportions for ongoing slots. This, in turn, would imply that the time between session would be prolonged (due to the lack of available slots) and in such a case the probability of the patient experiencing a crisis attack would increase. This increase in demand would then result in higher expected crisis times. However, this effect is only observed when the number of crisis slots is extremely low which is often not the case in practice. Again, these results can be used to identify values of p c that result in desirable performance. For example, if the target expected waiting time is 15 minutes, then, based on our analysis, a p c value of approximately 9.3% would be needed to attain this performance. performance. This is especially true when the system is overloaded with patients. Our results can be used by CAPS directors to identify the proportions for each of the service types that lead to the desired system performance. For the specific case of TAMU CAPS, it was identified that a first-time proportion of 12.5% and a crisis proportion of 9.3% would result in a system that meets their desired performance. It is important to note that such proportions will negatively impact the time to next session. This negative effect, however, can be mitigated by either increasing the proportion of external referrals or imposing a maximum session limit. This work can be extended in several important directions. For example, a more rigorous investigation of the composition of optimal schedule topologies could result in even better system performance. This is valuable as our numerical experiments reveal, even for fixed proportions, the substantial impact of the topology on the performance of the system. Part of this analysis would include the investigation of time-varying schedule topologies in which the proportion dedicated to each service type can potentially change over time. This is of particular relevance to college counsel-ing centers because of the predictable cyclical nature of the demand. Another interesting research direction is to enrich the simulation with group-based treatment options. These have been recently introduced by some CAPS facilities to handle the surge in demand. While group-based treatment options have the potential to improve system performance, their addition also results in a number of scheduling challenges. Within this context, the considered simulation model can be used to assess and quantify the impact of such options on the system. Lastly, this paper focuses on addressing the resource-level challenges facing CAPS. Another interesting dimension of the problem is the patient-level challenges such as the effectiveness of treatment plans. This is especially important when considering some of the policies being implemented by CAPS facilities. For example, does imposing a maximum session limit impact the effectiveness of treatments? Are group-based treatment options as effective as individual therapy? Establishing a quantitative framework to address these questions, in conjunction with the considered simulation model, can result in a counseling system with both a desirable system performance and patient outcome. We hope that this work can lead to further research in the aforementioned directions and potentially aid counseling centers in developing data-driven policies. Figure 2 2provides a high-level schematic summary of the operational flow at CAPS. Starting Figure 2 : 2Schematic summary of the operational flow at CAPS. We run the numerical experiment for the Fall 2019 semester which spans across 18 weeks starting from 09/2/2019 up until 12/30/2019. While the semester officially ends a couple of weeks prior to 12/30/2019, we decided to include this time period within our simulation experiment as the data revealed student arrivals during this time period. The Fall 2019 semester was chosen in order to avoid any anomalies in the data during the onset of COVID-19. However, it is possible to run the numerical experiments for later semesters to more rigorously quantify the impact of COVID-19 Figure 3 : 3Number of new patient arrivals per week at TAMU CAPS in Fall 2019. component of the simulation is the arrival rate of patients as it will play a vital role in describing the dynamics of the simulation model. To estimate the arrival rate, we use data on the total number of new patient arrivals on a per week basis. This data is summarized in Figure 4 : 4Distribution of total number of sessions for regular and crisis patients. is the number of sessions that the patient has to attend. Of course, different patients can require a different total number of sessions, which is typically determined by their counselor. In order to allow for this variability in the total number of sessions, we use a probability distribution based on information obtained from the patient-level record dataset. The distributions of the total number of sessions for regular and crisis patients are shown inFigures 4a and 4b,respectively. Although somewhat similar, the two distributions have some unique characteristics which highlights the difference in treatment patterns between regular and crisis patients. For instance, it can be observed that crisis patients are more likely to utilize a higher number of sessions, which emphasizes their need for CAPS' services for a relatively longer period. Another important distinction is that regular patients have a higher probability of requiring just one session, while crisis patients have a higher tendency of requiring more sessions. This can be attributed to the fact that crisis patients arriving into the system during an emergency crisis attack typically schedule a follow-up session with the counselor. Using these distributions, and depending on the arriving patient type, a random number of total sessions is generated for each arrival. Once the initial features have been allotted, patients can book their first-time session and then continue their treatment cycle with the assigned counselor until they meet the total number of sessions that was assigned to them. However, during their course of sessions at CAPS, patients may cancel or not show up for their scheduled session. As mentioned in Section 4, the no-show and cancellation probabilities are two key features assigned to each patient entity upon their arrival into the system. In the same context, there are two other probabilities assigned to each patient:The no-show exit and the cancellation exit probabilities. Our discussions with CAPS, as well as an analysis of the patient-level record data, revealed that in the event of a no-show or a cancellation there are instances where patients do not reschedule their appointment and simply discontinue their visits to CAPS. For instance, the data reveals that roughly 60% of the time regular patients discontinue their visits to CAPS after a no-show, while crisis patients only do so about 40% of the time. This trend is captured in our simulation using two additional features: The no-show exit and cancellation exit probabilities. Once a no-show occurs, the probability that the patient exits the system without continuing their remaining sessions is represented by the no-show exit probability. The cancellation exist probability, on the other hand, represents the probability of the patient exiting the system following a cancellation. These probabilities can vary depending on the patient type, i.e., regular or crisis. To obtain the no-show (or cancellation) probabilities we divide the number of no-shows (or cancellations) by the total number of sessions. Similarly, the no-show exit (or cancellation exit) probabilities are obtained by dividing the number of cases where the patient had a no-show (or cancellation) as their last session by the total number of no-shows (or cancellations). This process is performed separately for regular and crisis patients. All four of the aforementioned probabilities are shown for each patient type in Figure 5 : 5Empirical distribution of the time between sessions. Figure 6 : 6Boxplot for expected access time under the currently implemented TAMU CAPS operating parameters. TAMU CAPS. Unfortunately, as discussed in Section 5.1, the counselor schedule dataset provided by CAPS does not fully characterize a schedule topology. Instead, it provides Figure 6 6shows the box and whisker plot for the expected access time across the 32 replications of our experiment. The Figure reveals a number of interesting behaviors: First, for a fixed proportion distribution, the expected access time can vary quite substantially with values ranging from 7.2 days to as high as 10.5 days. This indicates that, even when the proportion distribution is fixed, the schedule topology can have a significant impact on system performance and a carefully designed schedule topology can improve system performance by up to 46%. Second, the average expected access time is obtained to be 8.5 days (with a half width of 0.4 days), which closely aligns with the actual reported access time of 7.8 days. This observation is important as it indicates that Figure 7 : 7Expected access time (a) and crisis time (b) as a function of the proportion of external referrals. the access time and the crisis time. As mentioned in Section 1, in order to meet a rise in demand, various university counseling centers, including TAMU CAPS, often refer students to external off-campus mental health providers, especially if they require a higher level of specialization.Typically, it is only after a few sessions that the counselor is in a position to judge whether the patient needs to be externally referred. Once that decision has been made and conveyed to the patient, they still attend a few more sessions with their regular counselor at CAPS in order to ensure a smooth transition to their new provider. Based on an analysis of the Fall 2019 patient-level record data, the proportion of patients that were referred to external providers was identified to be 5.6%.Other relevant information, such as the average number of sessions attended prior to a referral as well as the average number of transition sessions after a referral, is also determined from the same dataset and incorporated into the simulation model. The natural question in this regard is whether changing the proportion of external referrals influences the two KPIs of interest, i.e., the access time and the crisis time. We run a series of experiments by varying the external referral proportion from half of its current value, i.e., 2.8% to double, i.e., 11.2%. Other input parameters, such as the proportion distribution of the schedule topology and the maximum session limit is maintained at the current operating values for consistency. The results of this numerical experiment are summarized inTable 3. The results reveal that neither the expected access time nor the expected crisis time are impacted by changes in the proportion of external referrals. This is more clearly depicted inFigure 7which plots the expected value of both KPIs (along with the confidence intervals) as function of the proportion of external referrals. ( 2 2to 10), and run a total of 16 replications for each case to observe its impact on the average values of the two KPIs. The other parameters, such as the topology proportion distribution and the proportion of external referrals, are set to the current TAMU CAPS operating values. The results for this experiment are summarized in Figure 8 : 8Expected access time (a) and crisis time (b) as a function of the maximum session limit. the expected access time by a substantial 50% (on average). Increasing p f to 9%, on the other hand, reduces the expected access time by an average of 26%. The sensitivity of the expected access time to p f , however, exhibits diminishing returns. To better see this, Figure 9a plots the expected access time when p c = 1% as a function of p f . Such an observation is important because increasing the value of p f beyond a certain point may not be worthwhile since it will only lead to a marginal improvement in the system performance. The results also reveal that the expected access time is not impacted by p c . This is somewhat expected since varying the proportion of the topology dedicated to crisis should not impact the access time. Such results can help CAPS directors determine topology proportions that meet their ideal performance. For example, if CAPS aims to have an expected access time of just 3 days, then our results can be used to determine a p f value that will attain this performance (p f ≈ 12.5%). Figure 9 : 9Expected access time when p c = 1% as a function of p f (a) and expected crisis time when p f = 5% as a function of p c (b). paper, we construct a discrete-event simulation model that mimics the operational flow of college counseling centers. The considered model is general and incorporates a number of realistic factors that are often ignored in the literature. This leads to a model that accurately depicts the operations of college counseling centers. To demonstrate the benefits of the simulation model, we use data from TAMU CAPS and perform a series of numerical experiments to investigate the impact of certain factors on the system's performance. Our experiments lead to key observations on the effect of different policy changes on the system-level resources at counseling centers. Firstly, increasing the proportion of external referrals or imposing maximum session limits do not result in the desired impact if the structure of the schedule topology is not considered. This is an important observation as many CAPS facilities have implemented such policies with the hope of improving the performance of the system. Secondly, our experiments on the schedule topology reveals that the proportions dedicated to each of the service types have a significant impact on the system first time. Ongoing patients represent patients that have been previously seen by a counselor and that are now part of an ongoing treatment plan. Crisis patients, on the other hand, are patients that are going through a mental health emergency and that need immediate attention. The fourthDay 1 Day 2 Counselor 1 Counselor 2 Other Crisis First-time Ongoing Slot 1 some have specialty in specific domains (e.g., psychiatric treatment), which makes them particularly suited to handle certain patients. Collectively, counselors represent the majority of the employee- base at CAPS and are the key operators of the facility. Consequently, appropriately managing their time and effort (which is the responsibility of the director) is key to achieve a good performing system that effectively meets student demand. The duties and time commitment of counselors can be broadly categorized into four main components (referred to as service types). The first three service types correspond to serving three different types of patients: (i) first-time, (ii) ongoing, and (iii) crisis. As the name implies, first-time patients are new patients that will be seen for the Table 1 : 1Example of the tabular format for the schedule topology presented inFigure 1.Counselor 1 Counselor 2 Day Slot Service type † Booked? ‡ Service type † Booked? ‡ 1 Table 2 : 2No-show and cancellation-related probabilities for regular and crisis patients.Regular patients Crisis patients Table 3 : 3Expected access and crisis times for varying proportion of external referrals.Proportion of external referrals Access time in days (HW) Crisis time in hours (HW) 2.8% 8.66 (0.32) 1.64 (0.11) 4.2% 8.48 (0.39) 1.65 (0.17) 5.6% 8.47 (0.37) 1.81 (0.20) 7.0% 8.81 (0.35) 1.71 (0.18) 8.4% 8.65 (0.30) 1.70 (0.16) 9.8% 8.60 (0.39) 1.72 (0.21) 11.2% 8.73 (0.38) 1.67 (0.14) This is apparently counter-intuitive since one would expect that external referrals would improve access to new patients. However, note that according to the current TAMU CAPS policy, students are only referred to external providers once they are done with a certain number of sessions. Thus, although external referrals may allow for extra available ongoing slots, they still exhaust valuable first-time slots. As a consequence, it does not impact the access time since even if we increase the referral proportion from 2.8% to 11.2%, it still ends up utilizing the same number of first-time slots on average. A similar observation can be made with regard to the crisis time. Such results demonstrate how solely changing the proportion of external Table 4 : 4Expected access and crisis times for varying maximum session limits.Maximum session limit Expected access time in days (HW) Expected crisis time in hours (HW) 2 8.68 (0.40) 0.97 (0.07) 4 8.74 (0.49) 1.01 (0.09) 6 8.62 (0.43) 1.21 (0.09) 8 8.57 (0.32) 1.41 (0.11) 10 8.47 (0.37) 1.81 (0.20) Table 5 : 5Comparison of expected access time (EAT) and expected crisis time (ECT) with varying proportion of first-time (p f ) and crisis (p c ) sessions. Each cell reports the mean KPI value (in days for EAT and hours for ECT) and the half width for the 95% confidence interval (in parenthesis).p f p c 1% 3% 5% 7% 9% 5% EAT 26.56 (0.67) 26.13 (0.76) 26.81 (0.81) 26.25 (0.80) 26.31 (0.88) ECT 61.24 (3.75) 4.05 (0.53) 0.87 (0.10) 0.47 (0.05) 0.30 (0.03) 7% EAT 12.46 (0.49) 12.75 (0.57) 12.82 (0.67) 12.93 (0.64) 12.92 (0.75) ECT 64.16 (3.76) 4.28 (0.82) 0.89 (0.08) 0.45 (0.04) 0.27 (0.03) 9% EAT 6.25 (0.30) 6.30 (0.29) 6.27 (0.30) 6.2 (0.28) 6.29 (0.33) ECT 66.09 (3.86) 5.05 (1.25) 0.93 (0.07) 0.46 (0.05) 0.26 (0.02) 11% EAT 3.74 (0.13) 3.72 (0.16) 3.79 (0.17) 3.7 (0.17) 3.71 (0.18) ECT 67.82 (3.07) 4.74 (0.82) 0.95 (0.10) 0.44 (0.05) 0.27 (0.02) 13% EAT 2.65 (0.10) 2.56 (0.08) 2.59 (0.09) 2.56 (0.06) 2.52 (0.09) ECT 70.17 (3.09) 3.75 (0.58) 0.94 (0.09) 0.45 (0.05) 0.29 (0.03) 15% EAT 1.96 (0.07) 1.95 (0.05) 1.97 (0.06) 1.93 (0.05) 1.97 (0.06) ECT 69.78 (3.51) 3.84 (0.85) 0.90 (0.08) 0.42 (0.04) 0.28 (0.03) 17% EAT 1.54 (0.04) 1.57 (0.05) 1.58 (0.05) 1.56 (0.05) 1.54 (0.04) ECT 71.06 (3.44) 3.34 (0.58) 0.81 (0.07) 0.42 (0.05) 0.29 (0.03) Mental health in the COVID-19 pandemic. Walter Cullen, Gautam Gulati, Brendan D Kelly, QJM: An International Journal of Medicine. 113Walter Cullen, Gautam Gulati, and Brendan D Kelly. "Mental health in the COVID-19 pandemic". In: QJM: An International Journal of Medicine 113.5 (2020), pp. 311-312. The COVID-19 pandemic and mental health impacts. Kim Usher, Joanne Durkin, Navjot Bhullar, International Journal of Mental Health Nursing. 29315Kim Usher, Joanne Durkin, and Navjot Bhullar. "The COVID-19 pandemic and mental health impacts". In: International Journal of Mental Health Nursing 29.3 (2020), p. 315. Student mental health in the midst of the COVID-19 pandemic: A call for further research and immediate solutions. Nicholas Grubic, Shaylea Badovinac, Amer M Johri , International Journal of Social Psychiatry. 66Nicholas Grubic, Shaylea Badovinac, and Amer M Johri. "Student mental health in the midst of the COVID-19 pandemic: A call for further research and immediate solutions". In: International Journal of Social Psychiatry 66.5 (2020), pp. 517-518. Addressing collegiate mental health amid COVID-19 pandemic. Yusen Zhai, Xue Du, Psychiatry research. 288113003Yusen Zhai and Xue Du. "Addressing collegiate mental health amid COVID-19 pandemic". In: Psychiatry research 288 (2020), p. 113003. Effects of COVID-19 on college students' mental health in the United States: Interview survey study. Changwon Son, Journal of medical internet research. 2221279Changwon Son et al. "Effects of COVID-19 on college students' mental health in the United States: Interview survey study". In: Journal of medical internet research 22.9 (2020), e21279. College students: mental health problems and treatment considerations. Paola Pedrelli, Academic psychiatry. 39Paola Pedrelli et al. "College students: mental health problems and treatment considerations". In: Academic psychiatry 39.5 (2015), pp. 503-511. Mental Health of College Students and Their Non-College-Attending Peers: Results From the National Epidemiologic Study on Alcohol and Related Conditions. Carlos Blanco, 10.1001/ARCHPSYC.65.12.1429Archives of General Psychiatry. 65Carlos Blanco et al. "Mental Health of College Students and Their Non-College-Attending Peers: Results From the National Epidemiologic Study on Alcohol and Related Conditions". In: Archives of General Psychiatry 65 (12 Dec. 2008), pp. 1429-1437. issn: 0003-990X. doi: 10.1001/ARCHPSYC.65.12.1429. The Association for University and College Counseling Center Directors. Peter Leviness, Peter LeViness et al. The Association for University and College Counseling Center Directors. 2020, pp. 1-50. The Mental Health Needs of Today's College Students: Challenges and Recommendations. A Martha, Kitzrow, 10.2202/1949-6605.5037In: NASPA Journal. 46Martha A Kitzrow. "The Mental Health Needs of Today's College Students: Challenges and Recommendations". In: NASPA Journal 46 (4 2009), pp. 646-660. issn: 1559-5455. doi: 10.2202/1949-6605.5037. Counseling and Mental Health Services on Campus: A Handbook of Contemporary Practices and Challenges. James ArcherJr, Stewart Cooper, Jossey-BassJames Archer Jr. and Stewart Cooper. Counseling and Mental Health Services on Campus: A Handbook of Contemporary Practices and Challenges. Jossey-Bass, Aug. 1998. More Than Inconvenienced: The Unique Needs of U.S. College Students During the COVID-19 Pandemic. Alyssa M Lederer, 10.1177/1090198120969372Health Education and Behavior. 4815526127Alyssa M. Lederer et al. "More Than Inconvenienced: The Unique Needs of U.S. College Students During the COVID-19 Pandemic". In: Health Education and Behavior 48 (1 Feb. 2021), pp. 14-19. issn: 15526127. doi: 10.1177/1090198120969372. Addressing collegiate mental health amid COVID-19 pandemic. Yusen Zhai, Xue Du, 10.1016/J.PSYCHRES.2020.113003doi: 10 . 1016 / J . PSYCHRES.2020.113003Psychiatry Research. 288Yusen Zhai and Xue Du. "Addressing collegiate mental health amid COVID-19 pandemic". In: Psychiatry Research 288 (June 2020), p. 113003. issn: 0165-1781. doi: 10 . 1016 / J . PSYCHRES.2020.113003. College counseling services: Meeting today's demands. Michi Fu, Alice W Cheng, In: Psychological services. 14403Michi Fu and Alice W Cheng. "College counseling services: Meeting today's demands." In: Psychological services 14.4 (2017), p. 403. Campus counseling model concerns in the age of COVID-19. Eric Wood, Eric Wood. Campus counseling model concerns in the age of COVID-19. https://universitybusiness. com/campus-counseling-model-concerns-in-the-age-of-covid-19/. 2021. Stress and Resilience Among Professional Counselors During the COVID-19 Pandemic. Arañez Stacey Diane, Clark D Litam, John J S Ausloos, Harrichand, 10.1002/JCAD.12391doi: 10.1002/ JCAD.12391Journal of Counseling & Development. 99Stacey Diane Arañez Litam, Clark D. Ausloos, and John J.S. Harrichand. "Stress and Re- silience Among Professional Counselors During the COVID-19 Pandemic". In: Journal of Counseling & Development 99 (4 Oct. 2021), pp. 384-395. issn: 1556-6676. doi: 10.1002/ JCAD.12391. Time-Limited Counseling Outcome in a Nationwide College Counseling Center Sample. Matthew R Draper, 10.1002/J.2161-1882.2002.TB00204.XJournal of College Counseling. 5Matthew R. Draper et al. "Time-Limited Counseling Outcome in a Nationwide College Coun- seling Center Sample". In: Journal of College Counseling 5 (1 Mar. 2002), pp. 26-38. issn: 1099-0399. doi: 10.1002/J.2161-1882.2002.TB00204.X. College Counseling Today: Contemporary Students and How Counseling Centers Meet Their Needs. Jon L Brunner, 10.1080/87568225.2014.948770/SUPPL_FILE/WCSP_A_948770_SM3622.DOCXJournal of College Student Psychotherapy. 28Jon L. Brunner et al. "College Counseling Today: Contemporary Students and How Coun- seling Centers Meet Their Needs". In: Journal of College Student Psychotherapy 28 (4 Oct. 2014), pp. 257-324. issn: 15404730. doi: 10.1080/87568225.2014.948770/SUPPL_FILE/ WCSP_A_948770_SM3622.DOCX. Are we in crisis? National mental health and treatment trends in college counseling centers. Henry Xiao, 10.1037/SER0000130Psychological Services. 1415411559Henry Xiao et al. "Are we in crisis? National mental health and treatment trends in college counseling centers". In: Psychological Services 14 (4 Nov. 2017), pp. 407-415. issn: 15411559. doi: 10.1037/SER0000130. Meeting the Mental Health Needs of College Students with ASD: A Survey of University and College Counseling Center Directors. Qin Hu, Tara Chandrasekhar, 10.1007/s10803-020-04530-3Journal of Autism and Developmental Disorders. 51Qin Hu and Tara Chandrasekhar. "Meeting the Mental Health Needs of College Students with ASD: A Survey of University and College Counseling Center Directors". In: Journal of Autism and Developmental Disorders 51 (2021), pp. 341-345. doi: 10.1007/s10803-020-04530-3. Dialectical Behavior Therapy in College Counseling Centers: Current Literature and Implications for Practice. Carla D Chugani, 10.1080/87568225.2015.1008368Journal of College Student Psychotherapy. 29Carla D. Chugani. "Dialectical Behavior Therapy in College Counseling Centers: Current Literature and Implications for Practice". In: Journal of College Student Psychotherapy 29 (2 Apr. 2015), pp. 120-131. issn: 8756-8225. doi: 10.1080/87568225.2015.1008368. Being a college counselor on today's campus : roles, contributions, and special challenges. Bruce S Sharkin, Routledge. Bruce S. Sharkin. Being a college counselor on today's campus : roles, contributions, and special challenges. Routledge, 2012. National survey of counseling center directors. Robert P Gallagher, Robert P. Gallagher. National survey of counseling center directors. 2012. A Collaborative Process Model for Promoting Successful Referrals in College Counseling. Melanie M Iarussi, Brian M Shaw, 10.1002/jocc.12048Journal of College Counseling. 1910990399Melanie M. Iarussi and Brian M. Shaw. "A Collaborative Process Model for Promoting Suc- cessful Referrals in College Counseling". In: Journal of College Counseling 19 (3 Oct. 2016), pp. 261-275. issn: 10990399. doi: 10.1002/jocc.12048. . Jesse Owen, Lavanya Devdas, Emil Rodolfa, 10.1300/J035v22n02_03University Counseling Center Off-Campus Referrals". In: Journal of College Student Psychotherapy. 22Jesse Owen, Lavanya Devdas, and Emil Rodolfa. "University Counseling Center Off-Campus Referrals". In: Journal of College Student Psychotherapy 22 (2 Dec. 2007), pp. 13-29. issn: 8756-8225. doi: 10.1300/J035v22n02_03. Operations Research in Healthcare: a survey. Abdur Rais, Ana Vianaa, 10.1111/J.1475-3995.2010.00767.Xissn: 1475-3995. doi: 10. 1111/J.1475-3995.2010.00767.XInternational Transactions in Operational Research. 18Abdur Rais and Ana Vianaa. "Operations Research in Healthcare: a survey". In: International Transactions in Operational Research 18 (1 Jan. 2011), pp. 1-31. issn: 1475-3995. doi: 10. 1111/J.1475-3995.2010.00767.X. Optimization in Healthcare Delivery Modeling: Methods and Applications. Sakine Batun, Mehmet A Begen, 10.1007/978-1-4614-5885-2_4SpringerNew YorkSakine Batun and Mehmet A. Begen. Optimization in Healthcare Delivery Modeling: Methods and Applications. Springer New York, Jan. 2013, pp. 75-119. doi: 10.1007/978-1-4614- 5885-2_4. Appointment Scheduling with Discrete Random Durations. A Mehmet, Maurice Begen, Queyranne, 10.1287/moor.1110.0489In: Mathematics of Operations Research. 362Mehmet A. Begen and Maurice Queyranne. "Appointment Scheduling with Discrete Random Durations". In: Mathematics of Operations Research 36 (2 May 2011), pp. 240-257. issn: 0364-765X. doi: 10.1287/moor.1110.0489. A Sequential Bounding Approach for Optimal Appointment Scheduling. Brian Denton, Diwakar Gupta, 10.1080/07408170304395IIE Transactions. 35Brian Denton and Diwakar Gupta. "A Sequential Bounding Approach for Optimal Appoint- ment Scheduling". In: IIE Transactions 35 (11 2010), pp. 1003-1016. issn: 15458830. doi: 10.1080/07408170304395. OUTPATIENT SCHEDULING IN HEALTH CARE: A REVIEW OF LITERATURE. Tugba Cayirli, Emre Veral, 10.1111/J.1937-5956.2003.TB00218.XProduction and Operations Management. 12Tugba Cayirli and Emre Veral. "OUTPATIENT SCHEDULING IN HEALTH CARE: A REVIEW OF LITERATURE". In: Production and Operations Management 12 (4 Dec. 2003), pp. 519-549. issn: 1937-5956. doi: 10.1111/J.1937-5956.2003.TB00218.X. A probabilistic model for predicting the probability of no-show in hospital appointments. Adel Alaeddini, 10.1007/S10729-011-9148-9/FIGURES/9In: Health Care Management Science. 142Adel Alaeddini et al. "A probabilistic model for predicting the probability of no-show in hospital appointments". In: Health Care Management Science 14 (2 June 2011), pp. 146-157. issn: 13869620. doi: 10.1007/S10729-011-9148-9/FIGURES/9. An optimization based on simulation approach to the patient admission scheduling problem using a linear programing algorithm. C Granja, 10.1016/J.JBI.2014.08.007Journal of Biomedical Informatics. 52C. Granja et al. "An optimization based on simulation approach to the patient admission scheduling problem using a linear programing algorithm". In: Journal of Biomedical Infor- matics 52 (Dec. 2014), pp. 427-437. issn: 1532-0464. doi: 10.1016/J.JBI.2014.08.007. Dynamic scheduling of outpatient appointments under patient no-shows and cancellations. Nan Liu, Serhan Ziya, Vidyadhar G Kulkarni, 10.1287/MSOM.1090.0272In: Manufacturing and Service Operations Management. 1215234614Nan Liu, Serhan Ziya, and Vidyadhar G. Kulkarni. "Dynamic scheduling of outpatient ap- pointments under patient no-shows and cancellations". In: Manufacturing and Service Op- erations Management 12 (2 Mar. 2010), pp. 347-364. issn: 15234614. doi: 10.1287/MSOM. 1090.0272. The performance of a generalized Bailey-Welch rule for outpatient appointment scheduling under inpatient and emergency demand. Sabine Sickinger, Rainer Kolisch, In: Health care management science. 12Sabine Sickinger and Rainer Kolisch. "The performance of a generalized Bailey-Welch rule for outpatient appointment scheduling under inpatient and emergency demand". In: Health care management science 12.4 (2009), pp. 408-419. Discrete-Event Simulation. George S Fishman, 10.1007/978-1-4757-3552-9Discrete-Event Simulation. George S. Fishman. "Discrete-Event Simulation". In: Discrete-Event Simulation (2001). doi: 10.1007/978-1-4757-3552-9. An Appointment Scheduling Optimization Method in Healthcare with Simulation Approach. Ali Ala, Feng Chen, 10.1109/ICIEA49774.2020.91019952020 IEEE 7th International Conference on Industrial Engineering and Applications. 2020Ali Ala and Feng Chen. "An Appointment Scheduling Optimization Method in Healthcare with Simulation Approach". In: 2020 IEEE 7th International Conference on Industrial Engi- neering and Applications, ICIEA 2020 (Apr. 2020), pp. 833-837. doi: 10.1109/ICIEA49774. 2020.9101995. Improving Performance in Outpatient Appointment Services with a Simulation Optimization Approach. J Kenneth, Reena Klassen, Yoogalingam, 10.1111/J.1937-5956.2009.01021.XProduction and Operations Management. 18Kenneth J. Klassen and Reena Yoogalingam. "Improving Performance in Outpatient Ap- pointment Services with a Simulation Optimization Approach". In: Production and Opera- tions Management 18 (4 July 2009), pp. 447-458. issn: 1937-5956. doi: 10.1111/J.1937- 5956.2009.01021.X. Application of discrete event simulation in health care: A systematic review. Xiange Zhang, 10.1186/S12913-018-3456-4/FIGURES/5issn: 14726963. doi: 10.1186/ S12913-018-3456-4/FIGURES/5BMC Health Services Research. 181SeptXiange Zhang. "Application of discrete event simulation in health care: A systematic review". In: BMC Health Services Research 18 (1 Sept. 2018), pp. 1-11. issn: 14726963. doi: 10.1186/ S12913-018-3456-4/FIGURES/5. Discrete-event simulation and scheduling for Mohs micrographic surgery. Patrick Burns, Sailesh Konda, Michelle Alvarado, 10.1080/17477778.2020.1750315Journal of Simulation. 16Patrick Burns, Sailesh Konda, and Michelle Alvarado. "Discrete-event simulation and schedul- ing for Mohs micrographic surgery". In: Journal of Simulation 16 (1 Jan. 2022), pp. 43-57. issn: 1747-7778. doi: 10.1080/17477778.2020.1750315. A model to prioritize access to elective surgery on the basis of clinical urgency and waiting time. Roberto Valente, 10.1186/1472-6963-9-1BMC Health Services Research. 9Roberto Valente et al. "A model to prioritize access to elective surgery on the basis of clinical urgency and waiting time". In: BMC Health Services Research 9 (1 Dec. 2009), p. 1. issn: 1472-6963. doi: 10.1186/1472-6963-9-1. A simulation study to improve the performance of an emergency medical service: Application to the French Val-de-Marne department. Lina Aboueljinane, 10.1016/J.SIMPAT.2014.05.007doi: 10.1016/J. SIMPAT.2014.05.007Simulation Modelling Practice and Theory. 47Lina Aboueljinane et al. "A simulation study to improve the performance of an emergency medical service: Application to the French Val-de-Marne department". In: Simulation Mod- elling Practice and Theory 47 (Sept. 2014), pp. 46-59. issn: 1569-190X. doi: 10.1016/J. SIMPAT.2014.05.007. Reducing patient wait times and improving resource utilization at British Columbia Cancer Agency's ambulatory care unit through simulation. Pablo Santibáñez, 10.1007/S10729-009-9103-1/FIGURES/9In: Health Care Management Science. 1213869620Pablo Santibáñez et al. "Reducing patient wait times and improving resource utilization at British Columbia Cancer Agency's ambulatory care unit through simulation". In: Health Care Management Science 12 (4 Nov. 2009), pp. 392-407. issn: 13869620. doi: 10.1007/S10729- 009-9103-1/FIGURES/9. Model Goodness: Verification and Validation. Tayfur Altiok, Benjamin Melamed, 10.1016/B978-012370523-5/50009-2Academic PressTayfur Altiok and Benjamin Melamed. Model Goodness: Verification and Validation. Aca- demic Press, Jan. 2007, pp. 141-163. isbn: 978-0-12-370523-5. doi: 10.1016/B978-012370523- 5/50009-2. . A&amp;m Texas, Today, Texas A&M Reports First Day Enrollment Totals. Texas A&M Today. Texas A&M Reports First Day Enrollment Totals. https : / / today . tamu.edu/2021/08/31/texas-am-reports-first-day-enrollment-totals/. 2021.
[]
[ "A RANGE-DOPPLER-ANGLE ESTIMATION METHOD FOR PASSIVE BISTATIC RADAR", "A RANGE-DOPPLER-ANGLE ESTIMATION METHOD FOR PASSIVE BISTATIC RADAR" ]
[ "Member, IEEEWan ⋆ Liangtian \nSchool of Software\nKey Laboratory for Ubiquitous Network and Service Software of Liaoning Province\nDalian University of Technology\n116620DalianChina\n", "Xianpeng Wang \nState Key Laboratory of Marine Resource Utilization in South China Sea\nHainan University\n570228HaikouChina\n\nCollege of Information Science and Technology\nHainan University\n570228HaikouChina\n", "♯ ", "Member, IEEEGuoan Bi \nSchool of Electrical and Electronic Engineering\nNanyang Technological University\n639798Singapore\n" ]
[ "School of Software\nKey Laboratory for Ubiquitous Network and Service Software of Liaoning Province\nDalian University of Technology\n116620DalianChina", "State Key Laboratory of Marine Resource Utilization in South China Sea\nHainan University\n570228HaikouChina", "College of Information Science and Technology\nHainan University\n570228HaikouChina", "School of Electrical and Electronic Engineering\nNanyang Technological University\n639798Singapore" ]
[]
In this paper, an effective target detection and localization method is proposed for a passive bistatic radar (PBR) system. The PBR system consists of a commercial FM radio station, which is a noncooperative illuminator of opportunity (IO), referred to as the transmitter antenna and multiple surveillance antennas that form an antenna array, e.g., uniform linear array (ULA). Unlike other literatures where the reference signal is received by a directional antenna, here, the reference signal (direct path) is estimated by beamforming method. Then a modified extensive cancellation algorithm (MECA) based on (least squares) LS method is proposed to solve the disturbance cancellation. After cancelling the disturbance, the matched filter (MF) and LS methods are used for range-Doppler estimation of targets, and then the angles of targets are estimated based on beamforming method. The proposed method is suitable for an antenna array. Simulation results are presented to illustrate the superiority of the proposed MECA disturbance cancellation method and parameter estimation method.Index Terms-passive bistatic radar system, range-Dopplerangle estimation, matched filter, least squares
10.1109/icsigsys.2018.8372669
[ "https://arxiv.org/pdf/1710.10387v1.pdf" ]
26,348,825
1710.10387
a93cf2fe6f2ddfc1e6d0f9ed184a4fb1a97956f1
A RANGE-DOPPLER-ANGLE ESTIMATION METHOD FOR PASSIVE BISTATIC RADAR 28 Oct 2017 Member, IEEEWan ⋆ Liangtian School of Software Key Laboratory for Ubiquitous Network and Service Software of Liaoning Province Dalian University of Technology 116620DalianChina Xianpeng Wang State Key Laboratory of Marine Resource Utilization in South China Sea Hainan University 570228HaikouChina College of Information Science and Technology Hainan University 570228HaikouChina ♯ Member, IEEEGuoan Bi School of Electrical and Electronic Engineering Nanyang Technological University 639798Singapore A RANGE-DOPPLER-ANGLE ESTIMATION METHOD FOR PASSIVE BISTATIC RADAR 28 Oct 2017 In this paper, an effective target detection and localization method is proposed for a passive bistatic radar (PBR) system. The PBR system consists of a commercial FM radio station, which is a noncooperative illuminator of opportunity (IO), referred to as the transmitter antenna and multiple surveillance antennas that form an antenna array, e.g., uniform linear array (ULA). Unlike other literatures where the reference signal is received by a directional antenna, here, the reference signal (direct path) is estimated by beamforming method. Then a modified extensive cancellation algorithm (MECA) based on (least squares) LS method is proposed to solve the disturbance cancellation. After cancelling the disturbance, the matched filter (MF) and LS methods are used for range-Doppler estimation of targets, and then the angles of targets are estimated based on beamforming method. The proposed method is suitable for an antenna array. Simulation results are presented to illustrate the superiority of the proposed MECA disturbance cancellation method and parameter estimation method.Index Terms-passive bistatic radar system, range-Dopplerangle estimation, matched filter, least squares INTRODUCTION Passive sensing has been widely used in many applications, such as radar, underwater acoustics, seismology, etc [1,2,3,4,5]. A typical passive bistatic radar (PBR) system exploits a single non-cooperative illuminator of opportunity (IO), referred to as the transmitted antenna. Compared with active sensing, it has the advantage of low hardware system cost, working without interference with existing wireless systems, etc. In general, the reference antenna is steered towards the transmitter to collect the direct path signal (reference signal), while a surveillance antenna is used to measure a potential target echo (received signal) [1,2,6]. However, the detection performance of the methods based on matched filter (MF) [6] would be degraded without considering strong multipath propagations or clutters. The extensive cancellation algorithm (ECA) method was proposed in [6,7] to cancel the direct path, clutters, and their delay, Doppler shifted versions. Then a least squares (LS) adaptive interference cancellation was proposed in [8], which has better cancellation performance than ECA. However, the method proposed in [6,8] requires that the length of reference signal is longer than that of the received signal, which is not convenient for batch processing of raw data. In addition, many methods have been proposed to estimate range-Doppler [9,10,11,12], but there are few reports about range-Doppler-angle estimation. In this paper, a novel range-Doppler-angle estimation method is proposed for PBR system. The reference signal (direct path) is estimated based on beamforming method. Then, the problem of cancellation of direct path, multipath and clutter signals in PBR system is examined. In real application, the sidelobes of clutters cannot be cancelled completely since the existing disturbance cancellation method can work only when the length of direct path (reference signal) is longer than that of the received array data. In addition, the existing disturbance cancellation method is only suitable for a single antenna. Thus a modified extensive cancellation algorithm (MECA) method base on LS is proposed for disturbance cancellation. After disturbance cancellation, MF and LS methods are used for range-Doppler estimation of target. Finally, the angle of target is estimated based on beamforming method. SYSTEM MODEL As shown in Fig. 1, the PBR system is equipped with a surveillance array (e.g., uniform linear array (ULA)) to receive the direct path signal (unknown source signal), as well as to acquire the reflected target echo. In addition, the surveillance array can inevitably receive the signals from other sources such as the multipath and clutter echoes reflected or refracted from the ground and surrounding buildings. Consider the case that the surveillance array forms a ULA with adjacent antennas spaced by δA. As mentioned above, the data model of the surveillance array should contain the direct path, multipath/clutter and reflected target echo signals, which can be expressed as x(t, l) =d(t)e j2πf DA l + N C i=1 cid(t − τci)e j2πf CAi l + N T m=1 amd(t − τm)e j2πf T Dm t e j2πf T Am l + n(t, l), 0 ≤ t < T0, 0 ≤ l < LA,(1) where x(t, l) and n(t, l) respectively stand for the received signal and measurement noise of the lth antenna at time t, d(t) is the direct path signal. NC and NT are the number of clutters and targets, respectively. ci and am stand for the complex amplitude of the ith clutter and the mth target, respectively. τci and tm stand for the bistatic delay of ith clutter and mth target, respectively. fDA, fCAi and fT Am respectively stand for the angle frequencies of the direct path, the ith clutter and the mth target, particularly fDA = δ A λ cos θDA, fCAi = δ A λ cos θCAi and fT Am = δ A λ cos θT Am with λ being the wavelength of the carrier signal. Their corresponding angles are θDA, θCAi and θT Am , respectively. fT Dm is the Doppler frequency PARAMETER ESTIMATION FOR PBR SYSTEM 3.1. Direct path estimation First of all, we need to estimate the direct path, which is crucial in the ensuing parameter estimation process. The angle of the direct path can be assumed to be known a priori. This assumption is reasonable since the IO is static. If it is not, it can be estimated by some high accuracy direction-of-arrival (DOA) estimation algorithm such as (multiple signal classification) MUSIC [13,14], etc. The direct path can be estimated by using beamforming method, i.e., the ULA is steered to the direction of IO. In particular, given the received array data X ∈ C L T ×L A , where LT is the length of a segment of the observation time. The direct path estimation can be expressed as s dp = Xa * (θDA),(2) where a(θDA) = [1, e j2πf DA , · · · , e j2π(L A −1)f DA ] T is the steering vector of the direct path, s dp is the direct path estimation. Disturbance cancellation method At present, several cancellation techniques have been proposed in the literature [15,16,17]. We will revisit the ECA method by using an LS method [16]. For the ECA method, a dictionary is constructed, and each column of the matrix, which corresponds to a potential source of disturbance, is a delay and Doppler shifted version of the direct path. The dictionary can be expressed as Y = B[Λ−P S ref · · · Λ−1S ref S ref Λ1S ref · · · Λ−P S ref ],(3) where S ref = [s dp Ds dp D 2 s dp · · · D Q−1 s dp ], D is a matrix that applies a delay of the direct path. Λp is a diagonal matrix that applies the phase shift corresponding to the pth Doppler bin and B is an incidence matrix that selects only the last N rows of the following matrix. The detail of the definition of D,Λp and B can be found in [6]. The columns of matrix Y define a basis for the M -dimensional clutter subspace, where M = (2P + 1)Q. To minimize residual signal power after disturbance cancellation based on LS error criterion, the cost function can be written as min W X − YW 2 .(4) The optimized LS solution of the weighting vector can be expressed as W = (Y H Y) −1 Y H X. Then the output of surveillance array after cancellation can be expressed as XECA = X − YW = PX,(5) where P = IN − Y(Y H Y) −1 Y H is the projection matrix which projects the received matrix X onto the orthogonal subspace of the disturbance subspace. A Modified Disturbance Cancellation Method It should be noted that the length of direct path s dp (reference signal) is longer than that of received array data in the ECA method proposed in [6,16]. In this paper, the estimated direct path s dp = [s dp [1] · · · s dp [T ]] is regarded as the reference signal. Thus not all the received array data can be used for disturbance cancellation. The first R − 1 array data samples should be discarded, thus the dimension of the effective array data is (LT − R + 1) × LA. R − 1 is the number of additional reference signal samples [6]. However, the different lengths could cause a problem in real data processing. In general, the time window for parameter estimation is an integer, and a common setting is 1s [18]. This is convenient for both the batch processing of raw data and the comparison between the estimated parameters and the ground truth received by the automatic dependent surveillance broadcast (ADS-B) [19] in the ensuing parameter estimation. When disturbance cancellation method mentioned above is used, one condition is that the duration of reference signal (direct path) is larger than 1s (e.g., 1.01s). This means we have to process the raw data by 1.01s once a time. If we collect 500s real data for testing our proposed algorithm, the time of the raw data cannot match that of ADS-B after processing dozens of times because of the time difference 0.01s. It is hard to compare the estimated parameters with the ground truth, thus we can hardly know if our method is efficient. The other condition is that the duration of reference signal equals to 1s. This means that the duration of the received array data must be shorter than 1s (e.g., 0.99s), which leads to the fact that the bistatic Doppler is not an integer (The resolution of bistatic Doppler is 1/0.99). When the disturbance cancellation method is used, the bistatic Doppler bins that we want to cancel are integers, thus some Doppler bins at this resolution may not be cancelled completely. This may lead to performance degradation of disturbance cancellation. Thus a novel disturbance cancellation method is proposed and the length of direct path s dp and received array data are identical, which means that the delay and Doppler shifted version of the direct path corresponding to the clutters can be cancelled completely when the duration of reference signal equals to 1s. According to matrix of equation (11) in [8], it can be known that the length of direct path s dp and received array data are still different. However, it gives us an inspiration that we can design a similar matrix as mentioned above. We redefine a LT × Q direct path matrix, where each column is a unique circle shift delay copy of the direct path signal, as S mref =      s dp [1] s dp [T0] · · · s dp [T0 − Q + 2] s dp [2] s dp [1] · · · s dp [T0 − Q + 3] . . . . . . . . . . . . s dp [T0] s dp [T0 − 1] · · · s dp [T0 − Q + 1]     (6) where Q is the number of cancellation weight for the finite impulse response (FIR) filter. The other process is the same as subsection 3.2. It can be seen that the dimension of S mref is identical with S ref , which means the computation and memory load of this modified ECA (MECA) method are the same as those of ECA method without the issues pointed out at the beginning of this part. THE PROPOSED PARAMETER ESTIMATION METHOD For range-Doppler estimation, the most common method is MF, which evaluates the cross-correlation function (2D-CCF) between the received array data X and the direct path estimation s dp . In real application, the range-Doppler can be estimated based on only a single antenna and the direct path estimation. Since multiple antennas can be used, the multiple results of 2D-CCF can be summed together to improve the signal-to-noise ratio (SNR), which can improve the detection performance of the targets. The sum of multiple delay-Doppler 2D-CCFs can be expressed as ξ(τ, fD) = L A l=1 L T t=1 XMECA[l, t]s * dp [t − τ ]e −j2πf D t/L T ,(7) where XMECA is the output of surveillance array after doing MECA method, which contains the target echoes. It can be seen from (7) that, if only a single echo is contained in XMECA, then the 2D-CCF would achieve the maximum at (τ, fD). However, if there are several targets contained in XMECA, the sidelobes of strong target may mask the weak targets due to the self-ambiguity characteristic of FM signals. Moreover, if some strong clutter echoes are not cancelled completely but remain in XMECA, the sidelobes of clutters would mask the targets of interest. Numerical simulations will be provided to show this phenomena in the next section. For the problem that sidelobes of strong targets may mask the weak targets, a sequential range-Doppler estimation for the targets of interest is proposed. Since the range-Doppler of strong targets can be detected after MF, the strong targets can be cancelled in the range-Doppler (RD) map which corresponds to the 2D-CCF. Then the weak target masked by the sidelobes of the strong targets can be detected. The same process can be applied until all the targets of interest are detected. Assume that the range-Doppler of the strong targets that have already been detected are (r k , f Dk ), k = 1, · · · Nst, where Nst denotes the number of strong targets. Then we can reconstruct a direct path matrix containing the strong targets as follows Sst[k] =      s dp [1] s dp [T0] · · · s dp [T0 − r k − r0 + 1] s dp [2] s dp [1] · · · s dp [T0 − r k − r0 + 2] . . . . . . . . . . . . s dp [T0] s dp [T0 − 1] · · · s dp [T0 − r k − r0]      ,(8) k = 1, · · · Nst (It is to be noted here that we will omit the index k in the following for notational convenience), where r0 is a small integer, whose goal is that the strong target can be cancelled in the range bin completely. The dictionary containing the strong targets can be constructed as Ys = [Λ f Dk −f 0 Sst · · · Λ f Dk Sst · · · Λ f Dk +f 0 Sst],(9) where f0 is a small integer, whose goal is that the strong target can be cancelled in the Doppler bin completely. In general, both r0 and f0 can be chosen as 3. Then the output of surveillance array containing the weak targets can be expressed as XW = PsXMECA, where Ps = IL T − Ys(Y H s Y) −1 s Y H s is the projection matrix which projects XMECA onto the orthogonal subspace of the subspace of strong targets. Then XW is used to do the MF method based on (7) until all the weak targets are detected. Remark 1: It should be noted that both range and Doppler have the resolution limitation. In general, r k , k = 1, · · · Nst cannot be directly used to form the direct path matrix, as well as the dictionary containing strong targets by using f Dk . For example, if the sampling rate is 250MHz, the range resolution is 3×10 8 /250×10 6 = 1.2Km. Thus the r k used for forming the direct path matrix should be revised as r k /1.2. If the time window is fixed at 1s, the Doppler resolution is 1Hz. If the time window is fixed at 0.5s, the Doppler resolution is 2Hz. At this time, the Doppler used for constructing the dictionary should be revised as (· · · f Dk − 2, f Dk , f Dk + 2, · · · ). After the range-Doppler estimation, the remaining task is angle estimation for targets of interest. The value corresponding to the target in the RDmap for each antenna contains the angle information of the target. The values corresponding to multiple antennas can be written as follows zm = [z1m, z2m · · · zL A m] T , m = 1, · · · , NT .(10) Then the conventional beamforming method can be used to estimate the angle of targets by searching the maximum of spectrum function Pm(θ) = a H (θ)zmzm H a(θ), m = 1, · · · , NT .(11) Remark 2: The angle of target can be estimated at first. Thus the estimation accuracy of angle can be improved, since no accumulative error of range-Doppler estimation can affect the estimation accuracy of angle. However, this method has a limitation, that the number of targets cannot exceed the number of antennas. Our proposed method does not have this limitation, the trade-off is that the estimation accuracy of angle in the proposed algorithm is lower than that of the method which estimates angle of the target at first. SIMULATION RESULTS Modeling of the Transmitted Signal The modeling of transmitted signal is very important, since it would affect the performance of disturbance cancellation and parameter estimation. The most common signals for PBRs in use today are noncooperative FM commercial radio stations since they offer a good trade-off between performance and the system cost [6]. In order to simulate the FM radio signal, we take a segment of a song and then do frequency modulation such that the modulated signal has a bandwidth of about 100kHz. The normalized spectra of modulating signal and FM signal used for simulated scenario are depicted in Fig. 2. It should be noted that the spectrum of the FM signal is similar to the one used in [6], which verifies the correctness of the modeling of the transmitted signal. Then we will evaluate the self-ambiguity property of the FM signal in the range-Doppler domain. The delay-Doppler auto-ambiguity function (AAF) of signal s(t) over a time window [0, T0) can be written as ξ0(τ, fD) = T 0 0 s(t)s * (t − τ )e −j2πf D t dt.(12) It can be seen from (12) that the AAF measures the ambiguity level of a signal subject to a time delay τ and a Doppler shift fD. We plot in Fig. 6(a) appear at around ±65Hz with small range. A further study shows that the peak-to-sidelobes ratio (PSLR) of the AAF is about 20dB which is similar to the practical scenario [6]. Numerical Simulation for Parameter Estimation We focus on range-Doppler-angle estimation. An example of the range-Doppler and angle domains are given in Fig. 4(a) and Fig. 4(b), respectively. The sampling rate is 400kHz, i.e., four times of the bandwidth of the transmitted signal. The time window for parameter estimation is fixed at 1s, which means the duration of the received array data that we can use must be shorter than 1s for ECA. Here it is fixed at 0.99s. First, the ECA method in [6] of disturbance cancellation is compared with our proposed method MECA. It can be seen from Fig. 5(a) that the spacing between two adjacent Doppler frequencies is not an integer, thus the sidelobes of clutters cannot be cancelled completely. However, from Fig. 5(b), it can be seen that the spacing between two adjacent Doppler frequencies is an integer, the MECA method is corresponding to integer Doppler frequencies, thus the sidelobes of clutters can be cancelled completely. In addition, it can be seen that the power of 2D-CCF of the second strong target in Fig. 5(b) is stronger than that in Fig. 5(a), which verifies the reason mentioned above and the superiority of our proposed MECA method. Next, the range-Doppler-angle are estimated after using MECA method. It can be seen from Fig. 5(b) that two strong targets appear in the RDmap. Then they should be cancelled based on MECA method to detect weak targets. It can be seen from Fig. 6(b) that the weak targets appear after cancelling the strong targets. Thus the range-Doppler estimation of the target of interest have been completed. Finally, the angle of the target of interest is estimated based on beamforming method as shown in Fig. 7 . CONCLUSIONS In this paper, a direct path-range-Doppler-angle estimation method is proposed in PBR system. The direct path is estimated based on beamforming method. Then the range-Doppler of targets are estimated by using MF and LS methods after the disturbance cancellation based on MECA. The angles of targets are estimated by using beamforming method. The proposed estimation method is relatively simple, and is suitable for parameter estimation in real application. ACKNOWLEDGMENT The author would thank Dr. Danny Kai Pin Tan for his useful comments on disturbance cancellation. Fig. 1 . 1A PBR system with a surveillance array and a noncooperative IO of the mth target. It should be noted that the clutter does not have Doppler shift. The problem of interest is to estimate the target parameters τm, fT Dm and fT Am , m = 1, 2, . . . , NT from the observation x(t, l), 0 ≤ t < T0, 0 ≤ l < LA. Fig. 2 . 2Normalized spectra of (a) modulating signal and (b) FM signal used for simulated scenario. Fig. 3 .Fig. 4 . 34the 2D AAF of the FM signal presented in Fig. 2 in the range-Doppler plane as well as its zero range cut and zero Doppler cut in Fig. 4, where T0 = 1s. The strongest sidelobes Normalized 2D AAF of FM signal used for simulated scenario. (a) Zero range cut. (b) Zero Doppler cut. Sketch of the reference scenario. (a) In the range-Doppler plane. (b) In the angle plane. θ is positive when OY turn in a clockwise; otherwise, negative. Fig. 5 .Fig. 6 .Fig. 7 . 567Normalized 2D CCF with (a) ECA and (b) MECA. (a)Normalized 2D AAF of FM signal used for simulated scenario in 2D representation on range-Doppler plane. Angle estimation results using beamforming. Passive coherent location radar systems. part 1: Performance prediction. H D Griffiths, C J Baker, IEE Proc. Radar Sonar Navig. 1523H. D. Griffiths and C. J. Baker, "Passive coherent location radar systems. part 1: Performance prediction," IEE Proc. Radar Sonar Navig., vol. 152, no. 3, pp. 124-132, 2005. Passive coherent location radar systems. part 2: Waveform properties. C J Baker, H D Griffiths, I Papoutsis, IEE Proc. Radar Sonar Navig. 1523C. J. Baker, H. D. Griffiths, and I. Papoutsis, "Passive coher- ent location radar systems. part 2: Waveform properties," IEE Proc. Radar Sonar Navig., vol. 152, no. 3, pp. 160-168, 2005. Reconstruction of the reference signal in dvb-t based passive radar. M K Baczyk, M Malanowski, Int. J. Electron. Telecommun. 571M. K. Baczyk and M. Malanowski, "Reconstruction of the reference signal in dvb-t based passive radar," Int. J. Electron. Telecommun., vol. 57, no. 1, pp. 43-48, 2011. A bayesian approach to passive sonar detection and tracking in the presence of interferers. B A Yocom, B R L Cour, T W Yudichak, IEEE J. Ocean. Eng. 363B. A. Yocom, B. R. L. Cour, and T. W. Yudichak, "A bayesian approach to passive sonar detection and tracking in the pres- ence of interferers," IEEE J. Ocean. Eng., vol. 36, no. 3, pp. 386-405, 2011. Imaging passive seismic data. B Artman, Geophys. 714B. Artman, "Imaging passive seismic data," Geophys., vol. 71, no. 4, pp. SI117-SI187, 2006. A multistage processing algorithm for disturbance removal and target detection in passive bistatic radar. F Colone, D W O&apos;hagan, P Lombardo, C J Baker, IEEE Trans. Aerosp. Electron. Syst. 452F. Colone, D. W. O'Hagan, P. Lombardo, and C. J. Baker, "A multistage processing algorithm for disturbance removal and target detection in passive bistatic radar," IEEE Trans. Aerosp. Electron. Syst., vol. 45, no. 2, pp. 698-722, 2009. Space-time constant modulus algorithm for multipath removal on the reference signal exploited by passive bistatic radar. F Colone, R Cardinali, P Lombardo, O Crognale, A Cosmi, A Lauri, T Bucciarelli, IET Radar, Sonar Navig. 33F. Colone, R. Cardinali, P. Lombardo, O. Crognale, A. Cosmi, A. Lauri, and T. Bucciarelli, "Space-time constant modulus al- gorithm for multipath removal on the reference signal exploited by passive bistatic radar," IET Radar, Sonar Navig., vol. 3, no. 3, pp. 253-264, 2009. Spacetime interference analysis and suppression for airborne passive radar using transmissions of opportunity. D K P Tan, M Lesturgie, H Sun, Y Lu, IET Radar, Sonar Navig. 82D. K. P. Tan, M. Lesturgie, H. Sun, and Y. Lu, "Spacetime interference analysis and suppression for airborne passive radar using transmissions of opportunity," IET Radar, Sonar Navig., vol. 8, no. 2, pp. 142-152, 2014. Applications of passive surveillance radar system using cell phone base station illuminators. H Sun, D K P Tan, M Lesturgie, Y Lu, IEEE Aerosp. Electron. Syst. Mag. 253H. Sun, D. K. P. Tan, M. Lesturgie, and Y. Lu, "Applications of passive surveillance radar system using cell phone base station illuminators," IEEE Aerosp. Electron. Syst. Mag., vol. 25, no. 3, pp. 10-18, 2010. Detection in passive mimo radar networks. D E Hack, L K Patton, B Himed, M A Saville, IEEE Trans. Signal Process. 6211D. E. Hack, L. K. Patton, B. Himed, and M. A. Saville, "De- tection in passive mimo radar networks," IEEE Trans. Signal Process., vol. 62, no. 11, pp. 2999-3012, 2014. Two target detection algorithms for passive multistatic radar. J Liu, H Li, B Himed, IEEE Trans. Signal Process. 6222J. Liu, H. Li, and B. Himed, "Two target detection algorithms for passive multistatic radar," IEEE Trans. Signal Process., vol. 62, no. 22, pp. 5930-5939, 2014. Joint delay and doppler estimation for passive sensing with direct-path interference. X Zhang, H Li, J Liu, B Himed, IEEE Trans. Signal Process. 643X. Zhang, H. Li, J. Liu, and B. Himed, "Joint delay and doppler estimation for passive sensing with direct-path interference," IEEE Trans. Signal Process., vol. 64, no. 3, pp. 630-640, 2016. Multiple emitter location and signal parameter estimation. R O Schmidt, IEEE Trans. Antennas Propag. 343R. O. Schmidt, "Multiple emitter location and signal parameter estimation," IEEE Trans. Antennas Propag., vol. 34, no. 3, pp. 276-280, 1986. Two decades of array signal processing research: The parametric approach. H Krim, M Viberg, IEEE Signal Process. Mag. 134H. Krim and M. Viberg, "Two decades of array signal process- ing research: The parametric approach," IEEE Signal Process. Mag., vol. 13, no. 4, pp. 67-94, 1996. Direct-path filtering of dab waveform from pcl receiver target channel. A Guner, M Temple, R Claypoole, Electron. Lett. 391A. Guner, M. Temple, and R. Claypoole, "Direct-path filtering of dab waveform from pcl receiver target channel," Electron. Lett., vol. 39, no. 1, pp. 118-119, 2003. Cancellation of clutter and multipath in passive radar using a sequential approach. F Colone, R Cardinali, P Lombardo, IEEE Radar Conf. F. Colone, R. Cardinali, and P. Lombardo, "Cancellation of clutter and multipath in passive radar using a sequential ap- proach," IEEE Radar Conf., pp. 24-27, 2006. Comparison of clutter and multipath cancellation techniques for passive radar. R Cardinali, F Colone, C Ferretti, P Lombardo, R. Cardinali, F. Colone, C. Ferretti, and P. Lombardo, "Com- parison of clutter and multipath cancellation techniques for passive radar," IEEE Radar Conf., pp. 469-474, 2007. Passive bistatic radar systems. P E Howland, H D Griffiths, C J Baker, Bistatic Radar: Emerging Technology. 394P. E. Howland, H. D. Griffiths, and C. J. Baker, "Passive bistatic radar systems," Bistatic Radar: Emerging Technology, p. 394, 2008. ADS-B exchange. "ADS-B exchange," http://www.adsbexchange.com/.
[]
[ "Orbifolds, Penrose Limits and Supersymmetry Enhancement", "Orbifolds, Penrose Limits and Supersymmetry Enhancement" ]
[ "Kyungho Oh [email protected] \nDepartment of Mathematics\nComputer Science\nPhysics and Astronomy\nUniversity of Missouri at St Louis\nSt LouisUSA\n", "Radu Tatar [email protected] \nInstitut fur Physik\nHumboldt Universitat Berlin\n10115Germany\n", "\nIntroduction\n\n" ]
[ "Department of Mathematics\nComputer Science\nPhysics and Astronomy\nUniversity of Missouri at St Louis\nSt LouisUSA", "Institut fur Physik\nHumboldt Universitat Berlin\n10115Germany", "Introduction\n" ]
[]
We consider supersymmetric PP-wave limits for different N = 1 orbifold geometries of the five sphere S 5 and the five dimensional Einstein manifold T 1,1 . As there are several interesting ways to take the Penrose limits, the PP-wave geometry can be either maximal supersymmetric N = 4 or half-maximal supersymmetric N = 2. We discuss in detail the cases AdS 5 × S 5 /Z 3 , AdS 5 × S 5 /(Z m × Z n ) and AdS 5 × T 1,1 /(Z m × Z n ) and we identify the gauge invariant operators which correspond to stringy excitations for the different limits.
10.1103/physrevd.67.026001
[ "https://arxiv.org/pdf/hep-th/0205067v2.pdf" ]
13,964,650
hep-th/0205067
37ece8dc8a85abf4d73b2626abfea4bcf6a699c7
Orbifolds, Penrose Limits and Supersymmetry Enhancement May 2002 May 2002 Kyungho Oh [email protected] Department of Mathematics Computer Science Physics and Astronomy University of Missouri at St Louis St LouisUSA Radu Tatar [email protected] Institut fur Physik Humboldt Universitat Berlin 10115Germany Introduction Orbifolds, Penrose Limits and Supersymmetry Enhancement May 2002 May 2002arXiv:hep-th/0205067v2 21 We consider supersymmetric PP-wave limits for different N = 1 orbifold geometries of the five sphere S 5 and the five dimensional Einstein manifold T 1,1 . As there are several interesting ways to take the Penrose limits, the PP-wave geometry can be either maximal supersymmetric N = 4 or half-maximal supersymmetric N = 2. We discuss in detail the cases AdS 5 × S 5 /Z 3 , AdS 5 × S 5 /(Z m × Z n ) and AdS 5 × T 1,1 /(Z m × Z n ) and we identify the gauge invariant operators which correspond to stringy excitations for the different limits. Introduction The duality between open strings and closed strings has been explored extensively over the last years. One important example is the AdS/CFT conjecture between the N = 4 field theory and type IIB strings on AdS 5 × S 5 [1,2,3]. The conjecture has been generalized to orbifolds of S 5 [4,5] and to conifolds [6]. The supergravity limit of the string has been mainly considered so far because of the difficulties in quantizing strings in the presence of RR-fluxes. On the other hand, another maximal supersymmetric background, the pp-wave, has been discussed recently in [7] and string theory on the pp-wave is an exactly solvable model, where one can identify all the string oscillators [8]. The pp-wave solution appear as a Penrose limit of the AdS 5 × S 5 solution [7,9] so it can be used to obtain information about the AdS/CFT correspondence. The authors of [9] have extended the AdS/CFT conjecture to the case of strings moving on a pp-wave backgrounds where the corresponding field theory operators are the ones with high R charge and in this case the field theory describe not only the supergravity but also the full closed string theory. The idea of [9] has been extended in many directions [10,19,15,16,17,20,21,35]. The direction we are pursuing in this work was initiated in a series of papers [19,15,16,17,20,21,35] and involves geometries more complicated than the S 5 . Especially interesting are the cases of orbifolds of S 5 or conifolds where one can take two kinds of pp-wave limits, one which preserves the supersymmetry and the other one which enlarges the supersymmetry. As discussed in [7], if one takes the Penrose limit on directions orthogonal to the orbifolding direction, then we expect to get the same amount of supersymmetry, but a Penrose limit along the orbifolding direction will get an increase of supersymmetry. One example of the second type was described in [15,16,17] for the case of D3 branes at a conifold singularity, where the Penrose limit gives a maximal supersymmetric solution. In this case we expect a supersymmetry enhancement in field theory, from N = 1 to N = 4 and the relevant N = 1 multiplets which give rise to an N = 4 multiplet have been identified. In [20,21] a similar discussion has been developed for the supersymmetry enhancement from N = 2 to N = 4 in the case of S 5 /Z k . In the present work we study the supersymmetry enhancement in the Penrose limit for several examples of orbifolds. As only the infinitesimal neighborhood of the null geodesic is probed in the pp-wave limit, the orbifold action disappears unless it is considered locally around the null geodesic. In other words, the orbifolding action also changes in the pp-wave limit. Thus, in general, it is not possible to build duals to string oscillators in the Penrose limit from gauge invariant operators of the original orbifold theory. We have found that, in the Penrose limit, one needs to consider operators from the covering space of the original space. We also comment on anomalous dimensions and correlation function for the orbifold theories and on the interpretation as a limit of a DLCQ theory with the light-cone momentum p + fixed. In section 2 we will describe examples of N = 1 orbifolds of S 5 . The first model is S 5 /Z 3 whose Penrose limit was outlined in [16] for which we describe the string/field theory matching. As a second example we consider different boostings for the S 5 /(Z k × Z l ) orbifold which can give an enlargement of supersymmetry from N = 1 to N = 2 or N = 4. In section 3 we consider Penrose limits of T 1,1 /(Z k × Z l ) along the fixed circles of the quotienting action. 2 N = 1 orbifolds of S 5 Review of the AdS 5 × S 5 result We start with a brief review of the result of [9], pointing out the features which we expect to get from the orbifold discussion. Consider AdS 5 × S 5 where the anti-de-Sitter space AdS 5 is represented as a universal covering of a hyperboloid of radius R in the flat R 2,4 and a sphere S 5 of radius R in the flat space R 0,6 . One may regard the AdS 5 (resp. S 5 ) as a foliation of a time-like direction and a three sphere Ω 3 . (resp. a circle parameterized by ψ and a three sphere Ω ′ 3 .) Then the induced metric on AdS 5 × S 5 becomes ds 2 = R 2 −dt 2 cosh 2 ρ + dρ 2 + sinh 2 ρdΩ 2 3 + dψ 2 cos 2 θ + dθ 2 + sin 2 θdΩ ′ 2 3 .(1) One now considers the pp-limit by boosting along the ψ direction around ρ = 0. The metric in this limit can be obtained by taking R → ∞ after introducing coordinates x + = 1 2 (t + ψ), x − = R 2 2 (t − ψ)(2) and rescaling ρ = r/R, θ = y/R as follows: ds 2 = −4dx + dx − − (r · r + y · y)dx +2 + dy 2 + dr 2(3) where y and r parameterize points on R 4 . Only the components of the RR 5-form F with a plus index survive in this limit. The energy is given by E = i∂ t and the angular momentum in the direction ψ is J = −i∂ ψ and the latter is seen as a generator that rotates a 2-plane inside the original R 6 . In terms of the dual N = 4 theory, the energy E is related to the conformal weight ∆ and the angular momentum to the R-charge. As discussed in [9], the relation between the oscillations of the string in the pp-wave geometry (3) and the field theory quantities is (∆ − J) n = 1 + 4πgNn 2 J 2(4) where N stands for the rank of the gauge theory and g is the string coupling constant. The vacuum has ∆ − J = 0. In the N = 4 field theory, the interpretation of the string vacuum and of the string oscillators is made in terms of the gauge invariant operators. Consider the N = 4 multiplet in terms of a triplet of N = 1 multiplets, denoted by Z, Y 1 , Y 2 , the dimension of each field being 1. The complex field Z is on the directions whose rotation generator is J, so the value of J for the field Z is 1, therefore for the field Z we have ∆ − J = 0. The other fields, Y 1 , Y 2 (and their complex conjugatesȲ 1 ,Ȳ 2 ) have J = 0 and ∆ − J = 1. We can proceed to compare the stringy results with the field theory results, the string vacuum is given by Tr[Z J ] and the stringy oscillators are given by inserting Y 1 , Y 2 ,Ȳ 1 ,Ȳ 2 , i.e. the operators: Tr[Z J−1 ]Y i , Tr[Z J−1 ]Ȳ i , i = 1, 2(5) We can also have gauge invariant operators Tr[Z J−1 ]Z, Tr[Z J−2 ]Y iȲ j , etc, but in [9] arguments have been given that such operators will get infinite mass. String Oscillators in the pp-limit of the AdS 5 × S 5 /Z 3 The geometry AdS 5 × S 5 /Z 3 is obtained as a near horizon geometry of N D3 branes placed at a C 3 /Z 3 orbifold. The generator g of Z 3 acts on C 3 by g · (z 1 , z 2 , z 3 ) → (ωz 1 , ωz 2 , ωz 3 ), ω 3 = 1.(6) We consider the boosting along the direction of the orbifolding which has been studied in [16]. We need to consider a metric for the 3-fold covering of S 5 . As [16], it is convenient to consider S 5 as a Hopf fibration over CP 2 . The metric could be written as: ds 2 = (3dψ + A) 2 + d 2 CP 2(7) where dA/2 gives the Kähler class of CP 2 . As ψ ranges from 0 to 2π, we get a 3-fold of S 5 . More generally , we may take an orbifold theory on C 3 /Z m where the generator g of Z m acts on C 3 by g · (z 1 , z 2 , z 3 ) → (ω a 1 , z 1 , ω a 2 z 2 , ω a 3 z 3 ), ω m = 1, a 1 + a 2 + a 3 = 0 (mod m), a i > 0. (8) Then the S 5 is a Hopf fibration over a weighted projective space CP(a 1 , a 2 , a 3 ). As long as the null geodesic does not lie over the singular locus of the weighted projective space CP(a 1 , a 2 , a 3 ), there will no change in the argument. We now choose the null coordinates as: x + = 1 2 (t + 1 3 ψ) x − = R 2 2 (t − 1 3 ψ)(9) In the limit R → ∞ and after rescaling the transversal direction CP 2 , we obtain the maximally supersymmetric pp-wave metric (3) as in [16]. The light-cone momenta can be written in terms of conformal weight ∆ and the angular momentum J = −i∂ ψ : 2p − = i∂ x + = i(∂ t + 3∂ ψ ) = ∆ − 3J 2R 2 p + = i∂ x − = i(∂ t − 3∂ ψ ) = ∆ + 3J(10) Before we describe the duality string/field theory in the Penrose limit, we recall the results of [4,5] concerning the field theory on D3 branes at C 3 /Z 3 singularities. By starting with 3N D3 branes in the covering space of C 3 /Z 3 orbifold, the SU(3N) gauge group is broken to SU(N) 3 by orbifold action on the Chan-Paton factors and there are three fields in the bifundamental representation for each pair of gauge groups, denoted by X i , Y i , Z i , i = 1, 2, 3 (they come as 3 N × N blocks inside each 3N × 3N matrices X, Y, Z describing the transversal motion of the D-branes). The surviving KK modes are of the form [18]: Tr(X m 1 i Y m 2 i+1 Z m 3 i+2 ), m 1 + m 2 + m 3 = 0 (mod 3), i = 1, 2, 3 (mod 3)(11) The quiver gauge theories have a quantum Z 3 symmetry and the surviving KK modes have to be invariant under it. In the Penrose limit, the effect of the Z 3 action on the transversal direction to the boosting direction disappears as the string probes an infinitesimally small neighborhood of the boosting circle parameterized by ψ. In the quantum vacua, the Z 3 action remains along the boosting direction as we see in (10). In the orbifold theory S 5 /Z 3 , the global symmetry SO(6) ≈ SU(4) is broken up into U(1) × Z 3 . Before the limit, the Hopf fibration is non-trivial, so even if the Z 3 acts only along the Hopf fiber, this does not imply the breaking of global SO(6) isometry. In the pp-limit, the fibration becomes trivial and it breaks the global symmetry SO (6) to SO(4) × SO (2), with the SO(2) being in the boosting direction and SO(4) in the transverse directions. To describe the string/field theory duality, we denote by Z the boosted direction and by X, Y the transverse direction where the orbifold does not act so X, Y do not enter in a gauge invariant form. 1 The action of Z 3 orbifold is only on the Hopf fiber parameterized by Z. We identify the scalar field along the Hopf fiber as Z = Z 1 Z 2 Z 3 where Z i are the above fields in the bifundamental representation of SU(N) i × SU(N) i+1 , i = 1, 2, 3. The field Z is in the adjoint representation of SU(N) and has angular momentum on the U(1) direction equal to 3. The fields X, Y are also in the adjoint representation of the same SU(N) and together with Z they form an N = 4 multiplet. The vacuum of the string in the presence of the Z 3 is 1 √ 3JN 3J/2 Tr[Z J ](12) The first excited states are obtained by insertions of X, Y,X,Ȳ , for string in the pp-wave background these states being obtained by acting with a single oscillator on the ground states. Because there are eight bosonic zero modes oscillators, we expect to find eight bosonic states with ∆ − 3J = 1. They are Tr[Z J X], Tr[Z JX ] or Tr[Z J Y ], Tr[Z J Y ](13) and the ones with the covariant derivative Tr[Z J D µ Z](14) The non-supergravity modes are obtained by acting with creation operators which imply the introduction of a position dependent phase, besides the above insertions [9]. Because we discuss the Z 3 orbifold, we do not have a DLCQ limit as in [20,21], which holds only for Z n with large n. Therefore, if we make the identification of the radius of the x − direction as in [20,21]: πR 2 n = 2πR −(15) where R 2 is approximatively N (the rank of the gauge group), we see that when n is small, the radius R − of the x − direction is infinite, so we are not allowed to use a Discrete Light Cone Quantization. There is no winding mode discussion for the Z 3 orbifold and the insertions corresponding to the non-supergravity modes are identical to the ones of [9]. An interesting case of supersymmetry enhancement was treated in [20,21] for N = 2 orbifolds S 5 /Z n . By boosting along the non-fixed directions of the orbifold, one gets a maximal N = 4 theory. One interesting related development would be to consider the supersymmetry enhancement when D3 branes probe backgrounds of D7/O7 planes [31,32]. The Penrose limit in the fixed direction(orthogonal to O7) was considered in [35] but the discussion of Penrose limits on the non-fixed directions still remains to be discussed. One step further in this direction would be to consider the Penrose limit for the case when D3 branes probe geometries with orthogonal D7 branes as in [32,33,34]. Z m × Z n Orbifolds of S 5 In this subsection we consider the geometry AdS 5 × S 5 /(Z m × Z n ) which is the nearhorizon limit of the D3 branes placed at the tip of C 3 /(Z m × Z n ) 2 . The coordinates of C 3 are z1, z 2 , z 3 and the generators g m , g n of Z m , Z n act on (z 1 , z 2 , z 3 ) as g m : (z 1 , z 2 , z 3 ) → (e 2πi/m z 1 , e −2πi/m z 2 , z 3 ) (16) g n : (z 1 , z 2 , z 3 ) → (e 2πi/n z 1 , z 2 , e −2πi/n z 3 ).(17) The singular points in the quotient are points left invariant under elements of the discrete group. The complex curve z 1 = z 2 = 0 , parameterized by z 3 , is invariant under the Z m and becomes a curve of A m−1 singularities, the complex curve z 1 = z 3 = 0 , parameterized by z 2 , is invariant under the Z n and becomes a curve of A n−1 singularities and the complex curve z 2 = z 3 = 0 , parameterized by z 1 , is invariant under the Z r , r = gcd(m, n) and becomes a curve of A r−1 singularities. The field theory on D3 branes at C 3 /(Z m ×Z n ) singularity is N = 1 theory with gauge group m i=1 n j=1 SU(N) (i,j) and chiral bifundamentals [24,25]. The gauge invariant operators are TrH (i,j)(i+1,j) D (i+1,j)(i,j−1) V (i,j−1)(i,j)(18) where H (i,j)(i+1,j) are in the bifundamental representation of SU(N) (i,j) × SU(N) (i+1,j) , V (i,j)(i,j+1) are in the bifundamental representation of SU(N) (i,j) × SU(N) (i,j+1) and D (i+1,j+1)(i,j) are in the bifundamental representation of SU(N) (i+1,j+1) × SU(N) (i,j) . If D3 branes move to the points of A m−1 , (resp. A n−1 or A r−1 ) singularities described above, the field theory on the D3 branes becomes N = 2 with gauge group SU(N) m (resp. SU(N) n or SU(N) r ). Hence there are flat directions in the N = 1 theory which connect it to an N = 2 theory. In the S 5 /(Z m × Z n ) geometry, there are many interesting directions along which we can consider the boosting and the amount of the supersymmetry enhancement will depend on both the direction and the locality of the trajectories. We now classify the different possibilities: Case 1. Boosting in the direction of the Z m orbifolding (the same discussion holds for the direction of the Z n or Z r orbifolding). We understand by the direction of Z m orbifolding a U(1) direction where Z m is embedded in. For this purpose, it is convenient to consider S 5 as a foliation of the S 3 in C 2 with coordinates z 1 , z 2 and the S 1 in C 1 with coordinates z 3 . Furthermore, we consider S 3 as a Hopf fibration over CP 1 after changing the complex structure z 2 toz 2 and the Z 3 will locally act along the Hopf fiber. From this geometric description of S 5 , we obtain the metric for the AdS 5 × S 5 as: dS 2 AdS = R 2 (− cosh 2 ρdt 2 + dρ 2 + sinh 2 ρdΩ 2 3 ) ds 2 S 5 = R 2 dθ 2 + sin 2 θd 2 S 1 + cos 2 θ (dτ + (cos χ − 1)dφ) 2 + dχ 2 + sin 2 χdφ 2(19) where τ is the coordinate for the fiber direction and dχ 2 + sin 2 χdφ 2 is the metric for the base CP 1 in the Hopf fibration of S 3 . As in the previous section, we need to consider an mn-fold cover of S 5 . As the string probes only an infinitesimal neighborhood of the boosting direction, the action on the transverse directions to the Hopf fiber is irrelevant. For simplicity we take an m-fold covering of the S 5 where the S 3 part of the metric changes to (m dτ + (cos χ − 1)dφ) 2 + (dχ 2 + sin 2 χdφ 2 )(20) We choose the null coordinates as x + = 1 2 (t + τ m ), x − = R 2 2 (t − τ m )(21) and consider a scaling limit R → ∞ around θ = χ = 0 with ρ = r R , θ = u R , χ = v R(22) In this limit, the metric becomes ds 2 = dr 2 + r 2 dΩ 2 3 − r 2 dx + 2 − 2dx + dx − +du 2 + u 2 d 2 S 1 − 2dx + dx − − v 2 dx + dφ − u 2 dx + 2 + dv 2 + v 2 dφ 2 = −4dx + dx − − (r 2 + u 2 )dx + 2 +dr 2 + r 2 dΩ 2 3 + du 2 + u 2 d 2 S 1 + dv 2 + v 2 dφ 2 − v 2 dx + dφ = −4dx + dx − − (r 2 + u 2 + v 2 4 )dx + 2 +dr 2 + r 2 dΩ 2 3 + du 2 + u 2 d 2 S 1 + dv 2 + v 2 dφ ′ 2(23) where φ ′ = φ − 1/2x + . After changing to the rectangular coordinate system, one may rewrite this as ds 2 = −4dx + dx − − (r 2 + u 2 + v 2 ) dx + 2 + dr 2 + du 2 + dv 2(24) The pp wave has a natural decomposition of the R 8 transverse space into R 4 × R 2 × R 2 where R 4 is parameterized by r and the R 2 × R 2 by u and v, respectively. The covariantly constant flux of the R-R field is on the (x + , r) and (x + , u, v). In this geometry, the light cone momenta are: 2p − = i∂ x + = i(∂ t + m∂ τ ) = ∆ − mJ (25) 2R 2 p + = i∂ x − = i(∂ t − m∂ τ ) = ∆ + mJ The effective angular momentum in the boosting direction is mJ and this is the quantity which should be large in the Penrose limit of the AdS/CFT correspondence. Therefore we have two options, the first one being to consider a discrete orbifold group Z m with a very large m and finite J and the second a discrete orbifold group Z m with finite and with a large value for J [20,21]. In the field theory, the supersymmetry is enlarged from N = 1 to N = 4 and the corresponding global symmetries are SO(2) in the boosted direction and SO(4) in the transverse direction to the boosting. To identify the gauge invariant operators, we need to use the fact that we boost along the direction of the Z m orbifolding and the rest of the space is invariant. The direction of the boosting is denoted by Z and, as in the previous subsection, we denote the transverse coordinates to the boosting by X and Y . In the terms of the fields of the N = 1 theory, Z should be in a gauge invariant form and is written as a product Z = m i=1 Z i where Z i are either H (i,j)(i+1,j) for fixed j, D (i+1,j+1)(i,j) for fixed j or V (i,j)(i,j+1) for fixed i. The above fields are in the bifundamental representation of SU(N) i × SU(N) i+1 , the field Z transforms in the adjoint representation of the group SU(N). Together with the scalar fields denoting the transverse direction, X and Y, they form an N = 4 multiplet. The coupling of the SU(N) gauge theory is of order g 2 Y M = g s m and the effective 't Hooft parameter is g 2 Y M N J 2 m 2 which is finite being of the order of g s Nm/R 4 which is finite. Therefore we can treat the SU(N) gauge theory perturbatively. We can now proceed to describe the gauge invariant operators corresponding to the stringy ground state and excitations. The gauge invariant operators Tr(Z J ) have angular momentum mJ in the boosted direction due to the action of Z m , and this corresponds to the vacuum of the string theory. To describe the excitations, we need to consider the two cases discussed above, i.e. when m is either small or large. For the case of small m and large J, the first level eight bosonic zero mode oscillators are Tr(Z J X), Tr(Z J Y ) , Tr(Z JX ), Tr(Z JȲ )(26) together with Tr(Z J D µ Z). In this case x − is not compact as is was for the S 5 /Z 3 case discussed in the previous section. The insertions of X, Y,X,Ȳ should be made as Tr(Z l XZ J−l ), etc. The non-supergravity oscillations are obtained by introducing extra phases in the above operators. More interesting is the case when m is very large and the light cone is compact with radius πR 2 m , the light cone momentum being quantized 2p + = m R − . The string theory has a matrix string description which mimics the one of the flat space as pointed out in [20,36]. In [20] string propagation in DLCQ pp-wave has been considered and the states were labeled by two quantum numbers, the first being the DLCQ momentum k and the second being the winding number m in the x − direction. The vacuum corresponds to Tr(Z J ) which has 2p + = m R − and zero winding number. As J is finite, we can consider J = 1. The insertions of the fields X, Y,X,Ȳ should now be made into the trace of the string of Z i fields. To do this, we also need to consider the splitting of the matrices X, Y into m N ×N blocks, each one being inserted in m different positions and then a summation over the position is required to ensure gauge invariance. In terms of the original N = 1 theory, if we chose Z i to be the fields H (i,j)(i+1,j) for fixed j, then the fields X and Y are build by mN × N blocks which can be either D (i+1,j+1)(i,j) for fixed j (we denote these by X i ) or V (i,j)(i,j+1) for fixed i (we denote these by Y i . By choosing Z i transform in the bifundamental representation of SU(N) i × SU(N) i+1 , it results that X i transform in the bifundamental of SU(N) i+1 × SU(N) i and Y i are in the adjoint representation of SU(N) i . Therefore the fields X i should be inserted in between Z i and Z i and the fields Y i should be inserted in between Z i−1 and Z i . The first oscillators with zero winding number will then be m i=1 Tr(Z 1 Z 2 · · · Z i X i Z i · · · Z m ), m i=1 Tr(Z 1 Z 2 · · · Z i−1 Y i Z i · · · Z m ),(27)and m i=1 Tr(Z 1 Z 2 · · · Z i−1Xi Z i+1 · · · Z m ), m i=1 Tr(Z 1 Z 2 · · · Z i−1Ȳi Z i · · · Z m ),(28) where the summation over i ensures the gauge invariance. The states which have winding numbers are built with an additional factor e 2πi m in the above formulas. In this form, the stringy operators have an expansion which is similar to the Kaluza Klein expansion of a generic field of five dimensional theory reduced on a circle used in [22,23] to conjecture the deconstruction of a five dimensional theory for large m quiver theories in four dimensions. Our S 5 /(Z m × Z n ) model should actually be related to a (1, 1) theory in six dimensions [23], but we expect to get a five dimensions theory as long as we boost along the orbifolding directions. The two directions needed to deconstruct a six dimensional theory are obtained in different boosting, one discussed in this subsection and the other discussed in the next subsection. The conclusion is that a fast moving particle in the τ direction reduces the gauge group to SU(N) and enhances the supersymmetry from N = 1 to N = 4. Case 2. Boosting in the direction of the fixed locus of the Z m orbifolding (the same discussion holds for the Z n or Z r orbifolding) We take the same form of the metric as in (19), we parameterize the angle of S 1 by ψ, the phase of z 3 and we boost along the ψ direction. Since Z n acts on z 3 , we take an n-fold covering of S 5 replacing ψ by nψ in the metric. We introduce the null coordinates x + = 1 2 (t + ψ n ), x − = R 2 2 (t − ψ n )(29) and consider a scaling limit R → ∞ around θ = π/2 with ρ = r R , θ − π 2 = u R .(30) The computation is essentially the same as in [19], the transversal S 3 part of the metric (dτ + (cos χ − 1)dφ) 2 + dχ 2 + sin 2 χdφ 2 is left intact in this limit and hence the Z m action remains. We now denote the scalar field parameterizing the boosted direction by z 3 = Z and the scalar fields parameterizing the transverse directions by z 1 = X, z 2 = Y . The Z m discrete group acts now on X, Y, Z as X → e 2πi/m X, Y → e −2πi/m Y, Z → Z(31) and there is also an action of Z n discrete group on the boosting direction: Z → e −2πi/n Z(32) Because of the latter action, the field Z should enter at the power n and this is obtained if we consider that Z is a product of the N = 1 fields V (ij)(ij+1) for fixed i. We introduce the notation Z n = V (i,j)(i,j+1) V (i,j+1)(i,j+2) · · · V (i,j+n−1)(i,j+n)(33) where j = j + n (modn). The field Z n is in the adjoint representation of SU(N) i,j for fixed i, j. For future use,we also introduce the notation Z j = V (i,j)(i,j+1) . In this case the field theory after the boosting becomes N = 2 m i=1 SU(N) i , the gauge coupling constants of the gauge groups are of order g 2 Y M = g s n and the effective 't Hooft parameters are Because of the Z m projection, the field Z is actually promoted to a m N × m N matrix, with m N × N blocks, each block being in the adjoint representation of an SU(N) i,j . Together with the corresponding vectors of SU(N) i,j , they form N = 2 multiplets. The effective angular momentum in the boosting direction for TrZ J being nJ, we again have two choices, one when n is small and the other when n is big. Consider first the case when n is small. The vacuum of the string theory corresponds to the Z m invariant operators: 1 √ mJ Tr[S q Z nJ ](34) where S = (1, e 2πi/m , ..., e 2πi(m−1)/m ) denotes the q − th twisted sector. The oscillations of the string belong to the untwisted modes which are of the type Tr[S q Z nJ D µ (Z)](35) and Tr[S q Z nJ χ](36) where D µ is the covariant derivative and χ is the supersymmetric partner of the scalar Z. The scalar fields X and Y are now mN × mN matrices with m N × N extra diagonal blocks denoted by X i and Y i , each one transforming in the bifundamental representation of the group SU(N) i j × SU(N) i+1 j For the twisted sectors we need to consider states built with oscillators with a fractional moding. These are obtained by multiplying with X and Y which are acted upon by the Z m group, together with a position independent phase factor e 2πi J n(q) when inserting X, Y and e 2πi J n(−q) for insertions ofX,Ȳ . The discussion changes when m is very large. In this case we have a compact light cone with radius πR 2 n and the light cone momentum is quantized 2p + = n R − . The vacuum and the oscillations of the string belonging to the untwisted modes are the same as before, but we have a change in the definition of the oscillations of the twisted sectors. The insertions of X and Y should be now made between Z j−1 and Z j . To do this, we have to consider all the blocks X i , Y i , i = 1, · · · , m as nN × nN matrices and to split each one of them into n diagonal N × N blocks denoted by X i j , Y i j , i = 1, · · · , n for fixed i. The insertions will then be n j=1 Tr(Z 1 · · · Z j−1 X i j Z j · · · Z n ) (37) or n j=1 Tr(Z 1 · · · Z j−1 Y i j Z j · · · Z n )(38) where j denotes the insertion and i denotes the twisted sector. The winding modes are obtained by using the same formulas with an extra e 1πi n factor. The conclusion is that a fast moving particle moving in the ψ direction reduces the gauge group to m i=1 SU(N) i and enhances the supersymmetry from N = 1 to N = 2. Case 3. Boosting in a general direction which is neither Case 1 nor Case 2. In this case both discrete groups Z m or Z n are on the direction of the boosting and the string probes only a small strip along this direction, therefore there is no orbifold action on the scalar fields and the result is a maximal supersymmetric Penrose limit. Because we do not have any orbifold projection on the three scalar fields Z, X, Y , the situation is similar to moving the D3 brane from the tip of the Z m × Z n orbifold in the bulk, when the supersymmetry is changed from N = 1 to N = 4. The string/field theory duality then reduces to the one of subsection 2.1. There is no change in the angular momentum on the boosted directions due to the orbifolding. We have identified several boosting directions which imply an enlargement of supersymmetry. Three directions are along the Z m , Z n or Z r orbifolding which give maximal supersymmetric pp-limits, three directions are along the fixed loci of Z m , Z n or Z r orbifolds which give N = 2 supersymmetry and an infinite number of boosting directions are along a general direction which would give N = 4. The discussion is different for large m, n as compared to the case of small m, n. For the first case we get compact a compact light cone and this can be used to describe the pp-wave as the limit of a DLCQ theory with fixed p + . In terms of the choice of boosting, we get a specific circle so we get a two dimensional torus when both m and n are large. These two directions are the ones used by [23] to describe the deconstruction of the six dimensional (1,1) theories. Correlation Functions and Supersymmetry Enhancement In [11], a detailed analysis has been made for the anomalous dimensions and three point functions for the chiral and almost chiral operators introduced by [9] (see also [12] for similar discussion). In particular, the authors of [11] have identified the parameter g 2 = J 2 /N as the genus counting parameter in the free Yang-Mills theory, such the correlation function of TrZ J and TrZJ has a contribution JN J g 2h 2 from the genus h Feynman diagram. We want to see what happens for the pp-wave limits of orbifold theories. To do this, we start from the observation that the correlation functions for the orbifold theories coincide with those of N = 4 theory, modulo the rescaling of the gauge coupling constant, as observed in [13] with string theory methods and [14] by using field theory methods. If we consider the case S 5 /Z n , in the N = 2 theory we have a factor of 1/n in front of the correlations functions. After the Penrose limit on the non-fixed direction, we go from the orbifolds theory to the covering space and therefore the factor 1/n disappears. The correlation functions for the orbifold theories are then expected to have a similar expansion in genus as in the S 5 case. It would be interesting to show this in detail, by analogous computations to [13,14]. 3 The N = 1 orbifolds of T 1,1 The case of D3 branes at the conifold or at orbifolds of the conifold has been discussed extensively in the literature [6,26,27,29]. The conifold is a three dimensional hypersurface singularity in C 4 defined by: z 1 z 2 − z 3 z 4 = 0(39) which is a metric cone over the 5-dimensional Einstein manifold T 1,1 = SU(2) × SU(2)/U(1). The conifold can be realized as a holomorphic quotient of C 4 by the C * action given by [6] ( A 1 , A 2 , B 1 , B 2 ) → (λA 1 , λA 2 , λ −1 B 1 , λ −1 B 2 ).(40) The map z 1 = A 1 B 1 , z 2 = A 2 B 2 , z 3 = A 1 B 2 , z 4 = A 2 B 1(41) provides an isomorphism between these two representations of the conifold. The horizon T 11 can be identified with |A 1 | 2 + |A 2 | 2 = |B 1 | 2 + |B 2 | 2 = 1 quotient by an U(1) action induced by (40). Following [28], we can parameterize A i , B i in terms of Euler angles of SU(2) × SU (2): A 1 = cos θ 1 2 exp i 2 (ψ 1 + φ 1 ) A 2 = sin θ 1 2 exp i 2 (ψ 1 − φ 1 ) B 1 = cos θ 2 2 exp i 2 (ψ 2 + φ 2 ) B 2 = sin θ 2 2 exp i 2 (ψ 2 − φ 2 ),(42) and the U(1) is diagonally embedded in SU(2) × SU (2). After taking a further quotient by the remaining U(1) factor of SU (2) × SU (2), we obtain a product of two projective spaces CP 1 1 × CP 1 2 and may identify the parameters θ i , φ i with the spherical coordinates of CP 1 i for i = 1, 2. Now T 1,1 is a U(1) fibration over CP 1 1 × CP 1 2 and the U(1) fiber can be parameterized by ψ := ψ 1 + ψ 2 . The Einstein metric on T 1,1 of radius R is ds 2 T 1,1 = R 2 1 9 (dψ + cos θ 1 dφ 1 + cos θ 2 dφ 2 ) 2 + 1 6 (dθ 2 1 + sin 2 θ 1 dφ 2 1 + dθ 2 2 + sin 2 θ 2 dφ 2 2 ) ,(43) Consider an orbifold theory of the conifold where the discrete group Z m × Z n acts on A i , B j by (A 1 , A 2 , B 1 , B 2 ) → (e −2πi/m A 1 , A 2 , e 2πi/m B 1 , B 2 ),(44) and (A 1 , A 2 , B 1 , B 2 ) → (e −2πi/n A 1 , A 2 , B 1 , e 2πi/n B 2 ).(45) The action (44) descends to the horizon T 1,1 and yields two fixed circles |A 2 | 2 = |B 2 | 2 = 1, A 1 = B 1 = 0 (mod U(1)) and |A 1 | 2 = |B 1 | 2 = 1, A 2 = B 2 = 0 (mod U (1)) [27]. Similarly, the action (45) yields two fixed circles |A 2 | 2 = |B 1 | 2 = 1, A 1 = B 2 = 0 (mod U(1)) and |A 1 | 2 = |B 2 | 2 = 1, A 2 = B 1 = 0 (mod U(1)). The horizon T 11 /(Z m ×Z n ) is singular along these circles, having an A m−1 singularity along the first two circles and an A n−1 singularity along the last two circles. The discrete group Z m × Z n breaks the SU(2) × SU(2) part of the isometry group SU(2) × SU(2) × U(1) of T 1,1 and the U(1) part remains as the global R symmetry. In terms of Euler angles of SU(2) × SU(2), the discrete group Z m × Z n action is given by (ψ 1 , φ 1 , ψ 2 , φ 2 ) → (ψ 1 − 2πi/m, φ 1 − 2πi/m, ψ 2 + 2πi/m, φ 2 + 2πi/m) (ψ 1 , φ 1 , ψ 2 , φ 2 ) → (ψ 1 − 2πi/n, φ 1 − 2πi/n, ψ 2 + 2πi/n, φ 2 − 2πi/n)(46) What we see from the above equations is that the coordinate of the U(1) fiber (ψ = ψ 1 + ψ 2 ) is left invariant under the action of Z m × Z n , as should be in order to preserve the N = 1 supersymmetry. Now we study the Penrose limits of AdS 5 ×T 1,1 /Z m ×Z n . The metric for AdS 5 ×T 1,1 is ds 2 = R 2 [− cosh 2 ρdt 2 + dρ 2 + sinh 2 ρdΩ 2 3 1 9 (dψ + cos θ 1 dφ 1 + cos θ 2 dφ 2 ) 2 + 1 6 (dθ 2 1 + sin 2 θ 1 dφ 2 1 + dθ 2 2 + sin 2 θ 2 dφ 2 2 )](47) The Penrose limit for the conifold has been studied in [15,16,17]. As in the previous section, there are many directions of boosting. We want to study the boosting along the fixed locus of the discrete group action. Consider first the boosting along the circle (1)) which is a fixed locus of Z m action. In terms of the parameters used in (43), this is located at θ 1 = θ 2 = 0 and can be parameterized by ψ + φ 1 + φ 2 . Because of the action of Z n , we are actually dealing with an n-covering of T 1,1 and the metric of T 1,1 changes into |A 1 | 2 = |B 1 | 2 = 1, A 2 = B 2 = 0 (mod Un 2 9 (dψ + cos θ 1 dφ 1 + cos θ 2 dφ 2 ) 2 + 1 6 (dθ 2 1 + sin 2 θ 1 dφ 2 1 + dθ 2 2 + sin 2 θ 2 dφ 2 2 )](48) We introduce the null coordinates x + = 1 2 t + 1 3 n (ψ + φ 1 + φ 2 ) x − = R 2 2 t − 1 3 n (ψ + φ 1 + φ 2 )(49) and consider a scaling limit R → ∞ around θ 1 = θ 2 = 0 with ρ = r R , θ i = √ 6 R ξ i , i = 1, 2(50) and in the limit R → ∞, the metric becomes: ds 2 = −4dx + dx − − r 2 dx +2 + dr 2 + r 2 dΩ 2 3 + i=1,2 (dξ 2 i + ξ 2 i dφ 2 i − 2ξ 2 i dφ i dx + ) = −4dx + dx − + dr 2 − (r · r + w ·w)dx +2 + dwdw(51) where w = (ξ 1 e i(φ 1 −x + ) , ξ 2 e i(φ 2 −x + ) ). The same discussion can be extended to the other fixed circles A 1 = B 1 = 0, A 1 = B 2 = 0 and A 2 = B 1 = 0, where the boosting is again on the direction ψ + φ 1 + φ 2 , but around θ 1 = θ 2 = π, θ 1 = π, θ 2 = 0 and θ 1 = 0, θ 2 = π, respectively. The Penrose limit will be identical with (51) after redefining w appropriately. The transverse space R 8 decomposes into a product of R 4 which is in the r i directions and C 2 whose coordinates is given by w 1 and w 2 . We now investigate the effect of the orbifolding on the geometry of the pp-limit. Note that if we project the conifold (39) in C 4 to C 3 by (z 1 , z 2 , z 3 , z 4 ) → (z 1 , z 3 , z 4 ), we can identify the the boosting direction of T 1,1 with the angular direction of z 1 which is parameterized by 1/2(ψ + φ 1 + φ 2 ) as in (41) and (42), and the transversal space C 2 can be parameterized by z 3 and z 4 . On the pp-limit, Z n acts on the boosting direction as z 1 → e −2πi/n z 1(52) and on the transversal direction trivially, and on the other hand, Z m acts on the transversal direction as (z 3 , z 4 ) → (e −2πi/m z 3 , e 2πi/m z 4 )(53) and acts trivially on the boosting direction z 1 which is along the circle of boosting. In terms of the coordinate of the boosting direction,there is an Z n action onψ = ψ +φ 1 +φ 2 asψ →ψ − πi n . We now identify the field theories gauge invariant operators which are dual to the strings modes on the above pp-wave geometry. The transverse space is S 5 /Z n or S 5 /Z m , so the field theory is N = 2, n i=1 SU(N) i or m i=1 SU(N) i . Before the boosting the field theory is N = 1 m i=1 n j=1 SU(N) ij × m i=1 n j=1 SU(N) ′ ij .(55) and there are bifundamental fields ( A 1 ) i,j;i,j in SU(N) i,j × SU(N) ′ i,j , (A 2 ) i+1,j+1;i,j in SU(N) i+1,j+1 ×SU(N) ′ i,j , (B 1 ) i,j;i,j+1 in SU(N) ′ i,j ×SU(N) i,j+1 and (B 2 ) i,j;i+1,j in SU(N) ′ i,j × SU(N) ′ i+1,j . The products of A i , B j , which enter in the definitions of z 1 , z 3 , z 4 are (A 1 B 1 ) i,j;i,j+1 in SU(N) i,j × SU(N) i,j+1 , (A 1 B 2 ) i,j;i+1,j in SU(N) i,j × SU(N) i+1,j , and (A 2 B 1 ) i+1,j+1;i,j+1 in SU(N) i+1,j+1 × SU(N) i,j+1 . We want to see the change in the field theory after the boosting. There are four possible particular cases of particular gauge groups which correspond to D3 branes at four dimensional A m−1 or A n−1 singularities, and the field theory becomes N = 2, m i=1 SU(N) ij or m i=1 SU(N) ′ ij for fixed j and n j=1 SU(N) ij or n j=1 SU(N) ′ ij for fixed i. The chiral primaries are constructed from sums of gauge invariant products of chiral superfields, modulo F-and D-flatness condition [29]. They are products of the form A i 1 B j 1 A i 2 B j 2 · · · A imn B jmn , symmetrized in A i and B j . For fixed i, a particular example of a chiral primary involving only A 1 and B 1 fields is: (56) where j +n = j (mod n) so the trace is taken over the adjoint representation of SU(N) i,j . The R-charges of the fields A i , B i are not changed by the quotienting so the R-charge of the gauge invariant operator (56) is n. Tr((A 1 ) i,j;i,j (B 1 ) i,j;i,j+1 (A 1 ) i,j+1;i,j+1 (B 1 ) i,j+1;i,j+2 · · · (A 1 ) i,j+n−1;i,j+n−1 (B 1 ) i,j+n−1;ij+n ) We now relate the field theory R-charge with the other U(1) charges that appear in the field theory and geometry. In the geometry we have two rotation charges for the U(1) × U(1) isometry group which are denoted by J 1 and J 2 and they are related to the Cartan generators of the SU(2) × SU (2) global symmetry of the dual superconformal field theory by [15,16]: J a = −i ∂ ∂φ a |x ± = −i ∂ ∂φ a |t,ψ + i ∂ ∂ψ |t,φ i = Q a − 1 2 R a = 1, 2(57) Because of the Z n action on the fixed circle the quotiented conifold, the above relation becomes: nJ a = nQ a − R 2(58) We use the convention that A 1 has Q 1 = 1 2 and B 1 has Q 2 = 1 2 . In [15,16,17], the vacuum of the string theory has been identified with the state J 1 = J 2 = 0 and the first oscillations of the strings with J 1 = ±1, J 2 = 0 and J 1 = 0, J 2 = ±1. Consider now the boosting along z 1 direction and we want to identify the gauge invariant operators which correspond to the string theory ground state and first oscillation modes. In the case of the conifold, the ground state J 1 = J 2 = 0 was identified with the the gauge theory operators: [15,16,17]: Tr(A 1 B 1 ) J ,(59) the first oscillations J 1 = −1, J 2 = 0 or J 1 = 0, J 2 = −1 were identified with multiplication by A 1 B 2 or A 2 B 1(60) and the first oscillations J 1 = 1, J 2 = 0 or J 1 = 0, J 2 = 1 were identified with multiplication by A 1Ā2 orB 2 B 1(61) where A i , B i are all N × N matrices. A 1Ā2 orB 2 B 1 came from the semi-conserved currents of the SU(2) groups and were introduced in [30]. When there is a quotient action on the SU(2) groups, A 1Ā2 orB 2 B 1 are not invariant so they do not appear in the spectrum. Because the supersymmetry in Penrose limit is N = 2, we do not need the semi-conserved currents to build N = 2 multiplets and we only need A 1 B 2 and A 2 B 1 in order to build the field theory duals to the twisted sectors of the string theory. For the quotiented conifold, the matrix A 1 B 1 is promoted to a m N × m N matrix which splits into m, N × N diagonal matrices in the adjoint representation SU(N) i,j , i = 1, · · · , m, for fixed j. The matrices A 1 B 2 and A 2 B 1 become m N × m N matrices which also split into m extra-diagonal N ×N and each block corresponds to fields transforming in the bifundamental representation of SU(N) i,j × SU(N) i+1,j . The boosted direction is acted upon by the discrete group Z n so the invariant quantity is a product as in (56) with n copies of A 1 and n copies of B 1 , of the form (A 1 ) i,j;i,j (B 1 ) i,j;i,j+1 · · · (A 1 ) i,j+n−1;i,j+n−1 (B 1 ) i,j+n−1;i,j+n , which is indeed in the adjoint representation of SU(N) i,j . Denoting this by (A 1 B 1 ) n , we see that the equation (58) implies that it has J 1 = J 2 = 0 and it is the ground state of the string. The vector field for all SU(N) i,j with fixed j, together with the field [(A 1 B 1 ) n ] i form an N = 2 multiplet. The ground state is given by m mutually orthogonal Z m invariant single trace operators Tr[S q (A 1 B 1 ) nJ ](63) where S is defined as S = (1, e 2πi/m , ..., e 2πi(m−1)/m ) denotes the q − th twisted sector. The first level untwisted sectors are built with derivatives and descendants of (A 1 B 1 ) n and are of the form: Tr[S q (A 1 B 1 ) nJ D µ (A 1 B 1 ) n ](64) and Tr[S q (A 1 B 1 ) nJ χ](65) where D µ is the covariant derivative and χ is the supersymmetric partner of the scalar (A 1 B 1 ) n . The first level twisted sectors are written with insertions of A 1 B 2 and A 2 B 1 , which are acted upon by Z m but are invariant under Z n . They have zero angular momentum in the boosted direction so they are used to build first level string oscillations. The discussion is similar to the one of [19]. As the effective angular momentum of the string states is nJ, we again have the choice of choosing n to be either small or large. For the case of large n, the insertions of A 1 B 2 and A 2 B 1 should be made between different (A 1 ) i,j;i,j (B 1 ) i,j;i,j+1 . The Penrose limits of quotiented conifold will then be the limit of a DLCQ theory with constant p + . Conclusions In this paper we studied the Penrose limits of different N = 1 orbifold geometries of S 5 and T 11 which lead to supersymmetric PP-wave backgrounds with enlarged supersymmetry. We have considered the gauge invariant chiral operators in the different Penrose limits and we have identified the string oscillations in terms of the gauge invariant operators. We discussed the different choices for the rank of the quotient groups. n 2 which are finite being of the order of g s Nm/R 4 . This set of X, Y, Z is different from the original complex coordinates of C 3 in(11). But a change of complex structures we may identify them as complex coordinates of the infinitesimal neighborhood of the boosting circle. This model was also discussed in[20] AcknowledgmentsWe would like to thank Sunil Mukhi for correspondence, Keshav Dasgupta for comments on the manuscript and especially Gianguido Dall'Agata for several important discussions. The large N limit of superconformal field theories and supergravity. J Maldacena, hep-th/9711200Adv. Theor. Math. Phys. 21113Int. J. Theor. Phys.J. Maldacena, "The large N limit of superconformal field theories and supergravity," Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1998)] [hep-th/9711200]. Gauge theory correlators from noncritical string theory. S S Gubser, I R Klebanov, A M Polyakov, hep-th/9802109Phys. Lett. B. 428105S. S. Gubser, I. R. Klebanov and A. M. Polyakov, "Gauge theory correlators from non- critical string theory," Phys. Lett. B 428, 105 (1998) [hep-th/9802109]. Anti-de Sitter space and holography. E Witten, hep-th/9802150Adv. Theor. Math. Phys. 2253E. Witten, "Anti-de Sitter space and holography," Adv. Theor. Math. Phys. 2, 253 (1998) [hep-th/9802150]. 4d conformal theories and strings on orbifolds. S Kachru, E Silverstein, hep-th/9802183Phys. Rev. Lett. 804855S. Kachru and E. Silverstein, "4d conformal theories and strings on orbifolds," Phys. Rev. Lett. 80, 4855 (1998) [hep-th/9802183]. On conformal field theories in four dimensions. A E Lawrence, N Nekrasov, C Vafa, hep-th/9803015Nucl. Phys. B. 533199A. E. Lawrence, N. Nekrasov and C. Vafa, "On conformal field theories in four dimen- sions," Nucl. Phys. B 533, 199 (1998) [hep-th/9803015]. Superconformal field theory on threebranes at a Calabi-Yau singularity. I R Klebanov, E Witten, hep-th/9807080Nucl. Phys. B. 536199I. R. Klebanov and E. Witten, "Superconformal field theory on threebranes at a Calabi- Yau singularity," Nucl. Phys. B 536, 199 (1998) [hep-th/9807080]. A new maximally supersymmetric background of IIB superstring theory. M Blau, J Figueroa-O&apos;farrill, C Hull, G Papadopoulos, hep-th/0202111Penrose limits, supergravity and brane dynamics. 020147Penrose limits and maximal supersymmetryM. Blau, J. Figueroa-O'Farrill, C. Hull and G. Papadopoulos, "A new maximally su- persymmetric background of IIB superstring theory," JHEP 0201, 047 (2002) [hep- th/0110242]; "Penrose limits and maximal supersymmetry," [hep-th/0201081]; "Penrose limits, supergravity and brane dynamics," [hep-th/0202111]. Type IIB Green-Schwarz superstring in plane wave Ramond-Ramond background. R R Metsaev, hep-th/0112044Nucl. Phys. B. 62570R. R. Metsaev, "Type IIB Green-Schwarz superstring in plane wave Ramond-Ramond background," Nucl. Phys. B 625, 70 (2002) [hep-th/0112044]. Strings in flat space and pp waves from N = 4 super Yang Mills. D Berenstein, J Maldacena, H Nastase, hep-th/0202021D. Berenstein, J. Maldacena and H. Nastase, "Strings in flat space and pp waves from N = 4 super Yang Mills," [hep-th/0202021]. On Solvable Models of Type IIB Superstrings in NSNS and RR Plane Wave Backgrounds. J G Russo, A Tseytlin, hep-th/0202179J.G. Russo, A. Tseytlin, "On Solvable Models of Type IIB Superstrings in NSNS and RR Plane Wave Backgrounds," [hep-th/0202179]. Super-PP-wave Algebra from Super-AdS x S Algebras in Eleven-dimensions. M Hatsuda, K Kamimura, M Sakaguchi, hep- th/0204002From Super-AdS 5 × S 5 Algebra to Super-PP-Wave AlgebraM. Hatsuda, K. Kamimura, M. Sak- aguchi,"From Super-AdS 5 × S 5 Algebra to Super-PP-Wave Algebra," hep-th/0202190; "Super-PP-wave Algebra from Super-AdS x S Algebras in Eleven-dimensions," [hep- th/0204002]. Boundary States for GS Superstrings in an Hpp Wave Background. M Billo, &apos; , I Pesando, hep-th/0203028M. Billo', I. Pesando, "Boundary States for GS Superstrings in an Hpp Wave Background," [hep-th/0203028]. CY 4-fold and HK Manifolds to AdS, Penrose Limits and PP-Waves. U Gursoy, C Nunez, M Schvellinger, hep-th/0203124RG Flows From Spin. U. Gursoy, C. Nunez and M. Schvellinger, "RG Flows From Spin(7), CY 4-fold and HK Manifolds to AdS, Penrose Limits and PP-Waves," [hep-th/0203124]. Toroidal Compactification of PP-Waves. J Michelson, hep-th/0203140TwistedJ. Michelson, " (Twisted) Toroidal Compactification of PP-Waves," [hep-th/0203140]. M-theory PP-Waves, Penrose Limits and Supernumerary Supersymmetries. M Cvetic, H Lü, C N Pope, hep-th/0203229Penrose Limits, PP-Waves and Deformed M2-branesM. Cvetic, H. Lü and C.N. Pope, "Penrose Limits, PP-Waves and De- formed M2-branes," hep-th/0203082; "M-theory PP-Waves, Penrose Limits and Super- numerary Supersymmetries," [hep-th/0203229]. Penrose Limit, Spontaneous Symmetry Breaking and Holography in PP-Wave Background. S R Das, C Gomez, S-J Rey, hep-th/0203164S. R. Das, C. Gomez, S-J. Rey, " Pen- rose Limit, Spontaneous Symmetry Breaking and Holography in PP-Wave Background," [hep-th/0203164]. Noncommutative D-brane and Open String in PP-Wave Background With B-Field. C S Chu, P M Ho, hep-th/0203186C.S. Chu and P.M. Ho, "Noncommutative D-brane and Open String in PP-Wave Background With B-Field," [hep-th/0203186]. Dp Branes in PP-wave Background. A Dabholkar, S Parvizi, hep-th/0203231A. Dabholkar, S. Parvizi, "Dp Branes in PP-wave Background," [hep-th/0203231]. Open Strings in PP-Wave Background from Defect Conformal Field Theory. P Lee, J Park, hep-th/0203257P. Lee, J. Park, "Open Strings in PP-Wave Background from Defect Conformal Field Theory," [hep-th/0203257]. Penrose Limits of Non-standard Brane Intersections. H Lu, J F Vazquez-Poritz, hep- th/0204001H. Lu, J.F. Vazquez-Poritz, " Penrose Limits of Non-standard Brane Intersections," [hep- th/0204001]. Strings in Homogeneous Gravitational Waves and Null Holography. E Kiritsis, B Pioline, hep-th/0204004E. Kiritsis, B. Pioline, "Strings in Homogeneous Gravitational Waves and Null Holography," [hep-th/0204004]. D-Brane Solutions in pp-wave Background. A Kumar, R R Nayak, ; R G Sanjay, K Leigh, M Okuyama, Rozali, hep-th/0204026PP-Waves and Holography. A. Kumar, R. R. Nayak, Sanjay, " D-Brane Solutions in pp-wave Background,"hep-th/0204025. R. G. Leigh, K. Okuyama, M. Rozali, "PP- Waves and Holography," [hep-th/0204026]. Supersymmetric Branes in PP Wave Background. D Bak, hep-th/0204033D. Bak, "Supersymmetric Branes in PP Wave Background,"[hep-th/0204033]. A semi-classical limit of the gauge/string correspondence. S S Gubser, I R Klebanov, A M Polyakov, hep-th/0204051S. S. Gubser, I. R. Klebanov and A. M. Polyakov, "A semi-classical limit of the gauge/string correspondence," [hep-th/0204051]. K Skenderis, M Taylor, hep-th/0204054Branes in AdS and pp-wave Spacetimes. K. Skenderis, M. Taylor, "Branes in AdS and pp-wave Spacetimes," [hep-th/0204054]. Superstring Interactions in a pp-wave Background. M Spradlin, A Volovich, hep-th/0204146M. Spradlin, A. Volovich, "Superstring Interactions in a pp-wave Background," [hep-th/0204146]. Large angular momentum closed strings colliding with D-branes. Y Imamura, hep- th/0204200Y. Imamura, "Large angular momentum closed strings colliding with D-branes," [hep- th/0204200]. Open Strings in Exactly Solvable Model of Curved Spacetime and PP-Wave Limit. H Takayanagi, T Takayanagi, hep-th/0204234H. Takayanagi and T. Takayanagi, "Open Strings in Exactly Solvable Model of Curved Spacetime and PP-Wave Limit," [hep-th/0204234]. PPwaves and logarithmic conformal field theories. I Bakas, K Sfetsos, hep-th/0205006I. Bakas, K. Sfetsos, "PP- waves and logarithmic conformal field theories," [hep-th/0205006]. More on Penrose Limit of AdS 4 xQ 1,1,1. C Ahn, hep-th/0205008C. Ahn, "More on Penrose Limit of AdS 4 xQ 1,1,1 ", [hep-th/0205008]. Penrose limit and string quantization in AdS 3 × S 3. A Parnachev, D A Sahakyan, hep-th/0205015A. Parnachev, D. A. Sahakyan, "Pen- rose limit and string quantization in AdS 3 × S 3 ,[hep-th/0205015]. M5-branes with 3/8 supersymmetry in pp-wave background. H Singh, hep-th/0205020H. Singh, "M5-branes with 3/8 supersymmetry in pp-wave background", [hep-th/0205020]. Correspondence Principle in a PP-wave Background. M Li, hep-th/0205043M. Li, "Correspon- dence Principle in a PP-wave Background", [hep-th/0205043]. PP-wave string interactions from perturbative Yang-Mills theory. N R Constable, D Z Freedman, M Headrick, S Minwalla, L Motl, A Postnikov, W Skiba, hep- th/0205089N. R. Constable, D. Z. Freedman, M. Headrick, S. Minwalla, L. Motl, A. Postnikov and W. Skiba, "PP-wave string interactions from perturbative Yang-Mills theory," [hep- th/0205089]. A New Double-Scaling Limit of N=4 Super Yang-Mills Theory and PP-Wave Strings. C Kristjansen, J Plefka, G W Semenoff, M Staudacher, hep-th/0205033C. Kristjansen, J. Plefka, G. W. Semenoff, M. Staudacher, "A New Double-Scaling Limit of N=4 Super Yang-Mills Theory and PP-Wave Strings", [hep-th/0205033]. On lightcone string field theory from super Yang-Mills and holography. D Berenstein, H Nastase, hep-th/0205048D. Berenstein and H. Nastase, "On lightcone string field theory from super Yang-Mills and holography," [hep-th/0205048]. Operators with large R charge in N = 4 Yang-Mills theory. D J Gross, A Mikhailov, R Roiban, hep-th/0205066D. J. Gross, A. Mikhailov and R. Roiban, "Operators with large R charge in N = 4 Yang-Mills theory," [hep-th/0205066]. String expansion as large N expansion of gauge theories. M Bershadsky, Z Kakushadze, C Vafa, hep-th/9803076Nucl. Phys. B. 52359M. Bershadsky, Z. Kakushadze and C. Vafa, "String expansion as large N expansion of gauge theories," Nucl. Phys. B 523, 59 (1998) [hep-th/9803076]. Large N limit of orbifold field theories. M Bershadsky, A Johansen, hep-th/9803249Nucl. Phys. B. 536141M. Bershadsky and A. Johansen, "Large N limit of orbifold field theories," Nucl. Phys. B 536, 141 (1998) [hep-th/9803249]. PP wave limit and enhanced supersymmetry in gauge theories. N Itzhaki, I R Klebanov, S Mukhi, hep-th/0202153JHEP. 020348N. Itzhaki, I. R. Klebanov and S. Mukhi, "PP wave limit and enhanced supersymmetry in gauge theories," JHEP 0203 (2002) 048 [hep-th/0202153]. Penrose limit of N = 1 gauge theories. J Gomis, H Ooguri, hep-th/0202157J. Gomis and H. Ooguri, "Penrose limit of N = 1 gauge theories," [hep-th/0202157]. On Penrose limits and gauge theories. L A Zayas, J Sonnenschein, hep- th/0202186L. A. Zayas and J. Sonnenschein, "On Penrose limits and gauge theories," [hep- th/0202186]. Orbifolds of AdS(5) x S(5) and 4d conformal field theories. Y Oz, J Terning, hep-th/9803167Nucl. Phys. B. 532163Y. Oz and J. Terning, "Orbifolds of AdS(5) x S(5) and 4d conformal field theories," Nucl. Phys. B 532, 163 (1998) [hep-th/9803167]. M Alishahiha, M M Sheikh-Jabbari, hep-th/0203018The PP-wave limits of orbifolded AdS(5) x S**5. M. Alishahiha and M. M. Sheikh-Jabbari, "The PP-wave limits of orbifolded AdS(5) x S**5," [hep-th/0203018]; Superstring on pp-wave orbifold from large-N quiver gauge theory. N W Kim, A Pankiewicz, S J Rey, S Theisen, hep-th/0203080N. w. Kim, A. Pankiewicz, S. J. Rey and S. Theisen, "Su- perstring on pp-wave orbifold from large-N quiver gauge theory," [hep-th/0203080]; Strings on orbifolded pp-waves. T Takayanagi, S Terashima, hep-th/0203093T. Takayanagi and S. Terashima, "Strings on orbifolded pp-waves," [hep-th/0203093]; Penrose limits of orbifolds and orientifolds. E Floratos, A Kehagias, hep- th/0203134E. Floratos and A. Kehagias, "Penrose limits of orbifolds and orientifolds," [hep- th/0203134]. Strings from Quivers, Membranes from Moose. S Mukhi, M Rangamani, E Verlinde, hep-th/0204147S. Mukhi, M. Rangamani and E. Verlinde, "Strings from Quivers, Membranes from Moose," [hep-th/0204147]. M Alishahiha, M M Sheikh-Jabbari, hep-th/0204174Strings in PP-Waves and Worldsheet Deconstruction. M. Alishahiha and M. M. Sheikh-Jabbari, "Strings in PP-Waves and Worldsheet Decon- struction" [hep-th/0204174]. De)Constructing Dimensions. N Arkani-Hamed, A G Cohen, H Georgi, hep-th/0104005Phys.Rev.Lett. 86N. Arkani-Hamed, A. G. Cohen, H. Georgi, "(De)Constructing Dimensions," Phys.Rev.Lett. 86 (2001) 4757, [hep-th/0104005]. Mother Moose: Generating Extra Dimensions From Simple Groups at Large N. C T Hill, S Pokorski, J Wang ; I. Rothstein, W Skiba, hep-th/0109175Phys.Rev. 6465002Phys.Rev.C. T. Hill, S. Pokorski, J. Wang, Phys.Rev. D64, 105005 (2001) [hep-th/0104035] I. Rothstein, W. Skiba, "Mother Moose: Generating Extra Dimensions From Simple Groups at Large N ," Phys.Rev. D65, 065002 (2002), [hep-th/0109175]. Deconstructing (2,0) and Little String Theories. N Arkani-Hamed, A G Cohen, D B Kaplan, A Karch, L Motl, hep-th/0110146N. Arkani-Hamed, A. G. Cohen, D. B. Kaplan, A. Karch, L. Motl, "Deconstructing (2,0) and Little String Theories," [hep-th/0110146]. On the realization of chiral four-dimensional gauge theories using branes. A Hanany, A Zaffaroni, arXiv:hep-th/9801134JHEP. 98051A. Hanany and A. Zaffaroni, "On the realization of chiral four-dimensional gauge theories using branes," JHEP 9805, 001 (1998) [arXiv:hep-th/9801134]. Brane boxes and branes on singularities. A Hanany, A M Uranga, arXiv:hep-th/9805139JHEP. 980513A. Hanany and A. M. Uranga, "Brane boxes and branes on singularities," JHEP 9805, 013 (1998) [arXiv:hep-th/9805139]. Non-spherical horizons. I. D R Morrison, M R Plesser, arXiv:hep-th/9810201Adv. Theor. Math. Phys. 31D. R. Morrison and M. R. Plesser, "Non-spherical horizons. I," Adv. Theor. Math. Phys. 3, 1 (1999) [arXiv:hep-th/9810201]. Brane configurations for branes at conifolds. A M Uranga, arXiv:hep-th/9811004JHEP. 990122A. M. Uranga, "Brane configurations for branes at conifolds," JHEP 9901, 022 (1999), [arXiv:hep-th/9811004]; K Dasgupta, S Mukhi, arXiv:hep-th/9811139Brane constructions, conifolds and M-theory. 551204K. Dasgupta and S. Mukhi, "Brane constructions, conifolds and M-theory," Nucl. Phys. B 551 (1999) 204 [arXiv:hep-th/9811139]; A family of N = 1 SU(N)**k theories from branes at singularities. E Lopez, arXiv:hep-th/9812025JHEP. 990219E. Lopez, "A family of N = 1 SU(N)**k theories from branes at singularities," JHEP 9902, 019 (1999) [arXiv:hep-th/9812025]. . R De Mello Koch, K Oh, R Tatar, arXiv:hep-th/9812097Nucl. Phys. B. 555457R. de Mello Koch, K. Oh and R. Tatar, Nucl. Phys. B 555, 457 (1999) [arXiv:hep-th/9812097]. Branes at generalized conifolds and toric geometry. R Von Unge, arXiv:hep-th/9901091JHEP. 990223R. von Unge, "Branes at generalized conifolds and toric geometry," JHEP 9902, 023 (1999) [arXiv:hep-th/9901091]; Mirror symmetries for brane configurations and branes at singularities. M Aganagic, A Karch, D Lust, A Miemiec, arXiv:hep-th/9903093Nucl. Phys. B. 569277M. Aganagic, A. Karch, D. Lust and A. Miemiec, "Mir- ror symmetries for brane configurations and branes at singularities," Nucl. Phys. B 569, 277 (2000) [arXiv:hep-th/9903093]; Brane constructions, fractional branes and anti-de Sitter domain walls. K Dasgupta, S Mukhi, arXiv:hep-th/9904131JHEP. 99078K. Dasgupta and S. Mukhi, "Brane constructions, fractional branes and anti-de Sitter domain walls," JHEP 9907 (1999) 008 [arXiv:hep- th/9904131]; Branes at orbifolded conifold singularities and supersymmetric gauge field theories. K Oh, R Tatar, arXiv:hep-th/9906012JHEP. 991031K. Oh and R. Tatar, "Branes at orbifolded conifold singularities and super- symmetric gauge field theories," JHEP 9910, 031 (1999) [arXiv:hep-th/9906012]; Orientifolding the conifold. J Park, R Rabadan, A M Uranga, arXiv:hep-th/9907086Nucl. Phys. B. 57038J. Park, R. Rabadan and A. M. Uranga, "Orientifolding the conifold," Nucl. Phys. B 570, 38 (2000) [arXiv:hep-th/9907086]; Conifolds with discrete torsion and noncommutativity. K Dasgupta, S Hyun, K Oh, R Tatar, arXiv:hep-th/0008091JHEP. 000943K. Dasgupta, S. Hyun, K. Oh and R. Tatar, "Coni- folds with discrete torsion and noncommutativity," JHEP 0009, 043 (2000) [arXiv:hep- th/0008091]; Renormalization group flows on D3 branes at an orbifolded conifold. K Oh, R Tatar, arXiv:hep-th/0003183JHEP. 000530K. Oh and R. Tatar, "Renormalization group flows on D3 branes at an orbifolded coni- fold," JHEP 0005, 030 (2000) [arXiv:hep-th/0003183]. Moduli Space Of Calabi-Yau Manifolds. P Candelas, X De La Ossa, Nucl. Phys. B. 355455P. Candelas and X. de la Ossa, "Moduli Space Of Calabi-Yau Manifolds," Nucl. Phys. B 355, 455 (1991). Generalized conifolds and four dimensional N = 1 superconformal theories. S Gubser, N Nekrasov, S Shatashvili, hep-th/9811230JHEP. 99053S. Gubser, N. Nekrasov and S. Shatashvili, "Generalized conifolds and four dimensional N = 1 superconformal theories," JHEP 9905, 003 (1999) [hep-th/9811230]; A Ceresole, G Dall&apos;agata, R , S Ferrara, hep- th/9905226Spectrum of type IIB supergravity on AdS 5 × T 11 : Predictions on N = 1 SCFT's. 6166001A. Ceresole, G. Dall'Agata, R. D'Auria and S. Ferrara, "Spectrum of type IIB supergravity on AdS 5 × T 11 : Predictions on N = 1 SCFT's," Phys. Rev. D 61 (2000) 066001, [hep- th/9905226]; KK spectroscopy of type IIB supergravity on AdS(5) x T(11). A Ceresole, G Dall&apos;agata, R , hep-th/9907216JHEP. 99119A. Ceresole, G. Dall'Agata and R. D'Auria, "KK spectroscopy of type IIB supergravity on AdS(5) x T(11)," JHEP 9911, 009 (1999) [hep-th/9907216]. Large N superconformal gauge theories and supergravity orientifolds. A Fayyazuddin, M Spalinski, hep-th/9805096Nucl. Phys. B. 535219A. Fayyazuddin and M. Spalinski, "Large N superconformal gauge theories and super- gravity orientifolds," Nucl. Phys. B 535, 219 (1998) [hep-th/9805096]. The large N limit of N = 2,1 field theories from three-branes in F-theory. O Aharony, A Fayyazuddin, J Maldacena, hep-th/9806159JHEP. 980713O. Aharony, A. Fayyazuddin and J. Maldacena, "The large N limit of N = 2,1 field theories from three-branes in F-theory," JHEP 9807, 013 (1998) [hep-th/9806159]. The large N limit of N = 1 field theories from F theory. C H Ahn, K Oh, R Tatar, hep-th/9808143Mod. Phys. Lett. A. 14369C. h. Ahn, K. Oh and R. Tatar, "The large N limit of N = 1 field theories from F theory," Mod. Phys. Lett. A 14, 369 (1999) [hep-th/9808143]. Field theory questions for string theory answers. O Aharony, J Sonnenschein, S Yankielowicz, S Theisen, hep-th/9611222Nucl. Phys. B. 493177O. Aharony, J. Sonnenschein, S. Yankielowicz and S. Theisen, "Field theory questions for string theory answers," Nucl. Phys. B 493, 177 (1997) [hep-th/9611222]. Open strings on plane waves and their Yang-Mills duals. D Berenstein, E Gava, J Maldacena, K S Narain, H Nastase, hep-th/0203249D. Berenstein, E. Gava, J. Maldacena, K. S. Narain and H. Nastase, "Open strings on plane waves and their Yang-Mills duals," [hep-th/0203249]. String Interactions in PP-Waves. R Gopakumar, hep-th/0205174R. Gopakumar, "String Interactions in PP-Waves", [hep-th/0205174].
[]
[ "Retrieval Based Time Series Forecasting", "Retrieval Based Time Series Forecasting" ]
[ "Baoyu Jing [email protected] \nUniversity of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n\n", "Si Zhang [email protected] \nUniversity of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n\n", "Meta Yada Zhu [email protected] \nUniversity of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n\n", "Baoyu Jing \nUniversity of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n\n", "Si Zhang \nUniversity of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n\n", "Yada Zhu \nUniversity of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n\n", "Bin Peng [email protected] \nUniversity of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n\n", "Kaiyu Guan \nUniversity of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n\n", "Andrew Margenot [email protected] \nUniversity of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n\n", "Hanghang Tong [email protected] \nUniversity of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n\n" ]
[ "University of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n", "University of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n", "University of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n", "University of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n", "University of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n", "University of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n", "University of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n", "University of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n", "University of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n", "University of Illinois at Urbana-Champaign\nIBM Research Bin Peng\nUniversity of Illinois at Urbana-Champaign\n" ]
[ "CIKM'22 workshop on Applied Machine Learning Methods for Time Series Forecasting (AMLTS)" ]
Time series data appears in a variety of applications such as smart transportation and environmental monitoring. One of the fundamental problems for time series analysis is time series forecasting. Despite the success of recent deep time series forecasting methods, they require sufficient observation of historical values to make accurate forecasting. In other words, the ratio of the output length (or forecasting horizon) to the sum of the input and output lengths should be low enough (e.g., 0.3). As the ratio increases (e.g., to 0.8), the uncertainty for the forecasting accuracy increases significantly. In this paper, we show both theoretically and empirically that the uncertainty could be effectively reduced by retrieving relevant time series as references. In the theoretical analysis, we first quantify the uncertainty and show its connections to the Mean Squared Error (MSE). Then we prove that models with references are easier to learn than models without references since the retrieved references could reduce the uncertainty. To empirically demonstrate the effectiveness of the retrieval based time series forecasting models, we introduce a simple yet effective two-stage method, called ReTime consisting of a relational retrieval and a content synthesis. We also show that ReTime can be easily adapted to the spatial-temporal time series and time series imputation settings. Finally, we evaluate ReTime on real-world datasets to demonstrate its effectiveness.
10.48550/arxiv.2209.13525
[ "https://export.arxiv.org/pdf/2209.13525v1.pdf" ]
252,545,150
2209.13525
d6841564925af20fc23a35627f07b7c1a12e96fb
Retrieval Based Time Series Forecasting 2022. October 17-21, 2022 Baoyu Jing [email protected] University of Illinois at Urbana-Champaign IBM Research Bin Peng University of Illinois at Urbana-Champaign Si Zhang [email protected] University of Illinois at Urbana-Champaign IBM Research Bin Peng University of Illinois at Urbana-Champaign Meta Yada Zhu [email protected] University of Illinois at Urbana-Champaign IBM Research Bin Peng University of Illinois at Urbana-Champaign Baoyu Jing University of Illinois at Urbana-Champaign IBM Research Bin Peng University of Illinois at Urbana-Champaign Si Zhang University of Illinois at Urbana-Champaign IBM Research Bin Peng University of Illinois at Urbana-Champaign Yada Zhu University of Illinois at Urbana-Champaign IBM Research Bin Peng University of Illinois at Urbana-Champaign Bin Peng [email protected] University of Illinois at Urbana-Champaign IBM Research Bin Peng University of Illinois at Urbana-Champaign Kaiyu Guan University of Illinois at Urbana-Champaign IBM Research Bin Peng University of Illinois at Urbana-Champaign Andrew Margenot [email protected] University of Illinois at Urbana-Champaign IBM Research Bin Peng University of Illinois at Urbana-Champaign Hanghang Tong [email protected] University of Illinois at Urbana-Champaign IBM Research Bin Peng University of Illinois at Urbana-Champaign Retrieval Based Time Series Forecasting CIKM'22 workshop on Applied Machine Learning Methods for Time Series Forecasting (AMLTS) 2022. October 17-21, 2022University of Illinois at Urbana-Champaign Kaiyu Guan [email protected] University of Illinois at Urbana-Champaign Andrew Margenot University of Illinois at Urbana-Champaign Hanghang Tong ACM Reference Format: Time series data appears in a variety of applications such as smart transportation and environmental monitoring. One of the fundamental problems for time series analysis is time series forecasting. Despite the success of recent deep time series forecasting methods, they require sufficient observation of historical values to make accurate forecasting. In other words, the ratio of the output length (or forecasting horizon) to the sum of the input and output lengths should be low enough (e.g., 0.3). As the ratio increases (e.g., to 0.8), the uncertainty for the forecasting accuracy increases significantly. In this paper, we show both theoretically and empirically that the uncertainty could be effectively reduced by retrieving relevant time series as references. In the theoretical analysis, we first quantify the uncertainty and show its connections to the Mean Squared Error (MSE). Then we prove that models with references are easier to learn than models without references since the retrieved references could reduce the uncertainty. To empirically demonstrate the effectiveness of the retrieval based time series forecasting models, we introduce a simple yet effective two-stage method, called ReTime consisting of a relational retrieval and a content synthesis. We also show that ReTime can be easily adapted to the spatial-temporal time series and time series imputation settings. Finally, we evaluate ReTime on real-world datasets to demonstrate its effectiveness. INTRODUCTION Time series analysis has received paramount interest in numerous real-world applications [1][2][3][4][5][6][7], such as smart transportation and environmental monitoring. Accurately forecasting time series provides valuable insights for making transport policies [8] and climate change policies [9]. One of the fundamental problems for time series analysis is time series forecasting. Despite the success of recent deep learning methods for time series forecasting [1][2][3][4][5], a sufficient observation of the input historical time series is required. Specifically, the ratio of the output length (i.e., forecasting horizon) to the sum of the input and output length should be sufficiently low (e.g., 0.3). In fact, this ratio is a special case of the missing rate used in time series imputation, and thus for clarity and consistency with the literature, we formulate the task of time series forecasting from the perspective of time series imputation in this paper. Please refer to Section 2 for details. When the missing rate increases to a high level (e.g., 0.8), the uncertainty of the forecasting accuracy will increase significantly. In real-world applications, it is common that one wants to forecast future values based on very limited observations. In smart transportation, government administrators might be interested in forecasting the traffic conditions of a road with broken sensors [10]. In environmental monitoring, geologists always have a desire of obtaining the temperature of a specific location, where sensors cannot be easily placed [11]. To address this problem, we first formally quantify the uncertainty of the forecasting based on the entropy of the ground truth conditioned on the predicted values, and show its connections to the Mean Squared Error (MSE). Then we theoretically prove that models with reference time series are easier to train than those without references, since the references could reduce uncertainties. Motivated by the theoretical analysis, we introduce a simple yet effective two-stage time forecasting method called ReTime. Given a target time series, in the first relational retrieval stage, ReTime retrieves the references from a database based on the relations among time series. We use relational retrieval rather than content based retrieval since the input historical values of the target time series could be very unreliable when the input length is very short. Thus, content-based methods could retrieve unreliable references. In comparison, the relational information is usually reliable and easy to obtain in practice [12][13][14][15], such as whether two sensors are adjacent to each other in traffic/environmental monitoring. In the second content synthesis stage, ReTime synthesizes the future values based on the content of the target and the references. Next, we show that the proposed ReTime could be easily applied to the time series imputation task and the spatial-temporal time series setting. Finally, we empirically evaluate ReTime on two real-world datasets to demonstrate its effectiveness. The main contributions of the paper are summarized as follows: • Theoretical Analysis. We theoretically quantify the uncertainty of the predicted values based on conditional entropy and show its connections to the MSE. We also theoretically demonstrate that models with references are easier to train than those without references. • Algorithm. We introduce a two-stage method ReTime for time series forecasting, which is comprised of relational retrieval and content synthesis. ReTime can also be easily applied to the spatial-temporal time series and time series imputation settings. • Empirical Evaluation. We evaluate ReTime on two realworld datasets and various settings (i.e., single/spatial temporal forecasting/imputation) to demonstrate its effectiveness. PRELIMINARY The ratio /( + ) of the output length to the sum of the input length and output length is a special case of the missing rate used in time series imputation, and thus we formulate tasks of time series forecasting from the perspective of time series imputation. We summarize the mathematical notations in Table 1. Definition 2.1 (Single Time Series Forecasting). Given an incomplete target time series X ∈ R × , where and are the numbers of time steps and variates, along with its indicator mask M ∈ {0, 1} × , which indicates the absence/presence of the data points, the task aims to generate a new target time seriesX ∈ R × to predict the missing values (1 − M) ⊙X. Definition 2.2 (Retrieval Based Single Time Series Forecasting). Given an incomplete target time series X ∈ R × , where and are the numbers of time steps and variates, along with its indicator mask M ∈ {0, 1} × , the task aims to generate a new target time seriesX ∈ R × to predict the missing values (1 − M) ⊙X based on the input target time series X and the reference time series {Y } =1 retrieved from a database {Y ′ ∈ R ′ × } =1 , where ≫ is the number of time series and ′ ≫ . Definition 2.3 (Spatial-Temporal Time Series Forecasting). Given an incomplete spatial-temporal time series X ∈ R × × , where , and are the numbers of time series, time steps, and variates, along with its indicator mask M ∈ {0, 1} × × , where 0/1 indicates the absence/presence of the data points, and its adjacency matrix A ∈ R × , the task is to generate a new time seriesX ∈ R × × to predict the missing values (1 − M) ⊙X. Note that for forecasting, missing points and zeros are concentrated at the end of the target X and mask M after a certain separation time step : M[ : ] = 0. ,^,˜and are random variables for incomplete, generated, complete, and retrieved time series. and are the generation and retrieval models. is the relation between^and . THEORETICAL MOTIVATION An illustration of graphical models for methods with or without references is presented in Figure 1, which shows the relations among random variables. ,˜,^and denote the random variables for the incomplete target, complete target, generated target, and retrieved reference time series respectively. and denote the generation and retrieval model respectively. is the relation between^and˜. Following the common practice for linear regression [16], which assumes a Gaussian noise between^and˜, we define the relation as: (x|x) = N (x|x, 2 I)(1) where N denotes the Gaussian distribution,x andx ∈ R denote the values ofX andX ∈ R ( − )× at a future time step ∈ [ , ], is the standard deviation, and I ∈ R × is the identity matrix. Note that X[: ] =X[: ] for historical steps ∈ [1, ), and thus we only consider the future time steps ∈ [ , ]. We first quantify the uncertainty Δ for the accuracy of^in Definition 3.1 as the entropy of˜conditioned on^. According to Equation 1 and the definition of conditional entropy, we can calculate the uncertainty Δ (Lemma 3.1). As the only parameter in Δ is the standard deviation in Equation 1, we can further prove that Δ is equivalent to the MSE between˜and^. Definition 3.1 (Uncertainty of^). Δ = (˜|^) (2) where denotes the entropy. Lemma 3.1 (Uncertainty Calculation). According to the definition of conditional entropy and Equation (1), we have: Δ = 2 (1 + log 2 2 )(3) Lemma 3.2 (Eqivalence between Uncertainty and MSE). The uncertainty is equivalent to MSE. Δ ⇔(4) Proof. The only parameter of Δ in Lemma 3.1 is the standard deviation , which can be estimated by: = 1 ∑︁ =1 ||x −x || 2(5) where is the total number of data pairs. The item under the square root is MSE between˜and^. □ We further study the relations among the inputs , , outputô f the generation model , and the complete time series˜based on their dependencies shown in Figure 1. Firstly, given˜and^, we show in Lemma 3.3 that minimizing their MSE loss is equivalent to maximizing their mutual information (˜;^). Secondly, we prove that adding the retrieved reference to the input could reduce the uncertainty for . Lemma 3.4 shows that methods with have a higher lower-bound than those without for the mutual information of the ground-truth˜and the predicted valueŝ : (˜;^) ≥ (˜; , ) ≥ (˜; ). Finally, due to the equivalence of MSE and MI (Lemma 3.3), we can conclude that models with ( Figure 1a) are easier to learn than models without (Figure 1b) under the MSE loss. Lemma 3.3 (Eqivalence between MSE and MI). Minimizing the MSE loss of˜and^is equivalent to maximize the mutual information of˜and^: (˜;^). min E (x,x) ||x −x|| 2 ⇔ max (˜,^)(6) where (x,x) indicates whetherx andx is a true pair. Proof. Minimizing MSE ofx andx is equivalent to maximizing the log-likelihood log (x|x) [16], where is given in Equation (1): min E (x,x) ||x −x|| 2 ⇔ max E (x,x) [log (x|x)](7) Therefore, we only need to prove max E (x,x) [log (x|x)] ⇔ max (˜;^)(8) In fact, (˜;^) = (˜) − (˜|^)(9) and (˜) is a constant since the ground-truth˜is fixed in the dataset. Thus, we have max (˜;^) ⇔ max − (˜|^)(10) According to the definition of conditional entropy, we have − (˜|^) = E (x,x) [log (x|x)](11) The proof is concluded by combining Equations (7)(10)(11). □ Lemma 3.4 (MI Monotonicity). The following MI inequalities hold for the graphical model shown in Figure 1a. (˜;^) ≥ (˜; , ) ≥ (˜; )(12) Sketch of Proof. The first inequality is derived based on the data processing inequality [17]. The second inequality holds since (˜; , ) = (˜; ) + (˜; | ) and (˜; | ) ≥ 0. □ METHODOLOGY Relational Retrieval When the missing rate of the target X is high (e.g., 0.8), it will be hard to accurately complete X merely based on the observed historical content of X. Even for recent forecasting methods [1], the uncertainty of the prediction accuracy could be very high under a high missing rate. To reduce the uncertainty, we propose to retrieve references {Y } =1 from the database {Y ′ } =1 , based on the relations R and A ′ . We choose relational retrieval over contentbased retrieval since the observed historical content could be noisy and unreliable when the missing rate is high. Figure 3: Content Synthesis Given A ′ ∈ R × and R, we first construct a new adjacency matrix A ∈ R ( +1)×( +1) by appending R to the last row and column of A ′ . Then we use the Random Walk with Restart (RWR) [18] to obtain the relational proximity scores between X and {Y ′ ∈ R ′ × } =1 , whose closed-form solution is given by: p = (1 − )(I −Ã) −1 e(13) whereà is the normalized adjacency matrix, I is the identity matrix and ∈ (0, 1) is the tunable damping factor; e ∈ {0, 1} +1 is the indicator vector of X, where e[ + 1] = 1 and e[ ] = 0 for ∀ ∈ [1, · · · , ]; p ∈ R +1 is the relational proximity vector where p[ ] describes the proximity between X and Y ′ . Given p, we retrieve the top time series {Y } =1 as the references. Instead of using the entire Y ∈ R ′ × as the reference, we only use a -length ( ≪ ′ ) snippet of it, since using the entire sequence could introduce much irrelevant noisy information. Let and ′ be the start time of the target X and the reference snippet respectively. Many time series have clear periodicity patterns, such as the weekly pattern of traffic monitoring data and the yearly pattern of the air temperature. In this paper, we exploit the periodicity patterns and set the time difference between the target and Δ = − ′ as the length of one period. Content Synthesis Despite the relational closeness of X and {Y } =1 , they usually have content discrepancies. For example, peaks and valleys of X and Y might be different, and Y might be noisy. Besides, there are also temporal dependencies among different time steps. Therefore, it is necessary to build a model to combine their content. Figure 3 presents an illustration of the content synthesis model, which is comprised of the input, aggregation, and output modules. The input module maps X ∈ R × and {Y ∈ R × } =1 into embeddings H ∈ R ( +1)× × , where is the size of hidden dimension. Then the aggregation module aggregates the content of H across the + 1 time series and time steps into the aggregated embeddings H. Finally, the output module generates the completed time seriesX based on the aggregated embeddings H. A -Input Module. The input module maps the time series into the embedding space. Given X and {Y } =1 , we first apply separate linear layers to them. Then we apply the position encoding [19] and layer norm [20] to obtain the embeddings H ∈ R × and {H ′ ∈ R × } =1 . Finally, they are concatenated intoĤ ∈ R ( +1)× × , whereĤ[1 : ] = [H ′ 1 , . . . , H ′ ] andĤ[ + 1] = H. B -Aggregation Module. The aggregation module jointly considers the content discrepancies between H and {H ′ } =1 at each time step and the temporal dependencies across time steps. Based on the multi-head self-attention [19], we build content and temporal attention models to handle the content discrepancies and temporal dependencies. The aggregation module sequentially applies the content and temporal attention models toĤ ∈ R ( +1)× × , which is comprised of blocks, and the structure of the block is shown in Figure 3. Within the block, firstly, the content attention computes content attention scores over the first dimension ( + 1 time series) ofĤ for each step ∈ [1, · · · , ], and produces the content embeddings Z ∈ R ( +1)× × . Secondly, the temporal attention computes temporal attention scores over the second dimension ( steps) of Z for each time series ∈ [1, · · · , + 1], and encodes temporal information into Z. Finally, a forward layer [19] is applied to Z to obtain the aggregated embeddingsĤ ∈ R ( +1)× × . C -Output Module. The output module maps the aggregated embeddingsĤ from the embedding space R back into the original space R of the time series. Specifically, the output module takes input asĤ[ + 1], which is the embeddings of the target X. Then a Multi-Layer Perceptron (MLP) is applied over H[ + 1] to generate the predicted target time seriesX. Adaptation to Other Settings A -Spatial-Temporal Time Series. Spatial-temporal time series is ubiquitous and have attracted a lot of attention [21][22][23][24]. Different from the retrieval-based single time series forecasting, where the target is a single time series out of the database X ∉ {Y } =1 , the target of the spatial-temporal time series is the entire dataset X = {Y } =1 . ReTime can be naturally adapted to spatial-temporal time series by treating each single time series of spatial-temporal time series as the target and the rest time series as the database. The relational retrieval can be interpreted as the diffusion graph kernel [18], which identifies the most important neighbors, and the content synthesis aggregates the information of the neighbors. B -Time Series Imputation. It is obvious that ReTime can be naturally applied to the time series imputation task for regularlysampled time series. Training During training, given a complete target time seriesX, we generate a binary mask M to obtain the incomplete target time series X = M⊙X, where ⊙ is the Hadamard product. For forecasting, the values after the pre-defined time step are set as zeros. For imputation, we randomly generate the values for the masks according to the predefined missing rate. Then we feed X with its references {Y } =1 , which are retrieved by the relational retrieval model, to the content synthesis model to obtainX. We use the standard Mean Squared Error (MSE) betweenX andX to train the model. EXPERIMENTS 5.1 Experimental Setup Datasets. We evaluate ReTime on two real-world datasets, where the shapes of the datasets are formulated as × × : Traffic dataset is collected from Caltrans Performance Measurement System (PeMS) 1 . It contains hourly average speed and occupancy collected from 2,000 sensor stations in District 7 of California during June 2018. The size of the dataset is 2000 × 720 × 2. The relation between two stations is whether they are adjacent. Temperature is a subset of version 3 of the 20th Century Reanalysis 2 data [25]. It contains the monthly average temperature from 2001 to 2015, which covers a 30 × 30 area of North America, from 30 • N to 60 • N, 80 • W to 110 • W. The shape is 900 × 180 × 1. The relation between two locations is whether they are adjacent. For the single time series setting, given time series, we randomly select 10%/10% time series for validation and testing. As a result, the train/validation/test splits are 1600/200/200 and 720/90/90 for the traffic and temperature datasets. For the spatial-temporal setting, we select 10%/10% snippets from the entire datasets for validation/test. After splitting data, we segment each time series into 24/12-length snippets for traffic/temperature datasets respectively, which cover one day/one year. Time series are normalized according to the mean and standard deviation of the training sets. 1 https://dot.ca.gov/programs/traffic-operations/mpr/pems-source 2 https://psl.noaa.gov/data/gridded/data.20thC_ReanV3.html Evaluation Tasks. We evaluate ReTime for both single time series and spatial-temporal time series, and on both forecasting and imputation tasks. For each setting, we fix the snippet length of the target and change the missing rate from 0.2 to 0.8. Comparison Methods. We compare ReTime with the following baselines. The methods for single time series setting include blockstyle methods: N-BEATS [26] and Informer [1], and RNN based methods: BRITS [27], E2GAN [28]. The methods for spatial-temporal time series setting include block-style methods: StemGNN [22] and MTGNN [23], and RNN based methods: DCRNN [29], NET 3 [24]. Additionally, we also use the reference (REF) with the highest relational score to the target as a baseline. Implementation Details. The hidden dimensions for the traffic/temperature datasets are 256/64. The numbers of blocks and attention heads are 8/4 for the traffic/temperature datasets. The learning rates are tuned within [0.001, 0.0001]. Early stopping is applied on the validation set to prevent over-fitting. is tuned within [1,5,10,20]. For the relational retrieval, we set the damping factor of RWR as = 0.9. Given a target time series snippet, which starts from the time , we select reference snippets starting from ′ , and denote their difference as Δ = − ′ . For forecasting, Δ is one week/one year for the traffic/temperature datasets, which is generally the length of one cycle. For imputation, we set Δ = 0, which has the best performance in experiments. When applying forecasting models to imputation tasks, we take the entire X as input and force the models to predict the entireX. Main Results We compare ReTime with various baselines for single/spatial-temporal time series forecasting/imputation tasks for different missing rates. Single Time Series. The results for single time series are presented in Figure 4, where the first and second rows show the Root Mean Squared Error (RMSE) for forecasting and imputation respectively. For both forecasting and imputation, compared with RNN based methods, i.e., BRITS and E2GAN, recent block-style methods, i.e., N-BEATS and Informer, have lower RMSE scores. ReTime has much lower RMSE scores than N-BEATS and Informer, demonstrating the effectiveness of the proposed strategy. An interesting observation for the temperature dataset is that the simple retrieval baseline REF performs significantly better than other state-of-the-art baselines, corroborating the power of the reference time series. To be more specific, firstly, as indicated by the performance of the state-of-the-art methods, there might exist some complex temporal patterns for the temperature data, which could not be easily captured by models merely based on the observed content of the targets. Secondly, the superior performance of REF over the state-of-the-art methods demonstrates that the retrieved time series can indeed significantly reduce uncertainty. Spatial-Temporal Time Series. The results for spatial-temporal time series are presented in Figure 5. The general observation is similar to the single time series that ReTime consistently performs better than the state-of-the-art methods. Ablation Study We study the impact of each component of ReTime on the traffic dataset, which is the largest dataset. Impact of the Attentions. As shown in Figure 6, compared with the full model ReTime , if we only use either the content or temporal attention, the performance will drop. Besides, spatial attention alone performs worse than temporal attention alone, showing the importance of modeling temporal dependencies. Beyond the First Order Neighbors. In Figure 6, the "1st-order" is ReTime using the 1st-order neighbors of the targets as references. ReTime is better than "1st-order", indicating that it is important to choose appropriate neighbors as references. RWR used in the relational retrieval stage could capture global proximity scores. Effect of the Content Synthesis Model. The "Retrieval Only" in Figure 6 uses the average of the retrieved references as the prediction. Compared with the retrieval-only model, ReTime performs better, demonstrating the effectiveness of the content synthesis. Effect of Using Prior Snippets for Forecasting. When the time series has clear periodic patterns, it is natural to resort to the historical snippets in the database for help. In Figure 7, "Δ = 0" means that the target and reference snippets have the same start time, and thus the values of both the targets and references are zeros after the separation time . "Δ = 1 week" means that the start time of the reference snippets is one week before the targets. Figure 7 shows that using prior snippets can significantly improve the model's performance. Impact of We study the performance of ReTime and its time/memory usage w.r.t. the number of references ∈ {0, 1, 5, 10, 20}. The experiments are conducted on the traffic dataset. Performance of ReTime. The performance of ReTime w.r.t. the number of references is presented in Figure 8, where denotes the missing rate. ReTime achieves the best performance when ∈ {5, 10} Note that for spatial-temporal forecasting, the top 1 reference snippet of a target is its own historical snippet. Time and Memory Usage. We fix the batch size as 100 and record the average training time (×10 −2 seconds) of each iteration and the GPU memory usage (×10 3 MegaBytes) for different . The results in Figure 9 show that the training time and memory usage grow linearly w.r.t. . Many deep learning methods have been proposed to generate time series, including Recurrent Neural Network (RNN) based methods [27,30] and block-style methods [1,26]. For example, Cao et al. [27] introduce BRITS which leverages the recurrent dynamics for both correlated and uncorrelated multivariate time series. Fortuin et al. [31] combines the Gaussian process to capture the temporal dynamics and reconstruct missing values by VAE [32]. Luo et al. [33] introduce GRUI and propose a two-stage GAN [34] model, the generator and discriminator of which are based on GRUI. E2GAN [28] further simplifies the generator by combining GRUI and denoising autoencoder such that the GAN-based imputation can be trained in an end-to-end manner. Gamboa et al. [2] explore different neural networks for time series analysis. Oreshkin et al. [26] introduce N-BEATS for explainable time series forecasting. Li et al. [3] propose an enhanced version of Transformer [19] for forecasting. Zhou et al. [1] propose Informer for long time series forecasting. Wu et al. [4] introduce AutoFormer, which reduces the complexity of Transformer. However, when the missing rate grows to a high level, the performance of these methods drops significantly. ReTime address this issue by retrieving relevant reference time series as an augmentation. Spatial-Temporal Time Series Methods Time series often co-evolve with each other. Networks/graphs are commonly used data structures to model the relations among objects [18,[35][36][37], which have also been used to model relations in spatial-temporal time series. Traditional methods leverage probabilistic graphical models to introduce graph regularizations [38,39]. Recently, Li et al. [29] introduce DCRNN combining the diffusion graph kernel with RNN. Yu et al. [40] and Zhao et al. [41] respectively propose STGCN and TGCN for modeling spatial-temporal time series. Jing et al. [24] introduce NET 3 which captures both explicit and implicit relations among time series. Cao et al. [22] introduce StemGNN which combines graph and discrete Fourier transform to jointly model spatial and temporal relations. Wu et al. [23] introduce MTGNN which learns the relation graph of time series. However, these methods do not allow models to refer to the historical snippets when forecasting future values, which perform worse than ReTime. Retrieval Based Generation The main idea underlying retrieval-based methods is to retrieve references from databases to guide generation. Cao et al. [42] propose Re 3 Sum to generate document summaries based on the retrieved templates. Song et al. [43] propose to generate dialogues based on the retrieved references. Lewis et al. [44] introduce Retrieval-Augmented Generation (RAG) for knowledge-intensive natural language processing tasks e.g., summarization [45]. Tseng et al. [46] propose RetrievalGAN to generate images by retrieving relevant images. Ordonez et al. [47] generate image descriptions based on the retrieved captions. To the best of our knowledge, we present the first retrieval-based deep generation model for time series data. CONCLUSION In this paper, we theoretically quantify the uncertainty of the predicted values and prove that retrieved references could help to reduce the uncertainty for prediction results. To empirically demonstrate the effectiveness of retrieval-based forecasting, we build a simple yet effective method called ReTime, which is comprised of a relational retrieval stage and a content synthesis stage. The experimental results on real-world datasets demonstrate the effectiveness of ReTime. Figure 1 : 1Graphical models for methods w. or w/o references. Figure 2 : 2Overview of ReTime. Given X and {Y ′ } =1 , ReTime first retrieves the top references {Y } =1 based on the relations A ′ and R, and then combines the content of X and {Y } =1 to generateX. Solid/dashed curves are observed/unobserved values. Figure 2 2is an overview of ReTime. In the first stage, ReTime retrieves the top references {Y } =1 for the target X from the database {Y ′ } =1 , based on the relations R between the target and database and the relation graph A ′ of the database. In the second stage, ReTime combines X and {Y } =1 to generateX. Figure 4 :Figure 5 : 45RMSE scores on single time series forecasting and imputation. The lower the better. RMSE scores on spatial-temporal time series forecasting and imputation. The lower the better. Figure 6 :Figure 7 : 67Ablation study on the traffic dataset. Effect of using prior snippets for forecasting. Figure 8 :Figure 9 : 89The performance of ReTime w.r.t. different . Time CIKM'22 AMLTS, October 17-21, 2022, Atlanta, GA, USA. Baoyu Jing et al. Informer: Beyond efficient transformer for long sequence time-series forecasting. Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hui Xiong, Wancai Zhang, AAAI. 2021Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hui Xiong, and Wancai Zhang. Informer: Beyond efficient transformer for long sequence time-series forecasting. In AAAI, 2021. John Cristian Borges Gamboa, arXiv:1701.01887Deep learning for time-series analysis. arXiv preprintJohn Cristian Borges Gamboa. Deep learning for time-series analysis. arXiv preprint arXiv:1701.01887, 2017. Enhancing the locality and breaking the memory bottleneck of transformer on time series forecasting. Shiyang Li, Xiaoyong Jin, Yao Xuan, Xiyou Zhou, Wenhu Chen, Yu-Xiang Wang, Xifeng Yan, NeurIPSShiyang Li, Xiaoyong Jin, Yao Xuan, Xiyou Zhou, Wenhu Chen, Yu-Xiang Wang, and Xifeng Yan. Enhancing the locality and breaking the memory bottleneck of transformer on time series forecasting. NeurIPS, 2019. Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting. Haixu Wu, Jiehui Xu, Jianmin Wang, Mingsheng Long, Advances in Neural Information Processing Systems. 34Haixu Wu, Jiehui Xu, Jianmin Wang, and Mingsheng Long. Autoformer: De- composition transformers with auto-correlation for long-term series forecasting. Advances in Neural Information Processing Systems, 34:22419-22430, 2021. Domain adaptive multi-modality neural attention network for financial forecasting. Dawei Zhou, Lecheng Zheng, Yada Zhu, Jianbo Li, Jingrui He, WWW '20: The Web Conference 2020. Yennun Huang, Irwin King, Tie-Yan Liu, and Maarten van SteenTaipei, TaiwanDawei Zhou, Lecheng Zheng, Yada Zhu, Jianbo Li, and Jingrui He. Domain adaptive multi-modality neural attention network for financial forecasting. In Yennun Huang, Irwin King, Tie-Yan Liu, and Maarten van Steen, editors, WWW '20: The Web Conference 2020, Taipei, Taiwan, April 20-24, 2020, pages 2230-2240. . Acm / Iw3c2, 10.1145/3366423.33802882020ACM / IW3C2, 2020. doi: 10.1145/3366423.3380288. URL https://doi.org/10.1145/ 3366423.3380288. Network of tensor time series. Yada Zhu, Hanghang Tong, Baoyu Jing, Jinjun Xiong, Nitin Gaur, Anna Wanda Topol, App. 17/184880US PatentYada Zhu, Hanghang Tong, Baoyu Jing, JinJun Xiong, Nitin Gaur, and Anna Wanda Topol. Network of tensor time series, September 8 2022. US Patent App. 17/184,880. Deep music analogy via latent representation disentanglement. Ruihan Yang, Dingsu Wang, Ziyu Wang, Tianyao Chen, Junyan Jiang, Gus Xia, arXiv:1906.03626arXiv preprintRuihan Yang, Dingsu Wang, Ziyu Wang, Tianyao Chen, Junyan Jiang, and Gus Xia. Deep music analogy via latent representation disentanglement. arXiv preprint arXiv:1906.03626, 2019. Exploring the impact of social economic variables on traffic safety performance in hong kong: A time series analysis. Xin Li, Liyu Wu, Xianfeng Yang, Safety science. Xin Li, Liyu Wu, and Xianfeng Yang. Exploring the impact of social economic variables on traffic safety performance in hong kong: A time series analysis. Safety science, 2018. Exploring the relationship between climate change and rice yield in bangladesh: An analysis of time series data. Md Abdur Rashid Sarker, Khorshed Alam, Jeff Gow, Agricultural Systems. 112Md Abdur Rashid Sarker, Khorshed Alam, and Jeff Gow. Exploring the relation- ship between climate change and rice yield in bangladesh: An analysis of time series data. Agricultural Systems, 112:11-16, 2012. Forecasting network data: Spatial interpolation of traffic counts from texas data. Xiaokun Wang, Kara M Kockelman, Transportation research record. 21051Xiaokun Wang and Kara M Kockelman. Forecasting network data: Spatial inter- polation of traffic counts from texas data. Transportation research record, 2105(1): 100-108, 2009. Spatial interpolation of climatic normals: test of a new method in the canadian boreal forest. A Ian, Nalder, Ross W Wein, Agricultural and forest meteorology. 924Ian A Nalder and Ross W Wein. Spatial interpolation of climatic normals: test of a new method in the canadian boreal forest. Agricultural and forest meteorology, 92(4):211-225, 1998. Towards sensor database systems. Philippe Bonnet, Johannes Gehrke, Praveen Seshadri, International Conference on mobile Data management. Philippe Bonnet, Johannes Gehrke, and Praveen Seshadri. Towards sensor data- base systems. In International Conference on mobile Data management, 2001. Gorilla: A fast, scalable, in-memory time series database. Tuomas Pelkonen, Scott Franklin, Justin Teller, Paul Cavallaro, Qi Huang, Justin Meza, Kaushik Veeraraghavan, Proceedings of the VLDB Endowment. the VLDB Endowment8Tuomas Pelkonen, Scott Franklin, Justin Teller, Paul Cavallaro, Qi Huang, Justin Meza, and Kaushik Veeraraghavan. Gorilla: A fast, scalable, in-memory time series database. Proceedings of the VLDB Endowment, 8(12):1816-1827, 2015. Littletable: A time-series database and its uses. Sean Rhea, Eric Wang, Edmund Wong, Ethan Atkins, Nat Storer, Proceedings of the 2017 ACM International Conference on Management of Data. the 2017 ACM International Conference on Management of DataSean Rhea, Eric Wang, Edmund Wong, Ethan Atkins, and Nat Storer. Littletable: A time-series database and its uses. In Proceedings of the 2017 ACM International Conference on Management of Data, pages 125-138, 2017. Deep r-th root of rank supervised joint binary embedding for multivariate time series retrieval. Dongjin Song, Ning Xia, Wei Cheng, Haifeng Chen, Dacheng Tao, KDD. Dongjin Song, Ning Xia, Wei Cheng, Haifeng Chen, and Dacheng Tao. Deep r-th root of rank supervised joint binary embedding for multivariate time series retrieval. In KDD, pages 2229-2238, 2018. Machine learning: a probabilistic perspective. P Kevin, Murphy, MIT pressKevin P Murphy. Machine learning: a probabilistic perspective. MIT press, 2012. Information, physics, and computation. Marc Mezard, Andrea Montanari, Oxford University PressMarc Mezard and Andrea Montanari. Information, physics, and computation. Oxford University Press, 2009. Fast random walk with restart and its applications. Hanghang Tong, Christos Faloutsos, Jia-Yu Pan, ICDM. IEEEHanghang Tong, Christos Faloutsos, and Jia-Yu Pan. Fast random walk with restart and its applications. In ICDM, pages 613-622. IEEE, 2006. Attention is all you need. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, Illia Polosukhin, NeurIPS. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In NeurIPS, pages 5998-6008, 2017. . Jimmy Lei Ba, Jamie Ryan Kiros, Geoffrey E Hinton, arXiv:1607.06450Layer normalization. arXiv preprintJimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. Temporal regularized matrix factorization for high-dimensional time series prediction. Hsiang-Fu Yu, Nikhil Rao, Dhillon, NeurIPS. Hsiang-Fu Yu, Nikhil Rao, and Inderjit S Dhillon. Temporal regularized matrix factorization for high-dimensional time series prediction. In NeurIPS, 2016. Spectral temporal graph neural network for multivariate time-series forecasting. Defu Cao, Yujing Wang, Juanyong Duan, Ce Zhang, Xia Zhu, Congrui Huang, Yunhai Tong, Bixiong Xu, Jing Bai, Jie Tong, Qi Zhang, NeurIPS. Defu Cao, Yujing Wang, Juanyong Duan, Ce Zhang, Xia Zhu, Congrui Huang, Yunhai Tong, Bixiong Xu, Jing Bai, Jie Tong, and Qi Zhang. Spectral temporal graph neural network for multivariate time-series forecasting. In NeurIPS, 2020. Connecting the dots: Multivariate time series forecasting with graph neural networks. Zonghan Wu, Shirui Pan, Guodong Long, Jing Jiang, Xiaojun Chang, Chengqi Zhang, In KDD. Zonghan Wu, Shirui Pan, Guodong Long, Jing Jiang, Xiaojun Chang, and Chengqi Zhang. Connecting the dots: Multivariate time series forecasting with graph neural networks. In KDD, 2020. Network of tensor time series. Baoyu Jing, Hanghang Tong, Yada Zhu, The WebConf 2021. Baoyu Jing, Hanghang Tong, and Yada Zhu. Network of tensor time series. In The WebConf 2021, pages 2425-2437, 2021. Towards a more reliable historical reanalysis: Improvements for version 3 of the twentieth century reanalysis system. C Laura, Slivinski, P Gilbert, Jeffrey S Compo, Whitaker, D Prashant, Sardeshmukh, S Benjamin, Chesley Giese, Rob Mccoll, Xungang Allan, Russell Yin, Holly Vose, Titchner, Quarterly Journal of the Royal Meteorological Society. 145724Laura C Slivinski, Gilbert P Compo, Jeffrey S Whitaker, Prashant D Sardeshmukh, Benjamin S Giese, Chesley McColl, Rob Allan, Xungang Yin, Russell Vose, Holly Titchner, et al. Towards a more reliable historical reanalysis: Improvements for version 3 of the twentieth century reanalysis system. Quarterly Journal of the Royal Meteorological Society, 145(724):2876-2908, 2019. N-beats: Neural basis expansion analysis for interpretable time series forecasting. Dmitri Boris N Oreshkin, Nicolas Carpov, Yoshua Chapados, Bengio, ICLRBoris N Oreshkin, Dmitri Carpov, Nicolas Chapados, and Yoshua Bengio. N-beats: Neural basis expansion analysis for interpretable time series forecasting. ICLR, 2020. Brits: Bidirectional recurrent imputation for time series. Wei Cao, Dong Wang, Jian Li, Hao Zhou, Lei Li, Yitan Li, NeurIPS. Wei Cao, Dong Wang, Jian Li, Hao Zhou, Lei Li, and Yitan Li. Brits: Bidirectional recurrent imputation for time series. NeurIPS, 2018. E2gan: End-to-end generative adversarial network for multivariate time series imputation. Yonghong Luo, Ying Zhang, Xiangrui Cai, Xiaojie Yuan, IJCAI. Yonghong Luo, Ying Zhang, Xiangrui Cai, and Xiaojie Yuan. E2gan: End-to-end generative adversarial network for multivariate time series imputation. In IJCAI, pages 3094-3100, 2019. Diffusion convolutional recurrent neural network: Data-driven traffic forecasting. Yaguang Li, Rose Yu, Cyrus Shahabi, Yan Liu, ICLRYaguang Li, Rose Yu, Cyrus Shahabi, and Yan Liu. Diffusion convolutional recurrent neural network: Data-driven traffic forecasting. ICLR, 2018. Recurrent neural networks for multivariate time series with missing values. Zhengping Che, Sanjay Purushotham, Kyunghyun Cho, David Sontag, Yan Liu, Scientific reports. 81Zhengping Che, Sanjay Purushotham, Kyunghyun Cho, David Sontag, and Yan Liu. Recurrent neural networks for multivariate time series with missing values. Scientific reports, 8(1):1-12, 2018. Gp-vae: Deep probabilistic time series imputation. Dmitry Vincent Fortuin, Gunnar Baranchuk, Stephan Rätsch, Mandt, International Conference on Artificial Intelligence and Statistics. PMLRVincent Fortuin, Dmitry Baranchuk, Gunnar Rätsch, and Stephan Mandt. Gp-vae: Deep probabilistic time series imputation. In International Conference on Artificial Intelligence and Statistics, pages 1651-1661. PMLR, 2020. Auto-encoding variational bayes. P Diederik, Max Kingma, Welling, arXiv:1312.6114arXiv preprintDiederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013. Multivariate time series imputation with generative adversarial networks. Yonghong Luo, Xiangrui Cai, Ying Zhang, Jun Xu, Xiaojie Yuan, In NeurIPS. Yonghong Luo, Xiangrui Cai, Ying Zhang, Jun Xu, and Xiaojie Yuan. Multivariate time series imputation with generative adversarial networks. In NeurIPS, 2018. Generative adversarial nets. Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio, NeurIPS. Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In NeurIPS, 2014. Hdmi: High-order deep multiplex infomax. Baoyu Jing, Chanyoung Park, Hanghang Tong, Proceedings of the Web Conference 2021. the Web Conference 2021Baoyu Jing, Chanyoung Park, and Hanghang Tong. Hdmi: High-order deep multiplex infomax. In Proceedings of the Web Conference 2021, pages 2414-2424, 2021. Graph communal contrastive learning. Bolian Li, Baoyu Jing, Hanghang Tong, Proceedings of the ACM Web Conference 2022. the ACM Web Conference 2022Bolian Li, Baoyu Jing, and Hanghang Tong. Graph communal contrastive learning. In Proceedings of the ACM Web Conference 2022, pages 1203-1213, 2022. Graph-mvp: Multi-view prototypical contrastive learning for multiplex graphs. Baoyu Jing, Yuejia Xiang, Xi Chen, Yu Chen, Hanghang Tong, arXiv:2109.03560arXiv preprintBaoyu Jing, Yuejia Xiang, Xi Chen, Yu Chen, and Hanghang Tong. Graph-mvp: Multi-view prototypical contrastive learning for multiplex graphs. arXiv preprint arXiv:2109.03560, 2021. Fast mining of a network of coevolving time series. Yongjie Cai, Hanghang Tong, Wei Fan, Ping Ji, SDM. SIAM. Yongjie Cai, Hanghang Tong, Wei Fan, and Ping Ji. Fast mining of a network of coevolving time series. In SDM. SIAM, 2015. Facets: Fast comprehensive mining of coevolving high-order time series. Yongjie Cai, Hanghang Tong, Wei Fan, Ping Ji, Qing He, KDD. Yongjie Cai, Hanghang Tong, Wei Fan, Ping Ji, and Qing He. Facets: Fast com- prehensive mining of coevolving high-order time series. In KDD, 2015. Spatio-temporal graph convolutional networks: a deep learning framework for traffic forecasting. Bing Yu, Haoteng Yin, Zhanxing Zhu, In IJCAI. Bing Yu, Haoteng Yin, and Zhanxing Zhu. Spatio-temporal graph convolutional networks: a deep learning framework for traffic forecasting. In IJCAI, 2018. T-gcn: A temporal graph convolutional network for traffic prediction. Ling Zhao, Yujiao Song, Chao Zhang, Yu Liu, Pu Wang, Tao Lin, Min Deng, Haifeng Li, IEEE Transactions on Intelligent Transportation Systems. 219Ling Zhao, Yujiao Song, Chao Zhang, Yu Liu, Pu Wang, Tao Lin, Min Deng, and Haifeng Li. T-gcn: A temporal graph convolutional network for traffic prediction. IEEE Transactions on Intelligent Transportation Systems, 21(9), 2019. Retrieve, rerank and rewrite: Soft template based neural summarization. Ziqiang Cao, Wenjie Li, Furu Wei, Sujian Li, In ACL. Ziqiang Cao, Wenjie Li, Furu Wei, Sujian Li, et al. Retrieve, rerank and rewrite: Soft template based neural summarization. In ACL, 2018. An ensemble of retrieval-based and generation-based human-computer conversation systems. Yiping Song, Cheng-Te Li, Jian-Yun Nie, Ming Zhang, Dongyan Zhao, Rui Yan, In IJCAI. Yiping Song, Cheng-Te Li, Jian-Yun Nie, Ming Zhang, Dongyan Zhao, and Rui Yan. An ensemble of retrieval-based and generation-based human-computer conversation systems. In IJCAI, 2018. Retrieval-augmented generation for knowledge-intensive nlp tasks. Patrick Lewis, Ethan Perez, Aleksandra Piktus, Fabio Petroni, Vladimir Karpukhin, Naman Goyal, Heinrich Küttler, Mike Lewis, Wen-Tau Yih, Tim Rocktäschel, NeurIPS. Patrick Lewis, Ethan Perez, Aleksandra Piktus, Fabio Petroni, Vladimir Karpukhin, Naman Goyal, Heinrich Küttler, Mike Lewis, Wen-tau Yih, Tim Rocktäschel, et al. Retrieval-augmented generation for knowledge-intensive nlp tasks. In NeurIPS, 2020. Multiplex graph neural network for extractive text summarization. Baoyu Jing, Zeyu You, Tao Yang, Wei Fan, Hanghang Tong, Proceedings of the 2021. the 2021Baoyu Jing, Zeyu You, Tao Yang, Wei Fan, and Hanghang Tong. Multiplex graph neural network for extractive text summarization. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing. Conference on Empirical Methods in Natural Language Processing, pages 133-139, 2021. Retrievegan: Image synthesis via differentiable patch retrieval. Hung-Yu Tseng, Hsin-Ying Lee, Lu Jiang, Ming-Hsuan Yang, Weilong Yang, ECCV. Hung-Yu Tseng, Hsin-Ying Lee, Lu Jiang, Ming-Hsuan Yang, and Weilong Yang. Retrievegan: Image synthesis via differentiable patch retrieval. In ECCV, 2020. Large scale retrieval and generation of image descriptions. Vicente Ordonez, Xufeng Han, Polina Kuznetsova, Girish Kulkarni, Margaret Mitchell, Kota Yamaguchi, Karl Stratos, Amit Goyal, Jesse Dodge, Alyssa Mensch, IJCV. Vicente Ordonez, Xufeng Han, Polina Kuznetsova, Girish Kulkarni, Margaret Mitchell, Kota Yamaguchi, Karl Stratos, Amit Goyal, Jesse Dodge, Alyssa Mensch, et al. Large scale retrieval and generation of image descriptions. In IJCV, 2016.
[]
[ "Nanoscale self-templating for oxide epitaxy with large symmetry mismatch", "Nanoscale self-templating for oxide epitaxy with large symmetry mismatch" ]
[ "Xiang Gao \nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Shinbuhm Lee \nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "John Nichols \nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Tricia L Meyer \nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Thomas Z Ward \nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Matthew F Chisholm \nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Ho Nyung Lee \nMaterials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n" ]
[ "Materials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Materials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Materials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Materials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Materials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Materials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Materials Science and Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA" ]
[]
Direct observations using scanning transmission electron microscopy unveil an intriguing interfacial bi-layer that enables epitaxial growth of a strain-free, monoclinic, bronze-phase VO 2 (B) thin film on a perovskite SrTiO 3 (STO) substrate. We observe an ultrathin (2-3 unit cells) interlayer best described as highly strained VO 2 (B) nanodomains combined with an extra (Ti,V)O 2 layer on the TiO 2 terminated STO (001) surface. By forming a fully coherent interface
10.1038/srep38168
[ "https://export.arxiv.org/pdf/1609.07966v1.pdf" ]
6,920,934
1609.07966
a9a2a08d7b11aa1307962dae7faee56e23a6cfd6
Nanoscale self-templating for oxide epitaxy with large symmetry mismatch Xiang Gao Materials Science and Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Shinbuhm Lee Materials Science and Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA John Nichols Materials Science and Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Tricia L Meyer Materials Science and Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Thomas Z Ward Materials Science and Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Matthew F Chisholm Materials Science and Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Ho Nyung Lee Materials Science and Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Nanoscale self-templating for oxide epitaxy with large symmetry mismatch 1Oxide epitaxyinterfacesVO 2STEM Direct observations using scanning transmission electron microscopy unveil an intriguing interfacial bi-layer that enables epitaxial growth of a strain-free, monoclinic, bronze-phase VO 2 (B) thin film on a perovskite SrTiO 3 (STO) substrate. We observe an ultrathin (2-3 unit cells) interlayer best described as highly strained VO 2 (B) nanodomains combined with an extra (Ti,V)O 2 layer on the TiO 2 terminated STO (001) surface. By forming a fully coherent interface with the STO substrate and a semi-coherent interface with the strain-free epitaxial VO 2 (B) film above, the interfacial bi-layer enables the epitaxial connection of the two materials despite their large symmetry and lattice mismatch. Epitaxial synthesis of complex oxides has stimulated considerable interest in creating novel functionalities and physical properties, through various means to control the close interactions among the order parameters, such as lattice, spin, charge, and orbital. [1][2][3][4] Heterostructures of oxide materials have also played an important role in discovering novel phenomena as they can produce well-defined interfaces to couple electronic and magnetic ground states, structure, lattice, crystallographic symmetry, etc. Most studies on the epitaxial growth of complex oxides have focused on isostructural materials, e.g. perovskites on perovskites. While for many binary oxides, such as TiO 2 and VO 2 , also offer intriguing physical properties, [5][6][7][8][9][10][11] there are only few substrates available with similar structures (lattice parameters and crystal symmetry). The fundamental insights into the epitaxial growth of binary oxides thin films on lattice and symmetry mismatched substrates are of vital importance for exploring their unprecedented potential. [12][13][14] Recently, high quality VO 2 polymorphs were successfully stabilized as epitaxial thin films using pulsed laser epitaxy (PLE) on perovskite substrates, such as SrTiO 3 . 15-17 Among VO 2 polymorphs, 17 bronze-phase VO 2 (B) has a monoclinic structure (with C2/m symmetry) whose lattice constants are a = 12.03, b = 3.69, c = 6.42 Å, and β = 106.6 o , 18 whereas SrTiO 3 (with Pm3m symmetry) has a cubic structure with the lattice constant of 3.905 Å. Note that while many previous studies focused on R and M1 phase VO 2 , recent studies in developing advanced energy storage found VO 2 (B) to be a promising cathode material for Li ion batteries. [19][20][21] It is rather surprising that VO 2 (B) films with corner-and edge-sharing oxygen octahedra (see Figures 1a and b) can be epitaxially grown on STO with corner-sharing octahedra, despite the different oxygen networks and the large biaxial lattice mismatch. In this work, we report how two very dissimilar materials can form an epitaxial heterostructure by aberration-corrected scanning transmission electron microscopy (STEM) imaging and electron energy-loss spectroscopy (EELS). We found an interfacial bi-layer at the VO 2 (B)/STO interface that enables epitaxial growth of a structurally complex, low symmetry film on a high symmetry substrate. Results and discussion High quality VO 2 (B) epitaxial films were grown on (001)-oriented STO by PLE under well-optimized growth conditions. The details on the epitaxial growth and crystal quality as well as associated physical properties can be found elsewhere. 17 indicates that the IL has a higher level of structural disorder, which leads to the electron dechannelling of the incident beam. [22][23][24] Based on a geometric phase analysis (GPA), it is also seen that the VO 2 in the IL undergoes a significant lattice expansion along the film surface normal as compared to the VO 2 (B) film (see Figure S2 in Supporting Information). This result is consistent with the rather large in-plane compression (-5.5%) of the VO 2 (B) film in the Spatially resolved STEM-EELS data from the interfacial region is presented in Figure 4. The topmost layer of this reconstructed surface was predicted to contain clustered quartets of edge-sharing square-pyramidal TiO 5 . It is probably that the extra (Ti,V)-O layer on TiO 2terminated STO can introduce edge sharing oxygen containing units, which is more consistent with the VO 2 (B) structure. To our knowledge, the formation of such an interface bi-layer is not ready to be rationalized by any the existing growth models that involve either phase transition [27][28][29] or phase separation 30-32 at film/substrate interfaces to accommodate inter-phase structural discontinuities. The observed results reveal unambiguously, at the initial growth stage, the formation of VO 2 (B)/STO heterostructure involves a structural reconstruction process at the substrate surface to facilitate the symmetry transition between the two distinct component structures, followed by the epitaxial growth of VO 2 (B) nanodomains. The VO 2 (B) nanodomains forms a fully coherent interface with the STO substrate and are subject to considerable lattice strain. Once the strain energy in the VO 2 (B) nanodomains exceeds some critical level, misfit dislocations are introduced and the VO 2 (B) film then continues to grow in a fully relaxed state. The much larger domain size in the relatively strain-free film is an expected result of increased adatom mobility on the relaxed surface. Formation of the interfacial VO 2 (B) nanodomains indicates a nanoscale self-templating process that enables the epitaxy of strain-free VO 2 (B) film on STO substrate. The results not only enable novel insights into atomic mechanism of complex heterostructure interface at an atomic scale, but also shed light on the epitaxial design of two materials with large symmetry and lattice mismatch. Methods Epitaxial synthesis. VO 2 (B) epitaxial films were deposited on (001) SrTiO 3 substrates by pulsed laser epitaxy. A sintered ceramic VO 2 target was ablated with a KrF excimer laser (λ = 248 nm) at a repetition rate of 5 Hz and laser fluence of 1 Jcm -2 . The optimized substrate temperature and oxygen pressure to grow high quality thin films were 500 o C and 20 mTorr, respectively, and the samples were in-situ post-annealed in 1 atm of O 2 for 1 hour at the growth temperature to ensure the oxygen stoichiometry. Detailed information on the synthesis of single-crystalline VO 2 (B) (V 4+ ) and V 2 O 3 (V 3+ ) thin films utilized for EELS analysis can be found elsewhere. 14 Scanning Transmission Electron Microscopy (STEM). Cross-sectional specimens oriented along the [100]STO direction for STEM analysis were prepared using ion milling after mechanical thinning and precision polishing (using water-free abrasive). High-angle annular dark-field (HAADF) and low-angle annular dark-field (LAADF) imaging and electron-energy loss spectroscopy (EELS) analysis were carried out in Nion UltraSTEM200 operated at 200 keV. The microscope is equipped with a cold field-emission gun and a corrector of third-and fifthorder aberrations for sub-angstrom resolution. Inner/outer detector angles of 78/240 mrad and 30/63 mrad were used for HAADF and LAADF imaging, respectively. The convergence semiangle for the electron probe was set to 30 mrad. Figure 1 1shows atomic structure projections and corresponding cross-sectional high-angle annular dark-field (HAADF) images taken along the [100] VO2(B) and [010] VO2(B) directions of VO 2 (B). In the Z-contrast HAADF images, the cation columns containing Ti (Z = 22), V (Z = 23), and Sr (Z = 38) are seen with intensities strongly dependent on their atomic number, while columns containing only light O (Z = 8) atoms are hardly visible. The image shown in Figure 1d provides the reason why VO 2 (B) is of particular interest for energy storage as the atomic structure seen from the [010] VO2(B) direction features an open framework that offers a good ionic diffusion pathway. The structural projection along [010] VO2(B) also reveals the clear symmetry mismatch between the film and substrate.Thus, we chose this orientation for the majority of the STEM investigations. Figures 2a and 2b show cross-sectional HAADF images of an epitaxial VO 2 (B) film grown on a STO substrate. The images were taken along the [100] STO direction. As shown in Figures 2a and 2b, the film is found to contain at least two domains aligned parallel to the [100] VO2(B) and [010] VO2(B) directions, i.e. orthogonally positioned with respect to the [100]direction of the STO substrate. In fact, the film contains two additional domains that are rotated 180 degrees about the surface normal from those imaged inFigure 2. A thin (typically ~2 nm thick) interfacial layer (IL) can be seen between the VO 2 (B) film and the STO substrate. Based on fast-Fourier transformation (FFT) analysis, an array of misfit dislocations has formed between the IL and the structurally relaxed VO 2 (B) film (as indicated inFigures 2a and 2b, and in the corresponding FFT images inFigure S1in Supporting Information). The interface between the STO substrate and the IL appears to be fully coherent. As shown inFigures 2a and 2b, the average spacing between dislocations observed along the [100] VO2(B) direction is 3.6 ± 0.9 nm, while it is 7.9 ± 1.1 nm when seen along the [010] VO2(B) direction. These spacings are in good agreement with calculated values of ~3.5 nm and ~7.2 nm obtained using the lattice mismatch of +5.5 % for the [100]VO 2 (B) || [100]STO projection and -2.7 % for the orthogonal [010]VO 2 (B) || [100]STO projection. This result reveals unambiguously that the large bi-axial lattice mismatch between the film and substrate is accommodated by the creation of dislocations at the VO 2 (B)/TL interface, i.e. strain-free VO 2 (B) epitaxial films are obtained. Figure 2c shows a low-angle annular dark-field (LAADF) image taken from the sample seen along the [010] VO2(B) /[100] STO direction. The LAADF image highlights the interlayer, which is substantially brighter than the film or the substrate. This brighter contrast in a LAADF image [ 100 ] 100VO2(B) /[100] STO projection that will cause the observed out-of-plane expansion. Figure 3a 3ashows a HAADF image of the IL taken along the [010] VO2(B) direction. While there is a region of the IL (Figure 3b) that clearly duplicates the projected structure of the relaxed VO 2 (B) film above it (except that it is rotated 180 degrees about the surface normal), most of this layer (and its FFT, Figures 3d,e) looks to be a superposition of [100], [-100], [010], and [0-10] projections of epitaxially strained VO 2 (B). The other important feature of the IL is the extra atomic layer between the STO and VO 2 (B) indicated with a black arrow in Figures 3b and 3c. The atomic layer shows a periodic, but different arrangement of B-site atoms than that of the TiO 2 -termined STO surface. The HAADF images show the out-of-plane lattice spacing between the topmost TiO 2 layer of STO and the extra (Ti,V)-O layer to be 2.4 ± 0.1 Å, which is significantly larger than the 1.9Å (001) plane spacing in STO. The intensity variations indicate that, in this [100] STO projection, the extra layer contains additional (Ti,V) columns with roughly 1/2 the B-site density of its neighboring atom columns. Figures 4a and 4b respectively show element maps using the Ti-L 2,3 and V-L 2,3 signals taken from the same interfacial region shown inFigure 4c. The V-L 2,3 signal inFigure 4ashows a chemically abrupt interface between the film and STO substrate. On the other hand, the Ti-L 2,3 signal is seen to extend into the IL.Figure 4d shows background-subtracted Ti-L, V-L and O-K EELS profiles obtained layer-by-layer across the IL. Standard spectra obtained from singlecrystalline VO 2 (B) (V 4+ ) and V 2 O 3 (V 3+ ) thin films are also included for comparison. The peak position of V-L 2,3 edges are seen to remain fixed indicating little to no change in the valence state of V in the IL and the VO 2 (B) film. The Ti-L 2,3 EELS fine structure obtained from the extra (Ti,V)-O layer on the STO substrate surface shows broadened L 3 and L 2 edges, as well as a shift of the e g peaks toward lower energy-loss (see Figure 4d). The observed electronic state and atomic structure of this extra layer are in good agreement with previous theoretical simulations 25 and STEM observations 26 of the c(4 × 2) reconstructed STO(001) surface composed of a double-layer TiO 2 . Figure Captions Figure 1 . 1Atomic structure of VO 2 (B). a,b) Schematics and c,d) corresponding cross-sectional HAADF images of VO 2 (B) seen along a,c) the [100] and b,d) [010] directions. The hexagon in d) indicates the large open channel in VO 2 (B) useful for ionic conduction. Figure 2 .Figure 3 . 23Microstructure of the VO 2 (B)/STO interface. HAADF images show two growth twins orthogonally oriented along a) the [100] VO2(B) and b) [010] VO2(B) directions with respect to the [100] STO direction. c) LAADF image taken from the image in b), showing an extra intensity from the IL associated with increased electron beam dechanneling and, thus, scattering of electrons due to increased atomic disorder. High-resolution observation of IL. a) HAADF image showing the IL consists of nanodomains, e.g. I and I'. b,c) Magnified HAADF images taken from nanodomains I and I' marked by the dashed rectangles in a) showing in greater detail the local atom arrangements. d,e) FFT electron diffraction patterns obtained the VO 2 (B) film and the IL (nanodomain-I), respectively. The red and black arrows between b) and c) indicate extra atom planes formed at the upper and lower sides of IL, respectively. Figure 4 . 4Layer-by-layer EELS analysis. Elemental maps for a) V-L 2,3 and b) Ti-L 2,3 signals obtained from the interface region shown in c). d) Back-ground subtracted Ti-L 2,3 , V-L 2,3 , and O-K spectra obtained across the interface. The EELS spectra numbered 1 through 9 are obtained from the local atomic planes indicated in c). EELS profile intensity is normalized using the O-K edge beyond the ionization transitions to discrete states. AcknowledgementsThis work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. We would like to thank Qian He and Erjia Guo for helpful discussions.Additional InformationCompeting financial interests: The authors declare no competing financial interests. Emergent phenomena at oxide interfaces. H Y Hwang, Nature materials. 11Hwang, H. Y. et al. Emergent phenomena at oxide interfaces. Nature materials 11, 103- 113 (2012). Interface physics in complex oxide heterostructures. P Zubko, S Gariglio, M Gabay, P Ghosez, J.-M Triscone, Annu. Rev. Condens. Matter Phys. 2Zubko, P., Gariglio, S., Gabay, M., Ghosez, P. & Triscone, J.-M. Interface physics in complex oxide heterostructures. Annu. Rev. Condens. Matter Phys. 2, 141-165 (2011). Reversible redox reactions in an epitaxially stabilized SrCoOx oxygen sponge. H Jeen, Nature materials. 12Jeen, H. et al. Reversible redox reactions in an epitaxially stabilized SrCoOx oxygen sponge. Nature materials 12, 1057-1063 (2013). Epitaxial stabilization of oxides in thin films. O Y Gorbenko, S Samoilenkov, I Graboy, A Kaul, Chemistry of materials. 14Gorbenko, O. Y., Samoilenkov, S., Graboy, I. & Kaul, A. Epitaxial stabilization of oxides in thin films. Chemistry of materials 14, 4026-4043 (2002). Oxides which show a metal-to-insulator transition at the Neel temperature. F Morin, Physical Review Letters. 334Morin, F. Oxides which show a metal-to-insulator transition at the Neel temperature. Physical Review Letters 3, 34 (1959). A low-cost, high-efficiency solar cell based on dyesensitized. B O&apos;regan, M Grfitzeli, nature. 353O'regan, B. & Grfitzeli, M. A low-cost, high-efficiency solar cell based on dye- sensitized. nature 353, 737-740 (1991). Nanostructured materials for advanced energy conversion and storage devices. A S Arico, P Bruce, B Scrosati, J.-M Tarascon, W Van Schalkwijk, Nature materials. 4Arico, A. S., Bruce, P., Scrosati, B., Tarascon, J.-M. & Van Schalkwijk, W. Nanostructured materials for advanced energy conversion and storage devices. Nature materials 4, 366-377 (2005). Efficient photochemical water splitting by a chemically modified n-TiO2. S U Khan, M Al-Shahry, W B Ingler, science. 297Khan, S. U., Al-Shahry, M. & Ingler, W. B. Efficient photochemical water splitting by a chemically modified n-TiO2. science 297, 2243-2245 (2002). Oxide electronics utilizing ultrafast metal-insulator transitions. Z Yang, C Ko, S Ramanathan, Annual Review of Materials Research. 41Yang, Z., Ko, C. & Ramanathan, S. Oxide electronics utilizing ultrafast metal-insulator transitions. Annual Review of Materials Research 41, 337-367 (2011). Measurement of a solid-state triple point at the metal-insulator transition in VO2. J H Park, nature. 500Park, J. H. et al. Measurement of a solid-state triple point at the metal-insulator transition in VO2. nature 500, 431-434 (2013). . ultrafast electron diffraction. science. 346ultrafast electron diffraction. science 346, 445-448 (2014). Suppression of metal-insulator transition in VO2 by electric field-induced oxygen vacancy formation. J Jeong, science. 339Jeong, J. et al. Suppression of metal-insulator transition in VO2 by electric field-induced oxygen vacancy formation. science 339, 1402-1405 (2013). Why is anatase a better photocatalyst than rutile?-Model studies on epitaxial TiO2 films. T Luttrell, Scientific reports. 4Luttrell, T. et al. Why is anatase a better photocatalyst than rutile?-Model studies on epitaxial TiO2 films. Scientific reports 4 (2014). Growth control of the oxidation state in vanadium oxide thin films. S Lee, T L Meyer, S Park, T Egami, H N Lee, Applied Physics Letters. 105223515Lee, S., Meyer, T. L., Park, S., Egami, T. & Lee, H. N. Growth control of the oxidation state in vanadium oxide thin films. Applied Physics Letters 105, 223515 (2014). Textured metastable VO2 (B) thin films on SrTiO3 substrates with significantly enhanced conductivity. A Chen, Applied Physics Letters. 10471909Chen, A. et al. Textured metastable VO2 (B) thin films on SrTiO3 substrates with significantly enhanced conductivity. Applied Physics Letters 104, 071909 (2014). Selective growth of single phase VO2 (A, B, and M) polymorph thin films. A Srivastava, APL materials. 326101Srivastava, A. et al. Selective growth of single phase VO2 (A, B, and M) polymorph thin films. APL materials 3, 026101 (2015). Epitaxial stabilization and phase instability of VO2 polymorphs. S Lee, I N Ivanov, J K Keum, H N Lee, Scientific reports. 6Lee, S., Ivanov, I. N., Keum, J. K. & Lee, H. N. Epitaxial stabilization and phase instability of VO2 polymorphs. Scientific reports 6 (2016). The concentration wave approach to the pairwise interaction model for predicting the crystal structures of ceramics, I. B Pokrovskii, A Khachaturyan, Journal of Solid State Chemistry. 61Pokrovskii, B. & Khachaturyan, A. The concentration wave approach to the pairwise interaction model for predicting the crystal structures of ceramics, I. Journal of Solid State Chemistry 61, 137-153 (1986). Rechargeable lithium batteries with aqueous electrolytes. W Li, J R Dahn, D S Wainwright, Science-AAAS-Weekly Paper Edition-including Guide to Scientific Information. 264Li, W., Dahn, J. R. & Wainwright, D. S. Rechargeable lithium batteries with aqueous electrolytes. Science-AAAS-Weekly Paper Edition-including Guide to Scientific Information 264, 1115-1117 (1994). Nanoscroll Buffered Hybrid Nanostructural VO2 (B) Cathodes for High-Rate and Long-Life Lithium Storage. L Mai, Advanced materials. 25Mai, L. et al. Nanoscroll Buffered Hybrid Nanostructural VO2 (B) Cathodes for High- Rate and Long-Life Lithium Storage. Advanced materials 25, 2969-2973 (2013). VO2 nanowires assembled into hollow microspheres for high-rate and longlife lithium batteries. C Niu, Nano letters. 14Niu, C. et al. VO2 nanowires assembled into hollow microspheres for high-rate and long- life lithium batteries. Nano letters 14, 2873-2878 (2014). De-channelling contrast in annular dark-field STEM. J Cowley, Y Huang, Ultramicroscopy. 40Cowley, J. & Huang, Y. De-channelling contrast in annular dark-field STEM. Ultramicroscopy 40, 171-180 (1992). Detector geometry, thermal diffuse scattering and strain effects in ADF STEM imaging. S Hillyard, J Silcox, Ultramicroscopy. 58Hillyard, S. & Silcox, J. Detector geometry, thermal diffuse scattering and strain effects in ADF STEM imaging. Ultramicroscopy 58, 6-17 (1995). Scanning transmission electron microscopy: imaging and analysis. S J Pennycook, P D Nellist, Springer Science & Business MediaPennycook, S. J. & Nellist, P. D. Scanning transmission electron microscopy: imaging and analysis. (Springer Science & Business Media, 2011). Surface structures of SrTiO3 (001): A TiO2-rich reconstruction with ac (4× 2) unit cell. N Erdman, Journal of the American Chemical Society. 125Erdman, N. et al. Surface structures of SrTiO3 (001): A TiO2-rich reconstruction with ac (4× 2) unit cell. Journal of the American Chemical Society 125, 10050-10056 (2003). Bonding and structure of a reconstructed (001) surface of SrTiO3 from TEM. G Zhu, G Radtke, G A Botton, nature. 490Zhu, G.-z., Radtke, G. & Botton, G. A. Bonding and structure of a reconstructed (001) surface of SrTiO3 from TEM. nature 490, 384-387 (2012). Straininduced interface reconstruction in epitaxial heterostructures. N Lazarides, V Paltoglou, P Maniadis, G Tsironis, C Panagopoulos, Physical Review B. 84245428Lazarides, N., Paltoglou, V., Maniadis, P., Tsironis, G. & Panagopoulos, C. Strain- induced interface reconstruction in epitaxial heterostructures. Physical Review B 84, 245428 (2011). Role of interfacial transition layers in VO2/Al2O3 heterostructures. H Zhou, M F Chisholm, T.-H Yang, S J Pennycook, J Narayan, Journal of Applied Physics. 11073515Zhou, H., Chisholm, M. F., Yang, T.-H., Pennycook, S. J. & Narayan, J. Role of interfacial transition layers in VO2/Al2O3 heterostructures. Journal of Applied Physics 110, 073515 (2011). Domain epitaxy in TiO2/α-Al2O3 thin film heterostructures with Ti2O3 transient layer. M Bayati, Applied Physics Letters. 100251606Bayati, M. et al. Domain epitaxy in TiO2/α-Al2O3 thin film heterostructures with Ti2O3 transient layer. Applied Physics Letters 100, 251606 (2012). Polar oxide interface stabilization by formation of metallic nanocrystals. V K Lazarov, S A Chambers, M Gajdardziska-Josifovska, Physical Review Letters. 90216108Lazarov, V. K., Chambers, S. A. & Gajdardziska-Josifovska, M. Polar oxide interface stabilization by formation of metallic nanocrystals. Physical Review Letters 90, 216108 (2003).
[]
[ "The amazing dynamics of stochastic pattern formation and growth models inspired by the Conway's Game of Life", "The amazing dynamics of stochastic pattern formation and growth models inspired by the Conway's Game of Life" ]
[ "Leonid P Yaroslavsky [email protected] \nDept. of Physical Electronics\nSchool of Electrical Engineering\nTel Aviv University\n69978Tel AvivIsrael\n" ]
[ "Dept. of Physical Electronics\nSchool of Electrical Engineering\nTel Aviv University\n69978Tel AvivIsrael" ]
[]
Several modifications of the famous mathematical "Game of Life" are introduced by making "Game of Life" rules stochastic and mutual influence of cells in their 8-neighborhood on a rectangular lattice spatially non-uniform. Results are reported of experimental investigation of evolutionary dynamics of the introduced models. A number of new phenomena in the evolutionary dynamics of the models and collective behavior of patterns they generate are revealed, described and illustrated: formation of maze-like patterns as fixed points of the models, "self-controlled" growth, "eternal life in a bounded space" and "coherent shrinkage".
null
[ "https://arxiv.org/pdf/1304.8104v1.pdf" ]
20,401,622
1304.8104
192a00a09a100e8929a190a631dae0f04292ec84
The amazing dynamics of stochastic pattern formation and growth models inspired by the Conway's Game of Life Leonid P Yaroslavsky [email protected] Dept. of Physical Electronics School of Electrical Engineering Tel Aviv University 69978Tel AvivIsrael The amazing dynamics of stochastic pattern formation and growth models inspired by the Conway's Game of Life 1 Several modifications of the famous mathematical "Game of Life" are introduced by making "Game of Life" rules stochastic and mutual influence of cells in their 8-neighborhood on a rectangular lattice spatially non-uniform. Results are reported of experimental investigation of evolutionary dynamics of the introduced models. A number of new phenomena in the evolutionary dynamics of the models and collective behavior of patterns they generate are revealed, described and illustrated: formation of maze-like patterns as fixed points of the models, "self-controlled" growth, "eternal life in a bounded space" and "coherent shrinkage". Introduction. In this paper we consider evolutional dynamics of pattern formation and growth models derived through modifications of the famous mathematical model known as Conway's "Game of Life" [1]. In this model, 2D arrays of binary, i.e. assuming values 1( "live") or 2 ("empty"), cells arranged in nodes of a rectangular lattice within a rectangular "vital space" of a finite size (see (i) each "empty" cell that has exactly 3 "alive" neighbor cells in its 3x3 neighborhood on the rectangular lattice gives a "birth", i.e. becomes "live"; (ii) each "live" cell that has less than 2 and more than 3 "alive" cells in the neighborhood dies, i.e. becomes "empty"; (iii) in all other cases nothing happens. Patterns generated by this model in course of evolutional steps ,....) 2 , 1 ( ,  t t can formally be described by the equation: is an arbitrary initial binary pattern used as a seed, and   .                         3 , 2 , ,, 1 8 1 .  denotes element-wise logical "OR" operation on arrays of binary numbers. The remarkable property of the "Game of life" is that, from an arbitrary seed patters, it produces, in course of evolution, three types of pattern: -stable patterns, which, once appeared, remain unchanged unless they collide with neighbor patterns, which can happen in course of evolution; -periodical patterns ("oscillators"), which repeat themselves after a certain number of the evolution steps; obviously, stable patterns can be regarded as a special case of periodical patterns with a period of one step. -self-replicating moving patterns ("gilders"), which move across the lattice and replicate themselves in a shifted position after a certain number of steps; this can be regarded as a general "space-time" periodicity. Since its invention, "Game of life" has been intensively studied experimentally by numerous enthusiasts, which have been competing between each other in discovery of new stable patterns and oscillators. As a result, very many types of stable patterns, "oscillators" and "gliders" have been discovered, including very sophisticated ones such as "Gosper Glider Gun" and "2C5 Space Ship Gun P690". These patterns as well as their classification issues and rates of appearance one can find elsewhere ([2-9]). "Game of life" is not only a splendid plaything for mathematicians and amateurs. It can also, after appropriate modifications, serve as a base for evolutionary 2D pattern formation and growth models, and, more generally, for models of 2D nonlinear dynamic systems with feedback. We pursued this option in Refs. [10][11][12][13]. Specifically, we -interpreted "Game of life" in terms of nonlinear dynamic systems with feedback, and showed that it can algorithmically be regarded as akin to pseudo-random number generators and to earlier stochastic growth models by M. ) -introduced a stochastic modification of the Conway's model, in which death of "live" cell with less than 2 and more than 3 neighbors occurs with a certain probability InPattern098; rate of "live" cells 0.02 "Circle" "Squares" "Star" "Star2" "Blobs" Conway's Game of Life modifications Considered in this paper modifications of the standard Conway's model given by Eq. 1 are made in two ways: (i) "Deaths" of "live" cells that have less than two and more than three "live" neighbors are made stochastic with a probability 1  death P . (ii) Counting the number of "live" cells in 8-neighborhood of each cell by means of "Isotropic" mask is introduced with a purpose of securing better than in the standard model correspondence of mutual influence of the cells to Euclidian distance between them in a real 2D space. "Diagonal mask" is a 45 o rotated "isotropic" mask. In the "Cross" mask, influence of more distant diagonal cells is further decreased comparing to the "Isotropic" mask, and in the "Cross4" mask it is completely eliminated. "Cross4diagonal" mask is a 45 o rotated version of the "Cross4" mask. summation     l k S t , Masks "Hex0" and "Hex1" are introduced in an attempt to simulate cell neighborhoods that consists of 6 cells instead of the standard 8 cell neighborhood. Mask "Hex2" is a 45 o rotated version of the mask "Hex1". Ordering of chaos: the standard model We start with the more or less known type of the evolutionary dynamics, the "ordering of chaos". By the "ordering of chaos" we mean formation, out of, generally, chaotic seed patterns, stable formations that are "fixed points" of the model, i.e. formations, which, once appeared, are either not changing or oscillating in space or in "time" in course of evolution . In this section we revisit the "ordering of chaos" phenomenon for the standard model. Obviously, fixed points of the model must be patterns that consist of cells with only two or three "live" neighbors, which would not die on the next step of evolution, and of "empty" cells with more than or less than three "live" neighbor cells, which would not come to life. Stable and oscillating patterns generated by the standard non -stochastic model, i.e. for the probability of "death" 1  death P , are well known and well reported. However, it turns out that such patterns are rather the exception than the rule because they appear only when death P is strictly equal to one. Experiments show that, as soon as death P becomes only a little less than one, oscillating formations characteristic for the non-stochastic standard Conway's model occasionally collapse, producing chaotic clouds of "live" and "dying" cells that collide with each other and do not seem stabilizing ever. As death P goes down further, these clouds are becoming denser in the "vital space", gradually fill in it keeping their "births/deaths" activity seemingly permanently and demonstrating kind of "eternal life". For the probability of "death" lower than 5 . 0  death P , substantial changes in the evolutionary dynamics of the model are becoming noticeable. After a certain number of evolutionary steps, in different parts of the "vital space" patches of stripy patterns of different size and orientation emerge and grow in the see of active chaos. Borders of the patches remain to be active until death P is higher than about 0.3. When death P becomes lower than about 0.3, patches borders tend to stabilize, after a certain number of the evolutionary steps, the higher the lower death P , into mature maze-like patterns consisting of patches of alternative stripes of "live" and "empty" cells chaotically interrupted by dislocations, in which the direction of stripes is either switched to the perpendicular one or stripes of "live" and "empty" cells switch their positions. An example of such patterns is shown in Figure 3, right image. These patterns are, obviously, fixed points of the standard non-stochastic model as well. Therefore, for the stochastic modification of the standard Conway's model "ordering of "chaos" is possible only if the probability of "death" is sufficiently small ( 3 . 0  death P ). For higher probabilities of "death" the models exhibits the "eternal life" dynamics. . Growth of maze-like patterns from sparse or solid seed patterns. In color coded images cells that will "die" on the next step are shown pink, cells, in which "birth" will take place on the next step, are shown green, "live" stable cells are shown blue and empty cells are shown black. Genesis of emerging and growing of the maze-like patters can be better seen when seed patterns are sparse or solid. For sparse pseudo-random seed patterns, few isolated seeds of growth occasionally emerge that start growing and merging each other and gradually fill in the entire vital space. For solid seed patterns, growth starts at the pattern borders and then quite rapidly, in terms of the number of evolutionary steps, propagates to empty parts of the vital space. These processes are illustrated in Emergence of maze-like patterns as fixed points of the standard Conways' model for low probability of "death" has been already reported earlier ( [10][11][12][13]). In the experiments reported in this paper, a new remarkable property of the maze-like stable patterns generated by the stochastic modification of the standard Conway's model was observed, their capability to grow and merge. If one takes, as a seed pattern, a fragment of a maze-like stable pattern or a maze-like stable pattern with a hole and let them evolve with the probability of "death" 25 . 0  death P , the former will grow until it fills in the entire vital space and the latter will grow to fill the hole, as it is illustrated in Figure 5. One can also implant fragments of one maze-like stable pattern into another and use the pattern with the implanted fragment as a seed pattern for further evolution of the model. After some number of evolutionary steps, implanted fragment perfectly tailors itself in the new "home" as it is shown in Figure 6. In color coded images, cells that will "die" on the next step are shown red, cells to give birth are shown green, "live" cells are shown blue and empty cells are shown black. In black and white images, "live" cells are shown white. Maze-like pattern SP1. Circular window CW (while -1, black-0) - Maze-like pattern SP2. Seed pattern SP1.*CW+SP2_2.*(1-CW) Stable pattern evolved from the seed pattern SP1.*CW+SP2.*(1-CW) . "Ordering of chaos": other models For 1  death P , the other models introduced in Sect. 2 demonstrate "ordering of chaos" dynamics similar to that of the standard stochastic Conway's model: their evolution ends up with a set of isolated stable formations illustrated in Figure 7 or oscillating patterns. Examples of oscillating patterns observed in the experiments are shown in Figure 8 (for Isotropic, Diag, Cross and Cross4 masks) and in Figure 9 (for Hex0, Hex1 and Hex2 masks). Standard mask Isotropic_mask Diagonal_mask Isotropic_mask oscillators Osc_isotrop1_1-8 (period 8) Osc_isotrop2_1-4 (period 4) Osc_isotrop3_1-2 (period 2) Osc_isotrop3_1-6 (period 6) Osc_isotrop4_1-2 (period 2) Osc_isotrop5_1-6 (period 6) Diagonal_mask oscillators Osc_Diag_1_1-4 (period 4) Osc_Diag_2_1-6 (period 6) Osc_Diag_3_1-2 (period 2) Osc_Diag_4_1-6 (period 6) Osc_Diag_5_1-4 (period 4) Osc_Diag_6_1-2 (period 2) Osc_Diag_7_1-2 (period 2) Osc_Diag_8_1-2 (period 2) Osc_Diag_9_1-2 (period 2) Cross_mask oscillators Osc_cross1-1 Osc_cross1-2 Osc_cross1-3 Osc_cross1-4 Osc_cross1_1-4, (Period 4) Cross4_mask oscilators Osc_Cross4_1_1-6 (Period 6 ) Osc_Cross4_2_1-2 (Period 2) Cross4dia_mask oscillators Osc_Cross4diag_1_1-2 (Period 2) Hex0_mask oscillators Osc_hex0_1_1 -Osc_hex0_1_4 (Period 4) Osc_hex0_2_1 -Osc_hex0_2_3 (Period 3) Osc_hex0_3_1 -Osc_hex0_3_4 (Period 4) Osc_hex0_4_1 -Osc_hex0_4_3 (Period 3) Osc_hex0_5_1-6 (Period 6) Osc_hex0_6_1-4 (Period 4) Hex1_mask oscillators , "ordering of chaos" type of the evolutionary dynamics similar to that for the standard model was observed only for models with "Isotropic"_mask and Osc_hex1_1_1 -4 (Period 4) Osc_hex1_2_1 -3 (Period 3) Osc_hex1_3_1-2 (Period2) Osc_hex1_4_1-2 (Period 2) Osc_hex1_5_1-3 (Period 3) Osc_hex1_6_1-4 (Period4) Hex2_mask oscillators Osc_hex2_1_1-4 Osc_hex2_2_1-2 Osc_hex2_3_1-3 (Period 3) Osc_hex2_4_1-2 (Period 2) Osc_hex2_5_1-2. (Period 2) Osc_hex2_6_1-4. (Period 4) Osc_hex2_7_1-4. (Period 4) Osc_hex2_8_1-2 Osc_hex2_9_1-4 Osc_hex2_10_1-2 Osc_hex2_11_1-4 (Period 4) Osc_hex2_12_1-2 (Period 2) Osc_hex2_13_1-2 (Period 2) Hex2_mask: for 1 85 . 0    death P , they end up with isolated individual stable formations and for 5 . 0 0   death P , they converge to stable maze-like patterns (see Figure 10). 5 . 0 0   death P ; Maze-like formations with "random" dislocation Seed pattern "InPattern05". P death =0.85; 2.1x10 2 evsteps; stable Seed pattern "InPattern085". P death =0.1; 2x10 2 evsteps;stable However, in distinction from the standard model, maze-like patterns generated by these modified models feature only limited potentials to grow. As one can see from Figure 11 and Figure 12, circular fragments of the stable maze-like patterns chosen as seed patterns do grow, but only until growing patterns reach a square, for isotropic_mask model, or a hexagon, for Hex2-mask model, shapes that circumscribe the shape of the seed pattern. Then the growth stops. Thus the growth is kind of "self-controlled". In what follows we will see more examples of such a "self-controlled growth" for other models. The limited growth capability is reflected also in the capability of the models to fill holes in seed patterns. While isotropic_mask model does fill the hole, as can be seen from Figure 11. Growth capability of maze-like patterns generated by the Isotropic_mask model. In color coded images in the middle column cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and empty cells are shown black. In black and white images "live" cells are shown white. Figure 12. Growth capability of maze-like patterns generated by the Hex2_mask model. In color coded images in the middle column cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and empty cells are shown black. In black and white images "live" cells are shown white. Figure 15. Examples of the evolution of the Hex2_mask model from solid seed patterns. In color coded images in the second column cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and empty cells are shown black. In black and white images "live" cells are shown white. Models with "Cross4 and "Cross4diag" masks do not produce maze-like patterns. Rather their fixed points are, for the entire range of the probability of "death" 1 0   death P , what can be called "Manhattan-like" patterns ( Figure 16). Mask "Cross4" Seed pattern "Inpattern05" Figure 16. Manhattan-type stable patterns generated by the model with masks " Cross4" and "Cross4diag". In color coded images (except two last images in the second and third rows), cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and empty cells are shown black. In black and white images "live" cells are shown white. "Coherent shrinkage" and "eternal life in a bounded space" dynamics modes Perhaps, the most amazing types of evolutionary dynamics observed in the experiments are, along with above-mentioned "self-controlled growth", "coherent shrinkage" and "eternal life in a bounded space". They were observed with some models for the probabilities of "death" in the middle of the range 0÷1. Figure 17 and Figure 18 illustrate the phenomenon of "Coherent shrinkage" observed for a dense chaotic seed pattern "InPattern05". The phenomena of "self-controlled growth" and "coherent shrinkage" can be comprehended even better using Figure 19, Figure Cross_model, seed pattern "InPattern05"; P death =0.55; 10 3 evsteps 5x10 3 evsteps 7x10 3 evsteps 7.75x10 3 evsteps, stable Figure 17. "Coherent shrinkage" of patterns emerged from seed pattern "InPattern05" for Isotropic_mask, Diagonal_mask and Cross_mask models. In color coded images, cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and empty cells are shown black. In last images of every row, "live" cells are shown white. 10 2 evsteps 10 5 evsteps 1.5. 10 6 evsteps 2.10 6 evsteps 2.14 10 6 evsteps, stable Figure 18. "Coherent shrinkage" of patterns emerged from seed pattern "InPattern05" for Hex0-, Hex1-and Hex2_mask models. Images in the second from the top row show evolution of a fragment of the pattern obtained after 3.10 4 evsteps planted in an empty space. In color coded images cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and empty cells are shown black. In last images of every row, "live" cells are shown white. Figure 19. "Self-controlled growth" and "Coherent shrinkage" of patterns emerged from solid seed patterns for Isotropic_mask. In color coded images cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and empty cells are shown black. In black and white images "live" cells are shown white. . "Self-controlled growth" and Coherent shrinkage" of patterns emerged from solid seed patterns for the Diagonal_mask model. In color coded images cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and empty cells are shown black. In last images of every column, "live" cells are shown white. Figure 21. "Self-controlled growth" and Coherent shrinkage" of patterns emerged from solid seed patterns for the Cross_mask model. In color coded images cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and empty cells are shown black. In last images of every column, "live" cells are shown white. 2.6x10 4 evsteps. Figure 22. "Self-controlled growth" and Coherent shrinkage" of patterns emerged from solid seed patterns for the Hex0_mask model. In color coded images cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and "empty" cells arte shown black. In black and white images "live" cells are shown white. . "Self-controlled growth" and "Coherent shrinkage" of patterns emerged from solid seed patterns for the Hex1_mask model. In color coded images cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and "empty" cells arte shown black. In black and white images "live" cells are shown white. Figure 24. "Self-controlled growth" and Coherent shrinkage" of patterns emerged from solid seed patterns for the Hex2_mask model. In color coded images cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue, "empty" cells arte shown black. In black and white images "live" cells are shown white. As one can see from the figures, in the "coherent shrinkage" mode the models pass, in course of evolution, through stages of a sort of entire "life cycle": -"Birth": loci of growth emerge in seed patterns. -"Childhood and adolescence": born formations grow in size, forming kind of "communities" of cells. For chaotic seed patterns, in which "live" cells fill more all less uniformly the entire "vital space", this growth goes on within the "vital space". For "solid" seed patterns, the growth is "self-controlled": it goes on till "communities" reach a shape bounded, depending on the model, by a square (Isotropic and Diagonal_mask models), an octagon (Cross_mask model) or a hexagon (Hex0, Hex1 and Hex2-mask models). The emerged shaped communities stop growing further unless they touch another neighbor "community". In this case touching communities merge to form larger "communities", which continue growing till they reach a similarly bounded shape of a larger size. In such a way "communities" reach a state of "maturity". -The state of "maturity": bounded shaped mature "communities" stay like islands in the "ocean" of empty cells and keep their activity ("births" and "deaths") and their overall size and shape during a certain number of evolution steps, which depends on the probability of death: the lower the probability of death the larger this "population" stability period. -"Senescence". After a certain period of relative stability in size of their populations, "communities" begin to gradually shrink. The shrinkage appears to be "coherent": the "communities" are coherently shrinking from their borders preserving isomorphism of their shapes till the very end, when they either completely disintegrate to nil or, most frequently, end up with one of stable formations. The speed of the shrinkage depends on the probability of death and of the "community" size: the lower the probability of death and the larger the "community" size the lower the shrinkage speed. Some experimental data on the number of evolution steps from the beginning of growth to reaching a stable point obtained for the Hex0_mask model and the probability of "death" 25 . 0  death P with seed patterns "Square" of different size are presented in Figure 25 along with their analytical approximation as a fourth-power function of the "shaped community" size in certain normalized units. It is remarkable that, as one can see in the lower row of Figure 17, "communities", in the course of the "coherent shrinkage", preserve a capability of growth: if one extracts a fragment of a shrinking "community" and plants it into an empty space, the planted fragment resumes growing until it reaches a maturity state in a bounded shape, characteristic for the given model; after that it starts shrinking in the same way as its "mother community" does. As we have already indicated, "coherent shrinkage" slows down with decreasing the probability of "death". For the standard, Isotropic_mask, "Cross_mask", Hex0_mask, and Hex1_mask models, this slowing down might be so substantial that the dynamics of the models appears as "eternal life": upon reaching the state of "maturity", communities stay active (in terms of "births" and "deaths") and keep their outer bounds during millions of evolution steps. We illustrate this in Figure 27, Figure 28 and Figure 28 for "chaotic" seed patterns and in P death =0.9; 10 6 evsteps P death =0.73; 10 6 evsteps P death =0.57; 10 6 evsteps P death =0.4; 10 6 evsteps Isotropic_mask model. Seed pattern "InPattern09"; 5 . 0 75 . 0    death P P death =0.75; 10 6 evsteps P death =0.67; 10 6 evsteps P death =0.59; 10 6 evsteps. P death =0.5; 10 6 evsteps. Figure 26. Standard_mask and Isotropic_mask models: "Eternal life" dynamics in the limits of the "vital space". Cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green and stable cells are shown blue; "empty" cells are shown black. Diagonal_mask model P death =0.6; seed pattern "InPattern05" 10 2 evsteps 10 3 evsteps 10 4 evsteps 10 5 evsteps 3x10 6 evsteps 5x10 6 evsteps 7x10 6 evsteps 10 7 evsteps Red: 10 7 evsteps, blue:5x10 6 evsteps Red: 10 7 evsteps, blue:7x10 6 evsteps Diagonal_mask model, seed pattern "InPattern09", P death =0.2 10 2 evsteps 10 3 evsteps 10 5 evsteps 10 6 evsteps Diagonal_mask model, seed pattern "InPattern09", P death =0.05 10 2 evsteps 10 3 evsteps 10 5 evsteps 10 6 evsteps Figure 27. Diagonal_mask models: "eternal life" in a bounded space for various probabilities of "death" and "chaotic" seed patterns with densities of "live" cells 50% (first two rows) and 10% (last two rows). Last two images in the second row are color-coded, as it is indicated in the caption, in order to demonstrate that the outer shape of the octagon does not shrink noticeably after 5x10 6 evolution steps. In other color coded images cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green and stable cells are shown blue; "empty" cells are shown black. 10 2 evsteps 10 3 evsteps 10 5 evsteps 5x10 5 evsteps 7.5x10 5 evsteps 10 6 evsteps 1.5x10 6 evsteps. 2.5x10 6 evsteps Hex0_model; Seed pattern "InPattern05"; P death =0.3 10 2 evsteps 10 4 evsteps 5x10 5 evsteps 10 6 evsteps Hex1_mask model; Seed pattern InPattern05; P death =0.5 10 2 evsteps 10 4 evsteps 5x10 5 evsteps 10 6 evsteps Figure 28. Cross, Hex0 and Hex1_mask models: "eternal life" in a bounded space for various probabilities of "death" and "chaotic" seed patterns. Cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green and stable cells are shown blue; "empty" cells are shown black. In color coded images in three middle columns cells that will die on the next step are shown pink, cell that will give a birth are shown green, stable cells are shown blue and "empty" cells are shown black. In color coded images of the right hand column seed patterns are shown red; and emerged patterns after 10 6 evsteps are shown blue. In black and white images "live" cells are shown white. In color coded images in three middle columns cells that will die on the next step are shown pink, cell that will give a birth are shown green, stable cells are shown blue and "empty" cells are shown black. In color coded images of the right hand column seed patterns are shown red; and emerged patterns after 10 6 evsteps are shown blue. In black and white images "live" cells are shown white. Cross_mask model; P death =0.1 Seed pattern "Star2" 10 2 evsteps 10 3 evsteps 10 6 evsteps Seed pattern 10 2 evsteps 10 3 evsteps 3x10 6 evsteps Seed pattern 10 2 evsteps 10 3 evsteps 10 4 evsteps 10 5 evsteps. Seed pattern 10 2 evsteps 10 3 evsteps 10 4 evsteps 10 5 evsteps 10 6 evsteps 2x10 6 evsteps 3x10 6 evsteps Red: seed pattern; blue: 3x10 6 evsteps Seed pattern "Star" 10 2 evsteps 10 3 evsteps 10 4 evsteps 3x10 6 evsteps Red: seed pattern; blue: 3x10 6 evsteps Figure 32. Cross_mask model: "eternal life" in a bounded space is possible when bounded "communities" have sufficiently large size (compare first two rows with last three rows). In color coded images except last images of each sequence cells that will die on the next step are shown pink, cell that will give a birth are shown green, stable cells are shown blue. "Empty" cells are everywhere shown black. In black and white images "live cells are shown white. One can see that for all seed patterns but "Blobs" one, outer bounds of emerged patterns circumscribe corresponding seed patter. This is not the case for seed pattern "Blobs", perhaps due to insufficient size of the blobs. 10 3 evsteps 10 6 evsteps Red: seed pattern, blue: 10 6 evsteps Seed pattern "Circle" 10 2 evsteps 10 6 evsteps 3x10 6 evsteps Red: seed pattern, blue: 3x10 6 evsteps Seed pattern 10 2 evsteps 10 3 evsteps 10 4 evsteps 10 5 evsteps 10 6 evsteps 2x10 6 evsteps 3x10 6 evsteps Red: seed pattern, blue: 10 6 evsteps Seed pattern 10 2 evsteps 10 3 evsteps 10 6 evsteps Red: seed pattern, blue: 10 6 evsteps Seed pattern "Star" 10 2 evsteps 10 3 evsteps 2x10 6 evsteps Red: seed pattern, blue: 2x10 6 evsteps Figure 33. Hex0_mask model: "eternal life" in a bounded space for solid seed patterns. In color coded images except last images in each sequence cells that will die on the next step are shown pink, cell that will give a birth are shown green and stable cells are shown blue. In all images "empty" cells are shown black. For sufficiently large seed patterns, outer bounds of emerged patterns circumscribe the corresponding seed pattern. 10 3 evsteps 10 6 evsteps Red: seed pattern, blue: 10 6 evsteps Seed pattern 10 2 evsteps 10 6 evsteps 3x10 6 evsteps Red: seed pattern, blue: 3x10 6 evsteps Seed pattern 10 2 evsteps 10 3 evsteps 10 4 evsteps 10 5 evsteps 10 6 evsteps 2x10 6 evsteps 3x10 6 evsteps Red: seed pattern, blue: 3x10 6 evsteps Seed pattern "Txtr2" 10 2 evsteps 10 3 evsteps 3x10 6 evsteps Red: seed pattern, blue: 3x10 6 evsteps Seed pattern "Star" 10 2 evsteps 10 4 evsteps 10 6 evsteps Red: seed pattern, blue: 2x10 6 evsteps Figure 34. Hex1_mask model: "eternal life" in a bounded space for solid seed patterns. In color coded images except last images in each sequence cells that will die on the next step are shown pink, cell that will give a birth are shown green and stable cells are shown blue. In all images "empty" cells are shown black. For sufficiently large seed patterns, outer bounds of emerged patterns circumscribe the corresponding seed pattern. One can see from the figures that whereas for Standard_mask and Isotropic_mask models formations growing from seed pattern propagate till they reach bounds of the "vital space" and then stay active in this space, which demonstrates a capability of "unlimited" expansion, dynamics of other models is different. For Diagonal_mask, Cross_mask, Hex0_mask, Hex1_mask and Hex2_mask models, growing formation reach shapes bounded by geometric figures characteristic for each particular model: rectangle or right angles (Isotropic_mask and Diagonal_mask models), octagon (Cross_mask model), hexagon (Hex0_mask ans Hex1_mask models). Remarkably, formation, from chaotic and sufficiently dense seed patterns, of stable in size and shape active "communities" starts from shrinkage of the "vital space" from corners and the shrinkage stops when the "community" reaches a certain "critical" size (see Figure 27, two upper rows, and Figure 28). It is also remarkable that, as a rule, outer bounds of formations emerged from "solid" and sufficiently large seed patterns circumscribe corresponding seed patterns. One can also see in Surprisingly, Hex2_mask model seems to be, as Figure 35 shows, incapable of generating patterns that "live" permanently. For very low probabilities of death, its evolution does not enter into the "coherent shrinkage" stage and ends up with maze-like stable patterns even faster than for larger probabilities of "death". In most of the experiments, when "eternal life in a bounded space" dynamics seemed to be observed, we ran models maximum (1÷3)x10 6 evolution steps. In order to be better convinced in the possibility of the "eternal life" dynamics, we ran the cross_mask model 2x10 7 evolution steps for P death =0.3425. The results illustrated in Figure 36 tell in favor of this possibility. Seed pattern 10 2 evsteps 10 2 evsteps 10 2 evsteps 10 2 evsteps 10 2 evsteps 4x10 2 evsteps 4x10 2 evsteps 2x10 2 evsteps 3x10 2 evsteps 10 3 evsteps Figure 35. Evolution of the Hex2_mask model for the probability of death 0.1. In color coded images cells that will "die" on the next step are shown pink, cells that will give a birth are shown green and stable cells are shown blue. In black and white images "live" cells are shown white. In all images "empty" cells are shown black. Seed pattern "Star" 10 2 evsteps 10 3 evsteps 10 6 evsteps 10 7 evsteps 1.6x10 7 evsteps 1.8x10 7 evsteps 2x10 7 evsteps Red: 2x10 7 ; green: 10 7; blue: 10 4 evsteps Red: seed pattern; blue: 2x10 7 Figure 36. Verification of the possibility of "eternal life" in a bounded space dynamics on an example of extralong evolution of the Cross_mask model. Fourth and fifth images in the second row are color coded combinations of patterns emerged after different number of evolution steps in order to demonstrate that patterns do shrink between 10 4 evsteps and 10 7 evsteps and practically do not shrink between 10 4 evsteps and 2x10 7 evsteps. The very last image in the second row shows, in red, the seed pattern and, in blue, the pattern after 2x10 7 evsteps and demonstrates that the outer shape of the pattern does not circumscribes the seed pattern as it was observed in the case of P death =0.1 (see Figure 32, lower row). Conclusion Several modifications of the standard Conway's Game of Life have been suggested and evolutionary dynamics of the introduced new models has been experimentally investigated. In the experiments, a number of new phenomena has been revealed. Specifically it has been found that -Standard Conway's model in its stochastic modification with the probability of "death" lower than one demonstrates two types of the evolutionary dynamics: (i) "eternal life" in the "vital space" in the range of the probabilities of "death"     1 3 . 0 death P , where  -is an arbitrarily small number. (ii) "ordering of chaos" into maze-like patterns with stochastic "dislocations" for lower probabilities of death 3 . 0  death P . A remarkable feature of these patterns is that, being stable in the "vital space", their fragments preserve a capability of growth and implantation into other maze-like patterns. These patterns remind patterns of magnetic domains, finger prints, zebra skin, tiger fur, fish skin patterning and alike, which can be frequently found in live as well as in inanimate nature. -Other introduced models with modified weights exhibit four types of evolutionary dynamics: (iii) For 1  death P , they, similarly to the standard non-stochastic Conway's model, exhibit "ordering of chaos" into certain stable or oscillating formation, specific for each model. (iv) For sufficiently large 1  death P , the models feature "ordering of chaos" into maze-like patterns or "Manhattan"-like patterns; however, unlike the standard Conway's model, fragments of these patterns extracted from "mother" patterns, have only limited potentials of growth. For sufficiently low probabilities of "death", the models feature dynamics that consist of three stages: (1) "self-controlled growth" into active (in terms of births and deaths) "communities" bounded by shapes, characteristic for each model, (2) "stabilized in shape state of maturity" and (3) "coherent shrinkage" when bounded formations gradually shrink to nil or to a few stable or oscillating formations keeping in this process isomorphism of their bounding shapes until the very end. Inter alia, these results are remarkable illustrations of how purely local connection between interactive cells can determine their collective behavior. Figure 1 ) 1are subjected to evolution. Figure 1 . 1Cells in a "vital space" on a rectangular lattice. 8-neigboring cells of a cell indicated by the 8-point star are indicated by donuts Each cell in this array has 8 neighbors ( Figure 1 ) 1. Rules of the evolution are as following. At each subsequent step of evolution, are integer indices of cells on the rectangular lattice, Figure 2 . 2Examples of seed patterns used in the experiments. "Live" cells are shown white, empty cells are shown black) binary values is replaced by a weighted summation with rounding up the summation result: Figure 3 . 3"Ordering of chaos": formation of individual stable patterns and maze-like stripy patterns. "Live" cells are shown white and "empty" cells are shown black. Figure 4 4Figure 4. Growth of maze-like patterns from sparse or solid seed patterns. In color coded images cells that will "die" on the next step are shown pink, cells, in which "birth" will take place on the next step, are shown green, "live" stable cells are shown blue and empty cells are shown black. Figure 4 . 4Figure 4. Figure 5 . 5Growth capability of the maze-like patterns generated by the standard model. Upper row: growth of a circular fragment of a stable stripy pattern; bottom row: filling a circular hole in a fragment of a stable maze-like pattern. Figure 6 . 6Implanting a fragment of one maze-like pattern into another. Initial patterns are shown in the upper row. Second pattern with implanted central circular fragment of the first pattern and the resulting from it as a seed pattern the new stable maze-like pattern are shown in the bottom row. Circular mask used for extracting the implanted pattern is shown in the middle. Figure 7 . 7Stable points of the considered models for 1  death P (pseudo random seed pattern "InPattern05"). "Live" cells are shown white, empty cells are shown black. Figure 8 . 8Samples of oscillating patterns and their phases for Isotropic, Diagonal, Cross, Cross4 and Cross4diag masks. "Live" cells are shown white, empty cells are shown black. Figure 9 . 9Samples of oscillating patterns and their phases for Hex0, Hex1 and Hex2 masks. "Live" cells are shown white, empty cells are shown black. Figure 10 . 10Two types of "ordering of chaos" dynamics for Isotropic_mask and Hex2_mask models. "Live" cells are shown white, empty cells are shown black. Figure 11 and 11Figure 11 and Figure 12 , 12Hex2-model fills the hole only partially leaving empty configuration with horizontal, vertical and diagonal-oriented borders similar to those of the above mentioned hexagon bounded maze-like formation. Figure 13 andFigureFigure 13 .Figure 14 . 13131414 evidence that maze-like patterns generated be Isotropic_mask as well as those of Hex2_mask model preserve certain compatibility and allow implantation of one to another, similarly to what was observed for the standard model. Isotropic_mask model: implantation of a fragment of one stripy pattern into another. Initial patterns SP1 and SP2 are shown in the left column. Pattern SP1 with implanted central circular fragment of the pattern SP2 and the resulting from it as a seed pattern the new stable maze-like pattern are shown in the right column. Circular mask used for extracting the implanted pattern is shown in the middle. Hex2_mask model: implantation of a fragment of one maze-like pattern into another. Initial patterns SP1 and SP2 are shown in the left column. Pattern SP1 with implanted central circular fragment of the pattern SP2 used as a seed pattern and the resulting from new stable maze-like pattern are shown in the right column. Circular mask used for extracting the implanted pattern is shown in the middle. Figure 15 provides 15further illustrations of the "self-controlled growth" property of the Figure 21 , Figure 22 , 2122Figure 23andFigure 24which illustrate evolution of "solid" seed patterns."Isotropic_mask", seed pattern "model, seed pattern "InPattern05", P death =0. Figure 20 20Figure 20. "Self-controlled growth" and Coherent shrinkage" of patterns emerged from solid seed patterns for the Diagonal_mask model. In color coded images cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and empty cells are shown black. In last images of every column, "live" cells are shown white. Figure 23 23Figure 23. "Self-controlled growth" and "Coherent shrinkage" of patterns emerged from solid seed patterns for the Hex1_mask model. In color coded images cells that will "die" on the next step are shown pink, cells that will give "birth" are shown green, stable cells are shown blue and "empty" cells arte shown black. In black and white images "live" cells are shown white. Figure 25. Experimental data on the number of evolution steps from birth to collapse for the Hex0_mask model with 25 . 0  death P and seed patterns in form of squares of different sizes and their numerical approximation Figure 30 ,Figure 31 ,Figure 32 , 303132Figure 33, Figure 34 and Figure 35 for solid seed patterns. Standard mask model. Seed pattern "InPattern05" -is an arbitrarily small number. Figure 29 .Figure 30 . 2930Standard_mask model: "eternal life" in the "vital space". In the last images of every row, shown blue are emerged after 10 6 evsteps patterns and shown red are corresponding seed patterns. In color coded images in three middle columns cells that will die on the next step are shown pink, cell that will give a birth are shown green and stable cells are shown blue, "empty" cells are shown black; in black and white images "live" cells are shown white.Isotropic_mask model, P death =0.Isotropic_mask model: "eternal life in a bounded space" for solid seed patterns. Last images in each row demonstrate that outer bounds of emerged patterns (shown blue) circumscribe corresponding seed patterns. Figure 31 . 31Diagonal_mask model: "eternal life" in a bounded space for solid seed patterns. Last images in each row demonstrate that outer bounds of emerged patterns (shown blue) circumscribe corresponding seed patterns. Figure 30 ,Figure 30 ,Figure 34 , 303034third row, third row, Figure 33, third and fourth rows, third and fourth rows that the "eternal life in a bounded space" dynamics is possible only if bounded formations reach a sufficiently large size. models exhibit "eternal life in a bounded space" type of dynamics within "communities" bounded by shapes specific for each model (by squares or right angles, octagons, hexagons oriented parallel to the model rectangular lattice axes or 45 o rotated with respect to the lattice axes) and reached through a process of "self-controlled growth"; bounded formations demonstrate seemingly permanent activity (cell births and deaths) while keeping their outer bounds. (vi) Introduced a "gray-scale" modification of the model, in which cells can assume arbitrary values between zero and one and logical Conway's Game of Life rules are replaced by fuzzy logics. In this paper, we introduce several new modifications of the standard Conway's model and describe results of computer simulation experiments, which reveal new phenomena in the evolutionary dynamics of the models.1  death P , the case of 1  death P being correspondent to the standard non-stochastic Conway's model, -demonstrated that modified in this way model tends to produce, in course of evolution, maze-like patterns with chaotic dislocations, which very much remind zebra skin and tiger fur patterns, fingerprints, magnetic domain patterns and alike; 1D version of the model produces patterns that remind those some see-shells develop in their life. -As seed patterns, several realizations of arrays of pseudo-random numbers with different rate of "live" cells as well as several "solid" seed patterns were used in the experiments. They are shown in Figure 2. InPattern05; rate of "live" cells 0.5 InPattern085; rate of "live" cells 0.15 Circular fragment of the stable mazelike pattern (P death =0.4; 2x10 2 evsteps) used as a seed pattern.This fragment after 25 evsteps Same fragment stable after 2x10 3 evsteps. Outer square circumscribes the circular shape of the seed pattern Circular hole un the stable maze-loke pattern (P death =0.4; 2x10 2 evsteps) used as a seed pattern. This hole after 25 evsteps Stable. Hole filled after 225evol. Isotropic-mask and Hex2_mask models.Isotropic_mask model. P death =0.25 Seed pattern 15 evsteps 10 2 evsteps, stable Red: seed pattern; blue: 10 2 evsteps Seed pattern 30 evsteps. 10 2 evsteps, stable Red: seed pattern; blue: 10 2 evsteps Seed pattern 30 evsteps 10 2 evsteps, stable Red: seed pattern; blue: 10 2 evsteps Hex2_mask, P death =0.1 Seed pattern 25 evsteps 2x10 2 evsteps, stable Red: seed pattern; blue: 2x10 2 evsteps. Seed pattern 25 evsteps 2x10 2 evsteps, stable Red: seed pattern; blue: 2x10 2 evsteps. Seed pattern 25 evsteps 10 2 evsteps; stable 10 2 evsteps; stable P death =0.75; 25 evsteps. Stable (for P death =1, all cells die out )1 0   death P P death =1; 10 2 evsteps. Stable with 3 oscillators P death =0.5; 80 evsteps. Stable P death =0.1; 10 4 evsteps. Stable (blue) with oscillators (pink pixels) Mask "Cross4" Seed patterns " Circle" and "Star" P death =0.25; 4x10 2 evsteps Stable with two oscillators (green and pink pixels) Red: seed pattern; blue: P death =0.25; 4x10 2 evsteps P death =0.75; 25 evsteps. Stable P death =0.25; 2x10 2 evsteps. Stable with two oscillators (green and pink pixels) Red: seed pattern; blue: 2x10 2 evsteps Mask "Cross4diag"; Seed pattern "Inpattern05" P death =1; 10 2 evol. evstepss. Stable with 1 oscillator P death =0.5; 50 evol. evstepss. Stable with two oscillators P death =0.1; 1.5x10 3 evol. evsteps. Stable (blue) with 5 oscillators (red and green) Hex0_mask model, seed pattern "InPattern075". P death =0.45 10 2 evsteps. 2.x10 3 evsteps. 3.10 4 evsteps. 5x10 4 evsteps.; 5.4x10 4 evsteps, stable A fragment of the above hexagonal "community" taken as a seed pattern 10 2 evsteps 9.4x10 3 evsteps, stable Hex1_mask_model, P death =0.625 10 2 -th evsteps 10 4 -th evsteps 10 5 -th evsteps 5x10 5 evsteps 6x10 5 evsteps, stable Hex2_mask model. Seed pattern InPattern05; P death =0.75 10 2 evsteps 10 3 evsteps 10 4 evsteps 2x10 4 evsteps 2.4x10 4 evsteps, stable Hex2_mask model; Seed pattern InPattern05; P death =0.6 Hex2_mask model; P death =0.1Seed pattern "Star2" Seed pattern Seed pattern Seed pattern Hex0_mask model, P death =0.1; Seed pattern "Star2" Hex1_mask model; P death =0.25, Seed pattern "Star2" M Gardner, Mathematical Puzzles and Diversions. PenguinM. Gardner, Mathematical Puzzles and Diversions, Penguin, 1971 L P Yaroslavsky, I F Siverguina, Textures, Game "Life" and Non-linear Dynamic Systems, Problems of Control and Information Theory. 19L. P. Yaroslavsky, I. F. Siverguina, Textures, Game "Life" and Non-linear Dynamic Systems, Problems of Control and Information Theory, Vol. 19(4), 1990, pp. 349-372 Textures, the Game "Life" and nonlinear dynamic systems. L P Yaroslavsky, I F Siverguina, R Thaller, L Dimitrov, Image Analysis and Synthesis. W. Poelzleitner and E. Wenger, R. OldenburgWien, MünchenL. P. Yaroslavsky, I.F. Siverguina, R. Thaller, L. Dimitrov, Textures, the Game "Life" and nonlinear dynamic systems, In: Image Analysis and Synthesis, Ed. by W. Poelzleitner and E. Wenger, R. Oldenburg, Wien, München, 1993, pp. 293-315. From Random Numbers to Stochastic Growth Models and Texture Images. L Yaroslavsky, Pattern Formation, M. Gromov and A. CarboneWorld Scientific Publishing CoSingaporeL. Yaroslavsky, From Random Numbers to Stochastic Growth Models and Texture Images, In: Pattern Formation, M. Gromov and A. Carbone, Eds., World Scientific Publishing Co., Singapore, 1999, pp. 42-64 Stochastic nonlinear dynamics pattern formation and growth models. L Yaroslavsky, Nonlinear Biomedical Physics. 14L. Yaroslavsky, Stochastic nonlinear dynamics pattern formation and growth models, Nonlinear Biomedical Physics, 2007, 1:4 (5 July 2007) A Probabilistic Model for Morphogenesis. M Eden, Pergamon PressN.Y.M. Eden, A Probabilistic Model for Morphogenesis, pp. 359-370, Pergamon Press, N.Y., 1958 M Eden, A Two-dimensional Growth Process, Proc. of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. Univ. of California PressM. Eden, A Two-dimensional Growth Process, Proc. of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Univ. of California Press, V. 4, 1961, pp. 223-239 History of a Stochastic Growth Model. M Eden, Ph The´venaz, -th Int. Workshop on Digital Image Processing and Computer Graphics (DIP-97. Wenger and L6M. Eden, Ph. The´venaz, History of a Stochastic Growth Model, in: 6-th Int. Workshop on Digital Image Processing and Computer Graphics (DIP-97),E. Wenger and L. . Dimitrov, SPIE Proceedings Series. 3346Dimitrov, Eds., 20-22 Oct. 1997, Vienna, Austria, SPIE Proceedings Series volume 3346, pp. 43-54
[]
[ "Moderate-density parity-check codes from projective bundles", "Moderate-density parity-check codes from projective bundles" ]
[ "Jessica Bariffi \nInstitute of Mathematics\nUniversity of Zurich\nZurichSwitzerland\n\nInstitute of Communications and Navigation\nGerman Aerospace CenterOberpfaffenhofen-WesslingGermany\n", "Sam Mattheus \nVrije Universiteit Brussel\nBrusselsBelgium\n", "Alessandro Neri \nMax-Planck-Institute for Mathematics in the Sciences\nLeipzigGermany\n", "Joachim Rosenthal \nInstitute of Mathematics\nUniversity of Zurich\nZurichSwitzerland\n", "\nJessica Bariffi\n\n" ]
[ "Institute of Mathematics\nUniversity of Zurich\nZurichSwitzerland", "Institute of Communications and Navigation\nGerman Aerospace CenterOberpfaffenhofen-WesslingGermany", "Vrije Universiteit Brussel\nBrusselsBelgium", "Max-Planck-Institute for Mathematics in the Sciences\nLeipzigGermany", "Institute of Mathematics\nUniversity of Zurich\nZurichSwitzerland", "Jessica Bariffi\n" ]
[ "Designs, Codes and Cryptography" ]
New constructions for moderate-density parity-check (MDPC) codes using finite geometry are proposed. We design a parity-check matrix for the main family of binary codes as the concatenation of two matrices: the incidence matrix between points and lines of the Desarguesian projective plane and the incidence matrix between points and ovals of a projective bundle. A projective bundle is a special collection of ovals which pairwise meet in a unique point. We determine the minimum distance and the dimension of these codes, and we show that they have a natural quasi-cyclic structure. We consider alternative constructions based on an incidence matrix of a Desarguesian projective plane and compare their error-correction performance with regards to a modification of Gallager's bit-flipping decoding algorithm. In this setting, our codes have the best possible error-correction performance after one round of bit-flipping decoding given the parameters of the code's parity-check matrix.
10.1007/s10623-022-01054-y
[ "https://arxiv.org/pdf/2103.09722v1.pdf" ]
232,257,623
2103.09722
58784033f36232e03abe982814dc2110ed991cde
Moderate-density parity-check codes from projective bundles 2022 Jessica Bariffi Institute of Mathematics University of Zurich ZurichSwitzerland Institute of Communications and Navigation German Aerospace CenterOberpfaffenhofen-WesslingGermany Sam Mattheus Vrije Universiteit Brussel BrusselsBelgium Alessandro Neri Max-Planck-Institute for Mathematics in the Sciences LeipzigGermany Joachim Rosenthal Institute of Mathematics University of Zurich ZurichSwitzerland Jessica Bariffi Moderate-density parity-check codes from projective bundles Designs, Codes and Cryptography 90202210.1007/s10623-022-01054-yReceived: 9 March 2021 / Revised: 27 April 2022 / Accepted: 2 May 2022 / Published online: 24 May 2022MDPC codes · Projective bundle · Projective plane · Bit-flipping decoding algorithm New constructions for moderate-density parity-check (MDPC) codes using finite geometry are proposed. We design a parity-check matrix for the main family of binary codes as the concatenation of two matrices: the incidence matrix between points and lines of the Desarguesian projective plane and the incidence matrix between points and ovals of a projective bundle. A projective bundle is a special collection of ovals which pairwise meet in a unique point. We determine the minimum distance and the dimension of these codes, and we show that they have a natural quasi-cyclic structure. We consider alternative constructions based on an incidence matrix of a Desarguesian projective plane and compare their error-correction performance with regards to a modification of Gallager's bit-flipping decoding algorithm. In this setting, our codes have the best possible error-correction performance after one round of bit-flipping decoding given the parameters of the code's parity-check matrix. Mathematics Subject Classification 11T71 · 51E05 Introduction The close interplay between coding theory and finite geometry has emerged multiple times in the last 60 years, starting from the works of Prange [27] and Rudolph [31], where they proposed to construct linear codes starting from projective planes. Their idea was to use the incidence matrix of the plane as a generator matrix or as a parity-check matrix of a linear code, showing that the underlying geometry can be translated in metric properties of the corresponding codes. Generalizations of these constructions have been studied since the 70's and are still the subject of active research (see [2]). The relations between these two research areas had also a strong impact in the opposite direction. The most striking example is certainly the non-existence proof of a finite projective plane of order 10 shown in [20]. This groundbreaking result came-with the help of a computer-after a series of papers analyzed the binary linear code coming from a putative projective plane of order 10. A very important class of codes which was sensibly influenced by geometric constructions is given by low-density parity-check (LDPC) codes, which were introduced by Gallager in his seminal 1962 paper [7]. LDPC codes, as originally proposed, are binary linear codes with a very sparse parity-check matrix. This sparsity property is the bedrock of efficient decoding algorithms. Already Gallager provided two of such algorithms whose decoding complexity per iteration is linear in the block length. However, LDPC codes came to fame much later, when in 2001 Richardson, Shokrollahi and Urbanke [29] were able to show that LDPC codes are capable to approach the Shannon capacity in a practical manner. The above authors derived this result using random constructions of very large and sparse parity-check matrices. Because of these random constructions the performance of the codes was only guaranteed with high probability and there was also the practical disadvantage that the storage of a particular parity-check matrix required a lot of storage space. There are several design parameters one wants to optimize when constructing LDPC codes. On the side of guaranteeing that the distance is reasonably large, it was realized early that it is desirable that the girth of the associated Tanner graph is large as well. This last property helps to avoid decoding failures in many decoding algorithms. Thus, in order to guarantee that an LDPC code had desirable design parameters, such as a large distance or a large girth of the associated Tanner graph, some explicit constructions were needed. Already in 1982 Margulis [23] used group theoretic methods to construct a bipartite Cayley graph whose girth was large. This line of research was extended by Rosenthal and Vontobel [30] using some explicit constructions of Ramanujan graphs, which have exceptionally large girth. Maybe the first time objects from finite geometry were used to construct explicitly some good LDPC codes was in the work of Kou, Lin and Fossorier [19]. These authors gave four different constructions using affine and projective geometries over finite fields which did guarantee that the resulting code had a good distance and the associated Tanner graph had a girth of at least 6. Using points and lines in F m q Kim, Peled, Perepelitsa, Pless and Friedland [18] came up with incidence matrices representing excellent LDPC codes. In the last 15 years there has been active research to come up with further explicit constructions of LDPC codes with desirable parameters based on combinatorial structures [12,19,22,35,36]. Moderate-density parity-check (MDPC) codes were first introduced by Ouzan and Be'ery [25]. Misoczki, Tillich, Sendrier and Barreto [24] showed that MDPC codes could still be decoded with low complexity as long as the row-weight of each row vector of the parity-check matrix was not much more than the square root of the length of the code. These authors also showed that MDPC codes are highly interesting for the use in the area of code based cryptography. Similar as for LDPC codes, it is an important task to come up with explicit constructions of MDPC codes where, e.g., a good minimum distance can be guaranteed. Already Ouzan and Be'ery [25] provided a construction using cyclotomic cosets. Further constructions using quasi-cyclic codes can be found in [11,24]. This paper adds another dowel to the theory of error-correcting codes arising from geometric objects. We propose a new construction of linear codes using projective bundles in a Desarguesian projective plane, resulting in a family of MDPC codes. Concretely, a projective bundle in a projective plane of order q is a collection of q 2 + q + 1 ovals which mutually intersect in a unique point. We consider the incidence structure consisting of the lines of a projective plane together with the ovals of a projective bundle. Such an incidence structure arises from studying the F q -sublines of a scattered linear set of pseudoregulus type in PG(1, q 3 ) [21]. The incidence matrix of this structure will serve as a parity-check matrix of the proposed binary codes. We completely determine their dimension and minimum distance for both q even and odd. In addition, we observe that we can design these codes to possess a quasi-cyclic structure of index 2. As a consequence, their encoding can be achieved in linear time and implemented with linear feedback shift registers. Moreover, also the storage space required is only half their length. We then generalize this construction and consider other variations. Their error-correcting performance with regards to Gallager's bit-flipping algorithm is discussed. The main motivation arises from [34], where the error-correction capability of the bitflipping decoding algorithm on the parity-check matrix of an MDPC code was analyzed. There, it was derived that its performance is inversely proportional to the maximum column intersection of the parity-check matrix, which is the maximum number of positions of ones that two distinct columns share. We show indeed that the maximum column intersection of the derived parity-check matrices is the smallest possible for the chosen parameters, implying in turn the best possible performance for one round of the bit-flipping algorithm. The paper is organized as follows: Sect. 2 consists of the coding theory background needed in the paper. In particular, we introduce the family of MDPC codes and we recall the result on the performance of the bit-flipping algorithm presented in [34], which was decisive for the idea of this construction. In Sect. 3 we give a brief overview on projective planes, studying the basic properties of codes arising from them. Section 4 is dedicated to the new proposed MDPC code design using projective bundles. Here, we study some of the code properties and we determine its dimension, minimum distance and minimum weight codewords. The paper is based on the master's thesis of the first author [5] and in this section we extend the results which were originally stated there. The goal of Sect. 5 is to generalize the results stated in Sect. 4 in order to have more flexibility in the choice of the parameters. This is done by using several disjoint projective bundles instead of only one. We then propose another construction of binary codes in Sect. 6, which only uses the incidence matrix of a projective plane and its transpose, and study minimum distance and minimum weight codewords. We then compare the error-correction performances of the new codes by running several experiments. Finally, we recap our findings and draw some conclusive remarks in Sect. 7. Coding theory and moderate-density parity-check codes Let us start by briefly recalling some basics of coding theory. Throughout the paper q will always be a prime power, and we will denote the finite field with q elements by F q . The set of vectors of length n over F q will be denoted by F n q . We consider the Hamming weight on F n q defined as wt(v) := |{i ∈ {1, . . . , n} | v i = 0}| . It is well-known that it induces a metric, namely the Hamming distance which is given by d H : F n q × F n q −→ N (u, v) −→ wt(u − v). Definition 2.1 A q-ary linear code C of length n and dimension dim(C) = k is a kdimensional linear subspace of F n q endowed with the Hamming metric. The minimum distance of C is the minimum among all the possible weights of the nonzero codewords and it is denoted by d(C), i.e. d(C) := min{wt(c) | c ∈ C, c = 0}. In general, finding the minimum distance of a linear code and classifying its nonzero codewords of minimum weight is not an easy task. Even for linear codes from geometric constructions, it is often highly non-trivial to find sharp bounds or a classification of the smallest weight words, see for example [1,3,17,26,35]. A q-ary linear code of length n and dimension k will be denoted for brevity by [n, k] q code, or by [n, k, d] q code if the minimum distance d is known. Any [n, k] q code C has a dual code which is defined as C ⊥ = {x ∈ F n q | x · c = 0, ∀c ∈ C}. A generator matrix of an [n, k] q code C is a matrix G ∈ F k×n q whose rows form a basis of C. A generator matrix H ∈ F (n−k)×n q for the dual code C ⊥ is called a parity-check matrix of C. Note that C can also be represented by a parity-check matrix H , since it corresponds to its right kernel, i.e. C = ker(H ) = {c ∈ F n q | c · H = 0}. A matrix A ∈ F r ×s q is said to have row-weight w, for some nonnegative integer w, if every row of A has Hamming weight equal to w. Similarly, we say that A has column-weight v, if each of its columns has Hamming weight v. In the following we will focus on the family of moderate-density parity-check (MDPC) codes. They are an extension of the well-known low density parity-check (LDPC) codes, and they are defined by the row-weight of a parity-check matrix. The terminology was first introduced in [25], and then these codes were reintroduced and further generalized in [24] for cryptographic purposes. Definition 2.2 Let (v i , w i ). MDPC codes have been constructed in various ways. In their seminal paper [25], Ouzan and Be'ery designed cyclic MDPC codes carefully choosing the idempotent generator of the dual code. This structure has been generalized in order to design quasi-cyclic MDPC codes (see [11,24]). A different approach has been proposed in [34], where a random model is considered. In the definition of an MDPC code the chosen parity-check matrix is very important. Indeed, as for LDPC codes, an MDPC code automatically comes together with a decoding algorithm-for instance the bit-flipping algorithm-whose performance depends on the chosen parity-check matrix. Thus, in order to study the error-correction performance, we introduce the following quantity. Definition 2.3 Let H be a binary matrix. The maximum column intersection is the maximal cardinality of the intersection of the supports of any pair of distinct columns of H . The following result was found by Tillich in 2018 (for more details and the proof see [34]). It states the amount of errors that can be corrected within one round of the bit-flipping decoding algorithm. It hence follows that, the smaller s H , the more errors can be corrected after one round of the bit-flipping decoding algorithm. Using a random construction as the one proposed by Tillich, the expected value for the maximum column intersection s H is O( log n log log n ), as shown in [34,Proposition 2]. We would like to design MDPC codes in such a way that s H is as small as possible and, more importantly, that s H is deterministic. With this we can ensure that the bit-flipping decoder is able to correct a given amount of errors, which we will discuss in Sects. 4.3 and 6.2. MDPC codes from projective planes The projective plane PG(2, q) is a point-line geometry constructed from a three-dimensional vector space V over F q . Its points and lines are the one-and two-dimensional subspaces of V , respectively and the containment relation in V defines the incidence relation of the plane. It has q 2 + q + 1 points and equally many lines. The geometry satisfies the following properties: 1. Any two distinct points are incident with exactly one common line; 2. Any two distinct lines are incident with exactly one common point; 3. There are four points such that no three of them are collinear. This means that PG(2, q) can also be regarded as a symmetric 2-(q 2 + q + 1, q + 1, 1)design, where the lines correspond to the blocks. Moreover, every line in PG(2, q) is incident with q + 1 points and dually, every point is incident with q + 1 lines. One way to represent PG(2, q) is by an incidence matrix. This is a matrix A whose rows and columns are indexed by points and lines, respectively such that (A) p = 1 if p is incident with 0 otherwise. Here we describe an alternative way to represent the projective plane PG(2, q). We can identify the set of points with the integers modulo q 2 + q + 1. For the description of the lines, we will follow the instruction presented by Hirschfeld in [10, p77-p79]. Let us therefore introduce the following set. Definition 3.1 A set D = {d 0 , . . . , d r } ⊆ Z/(r 2 + r + 1)Z is called a perfect difference set, if all differences (d i − d j ) with i = j, are distinct modulo r 2 + r + 1, for i, j ∈ {0, . . . , r }. Example 3.2 For instance, consider r = 2. One can show that the set D = {0, 1, 3} of r + 1 = 3 integers is indeed a perfect difference set, since any two differences between two distinct elements are pairwise disjoint modulo r 2 + r + 1 = 7. Hirschfeld showed in [10, Theorem 4.2.2 and its Corollary] that the set of lines of PG(2, q) is fully described by the circulant shifts modulo q 2 + q + 1 of a perfect difference set of q + 1 elements. In this way we obtain a circulant incidence matrix in which the support of the first column is D. In order to illustrate this, consider the Fano plane PG(2, 2) consisting of seven points and lines. We have seen, that the points will be identified with the integers modulo q 2 +q +1 = 7. For the set of lines we will use the cyclic shifts (modulo 7) of the set D = {0, 1, 3}, which we have seen is in fact a perfect difference set. Explicitly, we obtain the following set of points P and set of lines L P = {0, 1, 2, 3, 4, 5, 6}, L = {{0 + i, 1 + i, 3 + i} | i ∈ {0, . . . , 6}}. The defining properties of projective planes have made them a good source of errorcorrecting codes by taking their incidence matrices as the parity-check matrix, as was done already in the late 1950s, cf. [27] or [31]. Definition 3.3 Let H be an incidence matrix of Π = PG(2, q) over the binary finite field F 2 . We define the code C 2 (Π) ⊥ ⊆ F q 2 +q+1 2 via C 2 (Π) ⊥ = ker(H ). Codes from planes have been intensively studied and many properties have been derived thanks to the underlying geometric structure. Among the most relevant properties, Graham and MacWilliams [9] completely determined the dimension of the codes C p (Π) ⊥ over F p and their minimum distance when p = 2 was determined by Assmus and Key [2]. Here we state the two results, restricting ourselves only to the case p = 2. Theorem 3.4 The code C 2 (Π) ⊥ is a [q 2 + q + 1, k, d] 2 code, where (k, d) = (1, q 2 + q + 1) if q is odd , (2 2h − 3 h + 2 h , 2 h + 2) if q = 2 h . The first part just follows from the observation that if A is the incidence matrix of a projective plane of order q, then by definition A A = A A = q I + J , where I is the identity matrix and J the all-one matrix of size q 2 + q + 1. From Theorem 3.4 we can see that binary codes from PG(2, q) are only interesting whenever q is even. Moreover, one can see that the incidence matrix of Π has constant row and column weight equal to q + 1 which is O( q 2 + q + 1). Hence, codes from projective planes are very special examples of MDPC codes. With the aid of Theorem 2.4, we can show that one round of the bit-flipping algorithm on these codes permits to decode up to half the minimum distance with no failure probability, for any projective plane. Theorem 3.5 Let Π be a projective plane of even order and H its incidence matrix, which is the parity-check matrix of the code C 2 (Π) ⊥ . After performing one round of bit-flipping on H we can correct any error of weight up to d−1 2 , where d is the minimum distance of C 2 (Π) ⊥ . Proof Since a projective plane is in particular a symmetric 2-(q 2 + q + 1, q + 1, 1)-design, the maximum column intersection of H is 1. Moreover, the matrix H is of type (q +1, q +1). Hence, applying Theorem 2.4, we obtain that one round of the bit-flipping algorithm corrects every error of weight at most d−1 2 . Theorem 3.5 shows that codes from planes are really powerful, and have the best performance according to Theorem 2.4, for a given matrix of type (q + 1, q + 1) and size (q 2 + q + 1) × (q 2 + q + 1). However, we can only construct codes from projective planes of even order, resulting in [2 2h + 2 h + 1, 2 2h − 3 h + 2 h , 2 h + 2] 2 codes. This lack of choice of the parameters motivated many variations on this construction. In the last 50 years, many codes have been constructed based on underlying geometric objects: Euclidean and projective geometries over finite fields [6,19,33], linear representation of Desarguesian projective planes [26], (semi-)partial geometries [16,35], generalized quadrangles [17,36], generalized polygons [22], Ramanujan graphs [23,30], q-regular bipartite graphs from point line geometries [18] and other incidence structures coming from combinatorial designs [12][13][14][15]. For the same reason, we propose a new construction of (families of) MDPC codes based on a suitable system of conics in a Desarguesian projective plane that behaves itself like a projective plane. This is encapsulated in the concept of projective bundles, which we define, along with other notions from finite geometry, in the following section. MDPC codes from projective bundles In this section we present the new MDPC codes using projective bundles by constructing its parity-check matrix. We start off by introducing the relevant geometrical objects, which are ovals and projective bundles in PG(2, q). Definition 4.1 An oval in PG(2, q) is a set of q + 1 points, such that every line intersects it in at most two points. The classical example of an oval is a non-degenerate conic, i.e. the locus of an irreducible homogeneous quadratic equation. When q is odd, Segre's seminal result [32] shows that the converse is also true: every oval is a conic. Definition 4.2 A line in PG(2, q) is skew, tangent or secant to a given oval if it intersects it in zero, one or two points, respectively. We recall some properties of ovals which were first recorded by Qvist [28]. We include the proof as it will be relevant later. Lemma 4.3 An oval in PG(2, q) has q + 1 tangent lines, one in each point. -If q is odd, every point not on the oval is incident with zero or two tangent lines. -If q is even, then all tangent lines are concurrent. Proof Consider a point on the oval. Then there are q lines through this point intersecting the oval in one more point. This means that one line remains, which is necessarily a tangent line, hence proving the first part of the lemma. Now suppose that q is odd and consider a point on a tangent line, not on the oval. As the number of points of the oval, i.e. q + 1, is even, this point is incident with an odd number of tangent lines more. Since the point is arbitrary, and there are q + 1 tangent lines in total, this implies that every point on the tangent line (but not on the oval) is incident with exactly two tangent lines. When q is even, we consider a point on a secant line, but not on the oval and proceed in a similar fashion as before: the number of points on the oval but not on the secant is q − 1 and hence odd, so this point is incident with an odd number of tangents. Since this point is arbitrary, and there are q + 1 tangent lines, this implies that every point on the secant line is incident with exactly one tangent line. This also means that the intersection point of two tangent lines is necessarily the intersection of all tangent lines: this intersection point cannot be on a secant as we just saw, so the q + 1 lines through the point are either tangent or skew. Since the oval has q + 1 points, which are all contained in one of these lines, we deduce that they must be all tangent. When q is even, one can add the point of concurrency of the tangent lines, which is called the nucleus, to the oval to obtain a set of q + 2 points that has zero or two points in common with every line. This leads us to the following definition. Definition 4.4 A hyperoval is a set of q + 2 points in PG(2, q) such that every line has zero or two points in common. A dual hyperoval is a set of q + 2 lines such that every point is incident with zero or two lines. We will encounter these objects again later on. We are now in the position to define projective bundles. Definition 4.5 A projective bundle is a collection of q 2 + q + 1 ovals of PG(2, q) mutually intersecting in a unique point. Projective bundles were introduced by Glynn in his Ph.D. thesis [8] under the name 'packings of (q + 1)-arcs'. The original definition is a bit more general and applies to any projective plane instead of just PG(2, q). Since the only known projective bundles exist in PG(2, q), it suffices for our purposes to restrict ourselves to this case. It follows from the definition that one can consider the points of PG(2, q) and the ovals of a projective bundle as the points and lines of a projective plane of order q. We can then define the notion of secant, tangent and skew ovals (which belong to the projective bundle) with respect to a line. Moreover, one can interchange the role of lines and ovals in the proof of Lemma 4.3 and find the following statement, which we record for convenience. Lemma 4.6 Given a projective bundle, a line in PG(2, q) has q + 1 tangent ovals, one in each point. • If q is even, then all tangent ovals are concurrent. • If q is odd, every point not on the line is incident with zero or two tangent ovals to this line. When q is even, we can similarly as before define a hyperoval of ovals as a set of q + 2 ovals such that every point is contained in zero or two of them. An interesting property of projective bundles is that a third projective plane can be found. This result is due to Glynn [8, Theorem 1.1.1] and served as the motivation for projective bundles: to possibly find new projective planes from known ones. Theorem 4.7 Consider the ovals of a projective bundle and the lines of PG(2, q) as points and lines, respectively, with incidence defined by tangency. Then this point-line geometry is a projective plane of order q. We can rephrase this in terms of incidence matrices as follows: if A and B are the pointline incidence matrices of PG(2, q) and the projective plane whose lines are the ovals of a projective bundle, then A B (mod 2) is again the incidence matrix of a projective plane. However, for q even this idea to construct new projective planes does not work, since then all three projective planes are isomorphic [8, Corollary 1. 1.1]. Glynn showed that projective bundles indeed exist for any q, and his examples are all bundles of conics. When q is odd, he showed the existence of three distinct types of projective bundles in PG(2, q), by identifying them with planes in PG (5, q). It was shown in [4] that perfect difference sets can also be used to describe these projective bundles. In fact, given a perfect difference set D ⊆ Z/(q 2 + q + 1)Z and its circular shifts corresponding to the set of lines of PG(2, q), the three bundles are represented in the following way. We are now going to construct the parity-check matrix as mentioned at the beginning of this section. Let us denote the projective plane formed by the points and lines of PG(2, q) by Π and the one formed by the points and the ovals of a projective bundle of PG(2, q) by Γ . Then define H = ( A | B ),(1) where A and B are the incidence matrices of Π and Γ , respectively. Hence, we obtain a (q 2 + q + 1) × 2(q 2 + q + 1) binary matrix defined by the points, lines and ovals of a projective bundle of PG(2, q). (1) is called a projective bundle code and we will denote it by Definition 4.8 A binary linear code with parity-check matrix H given in C 2 (Π Γ ) ⊥ = ker(H ). Clearly, the matrix H given in (1) has constant row-weight w = 2(q + 1) and constant column-weight v = q + 1. Hence, C 2 (Π Γ ) ⊥ is an MDPC code of length n = 2(q 2 + q + 1) and type (q + 1, 2(q + 1)). Remark 4.9 The family of MDPC codes that we are considering is built upon a parity-check matrix as in (1). In such a matrix the number of columns is twice the number of rows and this coincides with the setting originally studied in [24]. Example 4.10 Let us give a short example of a projective bundle code for a relatively small parameter q = 3. Hence, we consider the projective plane PG (2,3). Recall, that the set of points P is given by the set of integers modulo q 2 + q + 1 = 13. The set of lines L is defined by the image of a perfect difference set D of four integers under repeated application of the Singer cycle S(i) = i + 1. It is easy to verify that D = {0, 1, 3, 9} is a perfect difference set, i.e. L = {{0 + i, 1 + i, 3 + i, 9 + i} | i ∈ Z/13Z}. At this point, let us choose an inscribed bundle B I in PG(2, 3). As shown above, this bundle is represented by the cyclic shifts of 2D = {0, 2, 5, 6}. Hence, we obtain B I = {{0 + i, 2 + i, 5 + i, 6 + i} | i ∈ Z/13Z}. Concatenating the two corresponding incidence matrices A and B yields the desired paritycheck matrix H = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 · · · 1 · · · · · 1 · 1 1 · · · · · · 1 1 · · 1 · 1 1 · · · 1 · · · · · 1 · · 1 · · · · · · 1 1 · · 1 · 1 1 · · · 1 · · · · · 1 1 · 1 · · · · · · 1 1 · · 1 · 1 1 · · · 1 · · · · · · 1 · 1 · · · · · · 1 1 · · 1 · 1 1 · · · 1 · · · · · · 1 · 1 · · · · · · 1 1 · · 1 · 1 1 · · · 1 · · · 1 · · 1 · 1 · · · · · · 1 · · · 1 · 1 1 · · · 1 · · 1 1 · · 1 · 1 · · · · · · · · · · 1 · 1 1 · · · 1 · · 1 1 · · 1 · 1 · · · · · · · · · · 1 · 1 1 · · · 1 · · 1 1 · · 1 · 1 · · · · 1 · · · · · 1 · 1 1 · · · · · · 1 1 · · 1 · 1 · · · · 1 · · · · · 1 · 1 1 · · · · · · 1 1 · · 1 · 1 · · · · 1 · · · · · 1 · 1 1 · · · · · · 1 1 · · 1 · 1 · · · · 1 · · · · · 1 · 1 1 · · · · · · 1 1 · · 1 · 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , where the zero entries in the parity-check matrix are represented by dots. Remark 4.11 Observe that the matrix H defined in (1) can be constructed from a perfect difference set D, by taking the circular shifts of D and s D, with s ∈ {−1, 2, 2 −1 }. Such a matrix has a double circulant structure. Thus, the resulting code C 2 (Π Γ ) ⊥ is quasi-cyclic of index 2, and encoding can be achieved in linear time and implemented with linear feedback shift registers. Furthermore, we can also deduce-because of the circular structure-that the number of bits required to describe the parity-check matrix is about (q + 1) log 2 (q 2 + q + 1), which is approximately 2(q + 1) log 2 (q). Remark 4.12 For q odd, when Γ is a circumscribed bundle, the incidence structure Π Γ given by the points of PG(2, q) and the set of lines together with the ovals in Γ has already appeared in the literature. Indeed, it coincides with the incidence structure given by a scattered linear set of pseudoregulus type in PG(1, q 3 ) and the set of its F q -sublines; see [21,Remark 20]. In the following subsections we will analyse the dimension, minimum distance and errorcorrection performance with respect to the bit-flipping decoding algorithm of C 2 (Π Γ ) ⊥ . Dimension Recall from Theorem 3.4 that a binary code C 2 (Π) ⊥ from a projective plane Π ∼ = PG(2, q), is either trivial of codimension 1-when q is odd-or it is non-trivial to determine its dimension-when q is even. In our case, the structure of our code allows to both have a non-trivial code and to determine the exact dimension for all q. To do so, recall that if A is the incidence matrix of a projective plane of order q, then A A = A A = q I + J , where J is the all-one matrix of appropriate size. Using this result we are able to state the dimension of C 2 (Π Γ ) ⊥ . Proposition 4.13 Let Π be a projective plane of order q and let Γ be a projective bundle in Π. Then, dim C 2 (Π Γ ) ⊥ = q 2 + q + 2 if q is odd, 2 2h+1 + 2 h+1 − 2(3 h ) + 1 if q = 2 h . Proof In order to determine the dimension of the code, we need to compute the rank of a parity-check matrix H = ( A | B ). Since H is of size (q 2 + q + 1) × 2(q 2 + q + 1), we can already say that the rank of H is at most q 2 + q + 1. Now we consider the two cases. Case I q odd We know from Theorem 3.4 that rk(A) = q 2 + q, which gives us the lower bound rk(H ) ≥ rk(A) = q 2 + q. The matrix H has full rank q 2 + q + 1 if and only if there exists no element in the left-kernel, i.e. if there is no nonzero vector x ∈ F q 2 +q+1 2 such that x H = 0.(2) However, if x is the all-one vector then Eq. (2) is satisfied. Hence, there is an element in the cokernel which implies that H cannot have full rank and we conclude that dim C 2 (Π Γ ) ⊥ = q 2 + q + 2. Case II q even In this case, we consider the matrix H H = A A A B B A B B = J A B (A B) J . By Theorem 4.7 and the discussion below, A B = C is again the incidence matrix of PG(2, q), and hence the sum of all its rows/columns is equal to the all-one vector. Therefore, by doing row operations on H H , we obtain the matrix 0 A B + J (A B) J , which has the same rank as H H . Hence, rk(H ) ≥ rk(H H ) = rk(A B) + rk(A B + J ) ≥ 2 rk(A B) − 1, where the last inequality comes from the fact that J has rank 1, and the rank satisfies the triangle inequality. On the other hand, we have that the all-one vector is in the column spaces of both A and B, showing that rk(H ) ≤ rk(A) + rk(B) − 1. Since A, B and A B are all incidence matrices of a Desarguesian plane, they all have the same rank. Therefore, combining the two inequalities, we obtain rk(H ) = 2 rk(A) − 1, and using Theorem 3.4, we can conclude that dim C 2 (Π Γ ) ⊥ = 2(q 2 + q + 1) − rk(H ) = 2(q 2 + q + 1) − 2 rk(A) + 1 = 2 dim(C 2 (Π) ⊥ )) + 1 = 2 2h+1 + 2 h+1 − 2(3 h ) + 1. We can thus already say that C 2 (Π Γ ) ⊥ is a [2(q 2 + q + 1), q 2 + q + 2] 2 MDPC code of type (q + 1, 2q + 1). Minimum distance As mentioned earlier, we are interested in the error-correction capability. A relevant quantity to give information about error-correction and also error-detection is the minimum distance of a linear code. In the following we will determine the exact value of the minimum distance of C 2 (Π Γ ) ⊥ . An important observation for the proof is that geometrically, the support of a codeword of C 2 (Π Γ ) ⊥ corresponds to a set of lines and ovals such that every point of PG(2, q) is covered an even number of times. Theorem 4.14 The minimum distance of C 2 (Π Γ ) ⊥ is q +2 and the supports of the minimum weight codewords can be characterized, depending on the parity of q. For q odd, the support of a minimum weight codeword is -an oval and its q + 1 tangent lines, or -a line and its q + 1 tangent ovals. On the other hand for q even, we find that the support of a minimum weight codeword is -a dual hyperoval, or -a hyperoval of ovals. Proof Take a codeword of minimum weight in C 2 (Π Γ ) ⊥ and consider its support. This is a set of r lines L and s ovals O such that every point in PG(2, q) is incident with an even number of these elements. We will show that r + s ≥ q + 2 and equality only holds for the two examples stated. Every point is incident with q + 1 lines and q + 1 conics, so let a i , 0 ≤ i ≤ 2q + 2, be the number of points that are covered i times by the r lines and s conics in the support of the minimum weight codeword. Then we can double count the tuples (P), (P, E 1 ), (P, E 1 , E 2 ), where P is a point and E 1 , E 2 ∈ L ∪ O are lines or ovals incident with this point. Remark that by assumption a i = 0 whenever i is odd. We find the following three expressions: 2q+2 i=0 a i = q 2 + q + 1 ( 3 ) 2q+2 i=0 ia i = (r + s)(q + 1) (4) 2q+2 i=0 i(i − 1)a i ≤ r (r − 1) + s(s − 1) + 2rs,(5) where the last inequality follows as a line and oval intersect in at most two points. From these equations, we can find 2q+2 i=0 i(i − 2)a i ≤ (r + s)(r + s − q − 2) and hence r + s ≥ q + 2, as the sum on the left-hand side has only non-negative terms. Moreover, in the case of equality, a i = 0 whenever i / ∈ {0, 2}. Now consider a codeword of weight r + s = q + 2, consisting of r lines L and s ovals O. We will investigate the cases q odd and even separately and show the characterisation. Case I q odd Since q + 2 is odd and hence one of r or s is, we can suppose without loss of generality that r is odd. The argument works the same when s is odd, by interchanging the roles of lines and ovals. Consider a line not in L. Then this line is intersected an odd number of times by the r lines in L. Therefore, it should be tangent to an odd number of ovals in O, recalling that every point is incident with zero or two elements from L ∪ O. In particular, any line not in L is tangent to at least one oval in O. So count the N pairs ( , c), where is a line not in L, c ∈ O and | ∩ O| = 1. By the previous observation, it follows that q 2 + q + 1 − r = q 2 − 1 + s ≤ N . On the other hand, a oval has q + 1 tangent lines so that N ≤ s(q + 1). Combining these two leads to s ≥ q, which implies that r = 1 and s = q + 1. Remark that this argument only depends on r being odd. If o ∈ O is one of these q + 1 ovals, we see that the other q ovals intersect O in q distinct points, as no point is incident with more than two elements from L ∪ O. This immediately implies that the unique line in L must be tangent to o. As O was arbitrary, we conclude that the support of the codeword consists of one line and q + 1 ovals tangent to it. By Lemma 4.6 this indeed gives rise to a codeword, as every point not on the line is incident with zero or two ovals. Case II q even The situation is slightly different. Since q + 2 is even now, either r and s are both odd, or both even. When r is odd, we can reuse the argument from before to find the configuration of q + 1 ovals tangent to a line. However, by Lemma 4.6 we know that these q + 1 ovals are all incident with a unique point, which is hence covered q + 1 times, a contradiction. So suppose that r and s are even. Any line in L is intersected by the r − 1 other lines in L, leaving q + 1 − (r − 1) points to be covered by the ovals in O, which is an even number. We see that we must have an even number of tangent ovals to this line. Similarly for a line not in L, we observe that it is intersected an even number of times by the r lines in L and hence it should have an even number of intersections with the ovals in O, leading again to an even number of tangent ovals. In summary, every line in PG(2, q) is incident with an even number of tangent ovals. Now consider any line and recall that the ovals tangent to are concurrent, say in the point N , by Lemma 4.6. However, as we saw before, N , like any other point, is covered zero or twice by the elements of O ∪ L. It follows that N is incident with zero or two ovals in O and hence that is tangent to zero or two ovals of O. So suppose that s > 0, meaning we have at least one oval in O and consider its q + 1 tangent lines. Then each of these lines should have one more tangent oval, and all of these are distinct by Corollary 4.7, which means we find s = q + 2 ovals forming a hyperoval of ovals. If s = 0, we find a dual hyperoval, concluding the theorem. Error-correction capability It is well-known that the minimum distance of a code gives information about the decoding radius. This means that it reveals an upper bound on the amount of errors that can be always detected and corrected. We would like to focus in this subsection here on the performance of the constructed MDPC code C 2 (Π Γ ) ⊥ within one round of the bit-flipping decoding algorithm. We now adapt and apply Theorem 2.4 to the parity-check matrix H of C 2 (Π Γ ) ⊥ given in (1). Proposition 4.15 The maximum column intersection of the matrix H defined in (1) is s H = 2. Thus, after performing one round of the bit-flipping algorithm on H we can correct all the errors of weight at most q+1 4 in the code C 2 (Π Γ ) ⊥ . Proof From the construction of H we have that H consists of two matrices A and B which are the incidence matrices of points and lines and points and ovals of a projective bundle in PG(2, q), respectively. Clearly, both matrices A and B have a maximum column intersection equal to 1 as two distinct lines in a projective plane intersect in exactly one point and a similar property holds for every pair of distinct ovals of a projective bundle by definition. Since every line intersects an oval in at most 2 points, the maximum column intersection of the matrix H is at most 2. On the other hand, if we consider any two distinct points on an oval in the projective bundle, there always exists a line passing through them. Hence, s H = 2. The second part of the statement then follows directly from Theorem 2.4. Remark 4.16 Observe that s H = 1 for a parity-check matrix of size (q 2 + q + 1) × c and column weight q + 1 implies c ≤ q 2 + q + 1. this can be seen by counting the tuples {(x, y, B) | x, y ∈ B} in two ways. Thus, the value s H = 2 is the best possible for c > q 2 + q + 1. Furthermore, compared to a random construction of MDPC code, our design guarantees a deterministic error-correction performance for one round of the bit-flipping decoding algorithm. In particular, for the random model proposed in [34] it was proved that the expected value of s H is O( log n log log n ). Hence, our construction guarantees an errorcorrection capability for one round of the bit-flipping algorithm which improves the random construction by a factor O( log n log log n ). Additionally, we have implemented the parity-check matrix for our MDPC code design as well as the bit-flipping decoding algorithm. We were interested if we could correct even more errors than the number guaranteed in Proposition 4.15. Since the bit-flipping decoding algorithm is only dependent on the syndrome and not on the actual chosen codeword, we took the all-zero codeword and added a pseudo-random error-vector of a fixed weight wt(e) ≥ q+1 4 . We have generated 10 5 distinct error vectors. Each of these error vectors then was used to run one round of the bit-flipping decoding algorithm for all the three different families of MDPC codes that we have constructed. It turned out that the codes constructed from each of the three types of projective bundles-circumscribed, inscribed, self-polar-showed exactly the same error-correction performance. Finally, we have computed the probability of successful error-correction for the parameters q ∈ {5, 7, 9, 11, 13, 17, 19, 23, 25}. The following results were obtained for the different error weights. Table 1 shows that the probability to correct even more errors grows as we increase q. This is due to the fact, that for small q we reach the unique decoding radius much faster. In the following we show some empirical results on the success probability after performing more than one round of the bit-flipping decoding algorithm (Tables 2, 3). Remark 4.17 In [34] the author analyzed also the error-correction performance after two rounds of the bit-flipping decoding algorithm. More precisely he estimated the probability that one round of the algorithm corrects enough errors so that in the second round all remaining errors will be correctable. Following the notation of that paper, let us denote by S the number of errors left after one round of the bit-flipping algorithm. Assuming that we have an MDPC code of length n and of type (v, w), where both v and w are of order ( √ n), the probability that S is at least a certain value t satisfies the following inequality: P S ≥ t ≤ 1 √ t e t v 4 ln(1−e − 4wt n )+ t 8 ln(n)+O(t ln(t /t)) ,(6) where t = ( √ n) is the initial amount of errors that were introduced. We have seen in Proposition 4.15, that performing one round of the bit-flipping algorithm to a parity-check matrix H of C 2 (Π Γ ) ⊥ we can correct q+1 4 errors. Therefore, a second round of the bit-flipping is able to correct completely if after one round there are no more than (1), we obtain that we can successfully correct every error of weight t = ( √ n) after two rounds of the bit-flipping decoding algorithm with probability e − (n) . Generalizations Since our aim is to have more flexibility in the parameters, here we generalize the approach of Sect. 4, by considering several disjoint projective bundles. Let t > 1 be a positive integer and let us fix a Desarguesian projective plane Π = PG(2, q). Let Γ 1 , . . . , Γ t be disjoint 1 projective bundles of conics in Π. Since we want s H to be low, we cannot take projective bundles of ovals in general, as for example two ovals in PG(2, q), q even, could intersect in up to q points: take any oval, add the nucleus and delete another point to find a second oval intersecting it in q points. In Proposition 5.3 we will see that by choosing conics, we find s H = 4. Furthermore, the number t of disjoint projective bundles cannot be chosen arbitrarily. However, the restrictions we have on t will not affect our intent to construct MDPC codes; see the discussion in Remark 5.4 for the admissible values of t. Let us denote by A the incidence matrix of Π and by B i the incidence matrix of the projective bundle Γ i , for each i ∈ {1, . . . , t}. We then glue together all these matrices and consider the code C 2 (Π Γ 1 . . . Γ t ) ⊥ to be the binary linear code whose parity-check matrix is H q,t := (A | B 1 | · · · | B t ) . As already discussed, it is important to specify which parity-check matrix of a code we consider when we study the decoding properties, since the bit-flipping algorithm depends on the choice of the parity-check matrix. We focus now on the parameters on the constructed codes. We first start with a result on the dimension of the code C 2 (Π Γ 1 . . . Γ t ) ⊥ Proposition 5.1 Let Π = PG(2, q) be a Desarguesian projective plane of order q and let Γ 1 , . . . , Γ t be t projective bundles in Π. Then, dim C 2 (Π Γ 1 . . . Γ t ) ⊥ = t(q 2 + q + 1) + 1 if q is odd, dim C 2 (Π Γ 1 . . . Γ t ) ⊥ ≥ (t + 1)(2 2h − 3 h + 2 h ) + t i fq= 2 h . Proof The proof goes as for Proposition 4.13. Case I q odd We know from Theorem 3.4 that rk(A) = q 2 + q, which gives us the lower bound rk(H q,t ) ≥ rk(A) = q 2 + q. On the other hand, since A and each matrix B i have the all-one vector in its left-kernel, we have that also H q,t has a nontrivial left-kernel, and hence rk(H q,t ) = q 2 + q, yielding dim C 2 (Π Γ 1 . . . Γ t ) ⊥ = (t + 1)(q 2 + q + 1) − rk(H q,t ) = t(q 2 + q + 1) + 1. Case II q even Let us write q = 2 h . In this case, we have that the all-one vector belongs to the column spaces of each matrix B i . Therefore, rk(H q,t ) ≤ rk(A) + t i=1 rk(B i ) − t. Thus, we obtain dim C 2 (Π Γ 1 . . . Γ t ) ⊥ = (t + 1)(q 2 + q + 1) − rk(H q,t ) ≥ (t + 1)(q 2 + q + 1) − rk(A) − t i=1 rk(B i ) + t = (t + 1) dim(C 2 (Π) ⊥ ) + t = (t + 1)(2 2h − 3 h + 2 h ) + t, where the last equality comes from Theorem 3.4. Also in this general case we can study the minimum distance of the code C 2 (Π Γ 1 . . . Γ t ) ⊥ , generalizing the result on the minimum distance obtained when t = 1 in Theorem 4.14. However, this time we are only able to give a lower bound. Proposition 5.2 The minimum distance of C 2 (Π Γ 1 . . . Γ t ) ⊥ is at least q+2 2 . Proof The proof goes in a similar way as the one of Theorem 4.14. Take a codeword of minimum weight in C 2 (Π Γ 1 . . . Γ t ) ⊥ and consider its support. This is a set L of r lines and a set O i of s i ovals for each i ∈ {1, . . . , t} such that every point in PG(2, q) is incident with an even number of these elements. We will show that r + s 1 + . . . + s t ≥ q+2 2 . Let a i , 0 ≤ i ≤ (t + 1)(q + 1), be the number of points that are covered i times, then we can double count the tuples (P), (P, E 1 ), (P, E 1 , E 2 ), where P is a point and E 1 , E 2 ∈ L ∪ O 1 ∪ . . . ∪ O t are lines or ovals incident with this point. Remark that by assumption a i = 0 whenever i is odd. We find the following three expressions: (t+1)(q+1) i=0 a i = q 2 + q + 1 ( 8 ) (t+1)(q+1) i=0 ia i = r + t i=1 s i (q + 1) (9) (t+1)(q+1) i=0 i(i − 1)a i ≤ r (r − 1) + t i=1 s i (s i − 1) + 2r t i=1 s i + 4 1≤i< j≤t s i s j ,(10) as two conics intersect in at most 4 points by Bézout's theorem. Subtracting (9) from (10) we obtain 0 ≤ (t+1)(q+1) i=0 i(i − 2)a i = r + t i=1 s i r − q − 2 + t i=1 s i + 2 1≤i< j≤t s i s j . One can easily check that this last quantity is in turn at most r + t i=1 s i 2r − q − 2 + 2 t i=1 s i , which then implies r + t i=1 s i ≥ q + 2 2 . As a direct consequence of Proposition 5.2 we have that in principle it should be possible to correct at least q 4 errors in the code C 2 (Π Γ 1 . . . Γ t ) ⊥ when q is even, and at least q+1 4 when q is odd. However, also in this case, when running one round of the bit-flipping algorithm on the matrix H q,t given in (7), we only correct a smaller fraction of them, as the following result shows. (7) is at most 4. Thus, after performing one round of the bit-flipping algorithm on H q,t we can correct all the errors of weight at most q+1 Proposition 5.3 The maximum column intersection of the matrix H q,t defined in 8 in the code C 2 (Π Γ 1 . . . Γ t ) ⊥ . Proof The maximum column intersection is given by the maximum number of points lying in the intersection of elements in Π Γ 1 . . . Γ t . Each pair of lines intersects in exactly a point, and the same holds for every pair of conics belonging to the same projective bundle, since each projective bundle is itself (ismorphic to) a projective plane. Moreover, every line intersects a conic in at most two points, and we have already seen that each pair of conics meets in at most 4 points. Hence, the maximum column intersection of H q,t is at most 4. The second part of the statement directly follows from Theorem 2.4. Remark 5.4 At this point it is natural to ask whether it is possible to construct disjoint projective bundles, and-if so-how many of them we can have. It is shown in [4, Theorem 2.2] that one can always find (q − 1) disjoint projective bundles when q is even, and q 2 (q−1) 2 of them when q is odd. We want to remark that this is not a restriction, since we still want that our codes C 2 (Π Γ 1 . . . Γ t ) ⊥ (together with the parity-check matrices H q,t of the form (7)) give rise to a family of MDPC codes. Thus, we are typically interested in family of codes where t is a constant and does not grow with q. Remark 5.5 This construction provides a better performance of (one round of) the bit-flipping algorithm compared to the one run on random constructions of MDPC codes explained in [34]. As already explained in Remark 4.16, the random construction of MDPC codes provides in average MDPC codes whose maximum column intersection is O( log n log log n ), and thus one round of bit-flipping algorithm corrects errors of weight at most O( √ n log log n log n ) in these random codes. Hence, also the generalized constructions of codes from projective bundles have asymptotically better performance in terms of the bit-flipping algorithm. Further variations and comparisons In the previous section we have shown that codes constructed using projective bundles have a very interesting combinatorial and geometric structure, which allows to determine the parameters and correction capability properties. In particular, we were able not only to determine the minimum distance of codes obtained using only a projective bundle, but we also classified their minimum weight codewords. In terms of parameters, it is clear that there exist better families of linear codes. Indeed, the codes presented in Sect. 4 have minimum distance O( √ n) and hence they are certainly not asymptotically good. Nevertheless, in addition to the geometric structure, the combinatorial characterization also allows a very efficient storage. In particular, let q = 2 h be even, and consider a projective plane Π ∼ = PG(2, 2 h ) and the inscribed projective bundle Γ in PG (2, q). Then, it is readily seen that B = A 2 . When instead one has that Γ is a circumscribed bundle, then A = B 2 . In both cases, we have that the code C 2 (Π Γ ) ⊥ is (equivalent to) a code with parity-check matrix of the form H = ( A | A 2 ). Since we have seen in the proof of Proposition 4.13 that for q even we have rk(H ) = 2 rk(A) − 1, then one can verify that the code C 2 (Π Γ ) ⊥ is made by two copies of C 2 (Π) ⊥ , together with the all-one vector. Formally, we have C 2 (Π Γ ) ⊥ = (c + λx | c + λx) | c, c ∈ C 2 (Π) ⊥ , λ ∈ F 2 , where x denotes the all-one vector of length (2 2h + 2 h + 1). Thus, for coding theoretic purposes, this code is not more interesting than just the code C 2 (Π) ⊥ itself. However, we remark that we included nevertheless the results also when q is even, in order to have a complete study and for their intrinsic geometrical interest. Another code construction Now, for the case of q odd, we present another construction that only deals with the incidence matrix A of a projective plane Π ∼ = PG(2, q). We define the code D p (Π) ⊥ to be the code over F p whose parity-check matrix is H := I A A I .(11) For the rest of this section, we will always use H to denote the matrix in (11). The following lemma is a straightforward computation using Gaussian elimination, whose proof is left to the reader. Lemma 6.1 Let q be an odd prime power and let Π be a projective plane of order q. The code D p (Π) ⊥ coincides with the right kernel of I A 0 J + (q + 1)I . In particular, if q is odd and p = 2 the following matrices are parity-check matrices for the code D 2 (Π) ⊥ : H 1 := I A 0 x , H 2 := A I x 0 , where x is the all-one vector of length q 2 + q + 1. Observe that the columns of the matrix H in (11), as well as those of H 1 and H 2 of Lemma 6.1, can be thought as corresponding to lines and points of Π. In particular, the first q 2 +q +1 columns of H (resp. H 1 and H 2 ) correspond to the points of Π and the last q 2 +q +1 columns of H (resp. H 1 and H 2 ) correspond to the lines of Π. Furthermore, as we did in Theorem 4.14, since we are only considering binary codes, we also identify the supports of the codewords with the corresponding sets of lines and/or points. By the definition of its parity-check matrix, it follows that a codeword in D 2 (Π) ⊥ corresponds to a set of points and lines such that every point and line in PG(2, q) is incident with an even number of elements, where a point (resp. line) is incident with itself but no other point (resp. line). Furthermore, by Lemma 6.1, we see from considering the last row that both the number of points and the number of lines in a subset corresponding to a codeword should be even. Theorem 6.2 Let q be an odd prime power, and let Π be a projective plane of order q. Then, D 2 (Π) ⊥ is a [2(q 2 + q + 1), q 2 + q, 2q + 2] 2 code. Furthermore, the codewords of minimum weight correspond to one of the following cases: -Any two distinct lines 1 and 2 of Π and the set of points on one of 1 or 2 but not both; -Dually, any two distinct points P 1 , P 2 and the set of lines through one of P 1 or P 2 but not both; -An oval and its q + 1 tangent lines. Proof The fact that dim(D 2 (Π) ⊥ ) = q 2 + q directly follows from Lemma 6.1, since it is immediate to see that the matrices H 1 and H 2 have full row rank equal to q 2 + q + 2. Thus, we only need to determine the minimum distance d of D 2 (Π) ⊥ . The three families of subsets mentioned above indeed define codewords of D 2 (Π) ⊥ , which can be seen by the discussion preceding the theorem. Therefore, the minimum distance is at most the weight of any of these three, which is 2q + 2 for all of them. We will show that this is the minimum possible and characterize the three types as the only codewords of this weight. So take a set of r points and s lines and assume r + s ≤ 2q + 2. We cannot have r = 0 or s = 0 as can be seen geometrically or by Lemma 6.1, so by their evenness we find r , s ≥ 2. Since any line contains q + 1 points and any two lines intersect in one point, we see that there are at least s(q + 1) − s(s − 1) points which are covered an odd number of times. This means that r ≥ s(q + 2 − s), and hence r + s ≥ s(q + 3 − s). The minimum is attained for s ∈ {2, q + 1} and equals 2q + 2 so the minimum distance is indeed 2q + 2. We now characterize the codewords of minimum weight. When s = 2, we have two lines 1 , 2 and we need to add the 2q points on ( 1 ∪ 2 ) \ ( 1 ∩ 2 ) as they are incident with an odd number of lines. This already adds up to 2q + 2 and this set indeed defines a codeword. Dually, when r = 2 the same argument shows that one finds the second type of minimum weight codeword. Finally when r = s = q + 1 we find equality in the argument above which implies that there are no three lines concurrent, and dually that no three points are collinear. In other words, we find the union of an oval and a dual oval, which can only define a codeword if the dual oval consists exactly of the tangent lines of the oval, which gives the last type. Corollary 6.3 The number of minimum weight codewords in D 2 (Π) ⊥ is equal to q(q 2 + q + 1)(q 2 + 1). Proof By Theorem 6.2, the codewords of minimum weight are of three types. The number of codewords of the first type equals the number of pair of lines, and by duality is also equal to the number of codewords of the second type. They clearly sum up to (q 2 + q + 1)q(q + 1). The number of codewords of the last type coincides with the number of ovals in PG(2, q), which by Segre's theorem [32] is equal to q 2 (q 3 − 1), i.e. the number of nondegenerate conics. This gives the desired result. Error-correction capability and experimental results In the following result we derive the error-correction capability of (one round of) the bitflipping algorithm. (11) Thus, we have that the correction capability of one round of the bit-flipping algorithm for H as in (11) is the same as the one for H in (1). Hence, with these two choices of paritycheck matrices and using one round of the bit-flipping algorithm, the codes C 2 (Π Γ ) ⊥ and D 2 (Π) ⊥ have the same error-correction capability, at least for what concerning all the error patterns. Proposition 6.4 The intersection number of the matrix H defined in Nevertheless, similar to the experiments seen in Sect. 4.3 we were curious on how many additional errors we can add to the number of errors found in Proposition 6.4 and still be able to correct after one and more rounds of the bit-flipping decoding algorithm. The simulations were run under the exact same circumstances as for the code C 2 (Π Γ ) ⊥ in order to be able to compare the results. The following tables show the success rate of correcting more than (q + 1)/4 errors within up to four rounds of bit-flipping. Table 4 already shows us that the code D 2 (Π) ⊥ based on a parity-check defined in (11) has a probability of 1 to correct one error more than the expected number of errors from Proposition 6.4 for every value of q presented in Table 4. If we increase the number of rounds of the bit-flipping algorithm we expect that more and more errors are correctable. This expectation is motivated by Theorem 6.2, stating that the minimum distance of D 2 (Π) ⊥ is given by d(D 2 (Π) ⊥ ) = 2q + 2. Hence the unique decoding radius is q + 1, which is about four times as large as the number of errors correctable within one round of bit-flipping. The Tables 5, 6 and 7 support this expectation. Comparisons In this final section let us compare the two code constructions C 2 (Π Γ ) ⊥ and D 2 (Π) ⊥ of MDPC codes. First of all, with both the constructions we are able to give deterministic results on the error-correction performance for one round of the bit-flipping decoder, which for the random construction in [34] is not possible. The two codes C 2 (Π Γ ) ⊥ and D 2 (Π) ⊥ have the same length and almost the same dimension. From Theorem 4.14 and Theorem 6.2, we know that the minimum distance of D 2 (Π) ⊥ is almost twice of the minimum distance of C 2 (Π Γ ) ⊥ . Hence D 2 (Π) ⊥ , in general, is able to correct almost twice as many errors as C 2 (Π Γ ) ⊥ . Nevertheless, applying Theorem 2.4 to both of them, yields the same result for one round of the bit-flipping decoding algorithm. Furthermore, we observe that the number of nonzero entries of the parity-check matrix (1) for C 2 (Π ) ⊥ is almost the same as the number of nonzero entries of the paritycheck matrix (11) for D 2 (Π) ⊥ . Since the complexity of the bit-flipping decoder is depending on the length n, the maximal number of iterations and the number of nonzero entries, we deduce that it will have roughly the same run time. To conclude we can say, that the construction using projective bundles shows interesting properties from a mathematical viewpoint. Nevertheless, from a coding theoretic perspective the code D 2 (Π) ⊥ , which has the same length and almost the same dimension as C 2 (Π Γ ) ⊥ , shows a higher error-correction performance with respect to the bit-flipping decoder. In fact, its minimum distance is almost twice as large and hence the unique decoding radius is larger. Conclusion In this paper we proposed a new construction of a family of moderate-density parity-check codes arising from geometric objects. Starting from a Desarguesian projective plane Π of order q and a projective bundle Γ in Π, we constructed a binary linear code whose paritycheck matrix is obtained by concatenating the incidence matrices of Π and Γ . We observed that we can construct these two matrices taking the circular shifts of two perfect difference sets modulo (q 2 + q + 1), providing a natural structure as a quasi-cyclic code of index 2. Hence, the storage complexity is linear in the length and the encoding can be achieved in linear time using linear feedback shift registers. Furthermore, the underlying geometry of Γ and Π allowed us to study the metric properties of the corresponding code, and we could determine its exact dimension and minimum distance, as well as its minimum weight codewords. We then analyzed the performance of the bit-flipping algorithm showing that it outperforms asymptotically the one of the random construction of codes obtained in [34]. We then generalized the construction of this family of codes by concatenating the incidence matrices of several disjoint projective bundles living in the Desarguesian projective plane Π. In this case we were able to provide lower bounds on the parameters of the obtained codes exploiting their geometric properties. Nevertheless, we could still show that one round of the bit-flipping algorithm has the best asymptotic performance in terms of error-correction capability for the given parameters of the defining parity-check matrix. Finally, we gave an alternative construction of binary codes whose parity check matrices only use the incidence matrix of Π and its transpose. We determined the parameters of these codes and characterized the minimum weight codewords. The error-correction performance was then empirically studied by implementing the parity-check matrix and running the bit-flipping decoding for several iterations. The empirical results showed that this alternative construction outperforms the construction using projective bundles in terms of being able to successfully decode more errors. Hence, adding redundant rows to the parity-check matrix might seem to be a promising tool to improve the error-correction performance. Future research, thus, might study the case, when adding additional linearly dependent rows to the parity-check matrix of C 2 (Π Γ ) ⊥ and analyzing its performance for the bit-flipping decoder. In particular, it would be very interesting to understand whether there is a systematic way to add redundant parity-check equations in order to maximize the bit-flipping decoder performance. 1 . 1Cirumscribed bundle: set of all circular shifts of −D. 2. Inscribed bundle: set of all circular shifts of 2D. 3. Self-polar bundle: set of all circular shifts of D/2. . Applying(6) for t = q+1 4 to the parity-check matrix H of C 2 (Π Γ ) ⊥ given in {C i } be a family of binary linear codes of length n i with parity-check matrix H i . If H i has row weight O( √ n i ), {C i } is called a (family of) moderate-density parity-check (MDPC) code. If, in addition, the weight of every column of H i is a constant v i and the weight of every row of the H i is a constant w i we say the MDPC code is of type Theorem 2.4 Let C be an MDPC code of type (v, w) with parity-check matrix H . Let s H denote the maximum column intersection of H . Performing one round of the bit-flipping decoding algorithm with respect to H , we can correct all errors of weight at most v 2·s H . Table 1 Probability 1to correctly decode a received word, with error-weight q+1 4 + i for i = 1, 2, 3, after one round of the bit-flipping decoding algorithm q q+1 4 + 1 errors q+1 4 + 2 errors q+1 4 + 3 errors 5 50.82% 0.16% - 7 7.02% 0.34% - 9 79.31 % 3.86% - 11 43.83% 0.19% - 13 90.4% 14.4% - 17 97.2% 57.8% 7.8% 19 91.8% 42.6% 10.7% 23 97.86 % 77.66% 31.3% 25 99.87% 95.3% 71.25% Table 2 Probability to correctly decode a received word, with error-weight q+1 4 + i for i = 1, 2, 3, 4, after two rounds of the bit-flipping decoding algorithm q q+1 4 + 1 q+1 4 + 2 q+1 4 + 3 q+1 4 + 4 5 64.52% 3.45% - - 7 31.58% 4.75% - - 9 93.41 % 30.77% - - 11 43.83% 0.19% - - 13 99.45% 84.15% 35.52% 0.55% 17 100% 99.35% 84.69% 36.81% Table 3 Probability to correctly decode a received word, with error-weight q+1 4 + i for i = 1, 2, 3, 4, after three rounds of the bit-flipping decoding algorithm q q+1 4 + 1 q+1 4 + 2 q+1 4 + 3 q+1 4 + 4 5 64.52% 3.45% - - 7 45.61% 5.26% - - 9 94.51% 32.97% - - 11 93.23% 33.84% 8.27% - 13 99.45% 94.54% 44.36% 1.64% 17 100% 100% 93.49% 50.81% is s H = 2. Thus, after performing one round of the bit-flipping algorithm on H we can correct all the errors of weight at most q+1 in the code D 2 (Π) ⊥ .4 Table 4 4Probability to correctly decode a received word, with error-weight q+1 + i for i = 1, 2, 3, 4, after one round of the bit-flipping decoding algorithm Proof It is clear that the intersection number is s H = 2. Hence, by Theorem 2.4, after one round of the bit-flipping algorithm on H we can correct all the errors of weight q+2 4 , which is equal to q+1 4 since q is odd.4 q q+1 4 + 1 q+1 4 + 2 q+1 4 + 3 q+1 4 + 4 5 100% 51.61% 9.68% 1.61% 7 100% 64.91% 13.16% 6.14% 9 100% 100% 74.73% 42.86% 11 100% 100% 86.84% 53.01% 13 100% 100% 100% 92.08% 17 100% 100% 100% 100% Table 5 Probability to correctly decode a received word, with error-weight q+1 4 + i for i = 1, 2, 3, 4, after two rounds of the bit-flipping decoding algorithm q q+1 4 + 1 q+1 4 + 2 q+1 4 + 3 q+1 4 + 4 5 100% 53.23% 51.61% 9.68% 7 100% 64.91% 63.16% 52.63% 9 100% 100% 84.62% 68.68% 11 100% 100% 99.25% 84.21% 13 100% 100% 100% 99.73% 17 100% 100% 100% 100% Table 6 6Probability to correctly decode a received word, with error-weight q+1 + i for i = 1, 2, 3, 4, after three rounds of the bit-flipping decoding algorithm4 q q+1 4 + 1 q+1 4 + 2 q+1 4 + 3 q+1 4 + 4 5 100% 100% 90.32% 32.26% 7 100% 100% 76.32% 57.02% 9 100% 100% 100% 80.22% 11 100% 100% 100% 84.21% 13 100% 100% 100% 100% 17 100% 100% 100% 100% Table 7 Probability for D 2 (Π) ⊥ to correctly decode a received word, with error-weight q+1 4 + i for i = 1, 2, 3, 4, after four rounds of the bit-flipping decoding algorithm q q+1 4 + 1 q+1 4 + 2 q+1 4 + 3 q+1 4 + 4 5 100% 100% 98.38% 43.55% 7 100% 100% 83.33% 57.89% 9 100% 100% 100% 82.42% 11 100% 100% 100% 84.59% 13 100% 100% 100% 100% 17 100% 100% 100% 100% Here with "disjoint" we mean that any two distinct projective bundles have no common oval. AcknowledgementsWe would like to thank the referees for their valuable comments. In particular, the code discussed in Sect. 6 is due to one of their suggestions. The work of A. Neri was supported by the Swiss National Science Foundation through Grant No. 187711. The work of J. Rosenthal was supported by the Swiss National Science Foundation through Grant No. 188430.Funding Open access funding provided by University of Zurich.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Small weight codewords of projective geometric codes. S Adriaensen, L Denaux, J. Comb. Theory Ser. A. 18034Paper No. 105395Adriaensen S., Denaux L.: Small weight codewords of projective geometric codes. J. Comb. Theory Ser. A, 180:Paper No. 105395, 34 (2021). E F AssmusJr, J D Key, Designs and Their Codes. CambridgeCambridge University Press103Assmus E.F. Jr., Key J.D.: Designs and Their Codes, vol. 103. Cambridge Tracts in Mathematics. Cam- bridge University Press, Cambridge (1992). Projective geometric codes. B Bagchi, S P Inamdar, J. Comb. Theory Ser. A. 991Bagchi B., Inamdar S.P.: Projective geometric codes. J. Comb. Theory Ser. A 99(1), 128-142 (2002). Projective bundles. R D Baker, J M N Brown, G L Ebert, J C Fisher, Bull. Belg. Math. Soc.-Simon Stevin. 13Baker R.D., Brown J.M.N., Ebert G.L., Fisher J.C.: Projective bundles. Bull. Belg. Math. Soc.-Simon Stevin 1(3), 329-336 (1994). A Finite Geometry Construction for MDPC-Codes. J Bariffi, University of ZurichMaster's thesisBariffi J.: A Finite Geometry Construction for MDPC-Codes. Master's thesis, University of Zurich. https:// www.math.uzh.ch/index.php?id=pmastertheses&key1=604 (2020). A geometric approach to a class of cyclic codes. P Delsarte, J. Comb. Theory. 64Delsarte P.: A geometric approach to a class of cyclic codes. J. Comb. Theory 6(4), 340-358 (1969). Low-density parity-check codes. R Gallager, IRE Trans. Inf. Theory IT. 8Gallager R.: Low-density parity-check codes. IRE Trans. Inf. Theory IT 8, 21-28 (1962). Finite projective planes and related combinatorial systems. D G Glynn, University of Adelaide AdelaidePhD thesisGlynn D.G.: Finite projective planes and related combinatorial systems. PhD thesis, University of Ade- laide Adelaide (1978). On the number of information symbols in difference-set cyclic codes. R L Graham, J Macwilliams, Bell Syst. Tech. J. 45Graham R.L., MacWilliams J.: On the number of information symbols in difference-set cyclic codes. Bell Syst. Tech. J. 45, 1057-1070 (1966). J Hirschfeld, Projective Geometries Over Finite Fields. OxfordOxford University PressHirschfeld J.: Projective Geometries Over Finite Fields. Oxford University Press, Oxford (1998). A Janoska, MDPC decoding algorithms and their impact on the McEliece cryptosystem. In: 2018 Federated Conference on Computer Science and Information Systems (FedCSIS). IEEEJanoska A.: MDPC decoding algorithms and their impact on the McEliece cryptosystem. In: 2018 Fed- erated Conference on Computer Science and Information Systems (FedCSIS), pp. 1085-1089. IEEE (2018). Low-density parity-check codes from combinatorial designs. S J Johnson, AustraliaUniversity of NewcastlePhD thesisJohnson S.J.: Low-density parity-check codes from combinatorial designs. PhD thesis, University of Newcastle, Australia (2004). Construction of low-density parity-check codes from Kirkman triple systems. S J Johnson, S R Weller, IEEE Global Telecommunications Conference. IEEE2Johnson S.J., Weller S.R.: Construction of low-density parity-check codes from Kirkman triple systems. In: IEEE Global Telecommunications Conference, vol. 2, pp. 970-974. IEEE (2001). Regular low-density parity-check codes from combinatorial designs. S J Johnson, S R Weller, Proceedings 2001 IEEE Information Theory Workshop. 2001 IEEE Information Theory WorkshopIEEEJohnson S.J., Weller S.R.: Regular low-density parity-check codes from combinatorial designs. In: Pro- ceedings 2001 IEEE Information Theory Workshop, pp. 90-92. IEEE (2001). Regular low-density parity-check codes from oval designs. S J Johnson, S R Weller, Eur. Trans. Telecommun. 145Johnson S.J., Weller S.R.: Regular low-density parity-check codes from oval designs. Eur. Trans. Telecom- mun. 14(5), 399-409 (2003). Codes for iterative decoding from partial geometries. S J Johnson, S R Weller, IEEE Trans. Commun. 522Johnson S.J., Weller S.R.: Codes for iterative decoding from partial geometries. IEEE Trans. Commun. 52(2), 236-243 (2004). Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles. J.-L Kim, K E Mellinger, L Storme, Des. Codes Cryptogr. 421Kim J.-L., Mellinger K.E., Storme L.: Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles. Des. Codes Cryptogr. 42(1), 73-92 (2007). Explicit construction of families of LDPC codes with no 4-cycles. J.-L Kim, U N Peled, I Perepelitsa, V Pless, S Friedland, IEEE Trans. Inf. Theory. 5010Kim J.-L., Peled U.N., Perepelitsa I., Pless V., Friedland S.: Explicit construction of families of LDPC codes with no 4-cycles. IEEE Trans. Inf. Theory 50(10), 2378-2388 (2004). Low-density parity-check codes based on finite geometries: a rediscovery and new results. Y Kou, S Lin, M P C Fossorier, IEEE Trans. Inf. Theory. 477Kou Y., Lin S., Fossorier M.P.C.: Low-density parity-check codes based on finite geometries: a rediscovery and new results. IEEE Trans. Inf. Theory 47(7), 2711-2736 (2001). The non-existence of finite projective planes of order 10. C W Lam, L Thiel, S Swiercz, Can. J. Math. 416Lam C.W., Thiel L., Swiercz S.: The non-existence of finite projective planes of order 10. Can. J. Math. 41(6), 1117-1123 (1989). On linear sets on a projective line. M Lavrauw, G Van De Voorde, Des. Codes Cryptogr. 562Lavrauw M., Van de Voorde G.: On linear sets on a projective line. Des. Codes Cryptogr. 56(2), 89-104 (2010). LDPC codes from generalized polygons. Z Liu, D A Pados, IEEE Trans. Inf. Theory. 5111Liu Z., Pados D.A.: LDPC codes from generalized polygons. IEEE Trans. Inf. Theory 51(11), 3890-3898 (2005). Explicit constructions of graphs without short cycles and low density codes. G A Margulis, Combinatorica. 21Margulis G.A.: Explicit constructions of graphs without short cycles and low density codes. Combinatorica 2(1), 71-78 (1982). MDPC-McEliece: New McEliece variants from moderate density parity-check codes. R Misoczki, J.-P Tillich, N Sendrier, P S Barreto, 2013 IEEE International Symposium on Information Theory. IEEEMisoczki R., Tillich J.-P., Sendrier N., Barreto P.S.: MDPC-McEliece: New McEliece variants from moderate density parity-check codes. In: 2013 IEEE International Symposium on Information Theory, pp. 2069-2073. IEEE (2013). S Ouzan, &apos; Be, Y , arXiv:0911.3262Moderate-density parity-check codes. arXiv preprintOuzan S., Be'ery Y.: Moderate-density parity-check codes. arXiv preprint arXiv:0911.3262 (2009). Small weight codewords in the LDPC codes arising from linear representations of geometries. V Pepe, L Storme, G Van De Voorde, J. Comb. Des. 171Pepe V., Storme L., Van de Voorde G.: Small weight codewords in the LDPC codes arising from linear representations of geometries. J. Comb. Des. 17(1), 1-24 (2009). The use of coset equivalence in the analysis and decoding of group codes. E Prange, Air Force Cambridge Research Labs Hanscom AFB MATechnical reportPrange E.: The use of coset equivalence in the analysis and decoding of group codes. Technical report, Air Force Cambridge Research Labs Hanscom AFB MA (1959). Some remarks concerning curves of the second degree in a finite plane. B Qvist, Ann. Acad. Sci. Fennicae Ser. A. I. Math.-Phys. 195213427Qvist B.: Some remarks concerning curves of the second degree in a finite plane. Ann. Acad. Sci. Fennicae Ser. A. I. Math.-Phys. 1952(134), 27 (1952). Design of capacity-approaching irregular low-density paritycheck codes. T Richardson, A Shokrollahi, R Urbanke, IEEE Trans. Inf. Theory. 472Richardson T., Shokrollahi A., Urbanke R.: Design of capacity-approaching irregular low-density parity- check codes. IEEE Trans. Inf. Theory 47(2), 619-637 (2001). Constructions of LDPC codes using Ramanujan graphs and ideas from Margulis. J Rosenthal, P O Vontobel, Proceedings of the 38-th Allerton Conference on Communication, Control, and Computing. the 38-th Allerton Conference on Communication, Control, and ComputingRosenthal J., Vontobel P.O.: Constructions of LDPC codes using Ramanujan graphs and ideas from Margulis. In: Proceedings of the 38-th Allerton Conference on Communication, Control, and Computing, pp. 248-257 (2000). A class of majority logic decodable codes (corresp). L Rudolph, IEEE Trans. Inf. Theory. 132Rudolph L.: A class of majority logic decodable codes (corresp). IEEE Trans. Inf. Theory 13(2), 305-307 (1967). Ovals in a finite projective plane. Can. B Segre, J. Math. 7Segre B.: Ovals in a finite projective plane. Can. J. Math. 7, 414-416 (1955). Codes on finite geometries. H Tang, J Xu, S Lin, K A Abdel-Ghaffar, IEEE Trans. Inf. Theory. 512Tang H., Xu J., Lin S., Abdel-Ghaffar K.A.: Codes on finite geometries. IEEE Trans. Inf. Theory 51(2), 572-596 (2005). The decoding failure probability of MDPC codes. J.-P Tillich, 2018 IEEE International Symposium on Information Theory (ISIT). IEEETillich J.-P.: The decoding failure probability of MDPC codes. In: 2018 IEEE International Symposium on Information Theory (ISIT), pp. 941-945. IEEE (2018). LDPC codes associated with linear representations of geometries. P Vandendriessche, Adv. Math. Commun. 43Vandendriessche P.: LDPC codes associated with linear representations of geometries. Adv. Math. Com- mun. 4(3), 405-417 (2010). Construction of codes based on finite generalized quadrangles for iterative decoding. P Vontobel, R Tanner, Proceedings. 2001 IEEE International Symposium on Information Theory. 2001 IEEE International Symposium on Information TheoryIEEE223Vontobel P., Tanner R.: Construction of codes based on finite generalized quadrangles for iterative decoding. In: Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No. 01CH37252), p. 223. IEEE (2001). Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
[]
[ "SDA-SNE: Spatial Discontinuity-Aware Surface Normal Estimation via Multi-Directional Dynamic Programming", "SDA-SNE: Spatial Discontinuity-Aware Surface Normal Estimation via Multi-Directional Dynamic Programming" ]
[ "Nan Ming [email protected] \nDepartment of Control Science & Engineering\nMIAS Research Group, Robotics & Artificial Intelligence Laboratory\nTongji University\n201804ShanghaiChina\n", "Yi Feng [email protected] \nDepartment of Control Science & Engineering\nMIAS Research Group, Robotics & Artificial Intelligence Laboratory\nTongji University\n201804ShanghaiChina\n", "Rui Fan [email protected] \nDepartment of Control Science & Engineering\nMIAS Research Group, Robotics & Artificial Intelligence Laboratory\nTongji University\n201804ShanghaiChina\n" ]
[ "Department of Control Science & Engineering\nMIAS Research Group, Robotics & Artificial Intelligence Laboratory\nTongji University\n201804ShanghaiChina", "Department of Control Science & Engineering\nMIAS Research Group, Robotics & Artificial Intelligence Laboratory\nTongji University\n201804ShanghaiChina", "Department of Control Science & Engineering\nMIAS Research Group, Robotics & Artificial Intelligence Laboratory\nTongji University\n201804ShanghaiChina" ]
[]
The state-of-the-art (SoTA) surface normal estimators (SNEs) generally translate depth images into surface normal maps in an end-to-end fashion. Although such SNEs have greatly minimized the trade-off between efficiency and accuracy, their performance on spatial discontinuities, e.g., edges and ridges, is still unsatisfactory. To address this issue, this paper first introduces a novel multi-directional dynamic programming strategy to adaptively determine inliers (co-planar 3D points) by minimizing a (path) smoothness energy. The depth gradients can then be refined iteratively using a novel recursive polynomial interpolation algorithm, which helps yield more reasonable surface normals. Our introduced spatial discontinuity-aware (SDA) depth gradient refinement strategy is compatible with any depth-to-normal SNEs. Our proposed SDA-SNE achieves much greater performance than all other SoTA approaches, especially near/on spatial discontinuities. We further evaluate the performance of SDA-SNE with respect to different iterations, and the results suggest that it converges fast after only a few iterations. This ensures its high efficiency in various robotics and computer vision applications requiring realtime performance. Additional experiments on the datasets with different extents of random noise further validate our SDA-SNE's robustness and environmental adaptability. Our source code, demo video, and supplementary material are publicly available at mias.group/SDA-SNE.
10.1109/3dv57658.2022.00060
[ "https://export.arxiv.org/pdf/2208.08667v1.pdf" ]
251,643,601
2208.08667
17b39fb191ce49b4d83441b2ca03e4d53827e0d3
SDA-SNE: Spatial Discontinuity-Aware Surface Normal Estimation via Multi-Directional Dynamic Programming Nan Ming [email protected] Department of Control Science & Engineering MIAS Research Group, Robotics & Artificial Intelligence Laboratory Tongji University 201804ShanghaiChina Yi Feng [email protected] Department of Control Science & Engineering MIAS Research Group, Robotics & Artificial Intelligence Laboratory Tongji University 201804ShanghaiChina Rui Fan [email protected] Department of Control Science & Engineering MIAS Research Group, Robotics & Artificial Intelligence Laboratory Tongji University 201804ShanghaiChina SDA-SNE: Spatial Discontinuity-Aware Surface Normal Estimation via Multi-Directional Dynamic Programming The state-of-the-art (SoTA) surface normal estimators (SNEs) generally translate depth images into surface normal maps in an end-to-end fashion. Although such SNEs have greatly minimized the trade-off between efficiency and accuracy, their performance on spatial discontinuities, e.g., edges and ridges, is still unsatisfactory. To address this issue, this paper first introduces a novel multi-directional dynamic programming strategy to adaptively determine inliers (co-planar 3D points) by minimizing a (path) smoothness energy. The depth gradients can then be refined iteratively using a novel recursive polynomial interpolation algorithm, which helps yield more reasonable surface normals. Our introduced spatial discontinuity-aware (SDA) depth gradient refinement strategy is compatible with any depth-to-normal SNEs. Our proposed SDA-SNE achieves much greater performance than all other SoTA approaches, especially near/on spatial discontinuities. We further evaluate the performance of SDA-SNE with respect to different iterations, and the results suggest that it converges fast after only a few iterations. This ensures its high efficiency in various robotics and computer vision applications requiring realtime performance. Additional experiments on the datasets with different extents of random noise further validate our SDA-SNE's robustness and environmental adaptability. Our source code, demo video, and supplementary material are publicly available at mias.group/SDA-SNE. Introduction Surface normal is an informative 3D visual feature used in various computer vision and robotics applications, such as object recognition and scene understanding [5,9,13]. To date, there has not been extensive research on surface normal estimation, as it is merely considered to be an auxiliary * Corresponding author functionality for other computer vision and robotics applications [7]. As such applications are typically required to perform robustly and in real time, surface normal estimators (SNEs) must be sufficiently accurate and computationally efficient [7]. The state-of-the-art (SoTA) SNEs [2, 4, 7, 12-14, 16, 19] generally select a set of 3D points and compute surface normals via planar fitting, geometric transformation, or statistical analysis. However, such approaches are infeasible to estimate surface normals near/on spatial discontinuities, e.g., edges and ridges (see Fig. 1), as they generally introduce adjacent 3D points on different surfaces unintentionally [10]. Bormann et al. [3] proposed the first edge-aware SNE, capable of adaptively selecting reasonable adjacent 3D points on the same surface. It performs significantly better than prior arts near edges. Nonetheless, their method focuses only on edges and requires a manually-set discontinuity awareness threshold. This results in low adaptability to different scenarios and datasets. Hence, there is a strong motivation to develop an SNE capable of tackling all types of spatial discontinuities to achieve greater robustness and environmental adaptability. The major contributions of this work are summarized as follows: (1) Spatial discontinuity-aware surface normal estimator (SDA-SNE), a highly accurate SNE, significantly outperforming all other SoTA SNEs, especially near/on spatial discontinuities. (2) Multi-directional dynamic programming (DP) for depth gradient refinement. It iteratively introduces inliers (co-planar pixels) by minimizing a smoothness energy. (3) Path discontinuity (PD) norm, a semi-norm evaluating the discontinuity of a given path. It reflects the depth gradient estimation error generated by the Taylor expansion. (4) Recursive polynomial interpolation (RPI), an ultrafast polynomial interpolation algorithm. Compared to Lagrange and Newton polynomial interpolation, it iteratively refines the first-order derivative of depth with a more efficient polynomial interpolation strategy. Related Work The existing SoTA SNEs can be categorized into four classes: (1) optimization-based [13], (2) averaging-based [14], (3) affine-correspondence-based [2], and (4) depth-tonormal [23]. The optimization-based SNEs, e.g., PlaneSVD [14], PlanePCA [12], VectorSVD [14], and QuadSVD [17,22], compute surface normals by fitting local planar or curved surfaces to an observed 3D point cloud, using either singular value decomposition (SVD) or principal component analysis (PCA). The averaging-based SNEs, e.g., AreaWeighted [13] and AngleWeighted [14], estimate surface normals by computing the weighted average of the normal vectors of the triangles formed by each given 3D point and its neighbors. However, these two categories of SNEs are highly computationally intensive and unsuitable for online robotics and computer vision applications [7]. The affine-correspondence-based SNEs exploit the relationship between affinities and surface normals [2,4,19]. Nevertheless, such SNEs are developed specifically for stereo or multi-view cases and cannot generalize to other cases where monocular depth images are used. Recently, enormous progress has been made in end-toend depth-to-normal translation [6,7,16]. Such SNEs have demonstrated superior performance in terms of both speed and accuracy. Fan et al. [7] proposed a fast and accurate SNE referred to as 3F2N, which can infer surface normal information directly from depth or disparity images, with two gradient filters and one central tendency measurement filter (a mean or median filtering operation), as follows: n x = ∂d/∂u, n y = ∂d/∂v, n z = −Φ (x i − x)n x + (y i − y)n y z i − z , i = 1, . . . , m,(1) where n = [n x , n y , n z ] is the surface normal of a given 3D point p C = [x, y, z] in the camera coordinate system, p C is projected to a 2D pixel p = [u, v] via z[p , 1] = Kp C (K is the camera intrinsic matrix), p C i = [x i , y i , z i ] is one of the m adjacent pixels of p C , d is disparity (or inverse depth), and Φ{·} represents the central tendency measurement filtering operation for n z estimation. Nakagawa et al. [16] presented an SNE based on cross-product of two tangent vectors (CP2TV): n(u, v) = t u × t v , where t u = ∂x ∂u , ∂y ∂u , ∂z ∂u t v = ∂x ∂v , ∂y ∂v , ∂z ∂v ,(2)∂x ∂u = z f u + u − c u f u ∂z ∂u ∂y ∂u = v − c v f v ∂z ∂u ∂x ∂v = u − c u f u ∂z ∂v ∂y ∂v = z f v + v − c v f v ∂z ∂v ,(3)c = [c u , c v ] is the principal point (in pixels), and f u and f v are the camera focal lengths (in pixels) in the u and v directions, respectively. Since the depth or disparity gradient estimation has a direct impact on the surface normal quality, this paper focuses thoroughly on the strategy to improve the accuracy of depth gradient ∇ẑ = [ẑ u ,ẑ v ] 1 . The proposed strategy can also be applied to improve disparity gradient estimation, as it is inversely proportional to depth. Methodology As discussed in Sec. 2, surface normal estimation was formulated as a depth-to-normal translation problem in SoTA approaches, which have demonstrated superior performances over other methods in terms of both speed and accuracy [7]. The accuracy of such approaches is subject to the depth gradient quality. Since depth gradients typically jump near/on discontinuities, e.g., edges and ridges, the estimated surface normals near/on such discontinuities are always significantly different from their ground truth. Therefore, we propose a multi-directional DP strategy to translate ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ∇̂( 0) ∇̂( 1) (0) (1) (0)(1)� (0) � (1)(2) (2) ∇̂( 2) � (2) ⋯ ⋯ ⋯ ⋯ * ∇̂ * � * Updating the Depth Gradient: ∇̂( ) = ∇̂− 1 , ( ) (21) Minimizing the Smoothness Energy: The relationship among smoothness energy, state variable, depth gradient, and surface normal. a depth or disparity image into a surface normal map in a coarse-to-fine manner. It first initializes ∇ẑ using finite difference interpolation and then optimizes ∇ẑ using our proposed multi-directional DP. Accurate surface normals can then be obtained by plugging the optimum depth gradients into either the central tendency measurement filtering operation used in 3F2N (see (1)) or the cross-product operation used in CP2TV (see (2)-(3)). ( ) = min −1 ,(15) Multi-Directional Dynamic Programming Smoothness Energy and State Variable As a general rule, only inliers (co-planar pixels) should be introduced when estimating the depth gradient of an observed pixel p. Such pixels are determined by minimizing the (path) smoothness energy E = [E u , E v ] of p using DP, where E u and E v represent the horizontal and vertical components of E, respectively. To minimize the computational complexity, we recursively introduce adjacent coplanar pixels. In each iteration, two desired pixels p u and p v (corresponding to the minimum E u and E v , respectively) are used to extend the DP path and update the depth gradient of p. We define a state variable S = [s u , s v ], where s u = p u − p and s v = p v − p denote the horizontal and vertical components of S, respectively 2 . In the k-th iteration, the desired pixels are determined by minimizing E (k) as follows: E (k) = min S T E (k−1) , S .(4) The state variable S (k) is transferred to S (k) = arg min S T E (k−1) , S ,(5) where T denotes the smoothness energy transfer function. Therefore, we can update the depth gradient with the desired pixels using: ∇ẑ (k) = R ∇ẑ (k−1) , S (k) ,(6) where R denotes the depth gradient update function. As shown in Fig. 2, after the initialization of depth gradient ∇ẑ and DP variables E and S (detailed in Sec. 3.1.2), inliers are iteratively introduced using the smoothness energy transfer function T (detailed in Sec. 3.2) and the depth gradients are iteratively updated using the depth gradient update function R (detailed in Sec. 3.3). The optimum depth gradients can be obtained when the smoothness energy converges to the minimum. The pseudo-code of multidirectional DP is given in Algorithm 1. Initialization of Depth Gradient, Smoothness Energy, and State Variable In the initialization stage, the coarse depth gradient ∇ẑ (0) of a given pixel is obtained by computing the finite difference (FD) between adjacent pixels. The FDs of depth are divided into ∆ f z = [∆ f z u , ∆ f z v ] and ∆ b z = [∆ b z u , ∆ b z v ] , where the subscripts u and v represent the horizontal and vertical directions, respectively, and ∆ f and ∆ b represent forward and backward FDs, respectively. In order to Algorithm 1 Multi-Directional DP Input: Depth map z, Initial depth gradient [ẑ (0) u ,ẑ (0) v ], Ini- tial smoothness energy components E (0) u and E (0) v , Ini- tial state variable components s (0) u and s (0) v . Output: Optimum depth gradient [ẑ u ,ẑ v ] 1: [ẑ u ,ẑ v ] ← [ẑ (0) u ,ẑ (0) v ], [E u , E v ] ← [E (0) u , E (0) v ], [s u , s v ] ← [s (0) u , s (0) v ] 2: repeat 3: Initialize energy set Ω u ← ∅, Ω v ← ∅ 4: for each p do 5: 6: Ω u [p] ← Ω u [p] ∪ T (E u , E v , p, p , s u )Ω v [p] ← Ω v [p] ∪ T (E u , E v , p, p , s v ) 7: end for 8: s u [p] ← arg min p ∈N1(p) {Ω u [p][p ]} − p 9: s v [p] ← arg min p ∈N1(p) {Ω v [p][p ]} − p 10: E u [p] ← Ω u [p][p + s u ] 11: E v [p] ← Ω v [p][p + s v ] 12: [ẑ u ,ẑ v ] ← [R(ẑ u ,ẑ v , s u ), R(ẑ u ,ẑ v , s v )] 13: until all (s u [:] = 0) and all (s v [:] = 0) 14: return [ẑ u ,ẑ v ] achieve the discontinuity awareness ability, we set different weights to these FD operators by comparing the local smoothness measured byẑ uu andẑ vv , the second-order partial derivatives of adjacent pixels, as follows: η = η u , η v = arg min [i,j] {|ẑ uu (u + i, v)| + |ẑ vv (u, v + j)|} ,(7) where i, j ∈ {−1, 0, 1}. We can then use the smoothest pixels to linearly interpolate ∆ f z and ∆ b z: ∇ẑ (0) = 1 2 (1 + η) • ∆ f z + 1 2 (1 − η) • ∆ b z,(8) where • denotes the Hadamard product and 1 = [1, 1] . Furthermore, we initialize the smoothness energy E (0) as min{|ẑ uu (u + i, v)|}, min{|ẑ vv (u, v + j)|} and the state variable S (0) as diag(η u , η v ). Smoothness Energy Transfer Function This section discusses the formulation of the smoothness energy transfer function T . Energy transfer can be divided into two categories: 1) collinear transfer (s u × e 1 = 0 or s v × e 2 = 0, where e 1 = [1, 0] and e 2 = [0, 1] are the unit orthogonal base of 2D Euclidean space); 2) noncollinear transfer (s u × e 1 = 0 or s v × e 2 = 0). As a general rule, the collinear pixels should be introduced to adapt to the convex surface if they are smooth enough; otherwise, non-collinear pixels should be introduced to deal with discontinuities. Depth Gradient Calculation Based on the gradient theorem [25], the relationship among z, z u , and z v can be written as follows: z(p ) − z(p) = L z u du + z v dv,(9)where p = [u 0 , v 0 ] is the given pixel, p = [u 1 , v 1 ] is its adjacent pixel to be introduced (satisfying |u 0 − u 1 | ≤ 1, |v 0 − v 1 | ≤ 1) , and L is the path from p to p . We can therefore obtain (i) the collinear transfer as follows: z(p ) − z(p) = u1 u0 z u du, or z(p ) − z(p) = v1 v0 z v dv,(10) and (ii) the non-collinear transfer as follows: z(p ) − z(p) = L1 z u du + z v dv = L2 z u du + z v dv,(11) where L 1 and L 2 are two different paths from p to p . In practice, however, the discreteẑ u andẑ v can result in errors in (10) and (11). The errors ofẑ u andẑ v can be estimated using the path integral of z uu and z vv based on the Taylor expansion (more details are provided in the supplement). Hence, in order to measure these errors in a more convenient way, we propose the PD norm, which denotes the path integral (sum) of a series of |z uu | and |z vv |. Path Discontinuity Norm Let z(u, v) be a function in R 2 defined on an open set Ω and L be a specific path contained in Ω. The discontinuity extent of the path can be measured using our proposed PD norm as follows: ||z|| PD = L |z uu du| + |z vv dv|.(12) After computing the PD norm, we can judge whether an adjacent pixel p should be introduced. Therefore, we define the smoothness energy component as the PD norm with respect to a given path. In each iteration, we introduce two adjacent pixels which respectively minimize the smoothness energy components as follows: s u , s v = arg min p ∈N1(p) {||z|| PD } − p, L : p → p , if s u × e 1 = 0 (or s v × e 2 = 0) p → p → p , if s u × e 1 = 0 (or s v × e 2 = 0),(13) where N 1 (p) = {p : ||p −p|| ∞ ≤ 1}, and the smoothness energy components are minimized accordingly: E u (p), E v (p) = min p ∈N1(p) {||z|| PD } L : p → p , if s u × e 1 = 0 (or s v × e 2 = 0) p → p → p , if s u × e 1 = 0 (or s v × e 2 = 0).(14) The determination of co-planar pixels is, therefore, converted into an energy minimization problem. Energy Minimization Strategy Since depth is discrete, we formulate the energy transfer function T in (4) as follows: E (k) u (p), E (k) v (p) = min p ∈N1(p) E (k−1) p (p), |ẑ pp (p p )| + I(p, p p ), 2 · |ẑ oo (p o )| + E (k−1) p (p o ) , |ẑ oo (p o )| + E (k−1) p (p o ) + |ẑ pp (p d )| + E (k−1) o (p d ) ,(15) where the subscripts p, o, and d respectively denote the variable which is parallel, orthogonal, and diagonal to the given axis (either u-axis or v-axis); the representations of E p , E o , z pp ,ẑ oo , p p , p o , and p d are given in the supplement; and the indicator function I(p, p p ) =      0, if (ẑ pp (p + s p ) ·ẑ pp (p p + s p ) > 0) ∧(s p (p) = s (k−1) p (p p )) ∞, otherwise(16) add constraints to the monotonicity and convexity of the smoothness energy transfer function, improving the DP adaptivity to convex surfaces. The four terms in (15), in turn, denote the smoothness energy components when we introduce 1) no other pixels, 2) a parallel pixel p p , 3) an orthogonal pixel p o , and 4) a diagonal pixel p d . Depth Gradient Update Function This subsection discusses the formulation of the depth gradient update function R. Similar to T , depth gradient update can be divided into two categories: 1) collinear update (s u × e 1 = 0 or s v × e 2 = 0) and 2) non-collinear update (s u ×e 1 = 0 or s v ×e 2 = 0). For the collinear update, we use all the collinear pixels which satisfy the constraint in (16) to estimate depth gradients, while on the other hand, for the non-collinear update, we replace the depth gradient of each given pixel with the ones on other smoother paths using (11). Depth Gradient Collinear Update with Recursive Polynomial Interpolation The optimum ∇ẑ can be obtained by interpolating the depth of n collinear pixels into an (n−1)-th order polynomial and subsequently computing its derivative. Lagrange or Newton polynomial interpolation has redundant computations, as each pixel is repeatedly used for polynomial interpolation. Furthermore, it is incredibly complex to yield the closedform solution of the polynomial's derivative. To simplify the depth gradient collinear update process, we introduce a novel RPI algorithm, which can update ∇ẑ recursively with only adjacent pixels until the farthest pixel is accessed. Asẑ u andẑ v can be computed in the same way, we only provide the details onẑ u computation in this paper. To computeẑ u of a given pixel p = [u 0 , v 0 ] in the k-th iteration, we use a (k+1)-th order polynomial to interpolate the depth of (k + 2) pixels, which have the same vertical coordinates v 0 . The (k + 1)-th order polynomial f (k+1) 0 (u) interpolated by {(u 0 , z(u 0 )), (u 0 +1, z(u 0 +1)), . . . , (u 0 +k+1, z(u 0 + k + 1))} can be expanded into a recursive form as follows: f (k+1) 0 (u) = u 0 + k + 1 − u k + 1 f (k) 0 + u − u 0 k + 1 f (k) 1 ,(17) where the k-th order polynomials f are respectively interpolated by the first and the last k + 1 pixels (the proof of the theorem is provided in the supplement). Sinceẑ (k−1) u (u 0 ) = df (k) 0 du (u 0 ) andẑ (k−1) u (u 0 + 1) = df (k) 1 du (u 0 + 1) have been computed in the last iteration, we can updateẑ u (u 0 ) using: z (k) u (u 0 ) = df (k+1) 0 du (u 0 ) =ẑ (k−1) u (u 0 ) − 1 k + 1 · z(u 0 ) − f (k) 1 (u 0 ) .(18) Substituting f (k) 1 (u 0 ) = z(u 0 +1)− df (k) 1 du (u 0 +1) = z(u 0 + 1) −ẑ (k−1) u (u 0 + 1) into (18) yieldŝ z (k) u (u 0 ) =ẑ (k−1) u (u 0 )− 1 k + 1 · z(u 0 ) − z(u 0 + 1) +ẑ (k−1) u (u 0 + 1) .(19) Depth Gradient Non-Collinear Update Replacing the integral in (11) with summation yields: z p (p) ±ẑ o (p p ) =ẑ p (p o ) ±ẑ o (p).(20) As the pixels p p are regarded as outliers in non-colinear update, we replaceẑ o (p p ) withẑ o (p d ) orẑ o (p) based on the state variable. The depth gradient update function R in (6) can be rewritten as follows: z (k) u (p),ẑ (k) v (p) =                     ẑ (k−1) p (p), if s (k) u , s (k) v = 0 z (k−1) p (p) ± 1 k+1 z(p p ) − z(p) − 1 k+1ẑ (k−1) p (p p ), if s (k) u , s (k) v = p p − p z (k−1) p (p o ), if s (k) u , s (k) v = p o − p z (k−1) p (p o )± ẑ (k−1) o (p) −ẑ (k−1) o (p d ) , if s (k) u , s (k) v = p d − p.(21) The representations ofẑ p andẑ o are given in the supplement. The four terms in (21), in turn, represent 1) the unchanged depth gradient, 2) parallel update via RPI algorithm, 3) orthogonal update, and 4) diagonal update. The orthogonal and diagonal updates are based on the gradient theorem [25]. Experiments Datasets and Evaluation Metrics In this paper, we followed [7] and conducted comprehensive experiments to qualitatively and quantitatively evaluate the performance of our proposed SDA-SNE. More details on the 3F2N datasets 3 are available in [7]. Moreover, since range sensor data are typically noisy, we add random Gaussian noise of different variances σ to the original 3F2N datasets to compare the robustness between our proposed SDA-SNE and other SoTA SNEs. Two evaluation metrics are used to quantify the accuracy of SNEs: the average angular error (AAE) [15]: e A (M) = 1 P P k=1 φ k , φ k = cos −1 n k ,n k ||n k || 2 ||n k || 2 ,(22) and the proportion of good pixels (PGP) [15]: e P (M) = 1 P P k=1 δ (φ k , ϕ k ) , δ = 0 (φ k > ϕ) 1 (φ k ≤ ϕ) ,(23) where P denotes the total number of pixels used for evaluation, ϕ denotes the angular error tolerance, M represents the given SNE, and n k andn k represent the ground-truth and estimated surface normals, respectively. In addition to the above-mentioned evaluation metrics, we also introduce a novel evaluation metric, referred to as cross accuracy ratio (CAR), to depict the performance comparison between two SNEs as follows: 3 sites.google.com/view/3f2n/datasets r A (M 1 , M 2 ) = e A (M 1 ) e A (M 2 ) .(24) M 2 outperforms M 1 in terms of e A if r A > 1, and vice versa if 0 < r A < 1. Implementation Details As discussed in Sec. 3, initial depth gradients can be estimated by convolving a depth image with image gradient filters, e.g., Sobel [21], Scharr [11], Prewitt [18], etc. The second-order filters, e.g., Laplace [20], can be used to estimate the local depth gradient smoothnessẑ uu and z vv . We utilize a finite forward difference (FFD) kernel, i.e., [0, −1, 1], as well as a finite backward difference (FBD) kernel, i.e., [−1, 1, 0] to initialize (coarse)ẑ Algorithm Convergence and Computational Complexity Our proposed SDA-SNE converges when the state variable and surface normal estimation remains stable. We can obtain the lower bound of the smoothness energy components as follows: E u (p), E v (p) ≥ min p ∈N1(p) |ẑ uu (p )|, |ẑ vv (p )| ≥ 0. (25) The smoothness energy decreases monotonously after each iteration. Therefore, all state variables stop transferring after finite iterations (the smoothness energy converges to a global minimum). The computational complexity of our proposed PRI algorithm is O(n) (n denotes the number of interpolated pixels), when computing the depth gradient using (21). Therefore, RPI is much more efficient than Lagrange or Newton polynomial interpolation whose computational complexity is O(n 2 ). Furthermore, when the image resolution is M ×N pixels, the computational complexity of our proposed SDA-SNE is O(lM N ), where l denotes the number of iterations in DP. In most cases, l M, N because DP automatically stops when |ẑ uu | and |ẑ vv | no longer decrease. If we set a limit on l, SDA-SNE's computational complexity can be reduced to O(M N ), which is identical to the computational complexities of 3F2N [7] and CP2TV [16]. Performance Evaluation As our proposed multi-directional DP algorithm mainly aims at improving the performance of depth gradient estimation, it is compatible with any depth-to-normal SNEs. The AAE scores of SoTA depth-to-normal SNEs and such SNEs using our depth gradient estimation strategy are given in Table 1. It can be observed that by using our proposed depth gradient estimation strategy, the AAE scores of such depth-to-normal SNEs decrease by about 20-60%. Since SDA-SNE based on CP2TV performs better than SDA-SNE based on 3F2N, we use CP2TV to estimate n z in the following experiments. The qualitative comparison among these SNEs is shown in Fig. 5. It can be observed that our proposed SDA-SNE significantly outperforms 3F2N and CP2TV near/on discontinuities. Additionally, we compare the performance of our proposed SDA-SNE w.r.t. a collection of maximum iterations. Fig. 3 shows the CAR scores achieved by CP2TV-based and 3F2N-based SDA-SNEs on the 3F2N easy, medium, and hard datasets. It can be observed that the performance of SDA-SNE saturates after only several iterations. The accuracy increases by less than 5% when the maximum iteration is set to infinity. Therefore, the maximum iteration of multi-directional DP can be set to 3 to minimize its tradeoff between speed and accuracy. Moreover, it can be observed that our proposed SDA-SNE converges (the CAR score reaches the maximum) with more iterations on the hard dataset than on the easy and medium datasets. This is probably due to the fact that the hard dataset possesses more discontinuities than the easy and medium datasets. As a result, more distant pixels are required to be introduced to yield better depth gradients. Moreover, we compare our proposed SDA-SNE with all other SoTA SNEs presented in Sec. 2. e A and e P of all SNEs on the 3F2N easy, medium, and hard datasets are given in Table 2. The e A scores achieved by SDA-SNE are less than 1 • (easy), 5 • (medium), and 9 • (hard), respectively. The e P scores (tolerance: 10 • ) achieved by SDA-SNE are about 100% (easy), 90% (medium), and 80% (hard), respectively. These results suggest that our proposed SDA-SNE performs significantly better than all other SoTA SNEs, no matter whether an iteration limit is added or not. Furthermore, we compare the robustness (to random Gaussian noise) of our proposed SDA-SNE with two SoTA depth-to-normal SNEs (3F2N [7] and CP2TV [16]), as shown in Fig. 4. The logarithms of AAE scores and Gaussian variances are used to include the results of different datasets into a single figure. It is evident that (1) the e A scores achieved by all SNEs increase monotonically with the increasing noise level, and (2) our proposed SDA-SNE outperforms 3F2N and CP2TV at different noise levels. In addition, the compared SNEs' performance on the easy dataset degrades more dramatically than the medium and hard datasets, as the original easy dataset contains much fewer discontinuities. Although the performance of these depth-to-normal SNEs becomes remarkably similar, with the increase in noise level, our proposed SDA-SNE consistently outperforms 3F2N and CP2TV. This further demonstrates the superior robustness of our algorithm over others. Conclusion This paper presented a highly accurate discontinuityaware surface normal estimator, referred to as SDA-SNE. Our approach computes surface normals from a depth image by iteratively introducing adjacent co-planar pixels using a novel multi-directional dynamic programming algorithm. To refine the depth gradient in each iteration, we introduced a novel recursive polynomial interpolation al- gorithm with high computational efficiency. Our proposed depth gradient estimation approach is compatible with any depth-to-normal surface normal estimator, such as 3F2N and CP2TV. To evaluate the accuracy of our proposed surface normal estimator, we conducted extensive experiments on both clean and noisy datasets. Our proposed SDA-SNE achieves the highest accuracy on clean 3F2N datasets (0.68 • , 4.38 • , 8.10 • on the easy, medium, and hard datasets, respectively), outperforming all other SoTA surface normal estimators. It also demonstrates high robustness to different levels of random Gaussian noise. Additional experimental results suggest that our proposed SDA-SNE can achieve a similar performance when reducing the maximum itera-tion of multi-directional dynamic programming to 3. This ensures the high efficiency of our proposed SDA-SNE in various computer vision and robotics applications requiring real-time performance. (10) and (11) results in the upper bound of the depth gradient estimation error w.r.t. collinear transfer and non-collinear transfer. The depth gradient estimation error can be associated with our introduced PD norm (detailed in Sec. 7.1 and 7.2). Furthermore, we compare PD norm with total variation (TV) norm [26], as detailed in Sec. 7.3. Acknowledgements Depth Gradient Error w.r.t. Collinear Transfer In collinear transfer, (10) can be rewritten as follows: z(p ) − z(p) = (u 1 − u 0 )z u (p) + u1 u0 (u − u 0 )z uu du, or z(p ) − z(p) = (v 1 − v 0 )z v (p) + v1 v0 (v − v 0 )z vv dv.(26) Omitting the second-order terms results in the following collinear transfer error expressions: ε C = u1 u0 (u − u 0 )z uu du , or ε C = v1 v0 (v − v 0 )z vv dv .(27) As |u 0 − u 1 | ≤ 1 and |v 0 − v 1 | ≤ 1, PD norm is the upper bound of the collinear transfer error ε C : ε C ≤ ||z|| PD , L : p → p .(28) Depth Gradient Error w.r.t. Non-Collinear Transfer In non-collinear transfer, (11) can be revised as follows: (u 1 − u 0 )z u (p) + (v 1 − v 0 )z v (p ) + L1 (u − u 0 )z uu du + (v − v 1 )z vv dv = (v 1 − v 0 )z v (p) + (u 1 − u 0 )z u (p ) + L2 (u − u 1 )z uu du + (v − v 0 )z vv dv.(29) PD norm is also the upper bound of the non-collinear transferring error ε N w.r.t. different DP paths: ε N = L1 (u − u 0 )z uu du + (v − v 1 )z vv dv− L2 (u − u 1 )z uu du + (v − v 0 )z vv dv ≤ L1 |z uu du| + |z vv dv| + L2 |z uu du| + |z vv dv| = ||z|| PD , L = L 1 + L 2 : p → p → p.(30) Therefore, the PD norm determines the upper bound of the depth gradient estimation error. Optimizing the PD norm in (14) is equivalent to minimizing the error of z u and z v in (27) and (30). Comparison between PD Norm and TV Norm TV norm [26] is widely used to measure depth discontinuity. Compared to TV norm, our proposed PD norm introduces the second-order partial derivative of depth, demonstrating better awareness of ridges. We conduct additional experiments on the 3F2N datasets to compare the accuracy of our method using PD and TV norms, respectively. As shown in Table 3, the accuracy achieved when using PD norm is higher than that achieved when using TV norm. RPI Theorem Theorem: If the n-th order polynomial f (u) (n) ∈ P n [u] is interpolated by {(u 0 , z 0 ), (u 1 , z 1 ), · · · , (u n , z n )}, it can be expanded into a recursive form as follows: f (n) = u − u n u 0 − u n f (n−1) 0 + u − u 0 u n − u 0 f (n−1) 1 ,(31) where f (b) Substituting u = u n into (31), we can obtain: u − u n u 0 − u n = 0, u − u 0 u n − u 0 = 1, f (n) (u n ) = f (n−1) 1 (u n ) = z n .(33) (c) Substituting u = u 1 , · · · , u n−1 into (31), we can obtain: u − u 0 u n − u 0 + u − u n u 0 − u n = 1, f (n) (u i ) = f (n−1) 0 (u i ) = f (n−1) 1 (u i ) = z i ,(34) where i is an integer between 1 and n − 1. Additional Experiments on Noisy Data We also conduct additional experiments on the IRS dataset 4 [24] to further validate our proposed SDA-SNE's robustness to noise. The IRS dataset contains four indoor scenarios (home, office, restaurant, and store) [24]. The comparison among 3F2N [7], CP2TV [16], and our proposed SDA-SNE in terms of e A is given in Table. 4. It can be observed that our proposed SDA-SNE also outperforms 3F2N and CP2TV on the IRS dataset (e A is lower by about 40% and 30%, respectively). These results suggest that SDA-SNE performs robustly on real noisy data under different illumination conditions. Qualitative comparison among 3F2N, CP2TV, and SDA-SNE is given in Fig. 6. It can be observed that compared to 3F2N and CP2TV, SDA-SNE performs much better near/on discontinuities. Table 4. Comparison of eA (degrees) among 3F2N, CP2TV, and our proposed SDA-SDA on the IRS dataset. Figure 1 . 1Comparison between 3F2N (recently published SoTA SNE)[7] and our proposed SDA-SNE. A significantly improved region is marked with a dashed blue circle. Figure 2 . 2(a) An illustration of the energy transfer process, where the inliers (desired pixels) are gradually introduced in each iteration; (b) finite Laplace (FL) kernel, i.e., [−1, 2, −1] to estimatê z uu andẑ vv . We also conduct experiments w.r.t. different number of iterations to quantify the performance of our proposed SDA-SNE. By implementing the optimum maximum iteration, the trade-off between the speed and accuracy of our algorithm can be significantly minimized. Figure 3 . 3CAR comparison between CP2TV-based and 3F2Nbased SDA-SNEs w.r.t. different maximum iterations. Figure 4 . 4AAE comparison among 3F2N, CP2TV, and SDA-SNE on the 3F2N datasets with different levels of Gaussian noise added. Figure 5 . 5Examples of experimental results: Columns (i)-(v) show the depth images, surface normal ground truth, and the experimental results obtained using 3F2N, CP2TV, and our proposed SDA-SNE, respectively. Rows (a)-(c) show the results of the easy, medium, and hard datasets, respectively. This work was supported by the National Key R&D Program of China, under grant No. 2020AAA0108100, awarded to Prof. Qijun Chen. This work was also supported by the Fundamental Research Funds for the Central Universities, under projects No. 22120220184, No. 22120220214, and No. 2022-5-YB-08, awarded to Prof. Rui Fan. Taylor expansions of z u and z v into interpolated by the first and the last n points.Proof (a) Substituting u = u 0 into (31), we can obtain: Figure 6 . 6Examples of experimental results: Columns (i)-(vi) show the RGB images, depth images, surface normal ground truth, and the experimental results obtained using 3F2N, CP2TV, and our proposed SDA-SNE, respectively. Rows (a)-(d) show the home, office, restaurant, and store scenarios, respectively. Table 1 . 1Comparison of eA (degrees) between 3F2N and CP2TV w/ and w/o our depth gradient estimation strategy leveraged.Datasets 3F2N CP2TV w/o SDA w/ SDA w/o SDA w/ SDA Easy 1.657 0.782 1.686 0.677 Medium 5.686 4.535 6.015 4.379 Hard 15.315 9.237 13.819 8.098 Table 2 . 2Comparison of eA and eP (with respect to different ϕ) among SoTA SNEs on the 3F2N datasets[7]. ϕ=20• ϕ=30 • ϕ=10 • ϕ=20 • ϕ=30 • ϕ=10 • ϕ=20 • ϕ=30 •Method e A (degrees) ↓ ep ↑ Easy Medium Hard Easy Medium Hard ϕ=10 • PlaneSVD [14] 2.07 6.07 17.59 0.9648 0.9792 0.9855 0.8621 0.9531 0.9718 0.6202 0.7394 0.7914 PlanePCA [12] 2.07 6.07 17.59 0.9648 0.9792 0.9855 0.8621 0.9531 0.9718 0.6202 0.7394 0.7914 VectorSVD [13] 2.13 6.27 18.01 0.9643 0.9777 0.9846 0.8601 0.9495 0.9683 0.6187 0.7346 0.7848 AreaWeighted [13] 2.20 6.27 17.03 0.9636 0.9753 0.9819 0.8634 0.9504 0.9665 0.6248 0.7448 0.7977 AngleWeighted [13] 1.79 5.67 13.26 0.9762 0.9862 0.9893 0.8814 0.9711 0.9809 0.6625 0.8075 0.8651 FALS [1] 2.26 6.14 17.34 0.9654 0.9794 0.9857 0.8621 0.9547 0.9731 0.6209 0.7433 0.7961 SRI [1] 2.64 6.71 19.61 0.9499 0.9713 0.9798 0.8431 0.9403 0.9633 0.5594 0.6932 0.7605 LINE-MOD [8] 2.64 6.71 19.61 0.8542 0.9085 0.9343 0.7277 0.8803 0.9282 0.3375 0.4757 0.5636 SNE-RoadSeg [6] 2.04 6.28 16.37 0.9693 0.9810 0.9871 0.8618 0.9512 0.9725 0.6226 0.7589 0.8113 3F2N [7] 1.66 5.69 15.31 0.9723 0.9829 0.9889 0.8722 0.9600 0.9766 0.6631 0.7821 0.8289 CP2TV [16] 1.69 6.02 13.82 0.9740 0.9843 0.9899 0.8512 0.9554 0.9755 0.6840 0.8099 0.8562 SDA-SNE (iteration = ∞) 0.68 4.38 8.10 0.9947 0.9982 0.9991 0.9075 0.9868 0.9939 0.8035 0.9254 0.9461 SDA-SNE (iteration = 3) 0.67 4.38 8.14 0.9947 0.9983 0.9992 0.9075 0.9867 0.9938 0.8027 0.9174 0.9453 (a) (b) (c) Angular error (degrees) (ⅰ) (ⅱ) (ⅲ) (ⅳ) (ⅴ) Surface normal reference Table 3 . 3Comparison of eA when using TV and PD norms.Method e A (degrees) ↓ Easy Medium Hard SDA-SNE (with TV norm) 1.37 5.14 10.05 SDA-SNE (with PD norm) 0.67 4.38 8.14 In this paper, the variables with and without hat symbols denote the estimated and theoretical values, respectively. su, sv ∈ [i, j] | i, j ∈ {−1, 0, 1} because only adjacent pixels are considered. The representations of E (k−1) p , E (k−1) o ,ẑ pp , andẑ oo in(15)are shown as follows:The representations ofẑ(21)are shown as follows:Given a pixel p = [u 0 , v 0 ] , its adjacent pixels p p , p o , and p d are denoted as follows:NomenclatureNotation DescriptionCoordinate of a 3D point in the camera coordinate systemCoordinate of a 2D pixel in the pixel coordinate system p ∈ R 2 Adjacent pixel of a given pixel p K Camera intrinsic matrix ∇z ∈ R 2 Theoretical depth gradient ∇ẑ ∈ R 2 Estimated depth gradient n ∈ R 3 Theoretical surface normal n ∈ R 3 Estimated surface normal · (k)The k-th iteration · *The optimal value Eu, Ev ∈ R + Smoothness energy components on the u-axis and v-axis su, svState variable components on the u-axis and v-axis T:Smoothness energy transfer function R:Forward finite difference of depth on the u-axis and v-axisBackward finite difference of depth on the u-axis and v-axis zu, zv ∈ R Theoretical first-order partial derivatives of depth on the u-axis and v-axiŝ zu,ẑv ∈ R Estimated first-order partial derivatives of depth on the u-axis and v-axis zuu, zvv ∈ R Theoretical second-order partial derivatives of depth on the u-axis and v-axiŝ zuu,ẑvv ∈ R Estimated second-order partial derivatives of depth on the u-axis and v-axis ·p, · d , ·o Parallel, orthogonal, or diagonal to a given axis, respectively || · || PD Path discontinuity norm f (n)The n-th order polynomial e A ∈ R + Average angular error e P Proportion of good pixels r A Cross accuracy ratio ε C ∈ R + Collinear transfer error of depth gradient estimation ε N ∈ R + Non-collinear transfer error of depth gradient estimationTable 5. Nomenclature Fast and accurate computation of surface normals from range images. Hernan Badino, Daniel Huber, Yongwoon Park, Takeo Kanade, 2011 IEEE Int. Conf. on Robotics and Automation. IEEEHernan Badino, Daniel Huber, Yongwoon Park, and Takeo Kanade. Fast and accurate computation of surface normals from range images. In 2011 IEEE Int. Conf. on Robotics and Automation, pages 3084-3091. IEEE, 2011. 8 Optimal multi-view surface normal estimation using affine correspondences. Dániel Baráth, Ivan Eichhardt, Levente Hajder, IEEE Transactions on Image Processing. 287Dániel Baráth, Ivan Eichhardt, and Levente Hajder. Opti- mal multi-view surface normal estimation using affine cor- respondences. IEEE Transactions on Image Processing, 28(7):3301-3311, 2019. 1, 2 Fast and accurate normal estimation by efficient 3d edge detection. Richard Bormann, Joshua Hampp, Martin Hägele, Markus Vincze, IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Richard Bormann, Joshua Hampp, Martin Hägele, and Markus Vincze. Fast and accurate normal estimation by ef- ficient 3d edge detection. In 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 3930-3937, 2015. 1 Optimal surface normal from affine transformation. Barath Daniel, Jozsef Molnar, Levente Hajder, International Conference on Computer Vision Theory and Applications. SciTePress2Barath Daniel, Jozsef Molnar, and Levente Hajder. Optimal surface normal from affine transformation. In International Conference on Computer Vision Theory and Applications, volume 2, pages 305-316. SciTePress, 2015. 1, 2 Pothole detection based on disparity transformation and road surface modeling. Rui Fan, Umar Ozgunalp, Brett Hosking, Ming Liu, Ioannis Pitas, IEEE Transactions on Image Processing. 291Rui Fan, Umar Ozgunalp, Brett Hosking, Ming Liu, and Ioannis Pitas. Pothole detection based on disparity trans- formation and road surface modeling. IEEE Transactions on Image Processing, 29:897-908, 2019. 1 SNE-RoadSeg: Incorporating surface normal information into semantic segmentation for accurate freespace detection. Rui Fan, Hengli Wang, Peide Cai, Ming Liu, European Conference on Computer Vision. Springer2Rui Fan, Hengli Wang, Peide Cai, and Ming Liu. SNE- RoadSeg: Incorporating surface normal information into se- mantic segmentation for accurate freespace detection. In European Conference on Computer Vision, pages 340-356. Springer, 2020. 2, 8 Three-Filters-to-Normal: An accurate and ultrafast surface normal estimator. Rui Fan, Hengli Wang, Bohuan Xue, Huaiyang Huang, Yuan Wang, Ming Liu, Ioannis Pitas, IEEE Robotics and Automation Letters. 6311Rui Fan, Hengli Wang, Bohuan Xue, Huaiyang Huang, Yuan Wang, Ming Liu, and Ioannis Pitas. Three-Filters-to- Normal: An accurate and ultrafast surface normal estima- tor. IEEE Robotics and Automation Letters, 6(3):5405-5412, 2021. 1, 2, 6, 7, 8, 11 Gradient response maps for real-time detection of textureless objects. Stefan Hinterstoisser, Cedric Cagniart, Slobodan Ilic, Peter Sturm, Nassir Navab, Pascal Fua, Vincent Lepetit, IEEE Trans on Pattern Analysis and Machine Intelligence. 345Stefan Hinterstoisser, Cedric Cagniart, Slobodan Ilic, Peter Sturm, Nassir Navab, Pascal Fua, and Vincent Lepetit. Gra- dient response maps for real-time detection of textureless ob- jects. IEEE Trans on Pattern Analysis and Machine Intelli- gence, 34(5):876-888, 2011. 8 Gradient response maps for real-time detection of textureless objects. Stefan Hinterstoisser, Cedric Cagniart, Slobodan Ilic, Peter Sturm, Nassir Navab, Pascal Fua, Vincent Lepetit, IEEE Transactions on Pattern Analysis and Machine Intelligence. 345Stefan Hinterstoisser, Cedric Cagniart, Slobodan Ilic, Peter Sturm, Nassir Navab, Pascal Fua, and Vincent Lepetit. Gra- dient response maps for real-time detection of textureless ob- jects. IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(5):876-888, 2012. 1 Depth variation and stereo processing tasks in natural scenes. Arvind V Iyer, Johannes Burge, Journal of vision. 186Arvind V. Iyer and Johannes Burge. Depth variation and stereo processing tasks in natural scenes. Journal of vision, 18(6):4-4, 2018. 1 Principles of filter design. Handbook of computer vision and applications. Bernd Jähne, Hanno Scharr, Stefan Körkel, 2Bernd Jähne, Hanno Scharr, Stefan Körkel, et al. Principles of filter design. Handbook of computer vision and applica- tions, 2:125-151, 1999. 6 A quantitative evaluation of surface normal estimation in point clouds. Krzysztof Jordan, Philippos Mordohai, IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEEKrzysztof Jordan and Philippos Mordohai. A quantitative evaluation of surface normal estimation in point clouds. In 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 4220-4226. IEEE, 2014. 1, 2, 8 Comparison of surface normal estimation methods for range sensing applications. Klaas Klasing, Daniel Althoff, Dirk Wollherr, Martin Buss, 2009 IEEE International Conference on Robotics and Automation. IEEEKlaas Klasing, Daniel Althoff, Dirk Wollherr, and Martin Buss. Comparison of surface normal estimation methods for range sensing applications. In 2009 IEEE International Conference on Robotics and Automation, pages 3206-3211. IEEE, 2009. 1, 2, 8 Realtime segmentation of range data using continuous nearest neighbors. Klaas Klasing, Dirk Wollherr, Martin Buss, 2009 IEEE International Conference on Robotics and Automation. IEEEKlaas Klasing, Dirk Wollherr, and Martin Buss. Realtime segmentation of range data using continuous nearest neigh- bors. In 2009 IEEE International Conference on Robotics and Automation, pages 2431-2436. IEEE, 2009. 1, 2, 8 Symps: Brdf symmetry guided photometric stereo for shape and light source estimation. Feng Lu, Xiaowu Chen, Imari Sato, Yoichi Sato, IEEE trans on PAMI. 401Feng Lu, Xiaowu Chen, Imari Sato, and Yoichi Sato. Symps: Brdf symmetry guided photometric stereo for shape and light source estimation. IEEE trans on PAMI, 40(1):221-234, 2017. 6 Estimating surface normals with depth image gradients for fast and accurate registration. Yosuke Nakagawa, Hideaki Uchiyama, Hajime Nagahara, Rin-Ichiro Taniguchi, 2015 International Conference on 3D Vision. 811Yosuke Nakagawa, Hideaki Uchiyama, Hajime Nagahara, and Rin-Ichiro Taniguchi. Estimating surface normals with depth image gradients for fast and accurate registration. In 2015 International Conference on 3D Vision, pages 640- 647, 2015. 1, 2, 6, 7, 8, 11 On the normal vector estimation for point cloud data from smooth surfaces. D Ouyang, H Y Feng, Computer-Aided Design. 3710D. Ouyang and Feng H. Y. On the normal vector estimation for point cloud data from smooth surfaces. Computer-Aided Design, 37(10):1071-1079, 2005. 2 Object enhancement and extraction. Picture processing and Psychopictorics. M S Judith, Prewitt, 10Judith MS Prewitt et al. Object enhancement and extraction. Picture processing and Psychopictorics, 10(1):15-19, 1970. 6 Accurate reconstruction of oriented 3d points using affine correspondences. Carolina Raposo, Joao P Barreto, European Conference on Computer Vision. Springer1Carolina Raposo and Joao P Barreto. Accurate reconstruc- tion of oriented 3d points using affine correspondences. In European Conference on Computer Vision, pages 545-560. Springer, 2020. 1, 2 Discrete laplace-beltrami operators for shape analysis and segmentation. Martin Reuter, Silvia Biasotti, Daniela Giorgi, Giuseppe Patane, Michela Spagnuolo, Martin Reuter, Silvia Biasotti, Daniela Giorgi, Giuseppe Patane, and Michela Spagnuolo. Discrete laplace-beltrami operators for shape analysis and segmentation, 2009. 6 An isotropic 3x3 image gradient operator. Presentation at Stanford A.I. Project. Irwin Sobel, Irwin Sobel. An isotropic 3x3 image gradient operator. Pre- sentation at Stanford A.I. Project 1968, 02 2014. 6 Cloud data modelling employing a unified, non-redundant triangular mesh. W Sun, C Bradley, Y F Zhang, H T Loh, Computer-Aided Design. 332W. Sun, C. Bradley, Y. F. Zhang, and H. T. Loh. Cloud data modelling employing a unified, non-redundant triangu- lar mesh. Computer-Aided Design, 33(2):183-193, 2001. 2 SNE-RoadSeg+: Rethinking depth-normal translation and deep supervision for freespace detection. Hengli Wang, Rui Fan, Peide Cai, Ming Liu, 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Hengli Wang, Rui Fan, Peide Cai, and Ming Liu. SNE- RoadSeg+: Rethinking depth-normal translation and deep supervision for freespace detection. In 2021 IEEE/RSJ In- ternational Conference on Intelligent Robots and Systems (IROS), pages 1140-1145. IEEE, 2021. 2 Irs: A large naturalistic indoor robotics stereo dataset to train deep models for disparity and surface normal estimation. Qiang Wang, Shizhen Zheng, Qingsong Yan, Fei Deng, Kaiyong Zhao, Xiaowen Chu, 2021 IEEE International Conference on Multimedia and Expo (ICME). 11Qiang Wang, Shizhen Zheng, Qingsong Yan, Fei Deng, Kaiyong Zhao, and Xiaowen Chu. Irs: A large naturalistic indoor robotics stereo dataset to train deep models for dis- parity and surface normal estimation. In 2021 IEEE Interna- tional Conference on Multimedia and Expo (ICME), pages 1-6, 2021. 11 Multivariable mathematics. Pearson. E Richard, Hale F Williamson, Trotter, 46Richard E Williamson and Hale F Trotter. Multivariable mathematics. Pearson, 2004. 4, 6 Fast global segmentation based on the dual formulation of tvnorm. Qiang-Jun Xie, Wen-Biao Jin, Li Ma, Di-Bo Hou, 2010 3rd International Congress on Image and Signal Processing. 310Qiang-Jun Xie, Wen-Biao Jin, Li Ma, and Di-Bo Hou. Fast global segmentation based on the dual formulation of tv- norm. In 2010 3rd International Congress on Image and Signal Processing, volume 3, pages 1382-1385, 2010. 10 Detailed Representations of Subscripts p, o, and d in Multi-Directional DP. Detailed Representations of Subscripts p, o, and d in Multi-Directional DP
[]
[ "Searches for New Physics in Photonic Final States at LEP", "Searches for New Physics in Photonic Final States at LEP" ]
[ "Marat Gataullin \nDepartment of Physics\nCalifornia Institute of Technology\n91125PasadenaCA\n" ]
[ "Department of Physics\nCalifornia Institute of Technology\n91125PasadenaCA" ]
[]
A brief review of searches for physics beyond the Standard Model in photonic final states at LEP is given here. These include searches for supersymmetry, large extra dimensions and contact interactions. Recent results from all four LEP experiments are presented, including improved limits on the new scale of gravity for models with large extra dimensions and the most precise direct measurement of the number of light neutrino species. Status and prospects of the LEP combined searches are also discussed. PACS. 13.66.Hk Production of non-standard model particles in e + e − interactions -04.50.+h Gravity in more than four dimensions -12.60.Jv Supersymmetric models -12.60.-i Models beyond the standard model -13.15.+g Neutrino interactions
10.1140/epjcd/s2003-03-828-2
[ "https://arxiv.org/pdf/hep-ex/0311014v1.pdf" ]
14,463,810
hep-ex/0311014
62b47363e201a3626d1cda6c8ee2cba9eec00e9e
Searches for New Physics in Photonic Final States at LEP 6 Nov 2003 Marat Gataullin Department of Physics California Institute of Technology 91125PasadenaCA Searches for New Physics in Photonic Final States at LEP 6 Nov 2003Received: date / Revised version: dateEPJ manuscript No. (will be inserted by the editor) (on behalf of the LEP Collaborations) A brief review of searches for physics beyond the Standard Model in photonic final states at LEP is given here. These include searches for supersymmetry, large extra dimensions and contact interactions. Recent results from all four LEP experiments are presented, including improved limits on the new scale of gravity for models with large extra dimensions and the most precise direct measurement of the number of light neutrino species. Status and prospects of the LEP combined searches are also discussed. PACS. 13.66.Hk Production of non-standard model particles in e + e − interactions -04.50.+h Gravity in more than four dimensions -12.60.Jv Supersymmetric models -12.60.-i Models beyond the standard model -13.15.+g Neutrino interactions Introduction Photonic final states were produced at LEP via the reactions e + e − → ννγ(γ) and e + e − → γγ(γ), leading to two distinct topologies: single-and multi-photon events with missing energy and events with collinear photons (diphotons), respectively. These experimental signatures are also predicted by a wide variety of theories with physics beyond the Standard Model. Single-and multi-photon events can be used in direct searches for new neutral particles, such as graviton production in models with extra dimensions and neutralino and gravitino production processes in supersymmetry. Whereas in the di-photon topology, New Physics can manifest itself through deviations in the measured total and differential cross sections. Results reviewed here are based on studies of the photonic final states by the four LEP collaborations, ALEPH, DELPHI, L3 and OPAL, using the highest centre-of-mass energy, √ s, and luminosity LEP data samples collected in 1998-2000 at √ s = 189 − 208 GeV with an integrated luminosity of about 650 pb −1 per experiment. 2 Single-and Multi-Photon Signatures Neutrino Production In the Standard Model of the electroweak interactions the reaction e + e − → ννγ(γ) proceeds through s-channel Z exchange and t-channel W exchange, where the photons are radiated mainly from the incoming electrons and positrons. The distribution of the recoil mass to the photon system, M rec , is expected to peak around the Z mass in the s-channel, whereas photons from the t-channel W Send offprint requests to: [email protected] exchange are expected to have a relatively flat energy distribution, peaked at low energies. A typical selection (L3) of single-and multi-photon events at LEP requires no charged tracks and the transverse momentum of the photon system, P γ t , greater than 0.02 √ s. The purity of the selected ννγ(γ) sample is 99% and the selection efficiency is estimated to be about 71% [1]. The expected total and differential cross sections depend on the number of light neutrino families, N ν . The M rec spectrum of the single-and multi-photon events selected by the L3 experiment is shown in Figure 1 together with the expectations for N ν = 2, 3 and 4. To determine N ν a maximum likelihood fit is performed to the twodimensional distribution of M rec vs. | cos θ γ |. Including lower energy data, N ν is determined to be [1] This result is more precise than the present world average of measurements relying on the single-photon method [2]. ALEPH [3] and DELPHI [4] have performed similar measurements of N ν , which are also consistent with the Standard Model value of N ν = 3. Searches for Extra Dimensions Models with large extra dimensions [5] predict a gravity scale, M D , as low as the electroweak scale, naturally solving the hierarchy problem. Gravitons, G, are then produced in e + e − collisions through the process e + e − → γG, and escape detection, leading to a single-photon signature. All LEP experiments have performed searches for this reaction using selected samples of single-photon events. Since the photon energy spectrum from the graviton production is expected to be soft, the L3 experiment has also extended its standard single-photon selection to accept photons with P γ t as low as 0.008 √ s [1]. In the low P γ t region the Standard Model background is increased due to the reaction e + e − → e + e − γ(γ), where both electrons have a very low polar angle and cannot be detected. However, this effect is compensated by a significant increase in the accepted signal cross section. Effects of extra dimensions on the L3 photon energy spectrum are shown in Figure 2. A good agreement with the Standard Model expectation is observed and limits on the parameter M D are derived from a fit to the photon energy and polar angle distributions. Recent limits obtained at LEP are detailed in Table 1. These are the best current collider limits for the number of extra dimensions below 6. The presence of the brane in theories with extra dimensions creates additional degrees of freedom. Brane fluctuations in the extra-space directions would then manifest themselves as new stable particles, called "branons", π Br [6]. If the brane tension is below the gravity scale, branons can be detected at LEP via the reaction e + e − → π Br π Br γ, leading to a single-photon signature. The signal properties are similar to those of the graviton production process. L3 has limited branon masses to be above Searches for SUSY Single-and multi-photon events can be also produced by a variety of processes predicted in different models with supersymmetry (SUSY) [8]. These processes involve production and decays of neutralinos and gravitinos. No evidence for such models is observed and corresponding limits on SUSY parameters are given in References [1,3,4]. Combined searches are also performed by the LEP SUSY Working group [9]. In particular, a search for pair-production of neutralinos, each decaying into a photon and a gravitino, is motivated by an interpretation [10] of the rare eeγγ event observed by CDF [11]. This reaction, e + e − →χ 0 1χ 0 1 → GγGγ, is predicted by models with gauge-mediated SUSY breaking when the lightest and next-to-lightest supersymmetric particles are gravitino,G, and neutralino,χ 0 1 , respectively. The experimental signature of this process is very clean, involving events with two energetic acoplanar photons. No anomalous production of such events has been observed, and Figure 3 shows the region excluded by LEP in the mχ0 1 vs. mẽ plane, where mχ0 1 and mẽ are the neutralino and scalar electron masses. The above interpretation of the rare CDF event is now excluded. Collinear Photons Events with collinear photons are typically selected by requiring two energetic back-to-back photon candidates and no matching charged tracks. The cross section of this process has been measured by all four LEP collaborations [3,12,13,14]. The individual measurements have been combined by the LEP Di-photon Working Group [15]. The measured total cross sections normalized to the QED predictions are shown in Figure 4. To search for possible signs of new physics a global fit to the measured total and differential cross sections is performed. Good agreement with the Standard Model expectation is observed, and preliminary LEP combined limits have been derived in the context of several New Physics Models, some of which are described below. The process e + e − → γγ(γ) has a simple QED description at tree level, and provides a benchmark test of the QED at e + e − colliders. A simple and convenient way of parameterizing possible deviations from QED is the introduction of the cut-off parameters Λ ± . In a similar way, bounds on the mass scale of e + e − γγ contact interactions can be derived in terms of a parameter Λ 7 . The corresponding 95% C.L. limits are: Λ + > 392 GeV, Λ − > 364 GeV, Λ 7 > 837 GeV. In models with extra dimensions, photon pair production via virtual graviton exchange can interfere with the Standard Model diagrams, leading to modifications of the differential cross section 1 . Deviations from QED can be then described in terms of a new mass scale, M S , and a parameter λ = ±1, which gives the sign of the interference. The derived 95% C.L. limits are given by: M S (λ = +1) > 933 GeV, M S (λ = −1) > 1010 GeV. 1 It should be noted here, that searches for manifestations of extra dimensions are performed not only in photonic final states but in many other final state topologies [16]. Conclusions and Discussions Searches for New Physics in photonic states have been performed at LEP using the complete LEP data sample. These include searches for supersymmetry, large extra dimensions and deviations from QED. No evidence of such models is found. Constraints on various New Physics theories are set by the LEP experiments separately and by preliminary LEP combinations. Several search strategies and results have been briefly reviewed in this paper. Individual references should be consulted for details. The LEP combinations are expected to be finalized in the near future, and an improvement in sensitivity of the LEP combined searches with single-multi-photon events is expected. The L3 experiment has also published the selection results in the form of tables [1,13], which can be used to test future models involving photonic final states at LEP. Fig. 1 . 1The recoil mass spectrum of the single-and multiphoton events selected by L3 compared to the expected spectra for Nν = 2, 3 and 4. Fig. 2 . 2Distribution of the ratio of the photon energy to the beam energy, for the single-photon sample selected by L3. Expected signals from the reaction e + e − → γG are also shown. Fig. 3 . 3Region excluded at 95% C.L. in the mχ0 1 vs. mẽ plane. Overlaid is the region favored by the interpretation of the CDF event in the scalar electron scenario. This result is preliminary. Fig. 4 . 4The ratio of the measured and QED predicted cross sections of the process e + e − → γγ(γ) as a function of √ s. Table 1 . 1Lower limits at 95% C.L. on the gravitational scale, MD, as a function of the number of extra dimensions, n, obtained by ALEPH[3], DELPHI [4] (preliminary) and L3[1].ALEPH DELPHI L3 n MD (TeV) 2 1.26 1.36 1.50 4 0.77 0.82 0.91 6 0.57 0.59 0.65 103 GeV for a scenario with small brane tensions. Alter- natively, under the assumption of light branon masses, brane tensions below 206 GeV are excluded [7]. N ν = 2.98 ± 0.05(stat) ± 0.04(syst). Acknowledgements . P L3 Collab, Achard, CERN-EP/2003-068Phys. Lett. B. Submitted toL3 Collab., P. Achard et al., CERN-EP/2003-068 (2003), Submitted to Phys. Lett. B. . K Hagiwara, Phys. Rev. D. 6610001K. Hagiwara et al., Phys. Rev. D 66, (2002) 010001. . A Aleph Collab, Heister, Eur. Phys. J. C. 28ALEPH Collab., A. Heister et al. , Eur. Phys. J. C 28, (2003) 1. . N Arkani-Hamed, Phys. Lett. B. 429263N. Arkani-Hamed et al., Phys. Lett. B 429, (1998) 263; . E A Mirabelli, Phys. Rev. Lett. 822236E.A. Mirabelli et al., Phys. Rev. Lett. 82, (1999) 2236. . A Dobado, A L Maroto, Nucl. Phys. B. 592203A. Dobado and A.L. Maroto, Nucl. Phys. B 592, (2001) 203. H E Haber, G L Kane, A review can be found for. 11775A review can be found for example in: H.E. Haber and G.L. Kane, Phys. Rep. 117, (1985) 75. . J L Lopez, D V Nanopoulos, Phys. Rev. D. 554450J.L. Lopez and D.V. Nanopoulos, Phys. Rev. D 55, (1997) 4450. . F Cdf Collab, Abe, Phys. Rev. Lett. 811791CDF Collab., F. Abe et al., Phys. Rev. Lett. 81, (1998) 1791. . P L3 Collab, Achard, Phys. Lett. B. 53128L3 Collab., P. Achard et al., Phys. Lett. B 531, (2002) 28. . G Opal Collab, Abbiendi, Eur. Phys. J. C. 26331OPAL Collab., G. Abbiendi et al., Eur. Phys. J. C 26, (2003) 331. . S Mele, these proceedingsS. Mele, these proceedings; . J Hewett, M Spiropulu, Ann. Rev. Nucl. Part. Sci. 52397J. Hewett and M. Spirop- ulu, Ann. Rev. Nucl. Part. Sci. 52, (2002) 397; . J Alcaraz, Phys. Rev. D. 6775010J. Al- caraz et al., Phys. Rev. D 67, (2003) 075010; . M Gataullin, hep-ex/0108008M. Gataullin, hep-ex/0108008, (2001).
[]
[ "Temperature Spectra of Interstellar Dust Grains Heated by Cosmic Rays. III. Mixed Composition Grains *", "Temperature Spectra of Interstellar Dust Grains Heated by Cosmic Rays. III. Mixed Composition Grains *" ]
[ "Juris Kalvāns \nEngineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nEngineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nInženieru 101, Inženieru 101, VentspilsLV-3601, 3601VentspilsLVLatvia, Latvia\n", "Juris Roberts \nEngineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nEngineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nInženieru 101, Inženieru 101, VentspilsLV-3601, 3601VentspilsLVLatvia, Latvia\n", "Kalnin \nEngineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nEngineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nInženieru 101, Inženieru 101, VentspilsLV-3601, 3601VentspilsLVLatvia, Latvia\n" ]
[ "Engineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nEngineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nInženieru 101, Inženieru 101, VentspilsLV-3601, 3601VentspilsLVLatvia, Latvia", "Engineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nEngineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nInženieru 101, Inženieru 101, VentspilsLV-3601, 3601VentspilsLVLatvia, Latvia", "Engineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nEngineering Research Institute \"Ventspils International Radio Astronomy Center\" of Ventspils University of Applied Sciences\nInženieru 101, Inženieru 101, VentspilsLV-3601, 3601VentspilsLVLatvia, Latvia" ]
[]
Icy grains in the interstellar medium and star-formation regions consist of a variety of materials. Such composite grains interact differently with cosmic-ray (CR) particles compared to simple singlematerial grains. We aim to calculate the spectra of energies and temperatures of mixed-composition grains undergoing whole-grain heating by CRs. The grains were assumed to consist of a mixture of carbon and olivine, covered by ices consisting of carbon oxides and water. The energy and temperature spectra for grains with radii 0.05; 0.1, and 0.2 microns impacted by CRs were calculated for eight values of column density, relevant to molecular clouds and star-forming cores. The approach takes into account changes in ice thickness and composition with increasing column density. These detailed data for CR interaction with interstellar grains are intended for applications in astrochemical models. The main finding is that the a more accurate approach on grain heat capacity and other factors prevent a frequent heating of 0.1 micron or larger icy grains to high temperatures.
10.3847/1538-4365/ac5830
[ "https://export.arxiv.org/pdf/2209.07812v1.pdf" ]
252,355,283
2209.07812
f45bcd38606d4a7a023021dfa708167159114aba
Temperature Spectra of Interstellar Dust Grains Heated by Cosmic Rays. III. Mixed Composition Grains * 16 Sep 2022 Juris Kalvāns Engineering Research Institute "Ventspils International Radio Astronomy Center" of Ventspils University of Applied Sciences Engineering Research Institute "Ventspils International Radio Astronomy Center" of Ventspils University of Applied Sciences Inženieru 101, Inženieru 101, VentspilsLV-3601, 3601VentspilsLVLatvia, Latvia Juris Roberts Engineering Research Institute "Ventspils International Radio Astronomy Center" of Ventspils University of Applied Sciences Engineering Research Institute "Ventspils International Radio Astronomy Center" of Ventspils University of Applied Sciences Inženieru 101, Inženieru 101, VentspilsLV-3601, 3601VentspilsLVLatvia, Latvia Kalnin Engineering Research Institute "Ventspils International Radio Astronomy Center" of Ventspils University of Applied Sciences Engineering Research Institute "Ventspils International Radio Astronomy Center" of Ventspils University of Applied Sciences Inženieru 101, Inženieru 101, VentspilsLV-3601, 3601VentspilsLVLatvia, Latvia Temperature Spectra of Interstellar Dust Grains Heated by Cosmic Rays. III. Mixed Composition Grains * 16 Sep 2022Draft version September 19, 2022 Typeset using L A T E X twocolumn style in AASTeX631Cosmic rays(329) -Astrochemistry(75) -Interstellar dust(836) -Interstellar molecules(849) -Dark interstellar clouds(352) Icy grains in the interstellar medium and star-formation regions consist of a variety of materials. Such composite grains interact differently with cosmic-ray (CR) particles compared to simple singlematerial grains. We aim to calculate the spectra of energies and temperatures of mixed-composition grains undergoing whole-grain heating by CRs. The grains were assumed to consist of a mixture of carbon and olivine, covered by ices consisting of carbon oxides and water. The energy and temperature spectra for grains with radii 0.05; 0.1, and 0.2 microns impacted by CRs were calculated for eight values of column density, relevant to molecular clouds and star-forming cores. The approach takes into account changes in ice thickness and composition with increasing column density. These detailed data for CR interaction with interstellar grains are intended for applications in astrochemical models. The main finding is that the a more accurate approach on grain heat capacity and other factors prevent a frequent heating of 0.1 micron or larger icy grains to high temperatures. INTRODUCTION Cosmic-ray (CR) induced whole-grain heating of grains is a process that is able to significantly affect the balance between gaseous and solid ice phases in dense molecular clouds and star-forming regions (Hasegawa & Herbst 1993). In particular, an efficient sublimation and the resulting retention of the volatile CO molecule in the gas-phase influences the whole chemical evolution of a dense cloud core. Such CR-induced desorption (CRD) has since been adopted in most astrochemical models in use. Therefore, an accurate description of this process is of high importance in theoretical astrochemistry. Whole-grain heating affects also Corresponding author: Juris Kalvāns [email protected] the rates of diffusion and chemical reactions in the icy mantles of interstellar grains. In the recent years, the underlying physics for the CR-induced grain heating process has received renewed interest. Ivlev et al. (2015) investigated the role of CRs as the triggering factor for chemical explosions in the mantles of icy grains. Detailed temperature and energy spectra of grains undergoing whole-grain heating by CRs were published by Kalvāns (2016, hereinafter Paper I), Kalvāns (2018a, Paper II), and its data correction Kalvāns (2022, Paper IIa). The studies by Iqbal & Wakelam (2018), Zhao et al. (2018), Sipilä et al. (2020), and Silsbee et al. (2021) are motivated to address in detail the heating and cooling of grains by their aim to investigate the chemical differentiation of ices across grain sizes. In Kalvāns & Kalnin (2019) and Kalvāns (2021) we employed the data of Paper II for quantifying whole-grain heating in astrochemical models assuming a single size of grains. A similar work with a different approach is that of Sipilä et al. (2021). A number of theoretical studies also consider the specifics of CR interaction with icy interstellar grains. Reboussin et al. (2014); Kalvāns (2015); Ivlev et al. (2015) and Shingledecker et al. (2017) investigate the chemical processing of ices caused by CRs. While these studies cannot be compared directly with our work, some data overlaps. For example, the energy loss function of protons in water ice from Shingledecker et al. (2020) agrees well with our data, despite the different methods employed. The above studies have demonstrated the need of reliable energy and temperature spectra of CR-heated interstellar grains. While Paper II provide such elaborated data, it has several deficiencies. First, it considers the composition of the grain core and its icy mantle in a simplified manner, by assuming only a single component (olivine) for the refractory grain core. The choice of grain materials affects the energy loss function L grain of the impacting CR particle and the heat capacity C of the grain, which affect the temperature spectra of CRheated grains and thus the efficiency of CRD. Second, in a recent work (Kalvāns & Kalnin 2020), we found that C of a Debye solid, used in Paper II, is not suited for the temperatures involved in whole-grain heating by CRs and that less-general and material-specific C data are more reliable. The aim of this paper is to provide accurate data on CR-induced interstellar whole-grain heating for calculating the efficiency of CRD. We consider grains of mixed composition, which affects their L grain and C. In addition, the composition of the icy mantle changes with different interstellar extinctions A V . While this may have a limited effect on L grain , the heat capacities of the primary ice components H 2 O, CO, and CO 2 are quite different, which affects the grain's initial peak temperature reached immediately after the CR hit. Corresponding to the above aim, the tasks of this study are choosing feasible chemical and physical properties of icy grains in dense cores, quantifying their interaction with CRs, and presenting the essence of the results in an understandable manner. The results will permit more accurate modeling of CRD and other CRinduced processes on interstellar grains by providing the energy received and temperature reached by the grains with ice mantle properties relevant to the hydrogen column density N H and interstellar extinction A V in their environment. METHODS Cosmic-ray particles of various chemical elements, each with an initial energy spectrum, travel large distances through the tenuous interstellar medium (ISM) consisting of H, He, and traces of other elements. The CR particles interact with the ISM, which shifts higher-energy particles to lower energies and absorbs the lowest-energy particles. We are interested in CRs that impact a grain after traveling a certain amount of N H . Observations show that the existence and properties of the ice layer on grains depend on the A V (Whittet et al. 2001, see also Table A1 of Kalvāns 2018b and references therein), which means that the icy mantle will have a composition and thickness that depends on N H . The CR particles may impact a single icy grain at different angles, traversing tracks of various lengths. Some of the CR tracks will affect only the ice layer not touching the refractory grain core at all. As a CR particle traverses, it interacts with grain material, losing a certain amount of energy. While some of the energy escapes in the form of fast electrons and ejected ice molecules, the rest eventually thermalizes and the grain attains an uniform temperature within ≈ 10 −9 s (Bringa & Johnson 2004). For most cases, it can be assumed that no cooling occurs up to this point because even at >120 K significant evaporation of volatiles commonly encountered in interstellar ices require longer timescales (Kalvāns & Kalnin 2020). The thermal energy of the grain E grain is the basic outcome of our calculation. With the help of C, it is transformed into T CR , which is the initial or peak temperature of CR-heated grains for astrochemical models considering CRD and can be considered as a more applicable representation of E grain . Because different parts of the grain are bombarded by CRs of various elements and energies, the obtained E grain and T CR can be most accurately presented in the form of energy and temperature spectra, respectively. Grain model Similarly to our previous studies, three grain radii a were considered -0.05, 0.1, and 0.2 µm. This means that the data are optimized for application in common, general astrochemical models that consider a single grain size of about 0.1 µm. The data are not meant for specialized simulations investigating chemistry with different grain sizes, such as those mentioned in Section 1, or other advanced models considering a grain size distribution (e.g. Pauly & Garrod 2018). Chemical modeling with different grain sizes results in a different ice composition and thickness for each grain size (Iqbal & Wakelam 2018;Zhao et al. 2018). This effect is yet to be quantified for the variety of A V and density combinations encountered in evolving dense cores. Similar ice thickness for all grain sizes can be expected in a case when desorption does little to hamper ice accretion (Acharyya et al. 2011;Pauly & Garrod 2016). Moreover, there is suspicion that small grains are depleted via grain-grain collisions in molecular clouds (Silsbee et al. 2020). Therefore we consider describing CR-induced grain heating for relatively large grains with similar ice thickness at similar A V as a safe choice. Observations and models indicate that at increasing interstellar extinctions A V , molecules locked in icy mantles covering the grains become more abundant and with a higher proportion of "hypervolatile" icy species, such as CO, N 2 , O 2 , and CH 4 . It is not possible to calculate T CR and E grain spectra for all possible combinations of grain mantle thickness and ice composition at different A V (and thus, column densities N H ). We overcome this problem by adopting feasible average, characteristic ice composition for several A V values. The derivation of ice properties is described in Section 2.2. The calculation of the interaction between mixedcomposition grains and CRs is a non-trivial undertaking when the full spectra of multiple CR species is considered. To our knowledge, all previous studies, starting with that of Leger et al. (1985, except the series starting with Paper I), have considered only a single path for CRs hitting the grain -that through the center of a spherical homogeneous grain. The length of this path is equal to grain diameter, while its area is the maximum cross-section of the spherical grain. Such assumptions mean that, in practice, a cylinder with radius a and length 2a is considered. The energy left by CRs in such a cylindrical grain is higher by a factor of 1.5 (the ratio of volumes) compared to that of a spherical grain. Moreover, this approach ignores CR paths passing through ice, without touching grain nucleus. Such paths add lower-temperature grain heating events. These errors can be avoided by considering more than one CR path through the grain. A simple yet efficient way to do this is to consider a grain model consisting of several cuboids. It is possible to adjust the size of the cuboids so that the key parameters -CR track length, determining the energy acquired by the grain, and grain cross section, determining collision frequency, are very close to those of a sphere. The track length and cross-sections of the cuboids produce the volume of the grain. In Paper I, the grain model considered two CR tracks through the grain core and one or two tracks through the ice layer, depending on ice thickness. This approach decreased the error to less than 10 % in CR track length, overall grain cross-section, and volume, compared to a spherical grain. Continuing the approach of Paper I, we consider several tracks for CR particles passing through cuboid grain models that have geometrical properties representative for spherical bodies. The model was constructed by attaching cuboids to the faces of a central cube, and allowing the CR particles to enter the grain model only at right angles. This time, to make the calculations more accurate, we consider three CR tracks through the grain core. Figure 1 depicts the basic grain core model to scale. The cross-section of the central cuboid with length 2a is 0.339 relative to that of a circle. This means that the frequency of CR hits that can heat the grain to maximum temperatures is reduced by about a factor of 3, compared to the above-described approach considering a grain cylinder. For icy grains, cuboids consisting of ice were added to the base model applying the five principles laid out in Section 2.2 of Paper I. Shortly, this means that the surface area, maximum CR track length through grain core and ice, and grain volume are similar to the corresponding values of spherical grains covered with icy mantles of uniform thickness. This required also includ-ing a CR track that passes through only icy matter, without touching the refractory grain core (cf. Figure 3 of Paper II). To maintain the deviation of the abovelisted grain model geometrical parameters within 3 % from those of spherical grains, we found it necessary to add two such icy CR tracks for the smaller 0.05 µm grains. Table 1 shows the number of CR tracks considered for each grain size at each A V value. Considering more than a few CR tracks further complicates calculations, while giving increasingly lower benefits in data accuracy and thus is not cost-effective. Each cuboid has its own specific cross section S and track length s for calculating the energy left by a variety of CR species with a range of energies (see Section 2.3). Even the simplest bare grain has close to 30,000 entries for CR-grain interactions. Impacts of CRs resulting in similar grain energies E grain and, thus, heating temperatures T CR (Section 2.4) were grouped together with their frequencies summed up, producing E grain and T CR spectra (Section 3). The properties of the different CR tracks are described in detail Appendix A. Grain composition The dust grain refractory cores are described as composite grains consisting of mixed silicate and carbonaceous materials. A similar approach for describing CR interaction with grains was employed by Shen et al. (2004). Such composite grains are supported by some dust models (Li & Greenberg 1997;Vaidya et al. 2007;Valencic & Smith 2015). Even when the grains in the ISM consist of separate carbonaceous and silicate grain populations, the energy budget of CR impacts is similar to that of composite grains as long as the proportions of the chemical materials are equal, although the maximum energies and temperatures can vary. Observations indicate that the fraction of carbon in interstellar dust is 0.3-0.5 (see Table 1 of Hocuk et al. 2017, and references therein). Here we assume the approximate average value of 0.40. Table 2 summarizes the properties of grain materials. Two separate components were considered -the grain core consisting of a homogeneous mixture the "astronomical silicate" olivine MgFeSiO 4 and carbon covered by a mantle consisting of a homogeneous mixture consisting of H 2 O, CO 2 , and CO ices with proportions listed in Table 1. The density of the mixtures was calculated as the weighed average density of their components. Heat capacity C was calculated with a different method for each material. For the refractory grain core materials, volumnic heat capacities were applied directly, while for the volatile ices, whose constituent proportions are varied at different A V , C calculation could be conveniently done by first obtaining C per molecule, which was multiplied by the supposed exact number of molecules of a given type on the grain. The C values for CO 2 and, in particular, CO ices are substantially higher than those of H 2 O ice, which is the main reason why these materials were considered in this study. Similarly to density, the C of the mixtures were calculated as the weighed average value. Figure 2 shows that C substantially differs for the ice constituents H 2 O, CO, and CO 2 . The high heat capacity of carbon oxides means a lower T CR for the icy grain, compared to our previous calculations, where ice was assumed to have a heat capacity similar to that of pure H 2 O. Given that dust grains are subjected to interstellar ionizing radiation, amorphous carbon (aC) is a candidate carbonaceous component (Koike et al. 1980). However, no specific approach for calculating C for aC was found in literature. Thus, C was calculated as for graphite, albeit with the density of aC (1.557 g cm −3 ). In considering the heating and subsequent cooling of icy grains, one has to take into account that CO has its critical point at 134 K. In our calculations, this T CR value is exceeded by 0.05 µm grains 0.01 µm thick ice layer. (It is also exceeded for bare 0.05 µm grains that do not contain CO.) For CO-rich grains exceeding this temperature, an orderly, layer-by-layer evaporative cooling is likely not possible. This means that some calcula- Table 1. Grain parameters for calculating the TCR and Egrain spectra for grains hit by CRs. Ice composition and thickness data adopted from Kalvāns (2021) and were assumed equal for all three grain sizes. Table 6 of Appendix A. Giauque (1932) tion results -part of the T CR spectra for 0.05 µm grains -can be inaccurate. Nevertheless, the E grain spectrum is accurate because energy deposition occurs on much shorter time-scales (Bringa & Johnson 2004) and may still allow investigating processes on such "overheated" grains. We consider water and carbon oxides as the components of interstellar ices. While there have been observed other components, such as methane, methanol, and ammonia, which are important for cloud chemistry, their abundances are lower and less constrained by observations. Therefore, their effects on grain properties are limited and less well quantifiable. Observations of interstellar ices give limited information on their exact composition in the context of their surrounding medium and history. Astrochemical models can be calibrated with the help of the observational data, such as Boogert et al. (2011). Such calibration, coupling ice abundances to A V , allows to interpolate the lacking details of the "average" H 2 O, CO, and CO 2 relative abundances at different A V values with an acceptable degree of credibility. For obtaining the ice thickness and exact H 2 O, CO, and CO 2 ice relative abundances, the latest version of the model Alchemic-Venta was adapted. A compre-hensive description of this model is provided by Kalvāns (2021). Shortly, this model simulates the chemistry in a contracting typical low-mass star-forming core with the rate-equation method. A single change was made to the model -the interstellar photon and CR flux were modified by a factor of 0.2 (instead by 0.1 and 1.0 in that study), which ensured a better agreement of calculated relative abundances of icy species to observational data. The CR-induced processes in this model are N Hdependent. For CRD, the attenuation of CR flux was considered with the data of Paper II. Ice properties from the model were sampled at A V values listed in Table 1. The ice thickness of 0.01, 0.02, 0.025, and 0.03 µm roughly correspond to 30, 60, 75, and 90 ice monolayers. The ice abundances listed in Table 1 generally agree with other astrochemical models (Taquet et al. 2016;Iqbal & Wakelam 2018;Aikawa et al. 2020;Wakelam et al. 2021;O'Donoghue et al. 2022;Garrod et al. 2022), although can be different from models, where the production of some major icy molecule (which often is CO 2 ) has not been solved, or models adjusted to specific tasks (e.g., Vasyunin et al. 2017;Sipilä et al. 2020;Rawlings 2022). The advantage for using the relative abundances of icy species from Alchemic-Venta is the above-mentioned A V -dependent calibration of ice composition with observational data (for an example, see Kalvāns 2018b). The value for N H /A V was taken to be 1.6 × 10 21 cm −2 in the astrochemical model. However, a literature study reveals that this is in the lowest end of possible values. The published N H /A V values in the ISM range from approximately 1.8×10 21 (Predehl & Schmitt 1995) to 5.1× 10 21 cm −2 (Predehl & Truemper 1994;Güver &Özel 2009). Many of these measurements have been made towards X-ray sources (e.g., Reina & Tarenghi 1973). Higher N H values, relevant to molecular clouds, tend to have a higher N H /A V (Valencic & Smith 2015). Other recent accurate measurements using HI observations sample sight lines with low column densities (N H < 10 20 cm −2 ; Liszt 2014; Chen et al. 2015;Lenz et al. 2017), which are less relevant for our dense cloud core. Therefore, as the most reliable data we found the "gold" sample of Zuo et al. (2021), who consider N H > 5 × 10 21 cm −2 and find N H /A V = 2.2 × 10 21 cm −2 . Table 1 lists ice thickness and proportions between H 2 O, CO 2 , and CO along with the respective N H values used for calculating CR spectra in Section 2.3. These ice mantle data are most relevant to low-mass interstellar starless and star-forming cores in a quiescent environment. The choice of A V from 1.5 to 160 mag corresponds to molecular clouds and their dense cores, from the onset of water ice formation to prestellar cores. These A V and their corresponding column density N H values describe attenuation from the edge of the cloud core to its center. Cosmic ray spectra We consider CR particles that proceed through a cloud core in a straight line. In real clouds, the CR paths and attenuation are much more complex. Such aspects were not considered in our calculations. They can result in overall change of CR flux in dense cores, or even just parts of such cores. The effects of magnetic focusing, mirroring, and scattering can have different outcomes (Padovani et al. 2013;Silsbee et al. 2018;Ivlev et al. 2018) and should be accounted for in astrochemical simulations, depending on what type of object is modeled and other considerations. Also, one can take into account that the overall intensity of CRs varies, depending on the location in the Galaxy (e.g. Indriolo et al. 2015). Initial H spectrum Following an established practice, the energy spectra of all CR elemental components was based on the initial energy spectrum of protons. Our main result is a data set, which employs a basic CR proton spectrum, which we deem most relevant to the ISM. As a default choice, we adapted the "model high" spectrum of Padovani et al. (2018, Alexei Ivlev, private communication). This spectrum produces a CR intensity for H atoms that is slightly lower by a factor of ≈ 1.5 than the intensity adopted in Papers I and II. Only the results with the "model high" spectrum were discussed, however, for completeness, Appendix B shows also the T CR and E grain frequencies for the "model low" CR spectra of Padovani et al. (2018). The initial spectra for all other elemental CR species was obtained by multiplying the spectrum of protons by their respective abundances relative to H. Besides H, 29 more chemical elements were considered, in addition to the important 2 D and 3 He isotopes, for a total of 32 CR species. The species' abundances are based mostly on Voyager 1 data and are listed in Table 2 of Paper II (see references therein). CR spectra at different column densities The CR particles with their initial spectra propagate through the ISM with the column densities N H listed in Table 1. The values of N H indicate only the amount of H atoms traversed by the CR particles, while the medium contains also other elements. Here, the gas was assumed to have the local ISM abundances, consisting of 70.6 % H, 28.0 % He and 1.4 % other elements by mass (Jenkins 2009, parameter F * taken to be 1). The energy loss functions of the 32 CR species L gas for traversing ISM with the above gas composition were calculated with the srim2013 package (Ziegler et al. 2010) and are shown in Figure 3. The srim program does not consider CR energy loss via pion production, which becomes important at GeV energies ) and the CR spectrum at the maximum N H /A V = 3.52 × 10 23 cm −2 is overestimated by a factor of ≈ 2 (Alexei Ivlev, private communication). This means that the calculated data must not be extrapolated and thus the results are not applicable, for example, to dense circumstellar disks or the central part of massive prestellar cores. However, in objects with such a high N H , the efficiency of CRD is negligible. Figure 4 shows an example of the calculated spectra deep into the cloud at an A V value of 40 mag for several CR species. Figure 5 shows an example with oxygen nuclei how the CR spectra changes with increasing A V . CR energy loss in grains When quantifying the interaction of CRs with interstellar grains, it was assumed that CRs may arrive from all directions with equal probability, i.e., the spectra (per steradian) was multiplied by a factor of 4π, similarly to Paper II. This means that the results are most relevant for grains residing in the center of a spherical cloud. In other geometric setups, the calculated frequencies f E (for CR impacts giving grains a certain amount of energy E grain ) and f T (for impacts raising grain temperature to a certain T CR ) may have to be modified. The energy spectra of grains hit by CRs characterizes how often a grain receives a certain amount of energy. These data are less directly applicable but are more general. For example, they may allow recalculat- ing the temperature spectra of CR-heated grains with different material heat capacity or density, or estimating the chemical processing rate of ices. When a CR particle with an energy E CR hits the grain, the particle loses part of its energy, E lost . Generally, E lost = sL grain , where s is the track length through the refractory core (s core ) and ice (s ice ) parts of the grain and L grain is the aggregate energy loss function in the grain for a given CR nucleus, made up from the loss functions for the separate grain materials. For lowenergy particles that come to a complete stop in the grain E CR = E lost . In a 0.1 µm bare grain, such stoppage occurs for ions with total energies up to 20 keV for protons to about 230 keV for iron group nuclei. The energy loss functions L grain for the 32 CR nuclei were calculated with the srim2013 package separately for each of the five constituent materials -olivine, carbon, and H 2 O, CO 2 , and CO ices. Figure 6 shows an example for the loss functions, in this case for Mg nuclei in grain constituent materials. We assume that olivine and carbon are homogeneously mixed in the grain core, while the three ice components are homogeneously mixed in the icy mantle. Thus, the energy loss functions for both the grain core and the mantle were obtained as the weighed average of the L grain for the respective chemical components of the core and mantle. The proportions of these components are given in Table 1 grain core and the mantle. The CR track lengths s were taken from the grain models described in Section 2.1. RESULTS 3.1. CR-heated grain energy spectra When the CR particle has deposited the energy E lost in the grain, a part of this energy can leave the grain with fast electrons. The remaining part x E was estimated with analytical functions derived from the data in Table B1 of Leger et al. (1985). The resulting energy E grain = E lost x E was then assumed to thermalize within the grain, raising its temperature to T CR . The E grain frequency spectra for all grain types are listed in Appendix B. The procedure described above in Section 2 leaves us with a grain that has just received thermal energy E grain,1 (Z, E CR , track) (subscript '1' indicates a single CR impact event), which depends on the impacting CR element Z (32 possible values), its energy E CR (261-308 values per element, according to srim tables), and the type of track along which the CR particle transits the grain (3-5 possibilities per grain type). An E grain,1 entry includes also its associated frequency f E,1 . E grain,1 is coupled to the respective temperature of a grain T CR,1 with the following relation: E grain = TCR 10 CdT . (1) All the T CR,1 obtained from the E grain,1 array via Equation (1) for a grain type were combined in a single list and arranged according to their values. (The entries also include also E grain,1 and frequency values, with the latter renamed f T,1 .) To produce the temperature and its conjugated grain energy spectra, entries with similar T CR,1 values across a specific temperature ∆T CR and energy ∆E grain interval were grouped together (see also below section). The corresponding T CR was obtained as a weighed average: T CR = T CR,1 f T,1 f T,1 ,(2) while the combined frequency is simply the sum of the frequencies of the separate entries f T = f T,1 . A number of E grain values with their corresponding f E constitute the energy spectrum of CR-hit grains, while T CR with f T constitute the temperature spectrum for wholegrain heating. In this work, for a certain grain type we employ only ∆T CR and ∆E grain that are mutually interchangeable via relation (1), which means that f T and f E always are equal for the given interval. The energy spectra for a single-size grain is characterized by two trends with increasing A V . First, the overall E grain increases because of the longer tracks in thicker icy mantles. Second, the overall frequency f E for CR impacts by particles of certain species and energy E decreases because of absorption of CRs in the cloud. Figure 7 shows that these effects partially cancel each other with the result that the frequency density for 0.05 µm grains at a given energy is similar for A V values up to 9.0 mag, where active ice mantle growth is assumed. For the 0.1 and 0.2 µm grains, this canceling of effects is limited because the ice layer contributes relatively much less to the increase of CR track length in the grain. CR-heated grain temperature spectra Within about 10 −9 s after the CR impact, the grain obtains an uniform temperature (Leger et al. 1985;Bringa & Johnson 2004). The highest T CR of 245 K is achieved by the smallest -0.05 µm bare grains (at A V = 1.5 mag), while 0.05 µm grains with a 0.01 µm ice mantle reach 185 K. Sublimation of the hypervolatiles from such grains starts at timescales of 10 −10 s (Kalvāns & Kalnin 2020). This means that sublimation and some grain cooling may occur, when the heat has not yet fully dissipated within the grain, and the high peak temperatures actually are not reached. This means that the peak temperatures exceeding ≈ 150 K in our data are not entirely correct. This discrepancy does not affect the usability of our results in estimating CRD yield because at temperatures higher than ≈ 40 K, the yield depends on the grain's thermal energy content E grain , independently of the exact value of T CR . Also, observations and astrochemical models indicate that the bare or thin-ice-covered grains at low extinctions are poor with the hypervolatile ices. Figure 8 shows graphically the principal output of this study: the temperature spectra of CR-heated grains at different cloud depths, taking into account grain growth due to accumulation of icy mantles. The grains achieve an initial peak temperature T CR with a frequency f T . These data are listed in numerical form in Appendix B. For grains with T CR in excess of 200 K, the ∆T CR interval for data points was taken to be 5 K, for T CR > 100 K the interval was taken to be 2 K, while for T CR < 100 K, a 1 K interval was used. The grain energies E grain were listed corresponding to each T CR value. As stated in Section 2.4, we consider grains in an spherically symmetric environment with an uniform CR intensity from all directions. Tables 3-5 show summarized output -the frequency of CR-induced whole-grain heating events that result in the grains exceeding temperature thresholds starting with 20 K and higher. These data can be directly applied in astrochemical modeling. Summary In Appendix B the temperature and energy spectra for interstellar grains affected by CR hits is given in numerical form. When applying these data in describing sublimation from 0.05 µm grains having up to 0.02 µm thick icy mantle one has to consider that these grains may exceed the 134 K critical temperature of CO ice (Section 2.2). If CO ice is abundant in the mantle, T CR > 134 K can be expected to affect the nature of sublimation. Another issue for the 0.05 µm grains having a thin icy mantle of ≤ 0.1 µm exceeding temperatures of ≈ 150 K because the sublimation timescale for CO is comparable to or shorter than the ≈ 10 −9 s −1 necessary for heat dissipation throughout the grains. This effect will have little effect because the number of sublimated molecules primarily depends on grain thermal energy, not its exact temperature (Section 3.2). The data in this paper covers a similar range of A V and N H values to those in Papers II and IIa. Differences include that here we consider a similar ice thick- ness for all grain sizes (Section 2.1). Moreover, here we avoid calculation of C for the grain core with the Debye method, which underestimates C at temperatures below 70 K and overestimates it at temperatures above (Kalvāns 2021). The consideration of different components in mixed-composition grains induced two additional changes, allowing for a more accurate data. First, considering the less-dense aC mixed with a silicate material reduced L grain (i.e., the grain receives less energy), compared to that calculated in Paper IIa. Second, the explicit consideration of carbon oxides produces a higher ice C, important for grains with thick mantles abundant with CO and CO 2 . As a result, for most heating events of 0.2 µm grains, which rarely exceed 70 K, C for grain core and ice is much higher than that calculated in Paper IIa. For the 0.05 µm grains, the lower of C for grain cores after 70 K comes into play and very high T CR can be achieved. However, at high A V , more important is This work LJO85* S+04* Figure 9. Comparison of CR-induced whole-grain heating frequency fT for TCR above 27 K and 70 K by different authors. References: K18/22: Paper IIa; HH93: Hasegawa & Herbst (1993); R+07: Roberts et al. (2007); LJO85: Leger et al. (1985); S+04: Shen et al. (2004). that such small grains with a thick ice layer contain a substantial proportion of carbon oxides, which reduces their ability to reach T CR in excess of 100 K. For 0.1 µm grains, the effects are mixed, resulting to higher T CR for bare grains and lower T CR for grains with thick icy mantles, compared to the data of Paper IIa. The calculated E grain values are lower than those of Paper IIa by a factor of ≈ 0.8 The differences between the two calculations are more pronounced for T CR and f T , mainly because of the different C. The maximum T CR for 0.1 µm grains with a 0.03 µm ice layer is 72.5 K in this study and 91.3 K in Paper IIa data. The corresponding frequencies f 70 are not directly comparable because of the different N H values in the two studies. However, it can be said that for grains with a thick ≈ 0.03 µm ice layer, this study predicts an order of magnitude lower T CR than that of Paper IIa for temperatures in excess of 30 K. Lower T CR generally means that large 0.1 or 0.2 µm grains with thick icy mantles deep into a dense cloud core can relatively rarely exceed the critical 40 K T CR threshold, where evaporative cooling starts dominating over radiative cooling, and CRD can be efficient. For bare and thinly-iced grains, the frequencies f T are comparable with Paper IIa, while T CR here are significantly higher (for example, 133.3 K versus 103.3 K in Paper IIa for 0.1 µm bare grains). Figure 9 compares the heating frequencies to 27 K and 70 K at different A V to results from previous studies. JK was been funded by ERDF postdoctoral grant No. 1.1.1.2/VIAA/I/16/194 "Chemical effects of cosmicray induced heating of interstellar dust grains" being implemented in Ventspils University of Applied Sciences. JRK was funded by ERDF project No. 1.1.1.1/16/A/213 "Physical and chemical processes in the interstellar medium". We also thank the Ventspils City Council for support and Kristaps Veitners for some of the figures in the paper. We are also grateful to the anonymous referee, whose comments helped to substantially improve the study. This research has made use of NASA's Astrophysics Data System. APPENDIX A. GEOMETRICAL GRAIN MODEL Here we offer more details on the parameters for the cuboid grain models employed in the calculations. Figure 10 shows the cross-sections of the three geometrical grain types: bare grains (a) (compare with Figure 1), icy grains (b) and small icy grains (c). For types (b) and (c), ice thickness can be varied so that the energy E grain and frequency f E = f T match those of a spherical grain with a given ice thickness B. In turn, E grain , which determines T CR depends on track length s of a CR particle passing through the grain, while, and frequencies f E and f T (equal in Appendix B), depend on grain cross-section area S. Models (a) and (c) permit CR entrance and have similar cross-sections along all three axis, while for model (b), CR entrance is permitted along one axis only (with its cross-section featured here), in order to attain S and s most similar to those of a spherical grain. Table 6 lists the values for S and s for each track. B. TEMPERATURE AND ENERGY SPECTRA OF INTERSTELLAR GRAINS HIT BY COSMIC RAYS. Here we present the main results of this study -the temperature and energy spectra of CR-heated mixed olivine-aC grains with radius a covered with mixed H 2 O-CO 2 -CO icy mantles (Section 2.2) in interstellar dark, cold cloud core conditions with initial, ambient grain temperature of 10 K and gas column density N H (Table 1). The whole-grain heating temperature T CR (Section 3.2) is the weighed average for its given T CR interval ∆T CR , which usually spans 1 to 5 K (Equation (2)). In a similar manner, the grain's respective thermal energy E grain (Equation (2)) is the weighed average of energy interval ∆E grain . CR hits imparting the grains with the energy E grain (and thus heating them to T CR ) occur with a frequency f T . The latter depends on the adopted CR spectrum (Section 2.3). Our primary results, likely more relevant for common interstellar conditions, were calculated with the "model high" spectrum from Padovani et al. (2018). As supplemental results, we also provide f T with the "model low" CR spectrum. The initial CR spectra were modified by an N H that is equal in all directions (spherical symmetry). Figure 10. Left to right: cross sections of bare (a) and icy interstellar grain models with thin (b) and thick (c) icy mantles (cf. Figure 1 for bare grain). Different possible CR tracks are identified using the convention 'g': track passes only through refractory nucleus of the grain; 'i': only through ice; 'gi': through ice and grain nucleus. Table 7. Bare grains with a = 0.05 µm, shielded by interstellar gas with NH = 3.30 × 10 21 H atoms cm −2 . For temperatures of 134.45 K and above, the critical point of CO ice is surpassed (see Section 2.2). Table 7 continued Table 6. Cross-section S, and track lengths through grain core (score) and its icy mantle (sice) of tracks for all grain types (one per line) considered. See Table 1 for the relevant AV and NH for each ice thickness B and consult Figure 10 for the placement of the CR tracks in the grain models. Table 8. Grains with a = 0.05 µm and 0.01 µm thick icy mantle, shielded by interstellar gas with NH = 6.60 × 10 21 H atoms cm −2 . For temperatures of 134.45 K and above, the critical point of CO ice is surpassed (see Section 2.2). Table 8 continued Table 8 continued Table 9 continued Table 9 continued Table 19. Grains with a = 0.1 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 4.40 × 10 22 H atoms cm −2 . g3 g3 g3 g3 g1 g2 (a) gi3 gi1 gi2 i4 gi3 gi3 gi3 i4 i4 i4 (b) gi3 gi1 gi2 gi3 gi3 gi3 i6 i5 i6 i6 i6 (c)No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 ,No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 ,No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low Table 19 continued Table 21. Grains with a = 0.1 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 1.76 × 10 23 H atoms cm −2 . No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low Figure 1 . 1Scale model of the core of the interstellar grain. The thick lines show three types of allowed CR tracks through the grain. Track types along the other two symmetry axis are also possible but are not shown for clarity. Figure 2 . 2Heat capacities C for grain constituents. Figure 7 . 7Energy Egrain spectra for grains with core sizes 0.05, 0.1, and 0.2 µm hit by cosmic rays at AV 1.5; 3.0; 6.0; 9.0; 20; 40; 80; and 160 mag, with NH/AV = 2.20 × 10 21 cm −2 . Figure 8 . 8Temperature spectra for grains with core sizes 0.05, 0.1, and 0.2 µm hit by cosmic rays at AV 1.5; 3.0; 6.0; 9.0; 20; 40; 80; and 160 mag, with NH/AV = 2.20 × 10 21 cm −2 . Table 2 . 2Properties of grain materials.No. Material density (g cm −3 ) Heat Capacity Reference 1 Olivine 3.32 Draine & Li (2001) and Xie et al. (2018) 2 Carbon 1.56 Draine & Li (2001) and Xie et al. (2018) 3 H2O ice 0.90 Shulman (2004) 4 CO2 ice 1.60 Giauque & Egan (1937) 5 CO ice 1.10 Clayton & Energy loss function Lgas for CR species traversing interstellar matter; E lost includes also losses due to ISM atoms other than H.1E-9 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 1E-6 1E-3 1E+0 1E+3 L gas , MeV (10 15 H atoms cm -2 ) -1 E CR , MeV amu -1 H Zn Figure 3. Figure 5. Example energy spectra for oxygen CR nuclei at different AV value for each curve, with NH/AV = 2.20 × 10 21 cm −2 .1.E-7 1.E-6 1.E-5 1.E-4 1.E-3 1E-6 1E-4 1E-2 1E+0 1E+2 1E+4 J, MeV -1 cm -2 s -1 sr -1 E O CR , MeV amu -1 1.5 3.0 6.0 9.0 20 40 80 160 . The energy E lost for the whole grain was then obtained by adding the CR energies lost in the0E+0 1E+6 2E+6 3E+6 4E+6 1E-6 1E-4 1E-2 1E+0 1E+2 1E+4 L grain , eV µm -1 E CR , MeV amu -1 carbon olivine CO 2 CO H 2 O Figure 6. Energy loss functions Lgrain for magnesium CR nuclei traversing different grain materials. Table 3 . 3Frequency fT , s −1 , of CR impacts that lift the temperature of 0.05 µm grains above a minimum TCR threshold.AV = NH /(2.2 × 10 21 ) Table 7 (continued) 7No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 5 32.07 30.01-35.00 5.00 2.98E+03 ...-3.913E+03 1.524E+03 9.26E-11 6.17E-11 6 37.09 35.01-40.00 5.00 4.73E+03 ...-5.977E+03 2.064E+03 4.04E-11 2.78E-11 7 42.07 40.01-45.00 5.00 7.04E+03 ...-8.653E+03 2.676E+03 1.54E-11 1.15E-11 8 47.13 45.01-50.00 5.00 1.00E+04 ...-1.199E+04 3.333E+03 8.97E-12 6.78E-12 9 52.44 50.01-55.00 5.00 1.39E+04 ...-1.603E+04 4.041E+03 4.25E-12 2.63E-12 10 57.45 55.01-60.00 5.00 1.83E+04 ...-2.082E+04 4.788E+03 2.09E-12 1.51E-12 11 62.46 60.01-65.00 5.00 2.35E+04 ...-2.638E+04 5.565E+03 1.44E-12 1.01E-12 12 67.36 65.01-70.00 5.00 2.93E+04 ...-3.275E+04 6.365E+03 1.10E-12 7.40E-13 13 72.25 70.01-75.00 5.00 3.59E+04 ...-3.991E+04 7.167E+03 6.19E-13 3.96E-13 14 77.34 75.01-80.00 5.00 4.36E+04 ...-4.792E+04 8.008E+03 5.06E-13 2.97E-13 15 82.60 80.01-85.00 5.00 5.25E+04 ...-5.676E+04 8.841E+03 3.85E-13 2.44E-13 16 87.50 85.01-90.00 5.00 6.16E+04 ...-6.643E+04 9.674E+03 2.71E-13 1.70E-13 17 92.44 90.01-95.00 5.00 7.15E+04 ...-7.694E+04 1.050E+04 2.17E-13 1.41E-13 18 97.36 95.01-100.00 5.00 8.22E+04 ...-8.826E+04 1.132E+04 1.69E-13 1.13E-13 19 102.24 100.01-105.00 5.00 9.36E+04 ...-1.004E+05 1.213E+04 1.21E-13 8.69E-14 20 107.31 105.01-110.00 5.00 1.06E+05 ...-1.133E+05 1.293E+04 1.08E-13 7.91E-14 21 112.52 110.01-115.00 5.00 1.20E+05 ...-1.270E+05 1.371E+04 8.00E-14 5.75E-14 22 117.36 115.01-120.00 5.00 1.34E+05 ...-1.415E+05 1.447E+04 5.25E-14 3.66E-14 23 122.21 120.01-125.00 5.00 1.48E+05 ...-1.567E+05 1.521E+04 5.07E-14 3.54E-14 Table 7 continued 7 Table 7 (continued) 7No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 24 127.17 125.01-130.00 5.00 1.64E+05 ...-1.727E+05 1.594E+04 4.12E-14 3.06E-14 25 132.46 130.01-135.00 5.00 1.81E+05 ...-1.893E+05 1.664E+04 2.72E-14 1.87E-14 26 137.47 135.01-140.00 5.00 1.98E+05 ...-2.066E+05 1.731E+04 2.40E-14 1.91E-14 27 142.22 140.01-145.00 5.00 2.15E+05 ...-2.246E+05 1.797E+04 1.70E-14 1.02E-14 28 147.52 145.01-150.00 5.00 2.34E+05 ...-2.432E+05 1.860E+04 2.01E-14 1.39E-14 29 153.02 150.01-155.00 5.00 2.55E+05 ...-2.624E+05 1.921E+04 1.47E-14 1.02E-14 30 157.88 155.01-160.00 5.00 2.74E+05 ...-2.822E+05 1.984E+04 1.18E-14 7.84E-15 31 162.39 160.01-165.00 5.00 2.92E+05 ...-3.026E+05 2.037E+04 8.22E-15 4.11E-15 32 166.75 165.01-170.00 5.00 3.10E+05 ...-3.235E+05 2.092E+04 1.12E-14 8.14E-15 33 171.93 170.01-175.00 5.00 3.32E+05 ...-3.450E+05 2.145E+04 3.61E-15 3.52E-15 34 177.37 175.01-180.00 5.00 3.55E+05 ...-3.669E+05 2.196E+04 5.75E-15 5.75E-15 35 183.12 180.01-185.00 5.00 3.81E+05 ...-3.894E+05 2.246E+04 3.00E-15 3.00E-15 36 188.20 185.01-190.00 5.00 4.04E+05 ...-4.123E+05 2.295E+04 2.28E-15 2.28E-15 37 192.52 190.01-195.00 5.00 4.24E+05 ...-4.357E+05 2.342E+04 2.67E-15 2.67E-15 38 198.12 195.01-200.00 5.00 4.51E+05 ...-4.596E+05 2.388E+04 2.74E-15 2.74E-15 39 202.87 200.01-205.00 5.00 4.74E+05 ...-4.718E+05 1.216E+04 1.60E-15 1.60E-15 40 207.87 205.01-210.00 5.00 4.98E+05 ...-4.842E+05 1.238E+04 2.52E-15 2.52E-15 41 212.85 210.01-215.00 5.00 5.23E+05 ...-4.968E+05 1.260E+04 1.30E-15 1.30E-15 42 217.15 215.01-220.00 5.00 5.45E+05 ...-5.096E+05 1.281E+04 1.78E-15 1.78E-15 43 222.58 220.01-225.00 5.00 5.73E+05 ...-5.226E+05 1.303E+04 1.70E-15 1.70E-15 44 227.75 225.01-230.00 5.00 6.00E+05 ...-5.358E+05 1.323E+04 2.29E-15 2.29E-15 45 233.25 230.01-235.00 5.00 6.29E+05 ...-5.493E+05 1.344E+04 1.67E-16 1.67E-16 46 235.37 235.01-240.00 5.00 6.41E+05 ...-5.629E+05 1.365E+04 1.68E-17 1.68E-17 47 243.04 240.01-245.01 5.01 6.83E+05 ...-5.768E+05 1.388E+04 3.51E-18 3.51E-18 Table 8 (continued) 8No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 16 41.02 40.01-42.00 2.00 2.44E+04 ...-2.595E+04 3.213E+03 7.16E-13 4.31E-14 17 42.99 42.01-44.00 2.00 2.77E+04 ...-2.940E+04 3.458E+03 6.07E-13 3.54E-14 18 45.01 44.01-46.00 2.00 3.12E+04 ...-3.307E+04 3.669E+03 5.28E-13 2.70E-14 19 46.98 46.01-48.00 2.00 3.50E+04 ...-3.697E+04 3.898E+03 4.68E-13 2.34E-14 20 49.13 48.01-50.00 2.00 3.93E+04 ...-4.110E+04 4.128E+03 2.62E-13 1.62E-14 21 51.02 50.01-52.00 2.00 4.33E+04 ...-4.546E+04 4.359E+03 2.60E-13 1.49E-14 22 52.96 52.01-54.00 2.00 4.77E+04 ...-5.005E+04 4.590E+03 2.24E-13 1.23E-14 23 54.97 54.01-56.00 2.00 5.24E+04 ...-5.487E+04 4.821E+03 1.78E-13 9.49E-15 24 56.86 56.01-58.00 2.00 5.70E+04 ...-5.992E+04 5.051E+03 1.52E-13 7.95E-15 25 58.86 58.01-60.00 2.00 6.22E+04 ...-6.520E+04 5.282E+03 1.47E-13 7.52E-15 26 61.03 60.01-62.00 2.00 6.80E+04 ...-7.071E+04 5.512E+03 1.30E-13 5.92E-15 27 63.12 62.01-64.00 2.00 7.39E+04 ...-7.646E+04 5.741E+03 1.11E-13 6.16E-15 28 65.04 64.01-66.00 2.00 7.95E+04 ...-8.242E+04 5.969E+03 7.75E-14 3.94E-15 29 66.87 66.01-68.00 2.00 8.51E+04 ...-8.862E+04 6.197E+03 7.54E-14 3.30E-15 30 68.81 68.01-70.00 2.00 9.12E+04 ...-9.504E+04 6.423E+03 6.83E-14 2.93E-15 31 70.94 70.01-72.00 2.00 9.81E+04 ...-1.017E+05 6.614E+03 6.36E-14 3.01E-15 32 72.99 72.01-74.00 2.00 1.05E+05 ...-1.085E+05 6.870E+03 5.09E-14 2.54E-15 33 74.96 74.01-76.00 2.00 1.12E+05 ...-1.156E+05 7.092E+03 4.88E-14 2.29E-15 34 77.11 76.01-78.00 2.00 1.20E+05 ...-1.229E+05 7.312E+03 4.02E-14 1.79E-15 35 79.02 78.01-80.00 2.00 1.27E+05 ...-1.305E+05 7.531E+03 3.34E-14 1.51E-15 36 81.00 80.01-82.00 2.00 1.34E+05 ...-1.382E+05 7.747E+03 2.98E-14 1.42E-15 37 82.78 82.01-84.00 2.00 1.41E+05 ...-1.462E+05 7.962E+03 2.15E-14 8.83E-16 38 84.81 84.01-86.00 2.00 1.50E+05 ...-1.543E+05 8.175E+03 2.55E-14 1.21E-15 39 86.88 86.01-88.00 2.00 1.58E+05 ...-1.627E+05 8.386E+03 1.74E-14 6.82E-16 40 88.89 88.01-90.00 2.00 1.67E+05 ...-1.713E+05 8.595E+03 2.19E-14 1.03E-15 41 91.06 90.01-92.00 2.00 1.76E+05 ...-1.801E+05 8.802E+03 1.58E-14 6.73E-16 42 92.81 92.01-94.00 2.00 1.84E+05 ...-1.891E+05 9.007E+03 1.43E-14 6.03E-16 43 94.78 94.01-96.00 2.00 1.93E+05 ...-1.983E+05 9.210E+03 8.94E-15 4.09E-16 44 96.66 96.01-98.00 2.00 2.02E+05 ...-2.078E+05 9.411E+03 1.15E-14 5.19E-16 45 98.73 98.01-100.00 2.00 2.11E+05 ...-2.174E+05 9.610E+03 1.04E-14 4.48E-16 46 100.82 100.01-102.00 2.00 2.21E+05 ...-2.272E+05 9.807E+03 8.27E-15 3.75E-16 47 102.96 102.01-104.00 2.00 2.32E+05 ...-2.372E+05 1.000E+04 8.47E-15 3.73E-16 48 104.84 104.01-106.00 2.00 2.41E+05 ...-2.474E+05 1.020E+04 5.10E-15 1.96E-16 49 106.82 106.01-108.00 2.00 2.52E+05 ...-2.578E+05 1.039E+04 7.43E-15 3.20E-16 50 109.36 108.01-110.00 2.00 2.65E+05 ...-2.683E+05 1.058E+04 6.32E-15 2.70E-16 51 111.09 110.01-112.00 2.00 2.74E+05 ...-2.791E+05 1.076E+04 3.33E-15 1.24E-16 52 112.93 112.01-114.00 2.00 2.84E+05 ...-2.900E+05 1.095E+04 5.88E-15 2.42E-16 53 114.84 114.01-116.00 2.00 2.95E+05 ...-3.012E+05 1.113E+04 3.19E-15 1.17E-16 54 116.89 116.01-118.00 2.00 3.06E+05 ...-3.125E+05 1.131E+04 4.93E-15 2.08E-16 55 118.98 118.01-120.00 2.00 3.18E+05 ...-3.240E+05 1.149E+04 3.23E-15 1.17E-16 56 121.36 120.01-122.00 2.00 3.32E+05 ...-3.357E+05 1.167E+04 3.97E-15 1.60E-16 57 123.21 122.01-124.00 2.00 3.43E+05 ...-3.475E+05 1.185E+04 2.14E-15 7.52E-17 58 124.72 124.01-126.00 2.00 3.52E+05 ...-3.595E+05 1.202E+04 3.18E-15 1.23E-16 59 126.79 126.01-128.00 2.00 3.64E+05 ...-3.717E+05 1.219E+04 2.54E-15 9.91E-17 60 129.18 128.01-130.00 2.00 3.79E+05 ...-3.841E+05 1.236E+04 2.50E-15 9.73E-17 61 130.93 130.01-132.00 2.00 3.90E+05 ...-3.966E+05 1.253E+04 1.87E-15 6.77E-17 62 133.01 132.01-134.00 2.00 4.03E+05 ...-4.093E+05 1.270E+04 2.73E-15 1.04E-16 Table 8 (continued) 8No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low63 134.76 134.01-136.00 2.00 4.14E+05 ...-4.217E+05 1.237E+04 8.73E-16 3.08E-17 64 136.30 136.01-138.00 2.00 4.24E+05 ...-4.341E+05 1.238E+04 1.52E-15 6.01E-17 65 138.88 138.01-140.00 2.00 4.40E+05 ...-4.466E+05 1.254E+04 5.02E-16 1.83E-17 66 140.45 140.01-142.00 2.00 4.49E+05 ...-4.593E+05 1.269E+04 1.34E-15 5.07E-17 67 142.45 142.01-144.00 2.00 4.62E+05 ...-4.721E+05 1.285E+04 8.20E-16 3.14E-17 68 144.43 144.01-146.00 2.00 4.75E+05 ...-4.851E+05 1.300E+04 6.98E-16 2.64E-17 69 146.62 146.01-148.00 2.00 4.89E+05 ...-4.983E+05 1.315E+04 7.93E-16 2.99E-17 70 149.02 148.01-150.00 2.00 5.05E+05 ...-5.116E+05 1.330E+04 7.28E-16 2.70E-17 71 151.33 150.01-152.00 2.00 5.21E+05 ...-5.250E+05 1.345E+04 6.99E-16 2.59E-17 72 152.67 152.01-154.00 2.00 5.30E+05 ...-5.386E+05 1.360E+04 1.58E-16 5.62E-18 73 154.32 154.01-156.00 2.00 5.41E+05 ...-5.524E+05 1.375E+04 7.01E-16 2.56E-17 74 156.78 156.01-158.00 2.00 5.58E+05 ...-5.663E+05 1.389E+04 7.04E-16 2.54E-17 75 159.40 158.01-160.00 2.00 5.76E+05 ...-5.804E+05 1.411E+04 5.25E-16 1.91E-17 76 161.55 160.01-162.00 2.00 5.91E+05 ...-5.946E+05 1.418E+04 4.63E-16 1.65E-17 77 163.05 162.01-164.00 2.00 6.02E+05 ...-6.089E+05 1.433E+04 3.53E-16 1.25E-17 78 164.97 164.01-166.00 2.00 6.16E+05 ...-6.234E+05 1.447E+04 4.40E-16 1.57E-17 79 167.22 166.01-168.00 2.00 6.32E+05 ...-6.380E+05 1.461E+04 4.06E-16 1.43E-17 80 169.24 168.01-170.00 2.00 6.47E+05 ...-6.527E+05 1.476E+04 4.53E-16 1.60E-17 81 171.10 170.01-172.00 2.00 6.61E+05 ...-6.676E+05 1.490E+04 4.46E-16 1.57E-17 82 172.75 172.01-174.00 2.00 6.73E+05 ...-6.827E+05 1.504E+04 5.28E-16 1.84E-17 83 175.28 174.01-176.00 2.00 6.92E+05 ...-6.979E+05 1.518E+04 3.50E-17 1.29E-18 84 176.93 176.01-178.00 2.00 7.05E+05 ...-7.132E+05 1.532E+04 4.62E-17 1.68E-18 85 179.23 178.01-180.00 2.00 7.23E+05 ...-7.286E+05 1.546E+04 4.87E-19 1.82E-20 86 181.04 180.01-182.00 2.00 7.37E+05 ...-7.442E+05 1.560E+04 4.00E-19 1.48E-20 87 182.95 182.01-184.00 2.00 7.52E+05 ...-7.600E+05 1.573E+04 7.61E-19 2.80E-20 88 184.50 184.01-184.98 0.98 7.64E+05 ...-7.677E+05 7.760E+03 7.63E-19 2.81E-20 Table 9 . 9Grains with a = 0.05 µm and 0.020 µm thick icy mantle, shielded by interstellar gas with NH = 1.32 × 10 22 H atoms cm −2 . The highest TCR values are close to the critical point of CO ice of 134.45 K (Section 2.2). . T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , LowNo1 10.41 10.01-12.00 2.00 1.30E+02 5.530-7.055E+02 7.000E+02 1.15E-08 2.65E-09 2 12.73 12.01-14.00 2.00 1.05E+03 ...-1.742E+03 1.037E+03 6.57E-10 2.18E-11 3 14.78 14.01-16.00 2.00 2.26E+03 ...-3.166E+03 1.424E+03 1.48E-10 5.04E-12 4 16.81 16.01-18.00 2.00 3.88E+03 ...-5.007E+03 1.841E+03 4.77E-11 1.84E-12 5 18.87 18.01-20.00 2.00 5.98E+03 ...-7.315E+03 2.308E+03 1.91E-11 8.52E-13 6 20.90 20.01-22.00 2.00 8.55E+03 ...-1.010E+04 2.788E+03 8.88E-12 4.59E-13 7 22.95 22.01-24.00 2.00 1.16E+04 ...-1.339E+04 3.282E+03 4.61E-12 2.72E-13 8 24.92 24.01-26.00 2.00 1.51E+04 ...-1.717E+04 3.783E+03 2.75E-12 1.86E-13 9 26.96 26.01-28.00 2.00 1.92E+04 ...-2.146E+04 4.287E+03 1.71E-12 1.20E-13 10 28.94 28.01-30.00 2.00 2.37E+04 ...-2.625E+04 4.791E+03 1.25E-12 9.01E-14 11 31.02 30.01-32.00 2.00 2.89E+04 ...-3.154E+04 5.293E+03 8.17E-13 6.09E-14 12 32.99 32.01-34.00 2.00 3.44E+04 ...-3.733E+04 5.793E+03 5.86E-13 3.96E-14 13 34.90 34.01-36.00 2.00 4.02E+04 ...-4.362E+04 6.289E+03 4.36E-13 3.12E-14 Table 9 (continued) 9No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low14 36.98 36.01-38.00 2.00 4.69E+04 ...-5.040E+04 6.781E+03 3.18E-13 2.24E-14 15 38.99 38.01-40.00 2.00 5.40E+04 ...-5.767E+04 7.268E+03 2.55E-13 1.76E-14 16 41.00 40.01-42.00 2.00 6.15E+04 ...-6.542E+04 7.752E+03 1.83E-13 1.19E-14 17 43.01 42.01-44.00 2.00 6.96E+04 ...-7.369E+04 8.273E+03 1.59E-13 1.07E-14 18 45.02 44.01-46.00 2.00 7.81E+04 ...-8.240E+04 8.706E+03 1.16E-13 7.22E-15 19 46.98 46.01-48.00 2.00 8.69E+04 ...-9.158E+04 9.175E+03 1.03E-13 6.40E-15 20 48.93 48.01-50.00 2.00 9.60E+04 ...-1.012E+05 9.637E+03 7.39E-14 4.52E-15 21 50.93 50.01-52.00 2.00 1.06E+05 ...-1.113E+05 1.009E+04 6.50E-14 4.06E-15 22 52.96 52.01-54.00 2.00 1.16E+05 ...-1.218E+05 1.054E+04 5.31E-14 3.20E-15 23 54.99 54.01-56.00 2.00 1.27E+05 ...-1.328E+05 1.099E+04 4.52E-14 2.77E-15 24 57.03 56.01-58.00 2.00 1.39E+05 ...-1.443E+05 1.142E+04 3.80E-14 2.15E-15 25 59.02 58.01-60.00 2.00 1.50E+05 ...-1.561E+05 1.185E+04 2.52E-14 1.52E-15 26 60.92 60.01-62.00 2.00 1.62E+05 ...-1.684E+05 1.227E+04 2.37E-14 1.39E-15 27 62.97 62.01-64.00 2.00 1.74E+05 ...-1.811E+05 1.269E+04 2.04E-14 1.19E-15 28 64.97 64.01-66.00 2.00 1.87E+05 ...-1.942E+05 1.310E+04 1.67E-14 9.28E-16 29 66.76 66.01-68.00 2.00 1.99E+05 ...-2.077E+05 1.350E+04 1.40E-14 7.48E-16 30 68.77 68.01-70.00 2.00 2.13E+05 ...-2.216E+05 1.389E+04 1.22E-14 7.19E-16 31 70.93 70.01-72.00 2.00 2.28E+05 ...-2.358E+05 1.421E+04 9.23E-15 5.20E-16 32 72.89 72.01-74.00 2.00 2.42E+05 ...-2.504E+05 1.466E+04 8.28E-15 4.49E-16 33 74.90 74.01-76.00 2.00 2.57E+05 ...-2.655E+05 1.503E+04 7.69E-15 4.27E-16 34 77.03 76.01-78.00 2.00 2.73E+05 ...-2.808E+05 1.540E+04 5.87E-15 3.24E-16 35 79.07 78.01-80.00 2.00 2.89E+05 ...-2.966E+05 1.576E+04 5.23E-15 2.85E-16 36 81.26 80.01-82.00 2.00 3.07E+05 ...-3.127E+05 1.611E+04 4.81E-15 2.60E-16 37 83.05 82.01-84.00 2.00 3.21E+05 ...-3.292E+05 1.646E+04 3.08E-15 1.52E-16 38 84.73 84.01-86.00 2.00 3.35E+05 ...-3.460E+05 1.680E+04 3.57E-15 1.94E-16 39 87.31 86.01-88.00 2.00 3.57E+05 ...-3.631E+05 1.714E+04 3.37E-15 1.77E-16 40 89.28 88.01-90.00 2.00 3.74E+05 ...-3.806E+05 1.747E+04 2.08E-15 1.07E-16 41 91.10 90.01-92.00 2.00 3.90E+05 ...-3.984E+05 1.780E+04 2.28E-15 1.17E-16 42 93.30 92.01-94.00 2.00 4.10E+05 ...-4.165E+05 1.813E+04 1.89E-15 9.77E-17 43 94.97 94.01-96.00 2.00 4.26E+05 ...-4.350E+05 1.844E+04 1.09E-15 5.46E-17 44 96.69 96.01-98.00 2.00 4.41E+05 ...-4.537E+05 1.876E+04 1.53E-15 7.92E-17 45 98.79 98.01-100.00 2.00 4.61E+05 ...-4.728E+05 1.907E+04 1.62E-15 8.36E-17 46 100.98 100.01-102.00 2.00 4.82E+05 ...-4.922E+05 1.938E+04 1.22E-15 6.22E-17 47 102.91 102.01-104.00 2.00 5.01E+05 ...-5.119E+05 1.969E+04 6.25E-16 3.21E-17 48 104.64 104.01-106.00 2.00 5.18E+05 ...-5.319E+05 1.999E+04 4.45E-16 2.27E-17 49 106.91 106.01-108.00 2.00 5.41E+05 ...-5.522E+05 2.029E+04 7.52E-16 3.86E-17 50 109.37 108.01-110.00 2.00 5.66E+05 ...-5.727E+05 2.059E+04 4.07E-16 2.06E-17 51 111.26 110.01-112.00 2.00 5.86E+05 ...-5.936E+05 2.089E+04 3.97E-16 2.01E-17 52 113.14 112.01-114.00 2.00 6.06E+05 ...-6.148E+05 2.118E+04 3.75E-16 1.88E-17 53 115.11 114.01-116.00 2.00 6.27E+05 ...-6.363E+05 2.147E+04 3.33E-16 1.67E-17 54 117.21 116.01-118.00 2.00 6.49E+05 ...-6.580E+05 2.176E+04 3.42E-16 1.71E-17 55 119.07 118.01-120.00 2.00 6.70E+05 ...-6.801E+05 2.205E+04 2.83E-16 1.42E-17 56 121.06 120.01-122.00 2.00 6.92E+05 ...-7.024E+05 2.233E+04 2.40E-16 1.20E-17 57 122.88 122.01-124.00 2.00 7.12E+05 ...-7.250E+05 2.261E+04 3.07E-16 1.53E-17 58 124.80 124.01-126.00 2.00 7.34E+05 ...-7.479E+05 2.289E+04 3.72E-16 1.85E-17 59 127.13 126.01-128.00 2.00 7.61E+05 ...-7.711E+05 2.316E+04 2.91E-17 1.51E-18 60 128.35 128.01-130.00 2.00 7.75E+05 ...-7.945E+05 2.343E+04 1.21E-17 6.30E-19 Table 9 (continued) 9No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low61 131.13 130.01-132.00 2.00 8.08E+05 ...-8.182E+05 2.369E+04 3.21E-19 1.68E-20 62 133.00 132.01-133.76 1.76 8.30E+05 ...-8.394E+05 2.118E+04 6.72E-19 3.51E-20 Table 10 . 10Grains with a = 0.05 µm and 0.025 µm thick icy mantle, shielded by interstellar gas with NH = 1.98 × 10 22 H atoms cm −2 . Table 10 continued 10 Table 10 (continued) 10No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low39 86.78 86.01-88.00 2.00 5.16E+05 ...-5.303E+05 2.414E+04 9.71E-16 6.04E-17 40 88.83 88.01-90.00 2.00 5.41E+05 ...-5.549E+05 2.455E+04 3.66E-16 2.25E-17 41 90.86 90.01-92.00 2.00 5.66E+05 ...-5.798E+05 2.496E+04 5.67E-16 3.51E-17 42 93.06 92.01-94.00 2.00 5.93E+05 ...-6.052E+05 2.536E+04 3.20E-16 1.97E-17 43 94.78 94.01-96.00 2.00 6.15E+05 ...-6.309E+05 2.575E+04 2.91E-16 1.78E-17 44 97.00 96.01-98.00 2.00 6.44E+05 ...-6.571E+05 2.614E+04 4.58E-16 2.80E-17 45 99.28 98.01-100.00 2.00 6.74E+05 ...-6.836E+05 2.652E+04 1.83E-16 1.11E-17 46 100.90 100.01-102.00 2.00 6.96E+05 ...-7.105E+05 2.690E+04 2.84E-16 1.74E-17 47 103.16 102.01-104.00 2.00 7.26E+05 ...-7.378E+05 2.728E+04 2.32E-16 1.42E-17 48 104.95 104.01-106.00 2.00 7.51E+05 ...-7.654E+05 2.766E+04 2.41E-16 1.47E-17 49 106.40 106.01-108.00 2.00 7.71E+05 ...-7.935E+05 2.803E+04 1.97E-16 1.20E-17 50 108.72 108.01-110.00 2.00 8.04E+05 ...-8.219E+05 2.841E+04 2.11E-17 1.34E-18 51 110.99 110.01-112.00 2.00 8.36E+05 ...-8.507E+05 2.878E+04 3.15E-19 2.03E-20 52 113.07 112.01-113.79 1.79 8.66E+05 ...-8.769E+05 2.622E+04 5.19E-19 3.32E-20 Table 11 . 11Grains with a = 0.05 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 4.40 × 10 22 H atoms cm −2 .No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 1 10.21 10.01-12.00 2.00 1.55E+02 12.842-1.622E+03 1.609E+03 1.34E-08 2.99E-09 2 12.67 12.01-14.00 2.00 2.34E+03 ...-3.951E+03 2.329E+03 1.38E-10 7.97E-12 3 14.80 14.01-16.00 2.00 5.14E+03 ...-7.080E+03 3.130E+03 2.45E-11 1.88E-12 4 16.86 16.01-18.00 2.00 8.71E+03 ...-1.105E+04 3.973E+03 8.21E-12 7.35E-13 5 18.90 18.01-20.00 2.00 1.32E+04 ...-1.596E+04 4.905E+03 3.75E-12 3.79E-13 6 20.88 20.01-22.00 2.00 1.85E+04 ...-2.181E+04 5.855E+03 1.87E-12 2.04E-13 7 22.89 22.01-24.00 2.00 2.48E+04 ...-2.864E+04 6.826E+03 1.19E-12 1.35E-13 8 24.92 24.01-26.00 2.00 3.22E+04 ...-3.645E+04 7.810E+03 6.47E-13 7.42E-14 9 26.86 26.01-28.00 2.00 4.02E+04 ...-4.525E+04 8.798E+03 4.49E-13 4.82E-14 10 28.93 28.01-30.00 2.00 4.98E+04 ...-5.503E+04 9.785E+03 2.85E-13 3.08E-14 11 30.93 30.01-32.00 2.00 6.00E+04 ...-6.580E+04 1.077E+04 1.97E-13 2.01E-14 12 32.91 32.01-34.00 2.00 7.11E+04 ...-7.754E+04 1.174E+04 1.32E-13 1.36E-14 13 34.90 34.01-36.00 2.00 8.32E+04 ...-9.023E+04 1.270E+04 1.05E-13 1.06E-14 14 36.97 36.01-38.00 2.00 9.68E+04 ...-1.039E+05 1.365E+04 7.27E-14 6.74E-15 15 38.93 38.01-40.00 2.00 1.11E+05 ...-1.185E+05 1.458E+04 5.46E-14 5.40E-15 16 41.03 40.01-42.00 2.00 1.26E+05 ...-1.340E+05 1.549E+04 3.93E-14 3.74E-15 17 42.98 42.01-44.00 2.00 1.42E+05 ...-1.504E+05 1.648E+04 2.76E-14 2.52E-15 18 44.88 44.01-46.00 2.00 1.58E+05 ...-1.677E+05 1.728E+04 2.18E-14 2.04E-15 19 46.90 46.01-48.00 2.00 1.76E+05 ...-1.858E+05 1.814E+04 1.71E-14 1.58E-15 20 48.90 48.01-50.00 2.00 1.94E+05 ...-2.048E+05 1.898E+04 1.17E-14 1.06E-15 21 50.92 50.01-52.00 2.00 2.14E+05 ...-2.246E+05 1.980E+04 1.11E-14 1.00E-15 22 52.97 52.01-54.00 2.00 2.35E+05 ...-2.452E+05 2.059E+04 6.91E-15 6.37E-16 23 55.01 54.01-56.00 2.00 2.56E+05 ...-2.666E+05 2.137E+04 5.90E-15 5.45E-16 24 56.99 56.01-58.00 2.00 2.77E+05 ...-2.887E+05 2.212E+04 4.41E-15 3.95E-16 25 58.82 58.01-60.00 2.00 2.98E+05 ...-3.116E+05 2.285E+04 3.11E-15 2.83E-16 26 60.75 60.01-62.00 2.00 3.20E+05 ...-3.351E+05 2.356E+04 3.17E-15 2.88E-16 Table 11 continued 11 Table 11 (continued) 11No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 27 62.74 62.01-64.00 2.00 3.44E+05 ...-3.594E+05 2.425E+04 2.41E-15 2.15E-16 28 64.84 64.01-66.00 2.00 3.70E+05 ...-3.843E+05 2.492E+04 2.23E-15 2.04E-16 29 67.01 66.01-68.00 2.00 3.97E+05 ...-4.099E+05 2.556E+04 1.57E-15 1.42E-16 30 68.96 68.01-70.00 2.00 4.22E+05 ...-4.360E+05 2.619E+04 1.21E-15 1.09E-16 31 71.01 70.01-72.00 2.00 4.50E+05 ...-4.627E+05 2.666E+04 8.35E-16 7.59E-17 32 73.02 72.01-74.00 2.00 4.77E+05 ...-4.901E+05 2.739E+04 7.52E-16 6.82E-17 33 74.99 74.01-76.00 2.00 5.04E+05 ...-5.181E+05 2.796E+04 6.37E-16 5.79E-17 34 76.94 76.01-78.00 2.00 5.32E+05 ...-5.466E+05 2.852E+04 5.73E-16 5.19E-17 35 78.67 78.01-80.00 2.00 5.56E+05 ...-5.756E+05 2.906E+04 3.45E-16 3.15E-17 36 81.00 80.01-82.00 2.00 5.91E+05 ...-6.052E+05 2.959E+04 2.89E-16 2.62E-17 37 83.03 82.01-84.00 2.00 6.21E+05 ...-6.353E+05 3.010E+04 1.58E-16 1.43E-17 38 85.09 84.01-86.00 2.00 6.52E+05 ...-6.659E+05 3.061E+04 2.34E-16 2.12E-17 39 87.15 86.01-88.00 2.00 6.84E+05 ...-6.970E+05 3.110E+04 1.20E-16 1.08E-17 40 89.09 88.01-90.00 2.00 7.14E+05 ...-7.286E+05 3.159E+04 1.66E-16 1.50E-17 41 91.21 90.01-92.00 2.00 7.48E+05 ...-7.607E+05 3.207E+04 1.11E-16 1.00E-17 42 93.14 92.01-94.00 2.00 7.79E+05 ...-7.932E+05 3.254E+04 1.40E-16 1.27E-17 43 94.72 94.01-96.00 2.00 8.05E+05 ...-8.262E+05 3.300E+04 1.03E-16 9.31E-18 44 96.83 96.01-98.00 2.00 8.40E+05 ...-8.597E+05 3.346E+04 9.85E-18 9.33E-19 45 99.04 98.01-100.00 2.00 8.78E+05 ...-8.936E+05 3.392E+04 1.75E-19 1.67E-20 46 100.81 100.01-101.34 1.34 9.08E+05 ...-9.168E+05 2.315E+04 2.08E-19 1.99E-20 Table 12 . 12Grains with a = 0.05 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 8.80 × 10 22 H atoms cm −2 .No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 1 10.18 10.01-12.00 2.00 1.35E+02 12.842-1.622E+03 1.609E+03 1.23E-08 2.95E-09 2 12.67 12.01-14.00 2.00 2.34E+03 ...-3.951E+03 2.329E+03 8.74E-11 6.97E-12 3 14.81 14.01-16.00 2.00 5.15E+03 ...-7.080E+03 3.130E+03 1.58E-11 1.69E-12 4 16.86 16.01-18.00 2.00 8.72E+03 ...-1.105E+04 3.973E+03 5.51E-12 6.57E-13 5 18.90 18.01-20.00 2.00 1.32E+04 ...-1.596E+04 4.905E+03 2.55E-12 3.38E-13 6 20.88 20.01-22.00 2.00 1.85E+04 ...-2.181E+04 5.855E+03 1.27E-12 1.81E-13 7 22.89 22.01-24.00 2.00 2.48E+04 ...-2.864E+04 6.826E+03 7.99E-13 1.19E-13 8 24.92 24.01-26.00 2.00 3.22E+04 ...-3.645E+04 7.810E+03 4.33E-13 6.42E-14 9 26.87 26.01-28.00 2.00 4.02E+04 ...-4.525E+04 8.798E+03 2.93E-13 4.09E-14 10 28.94 28.01-30.00 2.00 4.98E+04 ...-5.503E+04 9.785E+03 1.85E-13 2.57E-14 11 30.93 30.01-32.00 2.00 6.00E+04 ...-6.580E+04 1.077E+04 1.24E-13 1.64E-14 12 32.91 32.01-34.00 2.00 7.11E+04 ...-7.754E+04 1.174E+04 8.26E-14 1.10E-14 13 34.90 34.01-36.00 2.00 8.32E+04 ...-9.023E+04 1.270E+04 6.49E-14 8.48E-15 14 36.96 36.01-38.00 2.00 9.68E+04 ...-1.039E+05 1.365E+04 4.29E-14 5.21E-15 15 38.92 38.01-40.00 2.00 1.11E+05 ...-1.185E+05 1.458E+04 3.26E-14 4.18E-15 16 41.03 40.01-42.00 2.00 1.26E+05 ...-1.340E+05 1.549E+04 2.30E-14 2.86E-15 17 42.98 42.01-44.00 2.00 1.42E+05 ...-1.504E+05 1.648E+04 1.57E-14 1.89E-15 18 44.88 44.01-46.00 2.00 1.58E+05 ...-1.677E+05 1.728E+04 1.24E-14 1.53E-15 19 46.90 46.01-48.00 2.00 1.76E+05 ...-1.858E+05 1.814E+04 9.43E-15 1.16E-15 20 48.89 48.01-50.00 2.00 1.94E+05 ...-2.048E+05 1.898E+04 6.39E-15 7.74E-16 Table 12 continued 12 Table 12 (continued) 12No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 21 50.91 50.01-52.00 2.00 2.14E+05 ...-2.246E+05 1.980E+04 6.02E-15 7.32E-16 22 52.96 52.01-54.00 2.00 2.34E+05 ...-2.452E+05 2.059E+04 3.73E-15 4.63E-16 23 55.00 54.01-56.00 2.00 2.56E+05 ...-2.666E+05 2.137E+04 3.15E-15 3.94E-16 24 56.98 56.01-58.00 2.00 2.77E+05 ...-2.887E+05 2.212E+04 2.31E-15 2.83E-16 25 58.82 58.01-60.00 2.00 2.98E+05 ...-3.116E+05 2.285E+04 1.63E-15 2.02E-16 26 60.74 60.01-62.00 2.00 3.20E+05 ...-3.351E+05 2.356E+04 1.65E-15 2.05E-16 27 62.73 62.01-64.00 2.00 3.44E+05 ...-3.594E+05 2.425E+04 1.25E-15 1.53E-16 28 64.83 64.01-66.00 2.00 3.70E+05 ...-3.843E+05 2.492E+04 1.15E-15 1.45E-16 29 67.01 66.01-68.00 2.00 3.97E+05 ...-4.099E+05 2.556E+04 8.04E-16 1.00E-16 30 68.96 68.01-70.00 2.00 4.22E+05 ...-4.360E+05 2.619E+04 6.25E-16 7.74E-17 31 71.02 70.01-72.00 2.00 4.50E+05 ...-4.627E+05 2.666E+04 4.27E-16 5.36E-17 32 73.02 72.01-74.00 2.00 4.77E+05 ...-4.901E+05 2.739E+04 3.84E-16 4.81E-17 33 74.99 74.01-76.00 2.00 5.04E+05 ...-5.181E+05 2.796E+04 3.25E-16 4.08E-17 34 76.94 76.01-78.00 2.00 5.32E+05 ...-5.466E+05 2.852E+04 2.90E-16 3.65E-17 35 78.67 78.01-80.00 2.00 5.56E+05 ...-5.756E+05 2.906E+04 1.76E-16 2.22E-17 36 81.00 80.01-82.00 2.00 5.91E+05 ...-6.052E+05 2.959E+04 1.47E-16 1.84E-17 37 83.03 82.01-84.00 2.00 6.21E+05 ...-6.353E+05 3.010E+04 8.03E-17 1.01E-17 38 85.09 84.01-86.00 2.00 6.52E+05 ...-6.659E+05 3.061E+04 1.18E-16 1.49E-17 39 87.15 86.01-88.00 2.00 6.84E+05 ...-6.970E+05 3.110E+04 6.05E-17 7.55E-18 40 89.09 88.01-90.00 2.00 7.14E+05 ...-7.286E+05 3.159E+04 8.39E-17 1.05E-17 41 91.21 90.01-92.00 2.00 7.48E+05 ...-7.607E+05 3.207E+04 5.59E-17 7.04E-18 42 93.14 92.01-94.00 2.00 7.79E+05 ...-7.932E+05 3.254E+04 7.03E-17 8.86E-18 43 94.72 94.01-96.00 2.00 8.05E+05 ...-8.262E+05 3.300E+04 5.16E-17 6.52E-18 44 96.83 96.01-98.00 2.00 8.40E+05 ...-8.597E+05 3.346E+04 4.95E-18 6.52E-19 45 99.04 98.01-100.00 2.00 8.78E+05 ...-8.936E+05 3.392E+04 8.80E-20 1.17E-20 46 100.81 100.01-101.34 1.34 9.08E+05 ...-9.168E+05 2.315E+04 1.04E-19 1.39E-20 Table 13 . 13Grains with a = 0.05 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 1.76 × 10 23 H atoms cm −2 .No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 1 10.16 10.01-12.00 2.00 1.17E+02 12.842-1.622E+03 1.609E+03 1.10E-08 2.89E-09 2 12.67 12.01-14.00 2.00 2.34E+03 ...-3.951E+03 2.329E+03 5.45E-11 6.03E-12 3 14.81 14.01-16.00 2.00 5.16E+03 ...-7.080E+03 3.130E+03 1.03E-11 1.48E-12 4 16.86 16.01-18.00 2.00 8.72E+03 ...-1.105E+04 3.973E+03 3.65E-12 5.71E-13 5 18.89 18.01-20.00 2.00 1.32E+04 ...-1.596E+04 4.905E+03 1.68E-12 2.91E-13 6 20.88 20.01-22.00 2.00 1.85E+04 ...-2.181E+04 5.855E+03 8.35E-13 1.55E-13 7 22.89 22.01-24.00 2.00 2.48E+04 ...-2.864E+04 6.826E+03 5.15E-13 1.01E-13 8 24.92 24.01-26.00 2.00 3.22E+04 ...-3.645E+04 7.810E+03 2.78E-13 5.33E-14 9 26.88 26.01-28.00 2.00 4.03E+04 ...-4.525E+04 8.798E+03 1.83E-13 3.32E-14 10 28.95 28.01-30.00 2.00 4.99E+04 ...-5.503E+04 9.785E+03 1.14E-13 2.04E-14 11 30.93 30.01-32.00 2.00 6.00E+04 ...-6.580E+04 1.077E+04 7.42E-14 1.26E-14 12 32.90 32.01-34.00 2.00 7.11E+04 ...-7.754E+04 1.174E+04 4.90E-14 8.43E-15 13 34.90 34.01-36.00 2.00 8.32E+04 ...-9.023E+04 1.270E+04 3.78E-14 6.38E-15 14 36.96 36.01-38.00 2.00 9.68E+04 ...-1.039E+05 1.365E+04 2.39E-14 3.81E-15 Table 13 continued 13 Table 13 (continued) 13No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 15 38.90 38.01-40.00 2.00 1.10E+05 ...-1.185E+05 1.458E+04 1.83E-14 3.06E-15 16 41.03 40.01-42.00 2.00 1.26E+05 ...-1.340E+05 1.549E+04 1.26E-14 2.06E-15 17 42.98 42.01-44.00 2.00 1.42E+05 ...-1.504E+05 1.648E+04 8.37E-15 1.34E-15 18 44.88 44.01-46.00 2.00 1.58E+05 ...-1.677E+05 1.728E+04 6.63E-15 1.08E-15 19 46.90 46.01-48.00 2.00 1.76E+05 ...-1.858E+05 1.814E+04 4.93E-15 8.10E-16 20 48.89 48.01-50.00 2.00 1.94E+05 ...-2.048E+05 1.898E+04 3.32E-15 5.39E-16 21 50.90 50.01-52.00 2.00 2.14E+05 ...-2.246E+05 1.980E+04 3.11E-15 5.08E-16 22 52.95 52.01-54.00 2.00 2.34E+05 ...-2.452E+05 2.059E+04 1.92E-15 3.20E-16 23 55.00 54.01-56.00 2.00 2.56E+05 ...-2.666E+05 2.137E+04 1.61E-15 2.71E-16 24 56.98 56.01-58.00 2.00 2.77E+05 ...-2.887E+05 2.212E+04 1.17E-15 1.94E-16 25 58.81 58.01-60.00 2.00 2.98E+05 ...-3.116E+05 2.285E+04 8.24E-16 1.38E-16 26 60.74 60.01-62.00 2.00 3.20E+05 ...-3.351E+05 2.356E+04 8.28E-16 1.40E-16 27 62.72 62.01-64.00 2.00 3.44E+05 ...-3.594E+05 2.425E+04 6.25E-16 1.04E-16 28 64.83 64.01-66.00 2.00 3.70E+05 ...-3.843E+05 2.492E+04 5.75E-16 9.81E-17 29 67.01 66.01-68.00 2.00 3.97E+05 ...-4.099E+05 2.556E+04 4.00E-16 6.78E-17 30 68.96 68.01-70.00 2.00 4.22E+05 ...-4.360E+05 2.619E+04 3.11E-16 5.25E-17 31 71.02 70.01-72.00 2.00 4.50E+05 ...-4.627E+05 2.666E+04 2.12E-16 3.63E-17 32 73.02 72.01-74.00 2.00 4.77E+05 ...-4.901E+05 2.739E+04 1.90E-16 3.25E-17 33 74.99 74.01-76.00 2.00 5.04E+05 ...-5.181E+05 2.796E+04 1.60E-16 2.75E-17 34 76.94 76.01-78.00 2.00 5.32E+05 ...-5.466E+05 2.852E+04 1.43E-16 2.46E-17 35 78.67 78.01-80.00 2.00 5.56E+05 ...-5.756E+05 2.906E+04 8.70E-17 1.50E-17 36 81.00 80.01-82.00 2.00 5.91E+05 ...-6.052E+05 2.959E+04 7.27E-17 1.25E-17 37 83.03 82.01-84.00 2.00 6.21E+05 ...-6.353E+05 3.010E+04 3.97E-17 6.78E-18 38 85.09 84.01-86.00 2.00 6.52E+05 ...-6.659E+05 3.061E+04 5.84E-17 1.00E-17 39 87.15 86.01-88.00 2.00 6.84E+05 ...-6.970E+05 3.110E+04 2.98E-17 5.10E-18 40 89.09 88.01-90.00 2.00 7.14E+05 ...-7.286E+05 3.159E+04 4.13E-17 7.10E-18 41 91.21 90.01-92.00 2.00 7.48E+05 ...-7.607E+05 3.207E+04 2.75E-17 4.74E-18 42 93.14 92.01-94.00 2.00 7.79E+05 ...-7.932E+05 3.254E+04 3.45E-17 5.96E-18 43 94.72 94.01-96.00 2.00 8.05E+05 ...-8.262E+05 3.300E+04 2.54E-17 4.38E-18 44 96.83 96.01-98.00 2.00 8.40E+05 ...-8.597E+05 3.346E+04 2.42E-18 4.36E-19 45 99.04 98.01-100.00 2.00 8.78E+05 ...-8.936E+05 3.392E+04 4.31E-20 7.84E-21 46 100.81 100.01-101.34 1.34 9.08E+05 ...-9.168E+05 2.315E+04 5.11E-20 9.32E-21 Table 14 . 14Grains with a = 0.05 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 3.52 × 10 23 H atoms cm −2 .No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 1 10.13 10.01-12.00 2.00 1.01E+02 12.842-1.622E+03 1.609E+03 9.75E-09 2.81E-09 2 12.68 12.01-14.00 2.00 2.35E+03 ...-3.951E+03 2.329E+03 3.37E-11 5.14E-12 3 14.82 14.01-16.00 2.00 5.16E+03 ...-7.080E+03 3.130E+03 6.67E-12 1.27E-12 4 16.85 16.01-18.00 2.00 8.71E+03 ...-1.105E+04 3.973E+03 2.36E-12 4.78E-13 5 18.89 18.01-20.00 2.00 1.32E+04 ...-1.596E+04 4.905E+03 1.07E-12 2.42E-13 6 20.88 20.01-22.00 2.00 1.85E+04 ...-2.181E+04 5.855E+03 5.29E-13 1.27E-13 7 22.89 22.01-24.00 2.00 2.48E+04 ...-2.864E+04 6.826E+03 3.23E-13 8.29E-14 8 24.92 24.01-26.00 2.00 3.22E+04 ...-3.645E+04 7.810E+03 1.72E-13 4.23E-14 Table 14 continued 14 Table 14 (continued) 14No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 9 26.89 26.01-28.00 2.00 4.03E+04 ...-4.525E+04 8.798E+03 1.10E-13 2.55E-14 10 28.96 28.01-30.00 2.00 4.99E+04 ...-5.503E+04 9.785E+03 6.64E-14 1.52E-14 11 30.93 30.01-32.00 2.00 6.00E+04 ...-6.580E+04 1.077E+04 4.19E-14 9.20E-15 12 32.89 32.01-34.00 2.00 7.10E+04 ...-7.754E+04 1.174E+04 2.73E-14 6.05E-15 13 34.90 34.01-36.00 2.00 8.33E+04 ...-9.023E+04 1.270E+04 2.06E-14 4.50E-15 14 36.95 36.01-38.00 2.00 9.67E+04 ...-1.039E+05 1.365E+04 1.26E-14 2.63E-15 15 38.89 38.01-40.00 2.00 1.10E+05 ...-1.185E+05 1.458E+04 9.63E-15 2.10E-15 16 41.03 40.01-42.00 2.00 1.26E+05 ...-1.340E+05 1.549E+04 6.55E-15 1.41E-15 17 42.97 42.01-44.00 2.00 1.42E+05 ...-1.504E+05 1.648E+04 4.26E-15 9.02E-16 18 44.88 44.01-46.00 2.00 1.58E+05 ...-1.677E+05 1.728E+04 3.37E-15 7.29E-16 19 46.90 46.01-48.00 2.00 1.76E+05 ...-1.858E+05 1.814E+04 2.47E-15 5.39E-16 20 48.88 48.01-50.00 2.00 1.94E+05 ...-2.048E+05 1.898E+04 1.66E-15 3.58E-16 21 50.90 50.01-52.00 2.00 2.14E+05 ...-2.246E+05 1.980E+04 1.55E-15 3.37E-16 22 52.95 52.01-54.00 2.00 2.34E+05 ...-2.452E+05 2.059E+04 9.50E-16 2.11E-16 23 55.00 54.01-56.00 2.00 2.56E+05 ...-2.666E+05 2.137E+04 7.92E-16 1.78E-16 24 56.98 56.01-58.00 2.00 2.77E+05 ...-2.887E+05 2.212E+04 5.73E-16 1.27E-16 25 58.81 58.01-60.00 2.00 2.98E+05 ...-3.116E+05 2.285E+04 4.03E-16 9.06E-17 26 60.74 60.01-62.00 2.00 3.20E+05 ...-3.351E+05 2.356E+04 4.04E-16 9.14E-17 27 62.72 62.01-64.00 2.00 3.44E+05 ...-3.594E+05 2.425E+04 3.04E-16 6.82E-17 28 64.83 64.01-66.00 2.00 3.70E+05 ...-3.843E+05 2.492E+04 2.79E-16 6.39E-17 29 67.01 66.01-68.00 2.00 3.97E+05 ...-4.099E+05 2.556E+04 1.94E-16 4.42E-17 30 68.95 68.01-70.00 2.00 4.22E+05 ...-4.360E+05 2.619E+04 1.51E-16 3.43E-17 31 71.02 70.01-72.00 2.00 4.50E+05 ...-4.627E+05 2.666E+04 1.02E-16 2.36E-17 32 73.02 72.01-74.00 2.00 4.77E+05 ...-4.901E+05 2.739E+04 9.18E-17 2.11E-17 33 74.99 74.01-76.00 2.00 5.04E+05 ...-5.181E+05 2.796E+04 7.75E-17 1.79E-17 34 76.94 76.01-78.00 2.00 5.32E+05 ...-5.466E+05 2.852E+04 6.91E-17 1.60E-17 35 78.67 78.01-80.00 2.00 5.56E+05 ...-5.756E+05 2.906E+04 4.20E-17 9.73E-18 36 81.00 80.01-82.00 2.00 5.91E+05 ...-6.052E+05 2.959E+04 3.51E-17 8.09E-18 37 83.03 82.01-84.00 2.00 6.21E+05 ...-6.353E+05 3.010E+04 1.91E-17 4.41E-18 38 85.09 84.01-86.00 2.00 6.52E+05 ...-6.659E+05 3.061E+04 2.82E-17 6.51E-18 39 87.15 86.01-88.00 2.00 6.84E+05 ...-6.970E+05 3.110E+04 1.44E-17 3.31E-18 40 89.09 88.01-90.00 2.00 7.14E+05 ...-7.286E+05 3.159E+04 1.99E-17 4.60E-18 41 91.21 90.01-92.00 2.00 7.48E+05 ...-7.607E+05 3.207E+04 1.32E-17 3.07E-18 42 93.14 92.01-94.00 2.00 7.79E+05 ...-7.932E+05 3.254E+04 1.66E-17 3.86E-18 43 94.72 94.01-96.00 2.00 8.05E+05 ...-8.262E+05 3.300E+04 1.22E-17 2.84E-18 44 96.83 96.01-98.00 2.00 8.40E+05 ...-8.597E+05 3.346E+04 1.16E-18 2.80E-19 45 99.04 98.01-100.00 2.00 8.78E+05 ...-8.936E+05 3.392E+04 2.06E-20 5.03E-21 46 100.81 100.01-101.34 1.34 9.08E+05 ...-9.168E+05 2.315E+04 2.44E-20 5.97E-21 Table 15 . 15Bare grains with a = 0.1 µm, shielded by interstellar gas with NH = 3.30 × 10 21 H atoms cm −2 .No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 1 10.60 10.01-12.00 2.00 2.01E+02 6.3238-6.752E+02 6.689E+02 2.05E-08 5.25E-09 2 12.85 12.01-14.00 2.00 9.97E+02 ...-1.465E+03 7.900E+02 3.83E-09 1.34E-10 3 14.88 14.01-16.00 2.00 1.88E+03 ...-2.438E+03 9.729E+02 1.73E-09 4.00E-11 Table 15 continued 15 Table 15 (continued) 15No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 4 16.90 16.01-18.00 2.00 2.96E+03 ...-3.648E+03 1.210E+03 8.99E-10 1.74E-11 5 18.90 18.01-20.00 2.00 4.31E+03 ...-5.164E+03 1.516E+03 4.88E-10 9.16E-12 6 20.90 20.01-22.00 2.00 5.98E+03 ...-7.037E+03 1.873E+03 2.53E-10 5.08E-12 7 22.89 22.01-24.00 2.00 8.03E+03 ...-9.319E+03 2.282E+03 1.62E-10 3.18E-12 8 24.99 24.01-26.00 2.00 1.07E+04 ...-1.206E+04 2.741E+03 8.72E-11 1.89E-12 9 26.90 26.01-28.00 2.00 1.35E+04 ...-1.531E+04 3.248E+03 5.16E-11 1.18E-12 10 28.96 28.01-30.00 2.00 1.71E+04 ...-1.911E+04 3.800E+03 3.30E-11 8.84E-13 11 31.03 30.01-32.00 2.00 2.13E+04 ...-2.350E+04 4.395E+03 2.12E-11 6.43E-13 12 32.94 32.01-34.00 2.00 2.58E+04 ...-2.853E+04 5.031E+03 1.49E-11 4.67E-13 13 35.09 34.01-36.00 2.00 3.16E+04 ...-3.424E+04 5.705E+03 9.55E-12 3.31E-13 14 37.00 36.01-38.00 2.00 3.74E+04 ...-4.065E+04 6.415E+03 6.15E-12 2.85E-13 15 38.98 38.01-40.00 2.00 4.41E+04 ...-4.781E+04 7.160E+03 4.27E-12 1.89E-13 16 41.00 40.01-42.00 2.00 5.17E+04 ...-5.575E+04 7.936E+03 3.80E-12 1.66E-13 17 43.01 42.01-44.00 2.00 6.01E+04 ...-6.454E+04 8.788E+03 2.96E-12 1.18E-13 18 44.77 44.01-46.00 2.00 6.82E+04 ...-7.412E+04 9.581E+03 2.48E-12 9.81E-14 19 46.96 46.01-48.00 2.00 7.91E+04 ...-8.456E+04 1.044E+04 1.65E-12 7.93E-14 20 49.02 48.01-50.00 2.00 9.03E+04 ...-9.589E+04 1.133E+04 1.45E-12 6.26E-14 21 51.06 50.01-52.00 2.00 1.02E+05 ...-1.081E+05 1.223E+04 1.08E-12 4.26E-14 22 53.05 52.01-54.00 2.00 1.15E+05 ...-1.213E+05 1.316E+04 1.01E-12 4.03E-14 23 54.92 54.01-56.00 2.00 1.28E+05 ...-1.354E+05 1.411E+04 7.18E-13 2.88E-14 24 56.88 56.01-58.00 2.00 1.42E+05 ...-1.505E+05 1.507E+04 7.73E-13 3.08E-14 25 58.98 58.01-60.00 2.00 1.58E+05 ...-1.665E+05 1.606E+04 5.15E-13 1.86E-14 26 60.86 60.01-62.00 2.00 1.74E+05 ...-1.836E+05 1.705E+04 5.01E-13 1.78E-14 27 62.88 62.01-64.00 2.00 1.91E+05 ...-2.016E+05 1.806E+04 4.14E-13 1.58E-14 28 64.99 64.01-66.00 2.00 2.11E+05 ...-2.207E+05 1.908E+04 3.48E-13 1.29E-14 29 67.06 66.01-68.00 2.00 2.31E+05 ...-2.408E+05 2.011E+04 2.76E-13 9.15E-15 30 69.00 68.01-70.00 2.00 2.51E+05 ...-2.620E+05 2.115E+04 2.37E-13 8.17E-15 31 70.94 70.01-72.00 2.00 2.72E+05 ...-2.840E+05 2.208E+04 1.79E-13 6.23E-15 32 72.98 72.01-74.00 2.00 2.95E+05 ...-3.073E+05 2.324E+04 1.94E-13 6.53E-15 33 75.12 74.01-76.00 2.00 3.21E+05 ...-3.316E+05 2.430E+04 1.22E-13 4.33E-15 34 77.01 76.01-78.00 2.00 3.44E+05 ...-3.569E+05 2.536E+04 1.35E-13 4.53E-15 35 78.89 78.01-80.00 2.00 3.69E+05 ...-3.834E+05 2.642E+04 9.40E-14 3.05E-15 36 81.01 80.01-82.00 2.00 3.97E+05 ...-4.109E+05 2.749E+04 8.24E-14 3.00E-15 37 82.93 82.01-84.00 2.00 4.24E+05 ...-4.394E+05 2.856E+04 6.02E-14 2.06E-15 38 84.90 84.01-86.00 2.00 4.53E+05 ...-4.690E+05 2.962E+04 7.10E-14 2.22E-15 39 87.05 86.01-88.00 2.00 4.85E+05 ...-4.997E+05 3.069E+04 5.15E-14 1.66E-15 40 88.96 88.01-90.00 2.00 5.15E+05 ...-5.315E+05 3.175E+04 4.66E-14 1.47E-15 41 91.00 90.01-92.00 2.00 5.48E+05 ...-5.643E+05 3.282E+04 5.06E-14 1.53E-15 42 93.21 92.01-94.00 2.00 5.85E+05 ...-5.982E+05 3.387E+04 4.47E-14 1.31E-15 43 95.04 94.01-96.00 2.00 6.16E+05 ...-6.331E+05 3.493E+04 2.30E-14 5.88E-16 44 96.98 96.01-98.00 2.00 6.51E+05 ...-6.691E+05 3.598E+04 4.03E-14 1.13E-15 45 98.97 98.01-100.00 2.00 6.87E+05 ...-7.061E+05 3.702E+04 2.61E-14 8.05E-16 46 101.38 100.01-102.00 2.00 7.32E+05 ...-7.442E+05 3.806E+04 1.29E-14 3.92E-16 47 103.29 102.01-104.00 2.00 7.69E+05 ...-7.832E+05 3.909E+04 1.47E-14 4.35E-16 48 105.43 104.01-106.00 2.00 8.12E+05 ...-8.234E+05 4.011E+04 1.15E-14 3.43E-16 49 107.72 106.01-108.00 2.00 8.59E+05 ...-8.645E+05 4.113E+04 8.43E-15 2.46E-16 50 109.19 108.01-110.00 2.00 8.90E+05 ...-9.066E+05 4.214E+04 6.38E-15 1.81E-16 Table 15 continued 15 Table 15 (continued) 15No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 51 111.18 110.01-112.00 2.00 9.32E+05 ...-9.497E+05 4.313E+04 1.03E-14 2.91E-16 52 113.24 112.01-114.00 2.00 9.77E+05 ...-9.939E+05 4.412E+04 6.23E-15 1.66E-16 53 114.89 114.01-116.00 2.00 1.01E+06 ...-1.039E+06 4.510E+04 6.13E-15 1.64E-16 54 117.10 116.01-118.00 2.00 1.06E+06 ...-1.085E+06 4.607E+04 9.47E-15 2.49E-16 55 119.27 118.01-120.00 2.00 1.12E+06 ...-1.132E+06 4.703E+04 5.15E-15 1.30E-16 56 121.13 120.01-122.00 2.00 1.16E+06 ...-1.180E+06 4.798E+04 6.03E-15 1.50E-16 57 123.00 122.01-124.00 2.00 1.20E+06 ...-1.229E+06 4.892E+04 5.13E-15 1.27E-16 58 125.20 124.01-126.00 2.00 1.26E+06 ...-1.279E+06 4.985E+04 6.66E-15 1.62E-16 59 126.93 126.01-128.00 2.00 1.30E+06 ...-1.330E+06 5.077E+04 6.08E-15 1.45E-16 60 129.19 128.01-130.00 2.00 1.36E+06 ...-1.381E+06 5.168E+04 6.30E-16 1.57E-17 61 130.98 130.01-132.00 2.00 1.41E+06 ...-1.434E+06 5.257E+04 5.18E-18 1.33E-19 62 133.25 132.01-134.17 2.17 1.47E+06 ...-1.492E+06 5.831E+04 1.40E-17 3.55E-19 Table 16 . 16Grains with a = 0.1 µm and 0.01 µm thick icy mantle, shielded by interstellar gas with NH = 6.60 × 10 21 H atoms cm −2 .No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 1 10.46 10.01-12.00 2.00 2.81E+02 10.987-1.310E+03 1.299E+03 2.87E-08 6.52E-09 2 12.76 12.01-14.00 2.00 1.94E+03 ...-3.107E+03 1.797E+03 2.56E-09 7.01E-11 3 14.80 14.01-16.00 2.00 4.02E+03 ...-5.525E+03 2.418E+03 7.12E-10 1.75E-11 4 16.85 16.01-18.00 2.00 6.80E+03 ...-8.654E+03 3.129E+03 2.48E-10 6.52E-12 5 18.88 18.01-20.00 2.00 1.03E+04 ...-1.261E+04 3.960E+03 1.02E-10 3.12E-12 6 20.88 20.01-22.00 2.00 1.47E+04 ...-1.747E+04 4.852E+03 5.01E-11 1.62E-12 7 22.92 22.01-24.00 2.00 2.01E+04 ...-2.327E+04 5.804E+03 2.25E-11 9.60E-13 8 24.87 24.01-26.00 2.00 2.62E+04 ...-3.008E+04 6.807E+03 1.59E-11 6.65E-13 9 27.04 26.01-28.00 2.00 3.41E+04 ...-3.793E+04 7.851E+03 8.38E-12 4.16E-13 10 28.91 28.01-30.00 2.00 4.20E+04 ...-4.686E+04 8.931E+03 5.32E-12 3.05E-13 11 30.90 30.01-32.00 2.00 5.13E+04 ...-5.690E+04 1.004E+04 3.74E-12 2.18E-13 12 32.95 32.01-34.00 2.00 6.22E+04 ...-6.809E+04 1.119E+04 3.13E-12 1.66E-13 13 34.90 34.01-36.00 2.00 7.36E+04 ...-8.044E+04 1.235E+04 2.20E-12 1.15E-13 14 37.00 36.01-38.00 2.00 8.72E+04 ...-9.399E+04 1.355E+04 1.37E-12 7.81E-14 15 38.92 38.01-40.00 2.00 1.01E+05 ...-1.088E+05 1.477E+04 1.18E-12 6.40E-14 16 40.92 40.01-42.00 2.00 1.16E+05 ...-1.248E+05 1.602E+04 8.77E-13 4.55E-14 17 42.87 42.01-44.00 2.00 1.32E+05 ...-1.421E+05 1.737E+04 7.53E-13 3.73E-14 18 44.86 44.01-46.00 2.00 1.50E+05 ...-1.607E+05 1.857E+04 6.08E-13 2.89E-14 19 46.96 46.01-48.00 2.00 1.70E+05 ...-1.806E+05 1.987E+04 4.89E-13 2.53E-14 20 49.00 48.01-50.00 2.00 1.91E+05 ...-2.018E+05 2.119E+04 3.67E-13 1.62E-14 21 50.89 50.01-52.00 2.00 2.12E+05 ...-2.243E+05 2.253E+04 2.99E-13 1.39E-14 22 52.93 52.01-54.00 2.00 2.35E+05 ...-2.482E+05 2.387E+04 2.55E-13 1.21E-14 23 54.91 54.01-56.00 2.00 2.60E+05 ...-2.734E+05 2.523E+04 1.97E-13 8.25E-15 24 56.85 56.01-58.00 2.00 2.85E+05 ...-3.000E+05 2.659E+04 1.51E-13 6.97E-15 25 58.79 58.01-60.00 2.00 3.11E+05 ...-3.280E+05 2.797E+04 1.34E-13 5.82E-15 26 60.96 60.01-62.00 2.00 3.42E+05 ...-3.573E+05 2.935E+04 1.26E-13 5.89E-15 27 63.10 62.01-64.00 2.00 3.74E+05 ...-3.881E+05 3.073E+04 9.60E-14 4.03E-15 28 64.99 64.01-66.00 2.00 4.04E+05 ...-4.202E+05 3.212E+04 5.66E-14 2.43E-15 Table 16 continued 16 Table 16 (continued) 16No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 29 66.97 66.01-68.00 2.00 4.36E+05 ...-4.537E+05 3.350E+04 6.41E-14 2.89E-15 30 69.03 68.01-70.00 2.00 4.72E+05 ...-4.886E+05 3.489E+04 4.46E-14 1.86E-15 31 70.92 70.01-72.00 2.00 5.05E+05 ...-5.247E+05 3.610E+04 3.68E-14 1.55E-15 32 72.86 72.01-74.00 2.00 5.41E+05 ...-5.623E+05 3.766E+04 3.63E-14 1.48E-15 33 74.96 74.01-76.00 2.00 5.81E+05 ...-6.014E+05 3.905E+04 3.47E-14 1.38E-15 34 76.97 76.01-78.00 2.00 6.21E+05 ...-6.418E+05 4.043E+04 2.49E-14 9.98E-16 35 79.12 78.01-80.00 2.00 6.65E+05 ...-6.836E+05 4.180E+04 2.40E-14 9.20E-16 36 81.05 80.01-82.00 2.00 7.06E+05 ...-7.268E+05 4.317E+04 1.87E-14 6.99E-16 37 82.76 82.01-84.00 2.00 7.44E+05 ...-7.713E+05 4.453E+04 1.65E-14 6.21E-16 38 84.99 84.01-86.00 2.00 7.94E+05 ...-8.172E+05 4.589E+04 1.35E-14 5.43E-16 39 87.37 86.01-88.00 2.00 8.50E+05 ...-8.644E+05 4.724E+04 9.23E-15 3.53E-16 40 89.55 88.01-90.00 2.00 9.02E+05 ...-9.130E+05 4.858E+04 5.08E-15 1.96E-16 41 90.93 90.01-92.00 2.00 9.36E+05 ...-9.629E+05 4.991E+04 3.97E-15 1.50E-16 42 92.75 92.01-94.00 2.00 9.82E+05 ...-1.014E+06 5.124E+04 6.00E-15 2.26E-16 43 95.15 94.01-96.00 2.00 1.04E+06 ...-1.067E+06 5.255E+04 5.66E-15 2.10E-16 44 97.13 96.01-98.00 2.00 1.10E+06 ...-1.121E+06 5.385E+04 3.36E-15 1.23E-16 45 98.73 98.01-100.00 2.00 1.14E+06 ...-1.176E+06 5.514E+04 3.33E-15 1.20E-16 46 100.87 100.01-102.00 2.00 1.20E+06 ...-1.232E+06 5.643E+04 4.59E-15 1.65E-16 47 103.01 102.01-104.00 2.00 1.26E+06 ...-1.290E+06 5.770E+04 2.97E-15 1.06E-16 48 105.15 104.01-106.00 2.00 1.32E+06 ...-1.349E+06 5.896E+04 3.56E-15 1.25E-16 49 106.81 106.01-108.00 2.00 1.37E+06 ...-1.409E+06 6.021E+04 3.05E-15 1.07E-16 50 108.79 108.01-110.00 2.00 1.43E+06 ...-1.470E+06 6.144E+04 3.10E-16 1.13E-17 51 111.09 110.01-112.00 2.00 1.50E+06 ...-1.533E+06 6.267E+04 4.29E-18 1.59E-19 52 112.71 112.01-113.20 1.20 1.56E+06 ...-1.572E+06 3.851E+04 4.87E-18 1.79E-19 Table 17 . 17Grains with a = 0.1 µm and 0.020 µm thick icy mantle, shielded by interstellar gas with NH = 1.32 × 10 22 H atoms cm −2 . . T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , LowNo1 10.22 10.01-11.00 1.00 2.61E+02 21.603-1.215E+03 1.193E+03 3.11E-08 6.62E-09 2 11.38 11.01-12.00 1.00 1.74E+03 ...-2.669E+03 1.454E+03 2.32E-09 7.88E-11 3 12.42 12.01-13.00 1.00 3.37E+03 ...-4.408E+03 1.739E+03 6.29E-10 2.02E-11 4 13.43 13.01-14.00 1.00 5.26E+03 ...-6.457E+03 2.050E+03 2.46E-10 8.45E-12 5 14.45 14.01-15.00 1.00 7.50E+03 ...-8.843E+03 2.385E+03 1.17E-10 4.28E-12 6 15.47 15.01-16.00 1.00 1.01E+04 ...-1.159E+04 2.743E+03 5.66E-11 2.38E-12 7 16.49 16.01-17.00 1.00 1.31E+04 ...-1.467E+04 3.088E+03 3.48E-11 1.61E-12 8 17.49 17.01-18.00 1.00 1.64E+04 ...-1.819E+04 3.513E+03 2.04E-11 9.86E-13 9 18.47 18.01-19.00 1.00 2.00E+04 ...-2.211E+04 3.925E+03 1.22E-11 6.65E-13 10 19.48 19.01-20.00 1.00 2.42E+04 ...-2.646E+04 4.350E+03 9.65E-12 5.73E-13 11 20.48 20.01-21.00 1.00 2.88E+04 ...-3.125E+04 4.787E+03 6.28E-12 3.75E-13 12 21.51 21.01-22.00 1.00 3.40E+04 ...-3.648E+04 5.234E+03 4.62E-12 3.14E-13 13 22.51 22.01-23.00 1.00 3.94E+04 ...-4.217E+04 5.690E+03 3.47E-12 2.54E-13 14 23.48 23.01-24.00 1.00 4.52E+04 ...-4.833E+04 6.153E+03 2.95E-12 1.99E-13 15 24.53 24.01-25.00 1.00 5.19E+04 ...-5.495E+04 6.622E+03 2.08E-12 1.59E-13 16 25.53 25.01-26.00 1.00 5.87E+04 ...-6.205E+04 7.097E+03 1.65E-12 1.17E-13 Table 17 continued 17 Table 17 (continued) 17No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low17 26.50 26.01-27.00 1.00 6.58E+04 ...-6.962E+04 7.575E+03 1.34E-12 9.48E-14 18 27.45 27.01-28.00 1.00 7.33E+04 ...-7.768E+04 8.057E+03 1.24E-12 8.01E-14 19 28.48 28.01-29.00 1.00 8.19E+04 ...-8.622E+04 8.542E+03 9.02E-13 6.72E-14 20 29.52 29.01-30.00 1.00 9.10E+04 ...-9.525E+04 9.030E+03 7.09E-13 5.07E-14 21 30.49 30.01-31.00 1.00 1.00E+05 ...-1.048E+05 9.519E+03 6.12E-13 4.35E-14 22 31.46 31.01-32.00 1.00 1.09E+05 ...-1.148E+05 1.001E+04 4.75E-13 3.32E-14 23 32.45 32.01-33.00 1.00 1.20E+05 ...-1.253E+05 1.050E+04 4.56E-13 2.98E-14 24 33.50 33.01-34.00 1.00 1.31E+05 ...-1.363E+05 1.100E+04 3.99E-13 2.63E-14 25 34.48 34.01-35.00 1.00 1.42E+05 ...-1.478E+05 1.150E+04 2.85E-13 1.81E-14 26 35.46 35.01-36.00 1.00 1.53E+05 ...-1.598E+05 1.199E+04 3.10E-13 2.12E-14 27 36.48 36.01-37.00 1.00 1.66E+05 ...-1.723E+05 1.249E+04 2.43E-13 1.51E-14 28 37.48 37.01-38.00 1.00 1.79E+05 ...-1.853E+05 1.299E+04 1.98E-13 1.08E-14 29 38.41 38.01-39.00 1.00 1.91E+05 ...-1.987E+05 1.349E+04 1.84E-13 1.22E-14 30 39.43 39.01-40.00 1.00 2.05E+05 ...-2.127E+05 1.399E+04 1.44E-13 9.44E-15 31 40.49 40.01-41.00 1.00 2.20E+05 ...-2.272E+05 1.449E+04 1.65E-13 1.01E-14 32 41.57 41.01-42.00 1.00 2.36E+05 ...-2.422E+05 1.500E+04 1.24E-13 7.50E-15 33 42.55 42.01-43.00 1.00 2.51E+05 ...-2.577E+05 1.550E+04 9.42E-14 5.69E-15 34 43.50 43.01-44.00 1.00 2.66E+05 ...-2.739E+05 1.616E+04 9.42E-14 5.63E-15 35 44.38 44.01-45.00 1.00 2.80E+05 ...-2.904E+05 1.651E+04 6.33E-14 3.39E-15 36 45.35 45.01-46.00 1.00 2.96E+05 ...-3.074E+05 1.701E+04 7.91E-14 4.72E-15 37 46.46 46.01-47.00 1.00 3.15E+05 ...-3.249E+05 1.751E+04 6.48E-14 4.05E-15 38 47.50 47.01-48.00 1.00 3.34E+05 ...-3.429E+05 1.801E+04 4.53E-14 2.33E-15 39 48.46 48.01-49.00 1.00 3.51E+05 ...-3.614E+05 1.851E+04 5.70E-14 3.46E-15 40 49.54 49.01-50.00 1.00 3.72E+05 ...-3.805E+05 1.901E+04 3.67E-14 2.10E-15 41 50.46 50.01-51.00 1.00 3.89E+05 ...-4.000E+05 1.951E+04 4.24E-14 2.24E-15 42 51.36 51.01-52.00 1.00 4.07E+05 ...-4.200E+05 2.001E+04 2.18E-14 1.32E-15 43 52.33 52.01-53.00 1.00 4.27E+05 ...-4.405E+05 2.051E+04 3.14E-14 1.80E-15 44 53.29 53.01-54.00 1.00 4.47E+05 ...-4.615E+05 2.101E+04 2.33E-14 1.33E-15 45 54.31 54.01-55.00 1.00 4.68E+05 ...-4.830E+05 2.150E+04 2.32E-14 1.32E-15 46 55.44 55.01-56.00 1.00 4.93E+05 ...-5.050E+05 2.200E+04 2.32E-14 1.30E-15 47 56.63 56.01-57.00 1.00 5.19E+05 ...-5.275E+05 2.249E+04 1.86E-14 1.05E-15 48 57.47 57.01-58.00 1.00 5.38E+05 ...-5.505E+05 2.298E+04 1.18E-14 5.94E-16 49 58.56 58.01-59.00 1.00 5.64E+05 ...-5.739E+05 2.347E+04 1.66E-14 9.26E-16 50 59.52 59.01-60.00 1.00 5.86E+05 ...-5.979E+05 2.396E+04 7.17E-15 3.59E-16 51 60.31 60.01-61.00 1.00 6.05E+05 ...-6.223E+05 2.444E+04 1.49E-14 8.14E-16 52 61.47 61.01-62.00 1.00 6.34E+05 ...-6.473E+05 2.493E+04 8.38E-15 4.09E-16 53 62.31 62.01-63.00 1.00 6.55E+05 ...-6.727E+05 2.541E+04 1.22E-14 6.73E-16 54 63.50 63.01-64.00 1.00 6.86E+05 ...-6.986E+05 2.589E+04 5.11E-15 2.52E-16 55 64.38 64.01-65.00 1.00 7.09E+05 ...-7.249E+05 2.637E+04 1.00E-14 5.36E-16 56 65.52 65.01-66.00 1.00 7.39E+05 ...-7.518E+05 2.685E+04 7.51E-15 3.87E-16 57 66.59 66.01-67.00 1.00 7.68E+05 ...-7.791E+05 2.732E+04 7.53E-15 3.89E-16 58 67.46 67.01-68.00 1.00 7.92E+05 ...-8.069E+05 2.779E+04 2.87E-15 1.43E-16 59 68.35 68.01-69.00 1.00 8.17E+05 ...-8.352E+05 2.826E+04 6.28E-15 3.29E-16 60 69.58 69.01-70.00 1.00 8.52E+05 ...-8.639E+05 2.873E+04 6.06E-15 3.14E-16 61 70.43 70.01-71.00 1.00 8.76E+05 ...-8.931E+05 2.920E+04 2.94E-15 1.45E-16 62 71.24 71.01-72.00 1.00 9.00E+05 ...-9.224E+05 2.936E+04 3.94E-15 2.07E-16 63 72.77 72.01-73.00 1.00 9.46E+05 ...-9.526E+05 3.012E+04 3.03E-15 1.57E-16 Table 17 continued 17 Table 17 (continued) 17No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low64 73.74 73.01-74.00 1.00 9.76E+05 ...-9.831E+05 3.058E+04 2.16E-15 1.10E-16 65 74.67 74.01-75.00 1.00 1.00E+06 ...-1.014E+06 3.103E+04 1.95E-15 1.01E-16 66 75.79 75.01-76.00 1.00 1.04E+06 ...-1.046E+06 3.149E+04 1.83E-15 9.37E-17 67 76.87 76.01-77.00 1.00 1.07E+06 ...-1.078E+06 3.194E+04 1.38E-15 7.05E-17 68 77.36 77.01-78.00 1.00 1.09E+06 ...-1.110E+06 3.239E+04 5.64E-16 2.79E-17 69 78.18 78.01-79.00 1.00 1.12E+06 ...-1.143E+06 3.284E+04 1.52E-15 7.75E-17 70 79.32 79.01-80.00 1.00 1.15E+06 ...-1.176E+06 3.329E+04 1.46E-15 7.40E-17 71 80.41 80.01-81.00 1.00 1.19E+06 ...-1.210E+06 3.373E+04 1.47E-15 7.40E-17 72 81.60 81.01-82.00 1.00 1.23E+06 ...-1.244E+06 3.417E+04 1.21E-15 6.07E-17 73 82.67 82.01-83.00 1.00 1.27E+06 ...-1.279E+06 3.461E+04 9.42E-16 4.71E-17 74 83.50 83.01-84.00 1.00 1.30E+06 ...-1.314E+06 3.505E+04 1.09E-15 5.43E-17 75 84.58 84.01-85.00 1.00 1.33E+06 ...-1.349E+06 3.548E+04 8.92E-16 4.47E-17 76 85.67 85.01-86.00 1.00 1.37E+06 ...-1.385E+06 3.592E+04 9.17E-16 4.58E-17 77 86.53 86.01-87.00 1.00 1.40E+06 ...-1.421E+06 3.635E+04 8.96E-16 4.47E-17 78 87.54 87.01-88.00 1.00 1.44E+06 ...-1.458E+06 3.678E+04 1.47E-15 7.30E-17 79 88.55 88.01-89.00 1.00 1.48E+06 ...-1.495E+06 3.720E+04 8.60E-17 4.48E-18 80 89.49 89.01-90.00 1.00 1.51E+06 ...-1.533E+06 3.763E+04 1.00E-16 5.20E-18 81 90.55 90.01-91.00 1.00 1.55E+06 ...-1.571E+06 3.805E+04 9.77E-19 5.13E-20 82 91.61 91.01-92.00 1.00 1.60E+06 ...-1.610E+06 3.847E+04 1.47E-18 7.69E-20 83 92.63 92.01-92.99 0.99 1.63E+06 ...-1.648E+06 3.889E+04 2.28E-18 1.19E-19 Table 18 . 18Grains with a = 0.1 µm and 0.025 µm thick icy mantle, shielded by interstellar gas with NH = 1.98 × 10 22 H atoms cm −2 .No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 1 10.17 10.01-11.00 1.00 3.17E+02 33.247-1.878E+03 1.845E+03 3.23E-08 6.48E-09 2 11.36 11.01-12.00 1.00 2.65E+03 ...-4.136E+03 2.258E+03 1.22E-09 4.58E-11 3 12.41 12.01-13.00 1.00 5.20E+03 ...-6.834E+03 2.699E+03 2.85E-10 1.14E-11 4 13.42 13.01-14.00 1.00 8.14E+03 ...-1.001E+04 3.173E+03 9.92E-11 4.70E-12 5 14.44 14.01-15.00 1.00 1.16E+04 ...-1.368E+04 3.677E+03 4.68E-11 2.46E-12 6 15.46 15.01-16.00 1.00 1.56E+04 ...-1.789E+04 4.209E+03 2.42E-11 1.38E-12 7 16.44 16.01-17.00 1.00 2.00E+04 ...-2.261E+04 4.715E+03 1.37E-11 8.62E-13 8 17.47 17.01-18.00 1.00 2.51E+04 ...-2.795E+04 5.338E+03 8.95E-12 6.30E-13 9 18.44 18.01-19.00 1.00 3.05E+04 ...-3.388E+04 5.936E+03 5.58E-12 4.33E-13 10 19.47 19.01-20.00 1.00 3.69E+04 ...-4.043E+04 6.549E+03 4.43E-12 3.45E-13 11 20.49 20.01-21.00 1.00 4.40E+04 ...-4.761E+04 7.177E+03 3.04E-12 2.54E-13 12 21.46 21.01-22.00 1.00 5.12E+04 ...-5.543E+04 7.818E+03 2.37E-12 1.99E-13 13 22.49 22.01-23.00 1.00 5.96E+04 ...-6.389E+04 8.468E+03 1.70E-12 1.39E-13 14 23.52 23.01-24.00 1.00 6.87E+04 ...-7.302E+04 9.126E+03 1.43E-12 1.16E-13 15 24.47 24.01-25.00 1.00 7.76E+04 ...-8.281E+04 9.792E+03 9.52E-13 7.67E-14 16 25.49 25.01-26.00 1.00 8.80E+04 ...-9.328E+04 1.046E+04 8.39E-13 7.09E-14 17 26.48 26.01-27.00 1.00 9.87E+04 ...-1.044E+05 1.114E+04 6.16E-13 5.02E-14 18 27.48 27.01-28.00 1.00 1.10E+05 ...-1.162E+05 1.182E+04 5.27E-13 4.27E-14 19 28.47 28.01-29.00 1.00 1.22E+05 ...-1.287E+05 1.250E+04 4.09E-13 3.17E-14 20 29.45 29.01-30.00 1.00 1.35E+05 ...-1.419E+05 1.318E+04 3.53E-13 2.76E-14 Table 18 continued 18 Table 18 (continued) 18No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low21 30.49 30.01-31.00 1.00 1.49E+05 ...-1.558E+05 1.387E+04 3.39E-13 2.65E-14 22 31.50 31.01-32.00 1.00 1.63E+05 ...-1.703E+05 1.455E+04 2.14E-13 1.56E-14 23 32.43 32.01-33.00 1.00 1.77E+05 ...-1.856E+05 1.524E+04 2.18E-13 1.56E-14 24 33.41 33.01-34.00 1.00 1.92E+05 ...-2.015E+05 1.592E+04 1.76E-13 1.30E-14 25 34.35 34.01-35.00 1.00 2.07E+05 ...-2.181E+05 1.661E+04 1.38E-13 1.02E-14 26 35.39 35.01-36.00 1.00 2.25E+05 ...-2.354E+05 1.729E+04 1.51E-13 1.09E-14 27 36.49 36.01-37.00 1.00 2.44E+05 ...-2.534E+05 1.798E+04 1.09E-13 7.82E-15 28 37.45 37.01-38.00 1.00 2.62E+05 ...-2.720E+05 1.866E+04 9.64E-14 6.31E-15 29 38.48 38.01-39.00 1.00 2.82E+05 ...-2.914E+05 1.934E+04 8.99E-14 6.24E-15 30 39.49 39.01-40.00 1.00 3.01E+05 ...-3.114E+05 2.002E+04 5.80E-14 4.20E-15 31 40.45 40.01-41.00 1.00 3.21E+05 ...-3.321E+05 2.070E+04 6.05E-14 4.21E-15 32 41.57 41.01-42.00 1.00 3.44E+05 ...-3.535E+05 2.138E+04 5.30E-14 3.64E-15 33 42.47 42.01-43.00 1.00 3.64E+05 ...-3.755E+05 2.205E+04 3.33E-14 2.03E-15 34 43.50 43.01-44.00 1.00 3.87E+05 ...-3.985E+05 2.295E+04 4.47E-14 3.11E-15 35 44.62 44.01-45.00 1.00 4.13E+05 ...-4.219E+05 2.340E+04 3.51E-14 2.23E-15 36 45.51 45.01-46.00 1.00 4.34E+05 ...-4.459E+05 2.406E+04 1.77E-14 1.18E-15 37 46.48 46.01-47.00 1.00 4.58E+05 ...-4.707E+05 2.472E+04 2.44E-14 1.63E-15 38 47.43 47.01-48.00 1.00 4.82E+05 ...-4.960E+05 2.538E+04 1.84E-14 1.21E-15 39 48.39 48.01-49.00 1.00 5.06E+05 ...-5.221E+05 2.604E+04 1.83E-14 1.19E-15 40 49.48 49.01-50.00 1.00 5.35E+05 ...-5.488E+05 2.669E+04 1.59E-14 1.04E-15 41 50.59 50.01-51.00 1.00 5.65E+05 ...-5.761E+05 2.734E+04 1.52E-14 9.87E-16 42 51.41 51.01-52.00 1.00 5.87E+05 ...-6.041E+05 2.798E+04 7.94E-15 4.78E-16 43 52.46 52.01-53.00 1.00 6.17E+05 ...-6.327E+05 2.862E+04 1.27E-14 8.17E-16 44 53.49 53.01-54.00 1.00 6.47E+05 ...-6.620E+05 2.926E+04 7.48E-15 4.43E-16 45 54.31 54.01-55.00 1.00 6.71E+05 ...-6.919E+05 2.989E+04 9.62E-15 6.25E-16 46 55.47 55.01-56.00 1.00 7.06E+05 ...-7.224E+05 3.051E+04 4.62E-15 2.77E-16 47 56.30 56.01-57.00 1.00 7.32E+05 ...-7.535E+05 3.114E+04 8.55E-15 5.40E-16 48 57.42 57.01-58.00 1.00 7.67E+05 ...-7.853E+05 3.175E+04 6.69E-15 4.13E-16 49 58.39 58.01-59.00 1.00 7.98E+05 ...-8.176E+05 3.237E+04 4.83E-15 3.03E-16 50 59.47 59.01-60.00 1.00 8.33E+05 ...-8.506E+05 3.298E+04 4.88E-15 3.07E-16 51 60.58 60.01-61.00 1.00 8.70E+05 ...-8.842E+05 3.358E+04 4.77E-15 2.97E-16 52 61.56 61.01-62.00 1.00 9.03E+05 ...-9.184E+05 3.418E+04 2.61E-15 1.58E-16 53 62.30 62.01-63.00 1.00 9.29E+05 ...-9.531E+05 3.477E+04 4.30E-15 2.68E-16 54 63.45 63.01-64.00 1.00 9.69E+05 ...-9.885E+05 3.536E+04 3.16E-15 1.96E-16 55 64.48 64.01-65.00 1.00 1.01E+06 ...-1.024E+06 3.595E+04 1.83E-15 1.14E-16 56 65.36 65.01-66.00 1.00 1.04E+06 ...-1.061E+06 3.653E+04 1.49E-15 9.18E-17 57 66.21 66.01-67.00 1.00 1.07E+06 ...-1.098E+06 3.710E+04 1.48E-15 9.16E-17 58 67.24 67.01-68.00 1.00 1.11E+06 ...-1.136E+06 3.767E+04 1.30E-15 7.98E-17 59 68.22 68.01-69.00 1.00 1.14E+06 ...-1.174E+06 3.824E+04 1.16E-15 7.10E-17 60 69.26 69.01-70.00 1.00 1.18E+06 ...-1.213E+06 3.880E+04 1.11E-15 6.79E-17 61 70.27 70.01-71.00 1.00 1.22E+06 ...-1.252E+06 3.936E+04 1.12E-15 6.85E-17 62 71.34 71.01-72.00 1.00 1.27E+06 ...-1.292E+06 3.951E+04 8.66E-16 5.30E-17 63 72.45 72.01-73.00 1.00 1.31E+06 ...-1.332E+06 4.045E+04 1.02E-15 6.20E-17 64 73.44 73.01-74.00 1.00 1.35E+06 ...-1.373E+06 4.099E+04 6.98E-16 4.27E-17 65 74.43 74.01-75.00 1.00 1.39E+06 ...-1.415E+06 4.153E+04 6.93E-16 4.22E-17 66 75.44 75.01-76.00 1.00 1.43E+06 ...-1.457E+06 4.207E+04 8.83E-16 5.38E-17 67 76.48 76.01-77.00 1.00 1.48E+06 ...-1.499E+06 4.260E+04 1.07E-15 6.51E-17 Table 18 continued 18 Table 18 (continued) 18No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 68 77.62 77.01-78.00 1.00 1.53E+06 ...-1.542E+06 4.312E+04 6.58E-17 4.18E-18 69 78.30 78.01-79.00 1.00 1.56E+06 ...-1.586E+06 4.365E+04 5.23E-17 3.33E-18 70 79.41 79.01-80.00 1.00 1.60E+06 ...-1.630E+06 4.416E+04 1.07E-18 6.89E-20 71 80.55 80.01-81.00 1.00 1.66E+06 ...-1.675E+06 4.468E+04 1.38E-18 8.82E-20 72 81.14 81.01-81.30 0.30 1.68E+06 ...-1.689E+06 1.395E+04 8.83E-19 5.67E-20 Table 19 (continued) 19No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 36 45.67 45.01-46.00 1.00 5.82E+05 ...-5.924E+05 3.155E+04 8.29E-15 7.58E-16 37 46.56 46.01-47.00 1.00 6.10E+05 ...-6.248E+05 3.238E+04 4.95E-15 4.29E-16 38 47.42 47.01-48.00 1.00 6.39E+05 ...-6.580E+05 3.321E+04 6.95E-15 6.33E-16 39 48.63 48.01-49.00 1.00 6.79E+05 ...-6.920E+05 3.403E+04 6.06E-15 5.48E-16 40 49.56 49.01-50.00 1.00 7.11E+05 ...-7.269E+05 3.484E+04 3.09E-15 2.75E-16 41 50.47 50.01-51.00 1.00 7.44E+05 ...-7.625E+05 3.565E+04 4.13E-15 3.75E-16 42 51.53 51.01-52.00 1.00 7.82E+05 ...-7.990E+05 3.645E+04 4.05E-15 3.62E-16 43 52.46 52.01-53.00 1.00 8.16E+05 ...-8.362E+05 3.724E+04 2.78E-15 2.52E-16 44 53.46 53.01-54.00 1.00 8.54E+05 ...-8.743E+05 3.803E+04 2.52E-15 2.30E-16 45 54.55 54.01-55.00 1.00 8.95E+05 ...-9.131E+05 3.881E+04 2.84E-15 2.58E-16 46 55.65 55.01-56.00 1.00 9.39E+05 ...-9.526E+05 3.958E+04 2.15E-15 1.95E-16 47 56.51 56.01-57.00 1.00 9.73E+05 ...-9.930E+05 4.034E+04 1.17E-15 1.06E-16 48 57.46 57.01-58.00 1.00 1.01E+06 ...-1.034E+06 4.109E+04 2.31E-15 2.10E-16 49 58.46 58.01-59.00 1.00 1.05E+06 ...-1.076E+06 4.184E+04 1.54E-15 1.39E-16 50 59.46 59.01-60.00 1.00 1.10E+06 ...-1.118E+06 4.258E+04 8.46E-16 7.69E-17 51 60.41 60.01-61.00 1.00 1.14E+06 ...-1.162E+06 4.331E+04 8.07E-16 7.29E-17 52 61.26 61.01-62.00 1.00 1.17E+06 ...-1.206E+06 4.403E+04 5.91E-16 5.35E-17 53 62.51 62.01-63.00 1.00 1.23E+06 ...-1.251E+06 4.475E+04 9.08E-16 8.25E-17 54 63.80 63.01-64.00 1.00 1.29E+06 ...-1.296E+06 4.545E+04 4.73E-16 4.25E-17 55 64.72 64.01-65.00 1.00 1.33E+06 ...-1.342E+06 4.615E+04 3.24E-16 2.92E-17 56 65.42 65.01-66.00 1.00 1.36E+06 ...-1.389E+06 4.684E+04 3.81E-16 3.44E-17 57 66.50 66.01-67.00 1.00 1.41E+06 ...-1.437E+06 4.753E+04 4.36E-16 3.95E-17 58 67.59 67.01-68.00 1.00 1.46E+06 ...-1.485E+06 4.820E+04 4.23E-16 3.83E-17 59 68.58 68.01-69.00 1.00 1.51E+06 ...-1.534E+06 4.887E+04 5.25E-16 4.75E-17 60 69.52 69.01-70.00 1.00 1.56E+06 ...-1.583E+06 4.953E+04 3.75E-17 3.55E-18 61 70.25 70.01-71.00 1.00 1.60E+06 ...-1.633E+06 5.019E+04 2.41E-17 2.29E-18 62 71.59 71.01-72.00 1.00 1.66E+06 ...-1.684E+06 5.033E+04 4.27E-19 4.07E-20 63 72.54 72.01-72.93 0.93 1.71E+06 ...-1.732E+06 4.837E+04 9.02E-19 8.61E-20 Table 20 . 20Grains with a = 0.1 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 8.80 × 10 22 H atoms cm −2 .No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 1 10.11 10.01-11.00 1.00 2.99E+02 45.014-2.553E+03 2.508E+03 2.72E-08 5.82E-09 2 11.36 11.01-12.00 1.00 3.60E+03 ...-5.636E+03 3.083E+03 3.35E-10 2.47E-11 3 12.40 12.01-13.00 1.00 7.06E+03 ...-9.329E+03 3.693E+03 7.12E-11 6.39E-12 4 13.44 13.01-14.00 1.00 1.12E+04 ...-1.367E+04 4.342E+03 2.58E-11 2.78E-12 5 14.46 14.01-15.00 1.00 1.59E+04 ...-1.870E+04 5.027E+03 1.21E-11 1.34E-12 6 15.45 15.01-16.00 1.00 2.12E+04 ...-2.444E+04 5.745E+03 7.00E-12 8.09E-13 7 16.47 16.01-17.00 1.00 2.74E+04 ...-3.087E+04 6.424E+03 4.12E-12 5.40E-13 8 17.48 17.01-18.00 1.00 3.43E+04 ...-3.812E+04 7.257E+03 2.77E-12 3.83E-13 9 18.48 18.01-19.00 1.00 4.20E+04 ...-4.618E+04 8.052E+03 1.78E-12 2.73E-13 10 19.46 19.01-20.00 1.00 5.03E+04 ...-5.504E+04 8.866E+03 1.41E-12 2.04E-13 11 20.46 20.01-21.00 1.00 5.96E+04 ...-6.474E+04 9.697E+03 9.17E-13 1.37E-13 12 21.47 21.01-22.00 1.00 6.98E+04 ...-7.528E+04 1.054E+04 7.29E-13 1.02E-13 Table 20 continued 20 Table 20 (continued) 20No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 13 22.49 22.01-23.00 1.00 8.09E+04 ...-8.668E+04 1.140E+04 5.19E-13 7.67E-14 14 23.50 23.01-24.00 1.00 9.29E+04 ...-9.895E+04 1.226E+04 3.52E-13 4.95E-14 15 24.43 24.01-25.00 1.00 1.05E+05 ...-1.121E+05 1.314E+04 2.80E-13 3.79E-14 16 25.45 25.01-26.00 1.00 1.19E+05 ...-1.261E+05 1.402E+04 2.52E-13 3.65E-14 17 26.49 26.01-27.00 1.00 1.33E+05 ...-1.410E+05 1.490E+04 1.77E-13 2.23E-14 18 27.50 27.01-28.00 1.00 1.49E+05 ...-1.568E+05 1.579E+04 1.45E-13 1.97E-14 19 28.47 28.01-29.00 1.00 1.65E+05 ...-1.735E+05 1.668E+04 1.16E-13 1.46E-14 20 29.44 29.01-30.00 1.00 1.81E+05 ...-1.910E+05 1.757E+04 8.50E-14 1.08E-14 21 30.39 30.01-31.00 1.00 1.98E+05 ...-2.095E+05 1.846E+04 8.35E-14 1.06E-14 22 31.50 31.01-32.00 1.00 2.19E+05 ...-2.289E+05 1.935E+04 6.94E-14 8.85E-15 23 32.54 32.01-33.00 1.00 2.40E+05 ...-2.491E+05 2.025E+04 4.62E-14 5.63E-15 24 33.52 33.01-34.00 1.00 2.60E+05 ...-2.702E+05 2.113E+04 4.33E-14 5.45E-15 25 34.53 34.01-35.00 1.00 2.82E+05 ...-2.923E+05 2.202E+04 3.60E-14 4.28E-15 26 35.55 35.01-36.00 1.00 3.05E+05 ...-3.152E+05 2.291E+04 2.79E-14 3.44E-15 27 36.55 36.01-37.00 1.00 3.29E+05 ...-3.390E+05 2.379E+04 2.24E-14 2.76E-15 28 37.60 37.01-38.00 1.00 3.54E+05 ...-3.636E+05 2.467E+04 1.79E-14 2.20E-15 29 38.47 38.01-39.00 1.00 3.76E+05 ...-3.892E+05 2.554E+04 1.17E-14 1.37E-15 30 39.38 39.01-40.00 1.00 3.99E+05 ...-4.156E+05 2.641E+04 1.46E-14 1.80E-15 31 40.42 40.01-41.00 1.00 4.27E+05 ...-4.429E+05 2.728E+04 1.02E-14 1.25E-15 32 41.51 41.01-42.00 1.00 4.58E+05 ...-4.710E+05 2.814E+04 1.05E-14 1.30E-15 33 42.52 42.01-43.00 1.00 4.86E+05 ...-5.000E+05 2.900E+04 5.24E-15 6.37E-16 34 43.51 43.01-44.00 1.00 5.15E+05 ...-5.302E+05 3.016E+04 7.66E-15 9.44E-16 35 44.66 44.01-45.00 1.00 5.50E+05 ...-5.609E+05 3.071E+04 5.01E-15 6.15E-16 36 45.68 45.01-46.00 1.00 5.82E+05 ...-5.924E+05 3.155E+04 4.38E-15 5.44E-16 37 46.56 46.01-47.00 1.00 6.10E+05 ...-6.248E+05 3.238E+04 2.56E-15 3.04E-16 38 47.41 47.01-48.00 1.00 6.39E+05 ...-6.580E+05 3.321E+04 3.64E-15 4.52E-16 39 48.63 48.01-49.00 1.00 6.79E+05 ...-6.920E+05 3.403E+04 3.16E-15 3.91E-16 40 49.56 49.01-50.00 1.00 7.11E+05 ...-7.269E+05 3.484E+04 1.59E-15 1.94E-16 41 50.47 50.01-51.00 1.00 7.44E+05 ...-7.625E+05 3.565E+04 2.14E-15 2.67E-16 42 51.53 51.01-52.00 1.00 7.82E+05 ...-7.990E+05 3.645E+04 2.08E-15 2.56E-16 43 52.46 52.01-53.00 1.00 8.16E+05 ...-8.362E+05 3.724E+04 1.43E-15 1.78E-16 44 53.46 53.01-54.00 1.00 8.54E+05 ...-8.743E+05 3.803E+04 1.30E-15 1.63E-16 45 54.55 54.01-55.00 1.00 8.96E+05 ...-9.131E+05 3.881E+04 1.45E-15 1.82E-16 46 55.65 55.01-56.00 1.00 9.39E+05 ...-9.526E+05 3.958E+04 1.10E-15 1.37E-16 47 56.51 56.01-57.00 1.00 9.73E+05 ...-9.930E+05 4.034E+04 5.92E-16 7.41E-17 48 57.46 57.01-58.00 1.00 1.01E+06 ...-1.034E+06 4.109E+04 1.17E-15 1.48E-16 49 58.47 58.01-59.00 1.00 1.05E+06 ...-1.076E+06 4.184E+04 7.78E-16 9.78E-17 50 59.46 59.01-60.00 1.00 1.10E+06 ...-1.118E+06 4.258E+04 4.30E-16 5.41E-17 51 60.41 60.01-61.00 1.00 1.14E+06 ...-1.162E+06 4.331E+04 4.10E-16 5.13E-17 52 61.25 61.01-62.00 1.00 1.17E+06 ...-1.206E+06 4.403E+04 3.00E-16 3.76E-17 53 62.51 62.01-63.00 1.00 1.23E+06 ...-1.251E+06 4.475E+04 4.61E-16 5.80E-17 54 63.80 63.01-64.00 1.00 1.29E+06 ...-1.296E+06 4.545E+04 2.39E-16 2.99E-17 55 64.72 64.01-65.00 1.00 1.33E+06 ...-1.342E+06 4.615E+04 1.64E-16 2.05E-17 56 65.42 65.01-66.00 1.00 1.36E+06 ...-1.389E+06 4.684E+04 1.93E-16 2.41E-17 57 66.50 66.01-67.00 1.00 1.41E+06 ...-1.437E+06 4.753E+04 2.20E-16 2.77E-17 58 67.59 67.01-68.00 1.00 1.46E+06 ...-1.485E+06 4.820E+04 2.13E-16 2.69E-17 59 68.58 68.01-69.00 1.00 1.51E+06 ...-1.534E+06 4.887E+04 2.64E-16 3.33E-17 Table 20 continued 20 Table 20 (continued) 20No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 60 69.52 69.01-70.00 1.00 1.56E+06 ...-1.583E+06 4.953E+04 1.89E-17 2.48E-18 61 70.25 70.01-71.00 1.00 1.60E+06 ...-1.633E+06 5.019E+04 1.21E-17 1.60E-18 62 71.59 71.01-72.00 1.00 1.66E+06 ...-1.684E+06 5.033E+04 2.14E-19 2.84E-20 63 72.54 72.01-72.93 0.93 1.71E+06 ...-1.732E+06 4.837E+04 4.53E-19 6.02E-20 Table 21 continued 21 Table 21 (continued) 21No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 37 46.56 46.01-47.00 1.00 6.10E+05 ...-6.248E+05 3.238E+04 1.28E-15 2.08E-16 38 47.41 47.01-48.00 1.00 6.38E+05 ...-6.580E+05 3.321E+04 1.83E-15 3.09E-16 39 48.63 48.01-49.00 1.00 6.79E+05 ...-6.920E+05 3.403E+04 1.59E-15 2.66E-16 40 49.56 49.01-50.00 1.00 7.11E+05 ...-7.269E+05 3.484E+04 7.89E-16 1.32E-16 41 50.47 50.01-51.00 1.00 7.44E+05 ...-7.625E+05 3.565E+04 1.07E-15 1.81E-16 42 51.53 51.01-52.00 1.00 7.82E+05 ...-7.990E+05 3.645E+04 1.04E-15 1.74E-16 43 52.46 52.01-53.00 1.00 8.16E+05 ...-8.362E+05 3.724E+04 7.11E-16 1.21E-16 44 53.46 53.01-54.00 1.00 8.54E+05 ...-8.743E+05 3.803E+04 6.43E-16 1.10E-16 45 54.55 54.01-55.00 1.00 8.96E+05 ...-9.131E+05 3.881E+04 7.20E-16 1.23E-16 46 55.65 55.01-56.00 1.00 9.39E+05 ...-9.526E+05 3.958E+04 5.42E-16 9.28E-17 47 56.51 56.01-57.00 1.00 9.73E+05 ...-9.930E+05 4.034E+04 2.92E-16 5.00E-17 48 57.46 57.01-58.00 1.00 1.01E+06 ...-1.034E+06 4.109E+04 5.79E-16 9.96E-17 49 58.47 58.01-59.00 1.00 1.05E+06 ...-1.076E+06 4.184E+04 3.84E-16 6.59E-17 50 59.46 59.01-60.00 1.00 1.10E+06 ...-1.118E+06 4.258E+04 2.12E-16 3.65E-17 51 60.41 60.01-61.00 1.00 1.14E+06 ...-1.162E+06 4.331E+04 2.02E-16 3.46E-17 52 61.25 61.01-62.00 1.00 1.17E+06 ...-1.206E+06 4.403E+04 1.48E-16 2.54E-17 53 62.51 62.01-63.00 1.00 1.23E+06 ...-1.251E+06 4.475E+04 2.27E-16 3.91E-17 54 63.80 63.01-64.00 1.00 1.29E+06 ...-1.296E+06 4.545E+04 1.18E-16 2.02E-17 55 64.72 64.01-65.00 1.00 1.33E+06 ...-1.342E+06 4.615E+04 8.07E-17 1.38E-17 56 65.42 65.01-66.00 1.00 1.36E+06 ...-1.389E+06 4.684E+04 9.48E-17 1.63E-17 57 66.50 66.01-67.00 1.00 1.41E+06 ...-1.437E+06 4.753E+04 1.08E-16 1.86E-17 58 67.59 67.01-68.00 1.00 1.46E+06 ...-1.485E+06 4.820E+04 1.05E-16 1.81E-17 59 68.58 68.01-69.00 1.00 1.51E+06 ...-1.534E+06 4.887E+04 1.30E-16 2.24E-17 60 69.52 69.01-70.00 1.00 1.56E+06 ...-1.583E+06 4.953E+04 9.23E-18 1.66E-18 61 70.25 70.01-71.00 1.00 1.60E+06 ...-1.633E+06 5.019E+04 5.92E-18 1.07E-18 62 71.59 71.01-72.00 1.00 1.66E+06 ...-1.684E+06 5.033E+04 1.05E-19 1.91E-20 63 72.54 72.01-72.93 0.93 1.71E+06 ...-1.732E+06 4.837E+04 2.22E-19 4.04E-20 Table 22 . 22Grains with a = 0.1 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 3.52 × 10 23 H atoms cm −2 .No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 1 10.09 10.01-11.00 1.00 2.33E+02 45.014-2.553E+03 2.508E+03 2.09E-08 5.48E-09 2 11.35 11.01-12.00 1.00 3.60E+03 ...-5.636E+03 3.083E+03 1.28E-10 1.80E-11 3 12.41 12.01-13.00 1.00 7.10E+03 ...-9.329E+03 3.693E+03 2.79E-11 4.80E-12 4 13.44 13.01-14.00 1.00 1.12E+04 ...-1.367E+04 4.342E+03 1.09E-11 2.10E-12 5 14.45 14.01-15.00 1.00 1.59E+04 ...-1.870E+04 5.027E+03 5.11E-12 9.82E-13 6 15.44 15.01-16.00 1.00 2.11E+04 ...-2.444E+04 5.745E+03 2.96E-12 5.67E-13 7 16.47 16.01-17.00 1.00 2.74E+04 ...-3.087E+04 6.424E+03 1.72E-12 3.87E-13 8 17.47 17.01-18.00 1.00 3.43E+04 ...-3.812E+04 7.257E+03 1.16E-12 2.72E-13 9 18.48 18.01-19.00 1.00 4.20E+04 ...-4.618E+04 8.052E+03 7.40E-13 1.98E-13 10 19.45 19.01-20.00 1.00 5.02E+04 ...-5.504E+04 8.866E+03 5.69E-13 1.42E-13 11 20.46 20.01-21.00 1.00 5.96E+04 ...-6.474E+04 9.697E+03 3.74E-13 9.42E-14 12 21.48 21.01-22.00 1.00 6.98E+04 ...-7.528E+04 1.054E+04 2.80E-13 6.65E-14 13 22.50 22.01-23.00 1.00 8.10E+04 ...-8.668E+04 1.140E+04 2.04E-13 4.91E-14 Table 22 continued 22 Table 22 (continued) 22No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 14 23.52 23.01-24.00 1.00 9.31E+04 ...-9.895E+04 1.226E+04 1.31E-13 3.03E-14 15 24.43 24.01-25.00 1.00 1.05E+05 ...-1.121E+05 1.314E+04 1.00E-13 2.25E-14 16 25.46 25.01-26.00 1.00 1.19E+05 ...-1.261E+05 1.402E+04 9.25E-14 2.15E-14 17 26.50 26.01-27.00 1.00 1.34E+05 ...-1.410E+05 1.490E+04 5.78E-14 1.23E-14 18 27.50 27.01-28.00 1.00 1.49E+05 ...-1.568E+05 1.579E+04 4.89E-14 1.09E-14 19 28.47 28.01-29.00 1.00 1.65E+05 ...-1.735E+05 1.668E+04 3.65E-14 7.76E-15 20 29.43 29.01-30.00 1.00 1.81E+05 ...-1.910E+05 1.757E+04 2.64E-14 5.64E-15 21 30.38 30.01-31.00 1.00 1.98E+05 ...-2.095E+05 1.846E+04 2.54E-14 5.46E-15 22 31.49 31.01-32.00 1.00 2.19E+05 ...-2.289E+05 1.935E+04 2.08E-14 4.49E-15 23 32.53 32.01-33.00 1.00 2.40E+05 ...-2.491E+05 2.025E+04 1.31E-14 2.77E-15 24 33.51 33.01-34.00 1.00 2.60E+05 ...-2.702E+05 2.113E+04 1.25E-14 2.70E-15 25 34.54 34.01-35.00 1.00 2.82E+05 ...-2.923E+05 2.202E+04 9.89E-15 2.06E-15 26 35.55 35.01-36.00 1.00 3.05E+05 ...-3.152E+05 2.291E+04 7.61E-15 1.64E-15 27 36.55 36.01-37.00 1.00 3.29E+05 ...-3.390E+05 2.379E+04 6.04E-15 1.31E-15 28 37.60 37.01-38.00 1.00 3.54E+05 ...-3.636E+05 2.467E+04 4.74E-15 1.03E-15 29 38.47 38.01-39.00 1.00 3.76E+05 ...-3.892E+05 2.554E+04 3.00E-15 6.30E-16 30 39.38 39.01-40.00 1.00 3.99E+05 ...-4.156E+05 2.641E+04 3.85E-15 8.40E-16 31 40.42 40.01-41.00 1.00 4.27E+05 ...-4.429E+05 2.728E+04 2.62E-15 5.71E-16 32 41.51 41.01-42.00 1.00 4.57E+05 ...-4.710E+05 2.814E+04 2.68E-15 5.94E-16 33 42.53 42.01-43.00 1.00 4.86E+05 ...-5.000E+05 2.900E+04 1.33E-15 2.91E-16 34 43.51 43.01-44.00 1.00 5.15E+05 ...-5.302E+05 3.016E+04 1.91E-15 4.26E-16 35 44.66 44.01-45.00 1.00 5.50E+05 ...-5.609E+05 3.071E+04 1.25E-15 2.77E-16 36 45.68 45.01-46.00 1.00 5.82E+05 ...-5.924E+05 3.155E+04 1.09E-15 2.44E-16 37 46.56 46.01-47.00 1.00 6.10E+05 ...-6.248E+05 3.238E+04 6.27E-16 1.36E-16 38 47.41 47.01-48.00 1.00 6.38E+05 ...-6.580E+05 3.321E+04 8.95E-16 2.02E-16 39 48.63 48.01-49.00 1.00 6.79E+05 ...-6.920E+05 3.403E+04 7.73E-16 1.74E-16 40 49.55 49.01-50.00 1.00 7.11E+05 ...-7.269E+05 3.484E+04 3.83E-16 8.61E-17 41 50.47 50.01-51.00 1.00 7.44E+05 ...-7.625E+05 3.565E+04 5.19E-16 1.18E-16 42 51.53 51.01-52.00 1.00 7.82E+05 ...-7.990E+05 3.645E+04 5.03E-16 1.14E-16 43 52.46 52.01-53.00 1.00 8.16E+05 ...-8.362E+05 3.724E+04 3.45E-16 7.87E-17 44 53.46 53.01-54.00 1.00 8.54E+05 ...-8.743E+05 3.803E+04 3.11E-16 7.16E-17 45 54.55 54.01-55.00 1.00 8.96E+05 ...-9.131E+05 3.881E+04 3.48E-16 8.02E-17 46 55.65 55.01-56.00 1.00 9.39E+05 ...-9.526E+05 3.958E+04 2.62E-16 6.03E-17 47 56.51 56.01-57.00 1.00 9.73E+05 ...-9.930E+05 4.034E+04 1.41E-16 3.25E-17 48 57.46 57.01-58.00 1.00 1.01E+06 ...-1.034E+06 4.109E+04 2.80E-16 6.47E-17 49 58.47 58.01-59.00 1.00 1.05E+06 ...-1.076E+06 4.184E+04 1.85E-16 4.28E-17 50 59.46 59.01-60.00 1.00 1.10E+06 ...-1.118E+06 4.258E+04 1.02E-16 2.37E-17 51 60.41 60.01-61.00 1.00 1.14E+06 ...-1.162E+06 4.331E+04 9.76E-17 2.25E-17 52 61.26 61.01-62.00 1.00 1.17E+06 ...-1.206E+06 4.403E+04 7.15E-17 1.65E-17 53 62.51 62.01-63.00 1.00 1.23E+06 ...-1.251E+06 4.475E+04 1.09E-16 2.54E-17 54 63.80 63.01-64.00 1.00 1.29E+06 ...-1.296E+06 4.545E+04 5.69E-17 1.31E-17 55 64.72 64.01-65.00 1.00 1.33E+06 ...-1.342E+06 4.615E+04 3.89E-17 8.97E-18 56 65.42 65.01-66.00 1.00 1.36E+06 ...-1.389E+06 4.684E+04 4.57E-17 1.06E-17 57 66.50 66.01-67.00 1.00 1.41E+06 ...-1.437E+06 4.753E+04 5.20E-17 1.21E-17 58 67.59 67.01-68.00 1.00 1.46E+06 ...-1.485E+06 4.820E+04 5.04E-17 1.17E-17 59 68.58 68.01-69.00 1.00 1.51E+06 ...-1.534E+06 4.887E+04 6.24E-17 1.45E-17 60 69.52 69.01-70.00 1.00 1.56E+06 ...-1.583E+06 4.953E+04 4.42E-18 1.07E-18 Table 22 continued 22 Table 22 (continued) 22No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 61 70.25 70.01-71.00 1.00 1.60E+06 ...-1.633E+06 5.019E+04 2.83E-18 6.88E-19 62 71.59 71.01-72.00 1.00 1.66E+06 ...-1.684E+06 5.033E+04 5.01E-20 1.22E-20 63 72.54 72.01-72.93 0.93 1.71E+06 ...-1.732E+06 4.837E+04 1.06E-19 2.59E-20 Table 23 . 23Bare grains with a = 0.2 µm, shielded by interstellar gas with NH = 3.30 × 10 21 H atoms cm −2 .No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low 1 10.23 10.01-11.00 1.00 6.24E+02 50.5902-2.633E+03 2.582E+03 9.42E-08 2.14E-08 2 11.40 11.01-12.00 1.00 3.74E+03 ...-5.402E+03 2.769E+03 1.05E-08 2.57E-10 3 12.44 12.01-13.00 1.00 6.70E+03 ...-8.410E+03 3.008E+03 3.60E-09 6.91E-11 4 13.45 13.01-14.00 1.00 9.86E+03 ...-1.172E+04 3.312E+03 1.74E-09 3.28E-11 5 14.44 14.01-15.00 1.00 1.33E+04 ...-1.540E+04 3.678E+03 8.07E-10 1.59E-11 6 15.39 15.01-16.00 1.00 1.70E+04 ...-1.950E+04 4.105E+03 5.48E-10 1.05E-11 7 16.44 16.01-17.00 1.00 2.15E+04 ...-2.405E+04 4.544E+03 3.37E-10 7.08E-12 8 17.48 17.01-18.00 1.00 2.65E+04 ...-2.918E+04 5.133E+03 2.23E-10 4.60E-12 9 18.51 18.01-19.00 1.00 3.21E+04 ...-3.492E+04 5.736E+03 1.29E-10 3.25E-12 10 19.49 19.01-20.00 1.00 3.81E+04 ...-4.131E+04 6.394E+03 8.45E-11 2.47E-12 11 20.48 20.01-21.00 1.00 4.47E+04 ...-4.842E+04 7.108E+03 7.20E-11 2.05E-12 12 21.49 21.01-22.00 1.00 5.23E+04 ...-5.629E+04 7.874E+03 5.20E-11 1.52E-12 13 22.45 22.01-23.00 1.00 6.02E+04 ...-6.499E+04 8.693E+03 3.32E-11 1.16E-12 14 23.48 23.01-24.00 1.00 6.96E+04 ...-7.455E+04 9.562E+03 3.02E-11 9.34E-13 15 24.44 24.01-25.00 1.00 7.92E+04 ...-8.503E+04 1.048E+04 1.91E-11 9.35E-13 16 25.47 25.01-26.00 1.00 9.04E+04 ...-9.648E+04 1.145E+04 1.62E-11 6.70E-13 17 26.49 26.01-27.00 1.00 1.03E+05 ...-1.089E+05 1.246E+04 1.15E-11 5.10E-13 18 27.45 27.01-28.00 1.00 1.15E+05 ...-1.225E+05 1.352E+04 1.06E-11 4.82E-13 19 28.45 28.01-29.00 1.00 1.29E+05 ...-1.371E+05 1.463E+04 9.30E-12 3.52E-13 20 29.49 29.01-30.00 1.00 1.45E+05 ...-1.529E+05 1.577E+04 1.07E-11 3.71E-13 21 30.59 30.01-31.00 1.00 1.63E+05 ...-1.698E+05 1.697E+04 4.70E-12 2.39E-13 22 31.54 31.01-32.00 1.00 1.80E+05 ...-1.880E+05 1.820E+04 4.51E-12 2.02E-13 23 32.48 32.01-33.00 1.00 1.98E+05 ...-2.075E+05 1.947E+04 4.10E-12 1.65E-13 24 33.51 33.01-34.00 1.00 2.18E+05 ...-2.283E+05 2.078E+04 3.74E-12 1.57E-13 25 34.50 34.01-35.00 1.00 2.39E+05 ...-2.504E+05 2.213E+04 3.04E-12 1.20E-13 26 35.54 35.01-36.00 1.00 2.63E+05 ...-2.739E+05 2.351E+04 3.22E-12 1.33E-13 27 36.56 36.01-37.00 1.00 2.88E+05 ...-2.988E+05 2.493E+04 2.42E-12 8.75E-14 28 37.53 37.01-38.00 1.00 3.13E+05 ...-3.252E+05 2.639E+04 1.88E-12 7.19E-14 29 38.47 38.01-39.00 1.00 3.39E+05 ...-3.531E+05 2.788E+04 1.80E-12 6.59E-14 30 39.47 39.01-40.00 1.00 3.67E+05 ...-3.825E+05 2.940E+04 1.71E-12 6.10E-14 31 40.43 40.01-41.00 1.00 3.96E+05 ...-4.135E+05 3.095E+04 1.36E-12 4.99E-14 32 41.51 41.01-42.00 1.00 4.30E+05 ...-4.460E+05 3.254E+04 1.41E-12 5.26E-14 33 42.53 42.01-43.00 1.00 4.64E+05 ...-4.801E+05 3.415E+04 8.96E-13 3.11E-14 34 43.49 43.01-44.00 1.00 4.98E+05 ...-5.163E+05 3.616E+04 9.23E-13 3.21E-14 35 44.51 44.01-45.00 1.00 5.35E+05 ...-5.538E+05 3.748E+04 8.81E-13 2.85E-14 36 45.47 45.01-46.00 1.00 5.72E+05 ...-5.929E+05 3.917E+04 6.05E-13 2.22E-14 37 46.46 46.01-47.00 1.00 6.11E+05 ...-6.338E+05 4.089E+04 6.43E-13 2.25E-14 38 47.50 47.01-48.00 1.00 6.55E+05 ...-6.765E+05 4.264E+04 5.17E-13 1.86E-14 Table 23 continued 23 Table 23 (continued) 23No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low39 48.49 48.01-49.00 1.00 6.98E+05 ...-7.209E+05 4.441E+04 4.06E-13 1.14E-14 40 49.45 49.01-50.00 1.00 7.42E+05 ...-7.671E+05 4.620E+04 4.86E-13 1.67E-14 41 50.45 50.01-51.00 1.00 7.88E+05 ...-8.151E+05 4.802E+04 3.67E-13 1.18E-14 42 51.51 51.01-52.00 1.00 8.41E+05 ...-8.650E+05 4.985E+04 2.56E-13 8.79E-15 43 52.52 52.01-53.00 1.00 8.92E+05 ...-9.167E+05 5.171E+04 2.85E-13 1.04E-14 44 53.62 53.01-54.00 1.00 9.49E+05 ...-9.703E+05 5.359E+04 2.16E-13 6.93E-15 45 54.44 54.01-55.00 1.00 9.95E+05 ...-1.026E+06 5.548E+04 1.59E-13 4.51E-15 46 55.39 55.01-56.00 1.00 1.05E+06 ...-1.083E+06 5.740E+04 1.92E-13 6.28E-15 47 56.40 56.01-57.00 1.00 1.11E+06 ...-1.142E+06 5.933E+04 1.65E-13 5.31E-15 48 57.48 57.01-58.00 1.00 1.17E+06 ...-1.204E+06 6.127E+04 1.83E-13 5.60E-15 49 58.46 58.01-59.00 1.00 1.23E+06 ...-1.267E+06 6.323E+04 1.09E-13 2.72E-15 50 59.30 59.01-60.00 1.00 1.29E+06 ...-1.332E+06 6.521E+04 1.48E-13 4.58E-15 51 60.57 60.01-61.00 1.00 1.37E+06 ...-1.399E+06 6.720E+04 1.28E-13 3.75E-15 52 61.52 61.01-62.00 1.00 1.44E+06 ...-1.469E+06 6.921E+04 1.17E-13 3.14E-15 53 62.45 62.01-63.00 1.00 1.50E+06 ...-1.540E+06 7.122E+04 7.22E-14 2.10E-15 54 63.59 63.01-64.00 1.00 1.58E+06 ...-1.613E+06 7.325E+04 5.10E-14 1.58E-15 55 64.65 64.01-65.00 1.00 1.66E+06 ...-1.688E+06 7.529E+04 5.36E-14 1.58E-15 56 65.75 65.01-66.00 1.00 1.75E+06 ...-1.766E+06 7.734E+04 4.61E-14 1.37E-15 57 66.68 66.01-67.00 1.00 1.82E+06 ...-1.845E+06 7.940E+04 9.82E-15 2.66E-16 58 67.46 67.01-68.00 1.00 1.88E+06 ...-1.927E+06 8.147E+04 4.53E-14 1.33E-15 59 68.51 68.01-69.00 1.00 1.97E+06 ...-2.010E+06 8.354E+04 2.49E-14 6.97E-16 60 69.38 69.01-70.00 1.00 2.04E+06 ...-2.096E+06 8.563E+04 2.47E-14 6.78E-16 61 70.53 70.01-71.00 1.00 2.14E+06 ...-2.183E+06 8.772E+04 3.88E-14 1.05E-15 62 71.70 71.01-72.00 1.00 2.25E+06 ...-2.272E+06 8.891E+04 2.38E-14 6.27E-16 63 72.58 72.01-73.00 1.00 2.33E+06 ...-2.364E+06 9.190E+04 2.22E-14 5.70E-16 64 73.49 73.01-74.00 1.00 2.41E+06 ...-2.458E+06 9.401E+04 1.95E-14 4.93E-16 65 74.47 74.01-75.00 1.00 2.50E+06 ...-2.554E+06 9.613E+04 2.29E-14 5.72E-16 66 75.44 75.01-76.00 1.00 2.60E+06 ...-2.653E+06 9.825E+04 2.04E-14 5.06E-16 67 76.53 76.01-77.00 1.00 2.71E+06 ...-2.753E+06 1.004E+05 2.06E-14 5.02E-16 68 77.54 77.01-78.00 1.00 2.81E+06 ...-2.856E+06 1.025E+05 3.12E-14 7.43E-16 69 78.55 78.01-79.00 1.00 2.91E+06 ...-2.960E+06 1.046E+05 1.94E-15 4.87E-17 70 79.16 79.01-80.00 1.00 2.98E+06 ...-3.067E+06 1.068E+05 1.03E-15 2.57E-17 71 80.56 80.01-81.00 1.00 3.13E+06 ...-3.176E+06 1.089E+05 2.68E-17 6.80E-19 72 81.27 81.01-81.49 0.49 3.21E+06 ...-3.231E+06 5.525E+04 3.26E-17 8.18E-19 Table 24 . 24Grains with a = 0.2 µm and 0.01 µm thick icy mantle, shielded by interstellar gas with NH = 6.60 × 10 21 H atoms cm −2 . . T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , LowNo1 10.20 10.01-11.00 1.00 7.44E+02 68.357-3.686E+03 3.617E+03 1.02E-07 2.01E-08 2 11.38 11.01-12.00 1.00 5.20E+03 ...-7.819E+03 4.133E+03 6.67E-09 1.73E-10 3 12.41 12.01-13.00 1.00 9.72E+03 ...-1.255E+04 4.732E+03 1.77E-09 4.47E-11 4 13.43 13.01-14.00 1.00 1.48E+04 ...-1.798E+04 5.426E+03 7.40E-10 1.95E-11 5 14.47 14.01-15.00 1.00 2.08E+04 ...-2.419E+04 6.210E+03 3.33E-10 9.56E-12 6 15.49 15.01-16.00 1.00 2.76E+04 ...-3.127E+04 7.079E+03 1.91E-10 5.74E-12 Table 24 continued 24 Table 24 (continued) 24No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low7 16.49 16.01-17.00 1.00 3.51E+04 ...-3.921E+04 7.944E+03 9.65E-11 3.63E-12 8 17.46 17.01-18.00 1.00 4.33E+04 ...-4.825E+04 9.044E+03 6.13E-11 2.69E-12 9 18.43 18.01-19.00 1.00 5.26E+04 ...-5.839E+04 1.014E+04 4.67E-11 1.86E-12 10 19.45 19.01-20.00 1.00 6.35E+04 ...-6.970E+04 1.130E+04 3.07E-11 1.44E-12 11 20.51 20.01-21.00 1.00 7.61E+04 ...-8.222E+04 1.253E+04 2.24E-11 1.15E-12 12 21.51 21.01-22.00 1.00 8.93E+04 ...-9.603E+04 1.381E+04 1.51E-11 8.60E-13 13 22.51 22.01-23.00 1.00 1.04E+05 ...-1.112E+05 1.514E+04 1.13E-11 6.41E-13 14 23.50 23.01-24.00 1.00 1.19E+05 ...-1.277E+05 1.653E+04 9.06E-12 5.18E-13 15 24.48 24.01-25.00 1.00 1.36E+05 ...-1.457E+05 1.797E+04 7.92E-12 3.98E-13 16 25.40 25.01-26.00 1.00 1.53E+05 ...-1.651E+05 1.945E+04 6.41E-12 3.22E-13 17 26.47 26.01-27.00 1.00 1.75E+05 ...-1.861E+05 2.098E+04 4.61E-12 2.60E-13 18 27.47 27.01-28.00 1.00 1.97E+05 ...-2.086E+05 2.254E+04 3.80E-12 2.04E-13 19 28.49 28.01-29.00 1.00 2.21E+05 ...-2.328E+05 2.415E+04 3.24E-12 1.67E-13 20 29.46 29.01-30.00 1.00 2.45E+05 ...-2.586E+05 2.580E+04 2.44E-12 1.29E-13 21 30.47 30.01-31.00 1.00 2.72E+05 ...-2.861E+05 2.748E+04 2.47E-12 1.28E-13 22 31.46 31.01-32.00 1.00 3.00E+05 ...-3.153E+05 2.920E+04 1.83E-12 8.41E-14 23 32.44 32.01-33.00 1.00 3.29E+05 ...-3.462E+05 3.095E+04 1.61E-12 8.39E-14 24 33.48 33.01-34.00 1.00 3.62E+05 ...-3.790E+05 3.274E+04 1.41E-12 6.67E-14 25 34.52 34.01-35.00 1.00 3.97E+05 ...-4.135E+05 3.456E+04 1.24E-12 5.63E-14 26 35.52 35.01-36.00 1.00 4.33E+05 ...-4.499E+05 3.641E+04 9.36E-13 4.58E-14 27 36.47 36.01-37.00 1.00 4.68E+05 ...-4.882E+05 3.830E+04 7.72E-13 3.38E-14 28 37.48 37.01-38.00 1.00 5.08E+05 ...-5.284E+05 4.021E+04 7.54E-13 3.40E-14 29 38.52 38.01-39.00 1.00 5.51E+05 ...-5.706E+05 4.215E+04 6.37E-13 2.70E-14 30 39.48 39.01-40.00 1.00 5.92E+05 ...-6.147E+05 4.412E+04 4.32E-13 2.04E-14 31 40.42 40.01-41.00 1.00 6.34E+05 ...-6.608E+05 4.612E+04 4.43E-13 2.02E-14 32 41.43 41.01-42.00 1.00 6.82E+05 ...-7.090E+05 4.815E+04 3.72E-13 1.70E-14 33 42.50 42.01-43.00 1.00 7.35E+05 ...-7.592E+05 5.020E+04 3.41E-13 1.43E-14 34 43.56 43.01-44.00 1.00 7.89E+05 ...-8.120E+05 5.280E+04 3.47E-13 1.44E-14 35 44.59 44.01-45.00 1.00 8.44E+05 ...-8.664E+05 5.439E+04 1.84E-13 8.15E-15 36 45.47 45.01-46.00 1.00 8.93E+05 ...-9.229E+05 5.651E+04 1.63E-13 7.22E-15 37 46.43 46.01-47.00 1.00 9.48E+05 ...-9.815E+05 5.865E+04 1.71E-13 7.19E-15 38 47.43 47.01-48.00 1.00 1.01E+06 ...-1.042E+06 6.082E+04 1.47E-13 6.30E-15 39 48.45 48.01-49.00 1.00 1.07E+06 ...-1.105E+06 6.300E+04 1.32E-13 5.59E-15 40 49.44 49.01-50.00 1.00 1.13E+06 ...-1.171E+06 6.520E+04 1.04E-13 4.33E-15 41 50.46 50.01-51.00 1.00 1.20E+06 ...-1.238E+06 6.742E+04 1.18E-13 4.82E-15 42 51.63 51.01-52.00 1.00 1.28E+06 ...-1.308E+06 6.966E+04 1.13E-13 4.39E-15 43 52.53 52.01-53.00 1.00 1.35E+06 ...-1.380E+06 7.192E+04 4.76E-14 1.70E-15 44 53.47 53.01-54.00 1.00 1.41E+06 ...-1.454E+06 7.419E+04 7.77E-14 3.04E-15 45 54.50 54.01-55.00 1.00 1.49E+06 ...-1.530E+06 7.647E+04 7.13E-14 2.65E-15 46 55.26 55.01-56.00 1.00 1.55E+06 ...-1.609E+06 7.877E+04 4.33E-14 1.66E-15 47 56.31 56.01-57.00 1.00 1.63E+06 ...-1.690E+06 8.108E+04 3.32E-14 1.33E-15 48 57.34 57.01-58.00 1.00 1.72E+06 ...-1.773E+06 8.340E+04 3.15E-14 1.23E-15 49 58.30 58.01-59.00 1.00 1.80E+06 ...-1.859E+06 8.574E+04 2.62E-14 1.03E-15 50 59.48 59.01-60.00 1.00 1.90E+06 ...-1.947E+06 8.809E+04 2.03E-14 7.82E-16 51 60.51 60.01-61.00 1.00 1.99E+06 ...-2.038E+06 9.044E+04 2.37E-14 9.07E-16 52 61.55 61.01-62.00 1.00 2.09E+06 ...-2.131E+06 9.281E+04 1.42E-14 5.29E-16 53 62.41 62.01-63.00 1.00 2.17E+06 ...-2.226E+06 9.518E+04 1.37E-14 5.06E-16 Table 24 continued 24 Table 24 (continued) 24No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low54 63.51 63.01-64.00 1.00 2.27E+06 ...-2.323E+06 9.756E+04 2.10E-14 7.76E-16 55 64.60 64.01-65.00 1.00 2.38E+06 ...-2.423E+06 9.995E+04 1.22E-14 4.38E-16 56 65.50 65.01-66.00 1.00 2.47E+06 ...-2.526E+06 1.023E+05 1.10E-14 3.94E-16 57 66.54 66.01-67.00 1.00 2.58E+06 ...-2.630E+06 1.047E+05 1.45E-14 5.15E-16 58 67.48 67.01-68.00 1.00 2.68E+06 ...-2.737E+06 1.072E+05 7.76E-15 2.76E-16 59 68.51 68.01-69.00 1.00 2.79E+06 ...-2.847E+06 1.096E+05 1.35E-14 4.76E-16 60 69.46 69.01-70.00 1.00 2.90E+06 ...-2.959E+06 1.120E+05 1.28E-14 4.46E-16 61 70.57 70.01-71.00 1.00 3.02E+06 ...-3.073E+06 1.144E+05 1.30E-15 4.75E-17 62 71.52 71.01-72.00 1.00 3.13E+06 ...-3.189E+06 1.156E+05 1.36E-17 5.06E-19 63 72.66 72.01-73.07 1.07 3.27E+06 ...-3.318E+06 1.289E+05 2.55E-17 9.37E-19 Table 25 . 25Grains with a = 0.2 µm and 0.020 µm thick icy mantle, shielded by interstellar gas with NH = 1.32 × 10 22 H atoms cm −2 .No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low1 10.15 10.01-11.00 1.00 9.06E+02 106.165-5.855E+03 5.749E+03 9.93E-08 1.65E-08 2 11.36 11.01-12.00 1.00 8.23E+03 ...-1.265E+04 6.800E+03 2.65E-09 8.65E-11 3 12.40 12.01-13.00 1.00 1.58E+04 ...-2.062E+04 7.964E+03 6.38E-10 2.36E-11 4 13.45 13.01-14.00 1.00 2.47E+04 ...-2.988E+04 9.261E+03 2.15E-10 9.30E-12 5 14.43 14.01-15.00 1.00 3.43E+04 ...-4.056E+04 1.068E+04 9.99E-11 4.78E-12 6 15.44 15.01-16.00 1.00 4.58E+04 ...-5.278E+04 1.222E+04 5.39E-11 3.13E-12 7 16.42 16.01-17.00 1.00 5.85E+04 ...-6.649E+04 1.371E+04 3.43E-11 2.09E-12 8 17.48 17.01-18.00 1.00 7.39E+04 ...-8.207E+04 1.558E+04 2.21E-11 1.45E-12 9 18.45 18.01-19.00 1.00 9.00E+04 ...-9.947E+04 1.740E+04 1.40E-11 1.02E-12 10 19.46 19.01-20.00 1.00 1.08E+05 ...-1.188E+05 1.931E+04 1.04E-11 7.43E-13 11 20.48 20.01-21.00 1.00 1.29E+05 ...-1.401E+05 2.129E+04 7.68E-12 5.37E-13 12 21.46 21.01-22.00 1.00 1.51E+05 ...-1.634E+05 2.333E+04 6.33E-12 4.06E-13 13 22.52 22.01-23.00 1.00 1.77E+05 ...-1.888E+05 2.544E+04 4.28E-12 3.12E-13 14 23.53 23.01-24.00 1.00 2.03E+05 ...-2.164E+05 2.760E+04 3.10E-12 2.19E-13 15 24.51 24.01-25.00 1.00 2.32E+05 ...-2.462E+05 2.980E+04 2.52E-12 1.81E-13 16 25.48 25.01-26.00 1.00 2.62E+05 ...-2.783E+05 3.206E+04 1.95E-12 1.24E-13 17 26.49 26.01-27.00 1.00 2.95E+05 ...-3.126E+05 3.435E+04 1.83E-12 1.18E-13 18 27.55 27.01-28.00 1.00 3.33E+05 ...-3.493E+05 3.668E+04 1.39E-12 9.47E-14 19 28.55 28.01-29.00 1.00 3.71E+05 ...-3.884E+05 3.905E+04 1.04E-12 6.24E-14 20 29.53 29.01-30.00 1.00 4.11E+05 ...-4.298E+05 4.145E+04 8.51E-13 5.03E-14 21 30.49 30.01-31.00 1.00 4.51E+05 ...-4.737E+05 4.388E+04 7.52E-13 4.61E-14 22 31.54 31.01-32.00 1.00 4.99E+05 ...-5.200E+05 4.634E+04 6.33E-13 3.96E-14 23 32.51 32.01-33.00 1.00 5.45E+05 ...-5.689E+05 4.883E+04 4.43E-13 2.53E-14 24 33.42 33.01-34.00 1.00 5.91E+05 ...-6.202E+05 5.134E+04 3.90E-13 2.24E-14 25 34.45 34.01-35.00 1.00 6.45E+05 ...-6.741E+05 5.388E+04 3.72E-13 2.20E-14 26 35.54 35.01-36.00 1.00 7.05E+05 ...-7.305E+05 5.645E+04 2.89E-13 1.63E-14 27 36.57 36.01-37.00 1.00 7.64E+05 ...-7.896E+05 5.903E+04 2.16E-13 1.18E-14 28 37.44 37.01-38.00 1.00 8.17E+05 ...-8.512E+05 6.164E+04 2.00E-13 1.12E-14 29 38.53 38.01-39.00 1.00 8.86E+05 ...-9.155E+05 6.427E+04 1.72E-13 1.00E-14 30 39.55 39.01-40.00 1.00 9.53E+05 ...-9.824E+05 6.693E+04 1.08E-13 6.04E-15 Table 25 continued 25 Table 25 (continued) 25No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low31 40.45 40.01-41.00 1.00 1.01E+06 ...-1.052E+06 6.960E+04 1.04E-13 5.69E-15 32 41.43 41.01-42.00 1.00 1.08E+06 ...-1.124E+06 7.229E+04 9.00E-14 4.96E-15 33 42.39 42.01-43.00 1.00 1.15E+06 ...-1.199E+06 7.499E+04 7.63E-14 4.19E-15 34 43.43 43.01-44.00 1.00 1.23E+06 ...-1.278E+06 7.851E+04 7.92E-14 4.22E-15 35 44.55 44.01-45.00 1.00 1.32E+06 ...-1.358E+06 8.048E+04 6.77E-14 3.57E-15 36 45.51 45.01-46.00 1.00 1.40E+06 ...-1.441E+06 8.323E+04 2.90E-14 1.42E-15 37 46.39 46.01-47.00 1.00 1.48E+06 ...-1.527E+06 8.600E+04 5.82E-14 3.10E-15 38 47.52 47.01-48.00 1.00 1.57E+06 ...-1.616E+06 8.877E+04 3.94E-14 2.00E-15 39 48.43 48.01-49.00 1.00 1.66E+06 ...-1.708E+06 9.156E+04 2.78E-14 1.45E-15 40 49.46 49.01-50.00 1.00 1.75E+06 ...-1.802E+06 9.436E+04 1.87E-14 9.81E-16 41 50.42 50.01-51.00 1.00 1.84E+06 ...-1.899E+06 9.717E+04 1.76E-14 9.11E-16 42 51.48 51.01-52.00 1.00 1.95E+06 ...-1.999E+06 9.998E+04 1.16E-14 6.01E-16 43 52.42 52.01-53.00 1.00 2.04E+06 ...-2.102E+06 1.028E+05 1.35E-14 6.98E-16 44 53.39 53.01-54.00 1.00 2.14E+06 ...-2.208E+06 1.056E+05 7.52E-15 3.84E-16 45 54.42 54.01-55.00 1.00 2.25E+06 ...-2.316E+06 1.085E+05 1.25E-14 6.39E-16 46 55.49 55.01-56.00 1.00 2.37E+06 ...-2.428E+06 1.113E+05 6.51E-15 3.28E-16 47 56.52 56.01-57.00 1.00 2.49E+06 ...-2.542E+06 1.142E+05 1.05E-14 5.28E-16 48 57.69 57.01-58.00 1.00 2.62E+06 ...-2.659E+06 1.170E+05 5.78E-15 2.89E-16 49 58.55 58.01-59.00 1.00 2.72E+06 ...-2.779E+06 1.199E+05 6.11E-15 3.06E-16 50 59.56 59.01-60.00 1.00 2.85E+06 ...-2.901E+06 1.227E+05 5.33E-15 2.66E-16 51 60.47 60.01-61.00 1.00 2.96E+06 ...-3.027E+06 1.256E+05 7.89E-15 3.93E-16 52 61.57 61.01-62.00 1.00 3.10E+06 ...-3.155E+06 1.284E+05 6.56E-16 3.41E-17 53 62.53 62.01-63.00 1.00 3.22E+06 ...-3.286E+06 1.313E+05 6.23E-18 3.27E-19 54 63.49 63.01-63.85 0.85 3.35E+06 ...-3.400E+06 1.138E+05 1.27E-17 6.62E-19 Table 26 . 26Grains with a = 0.2 µm and 0.025 µm thick icy mantle, shielded by interstellar gas with NH = 1.98 × 10 22 H atoms cm −2 . . T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , LowNo1 10.12 10.01-11.00 1.00 1.02E+03 146.366-8.144E+03 7.997E+03 8.98E-08 1.26E-08 2 11.35 11.01-12.00 1.00 1.14E+04 ...-1.771E+04 9.569E+03 1.36E-09 5.56E-11 3 12.42 12.01-13.00 1.00 2.23E+04 ...-2.898E+04 1.127E+04 2.76E-10 1.42E-11 4 13.43 13.01-14.00 1.00 3.44E+04 ...-4.211E+04 1.313E+04 9.78E-11 5.78E-12 5 14.45 14.01-15.00 1.00 4.87E+04 ...-5.723E+04 1.512E+04 5.01E-11 3.46E-12 6 15.43 15.01-16.00 1.00 6.45E+04 ...-7.449E+04 1.725E+04 2.53E-11 1.92E-12 7 16.44 16.01-17.00 1.00 8.28E+04 ...-9.378E+04 1.930E+04 1.83E-11 1.49E-12 8 17.48 17.01-18.00 1.00 1.04E+05 ...-1.156E+05 2.184E+04 1.07E-11 8.87E-13 9 18.48 18.01-19.00 1.00 1.27E+05 ...-1.399E+05 2.430E+04 7.73E-12 6.49E-13 10 19.47 19.01-20.00 1.00 1.53E+05 ...-1.668E+05 2.685E+04 5.60E-12 4.17E-13 11 20.49 20.01-21.00 1.00 1.81E+05 ...-1.963E+05 2.948E+04 3.74E-12 3.21E-13 12 21.50 21.01-22.00 1.00 2.13E+05 ...-2.284E+05 3.218E+04 2.89E-12 2.32E-13 13 22.51 22.01-23.00 1.00 2.46E+05 ...-2.634E+05 3.495E+04 2.06E-12 1.60E-13 14 23.49 23.01-24.00 1.00 2.82E+05 ...-3.012E+05 3.778E+04 1.66E-12 1.28E-13 15 24.48 24.01-25.00 1.00 3.21E+05 ...-3.418E+05 4.065E+04 1.32E-12 9.97E-14 16 25.49 25.01-26.00 1.00 3.64E+05 ...-3.854E+05 4.358E+04 1.06E-12 8.02E-14 Table 26 continued 26 Table 26 (continued) 26No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low17 26.51 26.01-27.00 1.00 4.09E+05 ...-4.319E+05 4.654E+04 7.93E-13 5.55E-14 18 27.47 27.01-28.00 1.00 4.55E+05 ...-4.815E+05 4.955E+04 6.33E-13 4.39E-14 19 28.45 28.01-29.00 1.00 5.06E+05 ...-5.341E+05 5.258E+04 5.21E-13 3.84E-14 20 29.47 29.01-30.00 1.00 5.60E+05 ...-5.897E+05 5.565E+04 4.39E-13 2.86E-14 21 30.56 30.01-31.00 1.00 6.23E+05 ...-6.485E+05 5.875E+04 3.68E-13 2.70E-14 22 31.62 31.01-32.00 1.00 6.87E+05 ...-7.103E+05 6.187E+04 2.57E-13 1.68E-14 23 32.59 32.01-33.00 1.00 7.49E+05 ...-7.753E+05 6.502E+04 1.90E-13 1.25E-14 24 33.53 33.01-34.00 1.00 8.12E+05 ...-8.435E+05 6.818E+04 1.80E-13 1.14E-14 25 34.46 34.01-35.00 1.00 8.76E+05 ...-9.149E+05 7.137E+04 1.24E-13 8.10E-15 26 35.48 35.01-36.00 1.00 9.51E+05 ...-9.895E+05 7.458E+04 1.26E-13 8.51E-15 27 36.51 36.01-37.00 1.00 1.03E+06 ...-1.067E+06 7.780E+04 8.51E-14 5.43E-15 28 37.47 37.01-38.00 1.00 1.11E+06 ...-1.148E+06 8.104E+04 6.85E-14 4.47E-15 29 38.39 38.01-39.00 1.00 1.18E+06 ...-1.233E+06 8.429E+04 6.20E-14 3.96E-15 30 39.36 39.01-40.00 1.00 1.26E+06 ...-1.320E+06 8.755E+04 5.66E-14 3.57E-15 31 40.37 40.01-41.00 1.00 1.35E+06 ...-1.411E+06 9.083E+04 4.89E-14 3.09E-15 32 41.51 41.01-42.00 1.00 1.46E+06 ...-1.505E+06 9.411E+04 4.02E-14 2.51E-15 33 42.57 42.01-43.00 1.00 1.56E+06 ...-1.603E+06 9.740E+04 3.94E-14 2.45E-15 34 43.63 43.01-44.00 1.00 1.67E+06 ...-1.704E+06 1.017E+05 2.70E-14 1.68E-15 35 44.56 44.01-45.00 1.00 1.76E+06 ...-1.808E+06 1.040E+05 1.66E-14 1.04E-15 36 45.54 45.01-46.00 1.00 1.87E+06 ...-1.916E+06 1.073E+05 1.31E-14 8.18E-16 37 46.48 46.01-47.00 1.00 1.97E+06 ...-2.026E+06 1.107E+05 9.30E-15 5.79E-16 38 47.37 47.01-48.00 1.00 2.07E+06 ...-2.140E+06 1.140E+05 9.99E-15 6.21E-16 39 48.53 48.01-49.00 1.00 2.20E+06 ...-2.258E+06 1.173E+05 9.70E-15 5.99E-16 40 49.50 49.01-50.00 1.00 2.32E+06 ...-2.378E+06 1.206E+05 5.32E-15 3.27E-16 41 50.49 50.01-51.00 1.00 2.44E+06 ...-2.502E+06 1.239E+05 8.14E-15 4.99E-16 42 51.51 51.01-52.00 1.00 2.57E+06 ...-2.629E+06 1.272E+05 4.44E-15 2.70E-16 43 52.42 52.01-53.00 1.00 2.68E+06 ...-2.760E+06 1.305E+05 5.44E-15 3.31E-16 44 53.51 53.01-54.00 1.00 2.83E+06 ...-2.894E+06 1.338E+05 5.18E-15 3.16E-16 45 54.58 54.01-55.00 1.00 2.97E+06 ...-3.031E+06 1.371E+05 5.68E-15 3.45E-16 46 55.19 55.01-56.00 1.00 3.06E+06 ...-3.171E+06 1.404E+05 1.53E-15 9.38E-17 47 56.08 56.01-57.00 1.00 3.18E+06 ...-3.315E+06 1.437E+05 1.40E-16 8.92E-18 48 57.54 57.01-57.88 0.88 3.39E+06 ...-3.444E+06 1.292E+05 9.23E-18 5.91E-19 0 Table 27 . 27Grains with a = 0.2 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 4.40 × 10 22 H atoms cm −2 . . T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , LowNo1 10.09 10.01-11.00 1.00 1.01E+03 185.277-1.037E+04 1.019E+04 7.56E-08 1.02E-08 2 11.35 11.01-12.00 1.00 1.45E+04 ...-2.267E+04 1.230E+04 5.54E-10 3.58E-11 3 12.41 12.01-13.00 1.00 2.85E+04 ...-3.723E+04 1.455E+04 1.23E-10 9.60E-12 4 13.44 13.01-14.00 1.00 4.45E+04 ...-5.421E+04 1.699E+04 4.33E-11 4.12E-12 5 14.43 14.01-15.00 1.00 6.25E+04 ...-7.379E+04 1.958E+04 2.33E-11 2.31E-12 6 15.45 15.01-16.00 1.00 8.37E+04 ...-9.611E+04 2.232E+04 1.31E-11 1.46E-12 7 16.47 16.01-17.00 1.00 1.08E+05 ...-1.210E+05 2.493E+04 8.11E-12 9.34E-13 Table 27 continued 27 Table 27 (continued) 27No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low8 17.48 17.01-18.00 1.00 1.35E+05 ...-1.492E+05 2.816E+04 5.19E-12 5.60E-13 9 18.45 18.01-19.00 1.00 1.63E+05 ...-1.805E+05 3.127E+04 3.57E-12 3.89E-13 10 19.47 19.01-20.00 1.00 1.97E+05 ...-2.150E+05 3.448E+04 2.36E-12 2.55E-13 11 20.47 20.01-21.00 1.00 2.33E+05 ...-2.527E+05 3.777E+04 1.74E-12 1.85E-13 12 21.48 21.01-22.00 1.00 2.73E+05 ...-2.939E+05 4.115E+04 1.26E-12 1.29E-13 13 22.47 22.01-23.00 1.00 3.15E+05 ...-3.385E+05 4.459E+04 9.61E-13 9.46E-14 14 23.50 23.01-24.00 1.00 3.63E+05 ...-3.866E+05 4.810E+04 7.83E-13 7.84E-14 15 24.50 24.01-25.00 1.00 4.12E+05 ...-4.382E+05 5.166E+04 4.83E-13 4.45E-14 16 25.43 25.01-26.00 1.00 4.62E+05 ...-4.935E+05 5.526E+04 4.27E-13 4.06E-14 17 26.44 26.01-27.00 1.00 5.19E+05 ...-5.524E+05 5.892E+04 3.45E-13 3.27E-14 18 27.47 27.01-28.00 1.00 5.82E+05 ...-6.150E+05 6.260E+04 2.64E-13 2.53E-14 19 28.51 28.01-29.00 1.00 6.49E+05 ...-6.813E+05 6.633E+04 2.01E-13 1.82E-14 20 29.51 29.01-30.00 1.00 7.18E+05 ...-7.514E+05 7.008E+04 1.35E-13 1.22E-14 21 30.47 30.01-31.00 1.00 7.87E+05 ...-8.253E+05 7.386E+04 1.23E-13 1.12E-14 22 31.48 31.01-32.00 1.00 8.63E+05 ...-9.029E+05 7.765E+04 9.75E-14 8.77E-15 23 32.56 32.01-33.00 1.00 9.50E+05 ...-9.844E+05 8.147E+04 7.57E-14 7.01E-15 24 33.54 33.01-34.00 1.00 1.03E+06 ...-1.070E+06 8.531E+04 4.53E-14 4.07E-15 25 34.44 34.01-35.00 1.00 1.11E+06 ...-1.159E+06 8.916E+04 4.23E-14 3.84E-15 26 35.38 35.01-36.00 1.00 1.20E+06 ...-1.252E+06 9.303E+04 3.40E-14 3.08E-15 27 36.36 36.01-37.00 1.00 1.29E+06 ...-1.349E+06 9.690E+04 3.20E-14 2.87E-15 28 37.36 37.01-38.00 1.00 1.39E+06 ...-1.450E+06 1.008E+05 2.55E-14 2.31E-15 29 38.53 38.01-39.00 1.00 1.51E+06 ...-1.554E+06 1.047E+05 2.28E-14 2.07E-15 30 39.51 39.01-40.00 1.00 1.61E+06 ...-1.663E+06 1.086E+05 1.60E-14 1.42E-15 31 40.51 40.01-41.00 1.00 1.72E+06 ...-1.775E+06 1.125E+05 1.35E-14 1.23E-15 32 41.56 41.01-42.00 1.00 1.84E+06 ...-1.892E+06 1.164E+05 1.14E-14 1.04E-15 33 42.72 42.01-43.00 1.00 1.98E+06 ...-2.012E+06 1.202E+05 5.38E-15 4.88E-16 34 43.61 43.01-44.00 1.00 2.09E+06 ...-2.137E+06 1.254E+05 4.96E-15 4.51E-16 35 44.47 44.01-45.00 1.00 2.20E+06 ...-2.265E+06 1.281E+05 2.81E-15 2.54E-16 36 45.38 45.01-46.00 1.00 2.32E+06 ...-2.397E+06 1.320E+05 4.52E-15 4.09E-16 37 46.54 46.01-47.00 1.00 2.47E+06 ...-2.533E+06 1.358E+05 3.81E-15 3.45E-16 38 47.50 47.01-48.00 1.00 2.60E+06 ...-2.673E+06 1.397E+05 2.13E-15 1.92E-16 39 48.45 48.01-49.00 1.00 2.74E+06 ...-2.816E+06 1.436E+05 3.06E-15 2.77E-16 40 49.58 49.01-50.00 1.00 2.90E+06 ...-2.964E+06 1.474E+05 2.36E-15 2.14E-16 41 50.50 50.01-51.00 1.00 3.04E+06 ...-3.115E+06 1.512E+05 2.63E-15 2.38E-16 42 51.44 51.01-52.00 1.00 3.18E+06 ...-3.270E+06 1.551E+05 2.16E-16 2.05E-17 43 52.50 52.01-53.00 1.00 3.35E+06 ...-3.429E+06 1.589E+05 3.41E-18 3.26E-19 44 53.22 53.01-53.38 0.38 3.46E+06 ...-3.490E+06 6.136E+04 2.72E-18 2.60E-19 Table 28 . 28Grains with a = 0.2 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 8.80 × 10 22 H atoms cm −2 . . T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , LowNo1 10.08 10.01-11.00 1.00 8.94E+02 185.277-1.037E+04 1.019E+04 6.63E-08 9.86E-09 2 11.35 11.01-12.00 1.00 1.45E+04 ...-2.267E+04 1.230E+04 3.49E-10 3.17E-11 3 12.42 12.01-13.00 1.00 2.85E+04 ...-3.723E+04 1.455E+04 7.98E-11 8.55E-12 Table 28 continued 28 Table 28 (continued) 28No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low4 13.44 13.01-14.00 1.00 4.45E+04 ...-5.421E+04 1.699E+04 2.93E-11 3.67E-12 5 14.43 14.01-15.00 1.00 6.25E+04 ...-7.379E+04 1.958E+04 1.58E-11 2.05E-12 6 15.45 15.01-16.00 1.00 8.36E+04 ...-9.611E+04 2.232E+04 8.87E-12 1.30E-12 7 16.46 16.01-17.00 1.00 1.08E+05 ...-1.210E+05 2.493E+04 5.49E-12 8.19E-13 8 17.48 17.01-18.00 1.00 1.35E+05 ...-1.492E+05 2.816E+04 3.42E-12 4.83E-13 9 18.46 18.01-19.00 1.00 1.64E+05 ...-1.805E+05 3.127E+04 2.35E-12 3.32E-13 23 32.57 32.01-33.00 1.00 9.50E+05 ...-9.844E+05 8.147E+04 4.09E-14 5.09E-15 24 33.54 33.01-34.00 1.00 1.03E+06 ...-1.070E+06 8.531E+04 2.41E-14 2.93E-15 25 34.44 34.01-35.00 1.00 1.11E+06 ...-1.159E+06 8.916E+04 2.23E-14 2.75E-15 26 35.38 35.01-36.00 1.00 1.19E+06 ...-1.252E+06 9.303E+04 1.79E-14 2.20E-15 27 36.36 36.01-37.00 1.00 1.29E+06 ...-1.349E+06 9.690E+04 1.67E-14 2.04E-15 28 37.36 37.01-38.00 1.00 1.39E+06 ...-1.450E+06 1.008E+05 1.33E-14 1.64E-15 29 38.53 38.01-39.00 1.00 1.51E+06 ...-1.554E+06 1.047E+05 1.18E-14 1.47E-15 30 39.51 39.01-40.00 1.00 1.61E+06 ...-1.663E+06 1.086E+05 8.16E-15 1.00E-15 31 40.51 40.01-41.00 1.00 1.72E+06 ...-1.775E+06 1.125E+05 6.90E-15 8.68E-16 32 41.56 41.01-42.00 1.00 1.84E+06 ...-1.892E+06 1.164E+05 5.85E-15 7.37E-16 33 42.72 42.01-43.00 1.00 1.98E+06 ...-2.012E+06 1.202E+05 2.75E-15 3.44E-16 34 43.61 43.01-44.00 1.00 2.09E+06 ...-2.137E+06 1.254E+05 2.54E-15 3.18E-16 35 44.47 44.01-45.00 1.00 2.20E+06 ...-2.265E+06 1.281E+05 1.43E-15 1.79E-16 36 45.38 45.01-46.00 1.00 2.32E+06 ...-2.397E+06 1.320E+05 2.30E-15 2.88E-16 37 46.54 46.01-47.00 1.00 2.47E+06 ...-2.533E+06 1.358E+05 1.93E-15 2.43E-16 38 47.50 47.01-48.00 1.00 2.60E+06 ...-2.673E+06 1.397E+05 1.08E-15 1.35E-16 39 48.45 48.01-49.00 1.00 2.74E+06 ...-2.816E+06 1.436E+05 1.55E-15 1.94E-16 40 49.58 49.01-50.00 1.00 2.90E+06 ...-2.964E+06 1.474E+05 1.19E-15 1.50E-16 41 50.50 50.01-51.00 1.00 3.04E+06 ...-3.115E+06 1.512E+05 1.32E-15 1.66E-16 42 51.44 51.01-52.00 1.00 3.18E+06 ...-3.270E+06 1.551E+05 1.09E-16 1.43E-17 43 52.50 52.01-53.00 1.00 3.35E+06 ...-3.429E+06 1.589E+05 1.72E-18 2.28E-19 44 53.22 53.01-53.38 0.38 3.46E+06 ...-3.490E+06 6.136E+04 1.36E-18 1.81E-19 Table 30 . 30Grains with a = 0.2 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 3.52 × 10 23 H atoms cm −2 . . T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , LowNo1 10.06 10.01-11.00 1.00 6.99E+02 185.277-1.037E+04 1.019E+04 4.69E-08 8.90E-09 2 11.36 11.01-12.00 1.00 1.46E+04 ...-2.267E+04 1.230E+04 1.39E-10 2.38E-11 3 12.42 12.01-13.00 1.00 2.86E+04 ...-3.723E+04 1.455E+04 3.33E-11 6.27E-12 4 13.45 13.01-14.00 1.00 4.46E+04 ...-5.421E+04 1.699E+04 1.24E-11 2.66E-12 5 14.43 14.01-15.00 1.00 6.25E+04 ...-7.379E+04 1.958E+04 6.46E-12 1.44E-12 Kalvāns & Kalnin Grains with a = 0.2 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 1.76 × 10 23 H atoms cm −2. T Cr, ∆t Cr , K E Grain, Table 29. LoweV f T , s −1 , High f T , s −1Table 29. Grains with a = 0.2 µm and 0.03 µm thick icy mantle, shielded by interstellar gas with NH = 1.76 × 10 23 H atoms cm −2 . No. T CR , K interval, K ∆T CR , K E grain , eV interval, eV ∆E grain , eV f T , s −1 , High f T , s −1 , Low . K Acharyya, G E Hassel, E Herbst, 10.1088/0004-637X/732/2/73ApJ. 73273Acharyya, K., Hassel, G. E., & Herbst, E. 2011, ApJ, 732, 73, doi: 10.1088/0004-637X/732/2/73 . Y Aikawa, K Furuya, S Yamamoto, N Sakai, 10.3847/1538-4357/ab994aApJ. 897110Aikawa, Y., Furuya, K., Yamamoto, S., & Sakai, N. 2020, ApJ, 897, 110, doi: 10.3847/1538-4357/ab994a . A C A Boogert, T L Huard, A M Cook, 10.1088/0004-637X/729/2/92ApJ. 72992Boogert, A. C. A., Huard, T. L., Cook, A. M., et al. 2011, ApJ, 729, 92, doi: 10.1088/0004-637X/729/2/92 . E M Bringa, R E Johnson, 10.1086/381382ApJ. 603159Bringa, E. M., & Johnson, R. E. 2004, ApJ, 603, 159, doi: 10.1086/381382 . B Q Chen, X W Liu, H B Yuan, Y Huang, M S Xiang, 10.1093/mnras/stv103MNRAS. 4482187Chen, B. Q., Liu, X. W., Yuan, H. B., Huang, Y., & Xiang, M. S. 2015, MNRAS, 448, 2187, doi: 10.1093/mnras/stv103 . J O Clayton, W F Giauque, Journal of the American Chemical Society. 542610Clayton, J. O., & Giauque, W. F. 1932, Journal of the American Chemical Society, 54, 2610 . B T Draine, A Li, 10.1086/320227ApJ. 551807Draine, B. T., & Li, A. 2001, ApJ, 551, 807, doi: 10.1086/320227 . R T Garrod, M Jin, K A Matis, 10.3847/1538-4365/ac3131ApJS. 2591Garrod, R. T., Jin, M., Matis, K. A., et al. 2022, ApJS, 259, 1, doi: 10.3847/1538-4365/ac3131 . W F Giauque, C J Egan, 10.1063/1.1749929JChPh. 545Giauque, W. F., & Egan, C. J. 1937, JChPh, 5, 45, doi: 10.1063/1.1749929 . T Güver, F &amp;özel, 10.1111/j.1365-2966.2009.15598.xMNRAS. 4002050Güver, T., &Özel, F. 2009, MNRAS, 400, 2050, doi: 10.1111/j.1365-2966.2009.15598.x . T I Hasegawa, E Herbst, MNRAS. 26183Hasegawa, T. I., & Herbst, E. 1993, MNRAS, 261, 83 . S Hocuk, L Szűcs, P Caselli, 10.1051/0004-6361/201629944A&A. 60458Hocuk, S., Szűcs, L., Caselli, P., et al. 2017, A&A, 604, A58, doi: 10.1051/0004-6361/201629944 . N Indriolo, D A Neufeld, M Gerin, 10.1088/0004-637X/800/1/40ApJ. 80040Indriolo, N., Neufeld, D. A., Gerin, M., et al. 2015, ApJ, 800, 40, doi: 10.1088/0004-637X/800/1/40 . W Iqbal, V Wakelam, 10.1051/0004-6361/201732486A&A. 61520Iqbal, W., & Wakelam, V. 2018, A&A, 615, A20, doi: 10.1051/0004-6361/201732486 . A V Ivlev, V A Dogiel, D O Chernyshov, 10.3847/1538-4357/aaadb9ApJ. 85523Ivlev, A. V., Dogiel, V. A., Chernyshov, D. O., et al. 2018, ApJ, 855, 23, doi: 10.3847/1538-4357/aaadb9 . A V Ivlev, T B Röcker, A Vasyunin, P Caselli, 10.1088/0004-637X/805/1/59ApJ. 80559Ivlev, A. V., Röcker, T. B., Vasyunin, A., & Caselli, P. 2015, ApJ, 805, 59, doi: 10.1088/0004-637X/805/1/59 . E B Jenkins, 10.1088/0004-637X/700/2/1299ApJ. 7001299Jenkins, E. B. 2009, ApJ, 700, 1299, doi: 10.1088/0004-637X/700/2/1299 . J Kalvāns, 10.3847/0067-0049/224/2/42ApJS. 57342A&A. Paper IKalvāns, J. 2015, A&A, 573, A38 -. 2016, ApJS, 224, 42 (Paper I), doi: 10.3847/0067-0049/224/2/42 . 10.1093/mnras/sty1172doi: 10.1093/mnras/sty1172MNRAS. 23962753ApJS-. 2018a, ApJS, 239, 6 (Paper II), doi: 10.3847/1538-4365/aae527 -. 2018b, MNRAS, 478, 2753, doi: 10.1093/mnras/sty1172 . J Kalvāns, J R Kalnin, 10.1093/mnras/stz1010MNRAS. 4862050Kalvāns, J., & Kalnin, J. R. 2019, MNRAS, 486, 2050, doi: 10.1093/mnras/stz1010 . J Kalvāns, 10.3847/1538-4365/ac5830doi: 10.3847/1538-4365/ac5830ApJS. 91068ApJ. Paper IIaKalvāns, J. 2021, ApJ, 910, 54, doi: 10.3847/1538-4357/abe30d -. 2022, ApJS, 259, 68 (Paper IIa), doi: 10.3847/1538-4365/ac5830 . J Kalvāns, J R Kalnin, 10.1051/0004-6361/202037906A&A. 64149Kalvāns, J., & Kalnin, J. R. 2020, A&A, 641, A49, doi: 10.1051/0004-6361/202037906 . C Koike, H Hasegawa, A Manabe, 10.1007/BF00642401Ap&SS. 67495Koike, C., Hasegawa, H., & Manabe, A. 1980, Ap&SS, 67, 495, doi: 10.1007/BF00642401 . A Leger, M Jura, A Omont, A&A. 144147Leger, A., Jura, M., & Omont, A. 1985, A&A, 144, 147 . D Lenz, B S Hensley, O Doré, 10.3847/1538-4357/aa84afApJ. 84638Lenz, D., Hensley, B. S., & Doré, O. 2017, ApJ, 846, 38, doi: 10.3847/1538-4357/aa84af . A Li, J M Greenberg, A&A. 323566Li, A., & Greenberg, J. M. 1997, A&A, 323, 566 . H Liszt, 10.1088/0004-637X/780/1/10ApJ. 78010Liszt, H. 2014, ApJ, 780, 10, doi: 10.1088/0004-637X/780/1/10 . R O&apos;donoghue, S Viti, M Padovani, T James, 10.3847/1538-4357/ac7963ApJ. 93463O'Donoghue, R., Viti, S., Padovani, M., & James, T. 2022, ApJ, 934, 63, doi: 10.3847/1538-4357/ac7963 . M Padovani, P Hennebelle, D Galli, 10.1051/0004-6361/201322407A&A. 560114Padovani, M., Hennebelle, P., & Galli, D. 2013, A&A, 560, A114, doi: 10.1051/0004-6361/201322407 . M Padovani, A V Ivlev, D Galli, P Caselli, 10.1051/0004-6361/201732202A&A. 614111Padovani, M., Ivlev, A. V., Galli, D., & Caselli, P. 2018, A&A, 614, A111, doi: 10.1051/0004-6361/201732202 . T Pauly, R T Garrod, 10.3847/0004-637X/817/2/146ApJ. 817146Pauly, T., & Garrod, R. T. 2016, ApJ, 817, 146, doi: 10.3847/0004-637X/817/2/146 . 10.3847/1538-4357/aaa96aApJ. 85413-. 2018, ApJ, 854, 13, doi: 10.3847/1538-4357/aaa96a . P Predehl, J H M M Schmitt, A&A. 293Predehl, P., & Schmitt, J. H. M. M. 1995, A&A, 293 . P Predehl, J Truemper, A&A. 29029Predehl, P., & Truemper, J. 1994, A&A, 290, L29 . J M Rawlings, 10.1093/mnras/stac2154MNRAS. Rawlings, J. M. C. 2022, MNRAS, doi: 10.1093/mnras/stac2154 . L Reboussin, V Wakelam, S Guilloteau, F Hersant, 10.1093/mnras/stu462MNRAS. 4403557Reboussin, L., Wakelam, V., Guilloteau, S., & Hersant, F. 2014, MNRAS, 440, 3557, doi: 10.1093/mnras/stu462 . C Reina, M Tarenghi, A&A. 26257Reina, C., & Tarenghi, M. 1973, A&A, 26, 257 . J F Roberts, J M C Rawlings, S Viti, D A Williams, 10.1111/j.1365-2966.2007.12402.xMNRAS. 382733Roberts, J. F., Rawlings, J. M. C., Viti, S., & Williams, D. A. 2007, MNRAS, 382, 733, doi: 10.1111/j.1365-2966.2007.12402.x . C J Shen, J M Greenberg, W A Schutte, E F Van Dishoeck, 10.1051/0004-6361:20031669A&A. 415203Shen, C. J., Greenberg, J. M., Schutte, W. A., & van Dishoeck, E. F. 2004, A&A, 415, 203, doi: 10.1051/0004-6361:20031669 . C N Shingledecker, S Incerti, A Ivlev, 10.3847/1538-4357/abbb30ApJ. 904189Shingledecker, C. N., Incerti, S., Ivlev, A., et al. 2020, ApJ, 904, 189, doi: 10.3847/1538-4357/abbb30 . C N Shingledecker, R Le Gal, E Herbst, 10.1039/C7CP01472DPhysical Chemistry Chemical Physics (Incorporating Faraday Transactions). 1911043Shingledecker, C. N., Le Gal, R., & Herbst, E. 2017, Physical Chemistry Chemical Physics (Incorporating Faraday Transactions), 19, 11043, doi: 10.1039/C7CP01472D . L M Shulman, 10.1051/0004-6361:20031746A&A. 416187Shulman, L. M. 2004, A&A, 416, 187, doi: 10.1051/0004-6361:20031746 . K Silsbee, P Caselli, A V Ivlev, 10.1093/mnras/stab2546MNRAS. 5076205Silsbee, K., Caselli, P., & Ivlev, A. V. 2021, MNRAS, 507, 6205, doi: 10.1093/mnras/stab2546 . K Silsbee, A V Ivlev, M Padovani, P Caselli, 10.3847/1538-4357/aad3cfApJ. 863188Silsbee, K., Ivlev, A. V., Padovani, M., & Caselli, P. 2018, ApJ, 863, 188, doi: 10.3847/1538-4357/aad3cf . K Silsbee, A V Ivlev, O Sipilä, P Caselli, B Zhao, 10.1051/0004-6361/202038063A&A. 64139Silsbee, K., Ivlev, A. V., Sipilä, O., Caselli, P., & Zhao, B. 2020, A&A, 641, A39, doi: 10.1051/0004-6361/202038063 . O Sipilä, K Silsbee, P Caselli, 10.3847/1538-4357/ac23ceApJ. 922126Sipilä, O., Silsbee, K., & Caselli, P. 2021, ApJ, 922, 126, doi: 10.3847/1538-4357/ac23ce . O Sipilä, B Zhao, P Caselli, 10.1051/0004-6361/202038353A&A. 64094Sipilä, O., Zhao, B., & Caselli, P. 2020, A&A, 640, A94, doi: 10.1051/0004-6361/202038353 . V Taquet, K Furuya, C Walsh, E F Van Dishoeck, 10.1093/mnras/stw2176MNRAS. 46299Taquet, V., Furuya, K., Walsh, C., & van Dishoeck, E. F. 2016, MNRAS, 462, S99, doi: 10.1093/mnras/stw2176 . D B Vaidya, R Gupta, T P Snow, 10.1111/j.1365-2966.2007.11975.xMNRAS. 379791Vaidya, D. B., Gupta, R., & Snow, T. P. 2007, MNRAS, 379, 791, doi: 10.1111/j.1365-2966.2007.11975.x . L A Valencic, R K Smith, 10.1088/0004-637X/809/1/66ApJ. 80966Valencic, L. A., & Smith, R. K. 2015, ApJ, 809, 66, doi: 10.1088/0004-637X/809/1/66 . A I Vasyunin, P Caselli, F Dulieu, I Jiménez-Serra, 10.3847/1538-4357/aa72ecApJ. 84233Vasyunin, A. I., Caselli, P., Dulieu, F., & Jiménez-Serra, I. 2017, ApJ, 842, 33, doi: 10.3847/1538-4357/aa72ec . V Wakelam, E Dartois, M Chabot, 10.1051/0004-6361/202039855A&A. 65263Wakelam, V., Dartois, E., Chabot, M., et al. 2021, A&A, 652, A63, doi: 10.1051/0004-6361/202039855 . D C B Whittet, P A Gerakines, J H Hough, S S Shenoy, 10.1086/318421ApJ. 547872Whittet, D. C. B., Gerakines, P. A., Hough, J. H., & Shenoy, S. S. 2001, ApJ, 547, 872, doi: 10.1086/318421 . Y Xie, L C Ho, A Li, J Shangguan, 10.3847/1538-4357/aae2b0ApJ. 86791Xie, Y., Ho, L. C., Li, A., & Shangguan, J. 2018, ApJ, 867, 91, doi: 10.3847/1538-4357/aae2b0 . B Zhao, P Caselli, Z.-Y Li, 10.1093/mnras/sty1165MNRAS. 4782723Zhao, B., Caselli, P., & Li, Z.-Y. 2018, MNRAS, 478, 2723, doi: 10.1093/mnras/sty1165 . J F Ziegler, M D Ziegler, J P Biersack, 10.1016/j.nimb.2010.02.091Nuclear Instruments and Methods in Physics Research B. 2681818Ziegler, J. F., Ziegler, M. D., & Biersack, J. P. 2010, Nuclear Instruments and Methods in Physics Research B, 268, 1818, doi: 10.1016/j.nimb.2010.02.091 . W Zuo, A Li, G Zhao, 10.3847/1538-4365/abcc6dApJS. 25222Zuo, W., Li, A., & Zhao, G. 2021, ApJS, 252, 22, doi: 10.3847/1538-4365/abcc6d
[]
[ "Non-Stationary Bandits with Knapsack Problems with Advice", "Non-Stationary Bandits with Knapsack Problems with Advice" ]
[ "Lixing Lyu ", "Chi Wang ", "Cheung " ]
[]
[]
We consider a non-stationary Bandits with Knapsack problem. The outcome distribution at each time is scaled by a non-stationary quantity that signifies changing demand volumes. Instead of studying settings with limited non-stationarity, we investigate how online predictions on the total demand volume Q allows us to improve our performance guarantees. We show that, without any prediction, any online algorithm incurs a linear-in-T regret. In contrast, with online predictions on Q, we propose an online algorithm that judiciously incorporates the predictions, and achieve regret bounds that depends on the accuracy of the predictions. These bounds are shown to be tight in settings when prediction accuracy improves across time.Our theoretical results are corroborated by our numerical findings.
10.48550/arxiv.2302.04182
[ "https://export.arxiv.org/pdf/2302.04182v1.pdf" ]
256,662,434
2302.04182
4f9dca8949bf11223b3eb06292cdfbfa0b27902e
Non-Stationary Bandits with Knapsack Problems with Advice February 9, 2023 Lixing Lyu Chi Wang Cheung Non-Stationary Bandits with Knapsack Problems with Advice February 9, 2023 We consider a non-stationary Bandits with Knapsack problem. The outcome distribution at each time is scaled by a non-stationary quantity that signifies changing demand volumes. Instead of studying settings with limited non-stationarity, we investigate how online predictions on the total demand volume Q allows us to improve our performance guarantees. We show that, without any prediction, any online algorithm incurs a linear-in-T regret. In contrast, with online predictions on Q, we propose an online algorithm that judiciously incorporates the predictions, and achieve regret bounds that depends on the accuracy of the predictions. These bounds are shown to be tight in settings when prediction accuracy improves across time.Our theoretical results are corroborated by our numerical findings. Introduction The multi-armed bandit problem (MAB) is a classical model on sequential decision making. The MAB problem features the trade-off between exploration and exploitation, i.e., between exploring for new information about the underlying system and exploiting the potentially optimal solution based on current information. MAB problems have been studied extensively over many decades, with diverse applications such as recommendation systems, ad allocation, resource allocation, revenue management and network routing/scheduling. Many of these mentioned applications involve resource constraints. For example, a seller experimenting with product prices may have limited product inventory. This motivates the formulation of the bandits with knapsack (BwK) problem, an online knapsack problem involving model uncertainty introduced by Badanidiyuru et al. [2013]. The agent has d ≥ 1 types of resources. At each time step t, the agent pulls an arm, which generates an array of outcomes consisting of the random reward the amounts of the d resources consumed. If some resource(s) are exhausted, then the agent stops pulling any arm. The objective is to maximize the expected total reward over a known horizon T , subject to the budget constraints on the d resources. The BwK problem was first studied in a stationary stochastic setting, dubbed stochastic BwK, where the outcome of an arm follows a stationary but latent probability distribution. Badanidiyuru et al. [2013], Agrawal and Devanur [2014] provide online algorithms for stochastic BwK with regret sub-linear in T . The regret is the difference between the optimum and the expected cumulative reward earned by the algorithm, and a sub-linear-in-T regret implies the convergence to opimality as T grows. An alternative setting, adversarial BwK, is introduced by Immorlica et al. [2019]. Each arm's outcome distribution can change arbitrarily over time. Contrary to stochastic BwK, it is impossible to achieve a regret sub-linear in T , even when the outcome distribution is changed only once during the horizon [Liu et al., 2022]. It begs a question: could a non-stationary BwK problem be more tractable under a less adversarial model than Immorlica et al. [2019], Liu et al. [2022]? Practically, while stationary models could be a strong assumption, it could be too pessimistic to assume the underlying model to be adversarily changing and completely latent. Consider the example of ad allocation. On one hand, the internet traffic is constantly changing, leading to a non-stationary model. On the other hand, forecast information is often available. Some advertisements are intrinsically more attractive than others, regardless of the internet traffic, meaning that the click probability of an ad can be regarded as stationary. How could the platform harness the probem structure? Can the allocator utilize forecast information on the internet traffic to improve his/her decisions? Motivated by the discussions above, we consider the Non-Stationary BwK with Online Advice problem (NS-BwK-OA). An outcome involve an adversarial and a stochastic component. The mean reward and mean type-i resource consumption of pulling arm a at time t are equal to q t · r(a) and q t · c(a, i) respectively. The quantity q t ∈ R >0 is a seasonal term that represents the demand volume at time t, while the stationary quantities r(a), c(a, i) are the mean reward earned and resource consumed per demand unit under arm a. In a dynamic pricing setting, q t counts the customers arrivals at time t. The quantities r(a), c(a, i) respectively represent the revenue earned and type-i resource consumed per customer under arm a, which could represent a pricing scheme. Our proposed outcome model generalizes the unconstrained settings in [Tracà et al., 2021, Lykouris et al., 2020. We make the following novel contributions: Model. We incoporate a prediction oracle in NS-BwK-OA, a non-stationary BwK problem. Such an incorporation is novel compared to existing BwK works, which always assume full model uncertainty. In NS-BwK-OA, we identify the total demand volume Q = T t=1 q t as a crucial (but latent) parameter. The oracle provides a predictionQ t to Q at every time step t. The oracle corresponds to how a firm constantly acquires updated information about Q, and a variety of existing time series prediciton tools can be used to construct such an oracle. Regret Lower Bounds. We derive two regret lower bounds on NS-BwK-OA. First, without the access to a prediction oracle, we show that any online algorithm suffers a linear-in-T regret, even when r, c are known and q t is known before the arm at time t is to be chosen. Second, with the access of a prediction oracle, we establish regret lower bounds that depend on the accuracy of the estimations. When the estimates are equal to the groundtruth, the regret lower bounds reduce to that of the stochastic BwK problem. Algorithms and Regret Upper Bounds. We design an online algorithm OA-UCB to utilize the predictions judiciously. OA-UCB is novel in its incorporation of the predictionQ t and demand volume q t into the estimated opportunity costs of the resources, in relation to the predicted demand volumes. We derive a regret upper bound on OA-UCB that depends on the accuracy of the predictions, even though the accuracy of each prediction is not known to the algorithm. OA-UCB is shown to achieve near optimal regret bounds when the accuracy of the predictions improves across time. Numerical Validations. We perform numerical experiments when {q t } T t=1 is governed by a time series model. The experiment highlights the benefit of predicitons. We show that an online algorithm, such as OA-UCB, that harnesses predictions judiciously can perform empirically better than existing baselines, which only has access to the bandit feedback from the latent environment. Related Work The Bandits with Knapsacks (BwK) problem has been extensively studied. Badanidiyuru et al. [2013] first introduced the stochastic BwK problem, which bears applications in dynamic pricing Besbes andZeevi [2009, 2012] and ad allocation Mehta et al. [2007]. The BwK problem is generalized by Agrawal and Devanur [2019], Liu et al. [2022] and our forthcoming Lemma 2, for any online algorithm, there exists a non-stationary BwK instance where the outcome distribution only changes oncee during the horizon, for which the algorihtm incurs a linear-in-T regret. Non-stationary stochastic bandits with no resource constraints are studied in [Besbes et al., 2014, Cheung et al., 2019, Zhu and Zheng, 2020, who provide sub-linear-in-T regret bounds in less restrictive non-stationary settings than Liu et al. [2022], while the amount of non-stationariety, quantified as the variational budget or the number of change points, has to be sub-linear in T . Our work goes in an orthogonal direction. Instead of studying settings with limited non-stationariety, we seek an improved regret bound when the decision maker is endowed with information additional (in the form of prediction oracle) to the online observations. Our work is related to an active stream of research works on online algorithm design with machine Lastly, our prediction model is also related to a line of works online optimization with predictions, which concerns improving the performance guarantee with the help of predictions. These predictions are provided to the DM at the beginning of each round sequentially. A variety of full feedback settings are studied in Rakhlin and Sridharan [2013a,b], Steinhardt and Liang [2014], Jadbabaie et al. [2015], and the contextual bandit setting is studied in Wei et al. [2020]. We remark that the abovementioned works do not involve resource constraints, and they are fundamental different from ours, as shown in the forthcoming Lemma 2. Notation. For a positive integer d, we denote [d] = {1, . . . , d}. We adopt the O(·), o(·), Ω(·) notation in Cormen et al. [2009]. Problem Setup We consider the Non-Stationary Bandit with Knapsack problem with Online Advice (NS-BwK-OA). The nature specifies an NS-BwK-OA problem instance, represented by the tuple (A, B, T, {q t } T t=1 , {P a } a∈A ). We denote A as the set of K arms. There are d types of resources, and the decision maker (DM) is endowed with B i ≥ 0 units of resource i for each i ∈ [d]. The planning horizon consists of T discrete time steps. Following the convention in [Badanidiyuru et al., 2013], we assume for all i ∈ [d] that B i = B = bT , where b is the normalized budget. At time t, there are q t units of demands arriving at the DM's platform. For example, q t can be the number of customers visiting an online shop at time step t, and a time step can be a fifteen minute interval. We assume q t ∈ [q, q], where 0 < q < q. The sequence {q t } T t=1 is an arbitrary element of [q, q] T fixed by the nature before the online dynamics. The arbitrariness represents the exogenous nature of the demands. When the DM pulls arm a ∈ A, the nature samples a vector (R(a), C(a, 1), . . . , C(a, d)) ∼ P a of random outcomes. The quantity R(a) is the reward earned per demand unit, and C(a, i) is the amount of type i resource consumed per demand unit. The random variables R(a), C(a, 1), . . . , C(a, d) are supported in [0, 1], and they can be arbitrarily correlated. We denote r(a) = E[R(a)], c(a, i) = E[C(a, i)], and r = (r(a)) a∈A , c = (c(a, i)) a∈A,i∈ [d] . To ensure feasiblity, we assume there is a null arm a 0 ∈ A such that R(a 0 ) = C(a 0 , i) = 0 with certainty for all i ∈ [d]. At each time t, the DM is provided with a prediction oracle F t . The oracle is a function F t : [q, q] t−1 → [qT, qT ] that provides an estimationQ t = F t (q 1 , . . . , q t−1 ) on Q = T t=1 q t with the past observations {q s } t−1 s=1 . The DM knows A, B, T , and has the access to F t in a sequential manner. In contrast, the DM does not know {P a } a∈A , {q t } T t=1 , while the upper bound q is known to the DM. Dynamics. At each time t, three events happen. Firstly, the DM receives a predictionQ t = F t (q 1 , . . . , q t−1 ) on Q. Secondly, based onQ t and the history observation, the DM selects arm A t ∈ A. Thirdly, the DM observes the feedback consisting of (i) demand volume q t , (ii) reward earned q t R t , (iii) resources consumed {q t C t,i } i∈ [d] . Recall that (R t , C t,1 , . . . , C t,d ) ∼ P At . Then, the DM proceeds to time t + 1. If some resource is depleted, i.e. ∃j ∈ [d] such that t s=1 q s C s,j > B j , then the null arm a 0 is to be pulled in the remaining horizon t + 1, . . . , T . We denote the stopping time here as τ . The DM aims to maximize the total reward E[ τ −1 t=1 q t R t ], subject to the resource constraints and model uncertainty. On q t . Our feedback model on q t is more informative than Lykouris et al. [2020], where none of q 1 , . . . , q T is observed during the horizon. In contrast, ours is less informative than Tracà et al. [2021], where q 1 , . . . , q T are all observed at time 1. Our assumption of observing q t at the end of time t is mild in online retail settings. For example, the number of visitors to a website within a time interval can be extracted from the electronic records when the interval ends. While the nature sets {q t } T t=1 to be fixed but arbitrary, the sequence is set without knowing the DM's online algorithm and prediciton oracle F = {F t } T t=1 . Our model is milder than the oblivious adversary model, where the nature sets a latent quantity (in this case {q t } T t=1 ) with the knowledge of the DM's algorithm before the online dynamics. Our milder model allows the possibility ofQ t = F t (q 1 , . . . , q t−1 ) being a sufficiently accurate (to be quantified in our main results) estimate to Q for each t, for example when {q t } T t=1 is govenred by a latent time series model. In contrary, an oblivious adversary can set Q to be far away from the predictionŝ Q 1 , . . . ,Q T in response to the information on F. On F. Our prediction oracle is a general Black-Box model. We do not impose any structural or param-eteric assumption on F or {q t } T t=1 . It is instructive to understand F as a side information provided to the DM by an external source. In the dynamic pricing example,Q t could be an estimate on the customer base population provided by an external marketing research firm. A prime candidate of F is the cornucopia of time series prediction models proposed in decades of research works on time series [Shumway and Stoffer, 2017, Hyndman and Athanasopoulos, 2021, Lim and Zohren, 2021. These prediction models allow one step prediction, where for any t, the predictor P inputs {q s } t−1 s=1 and outputs an estimateq t on q t . The predictionQ t can be constructed by (1) iteratively applying P on {q s } t−1 s=1 ∪ {q} t+ρ−1 t to outputq t+ρ , for ρ ∈ {0, . . . , T − t}, (2) summing over q 1 , . . . , q t−1 ,q t , . . . ,q T and returnQ t . We provide an example in the forthcoming Section 5. Regret. To measure the performance of an algorithm, we define the regret of an algorithm as Regret T = OPT − E τ −1 t=1 q t R t ,(1) where OPT denotes the expected cumulative reward of an offline optimal dynamic policy given all latent information and all adversarial terms. For analytical tractabililty in our regret upper bound, we consider an alternative benchmark OPT LP = max u∈∆ |A| T t=1 q t r u s.t. T t=1 q t c u ≤ B1 d ,(2) where ∆ |A| = {w ∈ [0, 1] d : a∈A w a = 1}. The benchmark (2) is justified by the following Lemma: Lemma 1. OPT LP ≥ OPT. The proof of Lemma 1 is in Appendix. Regret Lower Bounds In this section, we provide impossibility results on the NS-BwK-OA in the form of regret lower bounds. Firstly, we show that a linear-in-T regret is inevitable in the absence of the prediction oracle F. Lemma 2. Consider a fixed but arbitrary online algorithm that knows {P a } a∈A , {(q s , q s R s , q s C s,1 , . . . q s C s,d )} t−1 s=1 and q t , but does not have any access to a prediction oracle when the action A t is to be chosen at each time t. There exists an instance such that the online algorithm suffers Regret T = Ω(T ). In view of Lemma 2, we seek to understand if the DM can avoid Regret T = Ω(T ) when s/he is endowed with an accurate prediction on Q. Certainly, if the DM only recieves an uninformative prediction, such as a worst case prediction qT , at every time step, Regret T = Ω(T ) still cannot be avoided. In contrast, if the DM received an accurate prediction at a time step, we demonstrate our first step for deriving a better regret bound, in the form of a more benign regret lower bound compared to Lemma 2. We formalize the notion of being accurate by the two notions below. For T 0 ∈ [T − 1] and T0+1 ≥ 0, an instance is said to be (T 0 + 1, T0+1 )-well estimated by oracle F, if the predictionQ T0+1 = F T0+1 (q 1 , . . . , Q T0 ) returned by the oracle at time T 0 + 1 satisfies |Q −Q T0+1 | ≤ T0+1 . For T 0 ∈ [T − 1] and a prediction oracle F T0+1 , we say that T0+1 is (T 0 + 1, F T0+1 )-sound if there exists {q t } T0 t=1 ∈ [q, q] T0 such that T0+1 ≤ min{Q T0+1 − T0 s=1 q s −q(T −T 0 ), q(T −T 0 )−(Q T0+1 − T0 t=1 q t ),Q T0+1 /2}, whereQ T0+1 = F T0+1 (q 1 , . . . , q T0 ). The soundness notion imposes an upper bound on T0+1 , which ensures a non-trivial bounding condition in |Q −Q T0+1 | ≤ T0+1 for the "well estimated" notion. Theorem 1. Consider the NS-BwK-OA setting, and consider a fixed but arbitrary online algorithm and prediciton oracle F = {F t } T t=1 . For any T 0 ∈ [T − 1] and any T0+1 > 0 that is (T 0 + 1, F T0+1 )-sound, there exists a (T 0 + 1, T0+1 )-well estimated instance I such that Regret T = Ω max 1 Q T0 t=1 q t T0+1 , Λ ,(3) where {q t } T t=1 is the underlying demand seqeunce of I, and Q = T t=1 q t , and Λ = min OPT, OPT qK B + qKOPT . Theorem 1 is proved in Appendix B.2. In (3), the regret lower bound Λ is due to the uncertainty on {P a } a∈A , and Λ is derived directly from Badanidiyuru et al. [2013]. The regret lower bound 1 Q T0 t=1 q t T0+1 is due to the oracle's estimation error onQ T0+1 . Theorem 1 demonstrates a more benign regret lower bound than Ω(T ), under the condition that the prediction on Q is sufficiently accurate (as formalized as (T 0 + 1, T0+1 )-well estimated). More specifically, let us consider the following accurate prediction condition at time T 0 : T0+1 Q = O(T −α 0 ) for some α > 0.(4) The condition implies that, for the predictionQ T0+1 made using T 0 data points q 1 , . . . , q T0 , it holds that |1 − (Q T0+1 /Q)| = O(T −α 0 ) , which in turn implies thatQ t converges to Q as t grows. For example, when {q t } T t=1 are i.i.d. generated, the accurate prediction condition holds with α = 1/2. Altogether, under the accurate prediction condition, the corollary presents a strictly smaller regret lower bound than that in Lemma 2, which has no prediction oracle available. In complement, we design and analyze an online algorithm in the next section that reaps the benefits of predictions, and in particular nearly matches the regret lower bound in Corollary 1 under the accurate prediction condition. Thus, a o(T )-regret is possible in a non-stationary environment given accurate predictions as prescribed above, even though the amount of non-stationarity in the underlying model is not bounded in general. Algorithm and Analysis We propose the Online-Advice-UCB (OA-UCB) algorithm, displayed in Algorithm 1, for solving NS-BwK-OA. The algorithm design involve constructing confidence bounds to address the model uncertainty on r, c, as discussed in Section 4.1. In Section 4.2, we elaborate on OA-UCB, which uses Online Convex Optimization (OCO) tools to balance the trade-off among rewards and resources. Crucially, at each time t, we incorporate the predictionQ t to scale the opportunity costs of the resources. In addition, both q t andQ t are judiciously integrated into the OCO tools to factor the demand volumes into the consideration of the abovemention trade-off. In Section 4.3, we provide a regret upper bound to OA-UCB, and demonstrate its near-optimality when the accurate prediction condition (4) holds and when capacity is large. In Section 4.4 we provide a sketch proof of the regret upper bound, where the complete proof is in Appendix C. Confidence Bounds We consider the following confidence radius function: rad(v, N, δ) = 2v log 1 δ N + 4 log 1 δ N .(5) The function (5) satisfies the following property: Lemma 3 (Babaioff et al. [2015], Agrawal and Devanur [2014]). Let random variables {V i } N i=1 be independently distributed with support in [0, 1]. DenoteV = 1 N N i=1 V i , then with probability ≥ 1 − 3δ, we have V − E[V ] ≤ rad(V , N, δ) ≤ 3rad(E[V ], N, δ)., {C s,i } i∈[d] } s∈[t−1] , we compute the sample meanŝ R t (a) = 1 N + t−1 (a) t−1 s=1 R s 1 {As=a} , ∀a ∈ A, C t (a, i) = 1 N + t−1 (a) t−1 s=1 C s,i 1 {As=a} , ∀a ∈ A, i ∈ [d], where N + t−1 (a) = max{ t−1 s=1 1 {As=a} , 1}. In line with the principle of Optimism in Face of Uncertatinty, we construct upper confidence bounds (UCBs) for the rewards and lower confidence bounds (LCBs) for resource consumption ammounts. For each a ∈ A, we set UCB r,t (a) = min R t (a) + rad(R t (a), N + t−1 (a), δ), 1 . (6) For each a ∈ A, i ∈ [d], we set LCB c,t (a, i) = max Ĉ t (a, i) − rad(Ĉ t (a, i), N + t−1 (a), δ), 0 .(7) The design of the UCBs and LCBs are justified by Lemma 3 and the model assumption that r(a), c(a, i) ∈ [0, 1] for all a ∈ A, i ∈ [d]: Lemma 4. With probability ≥ 1 − 3KT dδ, we have UCB r,t (a) ≥ r(a), LCB c,t (a, i) ≤ c(a, i) for all a ∈ A, i ∈ [d]. Lemma 4 is proved in Appendix D.2. Details on OA-UCB OA-UCB is presented in Algorithm 1. At each time step t, the algorithm first computes a composite reward term UCB r,t (a) −Q t B · µ t LCB c,t (a),(8) where UCB r,t (a),Q t and LCB c,t (a) are the surrogates for the latent r(a), Q, c(a) respectively. The termQ t B · µ t LCB c,t (a) can be interpreted as the opportunity cost of the resources. The scalarization µ t ∈ S = {µ : µ 1 ≤ 1, µ ≥ 0 d }(9) weighs the relative importance of the resources. The factorQ t /B reflects that the opportunity cost increases withQ t , since with a higher total demand volume, the DM is more likely to exhaust some of the resources during the horizon, and similar reasoning holds for B. Altogether, (8) balances the trade-off between the reward of an arm and the opportunity cost of that arm's resource consumption. We select an arm that maximizes (8) at time t. After receiving the feedback, We update the scalarization µ t via the Online Gradient Descent (OGD) Agrawal and Devanur [2014], Hazan et al. [2016] on the sequence of functions {f t } T t=1 , where f t (x) = q tQt B B Q t 1 d − LCB c,t (A t ) x.(10) While f t incorporates the predictionQ t for the purpose of accounting for the estimated opportunity cost similar to (8), f t also incoporates the actual demand q t for accounting the actual amounts of resources consumed. In the OGD update in (11), for a resource type i, the coefficient µ t+1 (i) increases with q t LCB c,t (A t , i), meaning that a higher amount of resource i consumption at time t leads to a higher weightage of resource i's opportunity cost at time t + 1. Performance Guarantees of OA-UCB The following theorem provides a high-probability regret upper bound for Algorithm 1: Algorithm 1 Online-advice-UCB (OA-UCB) 1: Initialize µ 1 = 1 d 1 d , M = q + q 2 b √ d, η t = √ 2 M √ t . 2: for t = 1, 2, ..., T do 3: ReceiveQ t = F t (q 1 , . . . , q t−1 ). 4: Compute UCB r,t (a), LCB c,t (a) for all a ∈ A by (6), (7), where LCB c,t (a) = (LCB c,t (a, i)) i∈[d] . 5: Select A t ∈ argmax a∈A UCB (r) t (a) −Q t B · µ t LCB c,t (a) . 6: if ∃j ∈ [d] such that t s=1 q s C t,j > B then 7: Break, and pull the null arm a 0 all the way. 8: Observe q t , receive reward q t R t , and consume q t C t,i for each resource i ∈ [d]. 9: Update µ t+1 with OGD. µ t+1 is set to be Π S µ t − η t q tQt B B Q t 1 d − LCB c,t (A t ) ,(11) where S is defined in (9). Theorem 2. Consider the OA-UCB algorithm, that is provided with predictions that satisfy |Q t − Q| ≤ t for all t ∈ [T ]. With probability ≥ 1 − 3KT dδ, OPT LP − τ −1 t=1 q t R t ≤ O OPT LP qK B + qKOPT LP log 1 δ (12) + 1 Q + 1 B τ −1 t=1 q t t + M √ T .(13) Theorem 2 is proved in the Appendix, and we provide a sketch proof in Section 4.4. The Theorem holds in the special case when we set t = |Q t − Q|, and t represents an upper bound on the estimation error ofQ t on Q, for example by certain theoretical guarantees. The term (12) represents the regret due to the learning on r(a), c(a, i). The first term in (13) represents the regret due to the prediction error of the prediction oracle, and the second term in (13) represents the regret due to the application of OGD. Comparison between regret lower and upper bounds. The regret term (12) matches the lower bound term Λ in Theorem 1 within a logarithmic factor. Next, we compare the regret upper bound term ( 1 Q + 1 B ) τ −1 t=1 q t t and the lower bound term 1 Q T0 t=1 q t T0+1 in Theorem 1. We first assure that the lower and upper bound results are consistent, in the sense that our regret upper bound is indeed in Ω( 1 Q T0 t=1 q t T0+1 ) on the lower bounding instances constructed for the proof of Theorem 1. In those instances, T 0 is set in a way that the resource is not fully exhausted at time T 0 under any policy, thus the stopping time τ of OA-UCB satisfies τ > T 0 with certainty. More details are provided in Appendix B.3. Next, we highlight that the regret upper and lower bounds are nearly matching (modulo multiplicative factors of log(1/δ) and q/q, as well as the additive O(M √ T ) term), under the high capacity condition B = Θ(Q) and the accurate prediction condition (4) for each t ∈ [T ]. The first condition is similar to the large capacity assumption in the literature Besbes and Zeevi [2009,2012], Liu et al. [2022], while the second condition is a natural conditon that signfies a non-trivial estimation by the prediciton oracle, as discussed in Section 3. On one hand, by setting T 0 = Θ(T ) for the highest possible lower bound in Corollary 1, we yield the regret lower bound Ω(max{qT 1−α , Λ}). On the other hand, the second term in (13) is upper bounded as 1 Q + 1 B τ −1 t=1 q t t = O τ −1 t=1 q t t Q = O τ −1 t=1 q t t −α = O(qT 1−α ). Altogether, our claim on the nearly matching bounds is established. Proof Sketch of Theorem 2 We provide an overview on the proof of Theorem 2, which is fully proved in Appendix C. We first provide bounds on the regret induced by the estimation errors of the UCBs and LCBs. Now, with probability ≥ 1 − 3KT dδ, the inequalities τ −1 t=1 q t UCB r,t (A t ) − τ −1 t=1 q t R t ≤ O   log 1 δ   qK τ −1 t=1 q t R t + qK log T K     ,(14)τ −1 t=1 q t LCB c,t (A t , i) − τ −1 t=1 q t C t,i ≤ O log 1 δ qKB + qK log T K ∀i ∈ [d](15) hold. Inequalities (14, 15) are proved in Appendix D.3. Next, by the optimality of A t in Line 5 in Algorithm 1, UCB r,t (A t ) −Q t B · µ t LCB c,t (A t ) ≥UCB r,t u * −Q t B · µ t LCB c,t u * , which is equivalent to UCB r,t u * − UCB r,t (A t ) +Q t B µ t B Q t 1 d − LCB c,t u * ≤Q t B · µ t B Q t 1 d − LCB c,t (A t ) . Multiply q t on both side, and sum over t from 1 to τ − 1. By applying the OGD performance guarantee in Hazan et al. [2016] with {f t } T t=1 , S defined in (10, 9) respectively, we argue that, for all µ ∈ S, τ −1 t=1 q t UCB r,t u * − τ −1 t=1 q t UCB r,t (A t ) + τ −1 t=1 q tQ t B · µ t B Q t 1 d − LCB c,t u * ≤ τ −1 t=1 q tQ t B · B Q t 1 d − LCB c,t (A t ) µ + O M √ T . If τ ≤ T , then there exists j 0 ∈ [d] such that τ t=1 q t C t,j0 > B. Take µ = OPT LP Q e j0 ∈ S (This is because OPT LP = Qr u * ≤ Q). Analysis yields OPT LP − τ −1 t=1 q t UCB r,t (A t ) ≤ O log 1 δ OPT LP qK B + 1 Q + 1 B τ −1 t=1 q t t + M √ T .(16) If τ > T , it is the case that τ − 1 = T , and no resource is exhausted at the end of the horizon. Take µ = 0. Similar analaysis to the previous case shows that OPT LP − τ −1 t=1 q t UCB r,t (A t ) ≤ O 1 Q τ t=1 q t t + M √ T .(17) Combine (14), (16), (17) and the fact that OPT LP ≥ τ −1 t=1 q t R t , the theorem holds. Numerical Experiments We present numerical results, and compare our algorithm with several existing algorithms on BwK. Demand sequence {q t } T t=1 : We apply an AR(1) model to generate {q t }: q t = α + βq t−1 + ε t , where ε 1 , . . . ε T ∼ N (0, σ 2 ) are independent. Estimations {Q t } T t=1 : At round t, given history observation {q s } t−1 s=1 , there are many time series prediction tools in Python, MatLab or R that perform predictions to yield {q s } T s=t , whereq s is a estimate on q s . We define the estimationQ t as t−1 s=1 q s + T s=tq s . To achieve time-efficiency, we consider a "power-of-two" policy for updating theQ t on Q, as shown in Algorithm 2. That is, we only recomputeQ t when t = 2 k for some k ∈ N + . The estimation error of Algorithm 2 in terms of additive gap is plotted in Figure 1. Algorithm 2 Estimation Generation Policy Input: Time step t, history observation {q s } t−1 s=1 , previous estimationQ t−1 . 1: if t = 2 k for some k ∈ N + then 2: Compute predictions {q s } T s=t , and updateQ t = t−1 s=1 q s + T s=tq s . 3: else 4: SetQ t =Q t−1 . 5: returnQ t . Benchmarks: We compare OA-UCB with three existing online algorithms for BwK. The first algorithm is the PrimalDualBwK algorithm in Badanidiyuru et al. [2013], which we call "PDB" in the following. The second algorithm is the UCB algorithm presented in Agrawal and Devanur [2014], which we call "AD-UCB". The third algorithm is the Sliding-Window UCB in Liu et al. [2022], which we call "SW-UCB". In implementing SW-UCB, we set the sliding window size according to the suggestion in Liu et al. [2022], and we input the required non-stationarity measures by computing them from the ground truth {q t } 15000 t=1 . In the experiment, we simulate our algorithm and the benchmarks on a family of instances, with K = 10, d = 3, b = 3, α = 2, β = 0.5, σ = 0.5, and T varies from 5000 to 15000. Each arms's per-unit-demand outcome (R(a), {C i (a)} d i=1 ) follows the standard Gaussian distribution truncated in [0, 1] d+1 , which has mean denoted as (r(a), c(a)). We perform two groups of the experiment. In each group, we first generate a sample of (r, c, {q t } 15000 t=1 ). Then, for each fixed T , we simulate each algorithm ten times with demand volume sequence {q t } T t=1 , and compute the regret based on the sample average. OA-UCB, PDB and SW-UCB could appear conservative, meaning that they focus too much on not exhausting resources. In contrast, AD-UCB seems a little aggressive, meaning that it prefers to choose the arm with high reward and resource consumption. Conclusion We study a non-stationary bandit with knapsack problem, in the presense of a prediction oracle on the latent total demand volume Q. Our oracle is novel compared to existing models in online optimization with A Proof for Section 2 A.1 Proof for Lemma 1 Let's first consider OPT LP = max xt∈∆ |A| , ∀t∈[T ] T t=1 q t r x t s.t. T t=1 q t c x t ≤ B1 d ,(18) where ∆ |A| = {w : a∈A w a = 1} is the probability simplex across all arms. It is evident that OPT LP ≥ OPT, since for a fixed policy π that achieves OPT, the solutionx = {x t,a } t∈[T ],a∈A defined as x t,a = E[1(action a is chosen at t under π)] is feasible to OPT , and the objective value ofx in OPT LP is equal to the expected revenue earned in the online process. Next, we claim that OPT LP = OPT LP . Indeed, for each feasible solution (x t ) t∈[T ] to OPT LP , the solution u = T t=1 q t x t T t=1 q t , is feasible to LP OPT and has the same objective value as (x t ) t∈ [T ] . Altogether, the Lemma is proved. B Proofs for Section 3, and Consistency Remarks In this section, we provide proofs to the lower bound results. In both proofs, we consider an arbitrary but fixed deterministic online algorithm, that is, conditioned on the realization of the history in 1, . . . , t − 1 and q t ,Q t , the chosen arm A t is deterministic. This is without loss of generality, since the case of random online algorithm can be similarly handled by replace the chosen arm A t with a probability distribution over the arms, but we focus on deterministic case to ease the exposition. Lastly, in Section B.3 we demonstrate that our regret upper and lower bounds are consistent on the lower bounding instances we constructed in Section B.2. B.1 Proof for Lemma 2 Our lower bound example involve two instances I (1) , I (2) with determinstic rewards and deterministic consumption amounts. Both instances involve two non-dummy arms 1, 2 in addition to the null arm a 0 , and there is d = 1 resource type. Instances I (1) , I (2) differ in their respective seqeunces of demand volumes {q (1) t } T t=1 , {q(2) t } T t=1 , but for other parameters are the same in the two instances. In both I (1) , I (2) , arm 1 is associated with (deterministc) reward r(1) = 1 and (deterministic) consumption amount c(1, 1) = 1, while arm 2 is associated with (deterministc) reward r(2) = 3/4 and (deterministic) consumption amount c(2, 1) = 1/2. Both instances share the same horizon T , a positive even integer, and the same capacity B = T /2. The sequences of demand volumes {q Then the optimal reward for I (1) is at least T 2 (always select the arm 1 until the resource is fully consumed), and the optimal reward for I (2) is 3T 4 (always select arm 2 until the resource is fully consumed). Consider the first T /2 rounds, and consider an arbitrary online algorithm that knows {P a } a∈A , the sequence {(q s , q s R s , q s C s,1 , . . . q s C s,d )} t−1 s=1 and the time t demand q t when the action A t is to be chosen at each time t. Under this setting, the DM recieves the same set of observations in the first T /2 time steps in each of instances I (1) , I (2) . Consequently, the sequence of arm pulls in the first T /2 time steps are the same. Now, we denote N a = T /2 t=1 1(A t = a) for a ∈ {1, 2}. By the previous remark, N a is the number of times arm a is pulled during time steps 1, . . . , T /2 in each of the two instances. Observe that N 1 + N 2 ≤ T 2 , which implies N 1 ≤ T 4 or N 2 ≤ T 4 . We denote Reward T (I (i) ), Regret T (I (i) ) as the expected reward and the expected regret of the policy in instance I (i) . In what follows, we demonstrate that max i∈{1,2} (1) t } T t=1 , {q(2)Regret T (I (i) ) ≥ T 32 ,(19) which proves the Lemma. Case 1: N 1 ≤ T 4 . We consider the algorithm on I (1) , which earns Reward T (I (1) ) ≤ T 4 · 1 + T 4 · 3 4 + T 2 1 16 = 15 32 T. Hence, Regret T (I (1) ) ≥ T 2 − Reward T (I (1) ) ≥ 1 32 T. Case 2: N 2 ≤ T 4 . We consider the algorithm on I (2) , which earns Reward T (I (2) ) = T 2 − N 2 · 1 + N 2 · 3 4 + T 2 − T 2 − N 2 · 1 − N 2 · 1 2 · 3 4 1 2 = T 2 + N 2 4 ≤ 9 16 T. Hence, Regret T (I (2) ) ≥ 3 4 T − Reward T (I (2) ) ≥ 3 16 T. Altogether, the inequality (19) is shown. B.2 Proof for Theorem 1 By the Theorem's assumption that T0+1 > 0 is (T 0 + 1, F T0+1 )-sound, we know that there exists {q t } T0 t=1 ∈ [q, q] T0 such that 0 < T0+1 ≤ min Q T0+1 − T0 s=1 q s − q(T − T 0 ), q(T − T 0 ) −Q T0+1 − T0 t=1 q t ,Q T0+1 2 ,(20) whereQ T0+1 = F T0+1 (q 1 , . . . , q T0 ). To proceed, we take a demand volume seqeunce {q t } T0 t=1 ∈ [q, q] T0 that satisfies the assumption. In what follows, we first construct two deterministic instances I (1) , I (2) which only differ in their respective seqeunces of demand volumes {q (1) t } T t=T0+1 , {q(2) t } T t=T0+1 , but the two instances are the same on other parameters, and that q (1) t = q (2) t = q t for t ∈ {1, . . . , T 0 }. Both I (1) , I (2) only involve one resource constraint. We estbalish the Theorem by showing three claims: 1. Both I (1) , I (2) are (T 0 + 1, T0+1 )-well-estimated by F, and the underlying online algorithm and prediction oracle (which are assumed to be fixed but arbitrary in the Theorem statement) suffer Regret T (I (i) ) ≥ T0 t=1 q t T0+1 6Q (i) for some i ∈ {1, 2}.(21) In (21), we define Regret T (I (i) ) as the regret of the algorithm on instance I (i) , and Q (i) = T t=1 q (i) t . Among the set of instances {J (i) c } i∈[K] (see Instances {J (i) c } i∈[K] ) , the online algorithm suffers Regret T (J (i) c ) ≥ 1 128 min 1, Kq B opt(J (i) c ) for some i ∈ [K],(22) where opt(I) denote the optimum of instance I, even when the DM has complete knowledge on q 1 , . . . , q T , andQ t is equal to the ground truth Q in each of the instances in {J (i) c } i∈[K] . Among the set of instances {J (i) r } i∈[K] (see Instances {J (i) r } i∈[K] ) , the online algorithm suffers Regret T (J (i) r ) ≥ 1 20 qKopt(J (i) r ) for some i ∈ [K],(23) even when the DM has complete knowledge on q 1 , . . . , q T , andQ t is equal to the ground truth Q in each of the instances in {J (i) r } i∈[K] . Once we establish inequalities (21,22,23), the Theorem is shown. We remark that (22, 23) are direct consequences of Badanidiyuru et al. [2013]. We first extract the instances {J Badanidiyuru et al. [2013], then we construct the instances I (1) , I (2) . After that, we prove (21), which establish the Theorem. (i) c } i∈[K] , {J (i) r } i∈[K] that are constructed in Instances {J (i) c } i∈ [K] . These instances are single resource instances, with determinsitic rewards but stochastic consumption. According to Badanidiyuru et al. [2013], we first set parameters η = 1 32 min 1, K B , T = 16B η(1/2 − η) , and set q t = q for all t ∈ [T ]. The instances J (1) c , . . . , J (K) c share the same B, T, {q t } T t=1 , and the instances share the same (deterministic) reward function: R(a) = r(a) =      1 if a ∈ [K] \ {a 0 } 0 if a = a 0 . In contrast, instances J c . The probability distribution of C (i) (a) for each a, i ∈ [K] is defined as follow: C (i) (a) ∼            Bern(1/2) if a ∈ [K] \ {a 0 , i} Bern(1/2 − η) if a = i Bern(0) if a = a 0 , where Bern(p) denotes the Bernoulli distribution with mean d. The regret lower bound (22) is a direct consequence of Lemma 6.10 in Badanidiyuru et al. [2013], by incorporating the scaling factorq into the rewards earned by the DM and the optimal reward. Instances {J (i) r } i∈ [K] . These instances are single resource instances, with random rewards but deterministic consumption. These instances share the same B, T > K (set arbitrarily), the same demand volume seqeunce, which is q t = q for all t ∈ [T ], and the same resource consumption model, in that c(a) = 0 for all a ∈ A. These instances only differ in the reward distributions. We denote R (i) (a) as the random reward of arm a in instance J (i) r . The probability distribution of R (i) (a) for each a, i ∈ [K] is defined as follow: R (i) (a) ∼            Bern 1 2 − 1 4 K T if a ∈ [K] \ {a 0 , i} Bern(1/2) if a = i Bern(0) if a = a 0 . The regret lower bound (23) is a direct consequence of Claim 6.2a in Badanidiyuru et al. [2013], by incorporating the scaling factorq into the rewards earned by the DM and the optimal reward. Construct I (1) , I (2) . We first describe {q (1) t } T t=1 , {q (2) t } T t=1 . As previously mentioned, for t ∈ {1, . . . , T 0 }, we have q (1) t = q (2) t = q t . To define q (1) t , q (2) t for t ∈ {T 0 + 1, . . . , T }, first recall that |Q T0+1 − Q| ≤ T0+1 , where T0+1 satisfies (20). By (20), we know thatQ T0+1 − T0+1 − T0 t=1 q t ≥ q(T − T 0 ), and thatQ T0+1 + T0+1 − T0 t=1 q t ≤ q(T − T 0 ). We set q (1) T0+1 = . . . = q (1) T ∈ [q, q] and q (2) T0+1 = . . . = q (2) T ∈ [q, q] such that Q (1) = T t=1 q (1) t =Q T0+1 − T0+1 , Q (2) = T t=1 q (2) t =Q T0+1 + T0+1 , which is valid by the stated inequalities. Next, we define the parameters {r(a)} a∈A , {c(a, 1)} a , B. (recall d = 1) Similar to the proof for Lemma 2, we only consider deterministic instances, so it is sufficient to define the mean rewards and consumption amounts. To facilitate our discussion, we specify A = [K] = {1, 2, . . . , K}, with K ≥ 3 and arm K being the null arm. The parameters {r(a)} a∈A , {c(a, 1)} a , B shared between instances I (1) , I (2) are defined as follows: r(a) =            1 if a = 1, (1 + c)/2 if a = 2, 0 if a ∈ {3, . . . , K}, and c(a, 1) =            1 if a = 1, c if a = 2, 0 if a ∈ {3, . . . , K}, where c =Q T0+1 − T0+1 Q T0+1 + T0+1 . Finally, we set B =Q T0+1 − T0+1 . Inequality (20) ensures that c, B > 0. Proving (21). To evaluate the regrets in the two instances, we start with the optimal reawrds. The optimal reward in I (1) isQ T0+1 − T0+1 , which is achieved by pulling arm 1 until the resource is exhasuted. The optimal reward for I (2) isQ T0+1 , which is achieved by pulling arm 2 until the resource is exhasuted. Consider the execution of the fixed but arbitrary online algorithm during time steps 1, . . . , T 0 in each of the instances. The prediction oracle returns the same predictionQ t for t ∈ {1, . . . , T 0 } in both instances, since both instances share the same r, c, B, T and q (1) t = q (2) t for t ∈ {1, . . . , T 0 }. Consequently, the fixed but arbitrary online algorithm has the same sequence of arm pulls A 1 , . . . , A T0 during time steps 1, . . . , T 0 in both instances I (1) , I (2) . Now, for each arm i ∈ {1, 2}, we define N i = {t ∈ {1, . . . , T 0 } : A t = i}, which has the same realization in instances I (1) , I (2) . Since N 1 ∪ N 2 ⊆ [T 0 ], at least one of the cases t∈N1 q t ≤ 1 2 T0 s=1 q s or t∈N2 q t ≤ 1 2 T0 s=1 q s holds. We denote Reward T (I (i) ) as the expected reward of the online algorithm in instance I (i) . We proceed with the following case consideration: Case 1: t∈N1 q t ≤ 1 2 T0 s=1 q s . We consider the online algorithm's execution on I (1) , which yields Reward T (I (1) ) ≤ T0 s=1 q s 2 · 1 + T0 s=1 q s 2 · 1 2 (1 + c) + Q T0+1 − T0+1 − T0 s=1 q s · 1 = T0 s=1 q s − 1 4 + 1 4 c +Q T0+1 − T0+1 . Hence, Regret T (I (1) ) ≥ T0 s=1 q s · 1 4 (1 − c) = T0 s=1 q s T0+1 2(Q T0+1 + T0+1 ) ≥ T0 s=1 q s T0+1 6(Q T0+1 − T0+1 ) = T0 s=1 q s T0+1 6Q (1) , where the last inequality is by item (1) in property ( †). To this end, recallQ T0+1 − T0+1 = T t=1 q (1) t . Case 2: t∈N2 q t ≤ 1 2 T0 s=1 q s . We consider the online algorithm's execution on I (2) , which yields Reward T (I (2) ) ≤ T0 s=1 q s − t∈N2 q t · 1 + t∈N2 q t · 1 2 (1 + c) + B − T0 s=1 q s − t∈N2 q t − t∈N2 q t · c · 1 2 (1 + c) c = T0 s=1 q s 1 − 1 + c 2c + s∈N2 q s −1 + 1 + c 2 + 1 + c 2c − 1 + c 2 + B · 1 + c 2c = − T0 s=1 q s T0+1 Q T0+1 − T0+1 + s∈N2 q s · T0+1 Q T0+1 − T0+1 +Q T0 ≤ − T0 s=1 q s T0+1 2(Q T0+1 − T0+1 ) +Q T0+1 . Hence, Regret T (I (2) ) ≥ T0 s=1 q s T0+1 2(Q T0+1 − T0+1 ) ≥ T0 s=1 q s T0+1 2(Q T0+1 + T0+1 ) = T0 s=1 q s T0+1 2Q (2) . Altogether, the Theorem is proved. B.3 Consistency Between Regret Upper and Lower Bounds Recall that in the proof of Theorem 1, we constructed two instances I (1) , I (2) such that (see (21): Regret T (I (i) ) ≥ T0 t=1 q t T0+1 6Q (i) for some i ∈ {1, 2},(24) where Regret T (I (i) ) is the regret of an arbitrary but fixed online algorithm on I (i) , with its prediction oracle satisfying that |Q (i) −Q t | ≤ T0+1 for each i ∈ {1, 2}.(25) In the lower bound analysis on I (1) , I (2) , we establish the regret lower bound (24) solely hinging on the model uncertainty on Q (1) , Q (2) , and the bound (24) still holds when the DM knows {P a } a∈A . In particular, we can set the online algorihtm to be OA-UCB, with an oracle that satisfies the property (25) above. Now, also recall in our construction that q (1) t = q (2) t = q t for all t ∈ [T 0 ], thus the predictionsQ t for t ∈ [T 0 ] are the same in the two instances, whereas Q (1) =Q T0+1 − T0+1 but Q (2) =Q T0+1 + T0+1 , while we still have Q (2) ≤ 3Q (1) , so that Q (1) = Θ(Q (2) ). Therefore, (24) is equivalent to max i∈{1,2} {Regret T (I (i) )} ≥ Ω T0 t=1 q t T0+1 Q (1) .(26) To demonstrate the consistency, it suffices to show max i∈{1,2} 1 Q (1) τ −1 t=1 q t (i) t = Ω T0 t=1 q t T0+1 Q (1) .(27) where (i) t = |Q t − Q (i) | is the prediction error on instance I (i) at time t. Indeed, to be consistent, we should have Theorem 2 holds true for both instances, while (26) still holds true. We establish (27) as follows: max i∈{1,2} τ −1 t=1 q t (i) t ≥ τ −1 t=1 q t (1) t + (2) t 2 = τ −1 t=1 q t |Q t −Q T0+1 + T0+1 | + |Q t −Q T0+1 − T0+1 | 2 ≥ τ −1 t=1 q t 2 T0+1 2 (28) ≥ T0 t=1 q t T0+1 .(29) Step (28) is by the triangle inequality, and step (29) is by the fact that for any algorithm that fully exhausts the resource, its stopping time τ > T 0 (In the case when OA-UCB does not fully consume all the resource at the end of time T , by definition we have τ = T + 1 > T 0 ). By construction, the common budget B in both instances is strictly larger than T0 t=1 q t , thus the resource is always not exhasuted at T 0 , since at time t ∈ [T 0 ] the DM consumes at most q t units of resource. Altogether, (27) is shown and consistency is verified. C Proof of Theorem 2 Before we embark on the proof, we first state a well known result on online gradient descent: T t=1 f t (x t ) − min x * ∈S T t=1 f t (x * ) ≤ 3 2 GD √ T , where D = diam(S) and G = max t ∇f t . Algorithm 3 Online Gradient Descent 1: Initialize convex set S, x 1 ∈ K, step sizes {η t } T t=1 . 2: for t = 1, 2, ..., T do 3: Play x t and observe cost f t (x t ). 4: Update x t+1 = Π S (x t − η t ∇f t (x t )) . Now we begin the proof of Theorem 2. Denote UCB r,t = (UCB r,t (a)) a∈A , LCB c,t = (LCB c,t (a, i)) a∈A,i∈ [d] . We first claim that, at a time step t ≤ τ , e At ∈ arg max u∈∆ |A| UCB c,t u −Q t B · µ t LCB c,t u.(30) In fact, the following linear optimization problem max UCB r,t u −Q t B · µ t LCB c,t u s.t. u ∈ ∆ |A| has an extreme point solution such that u * = e a for some a ∈ A. According to the definition of A t , we know that u * = e At . Then the claim holds. Suppose u * is an optimal solution of (2), then we have OPT LP = Qr u * , Qc u * ≤ B1 and u * ∈ ∆ |A| . By the optimality of (30), we have UCB r,t (A t ) −Q t B · µ t LCB c,t (A t ) = UCB r,t e At −Q t B · µ t LCB c,t e At ≥ UCB r,t u * −Q t B · µ t LCB c,t u * , which is equivalent to UCB r,t u * − UCB r,t (A t ) +Q t B · µ t B Q t 1 d − LCB c,t u * ≤Q t B · µ t B Q t 1 d − LCB c,t (A t ) . Times q t on both side and sum all t from 1 to τ −1 and apply Lemma 5 with f t (x) = qtQt B B Qt 1 d − LCB c,t (A t ) x, S = {µ : µ 1 ≤ 1, µ ≥ 0 d }, D = diam(S) = √ 2, G = max t ∇f t = M , then we obtain τ −1 t=1 q t UCB r,t u * − τ −1 t=1 q t UCB r,t (A t ) + τ −1 t=1 q tQ t B · µ t B Q t 1 d − LCB c,t u * ≤ τ −1 t=1 q tQ t B · µ t B Q t 1 d − LCB c,t (A t ) ≤ min µ * ∈S τ −1 t=1 q tQ t B · B Q t 1 d − LCB c,t (A t ) µ * + O M √ T ≤ τ −1 t=1 q tQ t B · B Q t 1 d − LCB c,t (A t ) µ + O M √ T , ∀µ ∈ S.(31) Recap by lemma 4 that with probability ≥ 1 − 3KT dδ, we have LCB c,t ≤ c. Hence, with probability ≥ 1 − 3KT dδ, τ −1 t=1 q tQ t B · µ t B Q t 1 d − LCB c,t u * ≥ τ −1 t=1 q tQ t B · µ t B Q t 1 d − c u * (32a) = τ −1 t=1 q tQ t B · µ t B Q t 1 d − B Q 1 d + τ −1 t=1 q tQ t B · µ t B Q 1 d − c u * (32b) ≥ τ −1 t=1 q tQ t B · µ t B Q t 1 d − B Q 1 d (32c) = τ −1 t=1 q tQ t B B Q t − B Q µ t 1 ,(32d) where (32a) comes from Lemma 4, (32b) comes from rearranging the sum, and (32c) comes from the fact the definition of u * . We first consider the case τ ≤ T , which implies that there exists j 0 ∈ [d] such that τ t=1 q t C t,j0 > B ⇒ τ −1 t=1 q t C t,j0 > B − q.(33) Take µ = λe j0 , where λ ∈ [0, 1] is a constant that we tune later. In this case, with probability ≥ 1 − 3KT δ, τ −1 t=1 q t UCB r,t u * ≥ τ −1 t=1 q t r t u * = OPT LP Q τ −1 Q ,(34) and λ τ −1 t=1 q tQ t B B Q t 1 d − LCB c,t (A t ) e j0 = λ τ −1 t=1 q tQ t B B Q t − LCB c,t (A t , j 0 ) = λ τ −1 t=1 q tQ t B B Q t − B Q + λ τ −1 t=1 q tQ t B B Q − C t,j0 + λ τ −1 t=1 q tQ t B (C t,j0 − LCB c,t (A t , j 0 )) .(35) Then we deal with each term respectively: τ −1 t=1 q tQ t B B Q − C t,j0 = τ −1 t=1 q t Q B B Q − C t,j0 + τ −1 t=1 q tQ t − Q B B Q − C t,j0 (36a) ≤ Q τ −1 − Q B τ −1 t=1 q t C t,j0 + 1 B τ −1 t=1 q t t B Q − C t,j0 (36b) < Q τ −1 − Q + Q B q + 1 B τ −1 t=1 q t t B Q + 1 B τ −1 t=1 q t t C t,j0 (36c) ≤ Q τ −1 − Q + Q B q + 1 Q + 1 B τ −1 t=1 q t t ,(36d) where (36a) comes from rearranging the sum, (36c) comes from the (33), and (36d) comes from the assump- tion that C t,j0 is supported in [0, 1]. Similarly, τ −1 t=1 q tQ t B (C t,j0 − LCB c,t (A t , j 0 )) = τ −1 t=1 q t Q B (C t,j0 − LCB c,t (A t , j 0 )) + τ −1 t=1 q tQ t − Q B (C t,j0 − LCB c,t (A t , j 0 )) ≤ Q B τ −1 t=1 q t (C t,j0 − LCB c,t (A t , j 0 )) + 1 B τ −1 t=1 q t t ,(37) where the equality comes from rearranging the sum, and the inequality comes from the assuption that |Q t − Q| ≤ t , 0 ≤ LCB c,t (A t , j 0 ), C t,j0 ≤ 1. Combine (35), (36) and (37), we obtain λ τ −1 t=1 q tQ t B B Q t 1 d − LCB c,t (A t ) e j0 ≤ λ τ −1 t=1 q tQ t B B Q t − B Q + λ Q τ −1 − Q + Q B q + 1 Q + 1 B τ −1 t=1 q t t + λ Q B τ −1 t=1 q t (C t,j0 − LCB c,t (A t , j 0 )) + 1 B τ −1 t=1 q t t ≤ λ Q τ −1 − Q + Q B q + Q B τ −1 t=1 q t (C t,j0 − LCB c,t (A t , j 0 )) + τ −1 t=1 q tQ t B B Q t − B Q + O 1 Q + 1 B τ −1 t=1 q t t ,(38) where the second inequality comes from the assumption that λ ∈ [0, 1]. Finally, combine (31), (32), (34), (38), we obtain OPT LP Q τ −1 Q − τ −1 t=1 q t UCB r,t (A t ) + τ −1 t=1 q tQ t B B Q t − B Q µ t 1 ≤λ Q τ −1 − Q + Q B q + Q B τ −1 t=1 q t (C t,j0 − LCB c,t (A t , j 0 )) + τ −1 t=1 q tQ t B B Q t − B Q + O 1 Q + 1 B τ −1 t=1 q t t , which is equivalent to OPT LP − τ −1 t=1 q t UCB r,t (A t ) ≤ OPT LP 1 − Q τ −1 Q + λ Q τ −1 − Q + Q B q + Q B τ −1 t=1 q t (C t,j0 − LCB c,t (A t , j 0 )) + τ −1 t=1 q tQ t B B Q t − B Q (1 − µ t 1 ) + O 1 Q + 1 B τ −1 t=1 q t t + O M √ T . Let λ = OPT LP Q ≤ 1 (This is because OPT LP = Qr u * ≤ Q), then we can further derive with probability ≥ 1 − 3KT dδ, OPT LP − τ −1 t=1 q t UCB r,t (A t ) ≤ OPT LP 1 − Q τ −1 Q + OPT LP Q Q τ −1 − Q + Q B q + Q B τ −1 t=1 q t (C t,j0 − LCB c,t (A t , j 0 )) + τ −1 t=1 q tQ t B B Q t − B Q (1 − µ t 1 ) + O 1 Q + 1 B τ −1 t=1 q t t + O M √ T = OPT LP B q + OPT LP B τ −1 t=1 q t (C t,j0 − LCB c,t (A t , j 0 )) + τ −1 t=1 q tQ t B B Q t − B Q (1 − µ t 1 ) + O 1 Q + 1 B τ −1 t=1 q t t + O M √ T ≤ O log 1 δ OPT LP qK B + qK B log T K + 1 Q + 1 B τ −1 t=1 q t t + M √ T = O log 1 δ OPT LP qK B + 1 Q + 1 B τ −1 t=1 q t t + M √ T ,(39) where the second inequality comes from Lemma 15 and the following τ −1 t=1 q tQ t B B Q t − B Q (1 − µ t 1 ) ≤ τ t=1 q tQ t B B Q − B Q t = 1 Q τ t=1 q t Q t − Q ≤ 1 Q τ t=1 q t t . The above concludes our arguments for the case τ ≤ T . In complement, we then consider the case τ > T , which means that τ = T + 1, and no resource is fully exhausted during the horizon. With probability ≥ 1 − 3KT δ, we have T t=1 q t UCB r,t u * ≥ T t=1 q t r t u * = OPT LP .(40) Take µ = 0 and combine (31), (32), (40), with probablity ≥ 1 − 3KT δ, we have OPT LP − τ −1 t=1 q t UCB r,t (A t ) ≤ − τ −1 t=1 q tQ t B B Q t − B Q µ t 1 + O M √ T ≤ O 1 Q τ −1 t=1 q t t + M √ T .(41) Combine (41) and (39), for any stopping time τ , with probability ≥ 1 − 3KT dδ, we have OPT LP − τ −1 t=1 q t UCB r,t (A t ) ≤ O log 1 δ OPT LP qK B + 1 Q + 1 B τ −1 t=1 q t t + M √ T . By Lemma 14, we can further derive it to the high probability bound, that with probability ≥ 1 − 3KT dδ, OPT LP − τ −1 t=1 q t R t ≤ O   log 1 δ   OPT LP qK B + qK τ −1 t=1 q t R t + qK log T K   + 1 Q + 1 B τ −1 t=1 q t t + M √ T   ≤ O log 1 δ OPT LP qK B + qKOPT LP + 1 Q + 1 B τ −1 t=1 q t t + M √ T , where the second inequality comes from the fact that OPT LP ≥ τ −1 t=1 q t R t . Now we finish the proof of Theorem 2. D Proofs for Confidence Radii This section contains proofs for the confidence radius results, which largely follow the literature, but we provide complete proofs since we are in a non-stationary setting. Section D.1 provides the proof for Lemma 3, which allows us to extract the implicit constants in existing proofs in Babaioff et al. [2015], Agrawal and Devanur [2014]. Section D.2 provides the proof for Lemma 4. Finally, section D.3, we prove inequalities (14, 15). D.1 Proof for Lemma 3, due to Babaioff et al. [2015], Agrawal and Devanur [2014] In this subsection, we prove Lemma 3 by following the line of arguments in Babaioff et al. [2015]. We emphasize that a version of the Lemma has been proved in Babaioff et al. [2015]. We dervie the Lemma for the purpose of extracting the values of the constant coefficients. We first extract some relevant concentration inequalities in the following two Lemmas. Lemma 6 (Theorem 8 in Chung and Lu [2006]). Suppose {U i } n i=1 are independent random variables satisfying U i ≤ M , for 1 ≤ i ≤ n almost surely. Let U = n i=1 U i , U 2 = n i=1 E[U 2 i ]. With probability ≥ 1 − e −x , we have U − E[U ] ≤ 2 U 2 x + 2x 3 max{M, 0}. Lemma 7 (Theorem 6 in Chung and Lu [2006]). Suppose U i are independent random variables satisfying U i − E[U i ] ≤ M , M > 0, for 1 ≤ i ≤ n. Let U = n i=1 U i , Var(U ) = n i=1 Var(U i ), then with probability ≥ 1 − e −x , we have U − E[U ] ≤ 2Var(U )x + 2M x 3 . Using Lemma 7, we first derive the following Lemma that bounds the empirical mean: Lemma 8. Let {X i } n i=1 be independent random variables supported in [0, 1]. Let X = n i=1 X i and Var(X) = n i=1 Var(X i ). For any fixed x > 0, With probability ≥ 1 − 2e −x , we have |X − E[X]| ≤ 2Var(X)x + 2x 3 . Proof of Lemma 8. Apply Lemma 7 with U i = X i , U i = −X i , respectively, and M = 1, then with probability ≥ 1 − 2e −x , we have |X − E[X]| ≤ 2Var(X)x + 2x 3 . Next, we bound the difference between the ground truth variance and its empirical counterpart using Lemma 6: Lemma 9. Suppose X i are independent random variables supported in [0, 1]. Let X = n i=1 X i , Var(X) = n i=1 Var(X i ), V n = n i=1 (X i − E[X i ]) 2 then with probability ≥ 1 − 3e −x , we have Var(X) ≤ V n + 2 √ x. Proof of Lemma 9. The proof follows the line of argument in Audibert et al. [2009]. First, we apply Lemma 6 with U i = −(X i − E[X i ]) 2 and M = 0. With probability ≥ 1 − e −x , we have Var(X) ≤ V n + 2 n i=1 E (X i − E[X i ]) 4 x ≤ V n + 2 n i=1 E (X i − E[X i ]) 2 x = V n + 2Var(X)x.(42) Since X i ∈ [0, 1] almost surely for all i ∈ [n], we have Var(X i ) = E[X 2 i ] − E[X i ] 2 ≤ E[X i ] − E[X i ] 2 ≤ 1 4 . Now, observe that Var(X) = i=1 Var(X i ) ≤ n i=1 1 4 = n 4 ⇒ Var(X) ≤ √ n 2 . If 2 √ x ≥ √ n 2 , then the Lemma evidently holds. Otherwise, we assume 2 √ x ≤ √ n 2 , which is equivalent to x ≤ n 16 . Combining Lemma 8 and (42), with probability ≥ 1 − 3e −x , we have, Var(X) ≤ V n + 2Var(X)x + (X − E[X]) 2 n ≤ V n + 2Var(X)x + 1 n 2Var(X)x + 4 3 x 2Var(X)x + 4x 2 9 ≤ V n ++ 4 V n + x 36 ≤ V n + 2 √ x, which proves the Lemma. Lemma 10. Suppose X i are independent random variables supported in [0, 1]. Let X = n i=1 X i , then with probability ≥ 1 − 3e −x , we have |X − E[X]| ≤ √ 2Xx + 4x. Proof of Lemma 10. Apply Lemma 8 and Lemma 9, we directly derive that with probability ≥ 1 − 3e −x |X − E[X]| ≤ 2Var(X)x + 2x 3 ≤ 2V n x + 2 √ 2 + 2 3 x < 2V n x + 4x ≤ √ 2Xx + 4x, where the last inequality comes from the fact that for random variable whose support is [0, 1], then its variance is always smaller than its mean. P(X > R) ≤ 2 −R . Now we turn back to the proof of Lemma 3. Denote δ = e −x . Apply Lemma 10 then with probability ≥ 1 − 3δ, we have, N V − E V ≤ 2NV log 1 δ + 4 log 1 δ , which is equivalent to , V − E V ≤ rad V , N, δ .(43) Besides, P rad V , N, δ > 3rad E V , N, δ ≤ P V > 9E V + 32 log 1 δ ≤ 2 −9E[V ]−32 log( 1 δ ) ≤ δ.(44) Therefore, combining (43) and (44), the lemma holds. D.2 Proof for Lemma 4 By Lemma 3, with probability ≥ 1 − 3KT δ, we have |r(a) −R t (a)| ≤ rad(R t (a), N + t−1 (a), δ). Hence with probability ≥ 1 − 3KT δ, D.3 Proof for Inequalities 14, 15 We first provide the two lemmas: Lemma 12 (Theorem 1.6 in Freedman [1975]). Suppose {U i } n i=1 is a martingale difference sequence supported in [0, 1] with respect to the filtration {F i } n i=1 . Let U = n i=1 U i , and V = n i=1 Var(U i |F i−1 ) . Then for any a > 0, b > 0, we have P (|U | ≥ a, V ≤ b) ≤ 2e − a 2 2(a+b) . i } n i=1 . Since Var(U i |F i−1 ) = Var(X i |F i−1 ) = E[X 2 i |F i−1 ] − E[X i |F i−1 ] 2 ≤ E[X i |F i−1 ] = M i almost surely, we have V = n i=1 Var(U i |F i−1 ) ≤ n i=1 M i = M almost surely. Apply Lemma 12 with a = 2b log 1 δ + 2 log 1 δ for any b ≥ 1, it follows that with probability ≤ 2δ, |U | = n i=1 U i ≥ O b log 1 δ + log 1 δ & V ≤ b, Take the union bound over all integer b from 1 to n, noting that V ≤ M and b − 1 ≤ M ≤ b for some b ∈ {1, . . . , n} almost surely, with probability ≥ 1 − 2nδ we have n i=1 (X i − M i ) ≤ O M log 1 δ + log 1 δ . Altogether, the lemma holds. Now, we paraphrase inequalities 14, 15 as Lemmas 14, 15, and provide their proofs. Lemma 14. With probability ≥ 1 − 3KT δ, we have τ −1 t=1 q t UCB r,t (A t ) − τ −1 t=1 q t R t ≤ O   log 1 δ   qK τ −1 t=1 q t R t + qK log T K     . Lemma 15. With probability ≥ 1 − 3KT dδ, we have τ −1 t=1 q t LCB c,t (A t , i) − τ −1 t=1 q t C t,i ≤ O log 1 δ qKB + qK log T K , ∀i ∈ [d]. Proof of Lemma 14. First with probability ≥ 1 − 2T δ, we have τ −1 t=1 q t r(A t ) − τ −1 t=1 q t R t = q τ −1 t=1 q t q (r(A t ) − R t ) (45a) ≤ O   q log 1 δ τ −1 t=1 q t r(A t ) + q log 1 δ   (45b) ≤ O   q log 1 δ τ −1 t=1 q t UCB r,t (A t ) + q log 1 δ   ,(45c) where (45c) comes from Lemma 4. Inequality (45b) comes from Lemma 13, where we apply X t = qtRt q and F t−1 = σ({A s , q s , R s , {C s,i } d i=1 ,Q s } t−1 s=1 ∪ {q t }) . Then with probability ≥ 1 − 3KT δ, we also have qtUCBr,t(At) + 2qK log T K log 1 δ + 2qK log 1 δ   ,(46g) where • (46a) comes from the following, with probability ≥ 1 − 3KT δ, |UCB r,t (A t ) − r(A t )| ≤ R t−1 (A t ) − r(A t ) + rad(R t−1 (A t ), N + t−1 (A t ), δ) ≤ 2rad(R t−1 (A t ), N t−1 (A t ), δ) ≤ 6rad(r(A t ), N t−1 (A t ), δ). • (46b) comes from rearranging the sum. q n (a) means the n-th adversarial term that the algorithm selects a. • (46c) comes from the definition of rad(·, ·, ·). • (46d) comes from the following n i=1 w i √ i = n i=1 2w i 2 √ i ≤ n i=1 2w i i j=1 w j + i−1 j=1 w j = n i=1 2   i j=1 w j − i−1 j=1 w j   = 2 n i=1 w i , and n i=1 w i i ≤ n i=1 1 i ≤ (1 + log(n)). where w i ∈ (0, 1]. • In (46d) and (46e) Q t (a) = s∈[t],As=a q s . • (46e) comes from Jansen inequality. Combine (45) and (46), we have τ −1 t=1 q t UCB r,t (A t ) ≤ τ −1 t=1 q t r t + O   qK log 1 δ τ −1 t=1 q t UCB r,t (A t ) + qK log T K log 1 δ + qK log 1 δ   , which is equivalent to   τ −1 t=1 q t UCB r,t (A t ) − O qK log 1 δ   2 ≤ τ −1 t=1 q t r t + O qK log T K log 1 δ + qK log 1 δ , Hence, τ −1 t=1 q t UCB r,t (A t ) ≤ O qK log 1 δ + τ −1 t=1 q t r t + O qK log T K log 1 δ + qK log 1 δ ≤ τ −1 t=1 q t r t + O qK log T K log 1 δ + qK log 1 δ .(47) Combine (45) and (46), (47), we finish the proof. Proof of Lemma 15. The proof is quite similar to Lemma 14, so we omit the descriptive details. Similarly, with probability ≥ 1 − 2T dδ, we have τ −1 t=1 q t c(A t , i) − τ −1 t=1 q t C t,i = q τ −1 t=1 q t q (c(A t ) − C t,i ) ≤ O   q log 1 δ τ −1 t=1 q t c(A t , i) + q log 1 δ   ≤ O   q log 1 δ τ −1 t=1 q t UCB c,t (A t , i) + q log 1 δ   ,(48) Then with probability ≥ 1 − 3KT dδ, we also have 2 2c(a, i) Q τ −1 (a) q log 1 δ + 4 (1 + log(N τ −1 (a))) log 1 δ ≤ 12 2qK log 1 δ a∈A c(a, i)Q τ −1 (a) + 2qK log T K log 1 δ + 2qK log 1 δ ≤ 12   2qK log 1 δ τ −1 t=1 q t UCB c,t (A t , i) + 2qK log T K log 1 δ + 2qK log 1 δ   .(49) Similarly, τ −1 t=1 q t UCB c,t (A t , i) − τ −1 t=1 q t c(A t , i) ≤ 6 τ −1 t=1 q t rad(c(A t , i), N + t−1 (A t ), δ) ≤ O   qK log 1 δ τ −1 t=1 q t UCB c,t (A t , i) + qK log T K log 1 δ + qK log 1 δ   .(50) Combine (48) and (50), we have τ −1 t=1 q t UCB c,t (A t , i) ≤ τ −1 t=1 q t C t,i + O qK log 1 δ τ −1 t=1 q t UCB c,t (A t , i) + qK log T K log 1 δ + qK log 1 δ , which is equivalent to   τ −1 t=1 q t UCB c,t (A t , i) − O qK log 1 δ   2 ≤ τ −1 t=1 q t C t,i + O qK log T K log 1 δ + qK log 1 δ , Hence, τ −1 t=1 q t UCB c,t (A t , i) ≤ O qK log 1 δ + τ −1 t=1 q t C t,i + O qK log T K log 1 δ + qK log 1 δ ≤ τ −1 t=1 q t C t,i + O qK log T K log 1 δ + qK log 1 δ ≤ √ B + O qK log T K log 1 δ + qK log 1 δ ,(51) where the last inequality comes from the definition of the stopping time τ . Combine (48) and (49), (51), we finish the proof. [ 2014 ] 2014to incorporate convex constraints and concave reweards. Several variants are studied, such as the settings of contextual bandits Agrawal et al. [2016], Badanidiyuru et al. [2014], combinatorial semi-bandits Sankararaman and Slivkins [2018]. Non-stationary BwK problems, where the outcome distribution of each arm is changing over time, are studied recently. Immorlica et al. [2019] achieves a O(log T ) competitive ratio against the best fixed distribution benchmark in an adversarial setting. Rangi et al. [2018] consider both stochastic and adversarial BwK problems in the single resource case. Liu et al. [2022] design a sliding window learning algorithm with sublinear-in-T regret, assuming the amount of non-stationarity is upper bounded and known. A sub-linear-in-T regret non-stationary BwK is only possible in restrictive settings. For example, as shown in Immorlica et al. Lemma 2 is proved in Appednix B.1. Lemma 2 shows that even when all model information on time steps 1, . . . , t are revealed when A t is to be chosen, the DM still suffers Regret T = Ω(T ). Thus, NS-BwK-OA is fundamentally different from non-stationary bandits without resource constraints such as Besbes et al.[2015], and online optimization with predictions problems such as Rakhlin and Sridharan [2013a]. In these settings, we can achieve Regret T = 0 if all model information on time steps 1, . . . , t are available at the time point of choosing A t or the action at time t. Indeed, given all model information at time t, the DM achieve the optimum by choosing an arm or an action that maximizes the reward function of time t for every t ∈ [T ]. Corollary 1 . 0 , 10Consider the setting of Theorem 1. Suppose the accurate prediction condition (4) holds at T 0 , then the refined regret lower bound Regret T = Ω(max{qT 1−α Λ}) holds. We prove Lemma 3 in Appendix D.1 by following the line of argument in Babaioff et al. [2015] for the purpose of extracting the values of the coefficients in (5), which are implicit in Babaioff et al. [2015], Agrawal and Devanur [2014]. Based on the observation {R s Figure 1 : 1Estimation error Figure 2 :Figure 3 23Figures 2a and 2bplots the regret of each algorithm on different horizon lengths, in each of the two groups. The superiority in numerical performance for OA-UCB does not mean that our algorithm is strictly superior to the baselines. Indeed, our algorithm OA-UCB receives online advice by Algorithm 2, while the benchmarks do not. The numerical results instead indicate the benefit of prediciting the underlying nonstationary demand sequence, and showcase how a suitably designed algoirhtm such as OA-UCB could reap the benefit of predictions. In addition, we remark that the sliding window UCB algorithm proposed by Liu et al.[2022] is designed to handle arbitrarily changing mean outcome distributions, subject to constraints on the amount of temporal variations. On the one hand, the sliding window UCB algorithm has been shown to outperform the stationary BwK benchmarks in piece-wise stationary models where the mean outcome distributions change abruptly[Liu et al., 2022]. On the other hand, in our non-stationary scaling setting, the per-demand-unit mean outcomes r(a), c(a, 1), . . . , c(a, d) are time stationary for each arm a. Hence, historical data are useful for estimating these mean outcomes, which explain why the stationary benchmark could appear to out-perform sliding window UCB.(a) Regret on Experiment Group 1 (b) Regret on Experiment Group 2 Regret depict the trend of the accumulated rewards as time progresses, with horizon T = 10000. The black dotted lines indicate the stopping times of each algorithm respectively. Compared with our algorithm Figure 3 : 3Cumulative Reward growing. t } T t=1 of instances I (1) , I (2) for all t ∈ {1, . . . , T }. the resource consumption model. We denote C (i) (a) as the random consumption of arm a in instance J (i) Lemma 5 ( 5Theorem 3.1 in Hazan et al. [2016]). Suppose {f t } are convex functions, then Online Gradient Descent presented in Algorithm 3 applied on {f t } with step sizes {η t = D G √ t } guarantees the following for all T ≥ 1: 1 = 1a) ≤R t (a) + rad(R t (a), N + t−1 (a), δ)r(a) ≤ 1 ⇒ r(a) ≤ min R t (a) + rad(R t (a), N + t−1 (a), δ), UCB r,t (a).Similarly, with probability ≥ 1 − 3KT dδ, LCB c,t (a) c(a). Lemma 13 . 13Suppose {X i } n i=1 are random variables supported in [0, 1], where X i is F i -measurable and {F i } n i=1 is a filtration. Let M i = E[X i |F i−1 ] for each i ∈ {1,. . . , n}, and M = n i=1 M i . Then with probability ≥ 1 − 2nδ, Proof of Lemma 13. The proof follows the line of Theorem 4.10 in Babaioff et al. [2015]. Let U i = X i − M i for each i ∈ {1, . . . , n}. Clearly, {U i } n i=1 is a martingale difference sequence with respect to the filtration {F learned advice Piotr et al. [2019], Mitzenmacher and Vassilvitskii [2022]. While traditional online algorithm research focuses on worst case performance guarantee in full model uncertainty setting, this stream of works focuses on enhancing the performance guarantee when the decision maker (DM) is provided with a machine learned advice at the start of the online dynamics. A variety of results are derived in different settings, such as online caching Lykouris and Vassilvitskii [2021], rent-or-buy Purohit et al. [2018], scheduling Mitzenmacher [2019], Lattanzi et al. [2020], online set cover problems Bamas et al. [2020], Almanza et al. [2021], online matching Antoniadis et al. [2020]. Our research seeks to take a further step, by investigating the case when the DM receives pregressively updated predictions across the horizon, instead of being given a fixed prediction at the beginning. machine-learned advice (such as Lykouris and Vassilvitskii[2021]), in that ours returns a (possibly refined) prediction on Q every time step. There are many interesting future directions, such as investigating the modcles for NS-BwK-OA is also an interesting direction to pursuse Anand et al.[2020].Omar Besbes, Yonatan Gur, and Assaf Zeevi. Stochastic multi-armed-bandit problem with non-stationary rewards. Advances in neural information processing systems, 27, 2014.Wang Chi Cheung, David Simchi-Levi, and Ruihao Zhu. Learning to optimize under non-stationarity. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 1079-1087. PMLR, 2019. Fan Chung and Linyuan Lu. Concentration inequalities and martingale inequalities: a survey. Internet math-David A Freedman. On tail probabilities for martingales. the Annals of Probability, pages 100-118, 1975. Nicole Immorlica, Karthik Abinav Sankararaman, Robert Schapire, and Aleksandrs Slivkins. Adversarial bandits with knapsacks. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), Thodoris Lykouris and Sergei Vassilvitskii. Competitive caching with machine learned advice. Journal of the Thodoris Lykouris, Vahab Mirrokni, and Renato Paes Leme. Bandits with adversarial scaling. In Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th International Conference on Machine Learning, volume 119 Michael Mitzenmacher. Scheduling with predictions and the price of misprediction. arXiv preprint arXiv:1902.00732, 2019. Indyk Piotr, Singer Yaron, Vakilian Ali, and Vassilvitskii Sergei. Summer workshop on learning-based algorithms. 2019. URL http://www.mit.edu/˜vakilian/ttic-workshop.html. Purohit, Zoya Svitkina, and Ravi Kumar. Improving online algorithms via ml predictions. In Advances in Neural Information Processing Systems, volume 31, 2018. Alexander Rakhlin and Karthik Sridharan. Online learning with predictable sequences. In Conference on Learning Theory, pages 993-1019. PMLR, 2013a. Sasha Rakhlin and Karthik Sridharan. Optimization, learning, and games with predictable sequences. Advances in Neural Information Processing Systems, 26, 2013b. Anshuka Rangi, Massimo Franceschetti, and Long Tran-Thanh. Unifying the stochastic and the adversarial bandits with knapsack. arXiv preprint arXiv:1811.12253, 2018. Karthik Abinav Sankararaman and Aleksandrs Slivkins. Combinatorial semi-bandits with knapsacks. In International Conference on Artificial Intelligence and Statistics, pages 1760-1770. PMLR, 2018. R.H. Shumway and D.S. Stoffer. Time Series Analysis and Its Applications: With R Examples. Springer texts in statistics. Springer, 2017. URL https://github.com/nickpoison/tsa4/blob/master/ textRcode.md. Steinhardt and Percy Liang. Adaptivity and optimism: An improved exponentiated gradient algorithm. Stefano Tracà, Cynthia Rudin, and Weiyu Yan. Regulating greed over time in multi-armed bandits. J. Mach. Feng Zhu and Zeyu Zheng. When demands evolve larger and noisier: Learning and earning in a growing environment. In International Conference on Machine Learning, pages 11629-11638. PMLR, 2020.els Bamas et al. [2020], Lykouris and Vassilvitskii [2021], Purohit et al. [2018], Mitzenmacher [2019] in the presense of sequential prediction oracles similar to ours. It is also interesting to invenstigate other forms of predictions, such as confidence intervals containing Q in the spirit robust optimization Bertsimas and Sim [2004], or prediction with distributional information Diakonikolas et al. [2021]. Customizing prediction ora- References Shipra Agrawal and Nikhil R Devanur. Bandits with concave rewards and convex knapsacks. In Proceedings of the fifteenth ACM conference on Economics and computation, pages 989-1006, 2014. Shipra Agrawal, Nikhil R Devanur, and Lihong Li. An efficient algorithm for contextual bandits with knap- sacks, and an extension to concave objectives. In Conference on Learning Theory, pages 4-18. PMLR, 2016. Matteo Almanza, Flavio Chierichetti, Silvio Lattanzi, Alessandro Panconesi, and Giuseppe Re. Online facility location with multiple advice. Advances in Neural Information Processing Systems, 34:4661-4673, 2021. Keerti Anand, Rong Ge, and Debmalya Panigrahi. Customizing ML predictions for online algorithms. In Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 303-313. PMLR, 13-18 Jul 2020. Antonios Antoniadis, Themis Gouleakis, Pieter Kleer, and Pavel Kolev. Secretary and online matching prob- lems with machine learned advice. Advances in Neural Information Processing Systems, 33:7933-7944, 2020. Jean-Yves Audibert, Rémi Munos, and Csaba Szepesvári. Exploration-exploitation tradeoff using variance estimates in multi-armed bandits. Theoretical Computer Science, 410(19):1876-1902, 2009. Moshe Babaioff, Shaddin Dughmi, Robert Kleinberg, and Aleksandrs Slivkins. Dynamic pricing with limited supply, 2015. Ashwinkumar Badanidiyuru, Robert Kleinberg, and Aleksandrs Slivkins. Bandits with knapsacks. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 207-216. IEEE, 2013. Ashwinkumar Badanidiyuru, John Langford, and Aleksandrs Slivkins. Resourceful contextual bandits. In Conference on Learning Theory, pages 1109-1134. PMLR, 2014. Etienne Bamas, Andreas Maggiori, and Ola Svensson. The primal-dual method for learning augmented algorithms. Advances in Neural Information Processing Systems, 33:20083-20094, 2020. Dimitris Bertsimas and Melvyn Sim. The price of robustness. Operations Research, 52(1):35-53, 2004. Omar Besbes and Assaf Zeevi. Dynamic pricing without knowing the demand function: Risk bounds and near-optimal algorithms. Operations Research, 57(6):1407-1420, 2009. Omar Besbes and Assaf Zeevi. Blind network revenue management. Operations research, 60(6):1537-1550, 2012. Omar Besbes, Yonatan Gur, and Assaf Zeevi. Non-stationary stochastic optimization. Operations research, 63 (5):1227-1244, 2015. ematics, 3(1):79-127, 2006. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, 3rd Edition. MIT Press, 2009. ISBN 978-0-262-03384-8. Ilias Diakonikolas, Vasilis Kontonis, Christos Tzamos, Ali Vakilian, and Nikos Zarifis. Learning online al- gorithms with distributional advice. In Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pages 2687-2696. PMLR, 18-24 Jul 2021. Devdatt P Dubhashi and Alessandro Panconesi. Concentration of measure for the analysis of randomized algo- rithms. Cambridge University Press, 2009. Elad Hazan et al. Introduction to online convex optimization. Foundations and Trends® in Optimization, 2(3-4): 157-325, 2016. R.J. Hyndman and G. Athanasopoulos. Forecasting: principles and practice. OTexts, 2021. URL https:// otexts.com/fpp3/. pages 202-219. IEEE, 2019. Ali Jadbabaie, Alexander Rakhlin, Shahin Shahrampour, and Karthik Sridharan. Online optimization: Com- peting with dynamic comparators. In Artificial Intelligence and Statistics, pages 398-406. PMLR, 2015. Silvio Lattanzi, Thomas Lavastida, Benjamin Moseley, and Sergei Vassilvitskii. Online scheduling via learned weights. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1859- 1877. SIAM, 2020. Bryan Lim and Stefan Zohren. Time-series forecasting with deep learning: a survey. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 379(2194), 2021. Shang Liu, Jiashuo Jiang, and Xiaocheng Li. Non-stationary bandits with knapsacks. arXiv preprint arXiv:2205.12427, 2022. ACM (JACM), 68(4):1-25, 2021. of Proceedings of Machine Learning Research, pages 6511-6521, 2020. Aranyak Mehta, Amin Saberi, Umesh Vazirani, and Vijay Vazirani. Adwords and generalized online match- ing. Journal of the ACM (JACM), 54(5):22-es, 2007. Michael Mitzenmacher and Sergei Vassilvitskii. Algorithms with predictions. Commun. ACM, 65(7):33-35, jun 2022. Manish Jacob In International Conference on Machine Learning, pages 1593-1601. PMLR, 2014. Learn. Res., 22:3-1, 2021. Chen-Yu Wei, Haipeng Luo, and Alekh Agarwal. Taking a hint: How to leverage loss predictors in contextual bandits? In Conference on Learning Theory, pages 3583-3634. PMLR, 2020. Consequently, we can derive an upper bound for Var(X):2Var(X)x + 1 n 2 Var(X)x · √ n 2 √ n 4 + 4 3 2Var(X)x · n 16 + 4x 9 · n 16 = Var(X)x 13 12 √ 2 + 1 4 + V n + x 36 . Var(X) ≤ √ x 2 13 12 √ 2 + 1 4 + 1 2 x 13 12 √ 2 + 1 4 2 Lemma 11(Theorem 1.1 in Dubhashi and Panconesi [2009]). Suppose X i are independent random variables supported in [0, 1]. Let X = n i=1 X i , then for any R > 2eE[X], we have
[ "https://github.com/nickpoison/tsa4/blob/master/" ]
[ "Increasing trust in new data sources: crowdsourcing image classification for ecology", "Increasing trust in new data sources: crowdsourcing image classification for ecology" ]
[ "Edgar Santos-Fernandez [email protected] \nSchool of Mathematical Sciences\nQueensland University of Technology\nAustralia\n\nCentre for Data Science\nQueensland University of Technology\nAustralia\n", "Julie Vercelloni \nSchool of Mathematical Sciences\nQueensland University of Technology\nAustralia\n\nCentre for Data Science\nQueensland University of Technology\nAustralia\n", "Aiden Price \nSchool of Mathematical Sciences\nQueensland University of Technology\nAustralia\n\nCentre for Data Science\nQueensland University of Technology\nAustralia\n", "| Grace Heron \nSchool of Mathematical Sciences\nQueensland University of Technology\nAustralia\n\nCentre for Data Science\nQueensland University of Technology\nAustralia\n", "Bryce Christensen \nCorrespondence School of Mathematical Sciences\nVisualisation and Interactive Solutions for Engagement and Research (VISER) Lab\nQueensland University of Technology\nQueensland University of Technology\nAustralia\n", "Erin E Peterson \nSchool of Mathematical Sciences\nQueensland University of Technology\nAustralia\n\nCentre for Data Science\nQueensland University of Technology\nAustralia\n", "Kerrie Mengersen [email protected] \nSchool of Mathematical Sciences\nQueensland University of Technology\nAustralia\n\nCentre for Data Science\nQueensland University of Technology\nAustralia\n", "Australia " ]
[ "School of Mathematical Sciences\nQueensland University of Technology\nAustralia", "Centre for Data Science\nQueensland University of Technology\nAustralia", "School of Mathematical Sciences\nQueensland University of Technology\nAustralia", "Centre for Data Science\nQueensland University of Technology\nAustralia", "School of Mathematical Sciences\nQueensland University of Technology\nAustralia", "Centre for Data Science\nQueensland University of Technology\nAustralia", "School of Mathematical Sciences\nQueensland University of Technology\nAustralia", "Centre for Data Science\nQueensland University of Technology\nAustralia", "Correspondence School of Mathematical Sciences\nVisualisation and Interactive Solutions for Engagement and Research (VISER) Lab\nQueensland University of Technology\nQueensland University of Technology\nAustralia", "School of Mathematical Sciences\nQueensland University of Technology\nAustralia", "Centre for Data Science\nQueensland University of Technology\nAustralia", "School of Mathematical Sciences\nQueensland University of Technology\nAustralia", "Centre for Data Science\nQueensland University of Technology\nAustralia" ]
[]
Crowdsourcing methods facilitate the production of scientific information by non-experts. This form of citizen science (CS) is becoming a key source of complementary data in many fields to inform data-driven decisions and study challenging problems. However, concerns about the validity of these data often constrain their utility. In this paper, we focus on the use of citizen science data in addressing complex challenges in environmental conservation. We consider this issue from three perspectives. First, we present a literature scan of papers that have employed Bayesian models with citizen science in ecology. Second, we compare several popular majority vote algorithms and introduce a Bayesian item response model that estimates and accounts for participants' abilities after adjusting for the difficulty of the images they have classified. The model also enables participants to be clustered into groups based on ability. Third, we apply the model in a case study involving the classification of corals from underwater images from the Great Barrier Reef, Australia. We show that the model achieved superior results in general and, for difficult tasks, a weighted consensus method that uses only groups of experts and experienced participants produced better performance mea-1 arXiv:2305.01144v1 [stat.AP] 2 May 2023 2 Edgar Santos-Fernandez et al.sures. Moreover, we found that participants learn as they have more classification opportunities, which substantially increases their abilities over time. Overall, the paper demonstrates the feasibility of CS for answering complex and challenging ecological questions when these data are appropriately analysed. This serves as motivation for future work to increase the efficacy and trustworthiness of this emerging source of data.
10.1111/insr.12542
[ "https://export.arxiv.org/pdf/2305.01144v1.pdf" ]
258,437,148
2305.01144
038758b3749741be17c75b8da405e3c643d6534c
Increasing trust in new data sources: crowdsourcing image classification for ecology Edgar Santos-Fernandez [email protected] School of Mathematical Sciences Queensland University of Technology Australia Centre for Data Science Queensland University of Technology Australia Julie Vercelloni School of Mathematical Sciences Queensland University of Technology Australia Centre for Data Science Queensland University of Technology Australia Aiden Price School of Mathematical Sciences Queensland University of Technology Australia Centre for Data Science Queensland University of Technology Australia | Grace Heron School of Mathematical Sciences Queensland University of Technology Australia Centre for Data Science Queensland University of Technology Australia Bryce Christensen Correspondence School of Mathematical Sciences Visualisation and Interactive Solutions for Engagement and Research (VISER) Lab Queensland University of Technology Queensland University of Technology Australia Erin E Peterson School of Mathematical Sciences Queensland University of Technology Australia Centre for Data Science Queensland University of Technology Australia Kerrie Mengersen [email protected] School of Mathematical Sciences Queensland University of Technology Australia Centre for Data Science Queensland University of Technology Australia Australia Increasing trust in new data sources: crowdsourcing image classification for ecology O R I G I N A L A R T I C L EBayesian inferencecitizen scienceitem response model Crowdsourcing methods facilitate the production of scientific information by non-experts. This form of citizen science (CS) is becoming a key source of complementary data in many fields to inform data-driven decisions and study challenging problems. However, concerns about the validity of these data often constrain their utility. In this paper, we focus on the use of citizen science data in addressing complex challenges in environmental conservation. We consider this issue from three perspectives. First, we present a literature scan of papers that have employed Bayesian models with citizen science in ecology. Second, we compare several popular majority vote algorithms and introduce a Bayesian item response model that estimates and accounts for participants' abilities after adjusting for the difficulty of the images they have classified. The model also enables participants to be clustered into groups based on ability. Third, we apply the model in a case study involving the classification of corals from underwater images from the Great Barrier Reef, Australia. We show that the model achieved superior results in general and, for difficult tasks, a weighted consensus method that uses only groups of experts and experienced participants produced better performance mea-1 arXiv:2305.01144v1 [stat.AP] 2 May 2023 2 Edgar Santos-Fernandez et al.sures. Moreover, we found that participants learn as they have more classification opportunities, which substantially increases their abilities over time. Overall, the paper demonstrates the feasibility of CS for answering complex and challenging ecological questions when these data are appropriately analysed. This serves as motivation for future work to increase the efficacy and trustworthiness of this emerging source of data. Funding information Crowdsourcing methods facilitate the production of scientific information by non-experts. This form of citizen science (CS) is becoming a key source of complementary data in many fields to inform data-driven decisions and study challenging problems. However, concerns about the validity of these data often constrain their utility. In this paper, we focus on the use of citizen science data in addressing complex challenges in environmental conservation. We consider this issue from three perspectives. First, we present a literature scan of papers that have employed Bayesian models with citizen science in ecology. Second, we compare several popular majority vote algorithms and introduce a Bayesian item response model that estimates and accounts for participants' abilities after adjusting for the difficulty of the images they have classified. The model also enables participants to be clustered into groups based on ability. Third, we apply the model in a case study involving the classification of corals from underwater images from the Great Barrier Reef, Australia. We show that the model achieved superior results in general and, for difficult tasks, a weighted consensus method that uses only groups of experts and experienced participants produced better performance mea-1 arXiv:2305.01144v1 [stat.AP] 2 May 2023 sures. Moreover, we found that participants learn as they have more classification opportunities, which substantially increases their abilities over time. Overall, the paper demonstrates the feasibility of CS for answering complex and challenging ecological questions when these data are appropriately analysed. This serves as motivation for future work to increase the efficacy and trustworthiness of this emerging source of data. Keywords -Bayesian inference; citizen science; item response model | INTRODUCTION Citizen science (CS) and crowdsourcing involve volunteers and amateur scientists in the scientific process. Over the last decade the number of publications using CS data has increased dramatically (Fig 1). CS projects are now entrenched in many fields of science, including the notification and classification of astronomical events to learn about our universe, the identification and monitoring of environmental phenomena to learn about our world, and the reporting and assessment of health and medical outcomes to learn about our own selves. Quantitative information obtained from citizen scientists can be analysed in its own right, or it can fill gaps in professional data collection programs or experimental studies, or it can be used for complementary purposes such as training machine learning algorithms (Bradter et al., 2018). In the field of ecology, millions of citizen scientists are engaged in research projects worldwide, producing volumes of information in a cost-effective and timely manner and contributing substantively to aspirational targets such as the United Nations Sustainable Development Goals (Hsu et al., 2014;Fritz et al., 2019). These projects embrace a wide range of challenges, including assessments of climate change impacts, water and air quality measurements, monitoring of biodiversity trends and patterns in abundance, distribution, and richness of native and exotic species. Citizen science platforms have also become established. For example, iNaturalist, eButterfly and Zooniverse host hundreds of projects involving image classification. These online platforms have gained substantial recognition in recent years because they have the potential to reduce the workload of ecological experts and they engage large communities of contributors. Notwithstanding the popularity and benefits of CS, considerable data quality concerns remain regarding research involving data elicited by non-expert citizen scientists (Downs et al., 2021). These concerns primarily focus on the potential for bias arising from the unstructured nature of the data and the differing abilities of the participants. For example, while the individual classification accuracy is high in some CS image classification projects, ranging from 70% to 95% (e.g. Kosmala et al., 2016), aggregation via some form of consensus is often required to produce reliable classifications and estimates (Santos Fernandez et al., 2020;Santos-Fernandez and Mengersen, 2021). Many aggregation methods, including majority voting, enjoy great success in the literature. These methods rely on the assumption that participants have greater than 50% chance of answering correctly. However, such an assumption is not always satisfied in the presence of difficult tasks, in which the majority of the classifications can be incorrect (Raykar et al., 2010). In this paper, we focus on a particular set of problems that are of great interest for citizen science in ecology, namely the classification of objects in photos and videos. We investigate the feasibility of crowdsourcing methods for citizen science as a viable solution for manual classification of images when the task is difficult. This may occur, for example, if the images represent complex ecosystems, depicting aggregations of diverse species communities that change in space and time. Difficult tasks can also be related to images produced by different sources, such as environmental monitoring programs and cameras. We consider this problem through a Bayesian lens. To set the work in context, we first present a scan of papers that have employed Bayesian models with citizen science in ecology. The 84 exemplar papers are summarised with respect to the aim of the study, the statistical model/method and software employed, the data captured and the platform enlisted. Of particular interest is the acknowledgement and treatment of potential bias in the CS data. We then study how a suitable statistical model based on item response theory (IRT) can improve the quality of crowdsourced information and enable it to be used more confidently to help answer relevant ecological problems and estimate complex measures such as ecosystem health. This model can be used to assess changes in participants' abilities over time and whether they learn with more classification opportunities. It can also be used to identify those categories that are harder to classify and factors affecting the difficulties of the images. Following discussion of the model, we present a case study of the classification of underwater images from the Great Barrier Reef (GBR), Australia. In the experiment, we showed coral reef images to non-expert participants and asked them to classify five broad categories of benthic communities using the Amazon Mechanical Turk platform (MTurk). The statistical approach developed in this paper allows us to estimate the most difficult benthic categories, investigate different types of difficulty, including those related to images and cameras, and detect different groups of participants, based on their abilities to perform the required tasks. We also examine the relationship between the time that a participant needed to perform the classification and the participants' latent abilities. The rest of the paper is structured as follows. A review of Bayesian methods for citizen science is provided in Section 2 followed by the introduction of a Bayesian item response model which addresses biases in image classification in Section 3. Section 4 details the paper's case study in the Great Barrier Reef, Australia, including a results section summarising the applied models. Finally, Section 5 contains a detailed discussion centred around the quality of citizen science data for image classification specifically and for use in ecological research. It should be noted that we focus on images in this paper and often refer to images as 'items'. Additionally, we use the term 'label' to refer to the true underlying category of the target observation in any reference images. Despite dealing with the classification of images in this study, the proposed methodology can be extrapolated to other sources of citizen science-produced data such as audio recordings. | BAYESIAN METHODS FOR CITIZEN SCIENCE Scientists are increasingly employing Bayesian statistical methods to overcome some of the limitations of CS data. We conducted a scan of some of the most relevant developments, following the PRISMA guidelines (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) (Moher et al., 2009). We searched the Web of Science, Scopus, and Google Scholar databases using the keywords "citizen science" and "Bayes", and filtered articles belonging to Ecology and Conservation. The search was restricted to articles in the English language published in peer-reviewed articles from 2010 to 30-Apr-2022. We found a total of 84 articles meeting the inclusion criteria. For each article, we summarised the statistical method used, the most relevant findings, the data sources, the platform/project and the software/packages used for modelling (if it was mentioned explicitly). The articles are summarized in Table 3. In total, 36.4% of the articles used Bayesian regression and/or hierarchical models. A comparable percentage employed Bayesian species distribution models / Bayesian occupancy models. Approximately 20% of the models incorporate spatial and spatio-temporal dependence. Other statistical models not included in these categories accounted for 18%. In Fig 10 we have summarized the software used for Bayesian computation. Over half of the articles that specified the use of software mentioned the use of R (R Core Team, 2018), frequently in conjunction with dedicated software such as WinBUGS (Lunn et al., 2000), OpenBUGS (Spiegelhalter et al., 2007), Stan (Carpenter et al., 2017) and INLA (Rue et al., 2009). Among the scanned paper, the most popular R packages for Bayesian inference are R2jags (Su and Yajima, 2021), rjags (Plummer, 2022), R2OpenBUGS (Sturtz et al., 2005), rstan (Stan Development Team, 2018), unmarked (Fiske and Chandler, 2011), adegenet (Jombart and Ahmed, 2011), ape (Paradis and Schliep, 2019), jagsUI (Kellner, 2021), R2WinBUGS (Sturtz et al., 2005), rstanarm (Goodrich et al., 2022). A substantial proportion of the scanned articles deals with topics such as the estimation of the presence/absence of species or their abundance. Hence, many authors resort to Bayesian occupancy and species distribution models (e.g. Della Rocca and Milanesi, 2022;Sheard et al., 2021;Coomber et al., 2021;Ver Hoef et al., 2021;Erickson and Smith, 2021;Rodhouse et al., 2021). Ecological data is generally geo-referenced and many of the models account for spatial variation by using for instance conditional autoregressive models (CAR) (Arab et al., 2016;Croft et al., 2019;Dwyer et al., 2016;Purse et al., 2015;Santos Fernandez et al., 2020), covariance matrices (Reich et al., 2018), Gaussian processes (Sicacha-Parada et al., 2020), SPDE (Girardello et al., 2019;Humphreys et al., 2019) or simply incorporating spatially varying covariates. One of our main aims was to explore the treatment authors give to CS-produced data. We found that researchers often assume that no errors or biases are present in the data, which could affect statistical inference and decision making. For example, many CS projects in ecology use opportunistic data that are obtained without a sampling design or a professional survey, since traditional professional programs are costly and time-consuming and can fail to obtain accurate or precise estimates of infrequent species or other outcomes of interest. Geographical, spatial recording, and preferential sampling bias may arise from opportunistically collected data, with more observations from frequently visited locations such as around roads, with irregular frequencies across time, or as a result of preferential sampling spots with certain habitat types or where specific species are more likely to be found. See the discussion in Chevalier et al. (2021); Fournier et al. (2017); Zhu et al. (2020). In addition, observer bias may arise as the result of inexperience, or incorrect understanding or perception of a variable of interest. Detection bias may arise if a species or outcome of interest is missed or misidentified. Overall, we identified several strategies that have been proposed to cope with potential bias in CS data. A discussion around the use of opportunistic data can be found in Coron et al. (2018). Similarly, the temporal bias (observation efforts) arising from the seasonality of visitors to the area of study is addressed by Dwyer et al. (2016), specifically for sightings data. In a spatial context, van Strien et al. (2013b) suggested using post-stratification of sites and showed how opportunistic data can produce suitable estimates when adjusting for bias produced by geographical imbalance and unequal observation efforts. In another study, Humphreys et al. (2019) included human population density to account for the fact that sightings might be more likely in areas with larger populations. These authors also tackled observer bias by using observation effort expressed as observer hours as a predictor in their models. Imperfect detection is relevant, especially for data contributed by citizen scientists. Occupancy models generally involve a component for the presence/absence and another for the detection/non-detection. These models aim at There exists a fundamental gap in the literature regarding approaches to overcome other sources of bias. This includes potential biases resulting from the misclassification errors arising during the classification of objects on images by citizen scientists. We discuss this issue in the next section. | ADDRESSING CITIZEN SCIENCE BIAS IN IMAGE CLASSIFICATION In this section, we first discuss majority voting algorithms that are frequently used to estimate the true labels in manual image classification. This is followed by the introduction of a Bayesian item response model that is used to weight the evidence produced by citizen scientists. | Majority vote (MV) We consider a set of images j = 1, 2, · · · , J, each composed of k = 1, 2, · · · , K elicitation points selected using a spatially balanced random sampling approach. We deal with a binary classification task, in which a participant i is asked whether a category (e.g., coral) is present on a point belonging to a given image. Let Y ijk be the answer of the participant i for the k th point from the j th image. Y ijk = 1 if the participant considered the category to be present 0 Otherwise.(1) By design, multiple participants classify the same point k. Based on these answers, we obtain the majority vote by aggregating the answers, so that the category with the highest proportion of votes wins the vote, which is the mode.Ŷ jk = 1 if proportion of votes with "category present" > 0.5 0 Otherwise.(2) In general, this approach performs poorly for difficult tasks. This is because only expert participants are likely to respond correctly, and they can be outvoted by beginners (Raykar et al., 2010). A variation of this method is obtained using a weighted majority voting (WMV) which has been discussed, among others, by Littlestone et al. (1989);Lintott et al. (2008); Hines et al. (2015). In this method, each participant has a vote proportional to some weights, based on their knowledge, skills or past performance. | Bayesian IRT model The estimation of participants' abilities in crowdsourced data has been widely discussed in the literature (Whitehill et al., 2009;Welinder and Perona, 2010;Paun et al., 2018). We develop a Bayesian item response model with the aim of informing a weighted consensus voting approach. Reiterating for completeness the notation introduced earlier, let the binary response variable Y ijk represent whether a question associated with the k th point (k = 1, · · · , K) on the j th image (j = 1, · · · , J) taken using the l th camera is correctly answered or not by the i th participant (i = 1, · · · , I). We assume that Y ijkl follows a Bernoulli distribution with parameter p ijkl Y ijkl ∼ Bern p ijkl .(3) We use an extension of the item response model, namely the linear logistic test model (LLTM), which is formulated as follows p ijkl = η k + 1 − η k 1 1 + exp −α k θ i − β k − β l ,(4) where β k and β l are difficulties associated with the point and the camera. The parameter θ i represents the ability of the participant. The ability of an average participant can be anchored by setting it equal to zero to avoid identifiability issues with the model. Additionally, α k gives the slope of the logistic curve and η k is a pseudo-guessing associated with the point, indicating the probability of answering correctly due to guessing. We use Bayesian inference and therefore we need to define prior distributions for the parameters of interest in Eq4. θ i ∼ N 0, σ θ # hierarchical prior on the abilities σ θ ∼ uniform 0, 10 # flat prior on a weakly informative range for the s.d. of the users' abilities β k ∼ N µ bk , σ bk # hierarchical prior on the item difficulties µ βk ∼ N 0, 5 # weakly informative prior for the mean of the item difficulties σ βk ∼ Cauchy 0, 5 T 0, ∞ # informative prior for the sd of the item difficulty, allowing for substantially complex tasks β l ∼ N µ b , σ bl # hierarchical prior on the camera difficulties µ βl ∼ N 0, 5 # weakly informative prior for the mean of camera difficulty σ βl ∼ Cauchy 0, 5 T 0, ∞ # informative prior for the sd of the camera difficulties α k ∼ N 1, σ α # normal prior with mean 1 on the slope σ α ∼ Cauchy 0, 5 T 0, ∞ # half Cauchy prior on the slope sd truncated at 0 η k ∼ beta 1, 5 # weakly informative prior on the pseudoguessing | Changes in ability Several authors have suggested the implementation of dynamic item response models that account for temporal variation in the answers, under the principle that subjects' abilities change with time as a learning curve (e.g. Wang et al., 2013). To capture the learning in the process, we incorporated a temporally dependent component to the model p ijklt = η k + 1 − η k 1 1 + exp −α k φ t + θ i − β k − β l ,(5) where φ t is a common learning measure that captures the change in abilities according to the daily occasions that participants performed classifications t = 1, 2, · · · 15. For participant i, t = 1 represents the first classification day, t = 2 the second, and so on. | Consensus based on a Bayesian item response model The model described above is set in the context of a broader workflow, described as follows: 1. Produce a representative set of gold standard images e.g. 33% of the total number of images. In this set of images the true labels or answers will be obtained from expert elicitation or other another suitable method. Images in this set will be scored by most of the participants. 2. Fit an item response model and obtain estimates of the participants' abilities accounting for difficulties, guessing, etc. 3. For a weighted consensus, derive a weight for each participant proportional to their estimated ability using the posterior mean. We can compute the weights using w i = expθ i I i=1 expθ i . Alternatively, a fully Bayesian framework can be employed by using the draws from the posterior distribution of θ i to compute the distribution of w i . 4. Perform a weighted consensus vote to estimate the labels. Since our response variable is binary, the category with the largest K k=1 w i wins the weighted vote. | CASE STUDY: CLASSIFICATION OF IMAGES FROM CORAL REEFS "How inappropriate to call this planet Earth when it is quite clearly Ocean." -Arthur C. Clarke The Great Barrier Reef (GBR) is located on Australia's north eastern coast and is among the largest and most complex ecosystems in the world (Great Barrier Reef Marine Park Authority, 2009). Two impacts of climate change, including the increasing frequency of reef bleaching events and intensity of cyclones, are negatively affecting this ecosystem causing an unprecedented decline in the prevalence of hard corals (Hughes et al., 2017;De'ath et al., 2012;Ainsworth et al., 2016;Vercelloni et al., 2020). Estimation and assessment of this decline are difficult and expensive to quantify using traditional marine surveys considering the size of the GBR and the speed of decline (Gonzalez-Rivero et al., 2020). For this reason, some researchers are harnessing the strength of citizen science to produce estimates of reef-health indicators across large spatial and temporal scales Santos Fernandez et al., 2020). For instance, Santos Fernandez et al. (2020), utilized a spatial misclassification model, taking into account the proficiency of the participant in terms of sensitivity and specificity to account for bias in the data. Information from these models can then be used by reef managers and scientists to make data-enabled management decisions and inform future research. We performed an experiment using Amazon Mechanical Turk (https://www.mturk.com/) to assess the feasibility of using crowdsourced data for the estimation of hard coral cover, represented as the two-dimensional proportion of the seafloor covered in hard corals. Hard corals play an important role in reef ecosystems; their hard skeletons provide habitat for many organisms and they are vulnerable to a range of impacts that accumulate with climate change (Hughes et al., 2017). The dataset used in the study consisted of 514 geotagged images obtained from the XL Catlin Seaview Survey (González-Rivero et al., 2014) and the University of Queensland's Remote Sensing Research Centre (Roelfsema et al., 2018), which we used to assess the participants' abilities to identify hard corals. In practice, coral cover estimates from images are often based on a subset of classification points, rather than the whole image (Thompson et al., 2016;Sweatman et al., 2005). In these two programs, 40 to 50 spatially balanced, random classification points were selected on each image and classified by coral reef scientists (Gonzalez-Rivero et al., 2020;Roelfsema et al., 2021). We consider the classifications from the reef scientists as a gold standard (i.e. the ground truth). We engaged participants and provided instructions in an 11-page training document https://github.com/ EdgarSantos-Fernandez/reef_misclassification/HelpGuide_MTurk20200203.pdf, describing how to identify the different benthic categories, which included: hard and soft corals, algae, sand, water and other. Participants were also given the option to select unsure if they were uncertain about which category to select. Several image classification examples were included in this guide and the differences between commonly misclassified benthic groups were highlighted https://www.virtualreef.org.au/wp-content/uploads/VirtualReefDiver-Classification-HelpGuide-Part2.pdf. After studying the training document, a qualification task was used to assess the proficiency of the participants to accurately complete the task. More specifically, the participants were shown five images containing one classification point each and asked to select the correct class from the five possible choices. The qualification was granted to those scoring at least three correct classifications out of five. We designed a sampling protocol to select images representative of the GBR in terms of community composition (proportion of hard and soft corals, algae, and sand) and camera types (Canon, Lumix, Olympus, Sony, and Nikon). This produced a dataset composed of 514 images. We produced 514 human intelligence tasks (HITS) with a maximum number of 70 assignments per HIT (i.e. maximum number of times each image can be classified). Images were randomly assigned to the participants. We were concerned that classifying 40 or 50 points per image was too time consuming and that it would reduce participation. In addition, previous research has shown that accurate estimates of coral cover can be obtained with approximately 10 points Beijbom et al. (2015). Therefore, we asked participants to classify 15 classification points on each image, randomly selected from the 40-50 points previously classified by reef scientists. See the example in Fig.2. Participants were required to select a classification category for all of the points before submitting the classification. Every assignment (i.e. an image) was expected to take approximately one minute to complete. The payment was set to 0.10 USD per image and participants reported earning more than the federal minimum wage in the United States ($7.25 per hour) for their contributions. We monitored the quality of the classifications to prevent low-quality participants from contributing. | Performance measures Our category of interest in the analyses is hard corals. We used a suite of performance measures to describe the ability of the participants, which are based on the true positive (TP), true negative (TN), false positive (FP), and false negative (FN). In this context, TP are the points classified as hard coral given that the point is truly occupied by hard corals. A TN occurs when the point is correctly classified as something other than hard coral when there is no hard coral present. Similarly, FP represents points classified as hard coral when hard coral is absent, while FN occurs when hard coral is incorrectly classified as something else. This information was then used to generate other performance measures such as: • Sensitivity: measures the ability of a participant to identify a category when it is present: se = T P T P + F N . • Specificity: measures the ability of a participant to correctly identify a category when it is absent: sp = T N T N + F P . • Classification accuracy: the proportion of correctly identified classification points: acc = T P + T N T P + T N + F N + F P . • Precision: the probability of correctly classifying points that truly contain the category hard coral divided by the total number of points classified as containing it pre = T P T P + F P . • Matthews correlation coefficient (MCC) (Matthews, 1975): a measure of the effectiveness of the classifier using all the elements of the confusion matrix. • positive likelihood ratio (lr+): gives the number of true positives for every false positive. • negative likelihood ratio (lr − ): measures the number of false negatives for every true negative. We fitted the Bayesian item response model given in Eq 5. We extracted the posterior distribution of the abilities (θ i ) and clustered subjects. We then used these clusters to construct several model variations based on majority voting, restricted to experts or experts/experienced subjects. Three replicates were generated for different gold standard proportions and the results were averaged out. Participants were asked to classify what they observed within the points into multiple categories. | Results The data contributed by participants were aggregated using the five different methods. First, we considered the raw estimates obtained directly from the participants' classifications (i.e. raw), without any grouping or consensus applied (Table 1). The columns in this table represent the number of classification points (n) and the performance measures. We considered a traditional consensus and weighted variation based on item response model from Eq.5. The robustness of the results were assessed for the proposed consensus method using different proportions of images where the truth was known (10%, 20%, 33% and 50%); noting that images were randomly selected without replacement. Using the raw data without combining the subjects' answers produced relatively low-quality performance com- remarkably it was not the best-performing one. In Fig 4 we assess the dynamics of the rate of learning as the participants increased daily classification occasions (i.e. they work on the task a second, third, fourth time, etc). Fitting a linear regression to the posterior means at the occasion t produced a slope significantly different from 0 (p-value = 0.019), which indicates that the participants' skills increase with participation and they become better at classifying the points. After controlling for pseudo-guessing and discrimination, the average participant increased their probability of correct classification by approximately 4% after five occasions, and 8% and 12% in the 10 th and 15 th occasions, respectively. There were also differences in the difficulty of classifying each of the classes (Fig 5). The results showed that the soft coral category was the hardest to identify in the images since they are frequently misclassified as hard corals. As expected, points containing sand had the highest chances of correct classification. This category also exhibited the highest likelihood of being correctly identified by chance, as evidenced by the posterior distribution of the pseudoguessing parameter (Fig 6). Substantial differences were found among the cameras used to take the images. Images taken by the Nikon camera represented the vast majority and they were substantially more difficult (Fig 7). This camera was used to take images in the northern section of the GBR on a unique habitat as part of the XL Catlin Seaview survey (González-Rivero et al., 2014). Images from the Canon camera, taken as part of the Heron survey across different habitats, were easier F I G U R E 4 Posterior estimates of the learning parameter φ and 95% highest density interval as a function of the daily classification occasion. These estimates were obtained fitting the whole dataset (n = 514 images). The size of the dot is relative to the number of points classified. The error bars represent the 95% highest density interval of φ. to classify (Roelfsema and Phinn, 2010). However, the difficulty associated with the camera might be confounded by the regions where images were taken (North versus South of the GBR and habitats), where biodiversity differs. | Classification time and quality We evaluated the feasibility of using classification time as a straightforward indicator of classification quality. This is relevant in the context where participants are getting paid for completed images and thus trying to maximize their effort and it could be used as a straightforward way of detecting low-performing respondents. The boxplots in Fig 8 show the distributions of the classification times for each of the participant groups. We note that those participants identified as expert seem to require significantly higher median classification times compared to those in the beginner and competent groups as shown in Table 2. The comparison was made using the non-parametric Wilcoxon test, with an alternative hypothesis that the sampled elements from the group in the rows had substantially greater mean rank values than those in the columns (Table 2). However, due to the substantial overlap between these distributions and variability in the data, these results should be interpreted with caution. | DISCUSSION | Improving trust in citizen science data Citizen science (CS) has become an essential information source in many domains. However, the validity of research outputs using these data sources is often questioned, especially when participants with varying skills are involved in challenging tasks. The lack of trust in this new type of data hampers the full potential of CS programs to support management and data-driven decision-making. Our scan of recently published papers highlighted the broad use of CS data in addressing ecological questions such as mapping species abundance and understanding their associated drivers through time. Some, but not all, of the studies recognized the potential inherent bias in CS data, acknowledging its ecological implications and adapting methodological approaches accordingly. Notwithstanding these efforts, further work is required to advance current statistical practice with CS data. | Improving trust in citizen science data for image classification Our case study focused on the classification of objects in images and described a method that weights the evidence to produce the true latent labels, which are used to predict the health of coral populations along the Great Barrier Reef, Australia. This was achieved via estimation of participants' abilities after accounting for factors such as image difficulty. Applications based on image-based classification data are drawing increasing attention in many domains. It is therefore natural that there will be further opportunities to learn from the contribution of participants. However, new statistical methods are required if these opportunities are to be meaningfully realised. For example, the identification of benthic categories on images is currently deemed to be too challenging for most citizen scientists, who have little knowledge of what a hard coral looks like. Similarly, not all participants have the same commitment and skills and they engage differently in CS programs. This is critical when dealing with CS data and statistical models need to weigh the evidence, based on these factors. Obtaining the true classes using marine biologists or expert elicitation is expensive when large numbers of images need to be classified. Our results show that an item response model is a viable option when there are budget constraints. The item response modelling framework provides an effective way to assess participants' skills and cluster/group them by the level of expertise. This approach allows flexibility for the CS programs to select data and perform various assessments along the way (Santos- Fernandez and Mengersen, 2021). For example, our case study demonstrates increasing learning of the participants with time. This insight showcases the importance of retention in CS programs as people naturally learn, even with complicated tasks. These models also allowed us to identify careless or low-skilled respondents and detect software bots, who generally fall into the beginners' category. Responses from these participants are generally messy; therefore our voting algorithm does not include them. In our experience, this step was critical to achieving good classification performance. We also showed that multiple factors affect the difficulty of the task, including the underlying category on the images and the camera type. Identifying those categories and images that produce greater misclassification errors is critical to producing useful training and qualification materials. Identifying and combining expert responses is a suitable solution and this can be done even with a reduced gold standard dataset. Citizen science project managers can benefit from the approach in many ways: (1) Clustering participants allows weighting the evidence; (2) the ability to identify beginners means that they can be asked to re-qualify before contributing additional data; (3) gamification, such as leaderboards, can be constructed using the latent ability values and the number of classifications; and (4) a priori knowledge about which images are the most difficult could be used to assign them to the most skillful participants. Currently, paid platforms such as Amazon Mechanical Turk do not consider the contribution of the participants or their expertise. Instead, whoever requests the job can refuse payment for poor-quality work. A system based on ability scores, as developed in this research, constitutes a better approach to compensate participants and is more effective than the current binary system. We found that for easy classification tasks, with broad evenness in the responses of the participants, most approaches will perform relatively well. However, when the task is difficult, aggregating the answers of the participants using simple consensus tends to have poor performance. Using instead a weighted majority voting approach improves performance outcomes. | Improving trust in citizen science data for ecology The literature scan showed that most existing applications involve land-based ecosystem datasets. A substantial gap exists in the use of CS for marine ecology, especially in coral reef studies. Reef CS programs mostly exist to empower people and increase awareness, but few of them engage with robust and regular assessments of data quality which, in turn, reduce the trust of scientists and managers in using information collected by non-experts. However, marine research is evolving and collaborative CS programs such as Virtual Reef Diver combines modern statistical modelling and coral reef ecology to improve the integrity of CS data for decision-making. These methods can help marine ecology realise its full potential and contribute preservation of the health of the Great Barrier Reef by harnessing the strong engagement of several groups, including the online community. The valuable feedback we received in the study allowed us to improve project design, training, and compensation processes. An important feedback loop is also closed when measures of performance from the model are shared with the participants. Guidelines when analysing new data sources such as CS data are essential to increase the trust among the scientific community. Our study focuses on Bayesian methods to analyse CS data, we acknowledge that other quantitative approaches not considered here, especially from the general crowdsourcing literature, can also add important value to CS projects. The item response model or similar statistical techniques should be systematically applied to first quantify and assess the quality of CS data and second support choices of a further analytical framework for ecological purposes. For example, insights from the model can be combined into species distribution models to weight observations according to participant skills or other parameters of interest before estimations of occupancy probability. Data availability statement The dataset used in the case study can be found in the repository: https://github.com/EdgarSantos-Fernandez/ reef. Mathematical and Statistical Frontiers (ACEMS This research was supported by the Australian Research Council (ARC) Laureate Fellowship Program under the project "Bayesian Learning for Decision Making in the Big Data Era" (ID: FL150100150) Number of citizen science published articles per year since 2009 according to the Web of Science database and the forecast for the next five years with confidence intervals. minimizing the misclassification rate (number of false negatives and false positives). Variations of this models can be found in Strebel et al. (2014), Isaac et al. (2014), Petracca et al. (2018) and van Strien et al. (2013a). See Isaac et al. (2014) for a comparison of several Bayesian occupancy models in terms of the efficiency to detect trends based on different opportunistic data collection scenarios. Detection bias is also considered in Berberich et al. (2016) by accounting for false-positive detection (specificity) bias when spotting red wood ant nests. Viljugrein et al. (2019) also considered diagnostic sensitivity for the probability that an infected animal will be detected by testing. Similarly, Cumming and Henry (2019) corrected for observation effort to account for imperfect detection. More recently, Eisma et al. (2020) suggested a measurement error model to adjust for biases and errors in reports of rainfall measurement data produced by volunteers in Nepal. E 2 2Example of an underwater image from the Great Barrier Reef, Australia used for classification. pared to other aggregation methods, se = 0.642 and sp = 0.618 with an accuracy of 0.626 and M CC = 0.246. On average, the subjects identified 1.682 TP hard coral points for every FP hard coral classification. Using a traditional consensus approach combining all of the participants' responses substantially increased the performance measures compared to the raw data method e.g. se = 0.825 and M CC = 0.487. The majority voting methods using a selection of the participants outperform (M CC > 0.53) in all the variants explored. The ratio of TP/FP points identified under these methods is above 3. However, the likelihood ratio (FN/TN) was similar to the one obtained using the consensus approach. Fig 9 compares the statistical performance measures.In the case of the MV methods, we show four values per method representing the proportion of images in the gold standard set (10, 20, 33 and 50%). The last method (weighted variation) performed well compared to the approach based on experts and experienced, achieving marginal improvements in acc and pre. We found that the item response model captures well the abilities of the subjects, even with a small training dataset. This indicates that a minimal training set (e.g. 50 images) is enough to cluster subjects based on proficiency (i.e. expert, experienced, etc.).Four groups of participants were obtained from the quantiles of the posterior means of the abilities(Figure 3):beginners, competent, experienced and experts. The vertical axis gives the latent ability score (θ i ) and the x-axis the proportion of correctly classified points. Skilful participants have large ability score values. The size of the point gives the number of classification points and indicates the engagement in the project. The vertical bar is the 95% posterior highest density interval and represents the dispersion around the posterior ability estimate. In black, we represent a reference participant who self-identified with diving experience. This participant falls within the expert category, yet Abilities posterior estimates for four groups of participants (beginner, competent, experienced and experts) using a gold standard dataset (n = 171) and 95% highest density interval as a function of the proportion of correct answers. The size of the dot represents the number of points classified. The black dot represents a reference diver that engaged in the projects as a participant. Violin/box plots with difficulty parameters of the points on images (β k ) associated with the Violin/box plots of the posterior estimates of the pseudoguessing parameter associated with the benthic categories. Posterior estimates of the difficulties and 95% highest density interval of the five cameras used on the study as a function of the proportion of correct answers. The size of the dot represents the number of points classified in images taken from each camera. Box plots of the classification time as a function of the participants ability groups. Cheney, B.,Thompson, P. M., Ingram, S. N., Hammond, P. S., Stevick, P. T., Durban, J. W., Culloch, R. M., Elwen, S. H., Mandleberg, L., Janik, V. M. et al. (2013) Integrating multiple data sources to assess the distribution and abundance of bottlenose dolphins t ursiops truncatus in scottish waters. MammalReview, 43, 71-88. Chevalier, M., Broennimann, O., Cornuault, J. and Guisan, A. (2021) Data integration methods to account for spatial niche truncation effects in regional projections of species distribution. ECOLOGICAL APPLICATIONS, 31.Coomber, F. G., Smith, B. R., August, T. A., Harrower, C. A.,Powney, G. D. and Mathews, F. (2021) Using biological records to infer long-term occupancy trends of mammals in the UK. BIOLOGICAL CONSERVATION, 264.Coron, C., Calenge, C., Giraud, C. and Julliard, R. (2018) Bayesian estimation of species relative abundances and habitat preferences using opportunistic data. Environmental and ecological statistics, 25, 71-93.Crawford, B. A., Olds, M. J., Maerz, J. C. and Moore, C. T. (2020) Estimating population persistence for at-risk species using citizen science data. Biological Conservation, 243, 108489.Croft, S., Ward, A. I., Aegerter, J. N. and Smith, G. C. (2019) Modeling current and potential distributions of mammal species using presence-only data: A case study on british deer. Ecology and Evolution, 9, 8724-8735. Cumming, G. S. and Henry, D. A. (2019) Point counts outperform line transects when sampling birds along routes in south african protected areas. African Zoology, 54, 187-198. Della Rocca, F. and Milanesi, P. (2022) The new dominator of the world: Modeling the global distribution of the japanese beetle under land use and climate change scenarios. LAND, 11. Dennis, E. B., Morgan, B. J., Freeman, S. N., Ridout, M. S., Brereton, T. M., Fox, R., Powney, G. D. and Roy, D. B. (2017) Efficient occupancy model-fitting for extensive citizen-science data. PloS one, 12, e0174433. De'ath, G., Fabricius, K. E., Sweatman, H. and Puotinen, M. (2012) The 27-year decline of coral cover on the Great Barrier Reef and its causes. Proceedings of the National Academy of Sciences, 109, 17995-17999. Downs, R. R., Ramapriyan, H. K., Peng, G. and Wei, Y. (2021) Perspectives on citizen science data quality. Frontiers in Climate, 3, 25. Dwyer, R. G., Carpenter-Bundhoo, L., Franklin, C. E. and Campbell, H. A. (2016) Using citizen-collected wildlife sightings to predict traffic strike hot spots for threatened species: a case study on the southern cassowary. Journal of Applied Ecology, 53, 973-982. Eisma, J. A., Schoups, G., Davids, J. and Van de Giesen, N. (2020) A bayesian model for quantifying errors in citizen science data: application to rainfall observations from nepal. Erickson, K. D. and Smith, A. B. (2021) Accounting for imperfect detection in data from museums and herbaria when modeling species distributions: combining and contrasting data-level versus model-level bias correction. ECOGRAPHY, 44, 1341-1352. Eritja, R., Palmer, J. R., Roiz, D., Sanpera-Calbet, I. and Bartumeus, F. (2017) Direct evidence of adult aedes albopictus dispersal by car. Scientific Reports, 7, 1-15. Espeset, A. E., Harrison, J. G., Shapiro, A. M., Nice, C. C., Thorne, J. H., Waetjen, D. P., Fordyce, J. A. and Forister, M. L. (2016) Understanding a migratory species in a changing world: climatic effects and demographic declines in the Western Monarch revealed by four decades of intensive monitoring. OECOLOGIA, 181, 819-830. Farhadinia, M. S., Moll, R. J., Montgomery, R. A., Ashrafi, S., Johnson, P. J., Hunter, L. T. and Macdonald, D. W. (2018) Citizen science data facilitate monitoring of rare large carnivores in remote montane landscapes. Ecological indicators, 94, 283-291. Hultquist, C., Oravecz, Z. and Cervone, G. (2021) A bayesian approach to estimate the spatial distribution of crowdsourced radiation measurements around fukushima. ISPRS INTERNATIONAL JOURNAL OF GEO-INFORMATION, 10. Humphreys, J. M., Murrow, J. L., Sullivan, J. D. and Prosser, D. J. (2019) Seasonal occurrence and abundance of dabbling ducks across the continental united states: Joint spatio-temporal modelling for the genus anas. Diversity and Distributions. Isaac, N. J., van Strien, A. J., August, T. A., de Zeeuw, M. P. and Roy, D. B. (2014) Statistics for citizen science: extracting signals of change from noisy ecological data. Methods in Ecology and Evolution, 5, 1052-1060. Jombart, T. and Ahmed, I. (2011) adegenet 1.3-1: new tools for the analysis of genome-wide snp data. Bioinformatics. Kadoya, T. and Washitani, I. (2010) Predicting the rate of range expansion of an invasive alien bumblebee (Bombus Terrestris) using a stochastic spatio-temporal model. Biological Conservation, 143, 1228-1235. Kagawa, O., Uchida, S., Yamazaki, D., Osawa, Y., Ito, S. and Chiba, S. (2020) Citizen science via social media revealed conditions of symbiosis between a marine gastropod and an epibiotic alga. Scientific reports, 10, 1-10. Kellner, K. (2021) jagsUI: A Wrapper Around 'rjags' to Streamline 'JAGS' Analyses. URL: https://CRAN.R-project.org/ package=jagsUI. R package version 1.5.2. Kosmala, M., Wiggins, A., Swanson, A. and Simmons, B. (2016) Assessing data quality in citizen science. Frontiers in Ecology and the Environment, 14, 551-560. Lavariega, M. C., Ríos-Solís, J. A., Flores-Martínez, J. J., Galindo-Aguilar, R. E., Sánchez-Cordero, V., Juan-Albino, S. and Soriano-Martínez, I. (2020) Community-based monitoring of jaguar (panthera onca) in the chinantla region, mexico. Tropical Conservation Science, 13, 1940082920917825. Lintott, C. J., Schawinski, K., Slosar, A., Land, K., Bamford, S., Thomas, D., Raddick, M. J., Nichol, R. C., Szalay, A., Andreescu, D. et al. (2008) Galaxy zoo: morphologies derived from visual inspection of galaxies from the sloan digital sky survey. Monthly Notices of the Royal Astronomical Society, 389, 1179-1189. Littlestone, N., Warmuth, M. K. et al. (1989) The weighted majority algorithm. University of California, Santa Cruz, Computer Research Laboratory. Lopez, B., Minor, E. and Crooks, A. (2020) Insights into human-wildlife interactions in cities from bird sightings recorded online. LANDSCAPE AND URBAN PLANNING, 196. Lottig, N. R., Wagner, T., Henry, E. N., Cheruvelil, K. S., Webster, K. E., Downing, J. A. and Stow, C. A. (2014) Long-term citizen-collected data reveal geographical patterns and temporal trends in lake water clarity. PloS one, 9, e95769. Louvrier, J., Papaïx, J., Duchamp, C. and Gimenez, O. (2020) A mechanistic-statistical species distribution model to explain and forecast wolf (canis lupus) colonization in south-eastern france. Spatial Statistics, 100428. Lunn, D. J., Thomas, A., Best, N. and Spiegelhalter, D. (2000) WinBUGS -A Bayesian modelling framework: concepts, structure, and extensibility. Statistics and computing, 10, 325-337. Lyon, J. P., Bird, T. J., Kearns, J., Nicol, S., Tonkin, Z., Todd, C. R., O'Mahony, J., Hackett, G., Raymond, S., Lieschke, J. et al. (2019) Increased population size of fish in a lowland river following restoration of structural habitat. Ecological Applications, 29, e01882. Maguire, T. J. and Mundle, S. O. (2020) Citizen science data show temperature-driven declines in riverine sentinel invertebrates. Environmental Science & Technology Letters, 7, 303-307. Mair, L., Harrison, P. J., Jönsson, M., Löbel, S., Nordén, J., Siitonen, J., Lämås, T., Lundström, A. and Snäll, T. (2017) Evaluating citizen science data for forecasting species responses to national forest management. Ecology and evolution, 7, 368-378. Comparison of the raw performance measures and those obtained from different methods: consensus (con), item response consensus based on experts/experienced (exp) & experts participants (exp2), and using weighted consensus (con_w). (a) sensitivity (se) vs specificity (sp), (b) accuracy (acc) vs precision (pre), (c) negative likelihood ratio (lrn) vs positive likelihood ratio (lrp), (d) Matthews correlation coefficient (M CC). population estimated using combined structured and unstructured citizen science data sets R (rtrim, greta) Observation data for a range of birds (2012-2019, Germany) with sporadic sampling (2002-2018). Missing values modelled with spatial stream networks. R (h2o, SSN) Stonefly surveys lead by expert with citizen scientists (2002-2018) Scheme. ). We thank the editor and the reviewers for their valuable feedback and constructive suggestions. Thanks to the members of the VRD team (https://www.virtualreef.org.au/about/).We also like to thank all the participants who contributed to the classification of images. Ethical approval was granted for the collection of this data by the Research Ethics Advisory Team, Queensland University of Technology (QUT).Approval Number: 1600000830. Computations were performed through the QUT High Performance Computing (HPC) infrastructure. 6 | TABLES |TA B L E 1 Method 1. Performance measures obtained from the participants' classifications using raw data, and using classic consensus and item response consensus estimates. We considered several proportions of images TA B L E 2 p-values obtained using a pairwise Wilcoxon test comparing the classification times between groups.where the ground truth is known (10, 20, 33 and 50%). method n TP FP TN FN se sp acc pre M CC lr+ lr− raw 614,160 132,752 155,482 251,849 74,077 0.642 0.618 0.626 0.461 0.246 1.682 0.579 consensus 23,488 6,779 4,798 10,470 1,441 0.825 0.686 0.734 0.586 0.487 2.624 0.256 experts, GS:10%,n = 51 23,488 6,468 3,468 11,794 1,750 0.787 0.773 0.778 0.652 0.541 3.480 0.275 experts, GS:20%,n = 102 23,488 6,520 3,510 11,752 1,699 0.793 0.770 0.778 0.650 0.543 3.451 0.268 experts, GS:33%,n = 171 23,488 6,541 3,513 11,747 1,677 0.796 0.770 0.779 0.651 0.545 3.457 0.265 experts, GS:50%,n = 257 23,488 6,588 3,495 11,767 1,632 0.801 0.771 0.782 0.653 0.552 3.500 0.258 experts/experienced, GS:10%,n = 51 23,488 6,637 3,883 11,385 1,583 0.807 0.746 0.767 0.631 0.531 3.186 0.258 experts/experienced, GS:20%,n = 102 23,488 6,665 3,906 11,362 1,555 0.811 0.744 0.767 0.631 0.532 3.174 0.254 experts/experienced, GS:33%,n = 171 23,488 6,704 3,969 11,299 1,516 0.816 0.740 0.766 0.628 0.532 3.137 0.249 experts/experienced, GS:50%,n = 257 23,488 6,675 3,925 11,343 1,545 0.812 0.743 0.767 0.630 0.532 3.160 0.253 weighted, exp/exp,GS:10%,n = 51 23,488 6,155 3,043 12,219 2,063 0.749 0.801 0.783 0.671 0.539 3.810 0.311 weighted, exp/exp,GS:20%,n = 102 23,488 6,057 2,806 12,456 2,162 0.737 0.816 0.788 0.684 0.545 4.023 0.322 weighted, exp/exp,GS:33%,n = 171 23,488 6,182 2,852 12,409 2,036 0.752 0.813 0.792 0.684 0.554 4.029 0.305 weighted, exp/exp,GS:50%,n = 257 23,488 6,262 2,898 12,364 1,957 0.762 0.810 0.793 0.684 0.559 4.013 0.294 beginner competent experienced competent 1.0000 experienced 0.5932 0.2955 expert 0.0037 0.0009 0.0781 7 | SUPPLEMENTARY MATERIALS Models the occurrence of Phellinus ferrugineofuscus fungus used to forecast national forest projections in Sweden using presence and/or presence-absence data.Estimates the origins of species using stable isotopes from capture data while be-ing informed by a species distribution model obtained from CS data. environ-Hebridean Mink Project, which was set up with the objective of removing mink from North Uist, Benbecula and South Uist, while also trying to reduce the density from South Harris. Estimates source locations of invasions directly from spatial point pattern data without the need to specify dispersal parameters. Predict population size (abundance) and spatial and temporal patterns of Bicknells's Thrush (Catherus bicknelli). Assessing the reliability of citizen science data for mammals in the East African drylands. Volunteer citizen scientists from tourist lodges tagged and recorded wildlife counts that were modelled against expert sampling. It models spring arrival day in two species of migratory birds (Ruby-throated Hummingbird Archilochus colubris and Purple Martin Progne subis). It accounts for sampling effort, measurement error and a conditional Autoregressive (CAR) prior is used to model the mean Models population trends in the abundance of bird species and investigate its association with biological traits. OpenBUGS One person visiting one reference site and counting the number of basking tur-WinBUGS 1km-sq grid of sightings from Community for Coastal and Cassowary Conservation, 1999-2012. Traffic strikes. Yan et al. (2016) Bayesian belief network Assess ecosystem services related to water quality with Thames case study. Populations modelling in 21 species of birds for assessing the efficacy of pro-Wet Tropics in Queensland, Australia New Atlas of Australian Birds OpenBUGS initial data was collected by competent naturalists from the North American Bird Phenology Program (NABPP) and more recently by the Purple Martin Conservation Association (www.purplemartin.org). Models the month proportion of dietary items using long term monitoring data collected by citizens. The authors found the results of the Bayesian approach to be more consistent with other studies available literature. Naive Bayes with discrete Fourier transform Opportunistic data from two botanical gardens for flowering periods. //data.nbn.org.uk/ Data integration from profesional and CS programs to achienve improved estimates. from Great Britain from 1985 to 2004. Bayesian regression Assesses trends in lake-water clarity using a Bayesian hierarchical modeling. Estimates the average arrival and departure dates from a single breeding season for two species of long-distance migrant songbirds and two species of butterflies JAGS Simulated data with parameters obtained from previous studies.Author(s) Method Description Software / package Data Platform/ Project Sicacha-Parada et al. (2020) Bayesian Occupancy Accounting for spatial varying sampling effort due to accessibility of Citizen Sci- ence data. Inference about the important of terrain ruggedness index and solar ra- diation on Moose. INLA 472 observations of Moose in Norway GBIF https://www.gbif. org/ Zhu et al. (2020) Bayesian SDM Migratory connectivity of Swan Geese from SDMs, feather stable isotope assign- ment and satellite tracking MAXENT, R (maptools,raster) Collected 96 feathers from capturing swan geese in breeding and wintering grounds (2010-2017) eBird https://ebird. org/home, GBIF https: //www.gbif.org/ Santos Fer- nandez et al. (2020) Bayesian hierarchical model, Spatial beta regression Corrects misclassification errors in the an- notation of images. Adjust estimates of the proportion hard coral based on citi- zens abilities or performance measures. It accounts for spatial variation. R (stats, rstan), Stan 514 images from the Great Barrier Reef classified by 223 citizens. Amazon Me- chanical Turk classifications. Virtual Reef Diver https:// www.virtualreef.org.au Walker and Tay- lor (2020) Bayesian hierarchical model Population trajectory estimation for North American birds R (brms, rstan) Identification data for 28 species of songbirds and raptors (1928-2016, North America) eBird Benshemesh et al. (2020) Bayesian hierarchical model Estimations of trends and drivers of malleefowl breeding activity R (R2OpenBUGS, coda) Malleefowl mound activity data (1989- 2017, VIC SA, WA, and NSW, Australia) National Malleefowl Moni- toring Database (NMMD) Lopez et al. (2020) Bayesian generalised lin- ear model Using multiple online data platforms to determine which parameters impact the location of bird sightings R (sp, rjags, AER, spdep) Bird observation data (2015-2017, Chicago, North America) iNaturalist, eBird, Flickr Ryan et al. (2019) Approximate Bayesian Computation (ABC) based on Random Forests Modelling the genetic structure of the population fastSTRUCTURE (written in Python2.x), VCFtools, R (adegenet, poppr, ape) Small cabbage white butterfly (Pieris rapae). 32 countries. 150 volunteer scientists and citizens contributing a substantial proportion of the data. https://github.com/citscisean/ PierisrapaeInvasionHistory Pieris Project (http://www. pierisproject.org/) Author(s) Method Description Software / package Data Platform/ Project Humphreys et al. (2019) Bayesian regression with a hierarchical framework. Logistic and Poisson regression models in space and time Estimation of the occurrence and abun- dance. Effect of predictors such as pre- cipitation, continentality and wetness on dabbling ducks population distributions R (r-INLA, raster) 1 million records from ten ducks species in the United States from 2015- 2017 eBird project (https: //ebird.org/home) Croft et al. (2019) Hierarchical Bayesian model Species distribution models for presence- only data in six deer species. Accounts for spatial autocorrelation. R (hSDM) Great Britain (GB). National Biodiversity Network (NBN) Atlas. Data from 2012- 2016. Cumming and Henry (2019) Bayesian multispecies occupancy model Models bird point counts across South African National Parks. It accounts for im- perfect detection R (jagsUI, AHM- book) Girardello et al. (2019) Hierarchical Bayesian and spatial linear models Discusses observational and geographic bias INLA R (vegan, fossil) Biodiversity Informa- tion Facility (GBIF) https://www.gbif.org/ occurrence/download/ 0028634-181108115102211 Lyon et al. (2019) Bayesian hierarchical model and state-space Cormack-Jolly-Seber model Uses a state-space Cormack-Jolly-Seber approach to model population size of Murray cod and gold perch. JAGS, R Captured 7,312 Murray cod and 3,743 golden perch (2007-2013) from elec- trofishing census and tagging on 3,829 Murray cod and 3,316 golden perch with PIT and external tags and 689 Mur- ray cod and 466 golden perch with ra- dio tags. Anglers reported 1,338 tagged Murray cod and 275 golden perch. Outhwaite et al. (2019) Bayesian occupancy model Occupancy and species trends for bryophytes, lichens, and invertebrates in the UK. 5,293 UK bryophytes, lichens, and inver- tebrates (1970-2015) Author(s) Method Description Software / package Data Platform/ Project Viljugrein et al. (2019) Hierarchical change-in- ratio model, Stochastic scenario tree Models relationship between diagnostic test sensitivity, sample quality and dis- ease progression. Studies chronic wast- ing disease (CWD) infections in European reindeer populations. R (fitdistrplus, R2Jags) Citizen (hunters) collected heads for brain tissue extraction by experts Patten et al. (2019) Bayesian occupancy model Breeding thresholds in opportunistic Odonata records R (unmarked, rjags, segmented), ArcMap Citizen scientist image accompanied submissions from two online portals (2013 -2017) Odonata Central https:// www.odonatacentral.org/, iNaturalist https: //www.inaturalist.org/ Farhadinia et al. (2018) Bayesian hierarchical and Bayesian occupancy models Estimates of the occupancy and de- tection probabilities of Persian leopard species in remote landscapes R (R2jags), JAGS data collected from 80 sites on 2013 on Northeastern Iran. Platform Morii et al. (2018) Bayesian and PCA re- gression models Zero-inflated negative binomial model the frequencies of slug Limax maximus in Japan R, Stan Data collected from one participant from 2015-2016 Bradter et al. (2018) Bayesian site- occupancy-detection model. Logistic regres- sion Assesses habitat suitability models (HSMs) of Siberian jay (Perisoreus infaustus) in Sweden R (multispeciesPP), JAGS questionnaire answers from 60 subjects carried out from 2000-2013 Reich et al. (2018) Bayesian species occu- pancy models Two-stage optimal survey design by min- imization of the misclassification errors. Accounts for spatial variation. R (mgcv) Data from brown-headed nuthatch (Sitta pusilla) species collected in the Southeast United States eBird Petracca et al. (2018) Bayesian hierarchical oc- cupancy model Assesses the occurrence jaguars (Pan- thera onca) in Central America. Accounts for spatial and temporal dependence R, jagsUI, JAGS 3863 interviews from 2009-2014 Coron et al. (2018) Bayesian abundance model Estimates the abundance of birds correct- ing the bias introduced in the opportunis- tic data R (rjags) species of bird in Aquitaine (France) Zipkin and Saun- ders (2018) Bayesian population model Review integrative population models via data integration approach - Author(s) Method Description Software / package Data Platform/ Project Shima et al. (2018) Bayesian hierarchical model Assessing the impact of gender, age, weight, season, rainfall, traffic volumes, etc. on road kill risk of Lumholtz's tree kangaroo R (R2WinBUGS) Citizen sightings (1998-2000) and road- kill data (2012-2017), Atherton Table- lands, Australia Tree Kangaroo Mammal Group (data set) Mair et al. (2017) Bayesian species distri- bution model OpenBUGS, R (R2OpenBUGS, BRugs) 2000-2013 Swedish Lifewatch project www.analysisportal.se Dennis et al. (2017) Bayesian occupancy model Compares Bayesian and frequentist occu- pancy models for presence-only data R, Sparta UK Butterflies for the New Millennium database containing more than 11 mil- lion species occurrence. Fournier et al. (2017) Bayesian species distri- bution model mental variables ARCMAP Live traps 2005 -2009 and eBird re- ports (n = 3,632) from 2002 to 2012 of the Virginia rail Rallus limicola eBird Eritja et al. (2017) Bayesian logistic regres- sion Asian tiger mosquito (Aedes albopictus) alert probability (proxy for prevalence) in vehicles in Spain. R (rstanarm), Stan Expert validated citizen science reports of Aedes albopictus in vehicles in Spain (2014-2016) Mosquito Alert mo- bile app, http://www. mosquitoalert.com/ Faulkner et al. (2017) Extension of Dirichlet process mixture model R (rgeo) Trapping data from 2001-2005 with 409 captures. Citizen sightings from 2002-2005 with 125 sightings. Hebridean Mink Project Granroth- Wilding et al. (2017) Bayesian pedigree reconstruction Establishing pack dynamics of grey wolf (Canis Lupus) population in south-west Finland. R (MasterBayes, pe- gas, adegenet, pedi- gree) Local hunters and nature enthusiasts collected non-invasive samples (2013- 2016). Samples processed in laborato- ries by experts. Author(s) Method Description Software / package Data Platform/ Project Hill and Lloyd (2017) N-mixture models in hierarchical Bayesian framework JAGS, R (unmarked), ArcGIS 14,552 five minute points counts at 747 locations by citizen scientists on hiking trails Mountain Birdwatch (MBW) monitoring scheme Hof et al. (2017) Capture-mark-recapture (CMR), Bayesian growth models Citizen-science population abundance and growth rate estimates for green sea turtles foraging in the northern Great Barrier Reef, Australia R (Rmark) Capture Mark Recapture of Green Seas Turtles in the Northern Great Barrier Reef, Australia. 2003-2014, citizen (vol- unteers) teams lead by experts tagged 1316 turtles. Mang et al. (2017) Hierarchical Bayesian approach Models the spatio-temporal heterogene- ity in detection patterns of invasive rag- weed (Ambrosia artemisiifolia) Historical records in Austria (opportunis- tic data) Steger et al. (2017) Hierarchical Bayesian Abundance R Arab et al. (2016) Hierarchical Bayesian spatio-temporal model OpenBUGS Eastern North America from 2001 to 2010 hummingbirds.net and https://www.learner. org/jnorth/ database and Purple Martin Con- servation Association www.purplemartin.org Siddharthan et al. (2016) Naive Bayes classifier Bayesian online updating model for con- sensus voting. Compares efficiency the consensus labels obtained from Naive Bayes with the majority vote algorithm Weka classification of images from bumble- bees in the United Kingdom (22 species). 8,844 classification from 1,613 images classified and 763 users BEEWATCH (www.abdn.ac. uk/research/beewatch) Soykan et al. (2016) Bayesian log-linear hier- archical model OpenBUGS R (R2OpenBUGS) Birds counts by more than 70,000 ob- servers/year in North America. 551 species recorded annually from Dec 14 -Jan 5 between 1966 -2013 Author(s) Method Description Software / package Data Platform/ Project Armstrong (2016) Bayesian occupancy model Models three turtle species occupancy at one reference site tles Berberich et al. (2016) Nest-level data- augmentation patch- occupancy model Uses event-specific covariates and a plot- level Bayesian and maximum-likelihood model. Compared 3 different methods for quantifying detection probability of Red Wood Ant (RWA) nests (highly visi- ble, up to 2 metres) JAGS, R (vegan) 60 minutes for each of 16 3600 m2 plots, classified by 8 observers (2 expert, 6 inexperienced). GPS tracked, tagged and photos. R code available Harvard Forest Data Archive. Dwyer et al. (2016) Zero-inflated Poisson with conditional auto- regressive correlation structure Frequency of southern cassowary traffic strikes. Used traffic strike and citizen- collected sightings. R (gstat, ape, R2WinBUGS), Combines citizen science and researcher probe data. SQL, R (gRain) citizen science probe data (water quality essential parameters), researcher probe data (water quality essential parame- ters), citizen mobile data (waterbody features), water monitoring data from UK Environmental Agency, climate data from global model simulation (1993- 2012) Espeset et al. (2016) Bayesian hierarchical model Investigating flight patterns of western monarch butterflies over the past 40 years in California R (rjags) Identification data for western monarch butterflies (1972-2014, Northern Cali- fornia, North America) Art Shapiro's Butterfly Site, The Xerces Society, North American Butterfly Associa- tion, Monarch Net Purse et al. (2015) Bayesian spatial survival model Time to invasion from species as a func- tion of environmental predictors INLA R (survival) ladybird Harmonia axyridis in Great Britain UK Ladybird Survey (UKLS) http://www.harlequin- survey.org/ http://www. ladybird-survey.org/ Author(s) Method Description Software / package Data Platform/ Project Barnes et al. (2015) Bayesian logistic regres- sion tected areas R (R2JAGS, LISZT), JAGS, Arab and Courter (2015) Hierarchical Bayesian spatio-temporal model Spring arrival dates in bird migration from Purple Martins from (1905-1940) and (2001-2010). Error processes modelled with an auto-regressive model AR(1) and a Conditional Autoregressive (CAR) prior was used for the spatial dependency Yoshikawa and Osada (2015) Bayesian hierarchical time series model R, JAGS 15 bird species in Kanagawa, Japan Chapman et al. (2015) R (MCMCpack) Botanical Society of Britain and Ireland (BSBI) National Biodiversity Network (NBN) https: Pagel et al. (2014) Bayesian hierarchical spatio temporal ap- proach for modelling the abundance of species OpenBUGS butterfly species (Pyronia tithonus) New Millennium (BNM) project https://butterfly- conservation.org/our- work/recording-and- monitoring/butterflies- for-the-new-millennium Lottig et al. (2014) JAGS 239,741 Secchi depth measurements by citizens in the United States from 1938-2012 https://www.nalms.org/ secchidipin/ Strebel et al. (2014) Bayesian occupancy model Penalized thin-plate splines accounting for space and time Jags approx 1800 skilled citizens from the Swiss citizen-science bird recording scheme IS Author(s) Method Description Software / package Data Platform/ Project Roth et al. (2014) Bayesian site-occupancy model OpenBUGS, R Simulation Isaac et al. (2014) Bayesian occupancy model Compare 11 occupancy methods in terms of the efficiency to detect trends based on different opportunistic data collection scenarios Beck et al. Bayesian hierarchical model. Log linear model It models the abundance of Western Grebes in the Salish Sea. Species counts are described using a Poisson distribution and includes covariates to account for the sampling effort and and population effects. Detect widely known patterns of anthropogenic effects on Bush-Crickets. //vigienature.mnhn.fr van Strien et al. tunistic data from 5 countries to estimate the annual number of occupied sites per country. Countries are weighted by number of sites surveyed and range of species per country, this adjusts for unequal geographical distributions. JAGS, WinBUGS Opportunistic data -from many sources, no standardised sampling design. 1990-2008, Ireland, Great Britain, Netherlands, Belgium, France. Poisson log-linear model for modelling the species abundance with autorregressive component for temporal dependency . Compares abundance trends obtained from a professional and citizen science dataset using presence-only data. A mechanistic weighting is used to account for survey effort OpenBUGS Species Gateway and the Swedish Bird Survey from 2001-2009 in Sweden http://www.artportalen. Models the presence/absence of 269 species as a function of the year and the length of the list WinBUGS bird lists collected by Birds Queensland over 40 years around Brisbane, Aus-Using spatio-temporal presence/absence data to predict control of the near-future spread of the bumblebee Reported software utilized across the reviewed papers.WinBUGS Audubon Christmas Bird Count data (36 years) with almost 2.5 million records Author(s) Method Description Software / package Data Platform/ Project Penone et al. (2013) Bayesian model averag- ing (BMA) for general lin- earized models R (spdep), ArcGIS, Syrinx Citizen collect data, expert classified data. French Museum of Natural History http: (2013b) Bayesian occupancy model with hierarchi- cally coupled submodels for occupancy and detection Damselfy (Calioteryx splendens) oppor- Many Meiman et al. (2012) Bayesian hierarchical model Prediction of saltmarsh sparrow distribu- tion and nesting activity using multiple Bayesian hierarchical models WinBUGS Marsh classification data (2006-2008), saltmarsh sparrow presence data (2006- 2008), Connecticut, North America Snäll et al. (2011) Bayesian hierarchical re- gression model se Szabo et al. (2010) Bayesian logistic re- gression model for List Length Analysis tralia Kadoya and Washitani (2010) Bayesian ecological- niche model R, ArcGIS Precence/absence data for B. terrestris (1992-2010, Hokkaido, Japan) B. terrestris monitoring pro- gram 56.19% 11.43% 8.57% 6.67% 4.76% 2.86% 2.86% 6.67% 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 R JAGS OpenBUGS WinBUGS ArcGIS INLA Stan Other software frequency software R JAGS OpenBUGS WinBUGS ArcGIS INLA Stan Other F I G U R E 1 0 Edgar Santos-Fernandez et al. Climate change disables coral bleaching protection on the Great Barrier Reef. T D Ainsworth, S F Heron, J C Ortiz, P J Mumby, A Grech, D Ogawa, C M Eakin, W Leggat, Science. 352Ainsworth, T. D., Heron, S. F., Ortiz, J. C., Mumby, P. J., Grech, A., Ogawa, D., Eakin, C. M. and Leggat, W. (2016) Climate change disables coral bleaching protection on the Great Barrier Reef. Science, 352, 338-342. Counting cats for conservation: seasonal estimates of leopard density and drivers of distribution in the serengeti. M L Allen, S Wang, L O Olson, Q Li, M Krofel, Biodiversity and Conservation. 29Allen, M. L., Wang, S., Olson, L. O., Li, Q. and Krofel, M. (2020) Counting cats for conservation: seasonal estimates of leopard density and drivers of distribution in the serengeti. Biodiversity and Conservation, 29, 3591-3608. Spatio-temporal trend analysis of spring arrival data for migratory birds. A Arab, J R Courter, Communications in Statistics-Simulation and Computation. 44Arab, A. and Courter, J. R. (2015) Spatio-temporal trend analysis of spring arrival data for migratory birds. Communications in Statistics-Simulation and Computation, 44, 2535-2547. A spatio-temporal comparison of avian migration phenology using citizen science data. A Arab, J R Courter, J Zelt, Spatial Statistics. 18Arab, A., Courter, J. R. and Zelt, J. (2016) A spatio-temporal comparison of avian migration phenology using citizen science data. Spatial Statistics, 18, 234-245. Using reference sites to account for detection probability in occupancy surveys for freshwater turtles. D P Armstrong, Herpetological Conservation and Biology. 11Armstrong, D. P. (2016) Using reference sites to account for detection probability in occupancy surveys for freshwater turtles. Herpetological Conservation and Biology, 11, 505-518. Evaluating protected area effectiveness using bird lists in the australian wet tropics. M Barnes, J K Szabo, W K Morris, H Possingham, Diversity and Distributions. 21Barnes, M., Szabo, J. K., Morris, W. K. and Possingham, H. (2015) Evaluating protected area effectiveness using bird lists in the australian wet tropics. Diversity and Distributions, 21, 368-378. Using opportunistic photo-identifications to detect a population decline of killer whales (orcinus orca) in british and irish waters. S Beck, A D Foote, S Koetter, O Harries, L Mandleberg, P T Stevick, P Whooley, J W Durban, Journal of the Marine Biological Association of the United Kingdom. 94Beck, S., Foote, A. D., Koetter, S., Harries, O., Mandleberg, L., Stevick, P. T., Whooley, P. and Durban, J. W. (2014) Using opportunistic photo-identifications to detect a population decline of killer whales (orcinus orca) in british and irish waters. Journal of the Marine Biological Association of the United Kingdom, 94, 1327-1333. Towards automated annotation of benthic survey images: Variability of human experts and operational modes of automation. O Beijbom, P J Edmunds, C Roelfsema, J Smith, D I Kline, B P Neal, M J Dunlap, V Moriarty, T.-Y Fan, C.-J Tan, PloS one. 10130312Beijbom, O., Edmunds, P. J., Roelfsema, C., Smith, J., Kline, D. I., Neal, B. P., Dunlap, M. J., Moriarty, V., Fan, T.-Y., Tan, C.-J. et al. (2015) Towards automated annotation of benthic survey images: Variability of human experts and operational modes of automation. PloS one, 10, e0130312. Citizen scientists reveal nationwide trends and drivers in the breeding activity of a threatened bird, the Malleefowl (Leipoa Ocellata). J Benshemesh, D Southwell, R Barker, M Mccarthy, Biological Conservation. 108573Benshemesh, J., Southwell, D., Barker, R. and McCarthy, M. (2020) Citizen scientists reveal nationwide trends and drivers in the breeding activity of a threatened bird, the Malleefowl (Leipoa Ocellata). Biological Conservation, 246, 108573. Detection probabilities for sessile organisms. G M Berberich, C F Dormann, D Klimetzek, M B Berberich, N J Sanders, A M Ellison, Ecosphere. 71546Berberich, G. M., Dormann, C. F., Klimetzek, D., Berberich, M. B., Sanders, N. J. and Ellison, A. M. (2016) Detection probabilities for sessile organisms. Ecosphere, 7, e01546. Can opportunistically collected citizen science data fill a data gap for habitat suitability models of less common species?. U Bradter, L Mair, M Jönsson, J Knape, A Singer, T Snäll, Methods in Ecology and Evolution. 9Bradter, U., Mair, L., Jönsson, M., Knape, J., Singer, A. and Snäll, T. (2018) Can opportunistically collected citizen science data fill a data gap for habitat suitability models of less common species? Methods in Ecology and Evolution, 9, 1667-1678. Stan: A probabilistic programming language. B Carpenter, A Gelman, M D Hoffman, D Lee, B Goodrich, M Betancourt, M Brubaker, J Guo, P Li, A Riddell, Journal of statistical software. 76Carpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P. and Riddell, A. (2017) Stan: A probabilistic programming language. Journal of statistical software, 76. Unbiased inference of plant flowering phenology from biological recording data. D S Chapman, S Bell, S Helfer, D B Roy, Biological Journal of the Linnean Society. 115Chapman, D. S., Bell, S., Helfer, S. and Roy, D. B. (2015) Unbiased inference of plant flowering phenology from biological recording data. Biological Journal of the Linnean Society, 115, 543-554. Using geographic profiling to compare the value of sightings vs trap data in a biological invasion. S C Faulkner, R Verity, D Roberts, S S Roy, P A Robertson, M D Stevenson, Le Comber, S C , Diversity and Distributions. 23Faulkner, S. C., Verity, R., Roberts, D., Roy, S. S., Robertson, P. A., Stevenson, M. D. and Le Comber, S. C. (2017) Using geographic profiling to compare the value of sightings vs trap data in a biological invasion. Diversity and Distributions, 23, 104-112. unmarked: An R package for fitting hierarchical models of wildlife occurrence and abundance. I Fiske, R Chandler, Journal of Statistical Software. 43Fiske, I. and Chandler, R. (2011) unmarked: An R package for fitting hierarchical models of wildlife occurrence and abundance. Journal of Statistical Software, 43, 1-23. URL: https://www.jstatsoft.org/v43/i10/. Combining citizen science species distribution models and stable isotopes reveals migratory connectivity in the secretive virginia rail. A M Fournier, A R Sullivan, J K Bump, M Perkins, M C Shieldcastle, S L King, Journal of Applied Ecology. 54Fournier, A. M., Sullivan, A. R., Bump, J. K., Perkins, M., Shieldcastle, M. C. and King, S. L. (2017) Combining citizen science species distribution models and stable isotopes reveals migratory connectivity in the secretive virginia rail. Journal of Applied Ecology, 54, 618-627. Citizen science and the United Nations Sustainable Development Goals. S Fritz, L See, T Carlson, M M Haklay, J L Oliver, D Fraisl, R Mondardini, M Brocklehurst, L A Shanley, S Schade, Nature Sustainability. 2Fritz, S., See, L., Carlson, T., Haklay, M. M., Oliver, J. L., Fraisl, D., Mondardini, R., Brocklehurst, M., Shanley, L. A., Schade, S. et al. (2019) Citizen science and the United Nations Sustainable Development Goals. Nature Sustainability, 2, 922-930. Gaps in butterfly inventory data: A global analysis. M Girardello, A Chapman, R Dennis, L Kaila, P A Borges, A Santangeli, Biological conservation. 236Girardello, M., Chapman, A., Dennis, R., Kaila, L., Borges, P. A. and Santangeli, A. (2019) Gaps in butterfly inventory data: A global analysis. Biological conservation, 236, 289-295. Integrating community science research and space-time mapping to determine depth to groundwater in a remote rural region. A M Gomez, M Serre, E Wise, T Pavelsky, WATER RESOURCES RESEARCH. 57Gomez, A. M., Serre, M., Wise, E. and Pavelsky, T. (2021) Integrating community science research and space-time mapping to determine depth to groundwater in a remote rural region. WATER RESOURCES RESEARCH, 57. Monitoring of coral reefs using artificial intelligence: A feasible and cost-effective approach. M Gonzalez-Rivero, O Beijbom, A Rodriguez-Ramirez, D E Bryant, A Ganase, Y Gonzalez-Marrero, A Herrera-Reveles, E V Kennedy, C J Kim, S Lopez-Marcano, Remote Sensing. 12489Gonzalez-Rivero, M., Beijbom, O., Rodriguez-Ramirez, A., Bryant, D. E., Ganase, A., Gonzalez-Marrero, Y., Herrera-Reveles, A., Kennedy, E. V., Kim, C. J., Lopez-Marcano, S. et al. (2020) Monitoring of coral reefs using artificial intelligence: A feasible and cost-effective approach. Remote Sensing, 12, 489. The Catlin Seaview Survey-kilometre-scale seascape assessment, and monitoring of coral reef ecosystems. Aquatic Conservation: Marine and Freshwater Ecosystems. M González-Rivero, P Bongaerts, O Beijbom, O Pizarro, A Friedman, A Rodriguez-Ramirez, B Upcroft, D Laffoley, D Kline, C Bailhache, 24González-Rivero, M., Bongaerts, P., Beijbom, O., Pizarro, O., Friedman, A., Rodriguez-Ramirez, A., Upcroft, B., Laffoley, D., Kline, D., Bailhache, C. et al. (2014) The Catlin Seaview Survey-kilometre-scale seascape assessment, and monitoring of coral reef ecosystems. Aquatic Conservation: Marine and Freshwater Ecosystems, 24, 184-198. 2022) rstanarm: Bayesian applied regression modeling via Stan. B Goodrich, J Gabry, I Ali, S Brilleman, Goodrich, B., Gabry, J., Ali, I. and Brilleman, S. (2022) rstanarm: Bayesian applied regression modeling via Stan. URL: https: //mc-stan.org/rstanarm/. R package version 2.21.3. Non-invasive genetic monitoring involving citizen science enables reconstruction of current pack dynamics in a re-establishing wolf population. H Granroth-Wilding, C Primmer, M Lindqvist, J Poutanen, O Thalmann, J Aspi, J Harmoinen, I Kojola, T Laaksonen, BMC ecology. 1744Granroth-Wilding, H., Primmer, C., Lindqvist, M., Poutanen, J., Thalmann, O., Aspi, J., Harmoinen, J., Kojola, I. and Laaksonen, T. (2017) Non-invasive genetic monitoring involving citizen science enables reconstruction of current pack dynamics in a re-establishing wolf population. BMC ecology, 17, 44. Great barrier reef outlook report 2009: In brief. Great Barrier Reef Marine Park Authority. Great Barrier Reef Marine Park AuthorityGreat Barrier Reef Marine Park Authority (2009) Great barrier reef outlook report 2009: In brief. Great Barrier Reef Marine Park Authority. Model-based integration of citizen science data from disparate sources increases the precision of bird population trends. L R Hertzog, C Frank, S Klimek, N Roeder, H G S Boehner, J Kamp, DIVERSITY AND DISTRIBUTIONS. 27Hertzog, L. R., Frank, C., Klimek, S., Roeder, N., Boehner, H. G. S. and Kamp, J. (2021) Model-based integration of citizen science data from disparate sources increases the precision of bird population trends. DIVERSITY AND DISTRIBUTIONS, 27, 1106-1119. A fine-scale us population estimate of a montane spruce-fir bird species of conservation concern. J M Hill, J D Lloyd, Ecosphere. 81921Hill, J. M. and Lloyd, J. D. (2017) A fine-scale us population estimate of a montane spruce-fir bird species of conservation concern. Ecosphere, 8, e01921. Aggregating user input in ecology citizen science projects. G Hines, A Swanson, M Kosmala, C Lintott, Twenty-Seventh IAAI Conference. Hines, G., Swanson, A., Kosmala, M. and Lintott, C. (2015) Aggregating user input in ecology citizen science projects. In Twenty-Seventh IAAI Conference. First citizen-science population abundance and growth rate estimates for green sea turtles chelonia mydas foraging in the northern great barrier reef, australia. C A Hof, E Smallwood, J Meager, I P Bell, Marine Ecology Progress Series. 574Hof, C. A., Smallwood, E., Meager, J. and Bell, I. P. (2017) First citizen-science population abundance and growth rate estimates for green sea turtles chelonia mydas foraging in the northern great barrier reef, australia. Marine Ecology Progress Series, 574, 181-191. Development: Mobilize citizens to track sustainability. A Hsu, O Malik, L Johnson, D C Esty, Nature News. 50833Hsu, A., Malik, O., Johnson, L. and Esty, D. C. (2014) Development: Mobilize citizens to track sustainability. Nature News, 508, 33. Coral reefs in the anthropocene. T P Hughes, M L Barnes, D R Bellwood, J E Cinner, G S Cumming, J B Jackson, J Kleypas, I A Van De Leemput, J M Lough, T H Morrison, Nature. 546Hughes, T. P., Barnes, M. L., Bellwood, D. R., Cinner, J. E., Cumming, G. S., Jackson, J. B., Kleypas, J., Van De Leemput, I. A., Lough, J. M., Morrison, T. H. et al. (2017) Coral reefs in the anthropocene. Nature, 546, 82-90. Accounting for imperfect observation and estimating true species distributions in modelling biological invasions. T Mang, F Essl, D Moser, G Karrer, I Kleinbauer, S Dullinger, Ecography. 40Mang, T., Essl, F., Moser, D., Karrer, G., Kleinbauer, I. and Dullinger, S. (2017) Accounting for imperfect observation and estimating true species distributions in modelling biological invasions. Ecography, 40, 1187-1197. Comparison of the predicted and observed secondary structure of t4 phage lysozyme. B W Matthews, Biochimica et Biophysica Acta (BBA)-Protein Structure. 405Matthews, B. W. (1975) Comparison of the predicted and observed secondary structure of t4 phage lysozyme. Biochimica et Biophysica Acta (BBA)-Protein Structure, 405, 442-451. Comparing habitat models using ground-based and remote sensing data: Saltmarsh Sparrow presence versus nesting. S Meiman, D Civco, K Holsinger, C S Elphick, Wetlands. 32Meiman, S., Civco, D., Holsinger, K. and Elphick, C. S. (2012) Comparing habitat models using ground-based and remote sensing data: Saltmarsh Sparrow presence versus nesting. Wetlands, 32, 725-736. Preferred reporting items for systematic reviews and meta-analyses: the prisma statement. D Moher, A Liberati, J Tetzlaff, D G Altman, P Group, PLoS med. 61000097Moher, D., Liberati, A., Tetzlaff, J., Altman, D. G., Group, P. et al. (2009) Preferred reporting items for systematic reviews and meta-analyses: the prisma statement. PLoS med, 6, e1000097. Activity of invasive slug limax maximus in relation to climate conditions based on citizen's observations and novel regularization based statistical approaches. Y Morii, Y Ohkubo, S Watanabe, Science of the Total Environment. 637Morii, Y., Ohkubo, Y. and Watanabe, S. (2018) Activity of invasive slug limax maximus in relation to climate conditions based on citizen's observations and novel regularization based statistical approaches. Science of the Total Environment, 637, 1061-1068. Citizen science decisions: A Bayesian approach optimises effort. J Mugford, E Moltchanova, M Plank, J Sullivan, A Byrom, A James, ECOLOGICAL INFORMATICS. 63Mugford, J., Moltchanova, E., Plank, M., Sullivan, J., Byrom, A. and James, A. (2021) Citizen science decisions: A Bayesian approach optimises effort. ECOLOGICAL INFORMATICS, 63. Annual estimates of occupancy for bryophytes, lichens and invertebrates in the uk. C L Outhwaite, G D Powney, T A August, R E Chandler, S Rorke, O L Pescott, M Harvey, H E Roy, R Fox, D B Roy, Scientific data. 6Outhwaite, C. L., Powney, G. D., August, T. A., Chandler, R. E., Rorke, S., Pescott, O. L., Harvey, M., Roy, H. E., Fox, R., Roy, D. B. et al. (2019) Annual estimates of occupancy for bryophytes, lichens and invertebrates in the uk, 1970-2015. Scientific data, 6, 1-12. Quantifying range-wide variation in population trends from local abundance surveys and widespread opportunistic occurrence records. J Pagel, B J Anderson, R B O&apos;hara, W Cramer, R Fox, F Jeltsch, D B Roy, C D Thomas, F M Schurr, Methods in Ecology and Evolution. 5Pagel, J., Anderson, B. J., O'Hara, R. B., Cramer, W., Fox, R., Jeltsch, F., Roy, D. B., Thomas, C. D. and Schurr, F. M. (2014) Quan- tifying range-wide variation in population trends from local abundance surveys and widespread opportunistic occurrence records. Methods in Ecology and Evolution, 5, 751-760. ape 5.0: an environment for modern phylogenetics and evolutionary analyses in R. Bioinformatics. E Paradis, K Schliep, 35Paradis, E. and Schliep, K. (2019) ape 5.0: an environment for modern phylogenetics and evolutionary analyses in R. Bioinfor- matics, 35, 526-528. Breeding thresholds in opportunistic odonata records. M A Patten, E A Hjalmarson, B D Smith-Patten, J T Bried, Ecological Indicators. 106105460Patten, M. A., Hjalmarson, E. A., Smith-Patten, B. D. and Bried, J. T. (2019) Breeding thresholds in opportunistic odonata records. Ecological Indicators, 106, 105460. Comparing bayesian models of annotation. S Paun, B Carpenter, J Chamberlain, D Hovy, U Kruschwitz, M Poesio, Transactions of the Association for Computational Linguistics. 6Paun, S., Carpenter, B., Chamberlain, J., Hovy, D., Kruschwitz, U. and Poesio, M. (2018) Comparing bayesian models of anno- tation. Transactions of the Association for Computational Linguistics, 6, 571-585. Use of large-scale acoustic monitoring to assess anthropogenic pressures on orthoptera communities. C Penone, I Le Viol, V Pellissier, J.-F Julien, Y Bas, C Kerbiriou, Conservation Biology. 27Penone, C., Le Viol, I., Pellissier, V., JULIEN, J.-F., Bas, Y. and Kerbiriou, C. (2013) Use of large-scale acoustic monitoring to assess anthropogenic pressures on orthoptera communities. Conservation Biology, 27, 979-987. Monitoring through many eyes: Integrating disparate datasets to improve monitoring of the great barrier reef. E E Peterson, E Santos-Fernández, C Chen, S Clifford, J Vercelloni, A Pearse, R Brown, B Christensen, A James, K Anthony, Environmental Modelling & Software. 124104557Peterson, E. E., Santos-Fernández, E., Chen, C., Clifford, S., Vercelloni, J., Pearse, A., Brown, R., Christensen, B., James, A., Anthony, K. et al. (2020) Monitoring through many eyes: Integrating disparate datasets to improve monitoring of the great barrier reef. Environmental Modelling & Software, 124, 104557. Robust inference on large-scale species habitat use with interview data: The status of jaguars outside protected areas in central america. L S Petracca, J L Frair, J B Cohen, A P Calderón, J Carazo-Salazar, F Castañeda, D Corrales-Gutiérrez, R J Foster, B Harmsen, S Hernández-Potosme, Journal of applied ecology. 55Petracca, L. S., Frair, J. L., Cohen, J. B., Calderón, A. P., Carazo-Salazar, J., Castañeda, F., Corrales-Gutiérrez, D., Foster, R. J., Harmsen, B., Hernández-Potosme, S. et al. (2018) Robust inference on large-scale species habitat use with interview data: The status of jaguars outside protected areas in central america. Journal of applied ecology, 55, 723-734. 2022) rjags: Bayesian Graphical Models using MCMC. M Plummer, Plummer, M. (2022) rjags: Bayesian Graphical Models using MCMC. URL: https://CRAN.R-project.org/package=rjags. R package version 4-13. Landscape and climate determine patterns of spread for all colour morphs of the alien ladybird harmonia axyridis. B V Purse, R Comont, A Butler, P M Brown, C Kessel, H E Roy, Journal of Biogeography. 42Purse, B. V., Comont, R., Butler, A., Brown, P. M., Kessel, C. and Roy, H. E. (2015) Landscape and climate determine patterns of spread for all colour morphs of the alien ladybird harmonia axyridis. Journal of Biogeography, 42, 575-588. R Core Team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, AustriaR Core Team (2018) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL: https://www.R-project.org/. Learning from crowds. V C Raykar, S Yu, L H Zhao, G H Valadez, C Florin, L Bogoni, L Moy, Journal of Machine Learning Research. 11Raykar, V. C., Yu, S., Zhao, L. H., Valadez, G. H., Florin, C., Bogoni, L. and Moy, L. (2010) Learning from crowds. Journal of Machine Learning Research, 11, 1297-1322. Age and growth of the critically endangered flapper skate, Dipturus intermedius. Aquatic Conservation: Marine and Freshwater Ecosystems. T Régnier, J Dodd, S Benjamins, F M Gibb, P J Wright, 31Régnier, T., Dodd, J., Benjamins, S., Gibb, F. M. and Wright, P. J. (2021) Age and growth of the critically endangered flapper skate, Dipturus intermedius. Aquatic Conservation: Marine and Freshwater Ecosystems, 31, 2381-2388. Integrating auxiliary data in optimal spatial design for species distribution modelling. B J Reich, K Pacifici, J W Stallings, Methods in Ecology and Evolution. 9Reich, B. J., Pacifici, K. and Stallings, J. W. (2018) Integrating auxiliary data in optimal spatial design for species distribution modelling. Methods in Ecology and Evolution, 9, 1626-1637. Audible bats provide opportunities for citizen scientists. T J Rodhouse, S Rose, T Hawkins, R M Rodriguez, Conservation Science and Practice. 3435Rodhouse, T. J., Rose, S., Hawkins, T. and Rodriguez, R. M. (2021) Audible bats provide opportunities for citizen scientists. Conservation Science and Practice, 3, e435. Coral reef habitat mapping: A combination of object-based image analysis and ecological modelling. Remote sensing of environment. C Roelfsema, E Kovacs, J C Ortiz, N H Wolff, D Callaghan, M Wettle, M Ronan, S M Hamylton, P J Mumby, S Phinn, 208Roelfsema, C., Kovacs, E., Ortiz, J. C., Wolff, N. H., Callaghan, D., Wettle, M., Ronan, M., Hamylton, S. M., Mumby, P. J. and Phinn, S. (2018) Coral reef habitat mapping: A combination of object-based image analysis and ecological modelling. Remote sensing of environment, 208, 27-41. Benthic and coral reef community field data for heron reef, southern great barrier reef, australia. C Roelfsema, E M Kovacs, K Markey, J Vercelloni, A Rodriguez-Ramirez, S Lopez-Marcano, M Gonzalez-Rivero, O Hoegh-Guldberg, S R Phinn, 8Scientific dataRoelfsema, C., Kovacs, E. M., Markey, K., Vercelloni, J., Rodriguez-Ramirez, A., Lopez-Marcano, S., Gonzalez-Rivero, M., Hoegh- Guldberg, O. and Phinn, S. R. (2021) Benthic and coral reef community field data for heron reef, southern great barrier reef, australia, 2002-2018. Scientific data, 8, 1-7. Calibration and validation of coral reef benthic community maps derived from high spatial resolution satellite imagery. C Roelfsema, S Phinn, Journal of Applied Remote Sensing. 443527Roelfsema, C. and Phinn, S. (2010) Calibration and validation of coral reef benthic community maps derived from high spatial resolution satellite imagery. Journal of Applied Remote Sensing, 4, 043527. Estimating unbiased phenological trends by adapting site-occupancy models. T Roth, N Strebel, V Amrhein, Ecology. 95Roth, T., Strebel, N. and Amrhein, V. (2014) Estimating unbiased phenological trends by adapting site-occupancy models. Ecology, 95, 2144-2154. Approximate bayesian inference for latent gaussian models by using integrated nested laplace approximations. H Rue, S Martino, N Chopin, Journal of the royal statistical society: Series b (statistical methodology). 71Rue, H., Martino, S. and Chopin, N. (2009) Approximate bayesian inference for latent gaussian models by using integrated nested laplace approximations. Journal of the royal statistical society: Series b (statistical methodology), 71, 319-392. Global invasion history of the agricultural pest butterfly pieris rapae revealed with genomics and citizen science. S F Ryan, E Lombaert, A Espeset, R Vila, G Talavera, V Dincă, M M Doellman, M A Renshaw, M W Eng, E A Hornett, Proceedings of the National Academy of Sciences. 116Ryan, S. F., Lombaert, E., Espeset, A., Vila, R., Talavera, G., Dincă, V., Doellman, M. M., Renshaw, M. A., Eng, M. W., Hornett, E. A. et al. (2019) Global invasion history of the agricultural pest butterfly pieris rapae revealed with genomics and citizen science. Proceedings of the National Academy of Sciences, 116, 20015-20024. Understanding the reliability of citizen science observational data using item response models. E Santos-Fernandez, K Mengersen, Methods in Ecology and Evolution. 12Santos-Fernandez, E. and Mengersen, K. (2021) Understanding the reliability of citizen science observational data using item response models. Methods in Ecology and Evolution, 12, 1533-1548. Correcting misclassification errors in crowdsourced ecological data: A bayesian perspective. Santos Fernandez, E Peterson, E E Vercelloni, J Rushworth, E Mengersen, K , https:/rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssc.12453Journal of the Royal Statistical Society: Series C (Applied Statistics. Santos Fernandez, E., Peterson, E. E., Vercelloni, J., Rushworth, E. and Mengersen, K. (2020) Correcting misclassification errors in crowdsourced ecological data: A bayesian perspective. Journal of the Royal Statistical Society: Series C (Applied Statistics), n/a. URL: https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssc.12453. Flight plan for the future: floatplane pilots and researchers team up to predict invasive species dispersal in Alaska. T Schwoerer, R J Dial, J M Little, A E Martin, J M Morton, I Schmidt, J Ward, E J , BIOLOGICAL INVASIONS. 24Schwoerer, T., Dial, R. J., Little, J. M., Martin, A. E., Morton, J. M., Schmidt, I, J. and Ward, E. J. (2022) Flight plan for the future: floatplane pilots and researchers team up to predict invasive species dispersal in Alaska. BIOLOGICAL INVASIONS, 24, 1229-1245. Long-term trends in the occupancy of ants revealed through use of multi-sourced datasets. J K Sheard, C Rahbek, R R Dunn, N J Sanders, N J Isaac, Biology Letters. 1720210240Sheard, J. K., Rahbek, C., Dunn, R. R., Sanders, N. J. and Isaac, N. J. (2021) Long-term trends in the occupancy of ants revealed through use of multi-sourced datasets. Biology Letters, 17, 20210240. Factors affecting the mortality of Lumholtz's tree-kangaroo (Dendrolagus Lumholtzi) by vehicle strike. A L Shima, D S Gillieson, G M Crowley, R G Dwyer, L Berger, Wildlife Research. 45Shima, A. L., Gillieson, D. S., Crowley, G. M., Dwyer, R. G. and Berger, L. (2018) Factors affecting the mortality of Lumholtz's tree-kangaroo (Dendrolagus Lumholtzi) by vehicle strike. Wildlife Research, 45, 559-569. Accounting for spatial varying sampling effort due to accessibility in citizen science data: A case study of moose in norway. J Sicacha-Parada, I Steinsland, B Cretois, J Borgelt, Spatial Statistics. 100446Sicacha-Parada, J., Steinsland, I., Cretois, B. and Borgelt, J. (2020) Accounting for spatial varying sampling effort due to acces- sibility in citizen science data: A case study of moose in norway. Spatial Statistics, 100446. Crowdsourcing without a crowd: Reliable online species identification using bayesian models to minimize crowd size. A Siddharthan, C Lambin, A.-M Robinson, N Sharma, R Comont, E O&apos;mahony, C Mellish, R V D Wal, ACM Transactions on Intelligent Systems and Technology. 745TISTSiddharthan, A., Lambin, C., Robinson, A.-M., Sharma, N., Comont, R., O'mahony, E., Mellish, C. and Wal, R. V. D. (2016) Crowdsourcing without a crowd: Reliable online species identification using bayesian models to minimize crowd size. ACM Transactions on Intelligent Systems and Technology (TIST), 7, 45. Evaluating citizen-based presence data for bird monitoring. T Snäll, O Kindvall, J Nilsson, T Pärt, Biological conservation. 144Snäll, T., Kindvall, O., Nilsson, J. and Pärt, T. (2011) Evaluating citizen-based presence data for bird monitoring. Biological conservation, 144, 804-810. Population trends for north american winter birds based on hierarchical models. C U Soykan, J Sauer, J G Schuetz, G S Lebaron, K Dale, G M Langham, 1351Ecosphere, 7Soykan, C. U., Sauer, J., Schuetz, J. G., LeBaron, G. S., Dale, K. and Langham, G. M. (2016) Population trends for north american winter birds based on hierarchical models. Ecosphere, 7, e01351. Openbugs user manual. Version, 3. D Spiegelhalter, A Thomas, N Best, D Lunn, Spiegelhalter, D., Thomas, A., Best, N. and Lunn, D. (2007) Openbugs user manual. Version, 3, 2007. RStan: the R interface to Stan. Stan Development Team. Stan Development Team (2018) RStan: the R interface to Stan. URL: http://mc-stan.org/. R package version 2.18.2. Safari science: assessing the reliability of citizen science data for wildlife surveys. C Steger, B Butt, M B Hooten, Journal of Applied Ecology. 54Steger, C., Butt, B. and Hooten, M. B. (2017) Safari science: assessing the reliability of citizen science data for wildlife surveys. Journal of Applied Ecology, 54, 2053-2062. Studying phenology by flexible modelling of seasonal detectability peaks. N Strebel, M Kéry, M Schaub, H Schmid, Methods in Ecology and Evolution. 5Strebel, N., Kéry, M., Schaub, M. and Schmid, H. (2014) Studying phenology by flexible modelling of seasonal detectability peaks. Methods in Ecology and Evolution, 5, 483-490. Opportunistic citizen science data of animal species produce reliable estimates of distribution trends if analysed with occupancy models. A J Van Strien, C A Van Swaay, T Termaat, Journal of Applied Ecology. 50van Strien, A. J., van Swaay, C. A. and Termaat, T. (2013a) Opportunistic citizen science data of animal species produce reliable estimates of distribution trends if analysed with occupancy models. Journal of Applied Ecology, 50, 1450-1458. Occupancy modelling as a new approach to assess supranational trends using opportunistic data: a pilot study for the damselfly calopteryx splendens. A J Van Strien, T Termaat, V Kalkman, M Prins, G De Knijf, A.-L Gourmand, X Houard, B Nelson, C Plate, S Prentice, Biodiversity and Conservation. 22van Strien, A. J., Termaat, T., Kalkman, V., Prins, M., De Knijf, G., Gourmand, A.-L., Houard, X., Nelson, B., Plate, C., Prentice, S. et al. (2013b) Occupancy modelling as a new approach to assess supranational trends using opportunistic data: a pilot study for the damselfly calopteryx splendens. Biodiversity and Conservation, 22, 673-686. Strong differences in migratory connectivity patterns among species of Neotropical-Nearctic migratory birds revealed by combining stable isotopes and abundance in a Bayesian assignment analysis. C E Studds, J M WunderleJr, P P Marra, JOURNAL OF BIOGEOGRAPHY. 48Studds, C. E., Wunderle, Jr., J. M. and Marra, P. P. (2021) Strong differences in migratory connectivity patterns among species of Neotropical-Nearctic migratory birds revealed by combining stable isotopes and abundance in a Bayesian assignment analysis. JOURNAL OF BIOGEOGRAPHY, 48, 1746-1757. R2winbugs: A package for running winbugs from r. S Sturtz, U Ligges, A Gelman, Journal of Statistical Software. 12Sturtz, S., Ligges, U. and Gelman, A. (2005) R2winbugs: A package for running winbugs from r. Journal of Statistical Software, 12, 1-16. URL: http://www.jstatsoft.org. R2jags: Using R to Run JAGS. Y.-S Su, M Yajima, R package version 0.7-1Su, Y.-S. and Yajima, M. (2021) R2jags: Using R to Run JAGS. URL: https://CRAN.R-project.org/package=R2jags. R package version 0.7-1. Long-term monitoring of the great barrier reef. H Sweatman, S Burgess, A Cheal, G Coleman, J S C Delean, M Emslie, I Miller, K Osborne, A Mcdonald, A Thompson, Sweatman, H., Burgess, S., Cheal, A., Coleman, G., Delean, J. S. C., Emslie, M., Miller, I., Osborne, K., McDonald, A. and Thompson, A. (2005) Long-term monitoring of the great barrier reef. Regional avian species declines estimated from volunteercollected long-term data using list length analysis. J K Szabo, P A Vesk, P W Baxter, H P Possingham, Ecological Applications. 20Szabo, J. K., Vesk, P. A., Baxter, P. W. and Possingham, H. P. (2010) Regional avian species declines estimated from volunteer- collected long-term data using list length analysis. Ecological Applications, 20, 2157-2169. Marine monitoring program: Annual report for inshore coral reef monitoring. A Thompson, P Costello, J Davidson, M Logan, G Coleman, K Gunn, B Schaffelke, Thompson, A., Costello, P., Davidson, J., Logan, M., Coleman, G., Gunn, K. and Schaffelke, B. (2016) Marine monitoring pro- gram: Annual report for inshore coral reef monitoring 2014-2015. Species density models from opportunistic citizen science data. J M Ver Hoef, D Johnson, R Angliss, M Higham, METHODS IN ECOLOGY AND EVOLUTION. 12Ver Hoef, J. M., Johnson, D., Angliss, R. and Higham, M. (2021) Species density models from opportunistic citizen science data. METHODS IN ECOLOGY AND EVOLUTION, 12, 1911-1925. Forecasting intensifying disturbance effects on coral reefs. J Vercelloni, B Liquet, E V Kennedy, M González-Rivero, M J Caley, E E Peterson, M Puotinen, O Hoegh-Guldberg, K Mengersen, Global change biology. 26Vercelloni, J., Liquet, B., Kennedy, E. V., González-Rivero, M., Caley, M. J., Peterson, E. E., Puotinen, M., Hoegh-Guldberg, O. and Mengersen, K. (2020) Forecasting intensifying disturbance effects on coral reefs. Global change biology, 26, 2785- 2797. A method that accounts for differential detectability in mixed samples of long-term infections with applications to the case of chronic wasting disease in cervids. H Viljugrein, P Hopp, S L Benestad, E B Nilsen, J Våge, S Tavornpanich, C M Rolandsen, O Strand, A Mysterud, Methods in Ecology and Evolution. 10Viljugrein, H., Hopp, P., Benestad, S. L., Nilsen, E. B., Våge, J., Tavornpanich, S., Rolandsen, C. M., Strand, O. and Mysterud, A. (2019) A method that accounts for differential detectability in mixed samples of long-term infections with applications to the case of chronic wasting disease in cervids. Methods in Ecology and Evolution, 10, 134-145. Clustering community science data to infer songbird migratory connectivity in the western hemisphere. J G Vincent, R Schuster, S Wilson, D Fink, J R Bennett, ECOSPHERE13Vincent, J. G., Schuster, R., Wilson, S., Fink, D. and Bennett, J. R. (2022) Clustering community science data to infer songbird migratory connectivity in the western hemisphere. ECOSPHERE, 13. . Edgar Santos-Fernandez, Edgar Santos-Fernandez et al. Evaluating the efficacy of eBird data for modeling historical population trajectories of North American birds and for monitoring populations of boreal and Arctic breeding species. J Walker, P D Taylor, AVIAN CONSERVATION AND ECOLOGY. 15Walker, J. and Taylor, P. D. (2020) Evaluating the efficacy of eBird data for modeling historical population trajectories of North American birds and for monitoring populations of boreal and Arctic breeding species. AVIAN CONSERVATION AND ECOLOGY, 15. Bayesian analysis of dynamic item response models in educational testing. X Wang, J O Berger, D S Burdick, The Annals of Applied Statistics. 7Wang, X., Berger, J. O., Burdick, D. S. et al. (2013) Bayesian analysis of dynamic item response models in educational testing. The Annals of Applied Statistics, 7, 126-153. Online crowdsourcing: rating annotators and obtaining cost-effective labels. P Welinder, P Perona, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition-Workshops. IEEEWelinder, P. and Perona, P. (2010) Online crowdsourcing: rating annotators and obtaining cost-effective labels. In 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition-Workshops, 25-32. IEEE. Whose vote should count more: Optimal integration of labels from labelers of unknown expertise. J Whitehill, T.-F Wu, J Bergsma, J R Movellan, P L Ruvolo, Advances in neural information processing systems. Whitehill, J., Wu, T.-f., Bergsma, J., Movellan, J. R. and Ruvolo, P. L. (2009) Whose vote should count more: Optimal integration of labels from labelers of unknown expertise. In Advances in neural information processing systems, 2035-2043. Citizen science reveals an extensive shift in the winter distribution of migratory western grebes. S Wilson, E M Anderson, A S Wilson, D F Bertram, P Arcese, PLoS One. 865408Wilson, S., Anderson, E. M., Wilson, A. S., Bertram, D. F. and Arcese, P. (2013) Citizen science reveals an extensive shift in the winter distribution of migratory western grebes. PLoS One, 8, e65408. An informatics approach for smart evaluation of water quality related ecosystem services. W Yan, M Hutchins, S Loiselle, C Hall, Annals of Data Science. 3Yan, W., Hutchins, M., Loiselle, S. and Hall, C. (2016) An informatics approach for smart evaluation of water quality related ecosystem services. Annals of Data Science, 3, 251-264. Dietary compositions and their seasonal shifts in japanese resident birds, estimated from the analysis of volunteer monitoring data. T Yoshikawa, Y Osada, PloS one. 10119324Yoshikawa, T. and Osada, Y. (2015) Dietary compositions and their seasonal shifts in japanese resident birds, estimated from the analysis of volunteer monitoring data. PloS one, 10, e0119324. Migratory connectivity of swan geese based on species' distribution models, feather stable isotope assignment and satellite tracking. Q Zhu, K A Hobson, Q Zhao, Y Zhou, I Damba, N Batbayar, T Natsagdorj, B Davaasuren, A Antonov, J Guan, Diversity and DistributionsZhu, Q., Hobson, K. A., Zhao, Q., Zhou, Y., Damba, I., Batbayar, N., Natsagdorj, T., Davaasuren, B., Antonov, A., Guan, J. et al. (2020) Migratory connectivity of swan geese based on species' distribution models, feather stable isotope assignment and satellite tracking. Diversity and Distributions. Synthesizing multiple data types for biological conservation using integrated population models. E F Zipkin, S P Saunders, Biological Conservation. 217Zipkin, E. F. and Saunders, S. P. (2018) Synthesizing multiple data types for biological conservation using integrated population models. Biological Conservation, 217, 240-250.
[ "https://github.com/EdgarSantos-Fernandez/", "https://github.com/citscisean/" ]
[ "Observing topological charges and dynamical bulk-surface correspondence with ultracold atoms", "Observing topological charges and dynamical bulk-surface correspondence with ultracold atoms" ]
[ "Chang-Rui Yi \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nShanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina\n", "Long Zhang \nInternational Center for Quantum Materials\nSchool of Physics\nPeking University\n100871BeijingChina\n\nCollaborative Innovation Center of Quantum Matter\n100871BeijingChina\n", "Lin Zhang \nInternational Center for Quantum Materials\nSchool of Physics\nPeking University\n100871BeijingChina\n", "Rui-Heng Jiao \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nShanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina\n\nCollaborative Innovation Center of Quantum Matter\n100871BeijingChina\n", "Xiang-Can Cheng \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nShanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina\n", "Zong-Yao Wang \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nShanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina\n", "Xiao-Tian Xu \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nShanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina\n", "Wei Sun \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nShanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina\n", "Xiong-Jun Liu \nInternational Center for Quantum Materials\nSchool of Physics\nPeking University\n100871BeijingChina\n\nCollaborative Innovation Center of Quantum Matter\n100871BeijingChina\n", "Shuai Chen \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nShanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina\n", "Jian-Wei Pan \nHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nShanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina\n" ]
[ "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Shanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina", "International Center for Quantum Materials\nSchool of Physics\nPeking University\n100871BeijingChina", "Collaborative Innovation Center of Quantum Matter\n100871BeijingChina", "International Center for Quantum Materials\nSchool of Physics\nPeking University\n100871BeijingChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Shanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina", "Collaborative Innovation Center of Quantum Matter\n100871BeijingChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Shanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Shanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Shanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Shanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina", "International Center for Quantum Materials\nSchool of Physics\nPeking University\n100871BeijingChina", "Collaborative Innovation Center of Quantum Matter\n100871BeijingChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Shanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina", "Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina", "Shanghai Branch\nCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics\nUniversity of Science and Technology of China\n201315ShanghaiChina" ]
[]
In quenching a topological phase across phase transition, the dynamical bulk-surface correspondence emerges that the bulk topology of d-dimensional (dD) phase relates to the nontrivial pattern of quench dynamics emerging on (d − 1)D subspace, called band inversion surfaces (BISs) in momentum space. Here we report the first experimental observation of the dynamical bulk-surface correspondence through measuring the topological charges in a 2D quantum anomalous Hall model realized in an optical Raman lattice. The system can be quenched with respect to every spin axis by suddenly varying the two-photon detuning or phases of the Raman couplings, in which the topological charges and BISs are measured dynamically by the time-averaged spin textures. We observe that the total charges in the region enclosed by BISs define a dynamical topological invariant, which equals the Chern index of the post-quench band. The topological charges relate to an emergent dynamical field which exhibits nontrivial topology on BIS, rendering the dynamical bulk-surface correspondence. This study opens a new avenue to explore topological phases dynamically.Introduction.-Topological quantum matter [1, 2] has attracted intense interest due to the discovery of new fundamental phases [3-5] and broad potential applications[6,7]. Recent experimental advances in cold atoms highlight the realizations of various topological models, such as the one-dimensional (1D) Su-Schrieffer-Heeger model [8], 1D chiral topological phase[9], and 2D Chern insulator[10][11][12][13][14][15]. The studies commonly faces an important question: how to measure topological indices for cold atom systems? The 1D winding number can be detected by measuring Zak phase via Ramsey interferometry[8]. In 2D spin-orbit (SO) coupled Chern phase[14,15], the Chern number can be determined by measuring Bloch states at highly symmetric momenta[16]. These methods are however not generic or lack sufficient accuracy.Recently, a research focus has been drawn to nonequilibrium dynamics in topological quantum phases[9,[17][18][19][20][21]. Several theoretical works[22][23][24][25][26]proposed dynamical characterizations of topological phases by quantum quenches, with some predictions having been studied in experiment[27][28][29]. In particular, a dynamical bulk-surface correspondence was proposed[23], showing that the bulk topology of a dD topological phase universally corresponds to the nontrivial pattern of quench dynamics emerging on the (d − 1)D momentum subspace called band inversion surfaces (BISs), analogy to the wellknown bulk-boundary correspondence in real space[1,2]. A recent experiment observed the ring pattern of BISs dynamically[27]. However, the topological invariant of quench dynamics emerging on BISs was not observed, so the experimental verification of the essential dynamical
10.1103/physrevlett.123.190603
[ "https://export.arxiv.org/pdf/1905.06478v1.pdf" ]
155,099,829
1905.06478
f9d10a2692139fbcdf7bb80aa962b16389f11ecf
Observing topological charges and dynamical bulk-surface correspondence with ultracold atoms Chang-Rui Yi Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Shanghai Branch CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 201315ShanghaiChina Long Zhang International Center for Quantum Materials School of Physics Peking University 100871BeijingChina Collaborative Innovation Center of Quantum Matter 100871BeijingChina Lin Zhang International Center for Quantum Materials School of Physics Peking University 100871BeijingChina Rui-Heng Jiao Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Shanghai Branch CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 201315ShanghaiChina Collaborative Innovation Center of Quantum Matter 100871BeijingChina Xiang-Can Cheng Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Shanghai Branch CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 201315ShanghaiChina Zong-Yao Wang Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Shanghai Branch CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 201315ShanghaiChina Xiao-Tian Xu Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Shanghai Branch CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 201315ShanghaiChina Wei Sun Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Shanghai Branch CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 201315ShanghaiChina Xiong-Jun Liu International Center for Quantum Materials School of Physics Peking University 100871BeijingChina Collaborative Innovation Center of Quantum Matter 100871BeijingChina Shuai Chen Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Shanghai Branch CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 201315ShanghaiChina Jian-Wei Pan Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China 230026HefeiAnhuiChina Shanghai Branch CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics University of Science and Technology of China 201315ShanghaiChina Observing topological charges and dynamical bulk-surface correspondence with ultracold atoms (Dated: February 28, 2022) In quenching a topological phase across phase transition, the dynamical bulk-surface correspondence emerges that the bulk topology of d-dimensional (dD) phase relates to the nontrivial pattern of quench dynamics emerging on (d − 1)D subspace, called band inversion surfaces (BISs) in momentum space. Here we report the first experimental observation of the dynamical bulk-surface correspondence through measuring the topological charges in a 2D quantum anomalous Hall model realized in an optical Raman lattice. The system can be quenched with respect to every spin axis by suddenly varying the two-photon detuning or phases of the Raman couplings, in which the topological charges and BISs are measured dynamically by the time-averaged spin textures. We observe that the total charges in the region enclosed by BISs define a dynamical topological invariant, which equals the Chern index of the post-quench band. The topological charges relate to an emergent dynamical field which exhibits nontrivial topology on BIS, rendering the dynamical bulk-surface correspondence. This study opens a new avenue to explore topological phases dynamically.Introduction.-Topological quantum matter [1, 2] has attracted intense interest due to the discovery of new fundamental phases [3-5] and broad potential applications[6,7]. Recent experimental advances in cold atoms highlight the realizations of various topological models, such as the one-dimensional (1D) Su-Schrieffer-Heeger model [8], 1D chiral topological phase[9], and 2D Chern insulator[10][11][12][13][14][15]. The studies commonly faces an important question: how to measure topological indices for cold atom systems? The 1D winding number can be detected by measuring Zak phase via Ramsey interferometry[8]. In 2D spin-orbit (SO) coupled Chern phase[14,15], the Chern number can be determined by measuring Bloch states at highly symmetric momenta[16]. These methods are however not generic or lack sufficient accuracy.Recently, a research focus has been drawn to nonequilibrium dynamics in topological quantum phases[9,[17][18][19][20][21]. Several theoretical works[22][23][24][25][26]proposed dynamical characterizations of topological phases by quantum quenches, with some predictions having been studied in experiment[27][28][29]. In particular, a dynamical bulk-surface correspondence was proposed[23], showing that the bulk topology of a dD topological phase universally corresponds to the nontrivial pattern of quench dynamics emerging on the (d − 1)D momentum subspace called band inversion surfaces (BISs), analogy to the wellknown bulk-boundary correspondence in real space[1,2]. A recent experiment observed the ring pattern of BISs dynamically[27]. However, the topological invariant of quench dynamics emerging on BISs was not observed, so the experimental verification of the essential dynamical In quenching a topological phase across phase transition, the dynamical bulk-surface correspondence emerges that the bulk topology of d-dimensional (dD) phase relates to the nontrivial pattern of quench dynamics emerging on (d − 1)D subspace, called band inversion surfaces (BISs) in momentum space. Here we report the first experimental observation of the dynamical bulk-surface correspondence through measuring the topological charges in a 2D quantum anomalous Hall model realized in an optical Raman lattice. The system can be quenched with respect to every spin axis by suddenly varying the two-photon detuning or phases of the Raman couplings, in which the topological charges and BISs are measured dynamically by the time-averaged spin textures. We observe that the total charges in the region enclosed by BISs define a dynamical topological invariant, which equals the Chern index of the post-quench band. The topological charges relate to an emergent dynamical field which exhibits nontrivial topology on BIS, rendering the dynamical bulk-surface correspondence. This study opens a new avenue to explore topological phases dynamically. Introduction.-Topological quantum matter [1,2] has attracted intense interest due to the discovery of new fundamental phases [3][4][5] and broad potential applications [6,7]. Recent experimental advances in cold atoms highlight the realizations of various topological models, such as the one-dimensional (1D) Su-Schrieffer-Heeger model [8], 1D chiral topological phase [9], and 2D Chern insulator [10][11][12][13][14][15]. The studies commonly faces an important question: how to measure topological indices for cold atom systems? The 1D winding number can be detected by measuring Zak phase via Ramsey interferometry [8]. In 2D spin-orbit (SO) coupled Chern phase [14,15], the Chern number can be determined by measuring Bloch states at highly symmetric momenta [16]. These methods are however not generic or lack sufficient accuracy. Recently, a research focus has been drawn to nonequilibrium dynamics in topological quantum phases [9,[17][18][19][20][21]. Several theoretical works [22][23][24][25][26] proposed dynamical characterizations of topological phases by quantum quenches, with some predictions having been studied in experiment [27][28][29]. In particular, a dynamical bulk-surface correspondence was proposed [23], showing that the bulk topology of a dD topological phase universally corresponds to the nontrivial pattern of quench dynamics emerging on the (d − 1)D momentum subspace called band inversion surfaces (BISs), analogy to the wellknown bulk-boundary correspondence in real space [1,2]. A recent experiment observed the ring pattern of BISs dynamically [27]. However, the topological invariant of quench dynamics emerging on BISs was not observed, so the experimental verification of the essential dynamical bulk-surface correspondence is yet to be pursued. In this letter, we report the experimental observation in ultracold atoms of the dynamical bulk-surface correspondence [23] following a new scheme proposed in Ref. [25], and characterize the topological phases by dynamically detecting topological charges of monopoles in momentum space. The central idea of the new scheme is that through a sequence of quantum quenches along all spin axes in the topological system, the topology can be detected by measuring the quantum dynamics for only z-component spin polarization in each quench. We implement the study in a 2D quantum anomalous Hall (QAH) model in an optical Raman lattice, and quench the system along different spin axes by quickly varying the twophoton detuning or phases of Raman couplings based on the new scheme [25]. The complete information of bulk topology, including the topological charges and BISs, is extracted by measuring the z-component spin dynamics. We observe that the total charges in the region enclosed by BISs equal the Chern number of the postquench phase, and are related to an emergent dynamical field which exhibits nontrivial topological pattern on the BIS, rendering the dynamical bulk-surface correspondence in momentum space. Experimental setup.-Our experiment is based on a 2D QAH model [30] with C 4 symmetry proposed in theory [31] and realized for ultracold 87 Rb bosons trapped in a square optical Raman lattice [14,15,32]. The laser beam E x (E y ) with wavelength of λ = 787nm is incident from x (y) direction and reflected by mirror. The two beams pass through λ/2 wave plates and The quench process corresponds to changing the symmetry of Raman potentials (sinusoidal curves) with respect to lattice sites (yellow ellipsoids) in a spatial direction. The relative (anti)symmetry selects the hopping type: Before quench, only on-site spin flipping is permitted while spin-flipped hopping is forbidden; after quench the situation is reversed. (d) The s-band structure before and after quenching hx or hy. The color indicates the distribution of spin polarization σz with respect to eigenstates of the pre-or post-quench Hamiltonian. each splits into two orthogonally polarized components E x = E xz + E xy and E y = E yz + E yx [Fig. 1(a)], with E xy =ŷ |E xy | cos(k 0 x), E xz =ẑ |E xz | e iϕ1 cos(k 0 x − ϕ 1 ), E yx =x |E yx | cos(k 0 y) , and E yz =ẑ |E yz | e iϕ2 cos(k 0 y − ϕ 2 ), from which the scalar and vector potentials are generated and give the lattice and Raman potentials, respectively [15,31]. Here k 0 = 2π/λ and ϕ 1 (ϕ 2 ) is the relative phase between E xz (E yz ) and E xy (E yx ). As proposed in Ref. [25], the key technique here is that we apply electro-optic modulators (EOMs) to manipulate the relative symmetries between lattice and Raman potentials by tuning ϕ 1,2 . The spin-1/2 system, selected from the hyperfine states |↑ = |F = 1, m F = −1 and |↓ = |1, 0 , are coupled by the Raman transitions [see Fig. 1(b)]. In tight-binding regime the realized Bloch Hamiltonian H(q) = h(q) · σ reads (see more details in supplementary materials [33]) H(q) =(m x + 2t y so sin q y )σ x + (m y + 2t x so sin q x )σ y + (m z − 2t x 0 cos q x − 2t y 0 cos q y )σ z ,(1) where q = (q x , q y ), m i , σ i , and t i 0 (t i so ) are respectively the Bloch momentum, Zeeman constants, Pauli matrices acting on spin space, and the spin-conserved (flip) hopping coefficients, with i = x, y, z. The m z -term is related to the two-photon detuning by m z = δ/2. The Zeeman constants m x,y and spin-flip hopping coefficients t x,y so are induced by the Raman couplings and controlled by ϕ 1,2 . In this experiment, the quenches along different spin axes will be performed by quickly modulating m x,y,z [25,33]. The two Raman potentials are generated by two pairs of beams (E xz , E yx ) and (E yz , E xy ), respectively. For the setting with (ϕ 1 , ϕ 2 ) = (π/2, π/2), the former (or latter) potential ∝ sin(k 0 x) cos(k 0 y) (or sin(k 0 y) cos(k 0 x)) is symmetric with respect to every lattice site in y (or x) direction, but antisymmetric in x (or y) direction [33]. This is the configuration of 2D optical Raman lattice to realize QAH model [14,15,31,32], in which the nearestneighbor spin-flip hopping is induced by each Raman potential along the antisymmetric direction, while onsite spin-flip transition is forbidden. We then reach the Hamiltonian (S10) with m x = m y = 0. Further, for the other setting with (ϕ 1 , ϕ 2 ) = (0, π/2) (or (π/2, 0)), the former (or latter) Raman potential ∝ cos(k 0 x) cos(k 0 y), which is symmetric with respect to each lattice site in both x and y directions, and induces on-site spin-flip transition, but no nearest-neighbor spin-flip hopping. Then a nonzero m y (or m x ) is induced, while t x so ≈ 0 (or t y so ≈ 0) [25,33], and the Hamiltonian (S10) can describe a trivial system with large transverse Zeeman field. Quenches and measurements.-We implement the quenches with respect to all spin quantization axes. For simplicity we consider that t x so = t y so = t so and t x 0 = t y 0 = t 0 by setting the lattice depths (V 0 ) and the Raman potential amplitudes (Ω 0 ) to be isotropic for post-quench Hamiltonian. Quenching h z is realized by tuning the two-photon detuning δ [27,33], while quenching h x,y is performed by suddenly tuning (ϕ 1 , ϕ 2 ) between the aforementioned two settings [see Fig. 1(c)]. In each quench, the post-quench parameters are fixed. Quenching h x,y is performed as follows. First, the 87 Rb atoms are prepared slightly above the critical temperature of Bose-Einstein condensation, and then adiabatically loaded into the Raman lattice with (ϕ 1 , ϕ 2 ) = (0, π/2) or (π/2, 0) by ramping up the laser intensity in 100ms [33]. The atoms are then populated almost in the lowest trivial energy band of the pre-quench Hamiltonian [ Fig. 1(d)]. Second, using the EOM in the x (or y) direction we suddenly tune the phase to shift as (ϕ 1 , ϕ 2 ) → (π/2, π/2) within 2µs, corresponding to quenching h y (or h x ). The atoms are driven out of equilibrium and evolve under the post-quench Hamiltonian with the topologically nontrivial bands [ Fig. 1(d)]. Third, we hold the system for a certain time t, and measure the atom density n ↑,↓ (q, t) of each spin component in momentum space by time-offlight (TOF) expansion. The evolution of spin polariza- tion σ z (q, t) = [n ↑ (q, t) − n ↓ (q, t)]/[n ↑ (q, t) + n ↓ (q, t)] is obtained at each Bloch momentum. Figure 2: Time evolution of the spin polarization σz(q) y,z after quenching hy (a) and hz (b), respectively. Quenching hy is realized by tuning the phases from (ϕ1, ϕ2) = (0, π/2) to the setting (π/2, π/2). Quenching hz corresponds to varying the two-photon detuning from δ = −200Er to −0.2Er. In each case, the measured spin textures for different hold time t (upper) are compared with numerical calculations (lower). Spin textures after the two quenches exhibit distinctive patterns during the time evolution. The post-quench parameters are fixed at lattice depth V0 = 4.0Er, Raman coupling strength Ω0 = 1.0Er and δ = −0.2Er. Here Er is the recoil energy. Figure 2 shows the measured spin textures σ z (q, t) y,z at different time after quenching h y,z , which are compared with the numerical calculations. The spin evolution σ z (q, t) x is also measured by quenching h x but not shown here for simplicity. The polarizations σ z (q, t) y,z oscillate at frequencies equaling to local gap of the postquench bands at momentum q. In particular, for quenching h y in (a), there appears a spiral-like dynamical pattern, which gradually spreads to the whole BZ. For quenching h z in (b), the spin oscillations exhibit a ring structure with C 4 symmetry, which is the dynamical signature of the BIS emerging in quench dynamics. For each quench, we fit the spin oscillation at every momentum q [33], with which we further obtain the time-averaged spin textures σ z (q) x,y,z . These spin textures exhibit nontrivial patterns which characterize the topology of post-quench band, as elaborated below. -1 1 0 -1 1 0 -1 0 1 -1 0 1 -1 0 1-1 0 1-1 0 1 -1 0 1 -1 0 1 -1 0 1 qy(k 0 ) qx(k 0 ) -1 1 0 -1 1 0 -1 0 1 -1 0 1 -1 0 1-1 0 1-1 0 1 -1 0 1 -1 0 1 -1 0 1 qy(k 0 ) qx(k 0 ) t=0 0.2 0.4 0.6 0.8 1.0 1.2 ms t=0 0.2 0.4 0.6 0.8 1.0 1.2 ms (a) (b) Measuring topological charges.-As shown in theory [23,25], the bulk topology can be characterized by the total topological charges in the region enclosed by BISs. The information of topological charges and BISs is captured dynamically by the time-averaged spin textures σ z (q) i , which are shown in Fig.3(a). The momenta where all the polarizations σ z (q) x,y,z being zero form the BIS, which shapes a ring around the M point (q x = q y = k 0 ), and divide the whole BZ into V + (with h z > 0) and V − (with h z < 0) regions [33]. Be-sides the BIS, there are also other momenta satisfying σ z (q) x = 0 or σ z (q) y = 0, and form the curves denoted as L x1,x2 and L y1,y2 , respectively. As shown in Fig.3(b), the curves L x1,x2 and L y1,y2 intersect at four momentum points D 1,2,3,4 in the first BZ, which mark the locations of dynamical topological charges [25,34]: three in V − and one in V + . All our experimental measurements agree well with numerical calculations (Fig.3), which are based on the full-band model and take into account the finite temperature effect [33]. A special case is that if the pre-quench trivial phase is fully polarized by a large initial Zeeman term with |m i | |t 0,so |, the BIS refers to the 1D band-crossing ring with h z = 0, and topological charges are located at the nodes of SO coupling with h x = h y = 0, which are Γ, X 1,2 and M points in the BZ [25]. The charge value C = ±1 can be further characterized by the winding of the spin-texture field S(q) = (S y , S x ), with components given by [25] S y, x (q) = 1 N q sign[h z (q)] σ z (q) y,x .(2) Here N q denotes the normalization factor. Since the two curves that intersect at a charge divide the neighboring region into four parts, the charge value C can be simply determined by the signs of spin textures σ z (q) x,y in the four subregions, as illustrated in Fig. 3(c). The plus (minus) sign of σ z (q) x,y indicates that the field components S x,y point to the positive (negative) spin axes. The combined dynamical field S(q) characterizes the charge C = +1 if the field flows from or to the charge; C = −1 if it is "repulsed" by the charge. We finally find that three dynamical topological charges are enclosed in the region V − with one being C = −1 and two C = +1, whose summation characterizes the post-quench topological phase with the Chern number Ch = +1. Dynamical bulk-surface correspondence.-Similar to the well-known bulk-boundary correspondence in real space, there is also a dynamical bulk-surface correspondence emerging from quench dynamics in the momentum space [23]. It claims that the bulk topology can be characterized by the dynamical topological pattern on the lower-dimensional BISs. In Ref. [27], we have observed the quench dynamics that occurs on BISs, and accordingly determined the topological phase diagram, but the topological pattern can not be measured by quenching only h z . Here, using the time-averaged spin textures obtained by a sequence of quenches, we can observe the winding of an emergent dynamical field defined on the BIS, as an experimental observation of the dynamical bulk-surface correspondence. According to Refs. [23,25], the dynamical field g(q) = (g y , g x ) can be defined by where N q is normalization factor, and q ⊥ denotes the momentum perpendicular to the BIS and points from V − to V + . We calculate the field components g i (q) (i = x, y) by the variations of spin textures σ z (q) i along the direction q ⊥ , i.e., g i (q 0 ) = lim q ⊥ →0 [ σ z (q 0 + q ⊥ê⊥ ) i − σ z (q 0 − q ⊥ê⊥ ) i ] for every q 0 on the BIS. The results are shown in Fig. 4. A combination of the two components gives the emergent dynamical field g(q), which is observed to exhibit a nonzero winding along the ring despite local fluctuations induced by experimental errors. The winding number of the field g(q), resembling the flux of magnetic monopoles, counts the total charges C = +1 in the region V − , and characterizes the post-quench quantum phase with Chern number Ch = +1. g y,x (q) ≡ 1 N q ∂ q ⊥ σ z (q) y,x ,(3)-0.6 0.6 0 -1 1 0 -1 0 1 -1 0 1 -1 0 1 q x (k 0 ) q y (k 0 ) -1 1 0 q y (k 0 ) (a) -1 0 1 -1 1 0 -1 1 0 q x (k 0 ) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -0.6 -0.4 -0.2 0 0.2 0.4 -0. 6 0 0. 6 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -0.6 -0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -0.6 -0.4 -0.2 0 0.2 0.4 q y (k 0 ) q y (k 0 ) (b) BIS L x1 L y1 (c) S y S x S D 1 D 2 D 3 D 4 D 1 D 4 D 3 D 2 D 1 D 2 D 3 D 4 L x2 L y2 + - - in - in + in - in Conclusion.-We have detected by quench dynamics the topological charges and further the dynamical bulk-surface correspondence in a 2D QAH model through a sequence of quantum quenches. We observed that the bulk topology can be generally characterized by the total charges enclosed by a lower-dimensional momentum subspace called band inversion surface (BIS). Such topological charges are related to the emergent topological pattern of a dynamical field defined on the BIS, showing an essential dynamical bulk-surface correspondence. We emphasize that the scheme of quenching all the spin axes enables to extract the complete topological information by measuring only a single spin component, hence has great advantages in detecting topolog- Fig. 3(a)] across the BIS. The first (second) row are the experimental (numerical) results. The measured dynamical field g(q) exhibits a nonzero winding along the ring, which characterizes the bulk topology with Chern number Ch = +1. -1 0 1 -1 0 1 -1 0 1 -1 0 1 q y (k 0 ) q x (k 0 ) -1 0 1 - + D 2 D 1 D 4 D 3 D 1 D 2 D 4 D 3 ical states. This study clearly verifies the insight that a higher-dimensional topological system (e.g. 2D Chern insulator) can be characterized by lower-dimensional dynamical invariants (1D ring topology or 0D monopole charges), which has broad applications to dynamical classification of generic topological phases [23,25]. Note added.-In completing the manuscript, we notice a preprint which demonstrates the dynamical bulk-surface correspondence based on nitrogen-vacancy center [35]. In comparison, we first measured the topological charges with ultracold atoms, and provide an alternative powerful approach to verify the dynamical bulk-surface correspondence. Acknowledgement.-We thank Jin-Long Yu for fruitful discussions. Chang-Rui Yi and Long Zhang contribute equally to this work. Supplemental Material for: Observing topological charges and dynamical bulk-surface correspondence with ultracold atoms This supplemental material provides details of the experimental realization (Sec. I), data processing (Sec. II), numerical calculations (Sec. III) and theoretical background (Sec. IV). I. Experimental Realization Here we briefly explain the generation of lattice and Raman potentials (Sec. IA), the derivation of the Bloch Hamiltonian for different settings (Sec. IB), and the calibration of the two phases ϕ 1,2 (Sec. IC). More details can be found in Refs. [15,25,31]. IA. Raman lattice scheme The experimental setup for 87 Rb Bose gas is shown in Fig. 1a of the main text. Two magnetic sublevels | ↑ ≡ |F = 1, m F = −1 and | ↓ ≡ |1, 0 are selected by 10.2MHz Zeeman splitting generated via a bias magnetic field of 14.5G applied inẑ direction. The laser beam E x (E y ) with wavelength of λ = 787nm injected fromx (ŷ) direction passes through a high extinction-ratio polarization beam splitting and a λ/2 wave plates to generate two orthogonally polarized components E xy (E yx ) and E xz (E yz ). A electro-optic modulator (EOM) is placed in front of the mirror R x (R y ) to introduce a phase shift ϕ 1 (ϕ 2 ) between the two components E xy and E xz (E yx and E yz ). Hence, the laser beams E x and E y form the standing-wave fields for atoms: E x =ŷ |E xy | e i(α+α L /2) cos(k 0 x − α L /2) +ẑ |E xz | e i(α+α L /2+ϕ1) cos(k 0 x − α L /2 − ϕ 1 ), E y =x |E yx | e i(β+β L /2) cos(k 0 y − β L /2) +ẑ |E yz | e i(β+β L /2+ϕ2) cos(k 0 y − β L /2 − ϕ 2 ),(S1) where k 0 = 2π/λ, α and β denote the initial phases, and α L (β L ) is the phase acquired by E x (E y ) for an additional optical path from the atom cloud to mirror R x (R y ), then back to the atom cloud. As shown in Fig. S1, the lattice and Raman coupling potentials are contributed from both the D 2 (5 2 S 1/2 → 5 2 P 3/2 ) and D 1 (5 2 S 1/2 → 5 2 P 1/2 ) lines. The total Hamiltonian reads ( = 1) H = k 2 2m + V latt (x, y) ⊗ 1 + Ω x (x, y)σ x + Ω y (x, y)σ y + m z σ z ,(S2) where V latt (x, y) = 1 3 t 2 D2 ∆ D2 + t 2 D1 ∆ D1 (|E xz | 2 + |E xy | 2 + |E yz | 2 + |E yx | 2 ).(S3) is the square lattice potential, Ω x,y are the Raman potentials determined by the two Raman couplings Ω x = 1 6 t 2 D2 2∆ D2 − t 2 D1 ∆ D1 (ê z · E xz )(ê − · E yx ), Ω y = 1 6 t 2 D2 2∆ D2 − t 2 D1 ∆ D1 (ê + · E xy )(ê z · E yz ),(S4) and m z = δ/2 measures the two-photon detuning of Raman coupling. Hereê ± = (ê x ∓ iê y )/ √ 2, t D2 ≡ | J = 1/2||er||J = 3/2 |, t D1 ≡ | J = 1/2||er||J = 1/2 |, and t D2 ≈ √ 2t D1 ≈ 4.227ea 0 [36] , with a 0 being the Bohr radius. One can tune the phases ϕ 1,2 by EOMs to manipulate the Raman and lattice potentials. IB. Symmetric and asymmetric settings We consider three cases of the phases: (i) (ϕ 1 , ϕ 2 ) = (π/2, π/2); (ii) (ϕ 1 , ϕ 2 ) = (0, π/2); (iii) (ϕ 1 , ϕ 2 ) = (π/2, 0). (i) When (ϕ 1 , ϕ 2 ) = (π/2, π/2), the light fields read Figure S1: Light couplings of both D2 (5 2 S 1/2 → 5 2 P 3/2 ) and D1 (5 2 S 1/2 → 5 2 P 1/2 ) transitions for 87 Rb atoms. E x =ŷ|E xy |e i(α+α L /2) cos(k 0 x − α L /2) + iẑ|E xz |e i(α+α L /2) sin(k 0 x − α L /2), E y =x|E yx |e i(β+β L /2) cos(k 0 y − β L /2) + iẑ|E yz |e i(β+β L /2) sin(k 0 y − β L /2),(S5) which leads to the lattice potential V latt (x, y) = V 0x cos 2 (k 0 x − α L /2) + V 0y cos 2 (k 0 y − β L /2), (S6) with V 0x/0y = t 2 D1 3 1 |∆ D1 | − 2 |∆ D2 | (|E xy/yx | 2 − |E xz/yz | 2 ),(S7) and the Raman coupling potentials Ω x = Ω 0x sin(k 0 x − α L /2) cos(k 0 y − β L /2), Ω y = Ω 0y cos(k 0 x − α L /2) sin(k 0 y − β L /2),(S8) with Ω 0x/0y = t 2 D1 6 √ 2 1 |∆ D1 | + 1 |∆ D2 | |E xz/xy ||E yx/yz |. (S9) In the tight-binding limit and only considering s-bands, the Hamiltonian (S2) takes the form in the momentum space H = q Ψ † q HΨ q [25,31], where Ψ q = (c q↑ , c q↓ ) and H (i) = [m z − 2t 0 (cos q x a + cos q y a)]σ z + 2t so sin q x aσ y + 2t so sin q y aσ x (S10) is exactly the 2D QAH model. Here a is the lattice constant with a = λ/2, q is the Bloch momentum, and t 0 and t so are, respectively, the spin-conserved and spin-flipped hopping coefficients t 0 = −ˆdrφ s (x, y) k 2 2m + V latt (r) φ s (x − a, y), t so = Ω 0ˆd rφ s (x, y) cos(k 0 y) sin(k 0 x)φ s (x − a, y),(S11) where φ s (x, y) denotes the Wannier function and Ω 0x = Ω 0y = Ω 0 . (ii) When (ϕ 1 , ϕ 2 ) = (0, π/2), the light fields become E x =ŷ|E xy |e i(α+α L /2) cos(k 0 x − α L /2) +ẑ|E xz |e i(α+α L /2) cos(k 0 x − α L /2), E y =x|E yx |e i(β+β L /2) cos(k 0 y − β L /2) + iẑ|E yz |e i(β+β L /2) sin(k 0 y − β L /2). (S12) The lattice potential is V latt (x, y) = V 0x cos 2 (k 0 x − α L /2) + V 0y cos 2 (k 0 y − β L /2),(S13) with V 0y being defined as in Eq. (S7) and V 0x = Figure S2: Calibration of the phase ϕ1. Spin polarization P of BEC atoms is measured as a function of the phase ϕ1 with ϕ2 = π/2. The circles with error bars are experimental measurements and the red curve is the fitting curve. The phase ϕ1 is tuned to be 0 (or π/2) when P reaches the minimum (or maximum). The inset i) and ii) show the spin-resolved TOF imaging of the BEC in the case corresponding to ϕ1 = 0 (green square) and ϕ1 = π/2 (green triangle), respectively. The lattice depth V0y is varied from 4Er (ϕ1 = π/2) to 5.2Er (ϕ1 = 0), while other parameters are fixed at V0x = 4.0Er, Ω0 = 1.0Er and δ = −0.6Er. which generate spin-flipped hopping only inŷ-direction, accompanied with on-site spin flipping. In the tight-binding limit, the Bloch Hamiltonian reads [25] H (ii) = [m z − 2(t x 0 cos q x a + t y 0 cos q y a)]σ z + 2t x so sin q y aσ x + m y σ y ,(S15) where m y = Ω 0xˆd rφ s (r) cos(k 0 x) cos(k 0 y)φ s (r). (S16) (iii) When (ϕ 1 , ϕ 2 ) = (π/2, 0), the light fields become E x =ŷ|E xy |e i(α+α L /2) cos(k 0 x − α L /2) + iẑ|E xz |e i(α+α L /2) sin(k 0 x − α L /2), E y =x|E yx |e i(β+β L /2) cos(k 0 y − β L /2) +ẑ|E yz |e i(β+β L /2) cos(k 0 y − β L /2). (S17) The lattice potential is V latt (x, y) = V 0x cos 2 (k 0 x − α L /2) + V 0y cos 2 (k 0 y − β L /2),(S18) with V 0x being defined as in Eq. (S7) and V 0y = t 2 D 1 3 1 |∆ D 1 | − 2 |∆ D 2 | (|E yx | 2 + |E yz | 2 ) . The Raman potentials are Ω x = Ω 0x sin(k 0 x − α L /2) cos(k 0 y − β L /2) − Ω 0y cos(k 0 x − α L /2) cos(k 0 y − β L /2), Ω y = 0.(S19) which generate spin-flipped hopping only inx-direction, and also on-site spin flipping. In the tight-binding limit, the Bloch Hamiltonian reads [25] H (iii) = [m z − 2(t x 0 cos q x a + t y 0 cos q y a)]σ z + 2t y so sin q x aσ y + m x σ x ,(S20) where m x = Ω 0yˆd rφ s (r) cos(k 0 x) cos(k 0 y)φ s (r). IC. Calibration of the phases ϕ1,2 In experiment, we tune the phases ϕ 1,2 by the voltage applied on EOMs (−200 V ∼ 200 V). The instability of ϕ 1,2 , which is limited to approximately π/60 radians per 10 hours, is controlled by the thermoelectric cooler attached to EOMs. We calibrate the phases ϕ 1,2 by measuring the spin polarization of the Bose-Einstein condensate (BEC). We first prepare about 3 × 10 5 87 Rb atoms in the Raman lattice with some values of the phases ϕ 1,2 , which are cooled to condense at the ground state of the lowest band (q x = q y = 0). We then release the BEC atoms for 25 ms and take the spin-resolved TOF imaging to obtain the spin and momentum distribution of the atomic cloud. We calculate the spin polarization of the BEC P = N ↑ −N ↓ N ↑ +N ↓ , where N ↑ (N ↓ ) denotes the total atom number in the spin-up (-down) state, and accordingly calibrate the values of ϕ 1,2 . We show an example in Fig. S2, where the spin polarization P is measured as a function of the phase ϕ 1 with ϕ 2 being fixed at π/2. The measured variation of P reflects the generation of the transverse magnetization m y : when P takes the maximum, the magnetization m y 0, corresponding to ϕ 1 = π/2; when P is at the minimum, m y takes its largest value, corresponding to ϕ 1 = 0. II. Data processing Here, we explain the obtainment of the time-averaged spin textures from the time evolution of spin polarization (Sec. IIA) and the sign of h z (Sec. IIB). IIA. Data fitting After the measurement, we fit the time-evolved spin polarization σ z (q, t) at each momentum q into the combination function P z (q, t) = 3 i=1 A i (q) cos(2πf i (q)t + φ i )e − t τ 1 + B(q)e − t τ 2 + D(q).(S22) Here the first A i (q)-terms denote the damped oscillations with the characteristic damping time τ 1 , the second B(q)term represents a pure decay with the characteristic time τ 2 , and the last D(q)-term is an offset. Both the decay and damping are internal relaxation induced by the fluctuation of magnetic field and the atom-atom interaction [27,37]. In the derivation of time-averaged spin textures, we remove those non-ideal effects from Eq. (S22) and obtain σ z (q) x,y,z by σ z (q) x,y,z = 1 T T t=0 3 i=1 A i (q) cos(2πf i (q)t + φ i ) + B(q) + D(q).(S23) Here T is taken much longer than the oscillation period. As an example, the measured time-evolved spin polarization at q 1 = (0.66k 0 , 0.66k 0 ) is shown in Fig. S3 after quenching h y . Experimental data are fitted into Eq. (S22) (red curve). According to the fitting parameters, we have σ z (q 1 ) x ≈ −0.02. The time average of all momentum points in the first BZ are shown in Fig. 3(a) for three quenches. The deviation from numerical results comes from experimental errors induced by mechanical shaking and magnetic field fluctuation, as well as interaction induced decay or dephasing. IIB. The sign of hz The sign of h z = m z − 2t 0 (cos(q x ) + cos(q y )) can be determined by the size of the BISs from the experiments. In the theory [ Fig. S4], the region V − as well as the size of the BISs shrinks when the detuning increases from δ = −0.6 to δ = 0.6, from which we obtain h z < 0 (h z > 0) in the region V − (V + ). In the experiment, we measure the time evolution of spin polarization with different detuning and fixed lattice depth and Raman coupling strength in topological nontrivial regime. The measured results are found in Ref. [27]. The size of the BISs shrinks when the detuning increases, which is consistent with theoretical calculation. Therefore, h z < 0 in the region V − and h z > 0 in the region V + . III. Numerical calculations We compare our experimental observations to the numerical calculations. In the calculations, the post-quench unitary dynamics is described by the time evolution operator U (q, t) = exp(−iHt), where H is the full-band Hamiltonian (S2) with ϕ 1 = ϕ 2 = π/2. The time-dependent density matrix is then ρ(q, t) = U (q, t)ρ(q, t = 0)U (q, t) † . Here the initial state reads ρ(q, 0) = n p n |n n|, where |n denotes the n-th eigenstate of H with pre-quench parameters and p n is the population probability at the state |n determined by the Bose-Einstein statistics f (E n ; T, µ), with the temperature T being 150nk (100nk) for quenching h x,y (h z ), and the chemical potential µ being fixed by the particle density. The time-evolved spin textures are then given by σ z (q, t) = Tr[ρ(q, t)σ z ], and the time averages σ z (q) x,y,z are obtained by summing over a long enough time so that they approaches the ones averaged over infinite time. The time-averaged spin textures in topologically trivial region are also measured, as shown in Fig.S4 with V 0 = 4.0E r , Ω 0 = 1.0E r , δ = −1.0E r . One can see that there is no momentum satisfying σ z (q) x,y,z = 0, which characterizes the trivial phase with Chern number C = 0. The observations agree with the numeral calculations with the same experimental parameters (Fig.S4). IV. Theoretical background Here we present briefly the theoretical background of our experiment. Details can be found in Refs. [23,25,34]. In our dynamical classification theory, two essential concepts are introduced: One is band inversion surfaces (BISs) defined as the surfaces with h z (k) = 0; the other is topological charges located at nodes of the spin-orbit (SO) field h so (k) ≡ (h y , h x ) = 0. We find that the bulk topology is classified by either the total charges enclosed by BISs, or the winding of the SO field h so (k) on BISs. We make use of the time-averaged spin textures after quench, which is defined by (i = x, y, z) with E = h 2 x + h 2 y + h 2 z , to dynamically characterize the BISs, the SO field, the charges, and also the topology. Here ρ i (0) denotes the initial state for quenching h i and H is the post-quench Bloch Hamiltonian. In the following, we first give the description of dynamical characterization by deep quenches [23,25], and then discuss the shallow quench situation that our experiment belongs to. σ z (k) i = lim T →∞ 1 TˆT 0 dt Tr ρ i (0)e iHt σ z e −iHt = h z Tr [ρ i (0)H] /E 2 ,(S24) A. Deep quenches When the quench starts from the extremely deep trivial regime m i → −∞, the initial state ρ i (0) yields Tr [ρ i (0)H] = h i and the time-averaged spin texture reads σ z (k) i = h z (k)h i (k)/E 2 (k).(S25) One can see that no matter which axis is quenched, the spin polarization σ z (k) i always vanishes on BISs where h z (k) = 0. Hence, a BIS can be dynamically determined by vanishing time-averaged spin polarization independent of the quench axis: BIS = {k| σ z (k) x,y,z = 0}.(S26) In particular, σ z (k) z = 0 occurs only on BISs, so one can find out these surfaces simply by quenching h z . Besides, the spin texture σ z (k) x/y also vanish on the surfaces with h x/y (k) = 0 [see Eq. (S25)]. Accordingly, the nodes of the SO field can be found out by σ z (k) x,y = 0 but σ z (k) z = 0. Furthermore, near a node k = q c , the time-averaged spin texture directly reflects the SO field: σ z (k) i k→qc h i (k)/h z (q c ). Thus, we define a dynamical spin-texture field S(k), whose components are S i (k) ≡ sgn[h z (k)] N k σ z (k) i ,(S27) with N k being a factor. With this result, the topological charge can be dynamically determined by [25] C = sgn[J S (q c )]. where J S (k) ≡ det [(∂S i /∂k j )] is Jacobian determinant. Moreover, we define a dynamical directional derivative field g(k) = (g y , g x ), whose components are given by g i (k) = 1 N k ∂ k ⊥ σ z i ,(S29) where k ⊥ denotes the momentum perpendicular to BIS, and N k is the normalization factor. It can be shown that on the BIS g(k) =ĥ so (k), whereĥ so (k) ≡ h so (k)/|h so (k)| is the unit SO field. Thus the topological invariant can be characterized dynamically by W = j 1 2πˆB ISj [g y (k)dg x (k) − g x (k)dg y (k)] ,(S30) where the summation is over all the BISs. The result manifests itself a highly nontrivial dynamical bulk-surface correspondence [23]. B. Shallow quenches When the quench starts from a shallow trivial regime, i.e., m i is finitely large, we find that the dynamical characterization is not valid for any shallow quenches [34]. Here we give the validity condition for our quench experiment. For an shallow quench, the time-averaged spin texture read σ z (k) i = h z (k)Tr [ρ i (0)H] /E 2 (k).(S31) The BISs can still be identified by the vanishing spin polarizations via Eq. (S26). The directional derivative on BISs becomes ∂ k ⊥ σ z i = Tr [ρ i (0)H] /E 2 (S32) We denote h i ≡ Tr [ρ i (0)H], and consider the situation that a quench corresponds to adding an additional constant magnetization δm i in h i and tuning δm i from δm i < − max{|h i (k)|} to zero. To obtain the validity condition, we introduce local transformations U i (k) = e −iui(k) , with u i (k) = j=x,y,z u (i) j σ j , for each quench, which rotates the initial state ρ i around the axisû i ≡ u i /|u i | by an angle 2|u i | to the fully polarized one ρ i = U i (k)ρ i (k)U † i (k), with σ i ρ i = ρ i and σ j =i ρ i = 0. We further setû i to be normal to the plane spanned by the σ i axis and the pre-quench vector field h such that 0 ≤ 2|u i | < π/2 and u (i) i = 0. It can be checked that this transformation does not change the topology [34]. Note that after each quench, we return to the same post-quench Hamiltonian H post = j h j σ j . When quenching h i , we have the pre-quench Hamiltonian H (i) pre = H post + δm i σ i . We notice that the pre-quench vector field should be in the σ i axis after rotation, which gives H (i) pre = [(h i + δm i )/ cos 2|u i |] σ i .(S33) The relation (h i + δm i ) 2 / cos 2 2|u i | = (h i + δm i ) 2 + j =i h 2 j leads to δm i = − cot 2|u i | E 2 − h 2 i − h i .(S34) We use the equality U i σ i U † i = cos 2|u i |σ i + i sin 2|u i | |u i | j =i u (i) j σ i σ j ,(S35) and obtain the rotated post-quench Hamiltonian H (i) post = H (i) pre − δm i U i σ i U † i = h i + δm i sin 2 2|u i | cos 2|u i | σ i − iδm i sin 2|u i | |u i | j =i u (i) j σ i σ j .(S36) Figure 3 : 3Dynamical measurement of topological charges and Chern number. (a) Experimental measurements (upper) of time-averaged spin textures σz(q) i by quenching hi (i = x, y, z), compared with numerical calculations (lower). Dynamical characterization of σz(q) z = 0 identifies the BIS (green), which also emerges in σz(q) x,y . The BIS divides the BZ into two regions: V− with hz < 0 and V+ with hz > 0. In both σz(q) x and σz(q) y , other than the BIS, two additional curves are formed at momenta with vanishing spin polarization, denoted by Lx1,x2 (magenta) and Ly1,y2 (brown), respectively. (b) The four lines Lx1,x2and Ly1,y2 have four intersections D1,2,3,4, marking the locations of topological charges. Three charges are in the region V− and one in V+. (c) The charge value Ci = ±1 at each intersection Di (i = 1, 2, 3, 4) is determined by the signs of spin textures σz(q) x,y in the neighboring four subregions. The plus (minus) sign of σz(q) x,y indicates that the field components Sx,y point to the positive (negative) spin axes. When S flows from or to the charge, the charge value Ci = +1 (red); if S is "repulsed" by the charge, Ci = −1 (blue). Three topological charges are enclosed in V− with total charges being C = +1, giving the Chern number Ch = +1. The post-quench parameters are V0 = 4.0Er, Ω0 = 1.0Er and δ = −0.2Er. Figure 4 : 4Dynamical topological pattern emerging on the BIS. The field component gi(q) (i = x, y) is obtained by the variation of time-averaged spin texture σz(q) i [shown in This work was supported by the National Key R&D Program of China (under grants 2016YFA0301601 and 2016YFA0301604), National Natural Science Foundation of China (under grants No. 11674301, 11574008, 11825401, and 11761161003), the Chinese Academy of Sciences and the Anhui Initiative in Quantum Information Technologies (AHY120000) and the Chinese Academy of Sciences, and the Strategic Priority Research Program of Chinese Academy of Science (Grant No. XDB28000000). |∆ D 1 | − 2 |∆ D 2 | (|E xy | 2 + |E xz | 2 ). The Raman potentials are Ω x = 0, Ω y = Ω 0x cos(k 0 x − α L /2) cos(k 0 y − β L /2) + Ω 0y cos(k 0 x − α L /2) sin(k 0 y − β L /2). Figure S3 : S3Time evolution of spin polarization σz(q, t) at the momentum q1 after quenching hy. The green diamonds with error bars are experimental measurements with the post-quench parameters V0 = 4.0Er, Ω0 = 1.0Er and δ = −0.2Er. The red curve is the fitting curve. The inset is the spin texture at t = 0.4 ms, where q1 is marked. Figure S4 : S4hz for different detuning δ with t0 = 0.09. The dash curves are hz = 0, which is BIS. The BISs divide the first BZ into two regions: V− with hz < 0 and V+ with hz > 0. Figure S5 : S5Time-averaged spin textures in the topologically trivial phase. The first (second) row is the experimental measurements (numerical calculations) of σz(q) after quenching hx,y,z. The post-quench parameters are V0 = 4.0Er, Ω0 = 1.0Er, δ = −1.0Er. Thus, we have Tr[ρ i H post ] = Tr[ρ i H (i) post ] = h i , where we denote h i ≡ (h i + δm i sin 2 2|u i |)/ cos 2|u i | = h i cos 2|u i | − sin 2|u i | E 2 − h 2 i . Exy,xz or Eyx,yz for each beam, and EOMs are placed in front of the mirrors to tune the phase shift between the two components. (b) Level structure and Raman transitions. Two hyperfine states |1, −1 and |1, 0 are selected to form the double-Λ-type coupling configuration. (c)B x z y R x R y EOM lens lens coil E xy E xz E yx E yz λ/2 λ/2 E x E y (a) EOM -1 -0.5 0 0.5 1 -1 0 1 Raman potential quench (c) (d) quench E i E f |F, -1 |F, 0 2 P 3/2 2 P 1/2 ∆ 3/2 ∆ 1/2 E yz E yx E xy E xz δ 2 S 1/2 |1, 0 |1, -1 (b) Figure 1: Experimental scheme. (a) Experimental setup. Two laser beams Ex and Ey are incident on the atoms and then reflected by two mirrors Rx,y, simultaneously producing a 2D square lattice and two Raman coupling potentials. The λ/2 wave plates are used to generate two orthogonally polar- ized components On the BIS where h z (k) = 0, we have h x = h x cos 2|u x | − sin 2|u x ||h y |, h y = h y cos 2|u y | − sin 2|u y ||h x | (S38)It can be easily checked that the first terms {h x cos 2|u x |, h y cos 2|u y |} preserve the topological pattern of h so on BISs while the second terms {− sin 2|u x ||h y |, − sin 2|u y ||h x |} can change it. Here we assume δm x = δm y = δm, h x = 2t so sin q x , h y = 2t so sin q y , and h z = m z − 2t 0 cos q x − 2t 0 cos q y with |m z | < 4t 0 . In order to ensure that our dynamical characterization is valid, we should have 0 ≤ 2|u x |, 2|u y | < π/4 and δm < −2t so (| sin q x | + | sin q y |). Since cos q x + cos q y = m z /(2t 0 ) on the BIS, we haveIn our experiment, we have typically t so = 0.5t 0 and m x,y = −14t 0 , which satisfies Eq. (S39) very well. The locations of charges are dynamically determined by σ z (k) x,y = 0 but σ z (k) z = 0, which givesThese charges are obviously different from those monopole charges defined by h so (k) = 0, and hence dubbed as dynamical topological charges[34]. Moreover, the dynamical spin-texture field S(k) defined in Eq. (S27) characterizes these dynamical topological charges. We have proved that the winding of ( h y , h x ) along BISs classifies the bulk topology if the condition (S39) is satisfied. It is then expected that under the same condition, the total dynamical topological charges enclosed by BISs can be used for the dynamical characterization. . M Z Hasan, C L Kane, 10.1103/RevModPhys.82.3045Rev. Mod. Phys. 823045M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). . X.-L Qi, S.-C Zhang, 10.1103/RevModPhys.83.1057Rev. Mod. Phys. 831057X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). . M König, S Wiedmann, C Brüne, A Roth, H Buhmann, L W Molenkamp, X.-L Qi, S.-C Zhang, 10.1126/science.1148047Science. 318766M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buh- mann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007). . C.-Z Chang, J Zhang, X Feng, J Shen, Z Zhang, M Guo, K Li, Y Ou, P Wei, L.-L Wang, Z.-Q Ji, Y Feng, S Ji, X Chen, J Jia, X Dai, Z. Fang. C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. . K Zhang, Y He, L Wang, X.-C Lu, Q.-K Ma, Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. . Xue, 10.1126/science.1234414Science. 340167Xue, Science 340, 167 (2013). . B Q Lv, H M Weng, B B Fu, X P Wang, H Miao, J Ma, P Richard, X C Huang, L X Zhao, G F Chen, Z Fang, X Dai, T Qian, H Ding, 10.1103/PhysRevX.5.031013Phys. Rev. X. 531013B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X 5, 031013 (2015). . R J Bergholtz, Z Liu, 10.1142/S021797921330017XInt. J. Mod. Phys. B. 271330017R. J. Bergholtz and Z. Liu, Int. J. Mod. Phys. B 27, 1330017 (2013). . C Nayak, S H Simon, A Stern, M Freedman, S. Das Sarma, 10.1103/RevModPhys.80.1083Rev. Mod. Phys. 801083C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008). . M Atala, M Aidelsburger, J T Barreiro, D Abanin, T Kitagawa, E Demler, I Bloch, 10.1038/nphys2790Nat. Phys. 9795M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, Nat. Phys. 9, 795 (2013). . B Song, L Zhang, C He, T F J Poon, E Hajiyev, S Zhang, X.-J Liu, G.-B Jo, 10.1126/sciadv.aao4748Sci. Adv. 44748B. Song, L. Zhang, C. He, T. F. J. Poon, E. Hajiyev, S. Zhang, X.-J. Liu, and G.-B. Jo, Sci. Adv. 4, eaao4748 (2018). . M Aidelsburger, M Atala, M Lohse, J T Barreiro, B Paredes, I Bloch, 10.1103/PhysRevLett.111.185301Phys. Rev. Lett. 111185301M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Phys. Rev. Lett. 111, 185301 (2013). . H Miyake, G A Siviloglou, C J Kennedy, W C Burton, W Ketterle, 10.1103/PhysRevLett.111.185302Phys. Rev. Lett. 111185302H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Bur- ton, and W. Ketterle, Phys. Rev. Lett. 111, 185302 (2013). . G Jotzu, M Messer, R Desbuquois, M Lebrat, T Uehlinger, D Greif, T Esslinger, 10.1038/nature13915Nature. 515237G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2014). . M Aidelsburger, M Lohse, C Schweizer, M Atala, J T Barreiro, S Nascimbene, N Cooper, I Bloch, N Goldman, 10.1038/nphys3171Nat. Phys. 11162M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbene, N. Cooper, I. Bloch, and N. Goldman, Nat. Phys. 11, 162 (2015). . Z Wu, L Zhang, W Sun, X.-T Xu, B.-Z Wang, S.-C Ji, Y Deng, S Chen, X.-J Liu, J.-W Pan, 10.1126/science.aaf6689Science. 35483Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Science 354, 83 (2016). . W Sun, B.-Z Wang, X.-T Xu, C.-R Yi, L Zhang, Z Wu, Y Deng, X.-J Liu, S Chen, J.-W Pan, 10.1103/PhysRevLett.121.150401Phys. Rev. Lett. 121150401W. Sun, B.-Z. Wang, X.-T. Xu, C.-R. Yi, L. Zhang, Z. Wu, Y. Deng, X.-J. Liu, S. Chen, and J.-W. Pan, Phys. Rev. Lett. 121, 150401 (2018). . X.-J Liu, K T Law, T K Ng, P A Lee, 10.1103/PhysRevLett.111.120402Phys. Rev. Lett. 111120402X.-J. Liu, K. T. Law, T. K. Ng, and P. A. Lee, Phys. Rev. Lett. 111, 120402 (2013). . N Fläschner, D Vogel, M Tarnowski, M Tarnowski, B Rem, D.-S Lühmann, M Heyl, J Budich, L Mathey, K Sengstock, C Weitenberg, 10.1038/s41567-017-0013-8Nat. Phys. 14265N. Fläschner, D. Vogel, M. Tarnowski, M. Tarnowski, B. Rem, D.-S. Lühmann, M. Heyl, J. Budich, L. Mathey, K. Sengstock, and C. Weitenberg, Nat. Phys. 14, 265 (2018). . N Fläschner, B S Rem, M Tarnowski, D Vogel, D.-S Lühmann, K Sengstock, C Weitenberg, 10.1126/science.aad4568Science. 3521091N. Fläschner, B. S. Rem, M. Tarnowski, D. Vogel, D.- S. Lühmann, K. Sengstock, and C. Weitenberg, Science 352, 1091 (2016). . P Jurcevic, H Shen, P Hauke, C Maier, T Brydges, C Hempel, B P Lanyon, M Heyl, R Blatt, C F Roos, 10.1103/PhysRevLett.119.080501Phys. Rev. Lett. 11980501P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. P. Lanyon, M. Heyl, R. Blatt, and C. F. Roos, Phys. Rev. Lett. 119, 080501 (2017). . Y Hu, P Zoller, J C Budich, 10.1103/PhysRevLett.117.126803Phys. Rev. Lett. 117126803Y. Hu, P. Zoller, and J. C. Budich, Phys. Rev. Lett. 117, 126803 (2016). . J H Wilson, J C W Song, G Refael, 10.1103/PhysRevLett.117.235302Phys. Rev. Lett. 117235302J. H. Wilson, J. C. W. Song, and G. Refael, Phys. Rev. Lett. 117, 235302 (2016). . C Wang, P Zhang, X Chen, J Yu, H Zhai, 10.1103/PhysRevLett.118.185701Phys. Rev. Lett. 118185701C. Wang, P. Zhang, X. Chen, J. Yu, and H. Zhai, Phys. Rev. Lett. 118, 185701 (2017). . L Zhang, L Zhang, S Niu, X.-J Liu, 10.1016/j.scib.2018.09.018Sci. Bull. 631385L. Zhang, L. Zhang, S. Niu, and X.-J. Liu, Sci. Bull. 63, 1385 (2018). . J Yu, 10.1103/PhysRevA.99.043619Phys. Rev. A. 9943619J. Yu, Phys. Rev. A 99, 043619 (2019). . L Zhang, L Zhang, X.-J Liu, 10.1103/PhysRevA.99.053606Phys. Rev. A. 9953606L. Zhang, L. Zhang, and X.-J. Liu, Phys. Rev. A 99, 053606 (2019). . L Zhang, L Zhang, Y Hu, S Niu, X.-J Liu, arXiv:1903.09144L. Zhang, L. Zhang, Y. Hu, S. Niu, and X.-J. Liu, arXiv:1903.09144 (2019). . W Sun, C.-R Yi, B.-Z Wang, W.-W Zhang, B C Sanders, X.-T Xu, Z.-Y Wang, J Schmiedmayer, Y Deng, X.-J Liu, S Chen, J.-W Pan, 10.1103/PhysRevLett.121.250403Phys. Rev. Lett. 121250403W. Sun, C.-R. Yi, B.-Z. Wang, W.-W. Zhang, B. C. Sanders, X.-T. Xu, Z.-Y. Wang, J. Schmiedmayer, Y. Deng, X.-J. Liu, S. Chen, and J.-W. Pan, Phys. Rev. Lett. 121, 250403 (2018). . M Tarnowski, F N Ünal, N Fläschner, B S Rem, A Eckardt, K Sengstock, C Weitenberg, 10.1038/s41467-019-09668-yNat. Commun. 101728M. Tarnowski, F. N. Ünal, N. Fläschner, B. S. Rem, A. Eckardt, K. Sengstock, and C. Weitenberg, Nat. Com- mun. 10, 1728 (2019). . C.-R Yi, J.-L Yu, W Sun, X.-T Xu, S Chen, J.-W Pan, arXiv:1904.11656C.-R. Yi, J.-L. Yu, W. Sun, X.-T. Xu, S. Chen, and J.-W. Pan, arXiv:1904.11656 (2019). . X.-J Liu, K T Law, T K Ng, 10.1103/PhysRevLett.112.086401Phys. Rev. Lett. 11286401X.-J. Liu, K. T. Law, and T. K. Ng, Phys. Rev. Lett. 112, 086401 (2014). . B.-Z Wang, Y.-H Lu, W Sun, S Chen, Y Deng, X.-J Liu, 10.1103/PhysRevA.97.011605Phys. Rev. A. 9711605B.-Z. Wang, Y.-H. Lu, W. Sun, S. Chen, Y. Deng, and X.-J. Liu, Phys. Rev. A 97, 011605 (2018). . X.-T Xu, C.-R Yi, B.-Z Wang, W Sun, Y Deng, X.-J , X.-T. Xu, C.-R. Yi, B.-Z. Wang, W. Sun, Y. Deng, X.-J. . S Liu, J.-W Chen, 10.1016/j.scib.2018.11.002Pan, Sci. Bull. 631464Liu, S. Chen, and J.-W. Pan, Sci. Bull. 63, 1464 (2018). See the supplementary materials. See the supplementary materials. . L Zhang, L Zhang, X.-J Liu, to appear soonL. Zhang, L. Zhang, and X.-J. Liu, to appear soon. . Y Wang, W Ji, Z Chai, Y Guo, M Wang, X Ye, P Yu, L Zhang, X Qin, P Wang, arXiv:1904.09065Y. Wang, W. Ji, Z. Chai, Y. Guo, M. Wang, X. Ye, P. Yu, L. Zhang, X. Qin, P. Wang, et al., arXiv:1904.09065 (2019). Rubidium 87 d line data. D A Steck, D. A. Steck, "Rubidium 87 d line data," https://steck. us/alkalidata/rubidium87numbers.pdf, 2015. . N Kosugi, S Matsuo, K Konno, N Hatakenaka, 10.1103/PhysRevB.72.172509Phys. Rev. B. 72172509N. Kosugi, S. Matsuo, K. Konno, and N. Hatakenaka, Phys. Rev. B 72, 172509 (2005).
[]
[ "Dual-probe decoherence microscopy: Probing pockets of coherence in a decohering environment Dual-probe decoherence microscopy:Probing pockets of coherence in a decohering environment2", "Dual-probe decoherence microscopy: Probing pockets of coherence in a decohering environment Dual-probe decoherence microscopy:Probing pockets of coherence in a decohering environment2" ]
[ "Jan Jeske \nChemical and Quantum Physics\nSchool of Applied Sciences\nRMIT University\n3001MelbourneAustralia\n\nInstitut für Theoretische Festkörperphysik\nKarlsruhe Institute of Technology\nD-76128KarlsruheGermany\n\nDFG-Center for Functional Nanostructures (CFN)\nD-76128KarlsruheGermany\n", "Jared H Cole \nChemical and Quantum Physics\nSchool of Applied Sciences\nRMIT University\n3001MelbourneAustralia\n\nInstitut für Theoretische Festkörperphysik\nKarlsruhe Institute of Technology\nD-76128KarlsruheGermany\n\nDFG-Center for Functional Nanostructures (CFN)\nD-76128KarlsruheGermany\n", "Clemens Müller \nInstitut für Theorie der Kondensierten Materie\nKarlsruhe Institute of Technology\nD-76128KarlsruheGermany\n\nDépartement de Physique\nUniversité de Sherbrooke\nJ1K 2R1SherbrookeQuébecCanada\n", "Michael Marthaler \nInstitut für Theoretische Festkörperphysik\nKarlsruhe Institute of Technology\nD-76128KarlsruheGermany\n\nDFG-Center for Functional Nanostructures (CFN)\nD-76128KarlsruheGermany\n", "Gerd Schön \nInstitut für Theoretische Festkörperphysik\nKarlsruhe Institute of Technology\nD-76128KarlsruheGermany\n\nDFG-Center for Functional Nanostructures (CFN)\nD-76128KarlsruheGermany\n" ]
[ "Chemical and Quantum Physics\nSchool of Applied Sciences\nRMIT University\n3001MelbourneAustralia", "Institut für Theoretische Festkörperphysik\nKarlsruhe Institute of Technology\nD-76128KarlsruheGermany", "DFG-Center for Functional Nanostructures (CFN)\nD-76128KarlsruheGermany", "Chemical and Quantum Physics\nSchool of Applied Sciences\nRMIT University\n3001MelbourneAustralia", "Institut für Theoretische Festkörperphysik\nKarlsruhe Institute of Technology\nD-76128KarlsruheGermany", "DFG-Center for Functional Nanostructures (CFN)\nD-76128KarlsruheGermany", "Institut für Theorie der Kondensierten Materie\nKarlsruhe Institute of Technology\nD-76128KarlsruheGermany", "Département de Physique\nUniversité de Sherbrooke\nJ1K 2R1SherbrookeQuébecCanada", "Institut für Theoretische Festkörperphysik\nKarlsruhe Institute of Technology\nD-76128KarlsruheGermany", "DFG-Center for Functional Nanostructures (CFN)\nD-76128KarlsruheGermany", "Institut für Theoretische Festkörperphysik\nKarlsruhe Institute of Technology\nD-76128KarlsruheGermany", "DFG-Center for Functional Nanostructures (CFN)\nD-76128KarlsruheGermany" ]
[]
We study the use of a pair of qubits as a decoherence probe of a non-trivial environment. This dual-probe configuration is modelled by three two-level-systems which are coupled in a chain in which the middle system represents an environmental two-level-system (TLS). This TLS resides within the environment of the qubits and therefore its coupling to perturbing fluctuations (i.e. its decoherence) is assumed much stronger than the decoherence acting on the probe qubits. We study the evolution of such a tripartite system including the appearance of a decoherence-free state (dark state) and non-Markovian behaviour. We find that all parameters of this TLS can be obtained from measurements of one of the probe qubits. Furthermore we show the advantages of two qubits in probing environments and the new dynamics imposed by a TLS which couples to two qubits at once.
10.1088/1367-2630/14/2/023013
[ "https://arxiv.org/pdf/1110.1945v2.pdf" ]
54,621,787
1110.1945
64642a5f39f2dd2c2099b4b5861d33feab327f3c
Dual-probe decoherence microscopy: Probing pockets of coherence in a decohering environment Dual-probe decoherence microscopy:Probing pockets of coherence in a decohering environment2 8 Feb 2012 Jan Jeske Chemical and Quantum Physics School of Applied Sciences RMIT University 3001MelbourneAustralia Institut für Theoretische Festkörperphysik Karlsruhe Institute of Technology D-76128KarlsruheGermany DFG-Center for Functional Nanostructures (CFN) D-76128KarlsruheGermany Jared H Cole Chemical and Quantum Physics School of Applied Sciences RMIT University 3001MelbourneAustralia Institut für Theoretische Festkörperphysik Karlsruhe Institute of Technology D-76128KarlsruheGermany DFG-Center for Functional Nanostructures (CFN) D-76128KarlsruheGermany Clemens Müller Institut für Theorie der Kondensierten Materie Karlsruhe Institute of Technology D-76128KarlsruheGermany Département de Physique Université de Sherbrooke J1K 2R1SherbrookeQuébecCanada Michael Marthaler Institut für Theoretische Festkörperphysik Karlsruhe Institute of Technology D-76128KarlsruheGermany DFG-Center for Functional Nanostructures (CFN) D-76128KarlsruheGermany Gerd Schön Institut für Theoretische Festkörperphysik Karlsruhe Institute of Technology D-76128KarlsruheGermany DFG-Center for Functional Nanostructures (CFN) D-76128KarlsruheGermany Dual-probe decoherence microscopy: Probing pockets of coherence in a decohering environment Dual-probe decoherence microscopy:Probing pockets of coherence in a decohering environment2 8 Feb 2012numbers: 6837-d0365Wj0365Yz0540-a We study the use of a pair of qubits as a decoherence probe of a non-trivial environment. This dual-probe configuration is modelled by three two-level-systems which are coupled in a chain in which the middle system represents an environmental two-level-system (TLS). This TLS resides within the environment of the qubits and therefore its coupling to perturbing fluctuations (i.e. its decoherence) is assumed much stronger than the decoherence acting on the probe qubits. We study the evolution of such a tripartite system including the appearance of a decoherence-free state (dark state) and non-Markovian behaviour. We find that all parameters of this TLS can be obtained from measurements of one of the probe qubits. Furthermore we show the advantages of two qubits in probing environments and the new dynamics imposed by a TLS which couples to two qubits at once. Introduction The loss of coherence (decoherence) of quantum bits (qubits) due to environmental perturbations is an important obstacle on the way to large scale quantum electronics and quantum computation. Such perturbations at the same time contain information about the surrounding environment which generates them. The idea of using qubits as probes of their environment has recently generated interest [1,2,3,4,5] as an alternative application of qubit technology where the effects of decoherence are used, rather than suppressed. In general, when an environment acts on a qubit as a weakly coupled, fluctuating bath, the environmental effects can be simply expressed as a relaxation and an excitation rate as well as a pure dephasing rate. The decoherence process becomes much more complex when a qubit couples to any component of an environment strongly enough such that quantum mechanical levels within the environment need to be taken into account. In many systems, such partially coherent "pockets" are observed in the environment. Examples include two-level fluctuators in superconducting devices [6,7,8,9,10,11], impurity spins in semiconductors [12,13,14,15,16] and single-molecule magnets [17,18,19] such as 8 Fe and Ferritin. Valuable information about the quantum mechanical nature of the environment may be obtained by using qubits as a direct environmental probe, with potentially high sensitivity and high spatial resolution [2]. The concept of qubit probes has already been used to realise a nano-magnetometer using Nitrogen-Vacancy (NV) centres in diamond attached to the end of atomic force microscope cantilevers [20,21,22,23,24]. In this case the Zeeman splitting of electronic levels within the NV centre is optically probed to provide a direct measure of the local magnetic field to nanometre resolution [24]. Additional information which one may obtain from the decoherence processes has also been studied [25,26] in this context of classically fluctuating fields. We investigate a more general configuration in which a decoherence probe couples transversally to a localized pocket of coherence and can therefore exchange energy with, and extract information from, its immediate environment. We model such a pocket as an environmental two-level-system (TLS), for example a charge-or spin-impurity, which is then in turn coupled to its surrounding environment. We will show a full solution of the system dynamics in the relevant parameter regimes and demonstrate how one could use this information to identify and characterize an environmental TLS. The behaviour of one qubit interacting with such a partially coherent component of the environment is well understood [27]. In such systems ambiguity often arises between oscillatory evolution induced by partially coherent defects and oscillatory evolution which stems from the single qubit Hamiltonian itself. Two non-interacting qubits which couple to the same environment (figure 1A) enable the observation of any environmentally induced correlations between these qubits, removing the ambiguity. This indirect interaction is a good indicator for the existence of coherent regions in the closer environment. Solving the equations of motion for such a model numerically results in a variety of complex behaviours [28,29,30,31] in different regimes of parameter space. In this paper we focus on solving the system analytically in key regimes and then use these analytical solutions to understand the more general behaviour numerically. Our model of two qubits coupled to a common environmental TLS has many similarities to the well studied problem of two qubits coupled to a quantum harmonic oscillator. This classic Tavis-Cummings model has been widely investigated for its use in quantum information processing and studying atom-photon interactions. Interpreted in this context our results complement a variety of physical effects which appear in the Tavis-Cummings model, including dispersive coupling [32,33,34,35,36,37], ultrastrong coupling [38,39,40] and entanglement birth and death [41,42,43,44]. It is natural that similar effects appear in both systems due to their formal equivalence within the single-excitation subspace. There are however important physical differences, as an environmental TLS is typically more localised in space than a quantum harmonic oscillator and the symmetry of the TLS allows a more general coupling to its environment. After introducing the theoretical model in section 2 we present a general analysis of the system in section 3: In the weak decoherence regime we find an oscillation of energy between the qubits, i.e. an environmentally mediated coupling. We find a decoherence-free state (dark state), which leads to formation of stray entanglement between the qubits in the decoherence process independent of the decoherence strength of the TLS. In our analytical solutions we find a clear threshold between oscillations and decay. In order to clarify what constitutes an environmental pocket of coherence it is shown that the same threshold divides Markovian from non-Markovian system dynamics. We define an effective decay rate of the qubit dynamics which turns out to have a linear dependence on the decoherence rates of the TLS in the weak decoherence regime and a roughly inverse dependence in the strong decoherence regime. In section 4 we interpret our results in the context of dual-probe microscopy and we find that in the weak decoherence regime an environmental TLS can be fully characterized and located in a substrate. In the strong decoherence regime the TLS can only be located. Section 5 puts the theoretical model in context of present experimental qubit realizations. Model and methods In order to study how a qubit probe pair interacts with an environmental TLS we construct a simplified model. Consider an experimental setup in which the two qubits are attached to an atomic force microscope such that they can be positioned precisely (with a fixed distance between them) on top of a scanned substrate which contains the TLS (figure 1A). At several positions the population of the excited state of the qubit is measured as a function of time. In each position of the cantilever the coupling strengths between the qubits and the TLS, g 1 and g 2 , vary due to their relative position. We will Figure 1. A Experimental setup in which two qubits are attached to an atomic force microscope and moved along their connecting line y with a fixed distance between them. The TLS is located in the substrate underneath the qubits. B Illustration of the model system investigated in this paper: Two qubits with level splitting ω q coupled with individual coupling strengths g 1 , g 2 to an environmental TLS detuned by 2δ. The TLS in turn is coupled to a bath. C Energies of the Hamiltonian eigenstates and the corresponding decohering transitions. Dephasing transitions (∝ Γ ϕ ) transfer population between states of similar energies, relaxation transitions (∝ Γ 1 ) transfer population to subspaces of lower excitation number. Crosses indicate rates which disappear for g 1 = g 2 . The eigenstates are given in appendix Appendix C give a detailed model of this variation in Sec. 4. The full system Hamiltonian can be written as: Qubit 1 Qubit 2 2ω q 2ω t = 2ω q + 2δ TLS g 1 g 2 2ω q coupling to bath y A B ∝ Γ 1 ∝ Γ ϕ ∝ (g 1 − g 2 ) or (g 2 1 − g 2 2 ) 3ω q + δ ω q + δ 2 + 4g 2 ω q + δ ω q − δ 2 + 4g 2 −ω q + δ 2 + 4g 2 −ω q − δ −ω q − δ 2 + 4g 2 −3ω q − δ energy |8 |7 |6 |5 |4 |3 |2 |1 CH sys = ω q σ Q1 z + ω q σ Q2 z + (ω q + δ)σ T LS z + g 1 (σ Q1 x σ T LS x + σ Q1 y σ T LS y ) + g 2 (σ Q2 x σ T LS x + σ Q2 y σ T LS y )(1) where σ x , σ y and σ z are the respective Pauli operator which act on qubit 1 (Q1), qubit 2 (Q2) or the TLS (TLS). The first two terms describe the two qubits which have the same level splitting of 2ω q while the third term describes the TLS with a level splitting of 2(ω q +δ). Here, δ is the relative detuning between qubits and TLS. The last two terms in eq. (1) are transversal coupling terms between each of the qubits and the TLS with the respective coupling strengths g 1 and g 2 and we introduce g = g 2 1 + g 2 2 for later simplicity. We focus on the specific case of transversal coupling as we are particularly interested in direct energy exchange between the qubits and the TLS. Throughout this discussion, we use the terminology "qubit" and "TLS" to differentiate between the fabricated and controllable two-state-probes and the environmental two-level-system of interest. For ease of notation we will use the convention = k B = 1. As the system Hamiltonian H sys is block-diagonal, the coherent evolution is limited to the subspace states with equal excitation number. For the time evolution, we choose the state in which qubit 1 is in its excited state and the other two subsystems are in their ground state |Q1, Q2, T LS = | ↑↓↓ as the system's initial state. This is a state with a single excitation, and therefore we can neglect the subspaces of higher excitation numbers in the following calculations. A key advantage of probe qubits attached to a cantilever is that they can be calibrated while lifted away from the sample. This allows the intrinsic decoherence of the probes themselves to be accurately characterized. Once the probes are brought close to the sample, this intrinsic decoherence defines a limit in the sensitivity for detecting features within the sample. For our purposes, an ideal qubit probe is one with a very long intrinsic decoherence time, as this provides a large dynamic range for sensing. We assume the intrinsic contribution to decoherence to be small compared to the dynamics induced by the environmental TLS. Then we can ignore this contribution in what follows, i.e. assume that only the TLS is coupled to a fluctuating bath. For its coupling to the environment we take the operator H int =ŝB = (v ⊥ σ T LS x + v σ T LS z )B(2) where v ⊥ and v are the transversal and longitudinal coupling strengths respectively andB is an operator acting on the bath. We assume a low temperature bath ω q T . In most qubit architectures, the qubit's level splitting is significantly larger than the other energy scales in the problem. Typically this is a requirement to obtain coherence over long time scales as well as adequate control over the quantum system. We therefore assume throughout this discussion that ω q δ, g 1 , g 2 , T . Under this assumption we can neglect subspaces with more than one excitation. A large ω q guarantees a clear separation of these subspaces in energy while a low temperature bath guarantees the absence of spontaneous excitations from the bath. In the limit of large ω q one can often make an additional secular approximation which we discuss in detail later on. Breaking the assumption that the qubit's level splitting is the largest energy leads to the ultrastrong coupling regime, which is studied elsewhere [45]. We model the time evolution of the system's reduced density matrix ρ using the Bloch-Redfield equations [46] , [47]. Using the eigenvectors |1 to |8 (given in appendix Appendix C) of H sys as basis states, the Bloch-Redfield equations read element-wise: ρ nm = −iω nm ρ nm + n m R nmn m ρ n m(3) with the Redfield tensor: R Dual-probe decoherence microscopy:Probing pockets of coherence in a decohering environment6 Here ρ nm = n| ρ |m denotes the density matrix element at position n, m and ω nm := ω n − ω m is the energy difference of the Hamiltonian eigenstates |n and |m of the system. The system operator which couples to the environmentŝ, is defined in H int . In this approach the environment is solely characterized through its spectral function C(ω) := ∞ −∞ dτ e iωτ B (τ )B(0)(5) where the bath operatorB is taken in the interaction picture defined with respect to H int . This spectral function, eq. 5, is assumed to change slowly such that it does not change on the small scale of δ and g (but only on the much larger scale of 2ω q ). The transversal coupling to the low temperature bath leads to unidirectional population transfers to states with lower excitation numbers, i.e. relaxation. These transfers appear in the Bloch-Redfield equations as linear dependencies of the timederivatives of certain diagonal elements of the system's density matrix on other diagonal elements, each with a coefficient. These coefficients (i.e. relaxation transition rates) are all proportional to v 2 ⊥ C(2ω q ). For later use we define a general relaxation rate due to coupling of the TLS to the environment: Γ 1 := v 2 ⊥ C(2ω q )(6) The longitudinal bath coupling leads to two processes: First a loss of phase coherence between the states of the system, i.e. dephasing, which is mathematically represented by the decay of off-diagonal elements in the density matrix. Second a mutual population transfer between certain eigenstates [48] with the same excitation number: |2 , |4 and |5 , |7 (figure 1C). The corresponding decay rates and transition rates are all similarly proportional to: Γ ϕ := v 2 C(0)(7) An energy diagram of the eigenstates is depicted in figure 1C and all transitions are shown as arrows. When the TLS is decoupled from the qubits (g 1 = g 2 = 0) then Γ 1 is the decay rate of the population of its excited state and 2Γ ϕ is the additional decay rate of its two off-diagonal elements, i.e. its relaxation and pure dephasing rate respectively. Comparing the resulting Bloch-Redfield equations with an approach assuming Lindblad equations [49,50] with a phenomenological relaxation rate and dephasing rate on the TLS we find that the two sets of differential equations are equivalent when the following two conditions are met. First the spectral function should not change on the scale of δ and g. As the second condition one of the following three requirements has to be fulfilled: Either (i) the full secular approximation (explained in the next paragraph) is applied to both the Lindblad and Bloch-Redfield equations or (ii) we take only longitudinal TLS-bath coupling, i.e., v ⊥ = 0 or (iii) we assume only transversal TLS-bath coupling, v = 0, and choose an initial state which is confined to the single excitation subspace. In case of equivalence the two phenomenological rates in the Lindblad equations can be identified as our definitions Γ 1 and Γ ϕ . The full secular approximation neglects all dependencies between different elements of the system's density matrix if at least one of them is an off-diagonal element. The necessary and sufficient condition for this approximation is that the system's level splittings and their differences are large compared to the decoherence rates. Physically this means assuming ω q g Γ 1 , Γ ϕ and ω q |δ| i.e. the TLS is somewhat coherent. In the following sections, analytical solutions to the Bloch-Redfield equations in different regimes are discussed. For δ = 0 these solutions are given in appendix Appendix A. The first solution we show (appendix Appendix A.2) is obtained using the full secular approximation and is valid for what we call the weak decoherence regime, when the resulting decoherence rates are smaller than the coupling strength g between the qubits and the TLS. We obtain two further analytical solutions for purely transversal (appendix Appendix A.3) i.e. v = 0, and purely longitudinal (appendix Appendix A.4) i.e. v ⊥ = 0, TLS-environment coupling. The combination of our particular initial state, the large ω q limit and v ⊥ v = 0 allows us to solve the master equation without the secular approximation. That means no assumption about the relative sizes of g and Γ 1 , Γ 2 has to be made. These last two solutions are therefore also valid for strong decoherence, i.e., when the decoherence rates are bigger than the coupling strength g. Dynamics In this section we present the analytical solutions to the Bloch-Redfield equations for our system. We start the section by summarizing the results for a single qubit coupled to a TLS for later comparison. Following this, in section 3.2 we study the behaviour of two qubits coupled to a TLS in detail. A single qubit coupled to a TLS Before we study the more complicated case of two qubits, the dynamics of a single qubit coupled to an environmental TLS (see figure 2A) provides a clear overview of the relevant physics. This special case can be obtained from all solutions by setting g 2 = 0 and tracing out the second qubit. Performing this on the Hamiltonian, eq. (1), yields (for corresponding eigenstates see appendix Appendix C): H 1q = ω q σ Q1 z + (ω q + δ)σ T LS z + g 1 (σ Q1 x σ T LS x + σ Q1 y σ T LS y )(8) A comparison with the results in this section will later allow us to distinguish phenomena which depend purely on the existence of two qubits and those which are due to qubit-TLS coupling in general. This section strongly depends on previous work [27], which is reproduced in our notation. As observable we consider the expectation value σ Q1 z (which is proportional to the qubit's energy). Equivalently, one could use the probability to find the qubit in the excited state, P exc = 1 2 σ Q1 z + 1 . Dual-probe decoherence microscopy:Probing pockets of coherence in a decohering environment8 2ω q 2ω t = 2ω q + 2δ TLS coupling to bath Figure 2. A Illustration of the simplified model system of one qubit with level splitting ω q coupled with coupling strength g 1 to an environmental TLS detuned by δ. The TLS in turn is coupled to a bath. This system follows from figure 1A by setting g 2 = 0. B Expectation values for the case of a single qubit coupled to a TLS as a function of time for Γ 1 = Γ ϕ = 0.1g 1 ; g 2 = 0; δ = 0 C Real part of the one-sided Fourier transform of σ Q1 z (t) as a function of detuning δ with Γ 1 = Γ ϕ = 0.1g 1 ; g 2 = 0. This is a numerical solution of either the Bloch-Redfield equations, where Γ 1 and Γ ϕ are the definitions given by eq. 6 and 7 or a numerical solution of the Lindblad equations with phenomenological rates. Analytically we find the angular frequency of the oscillation: ω osc = 2 δ 2 + 4g 2 1 which is drawn as a dashed line. For details of the calculation, see section 3.1. g 1 Qubit 1 A B C Assuming weak decoherence (i.e. we take the full secular approximation in the Bloch-Redfield equations) and no detuning δ = 0 one finds the expectation values as a function of time: σ Q1 z (t) = −1 + e − Γ 1 2 t + e (− Γ 1 2 −Γϕ)t cos [4g 1 t](9)σ T LS z (t) = −1 + e − Γ 1 2 t − e (− Γ 1 2 −Γϕ)t cos [4g 1 t](10) The population oscillates between the qubit and the TLS (cf. Fig 2B) with the oscillation frequency proportional to their transversal coupling strength. The oscillations decay on the time scale corresponding to the decoherence rates of the TLS. Equivalently we can consider the evolution in Fourier space, where the frequency and the decay rate are equal to the position and width respectively of the corresponding frequency peak (for details see appendix Appendix A.5). In figure 2C the real part of the one-sided Fourier transform of σ Q1 z (t) is plotted as a function of detuning between the qubit and the TLS. The peak which starts at frequency 4g 1 diminishes with increasing detuning, indicating a shift from oscillatory behaviour to pure exponential decay. The analytical solution for σ Q1 z (t) with δ = 0 contains complicated coefficients [27], but the frequency of the oscillation is simply ω osc = 2 δ 2 + 4g 2 1 . This corresponds to the level splitting between the hybridized states |2 1q and |4 1q (given in appendix Appendix C) and is plotted as a dashed line in figure 2C. The oscillatory behaviour is described by this one frequency, which corresponds to the standard "generalized Rabi frequency" [51] from quantum optics. For large detuning δ the expectation value σ Q1 z (t) is dominated by one purely decaying term. A Taylor expansion for small g/δ on the corresponding decay rate shows that the rate vanishes with increasing δ/g as ( Γ 1 + 4Γ ϕ )g 2 /δ 2 → 0. So far only weak decoherence on the TLS was considered. We also want to consider the limit where the decoherence is stronger than the qubit-TLS coupling. Simplifying the equations to purely transversal bath coupling (v = 0 ⇒ Γ ϕ = 0) one finds analytical solutions for the system dynamics without the use of the secular approximation and therefore valid for stronger decoherence: σ Q1 z (t) = −1 + −64g 2 1 µ 2 e − tΓ 1 2 + 2e − tΓ 1 2 −32g 2 1 + Γ 2 1 µ 2 cosh tµ 2 + Γ 1 µ sinh tµ 2(11) where µ := Γ 2 1 − 64g 2 1 . From this expression we see that as the decoherence rate Γ 1 increases relative to the qubit-TLS coupling strength g 1 , the dynamics changes from oscillations to pure exponential decay. This becomes obvious by rewriting the hyperbolic cosine as: cosh 1 2 Γ 2 1 − 64g 2 1 t =        cos 1 2 64g 2 1 − Γ 2 1 t for 8g 1 > Γ 1 1 2 e +... + e −... for 8g 1 < Γ 1(12) and similarly for the hyperbolic sine functions. Therefore, we can identify the threshold between oscillations and decay in our approximations as precisely Γ 1 = 8g 1 . Two qubits coupled to a TLS Having reviewed the behaviour of a single qubit coupling to an environmental TLS, we now consider the behaviour of a dual-probe configuration. Such a system is of particular interest when there is no direct coupling between the qubits. This situation allows us to probe what we call coherent pockets of the environment. When such a pocket is present in the environment, probing simultaneously with two qubits shows qualitatively different behaviour to the standard weakly coupled, Markovian environment, which would affect each qubit independently. (t) as a function of detuning δ. Analytically we find the three oscillation frequencies −δ + δ 2 + 4g 2 , δ + δ 2 + 4g 2 , 2 δ 2 + 4g 2 which are given by the dashed lines in the plot. The parameters are: Γ 1 = Γ ϕ = 0.1g; g 1 = g 2 C Time evolution of the expectation values for strong detuning δ = 5g; Γ 1 = Γ ϕ = 0.1g; g 1 = g 2 . Even though the TLS is only minimally excited, the effective coupling mediated by it still leads to coherent exchange of energy between the two qubits. 3.2.1. Mediated coupling between the qubits in the weak decoherence regime We will first show how the coupling between qubits and TLS will mediate an effective interaction between the two qubits themselves. For simplicity we initially consider δ = 0, i.e. both qubits are resonant with the TLS. In this case, and for the initial state chosen in section 2 the energy from qubit 1 coherently oscillates between the two qubits via excitation of the TLS (figure 3A). In contrast to the simpler case presented in section 3.1, the oscillations now show two distinct frequencies, namely 4g and 2g. The smaller of the two frequencies corresponds to the oscillation of energy between the two qubits. The full analytical expression for σ Q1 z can be found in appendix Appendix A. Again the change in the system dynamics due to detuning δ = 0 can be understood best by regarding the Fourier transform of the expectation value σ Q1 z (t). This yields a peak for each term located at the corresponding frequency whose half-width-at-halfmaximum (HWHM) equals the decay rate of the corresponding oscillatory component. Figure 3B shows the result of a numerical solution of the Bloch-Redfield equations with dashed lines indicating the analytical expressions for the frequency shifts due to the detuning. With increasing detuning δ, the peak at frequency 2g splits into two different frequency peaks. The amplitude of the two high frequency contributions diminishes with stronger detuning δ, while the amplitude of the lower frequency peak increases. This means for stronger detuning δ > g, the energy oscillates between the qubits mainly at the lower frequency ω low = |δ| − δ 2 − 4g 2 . For sufficiently strong detuning, the TLS is largely unpopulated during this process ( figure 3C). This kind of off-resonant interaction with the TLS leads to an effective transversal coupling between the qubits. This is the usual dispersive coupling term [52,32,33,34,35], in this case due to virtual excitation of the TLS. Performing a Taylor expansion for g/δ 1 on both the lower oscillation frequency ω low and on the decay rate of this oscillating term in the weak decoherence solution with detuning yields ω low ≈ 2g 2 /|δ| and γ low ≈ g 2 (Γ 1 + 12Γ ϕ )/6δ 2 . The lower frequency can here be interpreted as the effective coupling strength between the qubits ω low = g ef f . This effective coupling approaches zero slower than the decay rate γ low as the magnitude of the detuning increases. The strongly detuned TLS therefore mediates an effective transversal coupling between the qubits with a weakened influence of the TLS' decoherence rates. However, ultimately the effective coupling strength (i.e. the frequency of the oscillation) approaches zero for δ/g → ∞. Here we see a fundamentally different behaviour as compared to a single qubit coupled to a TLS. The oscillations do not change to a pure decay for strong (but not yet infinite) detuning δ. Additionally the frequency of these oscillations approaches zero much slower (∝ g/δ) than the decay rate of the single qubit (∝ g 2 /δ 2 ). This result has important implications for future experimental designs involving several qubits in a closely confined space. There the occurrence of an environmental TLS, which couples to two qubits at once might have a realistic probability, especially in solid-state qubits. In that case the qubits are affected by the TLS over a wide range of detuning, causing effective coupling between the qubits. Fig 3A, the steady state of both qubits is not their respective ground state. Rather, they decay into a state with a finite probability of finding them excited. This behaviour can be attributed to the existence of a so called dark state in our system. The state |3 = g 2 g |↑↓↓ − g 1 g |↓↑↓ (appendix Appendix C) is an entangled state of both qubits with the TLS in its ground state. The amplitudes of the two states (with the respective qubit excited) have a relative complex phase of π in the time evolution which leads to a cancellation of the qubits' influence on the TLS. In our system, this state is thus not influenced by decoherence and the system will remain in it for a long time (i.e. for the intrinsic decoherence time of the qubits). This is simply a manifestation of the physics of super-and sub-radiance [53] due to the interfering pathways from the qubits to the TLS. Since our chosen initial state is a statistical mixture including the eigenstate |3 , the steady state of the system will still include this fraction of the dark state. The entanglement of the two qubits in the steady state depends on the interplay of two things: the "concurrence" [54] of the dark state by itself which is given by C = 2g 1 g 2 /g 2 (i.e. a Bell state for g 1 = g 2 ) and the fraction of the dark state in the mixed steady state. Taking both into consideration we find the maximal "entanglement of formation" [54] of the final state as E = 53% which is reached for Formation of stray entanglement As we can see in g 1 = g 2 / √ 3. 3.2.3. Threshold between weak and strong decoherence When the decoherence rates of the TLS are stronger than the qubits-TLS coupling Γ 1 , Γ ϕ > g the secular approximation (section 2) can no longer be fully applied. With increasing decoherence rates the dynamics of the three subsystems changes from an oscillating behaviour to a pure decay. This behaviour is analogous to a single qubit coupled to a strongly decoherent TLS. In section 3.1 we saw that in this case the crossover was defined by the point Γ 1 = 8g 1 . For two qubits, the crossover between the weak and strong decoherence regimes is investigated numerically. For oscillations to occur between the qubits and the environmental TLS there needs to be an instant in time in which the population of the TLS is larger than both qubits combined. We therefore define the maximum value in the evolution: M = max t σ T LS z (t) − σ Q1 z (t) + σ Q2 z (t)(13) as a measure of the strength of the oscillation. In the regime of strong decoherence the energy of the qubits decays via the TLS to the environment and our defined measure is always zero. In the weak decoherence (i.e., oscillating) regime the energy leaves the qubits and then partially returns via coherent oscillations from the TLS. This gives a positive value for the defined measure. Figure 4 is a logarithmic plot of this measure as a function of the two decoherence rates Γ 1 and Γ ϕ . There is a sudden drop in the oscillation strength to negligible values marking a clear threshold between the oscillating (weak decoherence) and the decaying (strong decoherence) regime. The oscillating regime (dark area) also marks precisely the parameter regime in which the full secular approximation is valid. For purely transversal (respectively purely longitudinal) TLS-bath coupling v = 0 ⇒ Γ ϕ = 0 (respectively v ⊥ = 0 ⇒ Γ 1 = 0) an analytical solution can be found. Analogous to eq. (11) and (12) we find the analytical threshold between strong and weak decoherence regime at the two points: Γ 1 = 8g, Γ ϕ = 0 and Γ ϕ = 4g, Γ 1 = 0(14) This corresponds to the point where the threshold in figure 4 crosses the two axes. The factor of two between Γ 1 and Γ ϕ stems directly from the definition of the rates (eq. . Log 10 of the oscillation strength M (eq. 13) as a function of the decoherence rates Γ 1 and Γ ϕ . The threshold between oscillations and decay can be seen as a drop in the oscillation strength by four orders of magnitude. We solved the full Bloch-Redfield equations numerically (without the secular approximation) and for simplicity we set g 1 = g 2 and δ = 0. The level splitting was ω q = 1000g. Figure 5. Two possibilities of placing the system/environment boundary: The left case was used to set up and solve the Bloch-Redfield equations for which the system dynamics is by default assumed to be Markovian. For the configuration on the right side, where the TLS is seen as a part of the environment, we investigate Markovianity of the system of two qubits, see text. 6 and 7) as they appear in the master equations i.e. the off-diagonal elements of the uncoupled TLS-density matrix decay with a rate 2Γ ϕ ). Markovianity The coherent coupling to environmental states usually leads to non-Markovian [49,55] dynamics in the system (excluding the environmental states). Using our model, we can choose where to draw the system/environment boundary (see figure 5) and therefore explore this behaviour in a systematic fashion. Regarding the TLS as part of the system, Markovian dynamics is assumed by default as this is a necessary condition to apply the Bloch-Redfield equations. However, tracing out the TLS we investigate Markovianity of the two qubit system. Several measures exist to theoretically quantify non-Markovianity[56, 57, 58] in qubit systems. For our purposes we use the connection between the degree of non-Markovianity and an increase in the time evolution of the distinguishability, calculated with the trace-distance [49] D(t) = T r [ρ 1 (t) − ρ 2 (t)] 2 of two different initial states. By tracing out the environmental TLS in the analytical solution for purely transversal TLS-bath coupling (thus reducing the density matrix to the system of the two qubits) one can find the trace-distance between the two time evolutions of the initial states |Q1, Q2 = |↑↓ and |Q1, Q2 = |↓↑ . We find that the trace-distance only increases in the weak decoherence (oscillating) regime. This means that it is the same analytically exact threshold between oscillating behaviour and decaying behaviour (compare eq. (14)) that divides non-Markovian from Markovian behaviour. The chronologically first and strongest increase ∆D ↑ in the trace-distance is ∆D ↑ =        exp − πΓ 1 64g 2 − Γ 2 1 for 8g ≥ Γ 1 0 for 8g < Γ 1(15) for the particular case of pure relaxation and equal qubit-TLS couplings (Γ ϕ = 0, g 1 = g 2 = g/ √ 2), where ∆D ↑ = 0 corresponds to a monotonically decreasing trace-distance, i.e. no increase in D(t). Here we see that our experimentally measurable definition of the oscillation strength M is in this case equivalent to a measure of non-Markovianity. We can therefore see the direct link between probing coherence in the environment and probing non-Markovianity. Effective decay rate In order to characterize the decohering influence of the TLS on our probe qubits, we want to introduce a single, effective decay rate, serving as a figure of merit towards determining the influence of the TLS' decoherence rates Γ 1 , Γ ϕ . In general, the time evolution of the qubits' expectation values are given by sums of exponential functions, where, for the weak decoherence regime, some of the terms will be decaying oscillations. In the strong decoherence regime, the effective decay rate is introduced by replacing the sum of exponentials f (t) = j c j exp(−γ j t) by a single exponential c exp(−γ ef f t) with the two conditions: 1) ∞ 0 dtf (t) = ∞ 0 dt c exp(−γ ef f t)(16) 2) f (0) = c (17) which leads to the simple formula: γ ef f = j c j j c j /γ j(18) This defines a single decay rate in the strong decoherence regime. From the two analytical solutions which are valid for strong decoherence given in appendix Appendix Dual-probe decoherence microscopy:Probing pockets of coherence in a decohering environment15 A (zero detuning and either purely longitudinal or purely transversal TLS-bath coupling) we see that γ ef f,⊥ = 16g 2 (g 2 1 + 2g 2 2 ) Γ 1 16g 2 g 2 1 + (g 2 1 + 4g 2 2 ) Γ 2 1 for Γ 1 > 8g, Γ ϕ = 0 (19) γ ef f, = 8g 2 (g 2 1 + 4g 2 2 ) (g 2 1 + 16g 2 2 ) Γ 2 for Γ ϕ > 4g, Γ 1 = 0(20) For purely transversal bath coupling the effective decay rate γ ef f,⊥ is plotted in figure 6. Interestingly the effective decay rate eq. (19) and (20) monotonically decreases with decreasing ratio g/Γ 1 (g/Γ ϕ respectively). The observation of weaker decoherence on the qubit with stronger decoherence rate of the TLS is due to "blocking" of dynamics of the TLS due to strong decoherence: The exchange of energy between the qubits and the TLS is slowed down and thereby also the loss of energy to the environment. Regarding decoherence as a measurement process this is analogous to the Zeno-effect [49]. This behaviour is in contrast to the weak decoherence regime, where both purely decaying exponents and decaying oscillations occur. We find two ways of defining an effective decay rate: either describe the decay of the envelope of the oscillations or the effective decay of their average. For details, we refer the reader to Ref. [27] and appendix Appendix B. Here we use the decay of the average and find the effective decay rate γ ef f to be linearly dependent on the relaxation rate Γ 1 : γ ef f = 1 2 Γ 1 for Γ 1 < 8g, Γ ϕ < 4g(21) In the intermediate regime 2 Γ 1 /g < 8 and 1 Γ ϕ /g < 4, where the lower bound is found empirically, the oscillations are slow on the timescale of the decay. In this case the first half oscillation, transmission of energy from the qubit into the TLS, dominates the behaviour, and the average decay rate does not reproduce the behaviour well. An effective decay rate is not a good description of the dynamics in this regime and the apparent discontinuity in figure 6 actually appears as a smooth transition in the time evolution of the qubits' expectation value. We have also plotted three numerical calculations in figure 6 for different level splittings ω q (while δ is always zero). As stated in section 2 the analytical result is obtained in the single excitation subspace which requires ω q g 1 , g 2 , δ. We see excellent agreement between the analytical solution and the numerical solution of the full Bloch-Redfield equations for ω q 1000g. Probing a single TLS with two qubits: Parameter extraction After studying the system and its dynamics in the previous sections, we now interpret the results in the context of decoherence microscopy. In particular we focus on the ability to obtain the TLS parameters with a dual probe and compare it with a single qubit-probe. For this purpose we only consider the Fourier transform of the evolution of the qubits' excited-state-population, as this conveniently represents the parameters of interest. Figure 6. Effective decay rate γ ef f of the energy of qubit 1 as a function of the TLS relaxation rate Γ 1 (both rates in units of g). In the weak decoherence regime they are linearly dependent (eq. 21), for strong decoherence the effective decay rate decreases with increasing relaxation rate Γ 1 (eq. 19). The other parameters of the plot are: figure 1A). The distance between the peaks is controlled by the distance between the qubits d qq . The width of the peaks is controlled by the height h above the sample. Here we chose d qq = 3h. The coupling strengths are normalized such that the maximum value is 1. Now we will give a more concrete form to the theoretical coupling parameters g 1 and g 2 from eq.1. If we assume the coupling strengths depend on the distances d 1 and d 2 between the qubits and the TLS, as g j ∝ 1/d 2 j and the qubits are moved along their connecting line above the TLS in the substrate (see figure 1) then the coupling strengths behave characteristically as a function of the position y (figure 7). g 1 = g 2 , Γ ϕ = 0 Weak decoherence regime -oscillating behaviour From an experimental point of view the parameters δ, Γ 1 , Γ ϕ of the environmental TLS and even the coupling strengths g 1 , g 2 are in general unknown. We first consider the weak decoherence regime when the qubits are close enough to the TLS (so that g Γ 1 , Γ ϕ ). Then the obtained information of a measurement of σ Q1 z (t) is equivalent to a horizontal Dual-probe decoherence microscopy:Probing pockets of coherence in a decohering environment17 line in figure 3B. The positions of the three peaks give the three frequencies in figure 3B i.e. the necessary information to obtain the level splitting of the TLS δ and the qubits-TLS coupling strength g uniquely. Measuring g at several positions above the sample allows the position of the environmental TLS to be obtained from the local minimum of g in figure 7 i.e. a single TLS in the substrate can be located. Γ1 2 + Γ ϕ g 4 1 (Γ1+2Γϕ)g 4 ⇒ g Γ 1 , Γ ϕ g 1 , g 2 The widths and heights of the peaks provide further parameters although they have very complicated dependencies. In the case of resonance (δ = 0) however we find three peaks which allow an enormously simplified parameter extraction shown in figure 8. Experimentally one such plot provides enough information to obtain all system parameters: g from the position of the peaks, Γ 1 and Γ ϕ from the HWHM, and (having obtained these three parameters) g 1 and g 2 from the heights of the peaks. All system parameters can be obtained from one measurement of the time evolution of the excited-state-population of one of the probe qubits on resonance with the TLS. To reach resonance experimentally the qubits could always be tuned to resonance with the TLS, once δ is obtained as explained in the previous paragraph. The major difference between the two qubits and a single qubit is the behaviour for detuning to the TLS. While the single qubit is effectively decoupled by detuning, the addition of a second qubit maintains an oscillating signal via the TLS-induced effective coupling between the qubits. As a result strong detuning and weak qubit-TLS coupling Figure 9. Expectation value of qubit 1 for the case of two qubits coupled to a TLS and g 1 = g 2 (top) and a single qubit coupled to a TLS (g 2 = 0, bottom). Both plots show two different cases: large detuning δ = 4.8g and weak qubit-TLS coupling g = 0.05Γ 1 . For all plots Γ ϕ = 0. These two different cases can only be clearly distinguished with two qubits. show two fundamentally different behaviours and can be distinguished with two qubits (figure 9). The system is also sensitive to TLS over a wider frequency range as the TLS-induced coupling decreases more slowly with detuning. Furthermore the additional two lower frequencies in figure 3B which correspond to oscillations between the two qubits make it possible to obtain the detuning without changing the level splitting of the qubits. Strong decoherence regime -decaying behaviour Scanning a substrate for isolated TLS one might find very different decoherence strengths for each TLS, some of which might be fluctuating so strongly (or coupled so weakly) that no coherent oscillations will occur even when the qubits are directly above it. In that case the above technique of parameter extraction is no longer applicable. However, the TLS can still be located (both with a single and two qubits) by monitoring the decay rate of the qubit at different locations along the y-axis in figure 1A. For pure relaxation the position dependency of the effective decay rate is shown in figures 10A (single qubit probe) and 10B (two qubit probes). The characteristic behaviour provides the position of the TLS. Experimental realisations Although in principle any qubit architecture can be adapted for performing decoherence microscopy, in order to study microscopic pockets of coherence, atomic scale qubits with long coherence times are ideal. In solid-state, this implies spin donors or defects, such as semiconductor donors [59,60,16] or colour centres in diamond [20,24,15]. As the NV centre in diamond is an experimentally established and well investigated system we discuss the requirements to use these centres in a dual-probe configuration. In A B Figure 10. A Characteristic behaviour of the effective decay rate of the energy of a single qubit (g 2 = 0) as a function of the position y. This plot is in the decaying regime Γ 1 = 10g 1 and Γ ϕ = 0 . B Characteristic behaviour of the effective decay rate of the energies of two qubits as a function of the position y. This plot is in the decaying regime Γ 1 = 10g 1 and Γ ϕ = 0 recent experiments the intrinsic decoherence of the NV centre is weak and dominated by the dephasing [20,15], which sets the sensitivity limit for probing environmental pockets of coherence. Isotopic purification of the diamond lattice [24] is currently investigated for quantum computation and sensing purposes and will result in much longer coherence times leading to a corresponding increase in sensitivity. In order to unambiguously identify coupling via the environment, we need to minimise or eliminate coupling between the probes. This can be achieved in two configurations. Using the nuclear spin states of the Nitrogen within each NV centre as the qubit probes provides a strong electron-nuclear coupling to electron spins in the environment whilst minimising the inter qubit (nuclear-nuclear) coupling. For both a probe-TLS distance and probe separation of 5 nm, we require a T 2 > 20 ms to reach the probe-TLS oscillation limit. Using current estimates [24,61] for the dephasing channels due to 13 C spins in the diamond lattice, this requires a 13 C concentration below 0.03%. For this probe separation, the inter-probe coupling is a factor of 1000 times smaller than the probe-TLS coupling and therefore provides no extra complication to the analysis. A second method of achieving strong probe-TLS coupling is to use a pair of NV centres whose crystallographic orientation is such that the natural NV-NV coupling is eliminated due to the angular dependence of the dipolar interaction. Although there are more serious fabrication challenges with this configuration, the maximum dephasing required (T 2 > 20µs) is considerably less due to the strength of the electronelectron interaction. Such dephasing times are well within the range currently seen in experiments using NV centers [24]. In either of these configurations, detecting sample impurities with large magnetic moments such as single molecule magnets ( 8 Fe, Ferritin) [17,18,19] is considerably easier, even with currently available intrinsic decoherence times. Depending on the background field configuration and qubit operating mode, these impurities will induce either dephasing dominated or energy exchange processes. Conclusion In this paper we have investigated the concept of a dual-probe decoherence microscope. Using a general model, we have studied the key characteristics of such a system analytically and numerically. Mapping out the temporal dynamics of the qubit probes provides detailed information about a TLS, the simplest example of a pocket of coherence contained within the sample. In addition to the TLS' level splitting and the coupling strengths to the probes one can obtain its dephasing and relaxation rate which implies the coupling strength to its surround environment and that environment's constitution. A dual-probe configuration simplifies the measurement process and increases sensitivity to detuned TLS. Furthermore we have shown how the oscillation amplitude of environmentally mediated coupling between two probes is largely unaffected by detuning and decoherence of the mediating TLS, although the frequency of oscillation still depends on detuning. The close relationship between environmentally mediated coupling and non-Markovian dynamics makes a dual-probe configuration ideal for both probing an environment's potential to induce non-Markovian dynamics in a system as well as detecting the spatial extent and interrogate pockets of coherence which sit within a more complex environment. Appendix A. Analytical understanding of the expectation values and their Fourier transforms In this appendix we give the three analytical solutions each for no detuning δ = 0 (i.e. δ g). As mentioned before all three solutions were obtained in the subspace of one excitation plus the ground state with the assumption ω q g 1 , g 2 . Furthermore a low temperature bath ω q T was assumed. First the secular approximation is explained, then the three solutions are given. The Bloch-Refield equations, eq. 3, can always be rewritten as a matrix multiplication. To do so, the elements of the density matrix need to be written in a vector. Here we choose the particular order: all diagonal elements first, then all off-diagonal elements. To solve the equations the resulting Redfield tensor R, which is now a matrix, needs to be diagonalized. The secular approximation means that for this diagonalisation process one can neglect off-diagonal elements in the Redfield tensor R when there is a large difference between the corresponding diagonal elements. In our chosen order we can regard separate blocks in the Redfield-tensor (see below). The diagonal elements in the lower right block each have a term −iω jk , whose magnitude is given by the energy difference of the two system states j and k. If these level splittings are large compared to the decoherence rates, then the two blocks linking diagonal and off-diagonal density matrix elements can be set to zero, i.e. the upper right block and the lower left block. If the differences of the energy differences ω jk −ω lm are also large compared to the decoherence rates, then all off-diagonal Redfield tensor elements in the lower right block can also be set to zero. This is what we call the full secular approximation. The resulting Redfield tensor is given by:               ρ 11 ρ 22 ρ 33 . . . ρ 12 ρ 13 . . .                =             R 1111 R 1122 R 1133 . . . R 2211 R 2222 R 2233 R 3311 R 3322 R 3333 . . . . . . 0 0 R 1212 − iω 12 0 0 0 R 1313 − iω 13 0 0 0 . . .                            ρ 11 ρ 22 ρ 33 . . . ρ 12 ρ 13 . . .                Mathematically the secular approximation is analogous to the rotating wave approximation: Separating the Redfield tensor into a coherent part (i.e. the −iω jk terms) and a decoherent part we can define an analogous "interaction picture" for the vector Dual-probe decoherence microscopy:Probing pockets of coherence in a decohering environment22 of density matrix elements ρ: R = R coh + R dec (A.1) ρ := exp(−R coh t) ρ (A.2) ⇒ d dt ρ = exp(−R coh t)R dec exp(R coh t) ρ =:R ρ (A.3) This results in a new Redfield tensorR in this "interaction picture" where all offdiagonal elements are multiplied by a rotating term with the rotation frequency equal to the difference of the corresponding diagonal R-elements. The secular approximation can then be justified analogously to the rotating wave approximation. Appendix A.2. Weak decoherence regime g Γ 1 , Γ ϕ The two qubits couple to the same environmental TLS strongly compared to the decoherence of the TLS. The full secular approximation is applied. σ Q1 z (t) = g 4 2 − g 4 1 − 2g 2 1 g 2 2 g 4 + g 4 1 g 4 e − Γ 1 t 2 + 4g 2 1 g 2 2 g 4 e − Γ 1 t 4 −Γϕt cos [2g t] + g 4 1 g 4 e − Γ 1 t 2 −Γϕt cos [4g t] σ Q2 z (t) = − g 4 1 + g 4 2 g 4 + g 2 1 g 2 2 g 4 e − Γ 1 t 2 − 4g 2 1 g 2 2 g 4 e (− Γ 1 4 −Γϕ)t cos [2g t] + g 2 1 g 2 2 g 4 e (− Γ 1 2 −Γϕ)t cos [4g t] σ T LS z (t) = −1 + g 2 1 g 2 e − Γ 1 t 2 − g 2 1 g 2 e − Γ 1 t 2 −Γϕt cos [4g t] (A.4) Appendix A.3. Purely transversal TLS-bath coupling v = 0 ⇒ Γ ϕ = 0 The TLS-bath coupling is purely transversal, i.e. there is relaxation only. The secular approximation need not be applied here, due to our particular choice of environmental coupling operator. Therefore this solution is also valid for strong relaxation. The TLS-bath coupling is purely longitudinal i.e. there is dephasing only. Again the secular approximation need not be applied here, due to our particular choice of initial state and environmental coupling operator. Therefore this solution is also valid for strong dephasing. Fitting such peaks allows us to experimentally obtain the frequency and decay rate in a precise and simple way, as displayed in figure 8. Additionally, the close correspondence to the parameters in the Fourier domain helps to depict frequencies and decay rates at the same time in figures 2C and 3B. σ Q1 z (t) = −g 4 1 − 2g 2 1 g 2 2 + g 4 2 g 4 − 64 g 4 1 µ 2 g 2 e − Γ 1 t 2 + 4g 2 1 g 2 2 µg 4 e − Γ 1 t 4 µ cosh µt 4 + Γ 1 sinh µt 4 + 2g 4 1 µ 2 g 4 e − Γ 1 t 2 Γ 2 1 − 32g 2 cosh µt 2 + Γ 1 µ sinh µt 2 (A.5) σ Q2 z (t) = − g 4 1 + g 4 2 g 4 − where k sums over all purely decaying terms and l sums over all oscillating terms. For the purely decaying terms it is important not to take the magnitude of the coefficients in case negative coefficients c k < 0 occur. In principle all terms with a non-zero imaginary part of the exponential rate are oscillations. However, when this imaginary part (which is the angular frequency of the oscillation) is smaller than the real part (which is the decay rate) then this term decays strongly before the time period of one oscillation, i.e. the term looks like a pure (nonexponential) decay. For a numerical criterion whether a term should be categorised as oscillating or purely decaying one could therefore measure the imaginary part relative to the real part for each individual term. However for simplicity of the criterion we categorize all terms with an imaginary part of the exponential rate below 0.1 (where g = 1) as purely decaying terms in our numerical calculations for figure 6. This is about one order of magnitude less than the decay rates plotted in figure 6. Appendix C. Hamiltonian eigenstates of the system For our system of two qubits coupled to one TLS the unnormalised eigenstates indicated in figure 1C are: |8 = | ↑↑↑ |7 = (−δ + δ 2 + 4g 2 )| ↑↑↓ + 2g 2 | ↑↓↑ + 2g 1 | ↓↑↑ |6 = −g 1 | ↑↓↑ + g 2 | ↓↑↑ |5 = (−δ − δ 2 + 4g 2 )| ↑↑↓ + 2g 2 | ↑↓↑ + 2g 1 | ↓↑↑ |4 = 2g 1 | ↑↓↓ + 2g 2 | ↓↑↓ + (δ + δ 2 + 4g 2 )| ↓↓↑ |3 = −g 2 | ↑↓↓ + g 1 | ↓↑↓ |2 = 2g 1 | ↑↓↓ + 2g 2 | ↓↑↓ + (δ − δ 2 + 4g 2 )| ↓↓↑ |1 = | ↓↓↓ (C.1) where ↑ indicates an excited state and ↓ a ground state of the two qubits and the TLS in the order |Q1, Q2, T LS . Setting g 2 = 0 and tracing out the second qubit yields the system of only one qubit coupled to a TLS discussed in section 3.1. Performing these operations on the above states we find: |8 → |8 1q = | ↑↑ |7 → |7 1q = (−δ + δ 2 + 4g 2 )| ↑↓ + 2g 1 | ↓↑ |6 → |6 1q = −g 1 | ↑↑ |5 → |5 1q = (−δ − δ 2 + 4g 2 )| ↑↓ + 2g 1 | ↓↑ |4 → |4 1q = 2g 1 | ↑↓ + (δ + δ 2 + 4g 2 )| ↓↑ |3 → |3 1q = g 1 | ↓↓ |2 → |2 1q = 2g 1 | ↑↓ + (δ − δ 2 + 4g 2 )| ↓↑ |1 → |1 1q = | ↓↓ (C.2) Several states become equivalent: |3 1q = g 1 |1 1q (C.3) |4 1q = 2g 1 −δ + δ 2 + 4g 2 |7 1q (C.4) |5 1q = −δ − δ 2 + 4g 2 1 2g 1 |2 1q (C.5) |6 1q = −g 1 |8 1q (C.6) We therefore need (as one should expect) only four states to describe this reduced system. nmn m := Λ m mnn +Λ nn m m − k (Λ nkkn δ mm +Λ kmm k δ nn ) Λ nmn m :=ŝ nmŝn m 1 2 C(ω = ω m n ) Λ nmn m :=ŝ nmŝn m 1 2 C(ω = ω n m ) . Figure 3 . 3A Expectation values of both qubits and the TLS. The excitation is shifted from one qubit via the TLS to the other qubit and back. The parameters are chosen as: Γ 1 = Γ ϕ = 0.1g; g 1 = g 2 ; δ = 0 B Real part of the one-sided Fourier transform of σ Q1 z Figure 4 4Figure 4. Log 10 of the oscillation strength M (eq. 13) as a function of the decoherence rates Γ 1 and Γ ϕ . The threshold between oscillations and decay can be seen as a drop in the oscillation strength by four orders of magnitude. We solved the full Bloch-Redfield equations numerically (without the secular approximation) and for simplicity we set g 1 = g 2 and δ = 0. The level splitting was ω q = 1000g. Figure 7 . 7Characteristic behaviour of the coupling strengths to the TLS in the substrate as a function of the position of the two qubits y (compare Figure 8 . 8Top: Real part of the one-sided Fourier transform of σ Q1 z (t) in the weak decoherence regime (eq. A.4) for δ = 0, Bottom: The table shows how the parameters could be obtained from a measurement of the plot above. Appendix A. 1 . 1Matrix form of the master equation and the secular approximation . 4 . 4Purely longitudinal TLS-bath coupling v ⊥ = 0 ⇒ Γ 1 = 0 in all expectation values contain constant and oscillating terms, with associated decay rates. Terms of this form are more easily understood in the Fourier domain.For measured signals of the form e −at cos[bt] the real part of its one-sided Fourier transform yields two Lorentzian peaks at the position of plus and minus the frequency b and with a half width at half maximum (HWHM) which equals the decay rate a:∞ 0 e −iωt e −at cos(bt) dt = a 2 (a 2 + (ω − b) 2 ) + a 2 (a 2 + (ω + b) 2 ) (A.11) AcknowledgmentsWe would like to thank N. Oxtoby, S. Huelga and N. Vogt for helpful discussions. This work was supported by the CFN of DFG, the EU project SOLID and the US ARO under contract no. W911NF-09-1-0336. We acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Karlsruhe Institute of Technology.Appendix B. Calculation of the effective decay rate of a sum of decaying oscillationsIn the strong decoherence regime the oscillations are a sum of several exponentials j c j exp(−γ j t). To find one effective decay rate we can simply use eq. 18. This procedure is sensible when the different decay rates are not too far (i.e. not orders of magnitude) apart. Note that eq. 18 can also be calculated from an integration:In the weak decoherence regime on the other hand we have additional oscillations for several terms, i.e. an expression of the formwhere some ω j might be zero and the cosine function might be replaced by a sine function for some terms. In such a case (as for example displayed in figure 2B or 3A) one needs to decide to either take the average or the envelope of the oscillations. The effective decay rate of the average neglects the oscillating terms completely and can therefore become zero when there are no purely decaying terms in the expression. On the other hand the average is unambiguous while the upper envelope and the lower envelope can lead to different effective decay rates. This is the reason why the average was chosen infigure 6for the weak decoherence (oscillating) regime. The calculation of the average is performed by neglecting all oscillating terms and calculating eq. 18 from the rest. Numerically that is easily done by rewriting all oscillations in eq. B.2 as exponentials, which yields an expression of the form:Then all terms with a non-zero imaginary part in the exponential rate can be neglected and the effective decay rate can be calculated as:average:where k sums over all purely decaying terms.The calculation of the envelope is performed by setting all oscillations (including the algebraic sign) to 1 (upper envelope) or -1 (lower envelope). Afterwards eq. 18 can be applied to all terms. Numerically that can easily be performed by taking the magnitude of the coefficients and real parts of the rates for all oscillating terms: upper envelope:γ ef f = k c k + l |c l | k c k /γ k + l |c l |/γ l (B.5) lower envelope: Josephson qubits as probes of 1/f noise. Alexander Shnirman, Gerd Schön, Ivar Martin, Yuriy Makhlin, Electron Correlation in New Materials and Nanosystems. Springernumber 241 in NATO Science SeriesAlexander Shnirman, Gerd Schön, Ivar Martin, and Yuriy Makhlin. Josephson qubits as probes of 1/f noise. In "Electron Correlation in New Materials and Nanosystems", number 241 in NATO Science Series, pages 343-356. NATO, Springer, 2007. Scanning quantum decoherence microscopy. Jared H Cole, C L Lloyd, Hollenberg, Nanotechnology. 2049495401Jared H. Cole and Lloyd C. L. Hollenberg. Scanning quantum decoherence microscopy. Nanotechnology, 20(49):495401, DEC 9 2009. Spin microscope based on optically detected magnetic resonance. M Boris, Gennady P Chernobrod, Berman, Journal of Applied Physics. 97114903Boris M. Chernobrod and Gennady P. Berman. Spin microscope based on optically detected magnetic resonance. Journal of Applied Physics, 97(1):014903, 2005. Fabrication of an array of single-electron transistors for a scanning probe microscope sensor. Jochen Weber, Juergen Weis, Maik Hauser, Klaus Von Klitzing, Nanotechnology. 1937375301Jochen Weber, Juergen Weis, Maik Hauser, and Klaus Von Klitzing. Fabrication of an array of single-electron transistors for a scanning probe microscope sensor. Nanotechnology, 19(37):375301, SEP 17 2008. International Workshop on Electron Spin Resonance and Related Phenomena in Low-Dimensional Structures. Sousa Rogerio De, Topics in Applied Physics. Fanciulli, M115Electron Spin as a Spectrometer of Nuclear-Spin Noise and Other FluctuationsRogerio de Sousa. Electron Spin as a Spectrometer of Nuclear-Spin Noise and Other Fluctuations. In Fanciulli, M, editor, Electron spin resonance and related phenomena in low-dimensional structures, volume 115 of Topics in Applied Physics, pages 183-220, 2009. International Workshop on Electron Spin Resonance and Related Phenomena in Low-Dimensional Structures, Sanremo, Italy, March 06-08, 2006. Decoherence in josephson qubits from dielectric loss. John M Martinis, K B Cooper, R Mcdermott, Matthias Steffen, Markus Ansmann, K D Osborn, K Cicak, Seongshik Oh, D P Pappas, R W Simmonds, Clare C Yu, Physical Review Letters. 95210503John M. Martinis, K. B. Cooper, R. McDermott, Matthias Steffen, Markus Ansmann, K. D. Osborn, K. Cicak, Seongshik Oh, D. P. Pappas, R. W. Simmonds, and Clare C. Yu. Decoherence in josephson qubits from dielectric loss. Physical Review Letters, 95:210503, Nov 2005. Process tomography of quantum memory in a josephson-phase qubit coupledto a two-level state. Matthew Neeley, M Ansmann, Radoslaw C Bialczak, M Hofheinz, N Katz, Erik Lucero, A O&apos;connell, H Wang, A N Cleland, John M Martinis, Nature Physics. 47Matthew Neeley, M. Ansmann, Radoslaw C. Bialczak, M. Hofheinz, N. Katz, Erik Lucero, A. O'Connell, H. Wang, A. N. Cleland, and John M. Martinis. Process tomography of quantum memory in a josephson-phase qubit coupledto a two-level state. Nature Physics, 4(7):523-526, JUL 2008. Tunneling spectroscopy of two-level systems inside a josephson junction. I Martin, L Bulaevskii, A Shnirman, Physical Review Letters. 95127002I. Martin, L. Bulaevskii, and A. Shnirman. Tunneling spectroscopy of two-level systems inside a josephson junction. Physical Review Letters, 95:127002, Sep 2005. Multiphoton spectroscopy of a hybrid quantum system. P Bushev, C Mueller, J Lisenfeld, J H Cole, A Lukashenko, A Shnirman, A V Ustinov, Physical Review B. 8213134530P. Bushev, C. Mueller, J. Lisenfeld, J. H. Cole, A. Lukashenko, A. Shnirman, and A. V. Ustinov. Multiphoton spectroscopy of a hybrid quantum system. Physical Review B, 82(13):134530, OCT 25 2010. Quantitative evaluation of defect-models in superconducting phase qubits. J H Cole, C Müller, P Bushev, G J Grabovskij, J Lisenfeld, A Lukashenko, A V Ustinov, A Shnirman, Applied Physics Letters. 9725252501J. H. Cole, C. Müller, P. Bushev, G. J. Grabovskij, J. Lisenfeld, A. Lukashenko, A. V. Ustinov, and A. Shnirman. Quantitative evaluation of defect-models in superconducting phase qubits. Applied Physics Letters, 97(25):252501, 2010. Rabi spectroscopy of a qubit-fluctuator system. Jürgen Lisenfeld, Clemens Müller, Jared H Cole, Pavel Bushev, Alexander Lukashenko, Alexander Shnirman, Alexey V Ustinov, Physical Review B. 8110100511Jürgen Lisenfeld, Clemens Müller, Jared H. Cole, Pavel Bushev, Alexander Lukashenko, Alexander Shnirman, and Alexey V. Ustinov. Rabi spectroscopy of a qubit-fluctuator system. Physical Review B, 81(10):100511, Mar 2010. Observation of coherent oscillations in a single electron spin. F Jelezko, T Gaebel, I Popa, A Gruber, J Wrachtrup, Physical Review Letters. 9276401F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup. Observation of coherent oscillations in a single electron spin. Physical Review Letters, 92:076401, Feb 2004. Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate. F Jelezko, T Gaebel, I Popa, M Domhan, A Gruber, J Wrachtrup, Physical Review Letters. 93130501F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup. Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate. Physical Review Letters, 93:130501, Sep 2004. Roomtemperature coherent coupling of single spins in diamond. T Gaebel, Domhan, C Popa, P Wittmann, F Neumann, Jelezko, Rabeau, Stavrias, Greentree, Prawer, J Meijer, Twamley, J Hemmer, Wrachtrup, Nature Physics. 26T Gaebel, M Domhan, I Popa, C Wittmann, P Neumann, F Jelezko, JR Rabeau, N Stavrias, AD Greentree, S Prawer, J Meijer, J Twamley, PR Hemmer, and J Wrachtrup. Room- temperature coherent coupling of single spins in diamond. Nature Physics, 2(6):408-413, JUN 2006. Coherent dynamics of coupled electron and nuclear spin qubits in diamond. L Childress, M V Dutt, J M Taylor, A S Zibrov, F Jelezko, J Wrachtrup, P R Hemmer, M D Lukin, Science. 3145797L. Childress, M. V. Gurudev Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin. Coherent dynamics of coupled electron and nuclear spin qubits in diamond. Science, 314(5797):281-285, OCT 13 2006. Single-shot readout of an electron spin in silicon. Andrea Morello, Jarryd J Pla, Floris A Zwanenburg, W Kok, Chan, Y Kuan, Hans Tan, Mikko Huebl, Christopher D Mottonen, Changyi Nugroho, Jessica A Yang, Andrew D C Van Donkelaar, David N Alves, Christopher C Jamieson, Escott, C L Lloyd, Robert G Hollenberg, Andrew S Clark, Dzurak, Nature. 4677316Andrea Morello, Jarryd J. Pla, Floris A. Zwanenburg, Kok W. Chan, Kuan Y. Tan, Hans Huebl, Mikko Mottonen, Christopher D. Nugroho, Changyi Yang, Jessica A. van Donkelaar, Andrew D. C. Alves, David N. Jamieson, Christopher C. Escott, Lloyd C. L. Hollenberg, Robert G. Clark, and Andrew S. Dzurak. Single-shot readout of an electron spin in silicon. Nature, 467(7316):687-691, OCT 7 2010. High-frequency resonant experiments in f e 8 molecular clusters. E Barco, J M Hernandez, J Tejada, N Biskup, R Achey, I Rutel, N Dalal, J Brooks, Physical Review B. 625E. del Barco, J. M. Hernandez, J. Tejada, N. Biskup, R. Achey, I. Rutel, N. Dalal, and J. Brooks. High-frequency resonant experiments in f e 8 molecular clusters. Physical Review B, 62(5):3018- 3021, Aug 2000. Macroscopic quantum tunneling in magnetic proteins. D D Awschalom, J F Smyth, G Grinstein, D P Divincenzo, D Loss, Physical Review Letters. 6820D. D. Awschalom, J. F. Smyth, G. Grinstein, D. P. DiVincenzo, and D. Loss. Macroscopic quantum tunneling in magnetic proteins. Physical Review Letters, 68(20):3092-3095, May 1992. Macroscopic resonant tunneling of magnetization in ferritin. J Tejada, X X Zhang, E Barco, J M Hernández, E M Chudnovsky, Physical Review Letters. 799J. Tejada, X. X. Zhang, E. del Barco, J. M. Hernández, and E. M. Chudnovsky. Macroscopic resonant tunneling of magnetization in ferritin. Physical Review Letters, 79(9):1754-1757, Sep 1997. Nanoscale imaging magnetometry with diamond spins under ambient conditions. I Y Gopalakrishnan Balasubramanian, Roman Chan, Mohannad Kolesov, Julia Al-Hmoud, Chang Tisler, Changdong Shin, Aleksander Kim, Philip R Wojcik, Anke Hemmer, Tobias Krueger, Alfred Hanke, Rudolf Leitenstorfer, Fedor Bratschitsch, Joerg Jelezko, Wrachtrup, Nature. 4557213Gopalakrishnan Balasubramanian, I. Y. Chan, Roman Kolesov, Mohannad Al-Hmoud, Julia Tisler, Chang Shin, Changdong Kim, Aleksander Wojcik, Philip R. Hemmer, Anke Krueger, Tobias Hanke, Alfred Leitenstorfer, Rudolf Bratschitsch, Fedor Jelezko, and Joerg Wrachtrup. Nanoscale imaging magnetometry with diamond spins under ambient conditions. Nature, 455(7213):648-U46, OCT 2 2008. Nanoscale magnetic sensing with an individual electronic spin in diamond. J R Maze, P L Stanwix, J S Hodges, S Hong, J M Taylor, P Cappellaro, L Jiang, M V Dutt, E Togan, A S Zibrov, A Yacoby, R L Walsworth, M D Lukin, Nature. 4557213J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M. Taylor, P. Cappellaro, L. Jiang, M. V. Gurudev Dutt, E. Togan, A. S. Zibrov, A. Yacoby, R. L. Walsworth, and M. D. Lukin. Nanoscale magnetic sensing with an individual electronic spin in diamond. Nature, 455(7213):644-647, October 2008. Scanning magnetic field microscope with a diamond single-spin sensor. C L Degen, Applied Physics Letters. 9224243111C. L. Degen. Scanning magnetic field microscope with a diamond single-spin sensor. Applied Physics Letters, 92(24):243111, 2008. High-sensitivity diamond magnetometer with nanoscale resolution. J M Taylor, P Cappellaro, L Childress, L Jiang, D Budker, P R Hemmer, A Yacoby, R Walsworth, M D Lukin, Nature Physics. 410J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin. High-sensitivity diamond magnetometer with nanoscale resolution. Nature Physics, 4(10):810-816, OCT 2008. Ultralong spin coherence time in isotopically engineered diamond. Gopalakrishnan Balasubramanian, Philipp Neumann, Daniel Twitchen, Matthew Markham, Roman Kolesov, Norikazu Mizuochi, Junichi Isoya, Jocelyn Achard, Johannes Beck, Julia Tissler, Vincent Jacques, Philip R Hemmer, Fedor Jelezko, Joerg Wrachtrup, Nature Materials. 85Gopalakrishnan Balasubramanian, Philipp Neumann, Daniel Twitchen, Matthew Markham, Roman Kolesov, Norikazu Mizuochi, Junichi Isoya, Jocelyn Achard, Johannes Beck, Julia Tissler, Vincent Jacques, Philip R. Hemmer, Fedor Jelezko, and Joerg Wrachtrup. Ultralong spin coherence time in isotopically engineered diamond. Nature Materials, 8(5):383-387, MAY 2009. Sensing of fluctuating nanoscale magnetic fields using nitrogen-vacancy centers in diamond. L T Hall, J H Cole, C D Hill, L C L Hollenberg, Physical Review Letters. 103220802L. T. Hall, J. H. Cole, C. D. Hill, and L. C. L. Hollenberg. Sensing of fluctuating nanoscale magnetic fields using nitrogen-vacancy centers in diamond. Physical Review Letters, 103:220802, Nov 2009. Monitoring ion-channel function in real time through quantum decoherence. Liam T Hall, Charles D Hill, Jared H Cole, Brigitte Staedler, Frank Caruso, Paul Mulvaney, Joerg Wrachtrup, Lloyd C L Hollenberg, Proceedings of the National Academy of Sciences of the United States of America. the National Academy of Sciences of the United States of America107Liam T. Hall, Charles D. Hill, Jared H. Cole, Brigitte Staedler, Frank Caruso, Paul Mulvaney, Joerg Wrachtrup, and Lloyd C. L. Hollenberg. Monitoring ion-channel function in real time through quantum decoherence. Proceedings of the National Academy of Sciences of the United States of America, 107(44):18777-18782, NOV 2 2010. Relaxation of Josephson qubits due to strong coupling to two-level systems. Clemens Mueller, Alexander Shnirman, Yuriy Makhlin, Physical Review B. 801394Clemens Mueller, Alexander Shnirman, and Yuriy Makhlin. Relaxation of Josephson qubits due to strong coupling to two-level systems. Physical Review B, 80(13):94, OCT 2009. Probing a composite spin-boson environment. Neil P Oxtoby, Angel Rivas, Susana F Huelga, Rosario Fazio, New Journal of Physics. 11663028Neil P. Oxtoby, Angel Rivas, Susana F. Huelga, and Rosario Fazio. Probing a composite spin-boson environment. New Journal of Physics, 11(6):063028, 2009. Decoherence by a spin thermal bath: Role of spin-spin interactions and initial state of the bath. Shengjun Yuan, Mikhail I Katsnelson, Hans De Raedt, Physical Review B. 7718184301Shengjun Yuan, Mikhail I. Katsnelson, and Hans De Raedt. Decoherence by a spin thermal bath: Role of spin-spin interactions and initial state of the bath. Physical Review B, 77(18):184301, May 2008. Quantum dynamics in nonequilibrium environments. Clive Emary, Physical Review A. 78332105Clive Emary. Quantum dynamics in nonequilibrium environments. Physical Review A, 78(3):032105, Sep 2008. Characterization of coherent impurity effects in solid-state qubits. E Paladino, M Sassetti, G Falci, U Weiss, Physical Review B. 77441303E. Paladino, M. Sassetti, G. Falci, and U. Weiss. Characterization of coherent impurity effects in solid-state qubits. Physical Review B, 77(4):041303, Jan 2008. Manipulating quantum entanglement with atoms and photons in a cavity. J M Raimond, M Brune, S Haroche, Review of Modern Physics. 733J. M. Raimond, M. Brune, and S. Haroche. Manipulating quantum entanglement with atoms and photons in a cavity. Review of Modern Physics, 73(3):565-582, Aug 2001. Quantum computation with ions in thermal motion. Anders Sørensen, Klaus Mølmer, Physical Review Letters. 829Anders Sørensen and Klaus Mølmer. Quantum computation with ions in thermal motion. Physical Review Letters, 82(9):1971-1974, Mar 1999. Efficient scheme for two-atom entanglement and quantum information processing in cavity qed. Guang-Can Shi-Biao Zheng, Guo, Physical Review Letters. 8511Shi-Biao Zheng and Guang-Can Guo. Efficient scheme for two-atom entanglement and quantum information processing in cavity qed. Physical Review Letters, 85(11):2392-2395, Sep 2000. Coupling superconducting qubits via a cavity bus. J Majer, J M Chow, J M Gambetta, Jens Koch, B R Johnson, J A Schreier, L Frunzio, D I Schuster, A A Houck, A Wallraff, A Blais, M H Devoret, S M Girvin, R J Schoelkopf, Nature. 4497161J. Majer, J. M. Chow, J. M. Gambetta, Jens Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. Coupling superconducting qubits via a cavity bus. Nature, 449(7161):443-447, September 2007. Two-qubit state tomography using a joint dispersive readout. S Filipp, P Maurer, P J Leek, M Baur, R Bianchetti, J M Fink, M Göppl, L Steffen, J M Gambetta, A Blais, A Wallraff, Phys. Rev. Lett. 102S. Filipp, P. Maurer, P. J. Leek, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, J. M. Gambetta, A. Blais, and A. Wallraff. Two-qubit state tomography using a joint dispersive readout. Phys. Rev. Lett., 102:200402, May 2009. Tunable coupling scheme for flux qubits at the optimal point. O Antti, Yasunobu Niskanen, Jaw-Shen Nakamura, Tsai, Phys. Rev. B. 7394506Antti O. Niskanen, Yasunobu Nakamura, and Jaw-Shen Tsai. Tunable coupling scheme for flux qubits at the optimal point. Phys. Rev. B, 73:094506, Mar 2006. Circuit quantum electrodynamics in the ultrastrong-coupling regime. T Niemczyk, F Deppe, H Huebl, E P Menzel, F Hocke, M J Schwarz, J J Garcia-Ripoll, D Zueco, T Hummer, E Solano, A Marx, R Gross, Nat Phys. 610T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hummer, E. Solano, A. Marx, and R. Gross. Circuit quantum electrodynamics in the ultrastrong-coupling regime. Nat Phys, 6(10):772-776, October 2010. Dressed collective qubit states and the tavis-cummings model in circuit qed. J M Fink, R Bianchetti, M Baur, M Göppl, L Steffen, S Filipp, P J Leek, A Blais, A Wallraff, Phys. Rev. Lett. 10383601J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff. Dressed collective qubit states and the tavis-cummings model in circuit qed. Phys. Rev. Lett., 103:083601, Aug 2009. Deep strong coupling regime of the jaynes-cummings model. J Casanova, G Romero, I Lizuain, J J García-Ripoll, E Solano, Phys. Rev. Lett. 105263603J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano. Deep strong coupling regime of the jaynes-cummings model. Phys. Rev. Lett., 105:263603, Dec 2010. Sudden death of entanglement. Ting Yu, J H Eberly, Science. 3235914Ting Yu and J. H. Eberly. Sudden death of entanglement. Science, 323(5914):598-601, 2009. Sudden death of entanglement of two jaynescummings atoms. Muhammed Yonac, Ting Yu, J H Eberly, Journal of Physics B: Atomic, Molecular and Optical Physics. 3915621Muhammed Yonac, Ting Yu, and J H Eberly. Sudden death of entanglement of two jaynescummings atoms. Journal of Physics B: Atomic, Molecular and Optical Physics, 39(15):S621, 2006. Negative entanglement measure, and what it implies. Ting Yu, J H Eberly, J.MOD.OPT. 542289Ting Yu and J. H. Eberly. Negative entanglement measure, and what it implies. J.MOD.OPT., 54:2289, 2007. Understanding entanglement sudden death through multipartite entanglement and quantum correlations. H Jared, Cole, Journal of Physics A: Mathematical and Theoretical. 4313135301Jared H Cole. Understanding entanglement sudden death through multipartite entanglement and quantum correlations. Journal of Physics A: Mathematical and Theoretical, 43(13):135301, 2010. Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states. S Ashhab, Franco Nori, Phys. Rev. A. 8142311S. Ashhab and Franco Nori. Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states. Phys. Rev. A, 81:042311, Apr 2010. Generalized theory of relaxation. F Bloch, Physical Review. 1054F. Bloch. Generalized theory of relaxation. Physical Review, 105(4):1206-1222, Feb 1957. On the theory of relaxation processes. A G Redfield, IBM Journal of Research and Development. 11A. G. Redfield. On the theory of relaxation processes. IBM Journal of Research and Development, 1(1):19 -31, 1957. Dispersive regime of circuit qed: Photon-dependent qubit dephasing and relaxation rates. Maxime Boissonneault, J M Gambetta, Alexandre Blais, Phys. Rev. A. 7913819Maxime Boissonneault, J. M. Gambetta, and Alexandre Blais. Dispersive regime of circuit qed: Photon-dependent qubit dephasing and relaxation rates. Phys. Rev. A, 79:013819, Jan 2009. The theory of open quantum systems. Heinz-Peter Breuer, Francesco Petruccione, Oxford University PressHeinz-Peter Breuer and Francesco Petruccione. The theory of open quantum systems. Oxford University Press, 2003. On the generators of quantum dynamical semigroups. G Lindblad, 1976.10.1007/BF01608499Communications in Mathematical Physics. 48G. Lindblad. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48:119-130, 1976. 10.1007/BF01608499. Quantum optics. O Marlan, M Scully, Zubairy, Cambridge Univ. PressMarlan O. Scully and M. Suhail Zubairy. Quantum optics. Cambridge Univ. Press, 1999. Introductory quantum optics. C C Gerry, P L Knight, Cambridge University PressC.C. Gerry and P.L. Knight. Introductory quantum optics. Cambridge University Press, 2005. Superradiance and subradiance. i. interatomic interference and symmetry properties in three-level systems. A Crubellier, Liberman, P Pavolini, Pillet, Journal of Physics B. 18183811A Crubellier, S Liberman, D Pavolini, and P Pillet. Superradiance and subradiance. i. interatomic interference and symmetry properties in three-level systems. Journal of Physics B, 18(18):3811, 1985. Entanglement of formation of an arbitrary state of two qubits. William K Wootters, Physical Review Letters. 80William K. Wootters. Entanglement of formation of an arbitrary state of two qubits. Physical Review Letters, 80:2245-2248, Mar 1998. Optimal decoherence control in non-markovian open dissipative quantum systems. Wei Cui, Yu Zai Rong Xi, Pan, Physical Review A. 7732117Wei Cui, Zai Rong Xi, and Yu Pan. Optimal decoherence control in non-markovian open dissipative quantum systems. Physical Review A, 77:032117, Mar 2008. Measure for the degree of non-markovian behavior of quantum processes in open systems. Heinz-Peter Breuer, Elsi-Mari Laine, Jyrki Piilo, Physical Review Letters. 10321210401Heinz-Peter Breuer, Elsi-Mari Laine, and Jyrki Piilo. Measure for the degree of non-markovian behavior of quantum processes in open systems. Physical Review Letters, 103(21):210401, Nov 2009. Entanglement and non-markovianity of quantum evolutions. Ángel Rivas, Susana F Huelga, Martin B Plenio, Physical Review Letters. 105550403Ángel Rivas, Susana F. Huelga, and Martin B. Plenio. Entanglement and non-markovianity of quantum evolutions. Physical Review Letters, 105(5):050403, Jul 2010. Quantum fisher information flow and nonmarkovian processes of open systems. Xiao-Ming Lu, Xiaoguang Wang, C P Sun, Physical Review A. 82442103Xiao-Ming Lu, Xiaoguang Wang, and C. P. Sun. Quantum fisher information flow and non- markovian processes of open systems. Physical Review A, 82(4):042103, Oct 2010. A silicon-based nuclear spin quantum computer. Be Kane, Nature. 3936681BE Kane. A silicon-based nuclear spin quantum computer. Nature, 393(6681):133-137, MAY 14 1998. Silicon quantum computation based on magnetic dipolar coupling. J D Rogerio De Sousa, S. Das Delgado, Sarma, Physical Review A. 70552304Rogerio de Sousa, J. D. Delgado, and S. Das Sarma. Silicon quantum computation based on magnetic dipolar coupling. Physical Review A, 70(5):052304, Nov 2004. Coherence of single spins coupled to a nuclear spin bath of varying density. N Mizuochi, P Neumann, F Rempp, J Beck, V Jacques, P Siyushev, K Nakamura, D J Twitchen, H Watanabe, S Yamasaki, F Jelezko, J Wrachtrup, Physical Review B. 80441201N. Mizuochi, P. Neumann, F. Rempp, J. Beck, V. Jacques, P. Siyushev, K. Nakamura, D. J. Twitchen, H. Watanabe, S. Yamasaki, F. Jelezko, and J. Wrachtrup. Coherence of single spins coupled to a nuclear spin bath of varying density. Physical Review B, 80(4):041201, Jul 2009.
[]
[ "Oceanic frontal divergence alters phytoplankton competition and distribution", "Oceanic frontal divergence alters phytoplankton competition and distribution" ]
[ "Abigail Plummer \nDepartment of Physics\nHarvard University\n02138CambridgeMA\n\nDepartment of Mechanical and Aerospace Engineering\nPrinceton University\n08544PrincetonNJ\n", "Mara Freilich \nMIT-Woods Hole Oceanographic Institution Joint Program in Oceanography and Applied Ocean Science\nWoods Hole02543MA\n\nScripps Institution of Oceanography\nUniversity of California San Diego\n92037San DiegoCA\n", "Roberto Benzi \nDepartment of Physics and Istituto Nazionale di Fisica Nucleare\nUniversity of Rome Tor Vergata\n00133RomeItaly\n", "Chang Jae Choi \nMonterey Bay Aquarium Research Institute\n95039Moss LandingCA\n\nGEOMAR Helmholtz Centre for Ocean Research\n24148KielGermany\n\nMarine Science Institute\nUniversity of Texas at Austin\n78373Port AransasTX\n", "Lisa Sudek \nMonterey Bay Aquarium Research Institute\n95039Moss LandingCA\n", "Alexandra Z Worden \nMonterey Bay Aquarium Research Institute\n95039Moss LandingCA\n\nGEOMAR Helmholtz Centre for Ocean Research\n24148KielGermany\n", "Federico Toschi \nDepartment of Applied Physics\nEindhoven University of Technology\n5600 MBEindhovenThe Netherlands\n\nIstituto per le Applicazioni del Calcolo\nConsiglio Nazionale delle Ricerche\n00185RomeItaly\n", "Amala Mahadevan \nDepartment of Physical Oceanography\nWoods Hole Oceanographic Institution\nWoods Hole02543MA\n" ]
[ "Department of Physics\nHarvard University\n02138CambridgeMA", "Department of Mechanical and Aerospace Engineering\nPrinceton University\n08544PrincetonNJ", "MIT-Woods Hole Oceanographic Institution Joint Program in Oceanography and Applied Ocean Science\nWoods Hole02543MA", "Scripps Institution of Oceanography\nUniversity of California San Diego\n92037San DiegoCA", "Department of Physics and Istituto Nazionale di Fisica Nucleare\nUniversity of Rome Tor Vergata\n00133RomeItaly", "Monterey Bay Aquarium Research Institute\n95039Moss LandingCA", "GEOMAR Helmholtz Centre for Ocean Research\n24148KielGermany", "Marine Science Institute\nUniversity of Texas at Austin\n78373Port AransasTX", "Monterey Bay Aquarium Research Institute\n95039Moss LandingCA", "Monterey Bay Aquarium Research Institute\n95039Moss LandingCA", "GEOMAR Helmholtz Centre for Ocean Research\n24148KielGermany", "Department of Applied Physics\nEindhoven University of Technology\n5600 MBEindhovenThe Netherlands", "Istituto per le Applicazioni del Calcolo\nConsiglio Nazionale delle Ricerche\n00185RomeItaly", "Department of Physical Oceanography\nWoods Hole Oceanographic Institution\nWoods Hole02543MA" ]
[]
Oceanic phytoplankton populations, which play an essential role in regulating the carbon and oxygen in our atmosphere and oceans, are shaped by their fluid environment. Ocean currents can alter the diversity and distribution of these populations. However, little is known about the ecological impact of submesoscale flows (1-10 km), which are challenging to observe and model but have a number of features that suggest the potential for outsized biological impact. Here we combine a highresolution spatial survey of closely related cyanobacterial populations with simulations and theory. Observations reveal that the relative abundance of Synechoccocus oligotypes varies on 1-10 km scales at an ocean front with submesoscale velocity gradients at the same scale. Simulations using an ocean model demonstrate that regions of divergence in the horizontal flow field can substantially modify ecological competition and dispersal on timescales of hours to days. Regions of positive (negative) divergence provide an advantage (disadvantage) to local populations, resulting in up to ∼ 20% variation in community composition in our model. We propose that submesoscale divergence is a plausible contributor to observed taxonomic variability at oceanic fronts, and can lead to regional variability in community composition.
null
[ "https://export.arxiv.org/pdf/2202.11745v2.pdf" ]
247,084,385
2202.11745
b94e17dba674cfa97c99db8d6f7d00cde7e83384
Oceanic frontal divergence alters phytoplankton competition and distribution Abigail Plummer Department of Physics Harvard University 02138CambridgeMA Department of Mechanical and Aerospace Engineering Princeton University 08544PrincetonNJ Mara Freilich MIT-Woods Hole Oceanographic Institution Joint Program in Oceanography and Applied Ocean Science Woods Hole02543MA Scripps Institution of Oceanography University of California San Diego 92037San DiegoCA Roberto Benzi Department of Physics and Istituto Nazionale di Fisica Nucleare University of Rome Tor Vergata 00133RomeItaly Chang Jae Choi Monterey Bay Aquarium Research Institute 95039Moss LandingCA GEOMAR Helmholtz Centre for Ocean Research 24148KielGermany Marine Science Institute University of Texas at Austin 78373Port AransasTX Lisa Sudek Monterey Bay Aquarium Research Institute 95039Moss LandingCA Alexandra Z Worden Monterey Bay Aquarium Research Institute 95039Moss LandingCA GEOMAR Helmholtz Centre for Ocean Research 24148KielGermany Federico Toschi Department of Applied Physics Eindhoven University of Technology 5600 MBEindhovenThe Netherlands Istituto per le Applicazioni del Calcolo Consiglio Nazionale delle Ricerche 00185RomeItaly Amala Mahadevan Department of Physical Oceanography Woods Hole Oceanographic Institution Woods Hole02543MA Oceanic frontal divergence alters phytoplankton competition and distribution (Dated: October 6, 2022) Oceanic phytoplankton populations, which play an essential role in regulating the carbon and oxygen in our atmosphere and oceans, are shaped by their fluid environment. Ocean currents can alter the diversity and distribution of these populations. However, little is known about the ecological impact of submesoscale flows (1-10 km), which are challenging to observe and model but have a number of features that suggest the potential for outsized biological impact. Here we combine a highresolution spatial survey of closely related cyanobacterial populations with simulations and theory. Observations reveal that the relative abundance of Synechoccocus oligotypes varies on 1-10 km scales at an ocean front with submesoscale velocity gradients at the same scale. Simulations using an ocean model demonstrate that regions of divergence in the horizontal flow field can substantially modify ecological competition and dispersal on timescales of hours to days. Regions of positive (negative) divergence provide an advantage (disadvantage) to local populations, resulting in up to ∼ 20% variation in community composition in our model. We propose that submesoscale divergence is a plausible contributor to observed taxonomic variability at oceanic fronts, and can lead to regional variability in community composition. I. Introduction Phytoplankton form the base of the marine food web and mediate ocean uptake of carbon and oxygen. Therefore, phytoplankton abundance and diversity are important determinants of the health of the ocean, and quantitative theories of marine ecology are required to make precise predictions about our changing climate. Although many factors such as nutrients, sunlight, and temperature are well-established drivers of community composition, they are often not sufficient to explain the observed spatial patterns of variability in phytoplankton community structure [1,2] or high levels of plankton diversity [3,4]. Some of the unaccounted-for variability is likely introduced by the physical flows transporting planktonic organisms [5][6][7]. Oceanic flow fields shape marine ecosystems because plankton swim much slower than the speed of ocean currents [8][9][10][11]. At the mesoscale (10 km and larger), ocean currents are primarily horizontal-the typ-ical magnitude of horizontal velocity is O(10 −1 ) m/s while the typical magnitude of vertical velocity is O(10 −5 ) m/s. Lateral stirring and mixing by ocean currents disperses organisms, which can increase diversity [12][13][14][15][16][17]. However, at the submesoscale (1-10 km spatial scales), there are a number of processes, including waves and frontal dynamics, that can increase the magnitude of the vertical velocity to O(10 −3 ) m/s and velocity divergence to the same order as the planetary vorticity (the Coriolis frequency, f ). Despite recent progress in the field of submesoscale dynamics [18], the ways that these dynamics influence ecological interactions are only beginning to be understood. For example, submesoscale flows can lead to higher productivity and alter community composition on timescales of days by facilitating the exchange of nutrients and organisms between the dark ocean interior and the sunlit surface layer [19][20][21][22]. Along with increased vertical velocity, submesoscale flows also display increased horizontal velocity divergence [23,24]. This property of submesoscale flows may be especially relevant to phytoplankton ecology, as recent theoretical work in population genetics has shown that even weak horizontal velocity divergence can affect long-term competition outcomes between organisms [25,26]. In some oceanographic observations, regions of velocity convergence have been shown to result in accumulation of phytoplankton populations. For example, convergence has been observed to impact biological populations in a coastal region of the Western Mediterranean Sea [27] and in open ocean regions [28][29][30]. These convergence zones may hold particular biogeochemical significance because they disproportionately cause accumulation of buoyant phytoplankton such as nitrogen-fixers [29,30]. However, while convergence zones are known to cause passive accumulation, the unique role these flows can play in ecological processes has not yet been appreciated in oceanographic studies. In this report, we investigate the ecological significance of submesoscale velocity divergence. We focus on a scenario of neutral competition between phytoplankton subpopulations with the same growth characteristics in both our observations and our model in order to isolate the effects of the physical flow field. We begin by presenting observational evidence of unexplained variations in phytoplankton community composition at a front with regions of divergence in the Western Mediterranean Sea. We then discuss in more detail how a phytoplankton population in a stratified environment may experience a flow field with a nonzero divergence, generating an effective compressibility. We demonstrate the relevance of effective compressibility with simulations of a twodimensional model for competition between two plankton populations in a realistic oceanographic flow field initialized with observational data. Finally, we extend and generalize the theoretical model presented in Plummer et al. [25] to two dimensions and short timescales to explain the behavior observed in simulations. The combined observational, computational, and theoretical evidence suggests that horizontal velocity divergence contributes to measurable variations in community composition through modulation of ecological competition. We conclude by contextualizing our results in terms of selective advantage and discussing implications for future simulations and observations of ocean ecology. II. Observations Here we present exceptionally high spatial resolution observations in a region of velocity convergence in the Western Mediterranean Sea. These observations establish that measureable genetic variations are present at the submesoscale. This dataset is unprecedented in its combined biological and physical resolution with horizontal resolution of 1 km and the use of approaches for resolving diversity and taxonomy at high phylogenetic resolution. The velocity convergence occurs at a front that is generated by the confluence of water from the Atlantic Ocean with the warmer and saltier water of the Mediterranean Sea. This confluence leads to a fast-flowing current at the boundary of the two water masses ( Figure 1A). Instabilities develop at the 100 km scale of the front and at the submesoscale as the water masses attempt to vertically stratify, with the Mediterranean water sinking below the Atlantic water, leading to increased vertical velocity, relative vorticity, and divergence. For more details, see SI Sec. A. We observe convergence of the velocity on the sampling track with a maximum value of ∼ 2.5 × 10 −4 s −1 calculated along the ship track (Fig. 1B). The variability in velocity gradients on kilometer scales (the along track velocity switches from divergent to convergent on 4 km scale, SI Fig. S5) and velocity gradients of the same order as the Coriolis frequency indicates the prevalence of surface-intensified submesoscale dynamics. During this sampling, surface divergence of up to 1.3 × 10 −4 s −1 was observed with drifters in this region [31]. The observed surface mixed layer is approximately 10 m and water parcels would not be adiabatically far from the surface mixed layer in this location, suggesting that vertical nutrient fluxes do not explain the observed population distributions. The vertical motion associated with the observed divergence likely results in perturbations of the depth of the mixed layer. In these observations, even as populations may move tens of kilometers in the horizontal, they are restricted to move only a few meters in the vertical due to the density stratification that restricts adiabatic exchange of water masses between the surface and interior [31]. Sampling of the biological community composition and nutrient concentrations at the sea surface occurred on May 30, 2018. Sampling for cell enumeration and phylogenetic characterization [32] was performed every 1 km while surveying in a V-shape pattern crossing approximately perpendicular to both the front and a chlorophyll filament (Fig. 1A). In the region under study we found that Synechococcus was the most abundant phytoplankter with 9,080-22,700 cells/ml compared with just 630-2,000 cells/ml Prochlorococcus and 2,200-5,500 cells/ml eukaryotes (SI Fig. S7). 90% of the Synechococcus cells were comprised of Synechococcus IV (SI Fig. S6), an ecotype previously observed in coastal and relatively cool waters [33]. Using oligotyping approaches to further discriminate genetic variants within this ecotype using V1-V2 16S rRNA gene amplicon sequences we detected a total of 16 Synechococcus IV oligotypes. Oligotyping is a method of quantifying microdiversity in microbial populations. An oligotype is a population defined based on subtle variations in nucleotide sequences [34]. Microdiversity can more powerfully explain both biotic and environmental associations, suggesting that although genetically highly similar, ecotypes have unique functional roles [35][36][37][38]. Moreover, genetic variability within species can increase net productivity [39]. All oligotypes within the Synechococcus IV ecotype likely have similar gross growth characteristics [15,40,41], although they may vary in their ecological function and food web interactions, including differential impacts from viruses [42,43]. Synechococcus ecotypes in particular have been , we show the concentration of Synechococcus cells in surface waters (circle size) and the percent difference of the relative abundance of the ATTT oligotype of Synechococcus ecotype IV relative to the sample with the bold outline, based on V1-V2 16S amplicon relative abundance data (colors). The arrows indicate the velocity of the flow at the shallowest depth measured by the vessel-mounted ADCP (16 m). (C) Variability in community composition is not correlated with environmental factors. Percent difference in the relative abundance of the ATTT oligotype is plotted against nitrate concentration (C1) and temperature (C2). The colors correspond to the percent differences given for each point in (B). observed to encompass groups that have substantial ecological variation [44]. We find approximately 10% variation in the abundance of the most abundant Synechococcus IV oligotype in our survey (oligotype ATTT) relative to the abundance of all other observed Synechococcus IV oligotypes on the scale of the front (which, as noted above, comprise 90% of all Synechococcus cells measured and approximately 75% of all phytoplankton cells). The relative abundance of the ATTT oligotype does not correlate with nutrients or temperature (Figs. 1 C1, C2). The observed variations in relative abundance did not necessarily originate locally. We can use satellite measurements of the geostrophic velocity to estimate where our samples were ten days prior to our observations-this calculation is shown in SI Fig. S10, and reveals no obvious correlation between mesoscale stirring and submesoscale patterns of genetic variation. However, we were not able to make any Lagrangian measurements of diversity, such as following a single water parcel as it transited a region of divergence. Therefore, we treat the distribution of oligotypes as a single high-resolution snapshot of a community that exists in a region with strong and variable divergence of the horizontal flow field. We conclude that the distribution of the Synechococcus IV oligotypes cannot be understood in terms of the measured abiotic factors alone. Instead, we query whether the observed submesoscale divergence may impact the distribution of oligotypes. However, we also note that the sampling and this analysis does not control for some possible mechanisms for generating genetic variability, including variable predation or alterations in host-viral encounter rates. III. Model In order to assess whether regions of divergence could have contributed to the observed genetic variability, we develop a simplified model that couples population dynamics in two dimensions to a time-varying flow field for a 24-hour period. Our model of competition between two populations allows for selective advantage, though we focus on neutral competition (populations have equivalent birth, death, and resource utilization rates). While the observations inform and motivate the model assumptions, due to the complexity of this system and limitations on data collection, we do not attempt to model all aspects of the observed system. Instead, by using a general, coarse-grained biological model, our study reveals broadly applicable principles driving ecological changes in frontal regions. A. Biophysical assumption: effectively 2D populations During the 24-hour period examined here, we assume that phytoplankton populations are restricted to live in a particular depth range within the three-dimensional flow field. This assumption is a reasonable approximation for several scenarios. The most intuitive scenario is positively buoyant organisms that cannot be subducted away from the sea surface [45,46]. Alternately, some organisms are restricted to live in narrow subsurface depth ranges due to motility and light dependent growth and predation [47]. The size and buoyancy characteristics of phytoplankton populations affect their depth ranges due to the impact of viscosity and physiological characteristics such as gas vesicles [48]. In our observations, it may be reasonable to model populations as restricted to a particular depth range because there is a strong depth partitioning of the community structure with reduced abundance of Synechococcus IV below the shallow mixed layer (SI Fig. S8). Of course, there are also many other important scenarios in which the assumption of a population restricted to a surface does not hold [22]. As discussed extensively in past work (see, e.g., [25,26,[49][50][51][52][53][54][55]), the insight that some populations are restricted to remain close to a particular depth surface is consequential because such populations can experience a velocity field with nonzero divergence. For a concrete example, we again consider positively buoyant organisms at the sea surface experiencing an upwelling-organisms are spread apart by an effectively compressible flow. This argument can be extended to organisms that experience a force confining them to a sub-surface depth [51]. Additionally, while we simulate a depth-restricted population for this study, the results can also be applied to populations restricted to density surfaces. Organisms may regulate their buoyancy to stay near a particular density surface [56]. In the models used here, the distribution of divergence for the along-isopycnal flows is similar to the distribution of divergence for flows on depth surfaces, with slightly weaker extreme values (SI Sec. C). Applying this assumption of a depth-restricted community, we consider two-dimensional populations experiencing a compressible flow field in our modeling. Exploring this simplified model allows us to isolate and quantify the impact of divergence on ecological competition. B. Model flow fields To generate realistic two-dimensional velocity fields for our model, we take horizontal slices from 3D nonhydrostatic ocean models, initialized with the hydrographic structure observed at the Almería-Oran front (SI Sec. A). Two models are used, one with a shallow mixed layer, as in the observations from May (called the "summer" model), and one in which the mixed layer has been deepened to generate surface-enhanced submesoscale dynamics (called the "winter" model). In the winter model the deep reaching front outcrops at the surface but in the summer model the surface layer is stratified and the density front does not outcrop. Using 24-hour periods from two different model runs allows us to examine a wider range of oceanographic conditions. A 24-hour period is long enough that we can observe population growth given the generation time used in simulations. C. Biological model: population dynamics We use a general biological model that can serve as a starting point for understanding a wide range of competition scenarios [57]. Specifically, we consider two populations, A and B, that compete with one another while being passively advected. We use the term "population" to refer to a group of organisms that shares common niche, competition, and growth characteristics (e.g. a species, ecotype, amplicon sequence variant, or oligotype). In terms of the observations of variations in the relative abundance of a Synechococcus IV oligotype reported in the previous section, population A would be the ATTT oligotype, the most abundant oligotype observed in the transect (approximately 10% of all Synechococcus IV cells), and population B would be all other oligotypes combined (approximately 90% of all Synechococcus IV cells). The population dynamics are modeled as a reactionadvection-diffusion system with logistic growth [57][58][59][60]. The coupled partial differential equations that describe this system can, for example, be derived by coarsegraining agent-based birth and death processes [54] and neglecting the noise terms due to the large population sizes (N 10 4 cells per mL in observations). These equations are ∂c A ∂t + ∇ · (uc A ) = D∇ 2 c A + µc A (1 − c A − c B ) + sµc A c B ,(1)∂c B ∂t + ∇ · (uc B ) = D∇ 2 c B + µc B (1 − c A − c B ) − sµc A c B .(2) Here, c A (x, t) and c B (x, t) describe the concentration of the population at position x as a fraction of the local carrying capacity of the respective population in the absence of competition and advection (i.e., when ∂c A /∂t = 0 with u = 0 and c B = 0). We note that the total concentration, c A + c B , is not required to be constant [54]. The diffusivity, D is assumed to have the same value as the carrier fluid and is 1 m 2 /s, unless otherwise noted, u(x, t) is a (compressible) two-dimensional velocity field, and µ is the growth rate when either population is dilute, set to 1 day −1 to approximate the growth rates of Synechococcus [61] unless otherwise noted. The parameter s is the selective advantage of population A-population A has a selective advantage s over population B due to differences in competition under crowded conditions when s > 0. Motivated by our observations of variations in the relative abundance of oligotypes which we assume to be neutral competitors, as well as our desire to isolate the effects of flow from the effects of selection, we set s = 0 for the results presented in the main text. Thus, populations A and B will behave identically when they are uniformly distributed and/or in the absence of flow. Selective advantage terms of various forms can be straightforwardly studied using this model and provide an intuitive means of quantifying divergence strength-this is discussed in Sec. IV C and described in detail in SI Sec. F, G, and H. D. Biological model: initial conditions In the main text, we analyze the following numerical experiment: We initialize the system such that population A is localized according to a Gaussian distribution centered on a particular x, y coordinate with a standard deviation of 4 km. We set the concentration of population B such that c B = 1 − c A everywhere. Therefore, the total concentration is everywhere equal to the equilibrium carrying capacity in the absence of flow. Each dot in Fig. 2 A,D represents an independent trial initialized in this manner, with the Gaussian population centered at a different location. This initialization allows us to systematically probe any advantage imparted by different areas of the dynamic flow field. Different types of initial conditions, including spatially extended communities, are examined in SI Sec. H. We evolve Eqs. 1 and 2 forward in time in the presence of the flow field, and measure changes in the distribution and proportion of c A and c B after a 24-hour period. Since we are working in the weak compressibility regime (see SI Sec. D for further discussion of this point), the total concentration will remain close to the carrying capacity value as time proceeds (c A + c B ≈ 1). E. Quantifying community change We next provide definitions and conceptual discussion of the two metrics we use to track local and regional changes in community composition. Change in relative abundance State of the art high throughput sequencing technologies quantify microbial community composition using relative abundance [62][63][64]. The relative abundance of population A in a total pop-ulation composed of both A and B is defined locally at every point x as f (x, t) = c A (x, t) c A (x, t) + c B (x, t) .(3) Normalizing by the initial relative abundance, f 0 , the change in the spatially-averaged fraction after time τ is defined as ∆ f f 0 = f (t = τ ) − f (t = 0) f (t = 0) ,(4) where brackets denote spatial averages. The change in the spatially-averaged relative abundance measures whether population A becomes more widespread relative to population B after a time τ . Tracking changes in the relative abundance includes the effects of both dispersal and growth/competition, and provides a local measurement of diversity related to αdiversity (the number of distinct populations within a local habitat) [19,65]. Change in global fraction We can gain more information about a population and its regional-scale influence if we also track changes in its biomass. Population A is successful on average over the whole region if its size (i.e. the number of A organisms) increases relative to that of population B. We define F avg as the fraction of the total biomass in population A over the whole domain, which we call the global fraction. F avg (t) = c A (x, t) c A (x, t) + c B (x, t) ,(5) where brackets denote spatial averages. Normalizing by the initial value, the change in this global fraction after a time τ is defined ∆F avg F avg 0 = F avg (t = τ ) − F avg (t = 0) F avg (t = 0) .(6) The change in the global fraction can only be nonzero when the growth rate µ is nonzero, and is unaffected by mixing within the domain. Therefore, tracking the global fraction allows us to evaluate if divergent flows affect the competition between populations and allow for differential growth. The global fraction is a global measurement of diversity related to γ-diversity (the diversity in a broader region). Resolving the absolute abundance is necessary to calculate the global fraction but this measurement is less commonly possible in microbial ecology. IV. Results We briefly summarize the main theoretical and computational results of this report. The community composition can be significantly affected by regions of divergence in a flow field over a 24-hour period. The regions of growth and decay of population A display similar spatial patterns to the regions of divergence in the flow (Fig. 2 A,D). The flow provides an advantage to some subset of organisms over others, which we quantify with a general theory for the influence of weakly compressible flows on community composition. Regions of positive divergence disperse organisms, locally decreasing competition and stimulating growth. Effective compressibility affects both the local community composition -quantified as the relative abundance (Eq. 4) -and regional community composition -quantified as the global fraction (Eq. 6) -through growth, competition, and dispersal. The regions of positive and negative divergence lead to ∼ ±20% changes in the relative abundance of population A without appreciably changing the total biomass (SI Sec. D). Alternately, we quantify the magnitude of this effect by measuring the ability of the flow to counteract a selective disadvantage; we find the strongest regions of positive divergence in our velocity fields allow a population with up to a 65% selective disadvantage to increase in relative abundance. We note that non-divergent flows cannot alter spatially averaged relative abundances or the global fraction in a closed system at its carrying capacity (c A + c B = 1) in the absence of noise and selection (SI Sec. E). Therefore, any growth or suppression of population A observed in simulations can be attributed to the effect of the regions of divergence. A. Local influence of divergence We find that changes in the spatially-averaged relative abundance have a linear relationship with the divergence experienced by that population, averaged over space and time (Fig. 3A). The linear relationship between relative abundance and divergence holds even when the organisms are not able to reproduce (µ = 0). In this case, changes in the relative abundance are solely due to dispersal. For an intuitive example of how Eq. 4 can be nonzero in the absence of growth, consider a region of positive divergence that is occupied solely by a non-reproducing population A. The flow will distribute A organisms throughout the system, increasing the relative abundance of A outside the source region. At the source itself, the relative abundance will remain locally equal to 1 as long as no B organisms are introduced, despite the local depletion in the amount of A organisms. Thus, the spatially averaged relative abundance can increase or decrease even when there is no growth. When µ = 0, the local depletion will be compensated by growth at the source, maintaining an approximately uniform distribution of biomass. Since the µ = 0 trials follow the same trend as the µ = 0 trials, we conclude that the observed increase in relative abundance over this 24-hour time period is primarily due to dispersal, rather than differential growth. Relative abundance in the weak compressibility limit We can calculate the relationship between velocity divergence and changes in the relative abundance in the weak compressibility limit. From the equations for the evolution of the concentrations of populations A and B (Eqs. 1 and 2), we obtain an equation for the evolution of the relative abundance of A (SI Sec. E). ∂f ∂t + u · ∇f = D∇ 2 f + 2D c ∇f · ∇c + sµcf (1 − f ), (7) where c(x, t) = c A (x, t) + c B (x, t) is total concentration, expressed as a fraction of the local carrying capacity. In this equation, the growth rate µ only appears directly in a logistic competition term, and implicitly as the relaxation rate of c. The second term on the right hand side of Eq. 7 is small for the case of a weakly compressible flow (SI Sec. C and [25]). Therefore, by integrating by parts and setting c = 1 in the selection term, the rate of change of the relative abundance integrated over space in a weakly compressible flow can be approximated ∂ ∂t Ω f dΩ ≈ Ω [f ∇ · u + sµf (1 − f )] dΩ,(8) where Ω is the area of the 2D domain. Note that the boundary terms can be neglected when population A is localized, as in our simulations (SI Sec. E). To compare the behavior of this equation with simulations, we integrate with respect to time and divide both sides by f 0 Ω ≡ Ω f (t = 0)dΩ. 1 Ω Ω f (t = τ )dΩ − f 0 f 0 ≈ τ 0 1 Ω Ω f f 0 ∇ · u + sµ f f 0 (1 − f ) dΩ dt. (9) For the case of neutral competition, we set s = 0, and Eq. 9 reduces to ∆ f f 0 ≈ τ f 0 f ∇ · u ,(10) where the brackets denote averages over all space, and the overbar denotes an average over the time interval t = 0 to t = τ . Agreement between simulations and theory There is excellent agreement between the theory (Fig. 3A, black line) and simulations for different flow fields, growth rates, and diffusivities (a diffusivity greater than the carrier fluid diffusivity could represent, for example, active dispersal). As expected from Eq. 10, there is no obvious dependence on the growth rate µ or diffusivity D. Over longer time periods we would expect that the results of the experiments with µ > 0 to differ more from those with µ = 0. Concentration gradients, ∇c, will become large for the µ = 0 simulations, violating the assumption of weak compressibility. Sufficiently large growth prevents the development of strong gradients in concentration. We note that we display relatively fewer data points for strong negative divergence, as these trials were most susceptible to numerical instability. B. Regional influence of divergence We examine the relationship between divergence and changes in the global fraction to disentangle the effects of dispersal from the effects of growth and competition. We find that changes in the global fraction have a linear dependence on the divergence experienced by that population (Fig. 3B), although the slope of the trend is smaller than for changes in the relative abundance. This discrepancy would not occur if c A + c B were strictly equal to 1 everywhere, in which case the global fraction and spatially-averaged relative abundance would be identical. However, even when c A + c B ≈ 1, as is the case for the weakly compressible flows considered here, there can be significant differences between these two measures. Even if on average the domain is uniformly occupied, especially strong convergences and divergences cause small local accumulations and deficits, which must be taken into account to understand the change in the global fraction, ∆F avg /F avg 0 , and the influence of competition and growth on changes in population distributions. As a result of these concentration fluctuations, the global fraction has a weaker dependence on the divergence than does the relative abundance. Fluctuations about the weak compressibility limit To understand how the global fraction can differ from the spatially-averaged relative abundance, we consider the equation for the change in the total concentration (SI Sec. E). ∂c ∂t + ∇ · (uc) = D∇ 2 c + µc(1 − c).(11) We model a small fluctuation in the total concentration, setting c = 1 + and assume that the growth is much larger than the divergence (µ ∇ · u). We neglect the time derivative as in ref. [52], and drop terms proportional to ∇ and (∇ · u) to obtain ≈ − 1 µ (∇ · u).(12) With this approximation, the spatially integrated relative abundance becomes Ω f dΩ ≈ Ω c A 1 − 1 µ (∇ · u) dΩ ≈ Ω c A + c A (∇ · u) µ dΩ. (13) Since we expect c A to depend on ∇ · u when c is allowed the fluctuate, the second term will not integrate to zero. Upon substituting this expression in to Eq. 8, integrating with respect to time, noting F avg (t) ≈ Ω c A dΩ/Ω, and taking the case of no selective advantage for simplicity, we find ∆F avg F avg 0 ≈ τ F avg 0 f ∇ · u − ∆ c A ∇ · u µF avg 0 ,(14) where ∆ indicates a difference between the initial and final time points. By modeling a fluctuation in the total concentration, we thus observe that the change in the global fraction, unlike the change in relative abundance, has an explicit dependence on µ that goes to zero when µ → ∞, at which point c is strictly equal to 1. Due to the approximations made, this relation breaks down for small µ. Agreement between simulations and theory We observe a µ dependence in the relationship between the global fraction and the weighted divergence, as expected from the fluctuation model of Eq. 14, but no clear dependence on the flow field (winter vs. summer) or diffusivity (Fig. 3B). There is no change in the global fraction when there is no growth (µ = 0), as in that case all covariance between divergence and c A is due to accumulation. Higher values of µ produce trends closer to the one-to-one relationship of Fig. 3A, as expected. The slopes of the lines of best fit for each set of parameters are given in the figure legend. Due to the approximations made in the fluctuation model, Eq. 14 cannot be used to quantitatively predict these slopes. C. Influence of selection The advantage provided to a population localized in a region of positive divergence can be directly compared to the advantage provided to a population by having a positive selective advantage. We explore the relationship between regions of divergence/effective compressibility and selective advantage in SI Secs. F, G, and H. The advantage provided by the strongest regions of positive divergence in our flow fields (the maximum effect size under idealized conditions) is equivalent to approximately a 65% selective advantage. Regions of positive divergence not only alter the outcomes of neutral competition but also allow populations with a selective disadvantage to increase their relative and global abundance. Numerical experiments that do not assume neutral competition with both idealized Gaussian initial conditions, as above, as well as spatially extended initial conditions based on commonly observed plankton biogeography are described in SI Secs. G and H and are consistent with theoretical results of SI Sec. F. V. Discussion Motivated and informed by observations, our coupled biophysical model considers realistic oceanic flow fields, resolved at the submesoscale, acting on phytoplankton populations restricted to live within a particular depth range. Phytoplankton living at regions of positive divergence enjoy the advantage of having would-be competitors constantly swept away by the flow, allowing offspring to easily spread. Those living at regions of negative divergence are instead challenged by a stream of new arrivals. The realistic oceanic flow fields are in a regime where the effect of divergence is primarily dispersal rather than loss of biomass [25,53], resulting in variations in relative abundance of up to 35% over one generation. Nonetheless, the effects of compressibility also affect growth and competition, resulting in variations in the global fraction of up to 20%. These simulated trends are consistent with theoretical expectations and of the same magnitude as the observed variations in community structure at a front in the Mediterranean Sea. The effects of divergence are integrated over time and are therefore stronger when a population resides in an area of positive divergence for a longer time. In these simulations, the summer flow field has a simpler divergence structure which leads to larger divergence when integrated over a day. The results suggest a mechanism that can affect plankton biogeography (SI Sec. H), alongside other established mechanisms such as fluctuations in light, temperature, and ecological interactions [66]. The impact of effective compressibility may vary on long space and time scales because velocity divergence patterns display seasonality and regional variation [67,68]. Our calculations and simulations assume a uniform nutrient distribution-if nutrients had been modeled explicitly, the advantage afforded by regions of positive divergence could be enhanced at the surface, with organisms born in these regions experiencing an even greater advantage due to the associated upwelling supplying nutrients [69]. We note that all of the effects discussed in this work would also arise in the more general oceanographic case where organisms are restricted to remain close to a fixed density surface rather than a fixed depth (realistic for organisms that may regulate their buoyancy, for example [70]). This scenario is discussed in SI Sec. C. The numerical experiments highlight that effective compressibility, unlike many mechanisms by which advection can affect competition, can be relevant even when populations are ecologically neutral (equally matched competitors) in the absence of a flow. Neutral theories in ecology emphasize the role of stochasticity and dispersal on population dynamics. These dynamics can be consequential because even if plankton types are neutral in their competition under the conditions at a given moment in time, they may differ in other ways, which means that the outcomes of the neutral competitions have biogeochemical implications [71]. The biogeochemical implications of competition in the presence of effective compressibility are likely most profound for biogeochemical processes carried out by positively buoyant populations, such as nitrogen fixers [29,30,72]. Incorporating divergence of the flow field into analysis of ecological processes may have relevance beyond the processes studied here. For example, positively buoyant artificial particles such as microplastics may accumulate in regions of convergence either at the mesoscale [73] or at the gyre scale [74]. These microplastics have microbial communities associated with them [75]. However, the framework of weak compressibility only applies when population growth is sufficiently fast to maintain an approximately uniform total concentration. Applying results from strong compressibility to populations on buoyant artificial particles may be a fruitful avenue for future research. Effective compressibility can either promote or suppress diversity, depending on the population structure. In SI Sec. G, for example, we demonstrate that a positive divergence can compensate for a competitive disadvantage. If rare species often occupy regions of positive divergence (for example, if they are brought to the surface by an upwelling event), effective compressibility should increase diversity. If instead rare plankton populations are drawn to downwellings, where their populations are more likely to shrink, diversity will be suppressed. The proposed mechanism and results presented here cannot be quantitatively validated using the existing observations. However, the sign and magnitude of the change in the community composition between the two observed transects is close in magnitude to the convergence observed in the transect in the Western Mediterranean Sea, supporting the idea that divergence could alter the community composition. The overall convergence is approximately 2 × 10 −5 s −1 (0.25f ) and the decrease in the relative abundance of the population from the upstream to the downstream transect (advection time of ∼ 3 hours between the transects) is 5-10%. More rigorous observational confirmation requires Lagrangian observations of microbial diversity, which will likely be technologically feasible in the near term. Observations should quantify divergence, population growth and selective advantage, and ecological effects. The processes discussed here are likely most important for positively buoyant populations and populations that are depth-stratified through other mechanisms in frontal regions where divergence is relatively large. VI. Conclusions At the scale of ocean fronts, phytoplankton can experience weakly divergent flows that disperse plankton populations and alter competition and growth. Both simulations and theory support the conclusion that regions of divergence significantly affect both the spatially-averaged relative abundance as well as the global fraction (when the growth rate is nonzero), though the details of the relationships differ. Regions of positive divergence support local populations, while regions of negative divergence suppress them. The regions of divergence in the flow fields examined here can lead to differences in relative abundances of up to 35% in ecologically neutral populations over a 24-hour period, which is consistent with the magnitude of population change in observations. These divergent flows provide an effective selective advantage of up to 65%. The effect of divergence is most likely to be a dominant driver of demographic change in locations with strong divergence, which occur over timescales of hours to days, and when oceanographic or physiological factors confine organisms to a given depth range. Divergence (i.e. effective compressibility) should be considered as a potential additional explanation for patchiness in community composition. Methods Samples were collected on May 30, 2018 from 20:00 UTC to 22:45pm UTC while underway from the sea surface using an oceanographic bucket. DNA samples were obtained by filtering 500 ml seawater through 47 mm 0.2 µm pore size polyethersulfone membrane filters (Supor 200, Pall Gelman). DNA was amplified targeting the V1-V2 hypervariable region of the 16S rRNA gene and paired-end library sequencing (2 × 300bp) was performed using the Illumina MiSeq platform (Illumina) [32]. After demultiplexing, merging reads, and quality control, cyanobacterial amplicons were parsed using the phylogenetic pipeline in PhyloAssigner v.6.166 [76] and then further classified using fine-scale cyanobacterial reference alignment and tree [32] according to protocols outlined in [77]. Oligotyping was then performed in Qiime using oligotyping pipeline version 3.1 on 108,088 reads classified as Synechococcus IV specifying 4 components [34]. Synechococcus abundance was quantified on preserved samples using a BD Influx flow cytometer. In Fig. 1, percent difference = fi−f0 f0 × 100 where f i is the abundance of the ATTT oligotype relative to all of the sequences identified as Synechococcus IV in sample i and f 0 is the same quantity in the reference sample. See SI Sec. B for additional details on biological sample analysis. Simultaneous with the biological sampling, we measured the depth structure of temperature and salinity with a towed profiler, the surface temperature and salinity with a thermosalinograph on the ship seawater intake, and the water velocity with a vessel-mounted acoustic Doppler current profiler (ADCP). To construct realistic oceanographic flow fields, we initialize the Process Study Ocean Model (PSOM) with density sections derived from observations of the Almería-Oran front sampled by a glider at 1 km horizontal resolution during the July 2017 IRENE research cruise. The boundaries of the data are interpolated to form an idealized domain that is 128 km by 206 km by 1 km in extent with a horizontal resolution of 500 m. This flow field has hydrographic and velocity gradient structure that is statistically similar to the summer season. This model summer flow field has a 5 m deep mixed layer that is lighter than any interior density surface, effectively isolating the surface from the interior. To construct the second initial condition, we deepen the mixed layer by cooling the surface and recomputing the surface density profile using convective adjustment until the maximum mixed layer depth is 70 m. This process leads to a flow field characteristic of the winter season. The winter model has a more active surface-enhanced submesoscale flow field, which results in smaller scale features in the velocity gradients [78]. The model is periodic in the east-west direction (parallel to the front) and has closed walls in the north and south. To perform simulations that couple the flow fields with the biological variables, we select 2D slices from the 3D PSOM fields. We use the surface layer in the summer model, and a slice at a depth of 52 m in the winter model, which is near the base of the mixed layer. See SI Secs. A and C for more information on the flow fields and further discussion of the validity of the constant depth approximation. We simulate the evolution of up to 1056 (33×32) initial conditions centered at different locations in the domain over a 24-hour period, all experiencing the same velocity field (Fig. 2). We simulate the population concentrations offline by stepping Eqs. 1 and 2 forward with a second-order Adams-Bashforth scheme and linearly interpolating the flow fields in time from model snapshots saved every 3 hours. The spatial derivatives in the diffusion operator are discretized using a central second-order finite-difference method. Acknowledgments We thank David R. Nelson and John Toner for useful discussions, Camille Poirier and Sebastian Sudek for assistance with biological sample processing, Eva Alou and Andrea Cabornero for providing the nutrient samples, John Allen for processing the VM-ADCP observations, Eric D'Asaro for serving as co-chief scientist of the research cruise, Mathieu Dever, Sebastian Essink, Kausalya Mahadevan, and Alex Beyer for sampling assistance at sea, and the captain and crew of the NRV Alliance for their assistance and expertise. Funding was provided by a Montrym grant and Martin Fellowship from MIT. contributed new reagants or analytic tools; A.P. and M.F. analyzed data; A.P. and M.F. wrote the paper; A.P., M.F., R.B., A.W., F.T., and A.M. revised the paper. Author Contributions Data Availability Upon publication, the model output will be made available with a DOI from Zenodo. The sequences will be available with BioSample accession numbers SAMN28021319-SAMN28021334 Code Availability The code to generate the oceanography flow fields is available at https://doi.org/10.5281/zenodo.3902273 (PSOM v1.0 with initial conditions in the released code). Upon publication, the biological model code will be made available with a DOI from Zenodo. The Almería-Oran front is a persistent density front between the Mediterranean and Atlantic water masses in the Western Mediterranean. The front is modified by instabilities at the mesoscale and submesoscale, which generate regions of divergence and convergence. There is a strong along-front flow as well as an ageostrophic cross-front flow [28]. Model To generate oceanographic flow fields, the Process Study Ocean Model (PSOM) [29,30] is initialized in thermal wind balance with hydrographic sections of salinity and temperature from Almería-Oran front observations. The model horizontal resolution is 500 m, except near the closed north and south walls where the cell length increases linearly to 2 km. The model is evolved with a horizontal diffusivity of 1 m 2 /s and a vertical diffusivity of 10 −5 m 2 /s. The flow fields develop meanders and smaller scale divergent features, which are mostly localized at the front. We wait until the total kinetic energy of the system has reached a steady state and record the 3D velocity fields. A snapshot of a 3D simulation can be found in Ref. [31]. The flow fields are dominated by an eastward flowing jet that is more variable in the winter model than in the summer model. The mean jet speed is therefore roughly twice as fast in the summer as in the winter (Fig. S1). Over a 24-hour period, the frontal jet present in the winter flow field moves a localized population at most a horizontal distance corresponding to approximately 14 % of the domain, and the summer flow field moves the population at most a distance of approximately 28 % of the domain. The large scale structure of the jet dictates the locations of divergent regions in the summer model (Fig. 2F). The regions of divergence in the winter model are also influenced by the meandering jet location, but have finer scale structure as well (Figs. 2C and S2). Given this spatial structure, the divergence has higher power spectral density in the summer model than the winter model at large spatial scales (low wavenumber) but falls off more rapidly such that the divergence has higher power spectral density at intermediate and small spatial scales in the winter model (Fig. S3). Observations Throughout the research cruise the depth structure of the hydrography and biogeochemistry were continuously being surveyed with an EcoCTD [32] and vessel-mounted ADCP. In the location where water samples were taken, the surface is stratified and has nearly uniform density across the section while the density contours slope downwards forming a front subsurface (Fig. S4). There are intrusions of high oxygen water as a result of vertical motion. There is a noticeable velocity convergence on the upstream (southern) leg of the observational transect (Fig. S5). DNA samples for characterizing the microbial community were obtained by filtering 500 ml seawater through 47 mm 0.2 µm pore size polyethersulfone membrane filters (Supor 200, Pall Gelman). Filters were placed into sterile cryovials, flash-frozen in liquid nitrogen where they were stored for the remainder of the research cruise. After the cruise, samples were stored at -80 • C until analysis. Sample DNA was extracted with a DNeasy Plant Kit (Qiagen), with a modification including a bead beating step [33]. DNA was amplified using the primers 27FB (5 -AGRGTTYGATYMTGGCTCAG-3 ) and 338RPL (5 -GCWGCCWCCCGTAGGWGT-3 ) as in [34,35] targeting the V1-V2 hypervariable region of the 16S rRNA gene with Illumina adapters. PCR reactions contained 25 ng of template, 5 µl of 10× buffer, 1 U of HiFi-Taq, 1.6 mM MgSO4 (Thermo Fisher) and 0.2 µM of each primer. The PCR cycling parameters were 94 • C for 2 min; 30×94 • C for 15 s, 55 • C for 30 s, 68 • C for 1 min, and a final elongation at 68 • C for 7 min. Paired-end library sequencing (2 × 300bp) was performed using the Illumina MiSeq platform (Illumina). Sequences were demultiplexed and assigned to samples using CASAVA (Illumina). A 10 bp running window was utilized to trim low-quality sequence ends at a Phred quality (Q) of 25 using Sickle 1.33 [36]. Paired-end reads were merged using USEARCH v10.0.240 [37] when reads had a ≥ 50 bp overlap with maximum 5% mismatch. The merged reads were then filtered to remove reads with maximum error rate > 0.001 or shorter than 200 bp. Only sequences with exact match to both primers were kept and primer sequences were trimmed using Cutadapt v.1.13 [38]. Cyanobacterial amplicons were initially parsed using the phylogenetic pipeline in PhyloAssigner v.6.166 [34] and then further classified using fine-scale cyanobacterial reference alignment and tree [35] according to protocols outlined in [39]. Oligotyping was then performed on aligned and trimmed samples in Qiime using oligotyping pipeline version 3.1 on 108,088 reads classified as Synechococcus IV specifying 4 components. This resulted in 16 oligotypes which represent 99.88% of all reads with a purity score of 1.0 [40]. Samples for quantifying the cyanobacteria abundance were preserved with EM grade 25% Glutaraldehyde (10 µl per 1 ml seawater). Samples were placed in sterile cryovials and flash frozen in liquid nitrogen and then stored in liquid nitrogen for the remainder of the research cruise after which they were stored at -80 • C until analysis. Samples were analyzed using a BD Influx flow cytometer equipped with a 488 nm laser. Calibration beds were added to each sample CSIC). Distribution of Synechococcus The phytoplankton community is dominated by a single ecotype of Synechococcus, clade IV. Synechococcus clade IV consistutes 89-92% of the Synechococcus sequences (Fig. S6). Previous observations suggest that Synechococcus IV is commonly the dominant Synechococcus in surface waters in the Alborán Sea [41]. Quantitatively, Synechococcus is the dominant biological population in these surface samples with 9,080-22,700 cells/ml compared with just 630-2,000 cells/ml Prochlorococcus and 2,200-5,500 cells/ml eukaryotes (Fig. S7). Although its abundance is relatively constant, the two easternmost samples have noticeably higher Synechococcus abundance (Fig. S7). There are 16 oligotypes of Synechococcus clade IV and although they are fairly evenly distributed, there is one oligotype that has the highest relative abundance across all samples, the ATTT oligotype (Fig. S6). Fig. S8 is provided for interpretation of the geographic locations of the samples in the two previous figures. While only the surface was sampled on this particular transect, in other samples taken on the same research cruise in nearby locations, Synechococcus populations are at highest abundance in the upper 50 m of the water column and decrease in abundance below that depth. In other oceanographic observations, dispersal and lateral mixing by mesoscale ocean currents has been shown to play an important role in shaping the distribution of phytoplankton populations [42]. To explore the influence of dispersal, and in particular of interleaving of nearby populations, on the spatial distribution of the observed populations, we trace the origin of the sampled populations backwards in time for 10 days using geostrophic velocity fields estimated from satellite altimetry (Fig. S10). These velocity fields have 1/4 • resolution. The sampled high chlorophyll filament appears to have originated 10 days earlier near the Spanish coast in a region with frequent upwelling [43]. The water masses were then advected by the Western Alborán Gyre and reached the sampled location on the western edge of the Eastern Alborán Gyre. Although there is some evidence of geostrophic straining with the populations converging on the edge of the Western Alborán Gyre, there is not evidence of interleaving. Instead, the water masses appear to have followed nearly parallel trajectories through the flow field. Interleaving could occur due to currents that are below the resolution of the altimeter or as a consequence to divergent currents that are not included in estimates of velocity from altimetry. C. Constant depth approximation In the main text, we simulate competition on constant depth fluid surfaces. We could have instead modeled competition on constant density, or isopycnal, surfaces. We would not expect this to weaken the impact of effective compressibility on the ecology, since the distribution of divergence on relevant isopycnal surfaces is similar to the distribution of divergence on the corresponding constant depth surfaces, as we discuss below. However, depth variations of isopycnal surfaces also lead to variations in light, which affect the carrying capacity of the fluid. Simulating the biological model on constant depth surfaces allows us to better isolate the effects of divergence. We present data for the flows tangent to isopycnal and constant depth surfaces in Fig. S11A. We compare the distribution of divergence on an isopycnal surface from the winter flow field at a density σ = 27.9 kg/m 3 , pictured in Fig. S11B, to the divergence on a fixed depth surface at the average depth of that isopycnal surface. Divergence on an isopycnal surface is calculated by first transforming the 3-dimensional velocity vector from depth to isopycnal coordinates then by computing the (2-dimensional) divergence of the flow the isopycnal surface. We see that the distributions are similar. The isopycnal surface used for comparison in Fig. S11 is on average deeper than both the winter flow field constant depth surface (52 m) and the summer flow field constant depth surface (0 m) used in the main text. We do not show isopycnal surfaces with these average depths because they outcrop at the sea surface. Instead, we note that isopycnals with shallower average depths in these models also span a smaller depth range and in general have even stronger regions of divergence due to the larger divergence at the sea surface. D. Weak compressibility regime The strength of the effective compressibility experienced by a population depends on both the magnitude of the divergence and the growth characteristics of the population, since rapidly reproducing phytoplankton can maintain a constant density even in the presence of a positively divergent flow [44]. In the main text, we often assume that we are in the weak compressibility regime, and the steady state concentration profile is close to the concentration profile in the absence of flow (c(x) ≈ 1). This assumption simplifies the analysis considerably, and also makes the uniformly occupied domains used as initial conditions in the simulations more reasonable (strong compressibility would lead to localization on downwellings). The weak compressibility regime was also the focus of Ref. [45], where it was defined to mean that Fisher population waves are able to propagate through regions of convergence without becoming trapped. We can estimate whether this condition holds using the data provided in We can also directly compare the time scale of the strongest divergence on the grid scale to the time scale of replication to find a dimensionless measure of flow divergence. For µ = 1 day −1 , |∇ · u| µ ≈ 0.35.(S2) The generation time, µ −1 , is therefore short relative to the source time. Organisms are able to reproduce multiple times while feeling the influence of even the strongest sources in the flow, if the sources are traveling with the mean flow, before being moved by the source itself away from the area of interest. These estimates, while useful, do not consider any effects resulting from sources and sinks moving relative to the mean flow. To be certain that the flow does not induce a significant reduction in the carrying capacity, we measure the total concentration over ten days of the simulation. If Fisher population waves can overcome the convergences, the system will remain approximately uniformly occupied. As we see in Fig. S12, the total concentration remains approximately equal to the concentration in the absence of flow (normalized to 1 in the figure) as the velocity field changes over time. Together, these arguments suggest that for both the summer and winter flows, we are comfortably in the weak compressibility regime. If we had instead found strong compressibility, we would still expect to observe profound effects on competition events. However, for strong compressibility, the population tends to localize on sinks [46], which makes the effect of regions of positive divergence more subtle, with number fluctuations becoming important. This regime would be interesting to study in the future. figure S8). E. Relative abundance in incompressible and compressible flows We begin with the evolution equations introduced in the main text. We are interested in how the fraction of population A changes in response to the external flow field. Therefore, we change variables and rewrite these equations in terms of f = c A c A +c B and c = c A + c B . We do this with the following steps. ∂c A ∂t + ∇ · (uc A ) = D∇ 2 c A + µc A (1 − c A − c B ) + sµc A c B ,(S3)∂c B ∂t + ∇ · (uc B ) = D∇ 2 c B + µc B (1 − c A − c B ) − sµc A c B .(S4 First consider the sum of Eqs. S3 and S4. The term concerning selection cancels (although it would not if we had assumed the selective advantage occurs due to a difference in growth rates, as we discuss in SI Sec. F). ∂c ∂t + ∇ · (uc) = D∇ 2 c + µc(1 − c).(S5) This equation gives the evolution of the total concentration field. To find the equation for the relative abundance, we take Eq. S3 and substitute c A = f c and c B = c − f c. The time derivative on the left hand side of Eq. S3 becomes ∂(f c) ∂t = c ∂f ∂t + f ∂c ∂t = c ∂f ∂t + f −∇ · (uc) + D∇ 2 c + µc(1 − c) . (S6) We simplify, distributing the divergence operator, to find ∂f ∂t + u · ∇f = D∇ 2 f + 2D c ∇f · ∇c + sµcf (1 − f ).(S7) Next, we consider the specific scenario treated in the main text. When the total carrying capacity is initially uniform in space, it will remain uniform for incompressible flows and approximately uniform for weakly compressible flows (SI Sec. D). Therefore, ∇c ≈ 0, and we neglect the ∇f · ∇c term and set c = 1 in the selection term. We integrate the remaining terms over the domain, integrate by parts, and apply Gauss's theorem. ∂ ∂t Ω f dΩ ≈ D S ∇f ·ndS − S f u ·ndS + Ω f (∇ · u)dΩ + sµ Ω f (1 − f )dΩ. (S8) The surface terms are zero for a sufficiently localized population of A (∇f = f = 0 at the boundary), leaving the condition ∂ ∂t Ω f dΩ ≈ Ω f (∇ · u)dΩ + sµ A f (1 − f )dΩ. (S9) Therefore, for an incompressible flow with ∇ · u = 0, the spatially averaged relative abundances remain constant in time for an initial condition with a localized population of type A embedded in a community of neutral competitors of type B such that c = c A + c B is uniform in space. Furthermore, if c = 1 initially, c = 1 for all time and f = c A . We can therefore attribute the changes we see in Figs. 2 and 3 of the main text to the influence of divergence. We note that this argument can also be applied for a carrying capacity that varies sufficiently slowly in space that the ∇c term can be neglected. The strength of the selection would then also become a function of space. F. Selective advantage Here, we consider the influence of selection using the governing equations. Relevant simulation results are given in the next sections, SI Appendices G and H. With selection, Eq. 10 of the main text gains an additional term, ∆ f f 0 ≈ τ f 0 f ∇ · u + sµ f (1 − f ) .(S10) For biologically realistic selection between closely related populations, s is small, making the characteristic time on which selection operates (sµ) −1 longer than one generation, µ −1 . Therefore, over our standard observation period τ =1 day= µ −1 , the difference between f (x, τ ) for s = 0 and f (x, τ ) for s = 0 is small, and we expect ∆ f /f 0 to be linear in s. In the simulations described in SI Appendices G and H, we use the same flow field and initial condition for each set of trials, only varying s, and we observe linear trends in Figs. S13 and S14, as expected. For sufficiently large |s| or at long times, the data deviate from the linear trend. While Eq. S10 is useful for testing the validity of our approximations and understanding how regions of divergence interact with community structure in general, as we discuss in the next two sections, we gain a more intuitive understanding by considering further approximations for a simplified case. In the main text and in SI Sec. G, we use a Gaussian initial condition with the fraction of organisms in the localized population given by f (x, y) = exp − x 2 + y 2 2σ 2 . (S11) We now assume that this is the population structure for all time. This approximation is reasonable when τ is small relative to the time scales of advection and selection. We also assume that the localized population is centered on a source or sink of the velocity field, such that u(x, y) = δ 2 (x + y).(S12) We can think of this as the first term in a Taylor series expansion, accurate sufficiently close to a source or sink. If δ > 0, this is a source (positive divergence). If δ < 0, this is a sink (negative divergence). This approximation is especially good for the simulations in SI Sec. G where we place the localized population on a region of high positive divergence. Integrating over all space (extending the bounds to ±∞ for simplicity), we find ∆ f f 0 ≈ τ δ + µs 2 . (S13) This result has a number of features that agree with our simulations. As we saw in Fig. 3 of the main text, the change in the relative abundance is linearly proportional to the strength of the divergence, and crosses zero at δ = 0 when s = 0. We do not have a dependence on the diffusivity, the localized population size, or the growth rate when s = 0. This argument also predicts that the change in relative abundance should be linearly proportional to the selective advantage with a slope of 0.50 for µ = 1 day −1 , and that the change in relative abundance will be zero when s = s * , with s * = − 2δ µ . (S14) Other forms of selective advantage As described in many places [45,[47][48][49], macroscopic equations for the concentration fields can be derived from microscopic, agent-based rules. The form of the selective advantage term in the (macroscopic) evolution equations used in this work (Eqs. S3 and S4) arises from reducing the microscopic rate of death-by-competition for population A, thereby giving population A an advantage. However, selective advantage is often instead modeled as a difference in growth rates. Coarse-graining with two different growth rates, µ and µ(1 + s), results in different evolution equations. Eq. S5 becomes ∂c ∂t + ∇ · (uc) = D∇ 2 c + µc(1 + f s − c).(S15) Eq. S7 becomes ∂f ∂t + u · ∇f = D∇ 2 f + 2D c ∇f · ∇c + sµf (1 − f ). (S16) Note that there is now a term in the equation for c that depends on s. Since type A has a faster growth rate, its equilibrium carrying capacity in isolation is greater than that of type B. Therefore, the total carrying capacity of the system now depends on the fraction f . Assuming weak compressibility, we set c = 1 + sf . If we make the same Gaussian population/linear source approximations as in the previous section, we find, to first order in s, ∂ ∂t Ω f dΩ ≈ (2δ + sµ) πσ 2 + 2Dsπ,(S17)∆ f f 0 ≈ τ δ + µs 2 + Ds σ 2 .(S18) Since D σ 2 < µ 2 (the amount of time it takes an organism to diffuse across the localized population is longer than two generations) in simulations, the correction due to birth-based selection is generally small. G. Disadvantaged intruder Using simulations, we can recast the effect of the flow field on the population structure as an effective selective advantage. We consider a simulation in which a localized population (normally distributed, as in the main text) is initialized to lie in a region of strong positive divergence, selected using the data from the trials in Fig. 3 of the main text. This population was found to experience growth and increased abundance relative to its competitor (i.e. it corresponds to a point in the top right corner of Fig. 3A). We now alter this competition by imposing a selective advantage/disadvantage on the population (nonzero s in Eqs. S3 and S4). For some negative value of s, s * , the selective disadvantage will exactly counterbalance the effect of the positive divergence, and the spatially averaged relative abundances will not change over our observation period. For s < s * , the relative abundance of species A will decrease. We therefore consider |s * | to be the effective selective advantage provided to the population by the flow field. This simulation can be thought of as tracking, for example, a low-light specialist organism arriving at the surface via an upwelling, where it is comparatively ill-suited to survive. A sufficiently strong upwelling underneath a disadvantaged ecotype could act as a lifeline, and allow it to avoid competitive exclusion in its newly harsh environment. Note that at the surface, divergent flow generates upwelling in the vertical, but this exact correspondence between strong upwelling (vertical velocity) and strong 2D divergence (vertical velocity gradient) does not necessarily hold subsurface. In Fig. S13, we observe that the winter flow field can compensate for a selective disadvantage of s * = −25%, and the summer flow field can compensate for a selective disadvantage of s * = −65%. Since we place the localized populations at sites that we know are particularly advantageous in these simulations, s * should be thought of as a maximum effect size under idealized conditions. In SI Sec. F, we show that we expect a slope of 0.5 for a Gaussian initial condition, as in this simulation. In Fig. S13, we find great agreement: a linear trend for both trials with a slope of 0.53 (R 2 = 0.99) for the summer flow field and a slope of 0.47 (R 2 = 0.99) for the winter flow field. The winter flow field initial condition is centered on a region of divergence with δ = 1.25 × 10 −6 s −1 (Fig. S13A). Substituting this value into Eq. S14, we estimate s * ≈ −0.22. The measured value of s * is −0.25. Agreement is somewhat worse for the summer flow field initial condition centered on a region of divergence with δ = 2.99 × 10 −6 s −1 . Eq. S14 gives an estimate of s * ≈ −0.52, while the measured s * is −0.65. Considering the severity of the approximations leading to Eq. S14, these estimates are perhaps surprisingly close to the measured values, which take into account nonlinearities and the time-dependent nature of the velocity field. H. Plankton biogeography In the ocean, populations are rarely spatially localized and instead display correlations with water masses and dynamics, due to physical, chemical, and ecological factors. To study the implications of effective compressibility on spatially extended plankton communities, we designed three initial conditions based on realistic phytoplankton biogeography for the summer flow field. These initial conditions exemplify a few ways that ecological communities may be distributed relative to a front, where the effects of compressibility are the largest. First, we might expect an upwelling to carry new species to a given depth level, as discussed in SI Sec. G [50]. The upwelling-inspired initial condition (shown in Fig. S14A) is constructed by placing population A in all areas with upwelling, while still requiring c A + c B = 1. We observe that the distribution of regions of positive divergence is much more complex than the simple Gaussian initial condition studied in the main text and SI Sec. G. Second, the water masses that meet at a front will often have distinct communities [51,52]. The distinct water mass initial condition, shown in Fig. S14B, is defined by placing population A in the region where the salinity is higher than 36.5 PSU and population B in the rest of the domain. Third, another possibility is that the front has a unique community due to influences of frontal currents on the rate of nutrient supply [53,54]. This frontal initial condition is constructed by placing population A near 36.5 PSU, as shown in Fig. S14C. We evolve these initial conditions as in SI Sec. G in the summer flow field, with the selective advantage/disadvantage of the community varied to measure the effective advantage conferred by the flow. Unlike Figs. 2, 3 and S13, these simulations were only evolved for 12 hours because the fine structure in the initial conditions made them more susceptible to numerical instabilities. We expect that doubling the period of the simulation would double the slope, while keeping s * constant. These initial populations experience a range of changes in their relative abundance and effective selective advantage due to the influence of the flow. Of the neutral populations (s = 0), the population initialized in an upwelling experiences the greatest change in relative abundance ( Fig. S14A; 9.7% change over 12 hours). The flow field is near the surface so there is a nearly linear relationship between upwelling and divergence. The two populations that are defined by salinity criteria, the frontal population (Fig. S14C) and the population on the dense side of the front (Fig. S14B) display distinct responses to the flow field. The neutral frontal population has nearly zero change in relative abundance ( Fig. S14C; 0.01% change over 12 hours). The population on the dense side of the front increases its relative abundance by 1.5% over 12 hours. The population initialized in an upwelling region has a large effective selective advantage (s * ≈ −0.3). However, we also find that the population initialized on the dense side of the front has a similar effective selective advantage (s * ≈ −0.32). Although the growth advantage imparted by the flow to the neutral population is relatively small in this case, the spatial population structure allows this population to overcome a large disadvantage. Referring back to Eq. S10, we know the slopes of the trends in Figs. S13 and S14 depend on the spatial distributions of the populations. The regions where f (1 − f ) is nonzero are where the two populations are in contact with each other and therefore competition is most important. The size of these regions sets the slope of ∆ f /f 0 versus s. The selective disadvantage at which the relative abundance does not change, s * , depends on both the spatial distributions of the populations and the divergence of the flow. For example, two populations with the same initial condition in different flow fields have the same dependence on the selective advantage (the slopes), but different values of s * (the x-intercepts) in Fig. S13. Populations in the same flow field with different initial conditions have different slopes and FIG. 1 . 1Relative abundances of Synechococcus IV oligotypes show no correlation with temperature or nitrate concentration across a front in the Mediterranean Sea with regions of strong divergence. (A) The yellow line ("V" shape) shows where water was sampled across the surface layer of a chlorophyll filament in the Western Mediterranean Sea . The background shading shows chlorophyll concentration measured by satellite (MODIS) with geostrophic velocity vectors from the AVISO satellite product on May 30, 2018. (B) Spatial variation in community composition and velocity. Plotted along the ship's track shown in (A) FIG. 2 . 2The spatial distribution of velocity divergence affects the growth (measured here as the change in the relative abundance) of local populations, shown in Figures A and D by the qualitative agreement between divergence contours and regions of positive/negative growth. Panels A-C use the winter flow field. Panels D-F use the summer flow field. (A,D) The change in the relative abundance, expressed as a percent (100 times Eq. 4) between the final and initial populations of type A after one day. Each dot is located at the spatial center of the localized population's Gaussian initial condition, with the color giving the magnitude of the change. The contours show divergence equal to 10 −6 s −1 (solid) and −10 −6 s −1 (dashed). The blue boxes show the spatial extent of the subdomains plotted in panels B,C and E,F. (B,E) Concentration of a example localized populations of type A as a function of space at the initial and final time. Black is concentration equal to one, and the black dashed lines contour where the population concentration cA is equal to 0.1. (C,F) Divergence as a function of position in the velocity field at the initial and final time, with the same black dashed lines as in panels B and E. In these trials, a diffusivity of 5 m 2 /s is used. FIG. 3 . 3Changes in the relative abundance and global fraction of a localized population are strongly dependent on the local flow conditions. For a given model and parameter combination, each point represents a different initial population location. (A) The normalized change in the relative abundance of population A (Eq. 4) over one day as a function of the integrated divergence experienced by that population. The solid black line shows the 1:1 line (our theoretical expectation, Eq. 10). (B) The change in the global fraction of population A (Eq. 6) over one day as a function of the integrated divergence experienced by that population. Inset shows the distribution of population-weighted integrated divergence experienced by populations initialized across the domain in the two different models. The µ = 0 populations evolved in the summer and winter flow fields are shown in yellow/orange and blue/indigo, respectively. The symbol shape denotes different trial parameters-diffusivity D (m 2 s −1 ) and population growth rate µ (day −1 ) are varied. The µ = 0 simulations are given grey markers. In (B), we see that ∆F avg = 0 for the µ = 0 trials, as expected. The integrated abundanceweighted divergence is computed using snapshots of the population and flow field taken every three hours. Between 144 and 1056 points for each model and parameter combination are shown. A.P., M.F., R.B., F.T., and A.M. designed research; A.P., M.F., C.J.C., and L.S. performed research; A.W. FIG. S1. Root mean-square jet speed in the periodic (east-west) direction, averaged over the 24-hour period used in simulations. Positive values are eastward.B. Biological observationsMethodsSamples were collected from the sea surface at 5 minute intervals using an oceanographic bucket while the ship was underway at 8 knots to obtain approximately 1 km lateral sampling resolution. A total of 16 samples were collected this way on May 30, 2018 from 20:00 UTC to 22:45pm UTC (22:00 May 30 to 00:45 May 31 local time). Samples were processed for later analysis. ) S2. Hovmöller diagrams of the divergence in (A) the winter flow field and (B) the summer flow field. FIG. S3. Wavenumber spectrum of the divergence during the simulation day from both models.immediately before analysis (0.75 µm yellow-green, Polysciences, Inc and 1.0-1.4 µm ultrarainbow, Spherotech). Each sample was run for 8 minutes at 25 µl min −1 after a pre-run of 2 minutes. Forward angle light scatter (FALS), side scatter, and autofluorescence at 692/20 nm, 572/13 nm, and 520/25 nm were recorded, with data collection triggered by FALS. Synechococcus cells were classified using red and orange autoflourescence and FALS.Samples to be analyzed for nitrate were frozen at -20 • C. and analyzed to determined the concentrations of nitrate, nitrite, silicate, and phosphate with a nutrient autoanalyzer at the Institute for Marine Sciences of Andalusia (ICMAN-FIG. S4. Depth structure of the northern transect on the V-shaped survey. The black lines are density contours in kg/m 3 . Oxygen is plotted as percent saturation. Depth is in meters. The red line shows the mixed layer depth. Observations collected in collaboration with Mathieu Dever. FIG. S5. Divergence of the along-track velocity on the southern observational transect. Fig. S2. At the grid scale of the fluid model, 500 m, the strongest regions of divergence in both winter and summer flow fields measure approximately 4 × 10 −6 s −1 . Upon comparing to the Fisher velocity for µ = 1 day −1 , we find (500 m)(4 × 10 −6 s −1 ) = 0.002 m/s < 2 Dµ ≈ 0.007 m/s. (S1) The inequality is even stronger for the µ = 2 day −1 and D = 5 m 2 /s simulations. FIG. S6. Community composition of bucket samples as relative abundance times 100, i.e. percent abundance. Top panel shows Synechococcus ecotypes and the lower panel shows oligotypes of the Synechococcus IV ecotype. FIG. S7 . S7Community composition of bucket samples. Bars show cells per ml of Prochlorococcus, Synechococcus, and picoeukaryotes enumerated by flow cytometery. FIG. S8. Locations of surface samples in Fig. S6. FIG. S9 .FIG. S10 . S9S10Vertical profile of Synechoccocus IV near the sampled region Stirring of biological communities. Each line traces the path of the sampled water parcels backwards in time using the satellite geostrophic velocity. The line color indicates the difference in the relative abundance of oligotype ATTT when compared with the population in sample 12 (sample numbers labeled in FIG . S11. A. Distribution of divergence at a fixed depth of 72 m (blue) and on an isopycnal surface corresponding to a density of σ = 27.9 kg/m 3 with a mean depth of 72 m (orange). Divergence is evaluated at the grid scale, 500 m, as inFig. S2. B. The isopycnal surface σ = 27.9 kg/m 3 , with the directions of velocities tangent to the surface given with the arrows. . S12. Total concentration (relative to the no-flow carrying capacity) as a function of time for the (A) winter flow field and (B) summer flow field, with µ = 1 day −1 . . S13. Change in relative abundance over 24 hours as a function of the selective (dis)advantage. The yellow and blue data correspond to a population that was initially a Gaussian centered at the location that yielded the greatest value of ∆ f /f0 in the neutral simulations ofFig. 3Afor the summer and winter flow fields, respectively. A selective disadvantage of approximately 65% for the summer flow field and 25% for the winter flow field is required to cancel out the effect of the divergence experienced by the population. The insets show the concentration of the focal population used in the neutral winter flow field simulations at the starting time (A) and after 24 hours (B). . S14. Change in relative abundance of the spatially extended populations over 12 hours of the summer flow field as a function of the selective (dis)advantage, as in Fig. S13. The initial concentration profiles of population A are pictured in the insets marked 1, and the concentration profiles after 12 hours are pictured in the insets marked 2. The inset axes and color bar are the same as in Fig. S13. Insets (A1,A2) show the upwelling community (slope = 0.32), (B1, B2) show distinct populations in distinct water masses (slope = 0.05), and (C1,C2) show a frontal community (slope = 0.18). . S Clayton, Y.-C Lin, M J Follows, A Z Worden, Limnology and Oceanography. 6275S. Clayton, Y.-C. Lin, M. J. Follows, and A. Z. Worden, Limnology and Oceanography 62, 75 (2017). . P Ramond, R Siano, S Schmitt, C Vargas, L Marié, L Mémery, M Sourisseau, Scientific Reports. 111P. Ramond, R. Siano, S. Schmitt, C. De Vargas, L. Marié, L. Mémery, and M. Sourisseau, Scientific Reports 11, 1 (2021). . G E Hutchinson, The American Naturalist. 95137G. E. Hutchinson, The American Naturalist 95, 137 (1961). . S J Biller, P M Berube, D Lindell, S W Chisholm, Nature Reviews Microbiology. 1313S. J. Biller, P. M. Berube, D. Lindell, and S. W. Chisholm, Nature Reviews Microbiology 13, 13 (2015). . D C Speirs, W S Gurney, Ecology. 821219D. C. Speirs and W. S. Gurney, Ecology 82, 1219 (2001). F Lutscher, E Mccauley, M A Lewis, Theoretical population biology. 71267F. Lutscher, E. McCauley, and M. A. Lewis, Theoretical population biology 71, 267 (2007). . J P Wares, J M Pringle, BMC Evolutionary Biology. 81J. P. Wares and J. M. Pringle, BMC Evolutionary Biol- ogy 8, 1 (2008). . A Mahadevan, J Campbell, Geophysical Research Letters. 29A. Mahadevan and J. Campbell, Geophysical Research Letters 29 (2002). . A Mahadevan, Annual Review of Marine Science. 8161A. Mahadevan, Annual Review of Marine Science 8, 161 (2016). . D J Mcgillicuddy, Annual Review of Marine Science. 8125D. J. McGillicuddy, Annual Review of Marine Science 8, 125 (2016). . F Herrerías-Azcué, V Pérez-Muñuzuri, T Galla, PLoS Computational Biology. 151007238F. Herrerías-Azcué, V. Pérez-Muñuzuri, and T. Galla, PLoS Computational Biology 15, e1007238 (2019). . F Ovidio, S De Monte, S Alvain, Y Dandonneau, M Lévy, Proceedings of the National Academy of Sciences. 10718366F. d'Ovidio, S. De Monte, S. Alvain, Y. Dandonneau, and M. Lévy, Proceedings of the National Academy of Sciences 107, 18366 (2010). . S Clayton, S Dutkiewicz, O Jahn, M J Follows, Limnology and Oceanography: Fluids and Environments. 3182S. Clayton, S. Dutkiewicz, O. Jahn, and M. J. Follows, Limnology and Oceanography: Fluids and Environments 3, 182 (2013). . A D Barton, S Dutkiewicz, G Flierl, J Bragg, M J Follows, Science. 3271509A. D. Barton, S. Dutkiewicz, G. Flierl, J. Bragg, and M. J. Follows, Science 327, 1509 (2010). . N Kashtan, S E Roggensack, S Rodrigue, J W Thompson, S J Biller, A Coe, H Ding, P Marttinen, R R Malmstrom, R Stocker, Science. 344416N. Kashtan, S. E. Roggensack, S. Rodrigue, J. W. Thompson, S. J. Biller, A. Coe, H. Ding, P. Marttinen, R. R. Malmstrom, R. Stocker, et al., Science 344, 416 (2014). . P Villa Martín, A Buček, T Bourguignon, S Pigolotti, Science Advances. 6P. Villa Martín, A. Buček, T. Bourguignon, and S. Pigolotti, Science Advances 6 (2020). . G Károlyi, Á Péntek, I Scheuring, T Tél, Z Toroczkai, Proceedings of the National Academy of Sciences. 9713661G. Károlyi,Á. Péntek, I. Scheuring, T. Tél, and Z. Toroczkai, Proceedings of the National Academy of Sciences 97, 13661 (2000). . J C Mcwilliams, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 47220160117J. C. McWilliams, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, 20160117 (2016). . M Lévy, P J Franks, K S Smith, Nature Communications. 91M. Lévy, P. J. Franks, and K. S. Smith, Nature Com- munications 9, 1 (2018). . T Uchida, D Balwada, R Abernathey, G Mckinley, S Smith, M Lévy, Nature Communications. 111T. Uchida, D. Balwada, R. P Abernathey, G. A McKin- ley, S. K Smith, and M. Lévy, Nature Communications 11, 1 (2020). . F Kessouri, D Bianchi, L Renault, J C Mcwilliams, H Frenzel, C A Deutsch, Global Biogeochemical Cycles. 34F. Kessouri, D. Bianchi, L. Renault, J. C. McWilliams, H. Frenzel, and C. A. Deutsch, Global Biogeochemical Cycles 34, e2020GB006578 (2020). . M A Freilich, G Flierl, A Mahadevan, Geophysical Research Letters. 49M. A. Freilich, G. Flierl, and A. Mahadevan, Geophysical Research Letters 49, e2021GL096180 (2022). . E A D&apos;asaro, A Y Shcherbina, J M Klymak, J Molemaker, G Novelli, C M Guigand, A C Haza, B K Haus, E H Ryan, G A Jacobs, Proceedings of the National Academy of Sciences. 1151162E. A. D'Asaro, A. Y. Shcherbina, J. M. Klymak, J. Mole- maker, G. Novelli, C. M. Guigand, A. C. Haza, B. K. Haus, E. H. Ryan, G. A. Jacobs, et al., Proceedings of the National Academy of Sciences 115, 1162 (2018). . R Barkan, M J Molemaker, K Srinivasan, J C Mcwilliams, E A , Journal of Physical Oceanography. 491593R. Barkan, M. J. Molemaker, K. Srinivasan, J. C. McWilliams, and E. A. D'Asaro, Journal of Physical Oceanography 49, 1593 (2019). A Plummer, R Benzi, D R Nelson, F Toschi, Proceedings of the National Academy of Sciences. the National Academy of Sciences116373A. Plummer, R. Benzi, D. R. Nelson, and F. Toschi, Proceedings of the National Academy of Sciences 116, 373 (2019). . G Guccione, R Benzi, A Plummer, F Toschi, Physical Review E. 10062105G. Guccione, R. Benzi, A. Plummer, and F. Toschi, Physical Review E 100, 062105 (2019). . I Hernández-Carrasco, A Orfila, V Rossi, V Garçon, Scientific Reports. 81I. Hernández-Carrasco, A. Orfila, V. Rossi, and V. Garçon, Scientific Reports 8, 1 (2018). . L Guidi, P H Calil, S Duhamel, K M Björkman, S C Doney, G A Jackson, B Li, M J Church, S Tozzi, Z S Kolber, Journal of Geophysical Research. 117BiogeosciencesL. Guidi, P. H. Calil, S. Duhamel, K. M. Björkman, S. C. Doney, G. A. Jackson, B. Li, M. J. Church, S. Tozzi, Z. S. Kolber, et al., Journal of Geophysical Research: Biogeosciences 117 (2012). . J B Palter, E J Ames, M Benavides, A Neto, J Granger, P H Moisander, K S Watkins-Brandt, A E White, Geophysical Research Letters. 47J. B. Palter, E. J. Ames, M. Benavides, A. Goncalves Neto, J. Granger, P. H. Moisander, K. S. Watkins-Brandt, and A. E. White, Geophysical Research Letters 47, e2020GL089103 (2020). . M Benavides, L Conradt, S Bonnet, I Berman-Frank, S Barrillon, A Petrenko, A Doglioli, ISME Communications. 11M. Benavides, L. Conradt, S. Bonnet, I. Berman-Frank, S. Barrillon, A. Petrenko, and A. Doglioli, ISME Com- munications 1, 1 (2021). . D R Tarry, S Essink, A Pascual, S Ruiz, P.-M Poulain, T Özgökmen, L R Centurioni, J T Farrar, A Shcherbina, A Mahadevan, Journal of Geophysical Research: Oceans. 126D. R. Tarry, S. Essink, A. Pascual, S. Ruiz, P.-M. Poulain, T.Özgökmen, L. R. Centurioni, J. T. Farrar, A. Shcherbina, A. Mahadevan, et al., Journal of Geo- physical Research: Oceans 126, e2020JC016614 (2021). . S Sudek, R C Everroad, A.-L M Gehman, J M Smith, C L Poirier, F P Chavez, A Z Worden, Environmental Microbiology. 173692S. Sudek, R. C. Everroad, A.-L. M. Gehman, J. M. Smith, C. L. Poirier, F. P. Chavez, and A. Z. Worden, Environ- mental Microbiology 17, 3692 (2015). . K Zwirglmaier, L Jardillier, M Ostrowski, S Mazard, L Garczarek, D Vaulot, F Not, R Massana, O Ulloa, D J Scanlan, Environmental Microbiology. 10147K. Zwirglmaier, L. Jardillier, M. Ostrowski, S. Mazard, L. Garczarek, D. Vaulot, F. Not, R. Massana, O. Ulloa, and D. J. Scanlan, Environmental Microbiology 10, 147 (2008). . A M Eren, L Maignien, W J Sul, L G Murphy, S L Grim, H G Morrison, M L Sogin, Methods in Ecology and Evolution. 41111A. M. Eren, L. Maignien, W. J. Sul, L. G. Murphy, S. L. Grim, H. G. Morrison, and M. L. Sogin, Methods in Ecology and Evolution 4, 1111 (2013). . K R Mackey, K Hunter-Cevera, G L Britten, L G Murphy, M L Sogin, J A Huber, Frontiers in Microbiology. 81496K. R. Mackey, K. Hunter-Cevera, G. L. Britten, L. G. Murphy, M. L. Sogin, and J. A. Huber, Frontiers in Microbiology 8, 1496 (2017). . D M Needham, R Sachdeva, J A Fuhrman, The ISME journal. 111614D. M. Needham, R. Sachdeva, and J. A. Fuhrman, The ISME journal 11, 1614 (2017). . A A Larkin, C A Garcia, K A Ingoglia, N S Garcia, S E Baer, B S Twining, M W Lomas, A C Martiny, Limnology and Oceanography. 65220A. A. Larkin, C. A. Garcia, K. A. Ingoglia, N. S. Garcia, S. E. Baer, B. S. Twining, M. W. Lomas, and A. C. Martiny, Limnology and Oceanography 65, S220 (2020). . N A Ahlgren, J N Perelman, Y.-C Yeh, J A Fuhrman, Environmental Microbiology. 212948N. A. Ahlgren, J. N. Perelman, Y.-C. Yeh, and J. A. Fuhrman, Environmental Microbiology 21, 2948 (2019). . C O Sjöqvist, A Kremp, The ISME journal. 102755C. O. Sjöqvist and A. Kremp, The ISME journal 10, 2755 (2016). . J Pittera, F Humily, M Thorel, D Grulois, L Garczarek, C Six, The ISME Journal. 81221J. Pittera, F. Humily, M. Thorel, D. Grulois, L. Gar- czarek, and C. Six, The ISME Journal 8, 1221 (2014). . C Six, M Ratin, D Marie, E Corre, Proceedings of the National Academy of Sciences. 118C. Six, M. Ratin, D. Marie, and E. Corre, Proceedings of the National Academy of Sciences 118 (2021). . E Jaspers, J Overmann, Applied and Environmental Microbiology. 704831E. Jaspers and J. Overmann, Applied and Environmental Microbiology 70, 4831 (2004). . M A Berry, J D White, T W Davis, S Jain, T H Johengen, G J Dick, O Sarnelle, V J Denef, Frontiers in microbiology. 8365M. A. Berry, J. D. White, T. W. Davis, S. Jain, T. H. Johengen, G. J. Dick, O. Sarnelle, and V. J. Denef, Fron- tiers in microbiology 8, 365 (2017). G K Farrant, H Doré, F M Cornejo-Castillo, F Partensky, M Ratin, M Ostrowski, F D Pitt, P Wincker, D J Scanlan, D Iudicone, Proceedings of the National Academy of Sciences. the National Academy of Sciences1133365G. K. Farrant, H. Doré, F. M. Cornejo-Castillo, F. Partensky, M. Ratin, M. Ostrowski, F. D. Pitt, P. Wincker, D. J. Scanlan, D. Iudicone, et al., Proceed- ings of the National Academy of Sciences 113, E3365 (2016). . J R Taylor, K M Smith, C A Vreugdenhil, Journal of Physical Oceanography. 501319J. R. Taylor, K. M. Smith, and C. A. Vreugdenhil, Jour- nal of Physical Oceanography 50, 1319 (2020). . J K Moore, T A Villareal, Marine Ecology Progress Series. 132203J. K. Moore and T. A. Villareal, Marine Ecology Progress Series 132, 203 (1996). . K J Benoit-Bird, T J Cowles, C E Wingard, Limnology and Oceanography. 541382K. J. Benoit-Bird, T. J. Cowles, and C. E. Wingard, Limnology and Oceanography 54, 1382 (2009). . E Litchman, C A Klausmeier, Evolution, and Systematics. 39615Annual Review of EcologyE. Litchman and C. A. Klausmeier, Annual Review of Ecology, Evolution, and Systematics 39, 615 (2008). . J Larkin, W Goldburg, M Bandi, Physica D: Nonlinear Phenomena. 2391264J. Larkin, W. Goldburg, and M. Bandi, Physica D: Non- linear Phenomena 239, 1264 (2010). . G Boffetta, J Davoudi, B Eckhardt, J Schumacher, Physical review letters. 93134501G. Boffetta, J. Davoudi, B. Eckhardt, and J. Schu- macher, Physical review letters 93, 134501 (2004). . M De Pietro, M A Van Hinsberg, L Biferale, H J Clercx, P Perlekar, F Toschi, Physical Review E. 9153002M. De Pietro, M. A. van Hinsberg, L. Biferale, H. J. Clercx, P. Perlekar, and F. Toschi, Physical Review E 91, 053002 (2015). . P Perlekar, R Benzi, D R Nelson, F Toschi, Physical Review Letters. 105144501P. Perlekar, R. Benzi, D. R. Nelson, and F. Toschi, Phys- ical Review Letters 105, 144501 (2010). . P Perlekar, R Benzi, D R Nelson, F Toschi, Journal of Turbulence. 14161P. Perlekar, R. Benzi, D. R. Nelson, and F. Toschi, Jour- nal of Turbulence 14, 161 (2013). . S Pigolotti, R Benzi, M H Jensen, D R Nelson, Physical Review Letters. 108128102S. Pigolotti, R. Benzi, M. H. Jensen, and D. R. Nelson, Physical Review Letters 108, 128102 (2012). S Pigolotti, R Benzi, P Perlekar, M H Jensen, F Toschi, D R Nelson, Theoretical population biology. 8472S. Pigolotti, R. Benzi, P. Perlekar, M. H. Jensen, F. Toschi, and D. R. Nelson, Theoretical population bi- ology 84, 72 (2013). . C Boyd, D Gradmann, Marine Biology. 141605C. Boyd and D. Gradmann, Marine Biology 141, 605 (2002). Mathematical biology: I. An introduction. J D Murray, Springer Science & Business Media17J. D. Murray, Mathematical biology: I. An introduction, Vol. 17 (Springer Science & Business Media, 2007). Z Neufeld, E Hernández-García, Chemical and Biological Processes in Fluid Flows: A Dynamical Systems Approach. World ScientificZ. Neufeld and E. Hernández-García, Chemical and Bi- ological Processes in Fluid Flows: A Dynamical Systems Approach (World Scientific, 2009). . T Tél, A De Moura, C Grebogi, G Károlyi, Physics reports. 41391T. Tél, A. de Moura, C. Grebogi, and G. Károlyi, Physics reports 413, 91 (2005). . R Benzi, D R Nelson, Physica D: Nonlinear Phenomena. 238R. Benzi and D. R. Nelson, Physica D: Nonlinear Phe- nomena 238, 2003 (2009). . A Z Worden, J K Nolan, B Palenik, Limnology and Oceanography. 49168A. Z. Worden, J. K. Nolan, and B. Palenik, Limnology and Oceanography 49, 168 (2004). . S Widder, R J Allen, T Pfeiffer, T P Curtis, C Wiuf, W T Sloan, O X Cordero, S P Brown, B Momeni, W Shou, The ISME Journal. 102557S. Widder, R. J. Allen, T. Pfeiffer, T. P. Curtis, C. Wiuf, W. T. Sloan, O. X. Cordero, S. P. Brown, B. Momeni, W. Shou, et al., The ISME Journal 10, 2557 (2016). . S Sunagawa, S G Acinas, P Bork, C Bowler, D Eveillard, G Gorsky, L Guidi, D Iudicone, E Karsenti, F Lombard, Nature Reviews Microbiology. 18428S. Sunagawa, S. G. Acinas, P. Bork, C. Bowler, D. Eveil- lard, G. Gorsky, L. Guidi, D. Iudicone, E. Karsenti, F. Lombard, et al., Nature Reviews Microbiology 18, 428 (2020). . L Vezzulli, J Martinez-Urtaza, R Stern, Current Opinion in Biotechnology. 7361L. Vezzulli, J. Martinez-Urtaza, and R. Stern, Current Opinion in Biotechnology 73, 61 (2022). . R H Whittaker, Taxon. 21213R. H. Whittaker, Taxon 21, 213 (1972). . P Richerson, R Armstrong, C R Goldman, Proceedings of the National Academy of Sciences. 671710P. Richerson, R. Armstrong, and C. R. Goldman, Pro- ceedings of the National Academy of Sciences 67, 1710 (1970). . J Callies, R Ferrari, J M Klymak, J Gula, Nature Communications. 61J. Callies, R. Ferrari, J. M. Klymak, and J. Gula, Nature Communications 6, 1 (2015). . J Choi, A Bracco, R Barkan, A F Shchepetkin, J C Mcwilliams, J M Molemaker, Journal of Physical Oceanography. 472361J. Choi, A. Bracco, R. Barkan, A. F. Shchepetkin, J. C. McWilliams, and J. M. Molemaker, Journal of Physical Oceanography 47, 2361 (2017). . C Perruche, P Rivière, G Lapeyre, X Carton, P Pondaven, Journal of Marine Research. 69105C. Perruche, P. Rivière, G. Lapeyre, X. Carton, and P. Pondaven, Journal of Marine Research 69, 105 (2011). . J S Guasto, R Rusconi, R Stocker, Annual Review of Fluid Mechanics. 44373J. S. Guasto, R. Rusconi, and R. Stocker, Annual Review of Fluid Mechanics 44, 373 (2012). . P Tréguer, C Bowler, B Moriceau, S Dutkiewicz, M Gehlen, O Aumont, L Bittner, R Dugdale, Z Finkel, D Iudicone, Nature Geoscience. 1127P. Tréguer, C. Bowler, B. Moriceau, S. Dutkiewicz, M. Gehlen, O. Aumont, L. Bittner, R. Dugdale, Z. Finkel, D. Iudicone, et al., Nature Geoscience 11, 27 (2018). . N A Held, J B Waterbury, E A Webb, R M Kellogg, M R Mcilvin, M Jakuba, F W Valois, D M Moran, K M Sutherland, M A Saito, Nature Microbiology. 7300N. A. Held, J. B. Waterbury, E. A. Webb, R. M. Kellogg, M. R. McIlvin, M. Jakuba, F. W. Valois, D. M. Moran, K. M. Sutherland, and M. A. Saito, Nature Microbiology 7, 300 (2022). . A Baudena, E Ser-Giacomi, I Jalón-Rojas, F Galgani, M L Pedrotti, Nature Communications. 131A. Baudena, E. Ser-Giacomi, I. Jalón-Rojas, F. Gal- gani, and M. L. Pedrotti, Nature Communications 13, 1 (2022). . K L Law, S Morét-Ferguson, N A Maximenko, G Proskurowski, E E Peacock, J Hafner, C M Reddy, Science. 3291185K. L. Law, S. Morét-Ferguson, N. A. Maximenko, G. Proskurowski, E. E. Peacock, J. Hafner, and C. M. Reddy, Science 329, 1185 (2010). . L A Amaral-Zettler, E R Zettler, T J Mincer, Nature Reviews Microbiology. 18139L. A. Amaral-Zettler, E. R. Zettler, and T. J. Mincer, Nature Reviews Microbiology 18, 139 (2020). . K L Vergin, B Beszteri, A Monier, J C Thrash, B Temperton, A H Treusch, F Kilpert, A Z Worden, S J Giovannoni, The ISME Journal. 71322K. L. Vergin, B. Beszteri, A. Monier, J. C. Thrash, B. Temperton, A. H. Treusch, F. Kilpert, A. Z. Wor- den, and S. J. Giovannoni, The ISME Journal 7, 1322 (2013). . C J Choi, C Bachy, G S Jaeger, C Poirier, L Sudek, V Sarma, A Mahadevan, S J Giovannoni, A Z Worden, Current Biology. 2715C. J. Choi, C. Bachy, G. S. Jaeger, C. Poirier, L. Sudek, V. Sarma, A. Mahadevan, S. J. Giovannoni, and A. Z. Worden, Current Biology 27, R15 (2017). . M Freilich, A Mahadevan, Journal of Geophysical Research: Oceans. 126M. Freilich and A. Mahadevan, Journal of Geophysical Research: Oceans 126, e2020JC017042 (2021). The impact of submesoscale physics on primary productivity of plankton. Amala Mahadevan, Annual Review of Marine Science. 8Amala Mahadevan. The impact of submesoscale physics on primary productivity of plankton. Annual Review of Marine Science, 8:161-184, 2016. A nonhydrostatic mesoscale ocean model. Part I: Well-posedness and scaling. Amala Mahadevan, Joseph Oliger, Robert Street, Journal of Physical Oceanography. 269Amala Mahadevan, Joseph Oliger, and Robert Street. A nonhydrostatic mesoscale ocean model. Part I: Well-posedness and scaling. Journal of Physical Oceanography, 26(9):1868-1880, 1996. A nonhydrostatic mesoscale ocean model. Part II: Numerical implementation. Amala Mahadevan, Joseph Oliger, Robert Street, Journal of Physical Oceanography. 269Amala Mahadevan, Joseph Oliger, and Robert Street. A nonhydrostatic mesoscale ocean model. Part II: Numerical implementation. Journal of Physical Oceanography, 26(9):1881-1900, 1996. Coherent pathways for subduction from the surface mixed layer at ocean fronts. Mara Freilich, Amala Mahadevan, Journal of Geophysical Research: Oceans. 1265Mara Freilich and Amala Mahadevan. Coherent pathways for subduction from the surface mixed layer at ocean fronts. Journal of Geophysical Research: Oceans, 126(5):e2020JC017042, 2021. EcoCTD for profiling oceanic physical-biological properties from an underway ship. Mathieu Dever, Mara Freilich, Thomas Farrar, Benjamin Hodges, Tom Lanagan, J Andrew, Amala Baron, Mahadevan, Journal of Atmospheric and Oceanic Technology. 375Mathieu Dever, Mara Freilich, J Thomas Farrar, Benjamin Hodges, Tom Lanagan, Andrew J Baron, and Amala Mahade- van. EcoCTD for profiling oceanic physical-biological properties from an underway ship. Journal of Atmospheric and Oceanic Technology, 37(5):825-840, 2020. Global distribution patterns of distinct clades of the photosynthetic picoeukaryote ostreococcus. Elif Demir-Hilton, Sebastian Sudek, Marie L Cuvelier, Jonathan P Chelle L Gentemann, Alexandra Z Zehr, Worden, The ISME Journal. 57Elif Demir-Hilton, Sebastian Sudek, Marie L Cuvelier, Chelle L Gentemann, Jonathan P Zehr, and Alexandra Z Worden. Global distribution patterns of distinct clades of the photosynthetic picoeukaryote ostreococcus. The ISME Journal, 5(7): 1095-1107, 2011. High-resolution SAR11 ecotype dynamics at the Bermuda Atlantic Time-series Study site by phylogenetic placement of pyrosequences. Bánk Kevin L Vergin, Adam Beszteri, Cameron Monier, Ben Thrash, Temperton, H Alexander, Fabian Treusch, Alexandra Z Kilpert, Stephen J Worden, Giovannoni, The ISME Journal. 77Kevin L Vergin, Bánk Beszteri, Adam Monier, J Cameron Thrash, Ben Temperton, Alexander H Treusch, Fabian Kilpert, Alexandra Z Worden, and Stephen J Giovannoni. High-resolution SAR11 ecotype dynamics at the Bermuda Atlantic Time-series Study site by phylogenetic placement of pyrosequences. The ISME Journal, 7(7):1322-1332, 2013. Cyanobacterial distributions along a physico-chemical gradient in the Northeastern Pacific Ocean. Sebastian Sudek, Craig Everroad, Alyssa-Lois M Gehman, Jason M Smith, Camille L Poirier, P Francisco, Alexandra Z Chavez, Worden, Environmental Microbiology. 1710Sebastian Sudek, R Craig Everroad, Alyssa-Lois M Gehman, Jason M Smith, Camille L Poirier, Francisco P Chavez, and Alexandra Z Worden. Cyanobacterial distributions along a physico-chemical gradient in the Northeastern Pacific Ocean. Environmental Microbiology, 17(10):3692-3707, 2015. Sickle: A sliding-window, adaptive, quality-based trimming tool for fastq files. N A Joshi, Fass, version 1.33)[softwareNA Joshi and JN2011 Fass. Sickle: A sliding-window, adaptive, quality-based trimming tool for fastq files (version 1.33)[software], 2011. Error filtering, pair assembly and error correction for next-generation sequencing reads. C Robert, Henrik Edgar, Flyvbjerg, Bioinformatics. 3121Robert C Edgar and Henrik Flyvbjerg. Error filtering, pair assembly and error correction for next-generation sequencing reads. Bioinformatics, 31(21):3476-3482, 2015. Cutadapt removes adapter sequences from high-throughput sequencing reads. Marcel Martin, EMBnet. journal. 171Marcel Martin. Cutadapt removes adapter sequences from high-throughput sequencing reads. EMBnet. journal, 17(1): 10-12, 2011. Newly discovered deep-branching marine plastid lineages are numerically rare but globally distributed. Chang Jae Choi, Charles Bachy, Spiro Gualtiero, Camille Jaeger, Lisa Poirier, Sudek, Amala Sarma, Mahadevan, J Stephen, Alexandra Z Giovannoni, Worden, Current Biology. 271Chang Jae Choi, Charles Bachy, Gualtiero Spiro Jaeger, Camille Poirier, Lisa Sudek, VVSS Sarma, Amala Mahadevan, Stephen J Giovannoni, and Alexandra Z Worden. Newly discovered deep-branching marine plastid lineages are numerically rare but globally distributed. Current Biology, 27(1):R15-R16, 2017. Oligotyping: differentiating between closely related microbial taxa using 16S rRNA gene data. Loïs Murat Eren, Maignien, Woo Jun, Sul, G Leslie, Sharon L Murphy, Hilary G Grim, Mitchell L Morrison, Sogin, Methods in Ecology and Evolution. 412A Murat Eren, Loïs Maignien, Woo Jun Sul, Leslie G Murphy, Sharon L Grim, Hilary G Morrison, and Mitchell L Sogin. Oligotyping: differentiating between closely related microbial taxa using 16S rRNA gene data. Methods in Ecology and Evolution, 4(12):1111-1119, 2013. Is the distribution of Prochlorococcus and Synechococcus ecotypes in the Mediterranean Sea affected by global warming?. Daniella Mella-Flores, Sophie Mazard, Florian Humily, Frédéric Partensky, Frédéric Mahé, Laetitia Bariat, Claude Courties, Dominique Marie, Joséphine Ras, Romain Mauriac, Biogeosciences. 89Daniella Mella-Flores, Sophie Mazard, Florian Humily, Frédéric Partensky, Frédéric Mahé, Laetitia Bariat, Claude Courties, Dominique Marie, Joséphine Ras, Romain Mauriac, et al. Is the distribution of Prochlorococcus and Synechococcus ecotypes in the Mediterranean Sea affected by global warming? Biogeosciences, 8(9):2785-2804, 2011. Fluid dynamical niches of phytoplankton types. Silvia Francesco D&apos;ovidio, Séverine De Monte, Yves Alvain, Marina Dandonneau, Lévy, Proceedings of the National Academy of Sciences. 10743Francesco d'Ovidio, Silvia De Monte, Séverine Alvain, Yves Dandonneau, and Marina Lévy. Fluid dynamical niches of phytoplankton types. Proceedings of the National Academy of Sciences, 107(43):18366-18370, 2010. Upwelling mechanisms in the northwestern Alboran Sea. Tarek Sarhan, Jesus Garcıa Lafuente, Manuel Vargas, M Juan, Francisco Vargas, Plaza, Journal of Marine Systems. 234Tarek Sarhan, Jesus Garcıa Lafuente, Manuel Vargas, Juan M Vargas, and Francisco Plaza. Upwelling mechanisms in the northwestern Alboran Sea. Journal of Marine Systems, 23(4):317-331, 2000. Cumulative compressibility effects on slow reactive dynamics in turbulent flows. Prasad Perlekar, Roberto Benzi, Federico David R Nelson, Toschi, Journal of Turbulence. 143Prasad Perlekar, Roberto Benzi, David R Nelson, and Federico Toschi. Cumulative compressibility effects on slow reactive dynamics in turbulent flows. Journal of Turbulence, 14(3):161-169, 2013. Fixation probabilities in weakly compressible fluid flows. Abigail Plummer, Roberto Benzi, Federico David R Nelson, Toschi, Proceedings of the National Academy of Sciences. the National Academy of Sciences116Abigail Plummer, Roberto Benzi, David R Nelson, and Federico Toschi. Fixation probabilities in weakly compressible fluid flows. Proceedings of the National Academy of Sciences, 116(2):373-378, 2019. Fisher equation with turbulence in one dimension. Roberto Benzi, David R Nelson, Physica D: Nonlinear Phenomena. 23819Roberto Benzi and David R Nelson. Fisher equation with turbulence in one dimension. Physica D: Nonlinear Phenomena, 238(19):2003-2015, 2009. The Fokker-Planck Equation. Hannes Risken, SpringerHannes Risken. The Fokker-Planck Equation. Springer, 1996. Stochastic Methods: A Handbook for the Natural and Social Sciences. Crispin Gardiner, Springer Series in Synergetics. Crispin Gardiner. Stochastic Methods: A Handbook for the Natural and Social Sciences. Springer Series in Synergetics, 2009. Population genetics in compressible flows. Simone Pigolotti, Roberto Benzi, H Mogens, David R Jensen, Nelson, Physical Review Letters. 10812128102Simone Pigolotti, Roberto Benzi, Mogens H Jensen, and David R Nelson. Population genetics in compressible flows. Physical Review Letters, 108(12):128102, 2012. Submesoscale hotspots of productivity and respiration: Insights from high-resolution oxygen and fluorescence sections. H R Rachel, Dennis J Stanley, Zoe O McgillicuddyJr, Haley M Sandwith, Pleskow, Deep Sea Research Part I: Oceanographic Research Papers. 130Rachel HR Stanley, Dennis J McGillicuddy Jr, Zoe O Sandwith, and Haley M Pleskow. Submesoscale hotspots of pro- ductivity and respiration: Insights from high-resolution oxygen and fluorescence sections. Deep Sea Research Part I: Oceanographic Research Papers, 130:1-11, 2017. Turbulent vertical kinetic energy in the ocean mixed layer. A D&apos; Eric, Asaro, Journal of Physical Oceanography. 3112Eric A D'Asaro. Turbulent vertical kinetic energy in the ocean mixed layer. Journal of Physical Oceanography, 31(12): 3530-3537, 2001. Dispersal, eddies, and the diversity of marine phytoplankton. Sophie Clayton, Stephanie Dutkiewicz, Oliver Jahn, Michael J Follows, Limnology and Oceanography: Fluids and Environments. 31Sophie Clayton, Stephanie Dutkiewicz, Oliver Jahn, and Michael J Follows. Dispersal, eddies, and the diversity of marine phytoplankton. Limnology and Oceanography: Fluids and Environments, 3(1):182-197, 2013. Multiscale routes to supply nutrients through the Kuroshio nutrient stream. Takeyoshi Nagai, Sophie Clayton, Yusuke Uchiyama, Kuroshio Current: Physical, Biogeochemical, and Ecosystem Dynamics. Takeyoshi Nagai, Sophie Clayton, and Yusuke Uchiyama. Multiscale routes to supply nutrients through the Kuroshio nutrient stream. Kuroshio Current: Physical, Biogeochemical, and Ecosystem Dynamics, pages 105-125, 2019. High N2 fixation in and near the Gulf stream consistent with a circulation control on diazotrophy. B Jaime, Elana J Palter, Mar Ames, Afonso Goncalves Benavides, Julie Neto, Granger, H Pia, Katie S Moisander, Angelicque E Watkins-Brandt, White, Geophysical Research Letters. 4716Jaime B Palter, Elana J Ames, Mar Benavides, Afonso Goncalves Neto, Julie Granger, Pia H Moisander, Katie S Watkins- Brandt, and Angelicque E White. High N2 fixation in and near the Gulf stream consistent with a circulation control on diazotrophy. Geophysical Research Letters, 47(16):e2020GL089103, 2020. The impact of submesoscale physics on primary productivity of plankton. Amala Mahadevan, Annual Review of Marine Science. 8Amala Mahadevan. The impact of submesoscale physics on primary productivity of plankton. Annual Review of Marine Science, 8:161-184, 2016. A nonhydrostatic mesoscale ocean model. Part I: Well-posedness and scaling. Amala Mahadevan, Joseph Oliger, Robert Street, Journal of Physical Oceanography. 269Amala Mahadevan, Joseph Oliger, and Robert Street. A nonhydrostatic mesoscale ocean model. Part I: Well-posedness and scaling. Journal of Physical Oceanography, 26(9):1868-1880, 1996. A nonhydrostatic mesoscale ocean model. Part II: Numerical implementation. Amala Mahadevan, Joseph Oliger, Robert Street, Journal of Physical Oceanography. 269Amala Mahadevan, Joseph Oliger, and Robert Street. A nonhydrostatic mesoscale ocean model. Part II: Numerical implementation. Journal of Physical Oceanography, 26(9):1881-1900, 1996. Coherent pathways for subduction from the surface mixed layer at ocean fronts. Mara Freilich, Amala Mahadevan, Journal of Geophysical Research: Oceans. 1265Mara Freilich and Amala Mahadevan. Coherent pathways for subduction from the surface mixed layer at ocean fronts. Journal of Geophysical Research: Oceans, 126(5):e2020JC017042, 2021. EcoCTD for profiling oceanic physical-biological properties from an underway ship. Mathieu Dever, Mara Freilich, Thomas Farrar, Benjamin Hodges, Tom Lanagan, J Andrew, Amala Baron, Mahadevan, Journal of Atmospheric and Oceanic Technology. 375Mathieu Dever, Mara Freilich, J Thomas Farrar, Benjamin Hodges, Tom Lanagan, Andrew J Baron, and Amala Mahade- van. EcoCTD for profiling oceanic physical-biological properties from an underway ship. Journal of Atmospheric and Oceanic Technology, 37(5):825-840, 2020. Global distribution patterns of distinct clades of the photosynthetic picoeukaryote ostreococcus. Elif Demir-Hilton, Sebastian Sudek, Marie L Cuvelier, Jonathan P Chelle L Gentemann, Alexandra Z Zehr, Worden, The ISME Journal. 57Elif Demir-Hilton, Sebastian Sudek, Marie L Cuvelier, Chelle L Gentemann, Jonathan P Zehr, and Alexandra Z Worden. Global distribution patterns of distinct clades of the photosynthetic picoeukaryote ostreococcus. The ISME Journal, 5(7): 1095-1107, 2011. High-resolution SAR11 ecotype dynamics at the Bermuda Atlantic Time-series Study site by phylogenetic placement of pyrosequences. Bánk Kevin L Vergin, Adam Beszteri, Cameron Monier, Ben Thrash, Temperton, H Alexander, Fabian Treusch, Alexandra Z Kilpert, Stephen J Worden, Giovannoni, The ISME Journal. 77Kevin L Vergin, Bánk Beszteri, Adam Monier, J Cameron Thrash, Ben Temperton, Alexander H Treusch, Fabian Kilpert, Alexandra Z Worden, and Stephen J Giovannoni. High-resolution SAR11 ecotype dynamics at the Bermuda Atlantic Time-series Study site by phylogenetic placement of pyrosequences. The ISME Journal, 7(7):1322-1332, 2013. Cyanobacterial distributions along a physico-chemical gradient in the Northeastern Pacific Ocean. Sebastian Sudek, Craig Everroad, Alyssa-Lois M Gehman, Jason M Smith, Camille L Poirier, P Francisco, Alexandra Z Chavez, Worden, Environmental Microbiology. 1710Sebastian Sudek, R Craig Everroad, Alyssa-Lois M Gehman, Jason M Smith, Camille L Poirier, Francisco P Chavez, and Alexandra Z Worden. Cyanobacterial distributions along a physico-chemical gradient in the Northeastern Pacific Ocean. Environmental Microbiology, 17(10):3692-3707, 2015. Sickle: A sliding-window, adaptive, quality-based trimming tool for fastq files. N A Joshi, Fass, version 1.33)[softwareNA Joshi and JN2011 Fass. Sickle: A sliding-window, adaptive, quality-based trimming tool for fastq files (version 1.33)[software], 2011. Error filtering, pair assembly and error correction for next-generation sequencing reads. C Robert, Henrik Edgar, Flyvbjerg, Bioinformatics. 3121Robert C Edgar and Henrik Flyvbjerg. Error filtering, pair assembly and error correction for next-generation sequencing reads. Bioinformatics, 31(21):3476-3482, 2015. Cutadapt removes adapter sequences from high-throughput sequencing reads. Marcel Martin, EMBnet. journal. 171Marcel Martin. Cutadapt removes adapter sequences from high-throughput sequencing reads. EMBnet. journal, 17(1): 10-12, 2011. Newly discovered deep-branching marine plastid lineages are numerically rare but globally distributed. Chang Jae Choi, Charles Bachy, Spiro Gualtiero, Camille Jaeger, Lisa Poirier, Sudek, Amala Sarma, Mahadevan, J Stephen, Alexandra Z Giovannoni, Worden, Current Biology. 271Chang Jae Choi, Charles Bachy, Gualtiero Spiro Jaeger, Camille Poirier, Lisa Sudek, VVSS Sarma, Amala Mahadevan, Stephen J Giovannoni, and Alexandra Z Worden. Newly discovered deep-branching marine plastid lineages are numerically rare but globally distributed. Current Biology, 27(1):R15-R16, 2017. Oligotyping: differentiating between closely related microbial taxa using 16S rRNA gene data. Loïs Murat Eren, Maignien, Woo Jun, Sul, G Leslie, Sharon L Murphy, Hilary G Grim, Mitchell L Morrison, Sogin, Methods in Ecology and Evolution. 412A Murat Eren, Loïs Maignien, Woo Jun Sul, Leslie G Murphy, Sharon L Grim, Hilary G Morrison, and Mitchell L Sogin. Oligotyping: differentiating between closely related microbial taxa using 16S rRNA gene data. Methods in Ecology and Evolution, 4(12):1111-1119, 2013. Is the distribution of Prochlorococcus and Synechococcus ecotypes in the Mediterranean Sea affected by global warming?. Daniella Mella-Flores, Sophie Mazard, Florian Humily, Frédéric Partensky, Frédéric Mahé, Laetitia Bariat, Claude Courties, Dominique Marie, Joséphine Ras, Romain Mauriac, Biogeosciences. 89Daniella Mella-Flores, Sophie Mazard, Florian Humily, Frédéric Partensky, Frédéric Mahé, Laetitia Bariat, Claude Courties, Dominique Marie, Joséphine Ras, Romain Mauriac, et al. Is the distribution of Prochlorococcus and Synechococcus ecotypes in the Mediterranean Sea affected by global warming? Biogeosciences, 8(9):2785-2804, 2011. Fluid dynamical niches of phytoplankton types. Silvia Francesco D&apos;ovidio, Séverine De Monte, Yves Alvain, Marina Dandonneau, Lévy, Proceedings of the National Academy of Sciences. 10743Francesco d'Ovidio, Silvia De Monte, Séverine Alvain, Yves Dandonneau, and Marina Lévy. Fluid dynamical niches of phytoplankton types. Proceedings of the National Academy of Sciences, 107(43):18366-18370, 2010. Upwelling mechanisms in the northwestern Alboran Sea. Tarek Sarhan, Jesus Garcıa Lafuente, Manuel Vargas, M Juan, Francisco Vargas, Plaza, Journal of Marine Systems. 234Tarek Sarhan, Jesus Garcıa Lafuente, Manuel Vargas, Juan M Vargas, and Francisco Plaza. Upwelling mechanisms in the northwestern Alboran Sea. Journal of Marine Systems, 23(4):317-331, 2000. Cumulative compressibility effects on slow reactive dynamics in turbulent flows. Prasad Perlekar, Roberto Benzi, Federico David R Nelson, Toschi, Journal of Turbulence. 143Prasad Perlekar, Roberto Benzi, David R Nelson, and Federico Toschi. Cumulative compressibility effects on slow reactive dynamics in turbulent flows. Journal of Turbulence, 14(3):161-169, 2013. Fixation probabilities in weakly compressible fluid flows. Abigail Plummer, Roberto Benzi, Federico David R Nelson, Toschi, Proceedings of the National Academy of Sciences. the National Academy of Sciences116Abigail Plummer, Roberto Benzi, David R Nelson, and Federico Toschi. Fixation probabilities in weakly compressible fluid flows. Proceedings of the National Academy of Sciences, 116(2):373-378, 2019. Fisher equation with turbulence in one dimension. Roberto Benzi, David R Nelson, Physica D: Nonlinear Phenomena. 23819Roberto Benzi and David R Nelson. Fisher equation with turbulence in one dimension. Physica D: Nonlinear Phenomena, 238(19):2003-2015, 2009. The Fokker-Planck Equation. Hannes Risken, SpringerHannes Risken. The Fokker-Planck Equation. Springer, 1996. Stochastic Methods: A Handbook for the Natural and Social Sciences. Crispin Gardiner, Springer Series in Synergetics. Crispin Gardiner. Stochastic Methods: A Handbook for the Natural and Social Sciences. Springer Series in Synergetics, 2009. Population genetics in compressible flows. Simone Pigolotti, Roberto Benzi, H Mogens, David R Jensen, Nelson, Physical Review Letters. 10812128102Simone Pigolotti, Roberto Benzi, Mogens H Jensen, and David R Nelson. Population genetics in compressible flows. Physical Review Letters, 108(12):128102, 2012. Submesoscale hotspots of productivity and respiration: Insights from high-resolution oxygen and fluorescence sections. H R Rachel, Dennis J Stanley, Zoe O McgillicuddyJr, Haley M Sandwith, Pleskow, Deep Sea Research Part I: Oceanographic Research Papers. 130Rachel HR Stanley, Dennis J McGillicuddy Jr, Zoe O Sandwith, and Haley M Pleskow. Submesoscale hotspots of pro- ductivity and respiration: Insights from high-resolution oxygen and fluorescence sections. Deep Sea Research Part I: Oceanographic Research Papers, 130:1-11, 2017. Turbulent vertical kinetic energy in the ocean mixed layer. A D&apos; Eric, Asaro, Journal of Physical Oceanography. 3112Eric A D'Asaro. Turbulent vertical kinetic energy in the ocean mixed layer. Journal of Physical Oceanography, 31(12): 3530-3537, 2001. Dispersal, eddies, and the diversity of marine phytoplankton. Sophie Clayton, Stephanie Dutkiewicz, Oliver Jahn, Michael J Follows, Limnology and Oceanography: Fluids and Environments. 31Sophie Clayton, Stephanie Dutkiewicz, Oliver Jahn, and Michael J Follows. Dispersal, eddies, and the diversity of marine phytoplankton. Limnology and Oceanography: Fluids and Environments, 3(1):182-197, 2013. Multiscale routes to supply nutrients through the Kuroshio nutrient stream. Takeyoshi Nagai, Sophie Clayton, Yusuke Uchiyama, Kuroshio Current: Physical, Biogeochemical, and Ecosystem Dynamics. Takeyoshi Nagai, Sophie Clayton, and Yusuke Uchiyama. Multiscale routes to supply nutrients through the Kuroshio nutrient stream. Kuroshio Current: Physical, Biogeochemical, and Ecosystem Dynamics, pages 105-125, 2019. High N2 fixation in and near the Gulf stream consistent with a circulation control on diazotrophy. B Jaime, Elana J Palter, Mar Ames, Afonso Goncalves Benavides, Julie Neto, Granger, H Pia, Katie S Moisander, Angelicque E Watkins-Brandt, White, Geophysical Research Letters. 4716Jaime B Palter, Elana J Ames, Mar Benavides, Afonso Goncalves Neto, Julie Granger, Pia H Moisander, Katie S Watkins- Brandt, and Angelicque E White. High N2 fixation in and near the Gulf stream consistent with a circulation control on diazotrophy. Geophysical Research Letters, 47(16):e2020GL089103, 2020.
[]
[ "POSITIVE JANTZEN SUM FORMULAS FOR CYCLOTOMIC HECKE ALGEBRAS", "POSITIVE JANTZEN SUM FORMULAS FOR CYCLOTOMIC HECKE ALGEBRAS" ]
[ "Andrew Mathas " ]
[]
[]
We prove a "positive" Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type A. That is, in the Grothendieck group, we show that the sum of the pieces of the Jantzen filtration is equal to an explicit non-negative linear combination of modules E ν f,e , which are modular reductions of simple modules for closely connected Hecke algebras in characteristic zero. The coefficient of E ν f,e in the sum formula is determined by the graded decomposition numbers in characteristic zero, which are known, and the characteristic of the field. As a consequence we see that the decomposition numbers of a cyclotomic Hecke algebra at an eth root of unity in characteristic p depend on the decomposition numbers of related cyclotomic Hecke algebras at ep r th roots of unity in characteristic zero, for r ≥ 0.Dedicated to Professor Gordon Douglass James, 31/12/1945-5/12/2020 This is the first paper that I have written about Jantzen sum formulas without my close collaborator and mentor Professor Gordon James. Gordon was a powerful and insightful mathematician who was a key player in shaping our current knowledge of the representation theory of the symmetric groups, Hecke algebras and Schur algebras. I feel privileged to have worked with him and I am proud to count him among my friends and colleagues. Even though Gordon did not contribute directly to this paper it is infused with his influence because it answers questions that Gordon and I discussed when we were working together.IntroductionJantzen filtrations have played a dominant role in representation theory since they were introduced by Jantzen in 1977[20]. In particular, stronger forms of the Lusztig and Kazhdan-Lusztig conjectures[24,33]say that the Jantzen filtrations and weight filtrations of Weyl modules coincide, results that are equivalent to (questions or) conjectures of Jantzen[21]. For quantum groups, complex semisimple Lie algebras, and Soergel bimodules these conjectures are now known to be true[3,7,12,25,44,45].Jantzen's observation for defining the Jantzen filtrations is extremely simple and elegant. Let p be a rational prime and M Z a finitely generated free Z-module that comes equipped with a non-degenerate bilinear form , Z . Then M Z has a filtration M ZLet G M be the Gram matrix of , Z with respect to some basis of M Z . By looking at the Smith normal form of G M , Jantzen observed the trivial but important fact thatwhere ν p is the usual p-adic valuation. In category O, a Weyl module ∆ λ is uniquely determined by the dimensions of its weight spaces. Applying this machinery to the weight spaces ∆ Λ µ of ∆ λ determines the dimensions of the µ-weight spaces in the Jantzen filtration of a Weyl module ∆ λ . This leads to the Jantzen sum formula of ∆ λ , which describes the sum k>0 J k (∆ λ ) in the Grothendieck group as an explicit Z-linear combination of more dominant Weyl module[22,II.8.19].One limitation of all of the Jantzen sum formulas in the literature is that they express k>0 J k (M ) in the Grothendieck group as a Z-linear combination of more dominant Weyl modules or Specht modules, 2000 Mathematics Subject Classification. 20G43, 20C08, 20C30, 05E10.
10.1007/s00209-021-02957-7
[ "https://arxiv.org/pdf/2106.15486v2.pdf" ]
235,669,582
2106.15486
23704086bf60ccda885e0df031f0f68139c75634
POSITIVE JANTZEN SUM FORMULAS FOR CYCLOTOMIC HECKE ALGEBRAS 2 Jul 2021 Andrew Mathas POSITIVE JANTZEN SUM FORMULAS FOR CYCLOTOMIC HECKE ALGEBRAS 2 Jul 2021 We prove a "positive" Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type A. That is, in the Grothendieck group, we show that the sum of the pieces of the Jantzen filtration is equal to an explicit non-negative linear combination of modules E ν f,e , which are modular reductions of simple modules for closely connected Hecke algebras in characteristic zero. The coefficient of E ν f,e in the sum formula is determined by the graded decomposition numbers in characteristic zero, which are known, and the characteristic of the field. As a consequence we see that the decomposition numbers of a cyclotomic Hecke algebra at an eth root of unity in characteristic p depend on the decomposition numbers of related cyclotomic Hecke algebras at ep r th roots of unity in characteristic zero, for r ≥ 0.Dedicated to Professor Gordon Douglass James, 31/12/1945-5/12/2020 This is the first paper that I have written about Jantzen sum formulas without my close collaborator and mentor Professor Gordon James. Gordon was a powerful and insightful mathematician who was a key player in shaping our current knowledge of the representation theory of the symmetric groups, Hecke algebras and Schur algebras. I feel privileged to have worked with him and I am proud to count him among my friends and colleagues. Even though Gordon did not contribute directly to this paper it is infused with his influence because it answers questions that Gordon and I discussed when we were working together.IntroductionJantzen filtrations have played a dominant role in representation theory since they were introduced by Jantzen in 1977[20]. In particular, stronger forms of the Lusztig and Kazhdan-Lusztig conjectures[24,33]say that the Jantzen filtrations and weight filtrations of Weyl modules coincide, results that are equivalent to (questions or) conjectures of Jantzen[21]. For quantum groups, complex semisimple Lie algebras, and Soergel bimodules these conjectures are now known to be true[3,7,12,25,44,45].Jantzen's observation for defining the Jantzen filtrations is extremely simple and elegant. Let p be a rational prime and M Z a finitely generated free Z-module that comes equipped with a non-degenerate bilinear form , Z . Then M Z has a filtration M ZLet G M be the Gram matrix of , Z with respect to some basis of M Z . By looking at the Smith normal form of G M , Jantzen observed the trivial but important fact thatwhere ν p is the usual p-adic valuation. In category O, a Weyl module ∆ λ is uniquely determined by the dimensions of its weight spaces. Applying this machinery to the weight spaces ∆ Λ µ of ∆ λ determines the dimensions of the µ-weight spaces in the Jantzen filtration of a Weyl module ∆ λ . This leads to the Jantzen sum formula of ∆ λ , which describes the sum k>0 J k (∆ λ ) in the Grothendieck group as an explicit Z-linear combination of more dominant Weyl module[22,II.8.19].One limitation of all of the Jantzen sum formulas in the literature is that they express k>0 J k (M ) in the Grothendieck group as a Z-linear combination of more dominant Weyl modules or Specht modules, 2000 Mathematics Subject Classification. 20G43, 20C08, 20C30, 05E10. where many of the coefficients are negative. This is "wrong" in the sense that k>0 J k (M ) can definitely be written as a non-negative linear combination of the images of the simple modules in the Grothendieck group. On the other hand, these negative coefficients in the sum formulas have to appear because, in general, the simple modules cannot be written as non-negative linear combinations of the Weyl modules in the Grothendieck group. It is natural to ask for a new version of the Jantzen sum formula that expresses k>0 J k (M ) as a non-negative linear combination of simple modules. Soon after Jantzen proved his sum formula, his student Schaper used Jantzen's sum formula for the general linear groups to prove a sum formula for the Specht modules of the symmetric groups [42]. Analogues of this result have since proved for the Specht modules of the Ariki-Koike algebras [18,19], which are cyclotomic Hecke algebras of type A that include the symmetric group and their Iwahori-Hecke algebras as special cases. In the literature, rather than working in the module category of the Hecke algebras, this result is always deduced by first proving a sum formula the Weyl modules of a related Schur algebra and then applying a Schur functor to deduce the sum formula for the Specht modules. In particular, the literature does not contain the proof of a sum formula for these algebras that takes place entirely inside the module category of Ariki-Koike algebras or symmetric groups. This paper proves a new positive Jantzen sum formulas for the Specht modules of the cyclotomic Hecke algebras of type A that writes k>0 J k (S λ ) as an explicit non-negative linear combination of certain modules E µ f,e . This sum formula is proved entirely within the module categories of the Hecke algebras. The modules E µ f,e that appear in our sum formulas are not, in general, simple and instead these modules are modular reductions of the simple modules of related cyclotomic Hecke algebras in characteristic zero. The coefficients that appear in the positive sum formula are described explicitly in terms of the (derivatives of graded) decomposition numbers. To state our main result we quickly introduce the notation that we need and refer the reader to later sections for the full definitions. Let F be a field of characteristic p ≥ 0 and let H Λ n (F ) be a cyclotomic Hecke algebra over F with a Hecke parameter ξ of quantum characteristic e (Definition 3.1). Let P Λ n be the set of ℓ-partitions. For λ ∈ P Λ n let S λ F be the corresponding Specht module for H Λ n (F ). If M is an H Λ n (F )-module let [M ] be its image in the Grothendieck group Rep H Λ n (F ) of H Λ n (F ). For f = ep r , for r ≥ 0, let ζ f ∈ C be a primitive f th root of unity and let {E ν C,f | ν ∈ K Λ f n } be the set of simple modules for a corresponding Hecke algebra H Λ f n (C) over C that has Hecke parameter ζ f ; see (4.17). By Proposition 4.18, there is a map of Grothendieck groups A f,e : Rep H Λ f n (C) −→ Rep H Λ n (F ) when p > 0. Let E ν f,e = A f,e E ν C,f and set E ν f,e = 0 if p = 0. Finally, let d C,f λµ (q) be the graded decomposition numbers for H Λ f n (C), for λ ∈ P Λ n and µ ∈ K Λ f n ; see (2.15). Then d C,f λµ (q) is a parabolic Kazhdan-Lusztig polynomial that, in principle, is known. Let (d C,f λµ ) ′ (1) be the derivative of d C,f λµ (q) evaluated at q = 1. Our main result is then the following: Main Theorem. Suppose that F is a field of characteristic p ≥ 0 and let λ ∈ P Λ n . In Rep H Λ n (F ), k>0 [J k (S λ F )] = µ∈K Λ n (d C,e λµ ) ′ (1)[E µ F,e ] + r>0 (p − 1)p r−1 f =ep r ν∈K Λ f n (d C,f λν ) ′ (1)[E ν f,e ]. Strikingly, in characteristic p the right hand side of this sum formula depends on modules for Hecke algebras at ep r th roots of unity. Moreover, by Lemma 4.13 below, all of the coefficients on the right hand side of the sum formula are non-negative integers, so this really is a positive Jantzen sum formula. When F is a field of characteristic zero the second sum over r > 0 on the right hand side of our main theorem vanishes vanishes. In this case, the new sum formula can be proved by assuming the deep fact that the Jantzen filtrations of graded Specht modules coincide with their grading filtrations. In fact, we do not prove our main result this way and, indeed, we cannot prove it this way because the Jantzen filtrations are only known to coincide with the grading filtrations for cyclotomic Hecke algebras of level 1, where this follows from work of Shan [43,Theorem 0.1]. (Shan actually proves that the Jantzen filtrations of the Weyl modules of the q-Schur algebras coincide with their grading filtrations. Using the Schur functor, it is possible to prove the corresponding result for the graded Specht modules.) The proof of our main theorem takes place entirely inside the module categories of the cyclotomic Hecke algebras. The main idea is to work with the formal characters of H Λ n (F )-modules (Definition 3.15). The key point is that character map is injective so the images of all modules in the Grothendieck group, including the images of the Jantzen filtration, are determined by their characters. This is essentially the same idea sketched above for computing the Jantzen sum formula of Weyl modules Using seminormal forms it is relatively easy to determine the "Jantzen characters" (Proposition 4.3). The proof of our main theorem follows by applying some combinatorial tricks using the formal characters of the graded Specht modules of the cyclotomic KLR algebras type A, which are isomorphic to the cyclotomic Hecke algebras. It is quite remarkable that the graded Specht modules enter this story, but their appearance in our arguments is almost accidental because instead of marking a direct connection between the Jantzen filtrations and the graded representation theory we only exploit some combinatorial shadows that are common to both settings. Here is a brief outline of the paper. Section 2 reviews the results that we need from the graded representation theory of the cyclotomic KLR algebras of type A. In particular, we introduce the qcharacter map for these algebras and use them to give a new proof that the graded decomposition matrices in positive characteristic factor through those in characteristic zero. Section 3 introduces the cyclotomic Hecke algebras and their character map. We define seminormal bases and use these to give a new factorisation of the Gram determinants that is compatible with taking characters. Section 4 contains the heart of the paper. We properly introduce the Jantzen filtrations and the Jantzen characters and then prove our main theorem by combining the results from the previous sections. Finally, section 5 shows how to prove the "classical" Jantzen sum formula using the framework of this paper. As the classical Jantzen sum formula is more explicit it takes more effort to prove than our positive sum formula, but it is worth the effort because the new "classical" sum formula that we obtain is nicer than the existing sum formula in the literature [19]. The definition of cyclotomic Hecke algebras used in this paper includes the degenerate cyclotomic algebras of [4] as the special case when the Hecke a parameter is one. Jantzen filtrations for these algebras have not appeared in the literature previously, so even the "classical" Jantzen sum formula that we obtain in the generate case is completely new. In this section we introduce the cyclotomic KLR algebras of type A, which are isomorphic to the cyclotomic Hecke algebras of type A. We will use these algebras to define the modules E µ f,e from the introduction and to prove some combinatorial identities that we use in Section 4.4 to compute characters for the Jantzen filtration. 2.1. Graded modules and algebras. In this paper all modules are finitely generated modules over a commutative ring with one. A graded module M is a module with a Z-grading M = d∈Z M d . Similarly, graded algebra A is a Z-graded algebra A = d∈Z A d with A d A e ⊆ A d+e , for all d, e ∈ Z. If A is a graded algebra then a graded A-module is a graded (right) A-module such that M d A e ⊆ M d+e , for d, e ∈ Z. If F is a field and M is a graded F -module then the graded dimension of M is dim q M = d∈Z (dim M d )q d ∈ N[q, q −1 ], where q is an indeterminate over Z. Given an integer s let q s M be the graded module obtained from M by shifting the grading of M by s so that (q s M ) d = M d−s . More generally, given a Laurent polynomial s(q) = d s d q d ∈ N[q, q −1 ] let s(q)M = d∈Z (q d M ) ⊕s d . Then dim q s(q)M = s(q) dim q M . Let A-grMod be the category of finitely generated graded (right) A-modules with degree preserving homomorphisms. If A is a graded algebra let A be the (ungraded) algebra obtained by forgetting the grading on A. Similarly, if M is a graded A-module let M be the (ungraded) A-module obtained by forgetting the grading on M . This defines an exact functor from the category of finite dimensional graded A-modules to the category of finite dimensional A-modules. 2.2. Graded cellar algebras. Cellular algebras, which were introduced by Graham and Lehrer [13], provide a convenient framework for constructing the simple modules of an algebra. Let R be a commutative ring with one. . Let A be a Z-graded R-algebra that is free and of finite rank. A graded cell datum for A is an ordered quadruple (P, T, C, deg) where (P, ≥) is a finite poset, T (λ) is a finite set for λ ∈ P, C : λ∈P T (λ) × T (λ) −→ A; (s, t) → c st and deg : λ∈P T (λ) −→ Z; t → deg t; are maps such that C is injective and the following hold: C 0 ) For s, t ∈ T (λ) and λ ∈ P, the element c st ∈ A is homogeneous of degree deg s + deg t. C 1 ) The set {c st | s, t ∈ T (λ), λ ∈ P} is a basis of A. C 2 ) If a ∈ A, s, t ∈ T (λ) and λ ∈ P then there exist scalars r tv (a) ∈ R that do not depend on s, such that c st a ≡ v∈T (λ) r tv (a)c sv (mod A >λ ) , where A >λ is the R-submodule of A spanned by {c uv | u, v ∈ T (µ) for µ > λ}. C 3 ) There is an unique algebra anti-isomorphism * : A −→ A such that c * st = c ts , for all s, t ∈ T (λ) and λ ∈ P. A graded cellular algebra is an algebra that has a graded cell datum. A cellular algebra is a graded cellular algebra that is concentrated in degree zero. That is, it has a graded cell datum (P, T, C, deg) with deg s = 0. for all s. In this case, (P, T, C) is a cell datum for A. In particular, note that if A is a graded cellular algebra then, forgetting the grading, A is a cellular algebra. Cellular algebras were defined by Graham and Lehrer [13] with the natural extension of their definitions and results to the graded setting being given in [14]. The proofs of all of the results in this section can be found in these two papers. Let A be a (graded) cellular algebra with cell datum (P, T, C, deg). Using the definitions it is easy to see that for each λ ∈ P there exist a (right) cell module C λ with basis {c t | t ∈ T (λ)} and with A-action c t a = v∈T (λ) r tv (a)c v , for a ∈ A, where the scalar r tv (a) ∈ R is from (C 2 ). Similarly, let C λ * be the left cell module indexed by λ, which is isomorphic to C λ as a vector space and where the A-action is given by ac t = v∈T (λ) r tv (a * )c v , for a ∈ A, Extending the notation from C 2 , let A ≥λ = c uv | u, v ∈ T (µ) for µ ≥ λ R and define the (A, A)bimodule map c λ by c λ : C λ * ⊗ C λ −→ A ≥λ /A >λ ; c s ⊗ c t → c st + A >λ , for s, t ∈ T (λ). Using (C 2 ) and (C 3 ), it follows that C λ has a homogeneous symmetric bilinear form , : C λ × C λ −→ R of degree zero such that (2.2) x · c λ (y ⊗ z) = x, y z, for x, y, z ∈ C λ . In particular, c t c uv = c t , c u c v , for t, u, v ∈ T (λ). This form is associative in the sense that xa, y = x, ya * , for all x, y ∈ C λ and a ∈ A. Hence, rad C λ = {x ∈ C λ | x, y = 0 for all y ∈ C λ } is a (graded) A-submodule of C λ . Set D λ = C λ / rad C λ . Then D λ is a (graded) A-module. Let D be an irreducible graded A-module. Then q k D is a non-isomorphic irreducible graded Amodule, for k ∈ Z \ {0}. If M is a graded A-module then the graded decomposition multiplicity of D in M is the Laurent polynomial (2.3) [M : D] q = k∈Z [M : q k D]q k ∈ N[q, q −1 ], where [M : q k D] is the multiplicity of q k D as a composition factor of M . The main results from the theory of (graded) cellular algebras are: 2.4. Theorem (Graham and Lehrer [13,14]). Let F be a field and suppose that A is a graded cellular algebra. Then: a) Let µ ∈ P and suppose that D µ = 0. Then D µ is a self-dual graded irreducible A-module b) Let P 0 = {µ ∈ P | D µ = 0}. Then {q k D µ | µ ∈ P 0 and k ∈ Z} is a complete set of pairwise non-isomorphic irreducible graded A-modules. c) If λ ∈ P and µ ∈ P 0 . Then [S µ : D µ ] q = 1 and [S λ : D µ ] q = 0 only if λ ≥ µ. We need the theory of graded cellular algebras only to explain the definition of the Specht modules of the cyclotomic Hecke algebras. The bilinear form on the Specht modules is the key to defining their Jantzen filtrations. Our main results also use the graded Specht modules, although we have to approach this indirectly because the bilinear form on the graded Specht modules is degenerate unless the graded Specht module is irreducible. 2.3. Cyclotomic KLR algebras. Fix an integer e ∈ {2, 3, 4, . . .} ∪ {∞} and let Γ = Γ e be the quiver with vertex set I = Z/eZ (we set eZ = {0} when e = ∞), and with edges i → i + 1, for i ∈ I. To the quiver Γ we attach the standard Lie theoretic data of a Cartan matrix (c ij ) i,j∈I , fundamental weights {Λ i | i ∈ I}, the positive weight lattice P + e = i∈I NΛ i , the positive root lattice Q + e = i∈I Nα i and we let (·, ·) be normalised invariant form determined by (α i , α j ) = c ij and (Λ i , α j ) = δ ij , for i, j ∈ I. Let Q + e,n = {β ∈ Q + e | i∈I (Λ i , β) = n} and for β ∈ Q + e,n let I β = {i ∈ I n | β = α i1 + · · · + α in }. Following Rouquier [40], let u and v be indeterminates and for i, j ∈ I define polynomials Q ij (u, v) =                (u − v)(v − u) if i ⇆ j, u − v if i → j, v − u if i ← j, 0 if i = j, 1 if i / -j. To define the cyclotomic KLR algebras of type A, fix a non-negative integer n ≥ 0 and a dominant weight Λ ∈ P + e and let ℓ = i∈I (Λ, α i ). 2.5. Definition (Khovanov and Lauda [26,27] and Rouquier [40]). Let e ∈ {2, 3, . . .} and fix β ∈ Q + e,n . The cyclotomic KLR algebra R Λ β of type Γ, determined by the weights (Λ, β), is the unital associative Z-algebra with generators {ψ 1 , . . . , ψ n−1 } ∪ {y 1 , . . . , y n } ∪ {e(i) | i ∈ I β } and relations y (Λ,αi 1 ) 1 e(i) = 0 e(i)e(j) = δ ij e(i), i∈I β e(i) = 1, y r e(i) = e(i)y r , ψ r e(i) = e(s r ·i)ψ r , y r y s = y s y r , ψ r ψ s = ψ s ψ r if |r − s| > 1, ψ r y s = y s ψ r if s = r, r + 1, ψ r y r+1 e(i) = (y r ψ r + δ irir+1 )e(i), y r+1 ψ r e(i) = (ψ r y r + δ ir ir+1 )e(i), ψ 2 r e(i) = Q irir+1 (y r , y r+1 )e(i) (ψ r ψ r+1 ψ r − ψ r+1 ψ r ψ r+1 )e(i) = δ ir ir+1 Q ir,ir+1 (y r+2 , y r+1 ) − Q ir ,ir+1 (y r , y r+1 ) y r − y r+2 e(i) for i, j ∈ I β and all admissible r and s. Set R Λ n = β∈Q + e,n R Λ β . The reader can check that the right hand side of the last relation is a polynomial in y r , y r+1 and y r+2 . Importantly, the algebras R Λ β and R Λ n are Z-graded, where the grading is determined by deg e(i) = 0, deg y r = 2 and deg ψ r e(i) = −(α ir , α ir+1 ), for all admissible i ∈ I n and 1 ≤ r ≤ n. If A is any ring let R Λ β (A) = R Λ β ⊗ Z A and R Λ n (A) = R Λ n ⊗ Z A be the corresponding cyclotomic KLR algebras over A. From the relations, R Λ n has a unique homogeneous anti-automorphism * of degree 0 that fixes all of the generators of R Λ n . Let S n be the symmetric group of degree n, which we consider as a Coxeter group with its standard set of Coxeter generators {s 1 , . . . , s n−1 }, where s r = (r, r + 1) for 1 ≤ r < n. For each w ∈ S n fix a reduced expression w = s r1 . . . s r k for w (that is, fix such a word with k minimal), and define ψ w = ψ r1 . . . ψ r k ∈ R Λ n . In general, the element ψ w depends on the choice of reduced expression but any fixed choice of reduced expression suffices for the results that follow. 2.4. Formal characters. Formal characters are a useful tool in the representation theory of the affine Hecke algebra. We need an analogue of these characters for R Λ n -modules. To this end, let A = Z[q, q −1 ] and let A[I n ] = A[i | i ∈ I n ] be the free A-module with basis {i | i ∈ I n }. If F is a field let Rep R Λ n (F ) be the Grothendieck group of finitely generated graded R Λ n (F )-modules and if M is an R Λ n -module let [M ] be its image in Rep R Λ n (F ). Then Rep R Λ n (F ) is the free A-module with basis {[D µ ] | µ ∈ K Λ n }, where q acts on Rep R Λ n (F ) by grading shift, so that [qM ] = q[M ], for M ∈ Rep R Λ n (F ) . Let L n be the positively graded commutative subalgebra of R Λ n that is generated by y 1 , . . . , y n and the idempotents e(i), for i ∈ I n . It is easy to see that the irreducible representations of L n are indexed by the n-tuples i ∈ I n such that e(i) = 0. Let Rep L n be the Grothendieck group of finitely generated graded L n -modules where q acts by grading shift, which we view as a free A-submodule of A[I n ]. 2.6. Definition. The graded (formal) character is the linear map ch q : Rep R Λ n −→ Rep L n given by ch q M = i∈I n dim q M e(i) i, for M ∈ Rep R Λ n . The map ch q can be viewed as an exact functor, namely the restriction functor, from the category of (graded) R Λ n -modules to the category of graded L n -modules. For our purposes it is enough to think of ch q as a linear map ch q : Rep R Λ n −→ A[I n ]. 2.7. Proposition. Let F be a field. Then the graded character map ch q : Rep R Λ n (F ) −→ A[I n ] is injective. Proof. Khovanov and Lauda [26,Theorem 5.17] state this result for the Grothendieck group of the corresponding affine KLR algebra, however, this immediately implies the result for the quotient cyclotomic KLR Hecke algebra R Λ n (F ). 2.5. Tableau combinatorics. Before we can define a homogeneous basis for R Λ n , and introduce the graded Specht modules for R Λ n , we need to introduce the combinatorics of standard tableaux. This combinatorics plays a key role in proofs of our main results. Recall that we fixed integers n and ℓ in the paragraph before Definition 2.5. A partition of m is a weakly decreasing sequence λ = (λ 1 , λ 2 , . . . ) of non-negative integers such that |λ| = λ 1 + λ 2 + · · · = m. An ℓ-partition of n is an ℓ-tuple λ = (λ (1) , . . . , λ (ℓ) ) of partitions such that |λ (1) | + · · · + |λ (ℓ) | = n. We identify the ℓ-partition λ with its diagram, which is the set of nodes {(l, r, c) | 1 ≤ c ≤ λ (l) r for 1 ≤ l ≤ ℓ}. We think of the diagram of λ as an ℓ-tuple of left-justified arrays of boxes. This allows us to talk of the components, rows and columns of λ. The set of ℓ-partitions of n is a poset under the dominance order , where λ ⊲ µ if l−1 k=1 |λ (k) | + i j=1 λ (l) j ≥ l−1 k=1 |µ (k) | + i j=1 µ (l) j , for 1 ≤ l ≤ ℓ and i ≥ 1. If λ µ and λ = µ then write λ ⊲ µ. Let P Λ n = P Λ ℓ,n be the set of ℓ-partitions of n, which we consider as a poset under dominance. Fix λ ∈ P Λ n . A λ-tableau is a bijective map t : λ −→ {1, 2, . . . , n}, which we identify with a labelling of (the diagram of) λ by {1, 2, . . . , n}. Let Shape(t) = λ be the shape of t. For example, are both λ-tableaux, where λ = (3, 1|1 2 |3) ∈ P Λ 13 , where ℓ = 3. A λ-tableau is standard if, in each component, its entries increase along rows and down columns. For example, both of the tableaux above are standard. Let Std(λ) be the set of standard λ-tableaux. If P is any set of ℓ-partitions let Std(P) = λ∈P Std(λ). Similarly set Std 2 (P) = {(s, t) | s, t ∈ Std(λ) for λ ∈ P}. If t is a λ-tableau set Shape(t) = λ and let t ↓m be the subtableau of t that contains the numbers {1, 2, . . . , m}. If t is a standard λ-tableau then Shape(t ↓m ) is a ℓ-partition for all m ≥ 0. Extend the dominance ordering to Std(P Λ n ) by defining s t if Shape(s ↓m ) Shape(t ↓m ) for 1 ≤ m ≤ n. As before, write s ⊲ t if s t and s = t. Even though we write P Λ n for the set of ℓ-partitions of n, so far none of the definitions in this section depend on Λ ∈ P + e . A multicharge for Λ is an ℓ-tuple κ = (κ 1 , . . . , κ ℓ ) ∈ Z ℓ such that (2.9) (Λ, α i ) = #{1 ≤ l ≤ ℓ | κ l ≡ i (mod e) }, for all i ∈ I. The residue of a node A = (l, r, c) ∈ λ is r(A) = κ l + c − r + eZ ∈ I. If t is a standard λ-tableau and t(A) = m, where 1 ≤ m ≤ n, then the residue of m in t is r m (t) = r(A). The residue sequence of t is r(t) = r 1 (t), r 2 (t), . . . , r n (t) ∈ I n . Set Std i (λ) = {t ∈ Std(λ) | r(t) = i}) and Std i (P Λ n ) = λ∈P Λ n Std i (λ). A node A is an addable node of λ if r(A) = i and A / ∈ λ and λ ∪ {A} is the (diagram of) a ℓ-partition of n + 1. Similarly, a node B is a removable node of λ if r(B) = i and B ∈ λ and λ \ {B} is a ℓ-partition of n − 1. Let Add i (λ) and Rem i (λ) be the sets of addable and removable i-nodes of λ. Let ≤ be the lexicographic order n the set of nodes. If A is a removable node of λ define d A,e (λ) = #{B ∈ Add i (λ) | A < B} − #{B ∈ Rem i (λ) | A < B}. Following Brundan, Kleshchev and Wang [6, §3.5], define the e-degree of a standard tableau t by (2.10) deg e (t) = deg e (t ↓(n−1) ) + d A,e (µ), if n > 0, 0, if n = 0, where A = t −1 (n) is the node in t that contains n, For convenience, set deg 0 (t) = 0 for all t. There is unique standard λ-tableaux t λ such that t λ t, for all t ∈ Std(λ). The tableau t λ has the numbers 1, 2, . . . , n entered in order from left to right along the rows of t λ (1) , and then t λ (2) , . . . , t λ (ℓ) . For example, the first tableau in (2.5) is t λ for λ = (3, 1|1 2 |3). Given a standard λ-tableau t define a permutation d(t) ∈ S n by t = t λ d(t). 2.6. Specht modules, simple modules and almost simple modules. The main results in this paper give explicit, cancellation free, descriptions of the Jantzen sum formula for the (ungraded) Specht modules. This section defines the modules that we use to describe these filtrations. As we now recall, the algebra R Λ n is a graded cellular algebra in the sense of Section 2.2. For λ ∈ P Λ n set i λ = r(t λ ) = (i λ 1 , . . . , i λ n ). For 1 ≤ m ≤ n set λ ↓m = Shape(t λ ↓m ) and let α λ m ∈ λ be the node such that t λ (α λ m ) = m. Define the polynomial y λ = n k=1 y a k (λ) r , where a k (λ) = #{α ∈ Add i λ k (λ ↓k ) | α < α λ k }. Then y λ is homogeneous of degree 2 deg t λ . For (s, t) ∈ Std 2 (λ) define ψ st = ψ * d(s) e(i λ )y λ ψ d(t) , By construction, ψ st is homogeneous and one can show that deg e ψ st = deg e s + deg t. Jun Hu and the author [14] proved that these elements give a graded cellular basis for R Λ n (F ), when F is a field. Building on this result, Ge Li [31] proved that R Λ n is a Z-free graded cellular algebra. 2.11. Theorem (Li [31]). The algebra R Λ n is free as an Z-module of rank ℓ n n!. Moreover, R Λ n is a graded cellular algebra with homogeneous cellular basis {ψ st | (s, t) ∈ Std 2 (P Λ n )}. Subsequently, Kang and Kashiwara [23,Theorem 4.5] generalised this result and proved that the cyclotomic KLR Hecke algebras indexed by symmetrisable Cartan matrices are Z-free. For this paper, the fact that R Λ n (Z) is a cellular algebra is very important. Since R Λ n is a cellular algebra, if a ∈ R Λ n and (s, t) ∈ Std 2 (λ), for λ ∈ P Λ n , then we can write ψ st a ≡ u∈Std(λ) r u ψ ut (mod R ⊲λ n ) , where the scalar r u ∈ Z depends only on a, s and u (and not t) and where R ⊲λ n is the two-sided ideal of R Λ n spanned by {ψ uv | (u, v) ∈ Std(µ) where µ ⊲ λ}. Define the graded Specht module S λ Z to the free Z-module with homogeneous basis {ψ s | s ∈ Std(λ)}, where deg ψ s = deg e s, and with R Λ n -action (2.12) ψ s a = u∈Std(λ) r u ψ u , for a ∈ R Λ n and s ∈ Std(λ). In particular, it follows that if i ∈ I n then ψ s e(i) = δ ii s ψ s . For any ring F let S λ F = S λ Z ⊗ Z F . Then S λ F is a graded R Λ n (F )-module. We abuse notation and write ψ s , instead of ψ s ⊗ Z 1 F , for the basis elements of S λ F . The cellular algebra axioms imply that S λ F has a bilinear form , : S λ F × S λ F −→ F determined by (2.13) ψ s , ψ t ψ u = ψ st ψ u , for s, t, u ∈ Std(λ). This form is associative in the sense that ax, y = x, a * y , for all a ∈ R Λ n and x, y ∈ S λ F . Hence, rad S λ F = {x ∈ S λ F | x, y = 0 for all y ∈ S λ F } is a graded R Λ n (F )-submodule of S λ F . Observe that these definitions make sense for any ring and, in particular, these definitions are valid even over F = Z. 2.14. Definition. Let µ ∈ P Λ n . Define E µ Z = S µ Z / rad S µ Z . For any ring F define E µ F = E µ Z ⊗ Z F and D µ F = S λ F / rad S λ F . By definition, E µ Z = D µ Z but in positive characteristic the R Λ n (F )-modules E µ F and D µ F are not necessarily isomorphic. By the theory of (graded) cellular algebras [14,Corollary 2.11], if F is a field then the module D µ F is either zero or absolutely irreducible and self-dual. Moreover, up to grading shift, every irreducible R Λ n (F )-module arises uniquely in this way. Let F be a field and set K Λ n = K Λ e,n = {µ ∈ P Λ n | D µ F = 0}. Then {D µ F d | µ ∈ K Λ n d d ∈ Z} is a complete set of pairwise non-isomorphism irreducible graded R Λ n -modules by [14,Corollary 5.11]. (The graded irreducible R Λ n (F )-modules were first classified in [5,Theorem 5.10].) Moreover, the set K Λ n is independent of the field F . For λ ∈ P Λ n and µ ∈ K Λ n define the graded decomposition number (2.15) d F,e λµ (q) = [S λ F : D µ F ] q = d∈Z [S λ F : D µ F d ]q d ∈ N[q, q −1 ]. In particular, d C,e λµ (q) is a graded decomposition number for R Λ n (C) in characteristic zero. The graded decomposition matrix of R Λ n (F ) is the unitriangular matrix D F,e (q) = d F,e λµ (q) , where λ ∈ P Λ n and µ ∈ K Λ n and where the rows are columns are ordered lexicographically. The following result plays a key role in this paper. We give the proof, because we need the underlying ideas below and because the proof is quite short, with the original reference [37] not being readily available. Given integers d and d ′ write d | d ′ if d ′ = ad, for some a ∈ Z. In particular, note that d | 0, for all d ∈ Z. 2.16. Theorem ( [37, Theorem 3.7.4 and Theorem 3.7.5]). Suppose that F is a field and µ ∈ K Λ n . Then: a) The module E µ Z is a Z-free Z-graded R Λ n (Z)-module b) If F = Q then E µ Q ∼ = D µ Q is an absolutely irreducible self-dual graded R Λ n (Q)-module c) If λ ∈ P Λ n then [S λ F : D µ F ] q = ν d C,e λν (q)[E ν F : D µ F ] q . Proof. Let G λ = ψ s , ψ t s,t∈Std(µ) be the Gram matrix of the integral graded Specht module S λ . Let N = # Std(µ) so that S λ Z is a free Z-module of rank N . Since Z is a principal ideal domain there exist positive integers d 1 , . . . , d N such that d 1 | d 2 | · · · | d N and two homogeneous bases {u i } and {v i } of the graded Specht module S λ Z such that u i , v j = δ ij d i , for 1 ≤ i, j ≤ N . That is, the diagonal matrix diag(d 1 , . . . , d N ) is the Smith normal form of G λ . Therefore, rad S λ Z is free as an Z-module with basis {u i | d i = 0} and E µ Z is free as a Z-module with basis {u i + rad S λ Z | d i = 0}, proving (a). (In the same way, observe that {v i | d i = 0} is a basis of rad S λ Z and {v i + rad S λ Z | d i = 0} is a basis of E µ Z .) By definition, E µ Q = D µ Q , so part (b) follows from the general theory of (graded) cellular algebras; specifically, see [14,Theorem 2.10]. The same result says that if F is a field then D µ F is a self-dual absolutely irreducible graded R Λ n -module. Let F be a field and consider (c). Taking graded characters is exact and commutes with base change, so ch q S λ F = ch q S λ Q = ν∈K Λ n d C,e λν (q) ch q D ν Q = ν∈K Λ n d C,e λν (q) ch q E ν Q = ν∈K Λ n d C,e λν (q) ch q E ν F = ν∈K Λ n d C,e λν (q) µ∈K Λ n [E ν F : D µ F ] q ch q D µ F = µ∈K Λ n ν∈K Λ n d C,e λν (q)[E ν F : D µ F ] q ch q D µ F . On the other hand, ch q S λ F = µ d F,e λ,µ (q) ch q D µ F , so part (c) follows by Proposition 2.7. Let a F,e νµ (q) = [E ν F : D µ F ] q , for µ, ν ∈ K Λ n . The matrix A F,e (q) = a F,e νµ (q) µ,ν∈K Λ n is the (graded) adjustment matrix of R Λ n (F ). Notice that Theorem 2.16(c) is equivalent to the following factorisation of the graded decomposition matrix: (2.17) D F,e (q) = D C,e (q)A F,e (q). The adjustment matrix A F,e (q) coincides with the matrix defined by Brundan and Kleshchev [5, Theorem 5.17] using different arguments. The arguments of [5] amount to carefully choosing a Z-lattice for the simple module D µ Q . The proof of Theorem 2.16 shows that the ψ-basis, by virtue of Theorem 2.11, automatically chooses a Z-lattice E µ Z for D µ Q . 2.18. Corollary. Suppose that λ ∈ P Λ n . Then ch q S λ F = µ∈K Λ n d C,e λµ (q) ch q E µ F . Let : Z[q, q −1 ] −→ Z[q, q −1 ] be the unique linear involution such that q = q −1 . Abusing notation, extend this to a map : Rep L n −→ Rep L n . 2.19. Corollary. Suppose that µ, ν ∈ K Λ n . Then ch q E ν F = ch q E ν F and a F,e νµ (q) = a F,e νµ (q). Proof. By construction and Theorem 2.16(b), ch q E ν F = ch q E ν Q = ch q D ν Q . Therefore, ch q E ν F = ch q E ν F since D ν Q is a self-dual graded R Λ n (Q)-module. This implies the second claim because ch q E ν F = ν∈K Λ n a F,e νµ (q) ch q D µ F and D µ F is self-dual. Let ν ∈ K Λ n . Since [E ν F : D ν F ] q = 1 it follows that E ν F is self-dual if and only if E ν F = D ν F . Cyclotomic Hecke algebras We are now ready to introduce the cyclotomic Hecke algebras of type A, and their Specht modules, which are the main focus of this paper. We define several different bases for these algebras and use them to compute the characters of the sum of modules in the Jantzen filtration of the Specht modules. 3.1. Cyclotomic Hecke algebras of type A. Let R be a commutative ring with 1 and fix nonnegative integers n, ℓ ≥ 0. If v ∈ R and k ∈ Z define the v-quantum integer [k] v = 1 + v + · · · + v k−1 , if k ≥ 0, −v −1 + v −2 − · · · − v −k , otherwise. Observe that if v = 1 then [k] v = (v k − 1)/(v − 1) and if v = 1 then [k] v = k. We use the following definition of the cyclotomic Hecke algebras of type A, following [16, §2]. 3.1. Definition. The cyclotomic Hecke algebraof type A with Hecke parameter v ∈ R × and ℓ-charge κ = (κ 1 , . . . , κ ℓ ) ∈ Z ℓ is the unital associative R-algebra H κ v = H κ v (A) with generators T 1 , . . . , T n−1 , L 1 , . . . , L n and relations ℓ l=1 (L 1 − [κ l ] v ) = 0, (T r − v)(T r + 1) = 0, L r L t = L t L r , T r T s = T s T r if |r − s| > 1, T s T s+1 T s = T s+1 T s T s+1 , T r L t = L t T r , if t = r, r + 1, L r+1 = T r L r T r + T r , where 1 ≤ r < n, 1 ≤ s < n − 1 and 1 ≤ t ≤ n. As is explained in [16, §2], if v = 1 these algebras are isomorphic to the Ariki-Koike algebras, which were introduced in [1], and when v = 1 they are isomorphic to the degenerate Ariki-Koike algebras. By definition, H κ v has a unique anti-isomorphism * that fixes every generator. Recall from Section 2.3 that S n is the symmetric group of degree n with distinguished Coxeter generators s 1 , . . . , s n−1 . If w ∈ S n let L(w) be the minimal length of w as a product of the Coxeter generators. A word w = s i1 . . . s i k is a reduced expression for w if k = L(w), for 1 ≤ i j < n. As the braid relations hold in H Λ n , if w = s i1 . . . s i k is reduced then T w = T i1 . . . T i k ∈ H Λ n depends only on w and not on the choice of reduced expression. These results are well-known and their proofs can be found, for example, in [34,Chapter 1]. By the results of [2,10], and in view of [16,Theorem 2.8], H κ v is cellular algebra with cellular basis indexed by Std 2 (P Λ n ), as in Theorem 2.11. Hence, exactly as before (except that we no longer have a grading), for each ℓ-partition λ ∈ P Λ n we have a Specht module S λ , which is a left H κ v -module. Let F be a field. A non-zero element ξ ∈ F has quantum characteristic e if e is the smallest positive integer such that [e] ξ = 0 -set e = ∞ if [k] ξ = 0, for k > 0. Given κ ∈ Z ℓ define a dominant weight Λ = Λ κ ∈ P + e by using (2.9). Define H Λ n (F ) = H κ ξ (F ). Unravelling the notation, the cyclotomic relation in H Λ n (F ) can be written as (3.2) 0 = ℓ l=1 (L 1 − [κ l ] ξ ) = i∈I (L 1 − [i] ξ ) (Λ,αi) , where [i] ξ has the obvious meaning, for i ∈ I. We can now give the relationship between the cyclotomic KLR and Hecke algebras of type A. 3.3. Theorem (Brundan and Kleshchev [5], Rouquier [40]). Let F be a field. Suppose that ξ ∈ F has quantum characteristic e and that Λ ∈ P + e . Then H Λ n (F ) ∼ = R Λ n (F ) as ungraded algebras. Notice that Theorem 3.3 implies that, up to isomorphism, H Λ n (F ) depends only on e, Λ and F , and not on the particular choice of ξ ∈ F with quantum characteristic e. As explained in Section 3.3 below, we will always assume that e is finite. 3.2. The Murphy basis. Following [10], this section defines a cellular basis of H Λ n that we will use to define Specht modules for H Λ n . Let λ ∈ P Λ n and define elements m λ = u λ w∈S λ T w , where ℓ l=2 |λ (1) |+···+|λ (l−1) | m=1 (L m − [κ l ]). For s, t ∈ Std(λ) define m st = T * d(s) m λ T d(t) . By definition, m * st = m ts . 3.4. Theorem (Dipper, James, Mathas [10,Theorem 3.26]). For any ring R, H Λ n is an R-free cellular algebra with cellular basis m st s, t ∈ Std(λ) for λ ∈ P Λ n . 3.5. Remark. The paper [10] only proves Theorem 3.4 in the case when ξ = 1. However, the argument given in [10] extends without change to the case when ξ = 1 because the result is bootstrapped from the Murphy basis of the Iwahori-Hecke algebra of the symmetric group [39], which is a cellular basis for all ξ by [34,Chapter 3]. Based on Theorem 3.4, all of the results in this paper hold for the degenerate cyclotomic Hecke algebras of type A, which correspond to the case when ξ = 1 (and e is the characteristic of R). As in Section 2.2, for each λ ∈ P Λ n let S λ be the Specht module determined by the cellular basis of Theorem 3.4. Then S λ is free as an R-module with basis {m t | t ∈ Std(λ)} and it comes equipped with an inner product , that is uniquely determined by m s m tv = m s , m t m v , for s, t, v ∈ Std(λ). Recall from (2.15) that d F,e λµ (q) = [S λ F : D µ F ] q is a graded decomposition number of R Λ n (F ), for λ ∈ P Λ n and µ ∈ K Λ n . By [6] or [14, §5.2], under the isomorphism of Theorem 3.3 the modules S λ F and S λ F coincide, once we forget the grading. In more detail, we have: 3.6. Corollary. Let λ ∈ P Λ n and µ ∈ K Λ n . Then S λ F ∼ = S λ F and D µ F ∼ = D µ F . Consequently, [S λ F : D µ F ] = d F,e λµ (1). Let D F,e = d F,e λµ be the decomposition matrix of H Λ n (F ). Then D F,e = D F,e (1) by Corollary 3.6, where D F,e (1) = D F,e (q)| q=1 is the graded decomposition matrix of Section 2.3 evaluated at q = 1. Seminormal forms. This section recalls the construction of the seminormal basis of H Λ n (K) in the semisimple case, so all of these results in this section, ultimately, go back to the seminal work of Young [47]. We follow the general framework from [36] but see also [1,16,35]. For the rest of this paper we assume that e < ∞ in Definition 2.5 or, equivalently, that ξ is not a root of unity in Definition 3.1. By [16,Corollary 2.10], up to isomorphism, we can always assume that e is finite, so there is no loss of generality in making this assumption. The advantage of assuming that e is finite is that this is allows us to adjust the multicharge of (2.9) modulo e in (3.7) below. Let F be a field. As in (2.9) we fix an ℓ-charge κ = (κ 1 , . . . , κ ℓ ) ∈ Z ℓ . For the rest of this paper, we impose the additional assumption that (3.7) κ 1 − 2(ℓ − 1)n ≥ · · · ≥ κ ℓ−1 − 2n ≥ κ ℓ ≥ n. Implicitly, this assumption fixes a choice of lattice in a modular system for H Λ n (F ) and, in principle, the Jantzen filtrations that we construct below depend on the choices in (3.7). In practice, we need (3.7) for the results on seminormal forms that we use below because (3.7) ensures that standard tableaux are uniquely determined by their content sequences. This said, assumption (3.7) is only a technical convenience because, by (2.9), the dominant weight Λ = Λ κ , and hence the algebra H Λ n (F ), depend only on the multiset {κ 1 , . . . , κ ℓ }. In particular, H Λ n (F ) does not depend on (3.7). As above, consider the cyclotomic Hecke algebra H Λ n = H Λ n (F ) with Hecke parameter ξ ∈ F of quantum characteristic e < ∞ and cyclotomic parameters [κ m ] ξ , for 1 ≤ m ≤ ℓ, where κ = (κ 1 , . . . , κ ℓ ) ∈ Z ℓ is a fixed choice of ℓ-charge. Let Let s, t ∈ Std(λ), for λ ∈ P Λ n . Define ([κ 1 ] z , . . . , [κ ℓ ] z ). Let H Λ n (K) = H Λ n (O) ⊗ O K. Then H Λ n (K) is a split semisimple algebra. Consider the field F as an O-algebra where x acts as 0. Then H Λ n (F ) ∼ = H Λ n (O) ⊗ O F . For λ ∈ P Λ n let S λ O be the (ungraded) Specht module for H Λ n (O) determined by the {m st } cellular basis and λ. Then S λ K = S λ O ⊗ O K is the Specht module for H Λ n (K). Let t = (t (1) | . . . |t (ℓ) ) ∈ Std(λ), for λ ∈ P Λ n . Supposef st = F s m st F t , where F t = n k=1 s∈Std(P Λ n ) c k (s) =c k (t) L k − [c k (s)] z [c k (t)] z − [c k (s)] z . Equation (3.8) implies that (3.9) f st ≡ m st + u⊲s,v⊲t r uv m uv (mod H ⊲λ n ) , for some r uv ∈ K. In particular, {f st } is a basis of H Λ n (K). Since the transition matrix between the two bases {m st } and {f st } is unitriangular it is not hard to see that {f st } is a cellular basis of H Λ n (K). Moreover, if λ ∈ P Λ n and s, t ∈ Std(λ) then S λ K ∼ = f st H Λ n (K). In fact, {f st } is a seminormal basis of H Λ n (K) in the sense of [16], which means that f st are simultaneous eigenvectors for L 1 , . . . , L n . Explicitly, in view of [19,Proposition 3.7], (3.10) L k f st = [c k (s)]f st and f st L k = [c k (t)]f st , for 1 ≤ k ≤ n. 1 This is the only place where assumption (3.7) is needed in the construction of the seminormal basis. That is, in this section (3.7) is only used to ensure that the content sequences separate the standard tableaux. The stronger form of (3.7) is used to define sequences of charged beta numbers in Chapter 5. These two formulas are equivalent since f * st = f ts . Using (3.10) and the definitions it follows that F u f st F v = δ su δ tv f st and hence that there exist scalars γ t ∈ K such that (3.11) f st f uv = δ tu γ t f sv , for all s, t ∈ Std(λ) and u, v ∈ Std(µ). In particular, it follows that F t = 1 γt f tt , for all t ∈ Std(P Λ n ). Suppose that s, tv ∈ Std(λ) and that v = t(r, r + 1). Then by [16,Proposition 3.11], f st T r = f sv − 1 [ρr (t)] f st , if t ⊲ v, [1+ρr (t)][1−ρr (t)] [ρr (t)] 2 f sv − 1 [ρr(t)] f st , if v ⊲ t, where ρ r (t) = c r (t) − c r+1 (t). Using this to compute γ v f vv = f vv f vv , when s = v, shows that: 3.12. Corollary ( [16, Corollary 3.10]). Suppose that t, v ∈ Std(λ), t ⊲ v and that v = t(r, r + 1), for some r with 1 ≤ r < n. Then γ v = [1 + ρ r (t)][1 − ρ r (t)] [ρ r (t)] 2 γ t . 3.4. Idempotents and characters. In view of Theorem 3.3, over a field the cyclotomic Hecke algebra H Λ n has analogues of the KLR idempotents e(i) ∈ R Λ n , for i ∈ I n . Rather than working over a field, we need these idempotents over O. Fortunately, these idempotents are easy to describe using the seminormal basis. For i ∈ I n define f i = t∈Std i (P Λ n ) F t = t∈Std i (P Λ n ) 1 γ t f tt . By definition, f i ∈ H Λ n (K). In fact, inspired by results of Murphy, we have: Finally, 3.13. Lemma. Suppose that i ∈ I n . Then f i = f * i is an idempotent in H Λ n (O). Moreover, if j ∈ I n then f i f j = δ ij f i . Proof. First observe that if t is a standard tableau then F * t = F t since, by definition, L * k = L k for 1 ≤ k ≤ n. Therefore, f * i = f i ,f i f j = δ ij f i because {F t | t ∈ Std(P Λ n )}M i = M f i . 3.15. Definition. Let M be an H Λ n (F )-module. the character of M is ch M = i∈I n (dim M i ) i ∈ Z[I n ]. Just as with the graded character ch q , we can view ch as an exact functor from the category of H Λ nmodules to the category of L ′ n -modules, where L ′ n = L 1 , . . . , L n . 3.5. Gram determinants of Specht modules. In this section we compute the Gram determinant of the Specht module with respect to the Murphy basis. Gram determinants are only well-defined up to a sign that depends on the ordering of the rows and columns. In this and later sections we fix a total ordering of the tableaux in Std(λ) that refines the dominance ordering and we use this ordering for both the rows and the columns of the Gram matrices. In fact, Corollary 3.18 below implies that if the same ordering is used for the rows and columns of the Gram matrix then the Gram determinant is independent of this choice of ordering. Throughout this section fix an ℓ-partition λ ∈ P Λ n . The Gram matrix of the Specht module S λ K is G λ = m s , m t s,t∈Std(λ) . For t ∈ Std(λ) define f t = m t F t . Then f t = m t + u⊲t s u m u by (3.9), for some s u ∈ K. Therefore, {f t | t ∈ Std(λ)} is a "seminormal" basis of S λ K . The following result is essentially a restatement of [19,Proposition 3.19], although the proof below is considerably easier because it exploits the defining property of the inner product on a cell module. 3.17. Lemma. Suppose that s, t ∈ Std(λ), for λ ∈ P Λ n . Then f s , f t = δ st γ t . Proof. In view of (2.2) and (3.11) , f s , f t f t = f s f tt = δ st γ s f s . Hence, f s , f t = δ st γ t as claimed. The transition matrix between the Murphy and the seminormal bases of S λ K is unitriangular, so: 3.18. Corollary. Suppose that λ ∈ P Λ n . Then det G λ = det f s , f t = t∈Std(λ) γ t . The Gram determinant is quite a crude statistic. To approach the Jantzen sum formula we need to embellish det G λ by adding enough data so that it determines a formal character. To do this we need a new cellular basis of H Λ n . Let s, t ∈ Std(λ) and set i = r(s) an j = r(t). The easiest way to see this is to notice that b t λ = m t λ , so that b t λ t = v t a tv m t λ v for some a tv ∈ O by the proof of Proposition 3.19. Hence, we can set b t = v a uv m v ∈ S λ , for t ∈ Std(λ). For each i ∈ I n define Gram matrices: G λ i = b s , b t s,t∈Std i (λ) . We adopt the convention that det G λ i = 1 if Std i (λ) = ∅. Then we have: 3.21. Lemma. Let λ ∈ P Λ n . Then det G λ = i∈I n det G λ i . Moreover, if i ∈ I n then det G λ i = t∈Std i (λ) γ t . Proof. The transition matrix from the Murphy basis to the b-basis is unitriangular, so det G λ = det b s , b t . Next, observe that if s ∈ Std i (λ) and t ∈ Std j (λ) then b s , b t = b s f i , b t f j = b s , b t f j f * i = δ ij b s , b t , where the last inequality follows by Lemma 3.13. Hence, det G λ = i∈I n det G λ i . Finally, det G λ i = f s , f t = t∈Std i (λ) γ t because, by going through the Murphy basis, the transition matrix from the b-basis to the seminormal basis of S λ K is unitriangular. Jantzen characters Jantzen filtrations of Specht modules underpin all of the results of this paper. Extending what we said in the introduction, Jantzen filtrations can be defined whenever we are given a module over a principal ideal domain that has a non-degenerate bilinear form. Therefore, to define the Jantzen filtrations of the Specht modules we need to work in the ungraded setting because the form on the graded Specht module S µ Q is degenerate unless S µ Q = D µ Q , as is evident from Theorem 2.16. On the other hand, we need to incorporate the idempotents e(i) from Section 2.3 into the Jantzen filtrations, which requires a careful choice of ground-ring. This section sets up the necessary machinery and then proves our main results. M O = J 0 (M O ) ⊇ J 1 (M O ) ⊇ J 2 (M O ) ⊇ . . . where J k (M O ) = {m ∈ M O | ν x (m) ≥ k}. Let M F = M O /xM O . Then the Jantzen filtration of M F is the filtration M F = J 0 (M F ) ⊇ J 1 (M F ) ⊇ J 2 (M F ) ⊇ . . . where J k (M F ) = J k (M O ) + xM O /xM O ∼ = J k (M O )/ J k (M O ) ∩ xM O , for k ≥ 0. Since M F is finite dimensional, J k (M F ) = 0 for k ≫ 0. The following easy observation of Jantzen's is the key to computing the Jantzen sum formula. Lemma (Jantzen). Suppose that M O has a non-degenerate bilinear form. Then k>0 dim J k (M F ) = ν x (G MO ). The proof is a straightforward calculation using the Smith normal form of G M , which exists because O is a principal ideal domain by assumption. For a proof that uses very similar language and notation to what we are using see [34,Lemma 5.30]. The cellular basis of Theorem 3.4 gives us a non-degenerate bilinear form on S λ O , so we can apply Jantzen's constructions to the Specht modules S λ O , for λ ∈ P Λ n . Rather than looking at the Jantzen filtration directly we consider their direct sum. 4.2. Definition. Suppose that λ ∈ P Λ n and let J λ F = ⊕ k>0 J k (S λ F ). The Jantzen character of S λ F is ch J λ F = k>0 ch J k (S λ F ). The main results of this paper follow from an explicit description of the Jantzen character ch J λ F . The next result is our first step towards describing this character. 4.3. Proposition. Let λ ∈ P Λ n . Then ch J λ F = i∈I n ν x (det G λ i )i.(S λ O )f i = J k (S λ O f i ) and J k (S λ F )f i = J k (S λ F f i ) , for all i ∈ I n . (As in Section 3.4, we are abusing notation and identifying f i ∈ H Λ n (O) and f i ⊗ 1 F ∈ H Λ n .) Hence, ch J λ F = k>0 ch J k (S λ F ) = i∈I n k>0 dim J k (S λ F )f i · i = i∈I n k>0 dim J k (S λ F f i ) · i = i∈I n ν x (det G λ i )i, where the last equality follows by applying Lemma 4.1 to S λ O f i and S λ F f i . Note that Lemma 4.1 applies because the bilinear form on S λ K f i is non-degenerate by Lemma 3.21. This completes the proof. Lemma 3.21 gives an explicit formula for det G λ i in terms of the γ-coefficients. The next step is to calculate ν x (γ t ), for t ∈ Std(λ). First, a small interlude on cyclotomic polynomials. 4.2. Cyclotomic polynomials. This section contains some brief reminders on cyclotomic polynomials that we will need to the compute the Jantzen characters. All of the facts that we quote are standard and can be found, for example, in [29,VI §3]. Let f ≥ 1. The f th cyclotomic polynomial in C[x] is Φ f (x) = 1≤d≤f gcd(d,f )=1 x − exp( 2πid f ) Since Φ f (x) is fixed by complex conjugation it follows that Φ f (x) ∈ R[x]. In fact, Φ f (x) ∈ Z[x]. Hence, by base change, we consider Φ f (x) as an element of F [x]. For this paper, a trivial but important observation is that x f − 1 = d|f Φ d (x). Therefore, (4.4) [f ] x = x f − 1 x − 1 = 1<d≤f d|f Φ d (x). In particular, if p is a prime integer then Φ p (x) = [p] x = 1 + x + · · · + x p−1 . Moreover, if f ∈ Z then (4.5) Φ f p (x) =    Φ f (x p ) Φ f (x) , if p ∤ f , Φ f (x p ), if p | f . This recurrence relation determines Φ d (x) uniquely as a polynomial in F [x]. Recall that z = x + ξ, that e > 0 is minimal such that [e] ξ = 0, and that F is a field of characteristic p ≥ 0. We can now state and prove the main fact that we need about cyclotomic polynomials. 4.6. Lemma. Suppose that F is an algebraically closed field of characteristic p ≥ 0 and that f ≥ 1. Then ν x Φ f (z) =      1, if f = e, (p − 1)p r−1 , if p > 0 and f = ep r for some k > 0, 0, otherwise. Proof. First suppose that p = 0. Since F is algebraically closed, it contains a complete set {ω 1 , . . . , ω φ f } of primitive f th roots of unity in F and Φ f (x) = φ f k=1 (x − ω k ). The constant term of Φ f (z) is Φ f (ξ) , which is non-zero if and only if e = f since ξ is a primitive eth root of unity. In particular, ν x (Φ f (z)) = 0 if f = e. On the other hand, if f = e then ξ = ω s , for some s, so the coefficient of x in Φ e (z) is k =s (ξ − ω k ) = 0, so ν x (Φ e (z)) = 1. Therefore, if p = 0 then ν x (Φ f (z)) = δ ef . Now suppose that p > 0. Write f = f ′ p r , for integers f ′ and r ≥ 0 such that gcd(f ′ , p) = 1. If r = 0, so that f = f ′ , then the constant term of Φ f (z) is Φ f (ξ) which is non-zero if and only if e = f = f ′ . Hence, by the argument of the last paragraph, ν x Φ f (z) = δ ef when r = 0. Finally, suppose that r > 0. By (4.5) and the fact that we are in characteristic p > 0, Φ f (z) = Φ f ′ (z) (p−1)p r−1 . Therefore, ν x Φ f (z) = (p − 1)p r−1 ν x Φ f ′ (z) = (p − 1)p r−1 δ ef ′ , completing the proof. Computing Jantzen characters. In [16,Theorem 3.13], Hu and the author gave a closed formula for det G λ using the KLR degree function (2.5) on standard tableaux deg f : Std(λ) −→ Z; t → deg f (t). We need to generalise this result to give a formula for det G λ i , for i ∈ I n . First, for λ ∈ P Λ n define [λ] ! z = ℓ l=1 r≥0 [λ (l) r ] ! z , where for k > 0 the t-quantum factorial of k is [k] ! z = [k] z [k −1] z . . . [1] z . For convenience, set [0] ! z = 1. An analogue of the next result is implicit in the proof of [16,Theorem 3.13]. 4.7. Lemma. Suppose that t ∈ Std(λ), for λ ∈ P Λ n . Then there exists an integer g t ∈ N such that γ t = z gt f >1 Φ f (x) deg f (t) . Proof. The proof is by induction on the dominance order on Std(λ). By [16,Proposition 3.11], which is a straightforward calculation using (3.8), (4.8) γ t λ = [λ] ! z 1≤l<m≤ℓ (l,r,c)∈[λ] [κ l + c − r] z − [κ m ] z = (l,r,c)∈[λ] [c] z l<m≤ℓ z κm [κ l + c − r − κ m ] z . Let A = (l, r, c) ∈ [λ] be a node in λ and set r = t λ (A) and µ = Shape(t λ ↓r ). Recalling (4.4), and the definition of the integer d A,f (µ) from (2.10), the contribution of the node A to γ t λ is [c] z l<m≤ℓ z κm [κ l + c − r − κ m ] z = z κ l+1 +···+κ ℓ f >1 Φ f (z) d A,f (µ) . Hence, the lemma holds when t = t λ . Now suppose that t = t λ . Then there exists a tableau s ∈ Std(λ) such that s ⊲ t = s(r, r + 1), where 1 ≤ r < n. By Corollary 3.12, and induction, γ t = [1 + ρ r (t)] z [1 − ρ r (t)] z [ρ r (t)] 2 z γ s = [1 + ρ r (t)] z [1 − ρ r (t)] z [ρ r (t)] 2 z z gs f >1 Φ f (x) deg f (s) . Let i = r(t). In the graded Specht module S λ , ψ t = ψ s ψ r with the degrees adding. Therefore, deg f (t) = deg f (s) − (α ir , α ir+1 ) = deg f (s) +      1, if 1 ± ρ r (t) ≡ 0 (mod f ) , −2, if ρ r (t) ≡ 0 (mod f ) , 0, otherwise. Hence, by induction, γ t can be written in the required form. This completes the proof. We are one definition away from our first description of the Jantzen characters. First, recall that e is the quantum characteristic of ξ ∈ F , so that e is the minimal positive integer such that [e] ξ = 0. That is, either ξ = 1 and e = char F , or ξ is a primitive eth root of unity in F . Consequently, either e = p or gcd(e, p) = 1. 4.9. Definition. Suppose that λ ∈ P Λ n and i ∈ I n . Define the (e, i)-degree of λ to be the integer deg e,i (λ) = t∈Std i (λ) deg e (t). Let p be the characteristic of the field F . Define the (e, p, i)-degree of λ to be pdeg e,i (λ) = deg e,i (i) + r≥1 (p − 1)p r−1 deg ep r ,i (λ). Recall from (2.10) that deg 0 (t) = 0, for all t ∈ Std(P Λ n ). Therefore, pdeg e,0,i (λ) = deg e,i (λ). The next result gives a complete description of the Jantzen characters. This formula has the advantage of being easy to compute in examples but it is not particularly useful in practice! 4.10. Proposition. Suppose that λ ∈ P Λ n . Then ch J λ F = i∈I n pdeg e,p,i (λ) i. Proof. Since H Λ n (F ) is a cellular algebra by Theorem 3.4, every field is a splitting field for so without loss of generality we can and do assume that F is algebraically closed. By Proposition 4.3 and Lemma 4.7, ch J λ F = i∈I n ν x (det G λ i )i by Proposition 4.3, = i∈I n ν x f >1 Φ f (z) deg f,i (λ) i by Lemma 4.7, = f >1 i∈I n deg f,i (λ)ν x Φ f (z) i = i∈I n pdeg e,p,i (λ) i, where the last equality follows by Lemma 4.6 and Definition 4.9. As the coefficient of i in the Jantzen character is a non-negative integer we obtain the following combinatorial statements, refining similar results from [16, §3.3]. 4.11. Corollary. Suppose that λ ∈ P Λ n and i ∈ I n . Then deg e,i (λ) ≥ 0 and pdeg e,p,i (λ) ≥ 0. 4.4. The positive Jantzen sum formula. Building on Proposition 4.10, we are now ready to prove our main results, which describe the Jantzen characters as explicit linear combinations of the characters of H Λ n (F )-modules. The next definition extends Corollary 3.6. 4.12. Definition. Suppose that F is a field. Set E µ F = E µ F , for µ ∈ K Λ n . Using Theorem 3.3, we view E µ F as an H Λ n (F )-module. The "traditional" way to define a module like E µ F is to construct a "decomposition map" by first choosing a modular system. We have sidestepped these additional complications by using the KLR algebra R Λ n (Z) to define E µ Z over Z. In effect, we used the triple (Q, Z, F ) as a "modular system". We return to this theme in Proposition 4.18 below. If f (q) ∈ A = Z[q, q −1 ] is a Laurent polynomial let f ′ (1) be the derivative of f (q) evaluated at q = 1. Define a linear map ∂ : A[I n ] −→ Z[I n ] by ∂ i∈I n f i (q)i = i∈I n f ′ i (1) i. By Remark 3.14, we can restrict ∂ to a linear map of Grothendieck groups ∂ : Rep L n −→ Rep L ′ n . For λ ∈ P Λ n and µ ∈ K Λ n , recall from (2.15) that d C,e λµ (q) = [S λ C : D µ C ] q ∈ N[q, q −1 ] is a characteristic zero graded decomposition number of R Λ n (C). 4.13. Lemma. Suppose that f = ep r , for r ≥ 0, and let λ ∈ P Λ n and µ ∈ K Proof. This is immediate because d C,e λµ (q) ∈ δ λµ + qN[q] by [5,Corollary 5.15]. (However, the proof of this fact from [5] is highly non-trivial.) According to the lemma, the coefficients in the character formula below are non-negative integers. 4.14. Theorem. Suppose that F is a field of characteristic zero and let λ ∈ P Λ n . Then ch J λ F = µ∈K Λ n (d C,e λµ ) ′ (1) ch E µ F . Proof. Let µ ∈ K Λ n and observe that ∂ ch q E µ F = 0 because ch q E µ F = ch q E µ F by Corollary 2.19. Therefore, ∂ d C,e λµ (q) ch q E µ F = (d C,e λµ ) ′ (1) ch E µ F by the chain rule. Next, notice that Corollary 2.18 implies that i∈I n t∈Std i (λ) q deg e (t) i = ch q S λ F = µ∈K Λ n d C,e λµ (q) ch q E µ F . Applying ∂ to both sides of this equation, using the previous remark, shows that i∈I n t∈Std i (λ) deg e (t) i = µ∈K Λ n d ′ λµ (1) ch E µ F . Since pdeg e,0,i (λ) = deg e,i (i) = t∈Std i (λ) deg e (t), for i ∈ I n , comparing the left hand side of the last displayed equation with Proposition 4.3 completes the proof. The characters of the graded Specht modules are crucial to the proof of Theorem 4.14, however, the proof is purely combinatorial and it does not use a representation theoretic connection between the Jantzen filtration of S λ F and the graded R Λ n (F )-modules E µ F . It would be interesting to find a direct connection between the modules J k (S λ F ) and E µ F . The Jantzen sum formula is usually stated in the Grothendieck group Rep H Λ n . In view of Proposition 3.16, Theorem 4.14 is equivalent to the following more "traditional" statement: 4.15. Corollary. Suppose that p = 0 and let λ ∈ P Λ n . Then [J λ F ] = k>1 [J k (S λ F )] = µ∈K Λ n (d C,e λµ ) ′ (1)[E µ F ]. 4.16. Remark. In characteristic zero, Ryom-Hansen [41, Theorem 1] proved an analogue of this result for Λ = Λ 0 by assuming a conjecture of Rouquier's [30, §9] that, in the post KLR world, says that the Jantzen filtration of S λ C coincides with its grading filtration. Yvonne [48, Theorem 2.11] extended Ryom-Hanson's work to arbitrary Λ ∈ P + e . Theorem 4.14 shows that the description of the Jantzen sum formula in terms of the integers (d C,e λµ ) ′ (1) is independent of Rouquier's conjecture. We next compute the Jantzen characters for fields of positive characteristic p > 0. As Proposition 4.10 and Definition 4.9 suggest, this involves looking at Hecke algebras at ep r th roots of unity for r ≥ 0. We first need to set up the combinatorial and representation theoretic machinery to do this. When r = 0 the tool that we need is provided by Theorem 2.16 but when r > 0 we need to work harder. If f = ep r , for r ≥ 0, let ζ f = exp( 2π f ) ∈ C. Then ζ f is a primitive f th root of unity in C. Let I f = Z/f Z. (In particular, I = I e . ) Since e divides f , there is a well-defined surjective map I f −→ I = I e ; i → i f given by "reducing modulo e". More explicitly, this map sends a + f Z to a + eZ, for a ∈ Z. Extending this notation, if i ∈ I n f let i f be the corresponding sequence in I n . This map induces an abelian group homomorphism L f,e : Z[I n f ] −→ Z[I n ]; i∈I n f c i i → i∈I n f c i i f , for c i ∈ Z. Recall from Section 2.3 that P + f and Q + f are the positive weight lattice and positive root lattice the attached to the quiver Γ f , for f ≥ 2. Abusing notation, let Λ i and α i be fundamental weights and simple roots in the corresponding weight lattices for any f ≥ 2, with the meaning being clear from the choice of index set I f . In Section 2.3 we fixed a dominant weight Λ = Λ e ∈ P + = P + e . For f = ep r fix a dominant weight Λ f ∈ P + f such that (4.17) (Λ, α i ) = j∈I f j f =i (Λ f , α j ), for all i ∈ I.d C,f λµ (q) = [S λ C,f : E µ C,f ] q , for λ ∈ P Λ n and µ ∈ K Λ f n . Armed with this notation we can state the result that we need. This result is partly motivated by the discussion after [ Rep H Λ f n (K) Rep H Λ n (F ) Z[I n f ] Z[I n ] A f,e L f,e ch ch Moreover, A f,e [S λ K,f ] = [S λ F ] , for all λ ∈ P Λ n . Proof. The two character maps are injective by Proposition 3.16, so once we show that A f,e exists it is automatically unique. The cyclotomic Hecke algebras are cellular, so every field is a splitting field. Hence, without loss of generality, we can assume that K = Q[ζ f ] and F = F p [ξ]. Let Z f = Z[ζ f ]. Then H Λ f n (Z f ) is a Z f - subalgebra of H Λ f n (K) and H Λ f n (K) ∼ = H Λ f n (Z f ) ⊗ Z f K. Since e divides f there is a unique surjective ring homomorphism π f : Z f → F such that π f (ζ f ) = ξ. Hence, we can consider F as a Z f -algebra. We claim that, as F -algebras, H Λ n (F ) ∼ = H Λ f n (Z f ) ⊗ Z f F. Since the cyclotomic Hecke algebras are free over any ring by Theorem 3.4, to prove this it is enough to check that tensoring with F p [ξ] respects the relations. The only relation that is not obviously preserved is the cyclotomic relation, which can be written in the form i∈I (3.2). This relation is preserved by tensoring with Z f by (4.17) because Proof. If λ ∈ P Λ n then [S λ f (L 1 − [i] ζ f ) (Λ f ,αi) = 0 by[i f ] ξ = π f ([i] ζ f ), for all i ∈ I f . Set p f = ker π f , a prime ideal in Z f , and let O f = (Z f ) p f be the localisation of Z f at p f . Then O f is a discrete valuation ring with maximal ideal pO f , quotient field K and residue field F ∼ = O f /pO f . Hence, the triple (K, O f , F ) is a p-modular system in the sense of [8, §16A]. Therefore, there is a well-defined decomposition map A f,e : Rep H Λ f n (K) −→ Rep H Λ n (F ) that is independent of the choice of O f by [8, Proposition 16.17]. To describe A f,e explicitly, if V is an H Λ f n (K)-module then A f,e (V ) is defined by choosing a full O f -lattice V O f in V and then setting A f,e (V ) = V O f ⊗ O f F .C,f ] = ν d C,f λν [E ν C,f ] in Rep H Λ f n (C),1 5 2, 1 3 2 2 , 1 1 5 1 2, 1 3 . 1 2 2 , 1 1 . 1 3, 1 2 2q . q 3, 2 q 2 . q 2 4, 1 . q . 5 q 2 . . 1 5 2, 1 3 2 2 , 1 3, 1 2 3, 2 4, 1 1 5 1 2, 1 3 . 1 2 2 , 1 q . 1 3, 1 2 . . . 1 3, 2 . . q . 1 4, 1 . . . . . 1 5 . . . . q . R Λ 5 = R Λ0 5 ∼ = F 2 S 5 with (e, p) = (2, 2) R Λ f 5 = R Λ0 5 with (e, p) = (4, 2) Comparing these two matrices shows that there is no graded adjustment matrix with entries in N[q, q −1 ] in this case. (In contrast, in agreement with Proposition 4.18, there is an adjustment matrix when we set q = 1, which corresponds to forgetting the grading.) Hence, when r > 0 we cannot expect the naive generalisations of Proposition 4.18 and Corollary 4.21 to hold in the graded setting. It would be interesting to find a connection between the algebras R Λ f n (F ) and R Λ n (F ). We are now able to prove what is really the main result of this paper. 4.23. Theorem. Suppose that F is a field of characteristic p > 0 and let λ ∈ P Λ n . Then ch J λ F = µ∈K Λ n (d C,e λµ ) ′ (1) ch E µ F + r>0 (p − 1)p r−1 f =ep r ν∈K Λ f n (d C,f λν ) ′ (1) ch E ν f,e . Moreover, all of the coefficients on the right hand side are non-negative integers. Proof. Arguing as in the proof of Theorem 4.14, if f = ep r then i∈I n f deg f (λ) i = ν∈K Λ f n (d C,f λν ) ′ (1) ch E ν C . Applying the map L f,e , and using Proposition 4.18 and Definition 4.19, this becomes i∈I n f deg f (λ) i f = ν∈K Λ f n (d C,f λν ) ′ (1) ch E ν f,e . Multiplying by (p − 1)p r−1 if r > 0 and then summing over r, as in Definition 4.9, i∈I n pdeg e,p,i (λ) i = µ∈K Λ n d ′ λµ (1) ch E µ F + r>0 (p − 1)p r−1 ν∈K Λ f n (d C,f λν ) ′ (1) ch E ν f,e . Hence, the character formula for ch J λ F follows by Proposition 4.10. Finally, the coefficients on the right hand side of the character formula for ch J λ F are non-negative integers by Lemma 4.13. Applying Proposition 3.16, just as we did after Theorem 4.14, we obtain the Main Theorem from the introduction: 4.24. Corollary. Suppose that F is a field of characteristic p > 0 and let λ ∈ P Λ n . Then k>0 [J k (S λ F )] = µ∈K Λ n (d C,e λµ ) ′ (1)[E µ F ] + r>0 (p − 1)p r−1 f =ep r ν∈K Λ f n (d C,f λν ) ′ (1)[E ν f,e ]. One of the applications of the classical Jantzen sum formula is to determined when S λ F = D λ F , for λ ∈ P Λ n . If F is a field of characteristic zero then Theorem 4.14 implies that S λ F is irreducible if and only if d C,e λµ = 0 for µ = λ, which is true but not very insightful. If p > 0 we obtain new information. 4.25. Corollary. Suppose that F is a field of characteristic p > 0 and let λ ∈ P Λ n . Then S λ F = D λ F if and only S λ C,f = D λ C,f , for all f = ep r and r ≥ 0. Although this result does not appear to be the literature it can be deduced from [19,Theorem 4.7(iii)]. In a similar vein, de Boeck et al [9] have shown that in positive characteristic the dimensions of the simple modules indexed by two column partitions for the Hecke algebra of the symmetric with ξ of quantum characteristic e depend upon the dimensions of the simple modules at ep r th roots of unity in characteristic zero. It would be interesting to find stronger connections between the representation theories of the Hecke algebras with quantum characteristic e in characteristic p > 0 and the Hecke algebras in characteristic zero at ep r th roots of unit, for r ≥ 0. A second application of the Jantzen sum formula is to give upper bound for decomposition numbers. For λ ∈ P Λ n and µ ∈ K Λ n define j F,e λµ = (d C,e λµ ) ′ (1)a λµ + r>0 (p − 1)p r−1 f =ep r ν∈K Λ f n (d C,f λν ) ′ (1)a f,e νµ . Then comparing the coefficient of ch D µ F on both sides of Theorem 4.23 shows that the following holds. 4.26. Corollary. Suppose that λ ∈ P Λ n and µ ∈ K Λ n , with µ = λ. Then d F,e λµ ≤ j F,e λµ . In particular, d F,e λµ = 0 only if j F,e λµ = 0. We find it striking that the upper bound on the decomposition numbers of H Λ n (F ) increases dramatically as soon as d C,f λν (q) is non-zero, for some f = ep r . In the special case when Λ = Λ 0 , so that H Λ n (F ) is an Iwahori-Hecke algebra of the symmetric group, Theorem 4.23 provides some evidence in favour of (the now disproved) James' conjecture [17, §4]. Both the statement and proof use the standard terminology of the weight, hooks and cores of partitions; see, for example, [34,Chapter 4]. 4.27. Corollary. Suppose that Λ = Λ 0 and that λ ∈ P Λ n is partition e-weight w < p. Then ch J λ F = ch J λ C = µ∈K Λ n (d C,e λµ ) ′ (1) ch E µ F . Proof. Let f = ep r , where r > 0. Since w < p, the partition λ cannot contain a removable f -hook. Therefore, λ is an f -core and the Specht module S λ C,f = E λ C,f = D λ C,f is irreducible. Hence, d C,f λµ (q) = δ λµ and Theorem 4.23 implies the result. In general we would not expect the character ch J λ F to determine the Jantzen filtration uniquely. Combining Corollary 4.27 with Williamson's counterexample to the James conjecture [46], shows that ch J λ F does not determine the Jantzen filtration uniquely even when w < p. The classical sum formula For completeness, this section shows how to use formal characters to prove a more "classical" version of the Jantzen sum formula [18,19]. In the cyclotomic case, the sum formula that we obtain is equivalent to, but slightly nicer than, that given by [19,Theorem 4.3]. Mostly this is because Definition 3.1 is implicitly invoking the Morita equivalence of [11] to restrict the "cyclotomic parameters" of H Λ n (F ) to a single ξorbit. Two other reasons why the results in this section are more elegant than the corresponding results in [19] is because we have already fixed the quantum characteristic e and because the combinatorial framework that we introduce directly links beta numbers and contents for ℓ-partitions. As in Section 3.3, throughout this section we fix a field F and a primitive eth root of unity ξ ∈ F , where e < ∞, and study the Hecke algebra H Λ n (F ). We continue assume that the charge κ satisfies (3.7) and we consider the algebras H Λ n (O) and H Λ n (K) with Hecke parameter z = x + ξ, where O = F [x] (x) and K = F (x). Charged beta numbers. We want an analogue of the branching rules for the Gram determinants G λ i . This requires a mild dose of new notation that is motivated by [19, §3]. The length of a partition λ = (λ 1 ≥ λ 2 ≥ · · · ≥ 0) is the smallest non-negative integer L(λ) such that λ L(λ) = 0. The length of an ℓ-partition λ ∈ P Λ m is L(λ) = max{L(λ (l) ) | 1 ≤ l ≤ ℓ}. If λ ∈ P Λ n then L(λ) ≤ n so, in what follows, we will always work with sequences of length n. Following Littlewood [32], a sequence of beta numbers of length n is a strictly decreasing sequence β 1 > · · · > β n ≥ 0 of n non-negative integers. It is straightforward to show that the set of beta numbers of length n are in bijection with the partitions of length at most n, where the partition λ corresponds to the sequence of beta numbers β λ 1 > · · · > β λ n ≥ 0, where β λ r = λ r + n − r, for r ≥ 1. (If λ is a partition of length L = L(λ) then the integers (β λ 1 + L − n > · · · > β λ l + L − n) are the first column hook lengths in λ.) Notice that if (β 1 , . . . , β n ) is a sequence of beta numbers of length n then (β 1 + 1, . . . , β n + 1, 0) is a sequence of beta numbers of length n + 1. Recall that we are assuming that the charge κ satisfies (3.7). A sequence of charged beta numbers (of length ℓn) is a sequence β = (β 1 , . . . , β ℓn ) ∈ N ℓn such that (5.1) κ l + n > β n(l−1)+1 > · · · > β nl ≥ κ l − n, for 1 ≤ l ≤ ℓ. That is, for 1 ≤ l ≤ n the subsequences (β n(l−1)+1 , . . . , β nl ) are beta numbers for the partitions λ (l) shifted by κ l − n ≥ 0. Assumption (3.7) ensures that β 1 > · · · > β ℓn ≥ 0 and, in particular, that the charged beta numbers are distinct. When the order is not important, call {β λ 1 , . . . , β λ ℓn } a set of (charged) beta numbers for λ. We frequently need to translate between ℓ-partitions and charged beta numbers. To facilitate this, given an integer s such that 1 ≤ s ≤ ℓn write s = (l s − 1)n + r s , where 1 ≤ l s ≤ ℓ and 1 ≤ r s ≤ n. Similarly, if 1 ≤ t ≤ ℓn then we write t = (l t − 1)n + r t etc. In this way, we refer to row s of λ as row r s of λ (ls) . Of course, row s of λ is empty if r s > L(λ (ls) ) or, equivalently, λ (ls) rs = 0. As with the beta numbers of partitions, if λ ∈ P Λ m and n ≥ L(λ) then λ can be associated with a unique sequence of charged beta numbers of a given length. Explicitly, λ ∈ P Λ m has sequence of charged beta numbers β λ = (β λ 1 , . . . , β λ ℓn ), where (5.2) β λ s = κ ls + λ (ls) rs − r s , where s = (l s − 1)n + r s as above. Even though our notation does not reflect this, β λ = β λ (κ, n) depends on λ and the choice of n and κ. Via (5.2), the beta number β λ s is naturally associated with the node (l s , r s , λ (ls) rs ) since, by definition, the content of this node is c(l s , r s , λ (ls) rs ) = β λ s . Notice that this node belongs to λ only if L(λ (ls) ) ≥ r s . If λ ∈ P Λ m then ℓn r=1 β λ r = n(κ 1 +· · ·+κ ℓ )− 1 2 ℓn(n+1)+m. Even though we do not need this (although compare Definition 5.4), it is not difficult to see that there is a bijection between the ℓ-partitions in P Λ m of length at most n and the sequences of charged beta numbers with sum n(κ 1 +· · ·+κ ℓ )− 1 2 ℓn(n+1)+m. 5.3. Example. Let e = 3 and Λ = Λ 0 + Λ 2 , with r(1, 1, 1) = 0. Let λ = (3, 2|1) so that m = |λ| = 6. The following table lists some of the infinitely many choices of sequence of charged beta numbers for λ, corresponding to different choices of κ and n ≥ L(λ) = 2. Notice that κ must be chosen in accordance with (3.7), so it depends on e, n and Λ, and that β λ depends on λ and κ. Below we will assume that n ≥ m, as in the last two example beta sequences, because this ensures that all partitions in P Λ m have a charged beta sequence of length ℓn. Consequently, these sequences of beta numbers can describe all possible ways of moving hooks (see Section 5.3), between the components of λ. ♦ The main results in this chapter are proved by induction on the size m of the ℓ-partitions using charged beta sequences of length ℓn. Accordingly, we work with an ℓ-partition λ ∈ P Λ m , for n ≥ m, and use the sequence of charged beta numbers β λ defined in (5.2), which has length ℓn. If i = (i 1 , . . . , i n ) ∈ I m and 1 ≤ k ≤ m let i ↓k = (i 1 , . . . , i k ) ∈ I k . If β ∈ N ℓn and w ∈ S ℓn then let w · β = (β w(1) , . . . , β w(ℓn) ). Since sequence of charged beta numbers are monotonically decreasing, w · β is a sequence of charged beta numbers if and only if w = 1. Abusing notation slightly, let L(w) be the minimal length of w ∈ S ℓn as a product of the standard Coxeter generators of S ℓn . 5.4. Definition. Suppose that i ∈ I m and β = (β 1 , . . . , β ℓn ) ∈ Z ℓn , where n ≥ m. Define d i (β) = (−1) L(w) dim S λ i , if β = w · β λ for some λ ∈ P Λ m and some w ∈ S ℓn , 0, otherwise, where S λ i = S λ F f i . Similarly, set d i (λ) = dim S λ i . In particular, d i (β) = 0 if β = w · β λ and S λ i = 0 for some w ∈ S ℓn , or if β s = β t for s = t, or if #{1 ≤ r ≤ ℓn | κ l + n > β r ≥ κ l − n} = n, for any l with 1 ≤ l ≤ ℓ. 5.2. Beta numbers and branching rules. The classical branching rules for the Specht modules of the symmetric groups are one of the cornerstones of modern representation theory. Analogues of these branching rules exist for the Specht modules of H Λ n [15,38]. For this paper we only need the following simple consequence of these branching rules. Let λ ∈ P Λ n and ν ∈ P Λ n−1 . Write ν → λ if λ = ν ∪ {α} for some addable node α of ν. If α is an i-node, for i ∈ I then we write ν i − → µ to emphasise the residue. If j ∈ I n−1 and i ∈ I let j ∨ i = (j 1 , . . . , j n−1 , i) ∈ I n . Extend ∨ i to a linear map Z[I n−1 ] ֒→ Z[I n ]. 5.5. Lemma. Suppose that λ ∈ P Λ n . Then ch S λ F = i∈I ν i − →λ ch S ν F ∨ i. Proof. By the main theorem of [38], the Specht module S λ F has a filtration as an H Λ n−1 (F )-module where the quotients are exactly the Specht modules S ν F , where ν → λ, with each Specht module appearing exactly once. As the character map ch : Rep H Λ n−1 −→ Rep L n−1 is exact the result follows. The advantage of using charged beta numbers, instead of ℓ-partitions, is that it is much easier to describe induction and restriction. The next corollary uses charged beta numbers to give an equivalent, but easier to apply, analogue of Lemma 5.5: 5.6. Corollary. Suppose that i ∈ I m and β = (β 1 , . . . , β ℓn ) ∈ N ℓn , where n ≥ m. Then Proof. First suppose that β = β λ , for some ℓ-partition λ ∈ P Λ m . It is well-known and easy to see that removing a node from a λ corresponds to decreasing one of the charged beta numbers of λ by 1. Now, β ′ = (β 1 , . . . , β s−1 , β s − 1, β s+1 , . . . , β ℓn ) is a sequence of charged beta numbers if and only if either β s > κ l and n | s, or β s > β r+1 + 1 and n ∤ s. As above, write s = (l s − 1)n + r s , where 1 ≤ l s ≤ ℓ and 1 ≤ r s ≤ n, then this corresponds to removing the node (l s , r s , λ (ls) rs ) from the ℓ-partition corresponding to β. When β = β λ the result now follows in view of Lemma 5.5. Finally, if β = w · β λ , for w ∈ S ℓn , then it is easy to see that β ′ is a sequence charged beta numbers if and only if β ′ = w · β ν , for some ℓ-partition ν ∈ P Λ m−1 , which implies the result. Beta numbers and hooks. This section shows that adding and subtracting numbers in a sequence of charged beta numbers corresponds to adding and removing rim hooks from the corresponding ℓpartitions. In order to describe this recall that if λ is a partition then λ ′ = (λ ′ 1 , λ ′ 2 , . . . ) is the partition that is conjugate to λ, where λ ′ c = #{r ≥ 1 | λ r ≥ c}. Equivalently, λ ′ c is the length of column c of λ. If α = (l, a, b) ∈ λ then the α-rim hook of λ is the following set of nodes: R λ α = {(l, c, d) ∈ λ | a ≤ c ≤ λ (l)′ b , b ≤ d ≤ λ (l) a , and (l, c + 1, d + 1) / ∈ λ}. By definition, λ \ R λ α is (the diagram of) an ℓ-partition. The α-hook length of R λ α is h α = |R λ α | and l α = #{(l, c, d) ∈ R λ α | c > a} = λ (l)′ b − a is the α-leg length of R λ α . Let r α = n(l − 1) + a be the row of λ that contains α. The foot of R λ α is the node f α = (l, a + l α , b) = (l, λ (l)′ b , b) ∈ λ, which is the leftmost node in the last row of R λ α . Finally, an h-hook is any rim hook of length h. 5.7. Example. Let λ = (4, 2, 1|6, 5, 3, 2|, 3, 1, 1), α = (2, 1, 3) and n = m = 28. Then f α = (2, 3, 3), h α = 6, l α = 2 and r α = n + 1 = 29. The rim hook R λ α can be pictured as follows:       α = (2, 1, 3) f α = (2, 3, 3) R λ α lα = 2 r α = 29       ♦ 5.8. Definition. Let λ ∈ P Λ m and suppose that α = (l s , r s , c s ) ∈ λ and that t > r α + l α = (l − 1)n + λ (l) ′ b , where λ (l) ′ is the partition conjugate to λ (l) . Define λ α,t to be the ℓ-partition of m obtained by wrapping a rim hook of length h α onto λ \ R λ α so that its foot node is the leftmost addable node in row t. If λ α,t is not an ℓ-partition, set λ α,t = ∅. If λ α,t = ∅ then set ε α,t = (−1) lα+l α ′ +1 , where α ′ ∈ λ α,t is the unique node such that λ α,t \ R λα,t α ′ = λ \ R λ α . Finally, set h ′ α = β λ s − β λ t − h α , where s = (l s − 1)n + r s = r α . By definition, if α ∈ λ then the integer h ′ α depends on λ, κ and n ≥ m. Notice also that if λ α,t = ∅ then ch S λα,t F ℓ-partition ν and the sign ε = ±1 are determined by d i (β λ that h = h ′ α , t > r α and d i (β λ st (h)) = ε α,t d i (λ α,t ). Hence, the result follows. We can now give the promised "classical" higher level Jantzen character formula. x be an indeterminate over F and consider the localisation O = F [x] (x) of the polynomial ring F [x] at the prime ideal (x) = xF [x]. Let K = F (x) be the field of fractions of F [x]. Let H Λ n (O) be the cyclotomic Hecke algebra over O with Hecke parameter z = x + ξ and with cyclotomic parameters that the integer m, with 1 ≤ m ≤ n, appears in row r and column c of t (k) . The content of m in t is c m (t) = κ m + c − r. Observe that the residue of m is r m (t) = c m (t) + eZ ∈ I. The content sequence of t is c(t) = c 1 (t), . . . , c n (t) . More generally, the content of the node (m, r, c) is c(m, r, c) = κ m + c − r. The key point of these definitions is that s = t if and only if c(s) = c(t), for s, t ∈ Std(P Λ n ). This is easily proved by induction on n using (3.7). 1 James and the author[19, Proposition 3.7] proved the following fundamental property of this basis:(3.8) m s L k = [c k (s)]m s + u⊲s r u m u , for s ∈ Std(λ), 1 ≤ k ≤ n and some r u ∈ R. for all i ∈ I n . By[16, Lemma 4.3], f i ∈ H Λ n (O). In order to apply this lemma from[16] we need to first show that (O, t) is an "idempotent subring of K" in the sense of[16, Definition 4.1]. In fact, there is nothing to do here because this is already established in[16, Example 4.2(b)]. is a set of pairwise orthogonal idempotents. 3.14. Remark. By definition, f i = 0 if and only if i = r(t) for some standard tableau t ∈ Std(P Λ n ). In contrast, it is not clear from Definition 2.5 when the KLR idempotent e(i) is non-zero. In fact, in view of Theorem 2.11, e(i) = 0 if and only if f i = 0. Since H Λ n ∼ = H Λ n (O) ⊗ O F we obtain pairwise orthogonal idempotents f i ⊗ 1 in H Λ n . We abuse notation and write f i for these idempotents in both H Λ n (O) and H Λ n . The meaning should always be clear from context. As a first consequence of the existence of the idempotents f i in H Λ n (O) and H Λ n , we can define an analogue of the graded characters of Definition 2.6 for these algebras. If M is an (ungraded) H Λ n -module then M = i M i as an O-module, when Define b st = f i m st f j . We can view b st as an element of H Λ n (O) or of H Λ n . By Lemma 3.13, b st = f i b st f j , where i = r(s) and j = r(t). 3.19. Proposition. The set {b st | s, t ∈ Std(λ), λ ∈ P Λ n } is a cellular basis of H Λ n (O). Proof. By (3.8), if u ∈ Std i (P Λ n ) then there exist scalars r tuv ∈ O such that m st F u ≡ δ tu m st + v⊲t r tuv m sv (mod H ⊲λ n ) . Hence, m st f i is equal to m st plus a linear combination of more dominant terms, so the transition matrix from the basis {m st } to the set {b st } is unitriangular. In particular, {b st } is a basis of H Λ n (O). It is now straightforward to check that {b st } is a cellular basis of H Λ n (O). As the cellularity of this basis is not used in what follows we leave these details to the reader. 3.20. Remark. The b-basis is compatible with the block decomposition of H Λ n . This basis first appeared in a more general context in [36, Theorem 4.5]. Instead of the b-basis we could use an analogue of the ψ-basis from Theorem 2.11 for H Λ n (O), which is constructed in [16]. There is no real gain in doing this, however, because the ψ-basis of H Λ n (O) takes considerably more effort to define. The b-basis {b st } of H Λ n gives a b-basis {b t } of the Specht module S λ O and, by extension of scalars, a b-basis of S λ K . 4. 1 . 1Jantzen filtrations. We recall Jantzen's construction of filtrations of modules that come equipped with a non-degenerate bilinear form in the special case when the ground ring is F [x] (x) . We then apply Jantzen's construction to the Specht modules of H Λ n (F ).As above, let O = F [x] (x) and let K = F (x). Let M O be a free O-module of finite rank and let M K = M O ⊗ O K. Suppose that there exists a non-degenerate bilinear form , : M O × M O −→ O.By definition, O is a discrete valuation ring with maximal ideal xO. Let ν x be the associated valuation. Explicitly, if 0 = a ∈ O then ν x (a) = max{k ≥ 0 | a ∈ x k O}. In particular, ν x (x + ξ) = 0.The Jantzen filtration of M O is the filtration n . Then (d C,f λµ ) ′ (1) ≥ 0 with equality if and only if λ = µ. These conditions determine the dominant weight Λ f uniquely if and only if r = 0. If r > 0 then the results below are independent of the choice of Λ f .Let H Λ f n (C) be the cyclotomic Hecke algebra over C with Hecke parameter ζ f and dominant weight Λ f . For λ ∈ P Λ n let S λ C,f be the graded Specht module for R Λ f n (C) indexed by λ. Similarly, given an ℓ-partition µ ∈ let E µ C,f = D µ C,f be the (almost) simple H Λ f n (C)-module defined by Definition 2.14 and let S λ C,f and D λ C,f be the corresponding Specht and simple modules for H Consequently, because the cellular basis {b st } of H Λ f n (K) from Proposition 3.19 is defined over any ring, if λ ∈ P Λ n then A f,e (S λ K ) = S λ F . It remains to show that the diagram in the proposition commutes. Since L f,e (ch S λ K ) = ch(S λ F ), we have commutativity on the Specht modules. The decomposition matrix D K,f is unitriangular, so{S λ F | λ ∈ K Λ f n } is a basis for Rep H Λ f n (K).Hence, this completes the proof.4.19.Definition. Suppose that f = ep r , for r ≥ 0, and let ν ∈ KΛ f n . Set E ν f,e = A f,e E ν C,f . Strictly speaking, E νf,e is the image of a module in the Grothendieck group of H Λ n (F ) but we abuse notation refer to it as the modular reduction of the almost simple module E µ C for HΛ f n (C). (By the proof of Proposition 4.18, the module E ν f,e is well-defined only up to a choice of lattice whereas [E ν f,e ] is independent of this choice.) In the special case when e = f or, equivalently, r = 0, comparing characters and applying Proposition 4.18 yields the following: 4.20. Corollary. Let λ ∈ P Λ n and µ ∈ K Λ n . Then [S λ F ] = A f,e (S λ C,e ) and [E µ F ] = [E µ F,e ]. This gives the following generalisation of the (ungraded analogue of) Theorem 2.16. Define the (f, e)-adjustment matrix A f,e = a f,e νµ , for ν ∈ K Λ f n and µ ∈ K Λ n , by a f,e νµ = [E ν f,e : D µ F ]. Proposition 4.18 gives the following generalisation of the ungraded analogue of (2.17) at q = 1. 4.21. Corollary. Suppose that f = ep r , for r ≥ 0. Then D F,e = D C,f A f,e . F ), however, it is not clear how to define a map between the Grothendieck groups of Rep R Λ f n (F ) and Rep R Λ n (F ). Using formal characters, it is straightforward to compute the following graded decomposition matrices corresponding to the choices e = p = 2, Λ = Λ 0 , f = 4 and Λ f = Λ 0 (so F = F 2 is the field of order 2): βs≡im (mod e) d i ′ (β 1 , . . . , β s−1 , β s − 1, β s+1 , . . . , β ℓn ), where i ′ = i ↓(m−1) . 2.1. Definition (Graham and Lehrer [13, Definition 1.1], Hu-Mathas [14, Definition 2.1]) Proposition. Let F be a field. The character map ch :Rep R Λ n (F ) −→ Rep Z[I n ] is injective.Proof. This is part of the folklore for H Λ n . It is proved in exactly the same way as Proposition 2.7 using the corresponding results for the affine Hecke algebra; see, for example,[28, Theorem 3.3.1]. Finally, note that if M is a graded R Λ n (F )-module then, in view of Theorem 3.3, M is naturally an H Λ n (F )-module. The proof Theorem 3.3 (or, more correctly, [16, Theorem A]), identifies the KLR idempotent e(i) ∈ R Λ n (F ) with f i ∈ H Λ n (F ), for all i ∈ I n . Therefore, ch M = ch q M q=1 . Hence, Proposition 3.16 implies Proposition 2.7.Let Rep H Λ n be the Grothendieck group of finitely generated H Λ n -modules. Then we can view ch as the induced map ch : Rep H Λ n −→ Rep L ′ n ֒→ Z[I n ]. Like the graded characters and Proposition 2.7, we have: 3.16. Proof. As anO-module, S λ O = i∈I n S λ O f i . By construction, {b s | s ∈ Std i (λ)} is a basis of S λ O f i and G λ iis the Gram matrix of the bilinear form on S λ O restricted to S λ O f i . Moreover, in view of Lemma 3.21, if i = j ∈ I n then the summands S λ O f i and S λ O f j are orthogonal with respect to the inner product on S λ O . Therefore, J k where the sum is over ν ∈ K obtained by setting q = 1 in the graded adjustment matrix A F,e (q) of (2.17).4.22.Remark. By Theorem 2.16, Corollary 4.21 extends to give graded adjustment matrices in the special case when e = f or, equivalently, r = 0. If r > 0 then RΛ f n . Applying the linear map A f,e , [S λ F ] = ν∈K Λ f n d C,f λν [E ν f,e ] = ν∈K Λ f n d C,f λν µ∈K Λ n a f,e νµ [D µ F ] = µ∈K Λ n ν∈K Λ f n d C,f λν a f,e νµ [D µ F ]. Since [S λ F ] = µ d F,e λµ [D µ F ] the result follows. In the special case when e = f , Corollary 4.20 implies that A F,e = A F,e (1), where A F,e (1) is the adjustment matrix Λ f n (C) ∼ = H Λ f n (C) by Theorem 3.3. By Definition 2.5 we have a cyclotomic KLR algebra R Λ f n (F ) defined over F , however, Definition 3.1 does not define a cyclotomic Hecke algebra H Λ f n (F ) because a field of characteristic p > 0 cannot contain a primitive f th root of unity when p divides f . Consequently, Theorem 3.3 does not apply to R Λ f n (F ) and, a priori, the algebra R Λ f n (F ) is not isomorphic to a cyclotomic Hecke algebra. Even so, Theorem 2.11 shows that R Λ f n (F ) is a graded cellular algebra that comes equipped with Specht modules S λ F , simple modules D µ F and almost simple modules E µ F , for λ ∈ P Λ n and µ ∈ K 5.18. Theorem. Suppose that λ ∈ P Λ n . Then ch J λ F = α∈λ t>rα ε α,t ν x ([h ′ α ] z ) ch S i∈I n α∈λ t>rα ε α,t ν x ([h ′ α ] z )d i (λ α,t ) iwhere the last equality follows by swapping the order of summation since, by definition, ch S λα,t F = i∈I n d i (λ α,t )i. This completes the proof.5.19. Remark. Using Lemma 4.6, if h ′ ∈ N then we can explicitly compute ν x ([h ′ ] z ) = f |h ′ ν x (Φ f (z)). appears in ch J λ F only if e divides h ′ α . Further, in view of Proposition 5.11(d), if 1 ≤ h α , h ′ α ≤ n then ch S λα,t F appears in ch J λ F if and only if ν x ([h α ] z ) = ν x ([h ′ α ] z ).λα,t F . Proof. Applying Proposition 4.3 and Corollary 5.17, ch J λ F = i∈I n ν x det G λ i i = = α∈λ t>rα ε α,t ν x ([h ′ α ] z ) ch S λα,t F , In particular, ch S λα,t F Acknowledgements. We thank Jun Hu for useful discussions about the results of this paper. This research was supported, in part, by the Australian Research Council.Cyclotomic KLR algebras and their modules= 0. 5.9. Example. Let λ = (3, 2|1) ∈ P Λ 6 . As in Example 5.3, take n = m = 6 and κ = (20, 6) so that β λ =(23,21,18,17,16,15,8,6,5,4,3,2). Using the notation of Definition 5.8, the following table gives the complete list of pairs (α, t) where s ≥ t. Here, as in Definition 5.8, if α = (l s , r s , c s ) then s = (l s − 1)n + r s = r α and h ′ α = β λ s − β λ t − h α . The right hand column identifies when a 2-partition appears more than once in the table as λ α,t , for some α and t. Proposition 5.11 below shows that no 2-partition appears more than twice and that if λ α,t = λ γ,u , for some α, γ ∈ λ and some integers t, u, then {h α , h ′ α } = {h γ , h ′ γ } and ε α,t = −ε γ,u . ♦ Given λ ∈ P Λ n and two integers 1 ≤ s, t ≤ ℓn, define the sequence:. . , β λ ℓn ) ∈ Z ℓn . These sequences will be used heavily in what follows. The next result records the combinatorial properties of sequence of charged beta numbers that we need. 5.11. Proposition. Let λ ∈ P Λ m , where n ≥ m and fix i ∈ I m and integers 1 ≤ s < t ≤ ℓn. Supposewhich is a contradiction. Hence, h = h ′ completing the proof of part (a).Before considering (b) and (c) recall that going back to Littlewood (cf.[34,Lemma 5.26]), it is wellknown that removing an h-rim hook from a partition corresponds to subtracting h from one of the beta numbers of the partition. This result easily translates into an analogous result for ℓ-partitions.In particular, it follows that h ≤ m if and only if h = h α for some α ∈ λ, so we have proved (b) since h ′ α = β s − β t − h α = h ′ . Next consider (c) and suppose that h > m. Then {β λ ls+1 , . . . , β λ s − h, . . . , β λ ls+n−1 } cannot be a set of (shifted) beta numbers for λ (ls) so l s < l t and, as in the first paragraph, {β λ ls+1 , . . . , β λ t +h ′ , . . . , β λ ls+n−1 } and {β λ lt+1 , . . . , β λ s − h ′ , . . . , β λ lt+n−1 } are (shifted) sets of beta numbers for λ (ls) and λ (lt) , respectively. Hence, by the last paragraph, h ′ = h α ′ for some α ′ ∈ λ. In particular, h ′ ≤ m. Moreover, by parts (a) and. This completes the proof of (c). Finally, as in (d), suppose that µ ∈ P Λ m and thatIn particular, there are exactly two ways to write µ in the form λ α,t if h ≤ m and h ′ ≤ m and, as in the last paragraph, there is only one to write µ in thisBeta numbers and Gram determinants.We are now ready to start proving our "classical" Jantzen sum formula. We start by giving a closed formula for the γ-coefficients from (3.11).Given two nodes α = (m, r, c) and α ′ = (m ′ , r ′ , c ′ ) define α to be below α ′ if either m ′ < m or m ′ = m and c ′ < c. Let Add k (t) to be the set of addable nodes of the tableau t ↓k that are below the node t −1 (k). Similarly, let Rem k (t) be the set of removable nodes of the tableau t ↓k that are below the node t −1 (k).5.12.Lemma. Suppose that t ∈ Std(λ), for λ ∈ P Λ m . ThenProof. This follows by induction on the dominance order. The base case is given by the closed formula for γ t λ given in(4.8)The next result is a "branching rule" for det G λ i using the charged beta numbers.Proof. If m = 1 then there is nothing to prove, so assume m > 1. By Lemma 3.21 and Lemma 5.12,Fix t ∈ Std(λ) and consider its contribution to det G λ i in the product above. As we read down the rows in each component, the addable and removable nodes in Add m (t) ∪ Rem m (t) alternate between addable and removable nodes, finishing with the addable node (k, L(λ (j) ) + 1, 1) at the bottom of the jth component, for 1 ≤ j ≤ ℓ. Set γ = t −1 (m) = (l s , r s , a) and let β = (l v , r v , c) ∈ Rem m (t) be a removable node and let α = (l u , r u , b) be the addable node directly above it. Write s = (l s − 1)n + r s , u = (l u −1)n+r u and v = (l v −1)n+r v . Note that r v > r u since β is below α, so v > u. The consecutive beta numbers between rows u and v all differ by 1, so he contribution from these three nodes toThe addable nodes (j, L(λ (j) ) + 1, 1) ∈ Add m (t) at the bottom of λ (j) , for l ≤ j ≤ ℓ, have not yet been included in this calculation. Via t, the contribution of such nodes to det G λ i is:For each j, all but one of the terms in the right hand product cancels out because β λ t = κ j − t, for (j − i)L(λ (j) ) < t ≤ jn. The point of adding the extra terms to the last two displayed equations is that this allows us to give a generic formula for these determinants.If s, t ∈ Std i (λ) and ν = Shape(s ↓(m−1) ) = Shape(t ↓(m−1) ) then Add m (s) = Add m (t) and Rem m (s) = Rem m (t), so such tableaux contribute the same factor to det G λ i . The number of these tableaux is dim S ν i ′ = d i ′ (β λ 1 , . . . , β λ s − 1, . . . , β λ ℓn ), since t −1 (m) = (l s , r s , a). Therefore, combining all of contributions to G λ i from all t ∈ Std i (λ) shows thatIn fact, in addition to the terms identified above the right hand product above introduces extra terms when β λ t ≡ i m or when the node corresponding to β λ t is not removable, in which case the exponent d i ′ (β λ 1 , . . . , β λ s − 1, . . . , β λ ℓn ) = 0. Hence, all of these "extra terms" either cancel out or are equal to 1, so adding them does not change the determinant. In addition, the condition n ∤ t in the denominator can be omitted because all of the extra terms that this introduces to the denominator are equal to 1 since they all have exponent zero.Write the right hand product as A λCombining the terms in the numerator and denominator in this way completes the proof.Armed with the inductive formula of Lemma 5.13 we can now give our first closed formula for the Gram determinant G λ i . This result should be compared with[19,Theorem 3.35]. The statement of the result, and its proof, require only that 1 ≤ s, t ≤ ℓn and β λ s − β λ t < h. In particular, in spite of what the notation might suggest, we do not assume that s and t are ordered, so both s < t and s > t are possible.Recall the definition of the sequence β λ st (h) (5.10). Given 1 ≤ s, t ≤ ℓn define δ lslt (n) = (1 − δ lslt )n. That is, δ lslt (n) = 0 if l s = l t and δ lslt (n) = n otherwise. Notice that the product formula in the next result does not require s < t although, implicitly, it does require l s ≤ l t . 5.14. Proposition. Suppose that i ∈ I m and λ ∈ P Λ m , and that n ≥ m. ThenProof. We argue by induction on m. If m = 0 then λ = (0| . . . |0) and β λ = (κ 1 − 1, κ 1 − 2, . . . , κ 1 − n, κ 2 − 1, . . . , κ 2 − n, . . . , . . . , κ ℓ − n).By convention empty products are 1, so det G λ i = 1. Similarly, since δ lslt (n) = 0 for all (s, t) the right hand product is equal to 1 in view of Proposition 5.11. Hence, the proposition holds when m = 0, giving us the base case of our induction. Now suppose that m > 0 and, as in Lemma 5.13, let i ′ = i ↓(m−1) . Recalling the notation and statement of Lemma 5.13 and applying induction,1≤s,t≤ℓn,r =s,t δ ls l t (n)<h<β λwhere the last equality follows using the formulas for A λ 1 and A λ 2 from Lemma 5.13. Applying Corollary 5.6 completes the proof.5.15.Remark. The proof of Proposition 5.14, and the notation that it uses, is different from the results in[18,19]because our definition of charged beta numbers corresponds to using first column hook lengths whereas these references use first row hook lengths. In the mathematics, this difference manifests itself through subtle sign changes. For example,[18]requires that h > β λ s − β λ t whereas we require that h < β λ s − β λ t . We use column hook lengths because of their direct connection with beta numbers.5.16.Example. Let e = 3, Λ = Λ 0 + Λ 2 and λ = (3, 2|1). As in Example 5.3 and Example 5.9, take n = 6 and κ = (20, 6) so that β λ =(23,21,18,17,16,15,8,6,5,4,3,2). The following table gives the different contributions to det G λ i , for i ∈ I n , given by Proposition 5.14. The entries in each row are complete;y determined by β λ and the integers s, t and h. In particular, h ′ = β λ s − β λ t − h and the 1 , . . . , β λ s − h, . . . , β λ t + h, . . . β λ ℓn ) = εd i (ν). where, in the product, δ lslt (n) < h ≤ β λ s − β λ t . By parts (b) and (c) of Proposition 5.11, if δ lslt (n) < h ≤ β λ s − β λ t then there exists a node α ∈ λ such A Hecke algebra of (Z/rZ) ≀ Sn and construction of its irreducible representations. S Ariki, K Koike, 10.1006/aima.1994.1057Adv. Math. 106Pages 9 and 11.S. Ariki and K. Koike, A Hecke algebra of (Z/rZ) ≀ Sn and construction of its irreducible representations, Adv. Math., 106 (1994), 216-243. [Pages 9 and 11.] Cyclotomic Nazarov-Wenzl algebras. S Ariki, A Mathas, H Rui, arXiv:math/0506467Special issue in honour of George Lusztig). 182Page 10.S. Ariki, A. Mathas, and H. Rui, Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J., 182 (2006), 47-134. (Special issue in honour of George Lusztig), arXiv:math/0506467. [Page 10.] A Belinson, J Bernstein, A proof of Jantzen conjectures, in I. M. Gel'fand Seminar. Providence, RIAmer. Math. Soc16Page 1.A. Belinson and J. Bernstein, A proof of Jantzen conjectures, in I. M. Gel'fand Seminar, Adv. Soviet Math., 16, Amer. Math. Soc., Providence, RI, 1993, 1-50. [Page 1.] Centers of degenerate cyclotomic Hecke algebras and parabolic category O. J Brundan, 10.1090/S1088-4165-08-00333-6Represent. Theory. 12Page 3.J. Brundan, Centers of degenerate cyclotomic Hecke algebras and parabolic category O, Represent. Theory, 12 (2008), 236-259. [Page 3.] Graded decomposition numbers for cyclotomic Hecke algebras. J Brundan, A Kleshchev, 10.1016/j.aim.2009.06.018Adv. Math. 222Pages 8, 9, 10, and 17.J. Brundan and A. Kleshchev, Graded decomposition numbers for cyclotomic Hecke algebras, Adv. Math., 222 (2009), 1883-1942. [Pages 8, 9, 10, and 17.] Graded Specht modules. J Brundan, A Kleshchev, W Wang, 10.1515/CRELLE.2011.033arXiv:0901.0218J. Reine Angew. Math. 655Pages 7 and 10.J. Brundan, A. Kleshchev, and W. Wang, Graded Specht modules , J. Reine Angew. Math., 655 (2011), 61-87. arXiv:0901.0218. [Pages 7 and 10.] Kazhdan-Lusztig conjecture and holonomic systems. J.-L Brylinski, M Kashiwara, 10.1007/BF01389272Invent. Math. 64Page 1.J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math., 64 (1981), 387-410. [Page 1.] C W Curtis, I Reiner, With applications to finite groups and orders. New York; New YorkWiley-Interscience PublicationIIMethods of representation theory. Page 19.C. W. Curtis and I. Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1987. With applications to finite groups and orders, A Wiley-Interscience Publication. [Page 19.] On bases of some simple modules of symmetric groups and Hecke algebras. M Boeck, A Evseev, S Lyle, L Speyer, 10.1007/s00031-017-9444-7Transform. Groups. 23Page 21.M. de Boeck, A. Evseev, S. Lyle, and L. Speyer, On bases of some simple modules of symmetric groups and Hecke algebras, Transform. Groups, 23 (2018), 631-669. [Page 21.] Cyclotomic q-Schur algebras. R Dipper, G James, A Mathas, 10.1007/PL00004665Math. Z. 229Page 10.R. Dipper, G. James, and A. Mathas, Cyclotomic q-Schur algebras, Math. Z., 229 (1998), 385-416. [Page 10.] Morita equivalences of Ariki-Koike algebras. R Dipper, A Mathas, 10.1007/s002090100371Math. Z. 240Page 21.R. Dipper and A. Mathas, Morita equivalences of Ariki-Koike algebras , Math. Z., 240 (2002), 579-610. [Page 21.] Towards the Kazhdan-Lusztig conjecture. O Gabber, A Joseph, Ann. Sci.École Norm. Sup. 4Page 1.O. Gabber and A. Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Sci.École Norm. Sup. (4), 14 (1981), 261-302. [Page 1.] Cellular algebras. J J Graham, G I Lehrer, 10.1007/BF01232365Invent. Math. 123Pages 4 and 5.J. J. Graham and G. I. Lehrer, Cellular algebras , Invent. Math., 123 (1996), 1-34. [Pages 4 and 5.] Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A. J Hu, A Mathas, 10.1016/j.aim.2010.03.002arXiv:0907.2985Adv. Math. 225Pages 4, 5, 7, 8, and 10.J. Hu and A. Mathas, Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A, Adv. Math., 225 (2010), 598-642. arXiv:0907.2985. [Pages 4, 5, 7, 8, and 10.] Graded induction for Specht modules. 10.1093/imrn/rnr058arXiv:1008.1462Int. Math. Res. Not. IMRN. Page 23., Graded induction for Specht modules , Int. Math. Res. Not. IMRN, 2012 (2012), 1230-1263. arXiv:1008.1462. [Page 23.] Seminormal forms and cyclotomic quiver Hecke algebras of type A. 10.1007/s00208-015-1242-8arXiv:1304.0906Math. Ann. 36416, Seminormal forms and cyclotomic quiver Hecke algebras of type A, Math. Ann., 364 (2016), 1189-1254. arXiv:1304.0906. [Pages 9, 10, 11, 12, 13, 16, and 17.] On the decomposition matrices of the symmetric groups. G James, J. Algebra. IIPage 21.G. James, On the decomposition matrices of the symmetric groups. II, J. Algebra, 43 (1976), 45-54. [Page 21.] A q-analogue of the Jantzen-Schaper theorem. G James, A Mathas, 10.1112/S0024611597000099Proc. London Math. Soc. 743Pages 2, 21, and 28.G. James and A. Mathas, A q-analogue of the Jantzen-Schaper theorem , Proc. London Math. Soc. (3), 74 (1997), 241-274. [Pages 2, 21, and 28.] The Jantzen sum formula for cyclotomic q-Schur algebras. 10.1090/S0002-9947-00-02492-2Trans. Amer. Math. Soc. 352Pages 2, 3, 11, 13, 21, 22, 26, 27, and 28., The Jantzen sum formula for cyclotomic q-Schur algebras, Trans. Amer. Math. Soc., 352 (2000), 5381-5404. [Pages 2, 3, 11, 13, 21, 22, 26, 27, and 28.] Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen. J C Jantzen, v+124Bonn. Math. Schr. 67Page 1.J. C. Jantzen, Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen, Bonn. Math. Schr., 67 (1973), v+124. [Page 1.] Moduln mit einem höchsten Gewicht. Lecture Notes in Mathematics. 750SpringerPage 1., Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, 750, Springer, Berlin, 1979. [Page 1.] Representations of algebraic groups. Mathematical Surveys and Monographs. 107American Mathematical Societysecond ed.. Page 1., Representations of algebraic groups, Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, second ed., 2003. [Page 1.] Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras. S.-J Kang, M Kashiwara, 10.1007/s00222-012-0388-1Invent. Math. 190Page 7.S.-J. Kang and M. Kashiwara, Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras , Invent. Math., 190 (2012), 699-742. [Page 7.] Representations of Coxeter groups and Hecke algebras. D Kazhdan, G Lusztig, 10.1007/BF01390031Invent. Math. 53Page 1.D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53 (1979), 165-184. [Page 1.] Affine Lie algebras and quantum groups. 10.1155/S1073792891000041Internat. Math. Res. Notices. Page 1., Affine Lie algebras and quantum groups, Internat. Math. Res. Notices, 1991, 21-29. [Page 1.] A diagrammatic approach to categorification of quantum groups. I , Represent. Theory. M Khovanov, A D Lauda, 10.1090/S1088-4165-09-00346-X13Pages 5 and 6.M. Khovanov and A. D. Lauda, A diagrammatic approach to categorification of quantum groups. I , Represent. Theory, 13 (2009), 309-347. [Pages 5 and 6.] A diagrammatic approach to categorification of quantum groups II. 10.1090/S0002-9947-2010-05210-9Trans. Amer. Math. Soc. 363Page 5., A diagrammatic approach to categorification of quantum groups II , Trans. Amer. Math. Soc., 363 (2011), 2685-2700. [Page 5.] Linear and projective representations of symmetric groups. A Kleshchev, 10.1017/CBO9780511542800Cambridge Tracts in Mathematics. 163Cambridge University PressPage 12.A. Kleshchev, Linear and projective representations of symmetric groups , Cambridge Tracts in Mathematics, 163, Cambridge University Press, Cambridge, 2005. [Page 12.] S Lang, Algebra , 10.1007/978-1-4613-0041-0Graduate Texts in Mathematics. New YorkSpringer-Verlag211third ed.. Page 15.S. Lang, Algebra, Graduate Texts in Mathematics, 211, Springer-Verlag, New York, third ed., 2002. [Page 15.] Hecke algebras at roots of unity and crystal bases of quantum affine algebras. A Lascoux, B Leclerc, J.-Y Thibon, Comm. Math. Phys. 181Page 18.A. Lascoux, B. Leclerc, and J.-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys., 181 (1996), 205-263. [Page 18.] Integral basis theorem of cyclotomic Khovanov-Lauda-Rouquier algebras of type A. G Li, 10.1016/j.jalgebra.2017.02.022arXiv:1412.3747J. Algebra. Page 7.G. Li, Integral basis theorem of cyclotomic Khovanov-Lauda-Rouquier algebras of type A, J. Algebra, 482 (2017), 1-101. arXiv:1412.3747. [Page 7.] Modular representations of symmetric groups. D E Littlewood, Proc. Roy. Soc. London. Ser. A. 209Page 22.D. E. Littlewood, Modular representations of symmetric groups, Proc. Roy. Soc. London. Ser. A., 209 (1951), 333-353. [Page 22.] On quantum groups. G Lusztig, 10.1016/0021-8693(90)90187-SJ. Algebra. Page 1.G. Lusztig, On quantum groups, J. Algebra, 131 (1990), 466-475. [Page 1.] Iwahori-Hecke algebras and Schur algebras of the symmetric group. A Mathas, 10.1090/ulect/015University Lecture Series. 15American Mathematical SocietyPages 10, 14, 18, 21, and 25.A. Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, 15, American Mathematical Society, Providence, RI, 1999. [Pages 10, 14, 18, 21, and 25.] Matrix units and generic degrees for the Ariki-Koike algebras. 10.1016/j.jalgebra.2004.07.021arXiv:math/0108164J. Algebra. 281Page 11., Matrix units and generic degrees for the Ariki-Koike algebras , J. Algebra, 281 (2004), 695-730. arXiv:math/0108164. [Page 11.] Seminormal forms and Gram determinants for cellular algebras. 10.1515/CRELLE.2008.042arXiv:math/0604108With an appendix by Marcos Soriano. 619Pages 11 and 13., Seminormal forms and Gram determinants for cellular algebras, J. Reine Angew. Math., 619 (2008), 141- 173. With an appendix by Marcos Soriano, arXiv:math/0604108. [Pages 11 and 13.] Cyclotomic quiver Hecke algebras of type A, in Modular representation theory of finite and p-adic groups. 10.1142/9789814651813_0005arXiv:1310.2142National University of Singapore Lecture Notes Series. G. W. Teck and K. M. Tan305World ScientificPage 8., Cyclotomic quiver Hecke algebras of type A, in Modular representation theory of finite and p-adic groups, G. W. Teck and K. M. Tan, eds., National University of Singapore Lecture Notes Series, 30, World Scientific, 2015, ch. 5, 165-266. arXiv:1310.2142. [Page 8.] Restricting Specht modules of cyclotomic Hecke algebras. 10.1007/s11425-016-9032-2arXiv:1610.09729Special Issue on Representation Theory. Page 23., Restricting Specht modules of cyclotomic Hecke algebras, Science China Mathematics, 61 (2018), 299-310. Special Issue on Representation Theory, arXiv:1610.09729. [Page 23.] The representations of Hecke algebras of type An. G E Murphy, 10.1006/jabr.1995.1079J. Algebra. 173Page 10.G. E. Murphy, The representations of Hecke algebras of type An, J. Algebra, 173 (1995), 97-121. [Page 10.] Quiver Hecke algebras and 2-Lie algebras. R Rouquier, Algebra Colloq. 19Pages 5 and 10.R. Rouquier, Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq., 19 (2012), 359-410. [Pages 5 and 10.] The Schaper formula and the Lascoux, Leclerc and Thibon algorithm. S Ryom-Hansen, 10.1023/A:1025761311221Lett. Math. Phys. 64Page 18.S. Ryom-Hansen, The Schaper formula and the Lascoux, Leclerc and Thibon algorithm , Lett. Math. Phys., 64 (2003), 213-219. [Page 18.] Charakterformeln für Weyl-Moduln und Specht-Moduln in Primcharacteristik, diplomarbeit. K.-D Schaper, BonnPage 2.K.-D. Schaper, Charakterformeln für Weyl-Moduln und Specht-Moduln in Primcharacteristik, diplomarbeit, Bonn, 1981. [Page 2.] Graded decomposition matrices of v-Schur algebras via Jantzen filtration. P Shan, 10.1090/S1088-4165-2012-00416-2Represent. Theory. Page 2.P. Shan, Graded decomposition matrices of v-Schur algebras via Jantzen filtration, Represent. Theory, 16 (2012), 212-269. [Page 2.] Andersen filtration and hard Lefschetz. W Soergel, 10.1007/s00039-007-0640-9Geom. Funct. Anal. 17Page 1.W. Soergel, Andersen filtration and hard Lefschetz , Geom. Funct. Anal., 17 (2008), 2066-2089. [Page 1.] Local Hodge theory of Soergel bimodules. G Williamson, 10.1007/s11511-017-0146-8Acta Math. 217Page 1.G. Williamson, Local Hodge theory of Soergel bimodules , Acta Math., 217 (2016), 341-404. [Page 1.] Schubert calculus and torsion explosion. 10.1090/jams/868arXiv:1309.5055J. Amer. Math. Soc. Alex Kontorovich and Peter J. McNamara30Page 21., Schubert calculus and torsion explosion , J. Amer. Math. Soc., 30 (2017), 1023-1046. With a joint appendix with Alex Kontorovich and Peter J. McNamara, arXiv:1309.5055. [Page 21.] On Quantitative Substitutional Analysis I. A Young, 10.1112/plms/s1-33.1.97Proc. London Math. Soc. 33Page 11.A. Young, On Quantitative Substitutional Analysis I , Proc. London Math. Soc., 33 (1900), 97-145. [Page 11.] A conjecture for q-decomposition matrices of cyclotomic v-Schur algebras. X , Yvonne , 10.1016/j.jalgebra.2006.03.048J. Algebra. 304Page 18.X. Yvonne, A conjecture for q-decomposition matrices of cyclotomic v-Schur algebras , J. Algebra, 304 (2006), 419- 456. [Page 18.]
[]
[ "Asymptotically optimal control for a multiclass queueing model in the moderate deviation heavy traffic regime *", "Asymptotically optimal control for a multiclass queueing model in the moderate deviation heavy traffic regime *" ]
[ "Rami Atar ", "Asaf Cohen " ]
[]
[]
A multi-class single-server queueing model with finite buffers, in which scheduling and admission of customers are subject to control, is studied in the moderate deviation heavy traffic regime. A risk-sensitive cost set over a finite time horizon [0, T ] is considered. The main result is the asymptotic optimality of a control policy derived via an underlying differential game. The result is the first to address a queueing control problem at the moderate deviation regime that goes beyond models having the so called pathwise minimality property. Moreover, despite the well known fact that an optimal control over a finite time interval is generically of a nonstationary feedback type, the proposed policy forms a stationary feedback, provided T is sufficiently large.
10.1214/16-aap1269
[ "https://arxiv.org/pdf/1510.04127v2.pdf" ]
27,630,596
1510.04127
5aff30d7b99f0a53fadfb50ddeb3dedfc769e595
Asymptotically optimal control for a multiclass queueing model in the moderate deviation heavy traffic regime * 11 Oct 2016 October 12, 2016 Rami Atar Asaf Cohen Asymptotically optimal control for a multiclass queueing model in the moderate deviation heavy traffic regime * 11 Oct 2016 October 12, 2016arXiv:1510.04127v2 [math.PR]AMS subject classifications: 49N70, 60F10, 60K25, 93E20 Keywords: moderate deviationsheavy trafficrisk-sensitive costdifferential games A multi-class single-server queueing model with finite buffers, in which scheduling and admission of customers are subject to control, is studied in the moderate deviation heavy traffic regime. A risk-sensitive cost set over a finite time horizon [0, T ] is considered. The main result is the asymptotic optimality of a control policy derived via an underlying differential game. The result is the first to address a queueing control problem at the moderate deviation regime that goes beyond models having the so called pathwise minimality property. Moreover, despite the well known fact that an optimal control over a finite time interval is generically of a nonstationary feedback type, the proposed policy forms a stationary feedback, provided T is sufficiently large. Introduction This paper continues a line of research started in [1] that aims at analyzing queueing control problems (QCPs) at the moderate deviation (MD) heavy traffic regime. The model under consideration consists of a server that serves customers from a number of classes, where allocation of the effort among classes is dynamically controlled. Customers are kept in buffers of finite size, one buffer for each class, and those that arrive to find a full buffer are lost. It is also possible to reject arrivals when buffers are not full. This control system is considered with a risk-sensitive (RS) cost, that accounts for holding of customers in the buffers as well as for rejections. At the heart of the analysis lies a differential game (DG) that has been analyzed in [2]. This paper proves the validity of the prediction of [2] that the DG governs the scaling limit, by showing that the QCP's value converges to the DG's value, and identifying an asymptotically optimal (AO) policy for the former that is constructed based on the latter. The limit result for the model treated in [1] was built on pathwise minimality, a property that considerably simplifies the analysis, which does not hold in our setting. Instead, the proof here is based on the Bellman (or the dynamic programming) equation and, specifically, a free boundary point characterized by it governs the asymptotic behavior. Traditionally, heavy traffic analysis of queueing models, and particularly QCPs, is carried out under the regime of diffusion-scale deviation (sometimes also referred to as ordinary deviation), but it is also relevant at the MD scale, where relatively few results exist [21], [20], [16]. The roots of large deviation (LD) analysis of control systems go back to Fleming [11], who studies the associated Hamilton-Jacobi equations. The connection of RS cost to DG was made by Jacobson [18]. Analyzing RS control by LD tools and the formulation of the corresponding maximum principle are due to Whittle [24]. Various aspects of this approach have been studied for controlled stochastic differential equations, for example in [10], [13], [15]. The treatment of a QCP at the MD scale is similar to that at the LD scale (in papers such as [3]) as far as the tools are concerned, but there are reasons to believe that the games obtained in the MD regime are solvable more often than in the LD regime. This statement is supported by the fact that the paper [1] solves a DG for the MD scale, whereas a solution of an analogous DG for the LD regime is not known in general (see [5] for a partial solution of the latter, and an open problem regarding its general solution). Similarly, the DG of this paper has been solved in [2], but an explicit solution for the LD analogue is not known. An additional advantage the MD regime has over LD is the invariance to the stochastic data, specifically, the arrival and service time distributions, as long as they possess exponential moments, where LD results are more sensitive. The combination of these properties provides a great deal of motivation for working at the MD scale. The aforementioned pathwise minimality property has been the basis for solving QCPs in diffusion scale heavy traffic asymptotics in various works in the past (for example, [17]). To describe this property, consider the simple diffusion control problem of minimizing a cost J(ζ) over all control processes ζ having R + -valued nondecreasing sample paths. The cost takes the form J(ζ) = E T 0 h(ξ t )dt, where h : R + → R + is nondecreasing, ξ t = x + w t + ζ t , x ≥ 0 and T > 0 are fixed, w is a standard Brownian motion, and the constraint ξ t ≥ 0 for all t ≥ 0 must be met. The solution is to set ζ t = − inf s≤t [(x + w s ) ∧ 0], making ξ a reflected Brownian motion starting from x, reflecting at 0. This follows by the well-known fact that a.s., for all t, x+w t +ζ t ≤ x+w t +ζ t , for any controlζ meeting the constraint (see, for example, Section 2 of [8]). Although this problem is simpler than typical diffusion control problems in the literature, pathwise solutions of these problems owe to this simple property (or sometimes multidimensional versions thereof). The DG of [1], identifying the MD asymptotics of a QCP, was also solved by such a consideration. A simplified version of this game, presented with one-dimensional instead of multidimensional dynamics, is as follows. It is a zero-sum game with payof J(λ,μ, ζ) = T 0 h(ϕ t )dt − T 0 (aλ 2 t + bμ 2 t )dt,(1) and dynamics ϕ t = x + t 0 (λ s −μ s )ds + ζ t .(2) Here, x, T , a and b are positive constants, and h is again nondecreasing. The function ζ is a control for the minimizing player, taking values in R + and is nondecreasing, while the functions λ andμ form a control for the maximizing player, and are nonnegative. The constraint ϕ t ≥ 0 for all t must be satisfied. In this game, the functions ϕ and ζ represent MD-scaled queue length and idleness processes, whileλ andμ stand for MD-scaled perturbations of the arrival and service processes. The function h is the running cost in the underlying RS cost, while the second term in (1) corresponds to penalty associated with changes of measure, and its form originates from the LD rate function (background on the structure of DGs governing RS control asymptotics appears in [12] and [14]). It is easy to see how pathwise minimality can be used once again to find an optimal strategy for the minimizer. Namely, for (λ,μ) given, setting ζ t = − inf s≤t (ψ s ∧ 0), where ψ t = x + t 0 (λ s −μ s )ds, results with ϕ that bounds from below any other dynamics adhering to the constraint. Significantly, this pathwise minimality property provides not only a solution to the game but also the basis of the AO proof in [1], as one can mimic the behavior of this strategy to come up with a policy for the queueing model that is automatically AO. It turns out that one cannot argue along the same lines for the game obtained under LD scaling. Indeed, note carefully that the solution method just presented uses the fact that the second term in (1) does not depend on the control for the minimizing player. However, under LD scaling, the corresponding penalty term, accounting for changes of measure, involves controls of both players. This makes it impossible to obtain a pathwise solution in the same fashion. This point is explained in detail in Section 1 of [1]. Although pathwise minimality is useful when it applies, it is not generic even under the diffusion and MD regimes. A natural approach to handle more general settings, that has been used in numerous papers on diffusion scale asymptotics, is to appeal to dynamic programming methods to solve diffusion control problems and then use these solutions as a vehicle for analyzing the QCP (for a small sample of these papers see [4], [7], [23]). This approach has not been considered before for MD asymptotics of QCPs. The model studied in this paper is indeed suitable for such an approach, and in fact constitutes a prototype for QCPs that are too complex to possess directly solvable DGs, while a solution via dynamic programming is available. The DG for our model differs from the one presented above. Again, we present it in a slightly simplified way; the precise details appear in Section 2. The payoff (1) has an additional term ̺ T , the dynamics (2) has an additional term −̺ t , and the constraint ϕ t ≥ 0 is strengthened to ϕ t ∈ [0, D] for all t, where D is a constant. The R + -valued nondecreasing function ̺ represents cumulative rejections, and is considered part of the control for the minimizing player; that is, in this game the minimizing player controls the pair (ζ, ̺). The constraint stems from the finiteness of the buffer, and the constant D is related to the buffer size (in fact, it is the buffer size measured in units of MD-scaled workload). In [2], this game was analyzed via a free boundary value problem, and solved for the value function and optimal strategy. The contribution of this paper is to substantiate the relation of the queueing model to the game in a rigorous manner, showing that the latter indeed governs the MD asymptotics. This is established by proving that the value of the RS QCP converges to that of the DG, and translating the DG's optimal strategy into an AO policy for the QCP. The proof of convergence of the RS value to the DG value is performed in two main steps, namely bounding the former from below and from above by the latter. We refer to them as the lower and upper bound, respectively. The proof of an asymptotic lower bound of a RS cost in models that do not have control is often based on the classical proof of (the lower bound in) Varadhan's lemma [9] for a sequence of processes satisfying the large deviations principle. This is the case for example for queueing models that are studied under a specified policy. It relies on the identification of a 'behavior' that contributes most to the cost, such as when the underlying stochastic dynamics (say, the suitably normalized multidimensional queue length) lie close to a specific path. In the case of controlled dynamics, this path is formulated as the control selected by the maximizing player in the DG. To obtain the lower bound one must consider an arbitrary sequence of policies, and then the challenge stems from the fact that different policies for the queueing model may give rise to different such paths. A brute force approach of identifying an optimal path for each arbitrary policy seems intractable. The argument uses instead properties of the DG studied in [2]. It has been shown that this DG, specified in terms of multi-dimensional dynamics, can be reduced to one dimension. The one-dimensional state corresponds to the (suitably normalized) total workload in the system. Moreover, there is a threshold, denoted by β 0 , dictating the behavior of both players. When the workload is below this threshold, there is a certain fixed path that guarantees attaining at least the game's value under any action of the minimizer (although it need not be optimal). When the threshold is exceeded, there is no such fixed path. However, the following fact can be used. As long as the workload remains above the level β 0 , the minimizer encounters an accumulated loss, which is higher than the cost of an immediate rejection to the level β 0 . We identify a suitable path for the maximizer that is effective until the time when the threshold is reached. We focus on this path when workload is above β 0 , and switch to the path alluded to above, when it is below β 0 . However, this switching time depends on the policy, and so it is random and varies with the scaling parameter. Accordingly, the argument uses time discretization, where each one of a finite collection of possible switching times is estimated separately. The upper bound is obtained by constructing a policy for the QCP for which the cost converges to the DG's value. There is a naive way of interpreting the DG solution as a control policy for the QCP. However, the two components of this policy, corresponding to rejection and service effort allocation, impose contradictory requirements. For rejections must occur only from a specific class, and only when the total workload in the system exceeds the aforementioned threshold. On the other hand, it stems from the game solution that the service allocation policy must cause the (suitably normalized) multidimensional queue length processes to evolve along a certain curve in state space, denoted in this paper by γ. One can express the fact that buffers are finite by requiring that these queueing processes always lie in a certain hyper-rectangular domain, denoted by X ; the curve γ happens to intersect the boundary of the domain X . When the multidimensional queueing process is on (certain parts of) that boundary, one of the buffers must be full, and then even small stochastic fluctuations require rejections so as to meet the buffer size constraint. As a consequence, rejections will not always adhere to the rejection policy alluded to above. We address this issue by aiming at a curve γ a , that approximates γ but does not intersect the boundary of the domain. The main body of work is to develop estimates that show that the queueing processes evolve along this approximate curve, up to negligible probabilities, and that as a result both elements of the policy are respected with sufficiently high probability. It is well known that an optimal control over a finite time interval is generically of a non-stationary feedback type (see, for example, Sections III.8-9 in [14], where the nonstationary optimal feedback is characterized by (8.5) and the stationary optimal feedback by (9.9), corresponding to a problem set on a finite time horizon and, respectively, an infinite time horizon with a discounted cost). Despite that, it was anticipated in [2], and established in this paper, that for the setting studied here, a RS cost set over a finite time horizon [0, T ] gives rise to a stationary feedback provided that T is sufficiently large. Indeed, the policy we present for the QCP has this feature, which makes it simple as compared to policies based on time-varying characteristics. We consider this as one of the main aspects of this paper's contribution. The organization of the paper is as follows. Section 2 presents the model, the MD scaling and the main result, which states that the MD QCP's value converges to that of the DG. Section 3 collects a few results from [2] required for the proof. Section 4 gives a lower bound on the QCP's value asymptotics in terms of the DG's value, and Section 5 finds a nearly optimal policy derived from the game's optimal strategy. Together, Sections 4 and 5 provide the proof of the main result. Some auxiliary results appear in the Appendix. We use the following notation. For a positive integer k and a, b ∈ R k , a · b denotes the usual scalar product, while · denotes Euclidean norm. {e 1 , . . . , e k } is the standard basis of R k . We denote [0, ∞) by R + . For 0 < T < ∞ and a function f : R + → R k , f T = sup [0,T ] f , while osc T (δ, f ) = sup{ f (u) − f (t) : 0 ≤ u ≤ t ≤ (u + δ) ∧ T }. Denote by AC([0, T ], R k ), C([0, T ], Model and results Model description We consider a model with I customer classes and a single server. A buffer with finite room is dedicated to each customer class, and upon arrival, customers are queued in the corresponding buffers, or rejected by the system administrator. Within each class, customers are served at the order of arrival, where the server may only serve the customer at the head of each line. Processor sharing is allowed, and so the server is capable of serving up to I customers of distinct classes simultaneously. The model is defined on a probability space (Ω, F, P). Expectation with respect to P is denoted by E. The parameters and processes we introduce depend on an index n ∈ N, serving as the scaling parameter. Arrivals occur according to independent renewal processes, and service times are independent and identically distributed across each class. Let I = {1, 2, . . . , I}. Let λ n i > 0, n ∈ N, i ∈ I be given parameters, representing the reciprocal mean inter-arrival times of class-i customers. Let {IA i (l) : l ∈ N} i∈I be I sequences of positive i.i.d. random variables with mean E[IA i (1)] = 1 and variance σ 2 i,IA = Var(IA i (1)) ∈ (0, ∞). With 0 1 = 0, the number of arrivals of class-i customers up to time t, for the n-th system, is given by A n i (t) := sup l ≥ 0 : l k=1 IA i (k) λ n i ≤ t , t ≥ 0.(3) For a collection ξ i , i ∈ I of stochastic processes we will always write ξ for (ξ i ) i∈I . Thus, in particular, A n is the I-dimensional process (A n i ) i∈I . Similarly we consider another set of parameters µ n i > 0, n ∈ N, i ∈ I, representing reciprocal mean service times. We are also given I independent sequences {ST i (l) : l ∈ N} i∈I of positive i.i.d. random variables (independent also of the sequences {IA i }) with mean E[ST i (1)] = 1 and variance σ 2 i,ST = Var(ST i (1)) ∈ (0, ∞). The time required to complete the service of the l-th class-i customer is given by ST i (l)/µ n i , and the potential service time processes are defined as S n i (t) := sup l ≥ 0 : l k=1 ST i (k) µ n i ≤ t , t ≥ 0. We consider the moderate deviations rate parameters {b n }, that form a sequence, fixed throughout, with the property that lim b n = ∞ while lim b n / √ n = 0, as n → ∞. The arrival and service parameters are assumed to satisfy the following conditions. As n → ∞, λ n i n → λ i ∈ (0, ∞), µ n i n → µ i ∈ (0, ∞),(4)λ n i := 1 b n √ n (λ n i − nλ i ) →λ i ∈ (−∞, ∞),μ n i := 1 b n √ n (µ n i − nµ i ) →μ i ∈ (−∞, ∞). (5) The system is assumed to be critically loaded in the sense that the overall traffic intensity equals 1, namely I 1 ρ i = 1 where ρ i = λ i /µ i for i ∈ I. For i ∈ I, let X n i be a process representing the number of class-i customers in the n-th system. Denote the number of rejection of class-i arrivals until time t in the n-th system by R n i (t). With S = {x = (x 1 , . . . , x I ) ∈ R I + : x i ≤ 1}, let B n be an S-values process, whose i-th component represents the fraction of effort devoted by the server to the class-i customer at the head of the line. Then the number of service completions of class-i jobs during the time interval [0, t] is given by S n i (T n i (t)), where T n i (t) := t 0 B n i (u)du(6) is the time devoted to class-i customers by time t. With an abuse of notation, we often write S n • T n for (S n 1 • T n 1 , . . . , S n I • T n I ). We have the balance equation X n i (t) = X n i (0) + A n i (t) − S n i (T n i (t)) − R n i (t).(7) For simplicity, the initial conditions X n (0) = (X n 1 (0), . . . , X n I (0)) are assumed to be deterministic. We also assumeX n (0) →x = (x 1 , . . . , x I ) as n → ∞, i ∈ I. Note that, by construction, the arrival and potential service processes have RCLL paths, and accordingly, so does X n . The MD-scaled version of the queue length process satisfies X n (t) := 1 b n √ n X n (t) ∈ X := I i=1 [0, D i ], t ≥ 0,(8) where D i > 0 are fixed constants. Thus the size of buffer i is given by b n √ nD i . Additional MD-scaled processes arẽ A n i (t) = 1 b n √ n (A n i (t) − λ n i t),S n i (t) = 1 b n √ n (S n i (t) − µ n i t),R n i (t) = 1 b n √ n R n i (t). The process U n := (B n , R n ) is regarded as a control, that is determined based on observations from the past events in the system. The precise definition of an admissible control is as follows. Given n, the processes U n and X n are said to be an admissible control and the corresponding queue length process if the sample paths of U n lie in D([0, ∞), S), (8) holds, and • U n is adapted to the filtration σ{A n i (u), S n i (T n i (u)), i ∈ I, u ≤ t}, where T n is given by (6); • For i ∈ I and t ≥ 0, one has X n i (t) = 0 implies B n i (t) = 0.(9) Denote the class of all admissible controls U n by U n . Note that this class depends on A n and S n , but we consider these processes to be fixed. For U n ∈ U n and the corresponding queue length process X n , the processesR n andX n are referred as the scaled rejection and queue length process corresponding to U n . Throughout, we assume the finite exponential moment condition, namely As shown in [22], under this condition, the scaled processes (à n ,S n ) satisfy a moderate deviation principle. Namely, for k = 1, 2, let J k (T, ·) be functions mapping D([0, T ], R I ) to [0, ∞] given by J k (T, ψ) = I i=1 s i,k T 0ψ i (u) 2 du if all ψ i ∈ AC 0 ([0, T ], R), ∞ otherwise, k = 1, 2,(10)for ψ = (ψ 1 , . . . , ψ I ) ∈ D([0, T ], R I ), where s i,1 = 1 2λ i σ 2 i,IA and s i,2 = 1 2µ i σ 2 i,ST , i ∈ I. Let J(T, ψ) = J 1 (T, ψ 1 ) + J 2 (T, ψ 2 ) for ψ = (ψ 1 , ψ 2 ) ∈ D([0, T ], R 2I ). Note that J is lower semicontinuous with compact level sets. Then one has Proposition 2.1 ( [22]) Under Assumption 2.1, for T > 0 fixed, the following holds. For every closed set F ⊂ D([0, T ], R 2I ), lim sup 1 b 2 n log P((à n ,S n ) ∈ F ) ≤ − inf ψ∈F J(T, ψ), and for every open set G ⊂ D([0, T ], R 2I ), lim inf 1 b 2 n log P((à n ,S n ) ∈ G) ≥ − inf ψ∈G J(T, ψ). To present the RS control problem, fixh,r ∈ (0, ∞) I . Given T ∈ (0, ∞) and n, the cost associated with a control U n ∈ U n is given by J n (T,X n (0), U n ) = 1 b 2 n log E T 0 e b 2 n [ t 0h ·X n (u)du+r·R n (t))] dt , whereX n andR n are the rescaled queue length and rejection processes corresponding to U n . In the above notation, we have emphasized the dependence on the initial stateX n (0). For background and a motivating discussion about this type of cost the reader is referred to [12]. The value of interest is given by V n (T,X n (0)) = inf U n ∈U n J n (T,X n (0), U n ). The differential game and main result Whereas the scaled queue length processX n is multidimensional, it is suggested in [2] that it is governed by a DG defined in terms of one-dimensional dynamics. The main result of this paper is the proof of this claim. Before presenting the formulation of this game, it is useful to draw attention to several stochastic processes that are themselves one-dimensional because the structure of the DG is closely related to them. Let θ n = n µ n DefineX n = θ n ·X n ,Ǎ n = θ n ·Ã n ,Š n = θ n ·S n , R n = θ n ·R n ,Y n = θ n ·Ỹ n ,Ž n = θ n ·Z n ,y n = θ n ·ỹ n . Also, letS n (t 1 , . . . , t I ) = (S n 1 (t 1 ), . . . ,S n I (t I )), S n (t 1 , . . . , t I ) = θ n ·S n (t 1 , . . . , t I ) = i θ n iS n i (t i ). Note that the sample paths ofS n [resp.,Š n ,S n ,Š n ] map R + → R I [resp., R + → R + , R I + → R I , R I + → R]. Next, the procesš Z n = i n µ n iZ n i is nonnegative and nondecreasing,(14) thanks to the fact that i B n i ≤ 1 while i ρ i = 1. Also, for every i ∈ I, the processR n i is nondecreasing. Thus by (11), X n =Y n +Ǎ n −Š n • T n +Ž n −Ř n ,(15) whereŽ n andŘ n are nonnegative, nondecreasing processes. Moreover, X n (t) ∈ [0, D n ], t ≥ 0.(16) It follows from the contraction principle that (Ǎ n (t),Š n (t)), t ∈ [0, T ], satisfy the MDP with the rate function I(T, ψ) = I 1 (T, ψ 1 ) + I 2 (T, ψ 2 ), ψ = (ψ 1 , ψ 2 ) ∈ D([0, T ], R 2 ), where I k (T, ψ k ) = s k T 0 (ψ k ) 2 (u)du if ψ k ∈ AC 0 ([0, T ], R), ∞ otherwise,(17)s 1 := I i=1 2ρ i σ 2 i,IA µ i −1 , and s 2 : = I i=1 2ρ i σ 2 i,ST µ i −1 .(18)See Lemma A.2. To define the DG, Denote x = θ ·x, y := limy n = i θ i (λ i − ρ iμi ), and y(t) = x + yt, t ∈ R + , and let P = C 0 ([0, ∞), R), E = {ξ ∈ D([0, ∞), R + ) : ξ is nondecreasing}.(19) Endow both spaces with the topology of uniform convergence on compacts. Given ψ = (ψ 1 , ψ 2 ) ∈ P 2 and (ζ, ̺) ∈ E 2 , the dynamics associated with the initial condition x and the data ψ, ζ, ̺ is defined as ϕ = y + ψ 1 − ψ 2 + ζ − ̺.(20) The game is played by a maximizing player that selects ψ = (ψ 1 , ψ 2 ) and a minimizing player that selects (ζ, ̺). We sometimes write the dependence of the dynamics on the data as ϕ[x, ψ, (ζ, ̺)]. There is an analogy between the above equation and equation (15), and between the condition that ζ and ̺ are nondecreasing and property (14). The control ζ stands for the scaled idle time processZ n and ̺ stands for the scaled rejection processR n . The following condition, analogous to property (16), will also be required, namely ϕ(t) ∈ [0, D], t ≥ 0,(21) where D = I i=1 θ i D i = lim n→∞ D n . A measurable mapping α : P 2 → E 2 is called a strategy for the minimizing player if it satisfies the causality property: for every ψ,ψ ∈ P 2 and t ∈ [0, ∞), ψ(u) =ψ(u) for every u ∈ [0, t] implies α[ψ](u) = α[ψ](u) for every u ∈ [0, t].(22) Given an initial condition x, a strategy α is said to be admissible for the initial condition x if, whenever (ψ 1 , ψ 2 ) ∈ P 2 and (ζ, ̺) = α[ψ] , the corresponding dynamics (20) satisfies the buffer constraint (21). We denote by A x the collection of admissible strategies for the initial condition x. We now describe the components of the cost function. For w ∈ R + , denote h(w) = inf{h · ξ : ξ ∈ X , θ · ξ = w}.(23) By the convexity of the set X , h is convex. Moreover, h(w) ≥ 0 for w ≥ 0 and equality holds if and only if w = 0. Therefore, h is strictly increasing on [0, D]. Let r = min{r · ξ : ξ ∈ R I + , θ · ξ = 1}. It is easy to see that r = min{r i µ i : i ∈ I} = r i * µ i * ,(24) where i * is an index (fixed throughout) that minimizes r i µ i . The index i * indicates the class that has lowest rejection cost per unit of workload. It plays an important role in Section 5 where our AO policy is presented; specifically, the policy is aimed at rejecting jobs from this class only. Given x ∈ [0, D], T ∈ R + , ψ = (ψ 1 , ψ 2 ) ∈ P 2 , and (ζ, ̺) ∈ E 2 , we define the cost until time T by c(x, T, ψ, ζ, ̺) = T 0 h(ϕ(t))dt + r̺(T ) − I(T, ψ),(25) where ϕ is the corresponding dynamics. The value of the game is defined by V (x) = inf α∈Ax sup ψ∈P 2 ,T ∈R + c(x, T, ψ, α[ψ]).(26) We call ψ the path control and the T a time control, or sometimes the termination time. Note that both are controlled by the maximizing player. We sometimes use the notation V h,r for V when we want to emphasize the dependence on the function h and the constant r. Namely, Vĥ ,r (x) is defined as V (x) with (ĥ,r) in place of (h, r) in (25). Recall that lim n→∞X n (0) = x. Our first main result is the following. Theorem 2.1 Let Assumption 2.1 hold. Then for all sufficiently large T , lim n→∞ V n (T,X n (0)) = V (x). The second main result of this paper is Theorem 5.1, that constructs a policy for the QCP, which is AO. Some useful properties of the game We briefly mention some results from [2] regarding the DG, to be used in the sequel. Set s = (s −1 1 + s −1 2 ) −1 . The following proposition follows by Lemma 3.1, Proposition 3.1, and Theorems 3.1, 3.2, and 3.5 in [2]. The contribution of the latter is to deduce part (i) below. Proposition 3.1 (i) For T ∈ R + , set V (T, x) = inf α∈Ax sup ψ∈P 2 ,t∈[0,T ] c(x, t, ψ, α[ψ]) (compare with (26)). Then V (T, x) = V (x) for all sufficiently large T . (ii) If −y < r/(4s) then for every x ∈ [0, D] one has V (x) = ∞. (iii) If −y ≥ r/(4s) then V (x) =        x 0 2s − y − y 2 − h(u) s du, 0 ≤ x ≤ β 0 , V (β 0 ) + r(x − β 0 ), β 0 < x ≤ D,(27) where, 1 β 0 =            h −1 −r 2 4s − ry , −h(D) ≤ r 2 4s + ry ≤ −h(0), D, r 2 4s + ry < −h(D).(28) We now present an optimal strategy for the minimizer. This strategy plays an important role in proving the upper bound and in finding an AO policy in the multidimensional stochastic problem. The minimizer's optimal strategy is of a β-barrier form. Informally, this is a strategy that uses the minimal control (ζ, ̺) so as to keep the dynamics ϕ in [0, β] at all times. In the definition that follows, and throughout the paper, we denote the Skorohod map on an interval [a, b] by Γ [a,b] ; see Appendix A.1. Definition 3.1 Fix (x, β) ∈ [0, D] 2 . The strategy α β = (α β,1 , α β,2 ) is called a β-barrier strat- egy if for every ψ ∈ P 2 one has (ϕ, α β,1 , α β,2 )[ψ] = Γ [0,β] (ψ). The next proposition follows by Proposition 3.1 and Theorems 3.1, 3.2, and 3.4 in [2]. Proposition 3.2 The β 0 -barrier strategy, α β 0 , is an optimal strategy. We provide two propositions that are useful in the proof of the lower bound. For this we present two path controls, ψ * and ψ ♯ x associated with an initial state in the intervals (β 0 , D] and [0, β 0 ) respectively. Fix x ∈ (β 0 , D]. Let ∆ > 0 be such that x > β 0 + ∆. Fix (ζ, ̺) ∈ P 2 . Define ψ ♯ (t) = (rt/(2s 1 ), −rt/(2s 2 )), 0 ≤ t ≤ τ ∆ , where τ ∆ = τ (ζ,̺),∆ := inf{t ≥ 0 : ϕ[x, ψ ♯ , (ζ, ̺)] ≤ β 0 + ∆} is the first time that the dynamics, ϕ := ϕ[x, ψ ♯ , (ζ, ̺)], cross β 0 +∆. The following proposition is Proposition 3.2 in [2]. Proposition 3.3 For every (ζ, ̺) ∈ P 2 such that ̺(0) − ζ(0) < x − (β 0 + ∆) one has τ ∆ 0 h(ϕ(t))dt + r̺(τ ∆ ) − I(τ ∆ , ψ ♯ ) > r(x − (β 0 + ∆)).(29) Note that the l.h.s. is the cost associated with x, ψ ♯ , and (ζ, ̺) incurred until the time the dynamics cross β 0 +∆, whereas the r.h.s. gives the cost of an immediate rejection of x−(β 0 +∆). The result thus implies that if x > β 0 then the minimizer will reject x − β 0 units of mass at time zero. Next, fix x ∈ [0, β 0 ). Let ψ * x = s s 1 ω * x , −s s 2 ω * x ,(30) where ω * x ∈ C([0, τ * x ), R) is the unique solution oḟ ω * x (t) =V (x + yt + ω * x (t)) 2s , t ≥ 0,(31) with ω * x (0) = 0 and τ * x = x 0 1/ y 2 − h(ξ)/s dξ.(32) Existence and uniqueness for ω * x over the time interval [0, τ * x ] are shown in Section 3.6.2 in [2]. The next proposition, which follows by Proposition 3.3, Theorem 3.5, and equation (87) in [2], states that by using the path control ψ * x and by choosing the time control to be the first time that the actual dynamics of the game hit zero the maximizer can guarantee that the cost will be at least V (x). The proposition is valid since the function h is convex. Proposition 3.4 Fix x ∈ [0, β 0 ). For every ̺ ∈ P one has c(x, τ x , ψ * x , (0, ̺)) ≥ V (x),(33) where τ x := τ [x, ψ * x , (0, ̺)] is the first time that the dynamics ϕ[x, ψ * x , (0, ̺)] hits zero. Moreover, τ x ≤ τ * x = inf{t ≥ 0 : y(t) + ψ * ,1 x (t) − ψ * ,2 x (t) = 0}.(34) Lower bound Recall that from Proposition 3.1.(i), for sufficiently large T , V (T, x) = V (x). Hence, now onwards we compare the value function of the QCP to V (x). Theorem 4.1 Let Assumption 2.1 hold. Then for T sufficiently large, lim inf n→∞ V n (T,X n (0)) ≥ V (x). We present two lemmas that together yield Theorem 4.1. The first provides a lower bound on the RS cost for an arbitrary sequence of policies in terms of an expression involving only the one-dimensional processes. The latter is further bounded by the DG value function, in the second lemma. Lemma 4.1 Fix a sequence of admissible controls {U n ∈ U n }, n ∈ N. Then for every T > 0, δ ∈ (0, T ) and ε > 0 one has lim inf n→∞ J n (T,X n (0), U n ) ≥ lim inf n→∞ 1 b 2 n log E e b 2 n T −δ 0 h(X n (u))du+r(1−ε)Ř n (T −δ) − ε. Lemma 4.2 Fix {U n } as in Lemma 4.1. Then there existsT > 0 such that for every 0 < ε < 1/2 one has lim inf n→∞ 1 b 2 n log E e b 2 n T 0 h(X n (u))du+r(1−ε)Ř n (T ) ≥ V h,r(1−ε) (x).(35) Proof of Theorem 4.1: Combining Lemma 4.1 and Lemma 4.2, for any T and δ > 0 such that T − δ >T , and any ε ∈ (0, 1/2), lim inf n→∞ J n (T,X n (0), U n ) ≥ lim inf n→∞ J n (T ,X n (0), U n ) ≥ V h,r(1−ε) (x) − ε. From (27)-(28) it follows that lim ε→0 V h,r(1−ε) (x) = V h,r (x) = V (x), and the result follows. ✷ Proof of Lemma 4.1: Fix {U n }, T > 0, δ > 0 and ε > 0. Then T 0 e b 2 n ( t 0h ·X n (u)du+r·R n (t)) dt ≥ e b 2 n T −δ 0h ·X n (u)du+r·R n (T −δ) δ,(36) where we used monotonicity of the integrand with respect to t. Next, by the definition of h and r,h ·X n ≥ h(θ ·X n ) andr ·R n ≥ rθ ·R n .(37) Since θ n → θ,X n takes values in a fixed, compact set, and h is uniformly continuous on this set, it follows that for sufficiently large n, T 0 h(θ ·X n (u))du ≥ T 0 h(X n (u))du − ε and θ i ≥ θ n i (1 − ε), i ∈ I.(38) Combining (36), (37) and (38) yields the result. ✷ In the rest of this section we prove Lemma 4.2. Proof of Lemma 4.2: Rather than working with general ε ∈ (0, 1/2), we consider a general r > 0. For every r > 0 we findT =T (r) (in (40)) that satisfies (35) with ε = 0. As we will see,T is continuous w.r.t. r and finite for r > 0. Hence, we obtain that (35) is valid with a fixedT and all ε ∈ (0, 1/2). We thus turn to proving lim inf 1 b 2 n log E e b 2 n T 0 h(X n (u))du+rŘ n (T ) ≥ V (x).(39) Sketch of the proof: The lemma relates the stochastic control problem to the DG. The strategies in the game are analogues of the policies in the stochastic problem, whereas the controls selected by the maximizing player play a similar role to the variational problem in Varadhan's lemma (Theorem 4.3.1 of [9]). Following the spirit of the proof of Varadhan's lemma, one focuses on the event that the paths (à n ,S n ), projected in the θ n direction, are in a neighborhood of a specific P 2 -path control. The latter is referred to as the reference path. One then shows that the processX n is in a neighborhood of the game dynamics obtained when the reference path is selected by the maximizing player. We now describe the reference path. First, recall the balance equation (15). Consider paths (à n ,S n ) such that (Ǎ n ,Š n • T n ) are close to some path ψ = (ψ 1 , ψ 2 ) ∈ P 2 . ThenX n is close to y + ψ 1 − ψ 2 +Ž n −Ř n . By the nonnegativity ofŽ n , we have thatX n is bounded from below by ϕ n = y + ψ 1 − ψ 2 −Ř n , up to a small error term. The proof proceeds by comparing the process ϕ n to the game dynamics with initial state x, maximizer's path control ψ, and minimizer's control given by (0,Ř n ). This process cannot be regarded game dynamics because of the stochasticity of the minimizer's control term, and the fact that it is not attained by a strategy in the sense of the game. Moreover, it is not guaranteed that ϕ n takes values in [0, D]. However, these obstacles can be treated. Let us first describe the case x ≤ β 0 , where the treatment is least complicated. In this case we consider the event, denoted by O n , that the paths (à n ,S n ) are close to the path ψ * x identified in Proposition 3.4 (in this rough sketch we do not quantify the term "close"). We then obtain lim inf 1 b 2 n log E e b 2 n T 0 h(X n (u))du+rŘ n (T ) ≥ lim inf 1 b 2 n log E e b 2 n T 0 h(X n (u))du+rŘ n (T ) 1 O n ≥ c(x, τ x , ψ * x , (0, ̺)) ≥ V (x), where the second inequality follows using the closeness of the data to ψ * x and the structure of the function c, while the last inequality follows from Proposition 3.4. Thus, Proposition 3.4 allows us to focus on data that is close to one fixed path, ψ * x , for every n ∈ N. The situation is more subtle in the case where x > β 0 , as we do not have an analogue of Proposition 3.4. That is, we are unable to identify a single path that guarantees V (x) as a lower bound on the cost under all strategies. Proposition 3.3 proposes how the maximizer in the game should act until the threshold β 0 is reached, namely to use the control ψ ♯ (note that the time when the threshold is reached depends on the minimizier's strategy). Therefore we focus on data (à n ,S n ) close to ψ ♯ until the workload in the stochastic model hits β 0 . The proposition then guarantees that up to this time the cost incurred is bounded below by r(x − β 0 ). Once β 0 is reached, ψ β 0 is used as explained above, and (27) is used to obtain V (x) as a lower bound. As we already mentioned, in the game, the time when one switches from ψ ♯ to ψ * β 0 depends on the strategy. As far as the QCP is concerned, this means that one has no control over the switching time, which may be random and vary with n. The argument therefore uses time discretization (see Lemma 4.4), with which each one of a finite collection of switching times is estimated separately. Proof in details: We first prove the lower bound for the case x > β 0 . The other case, which is simpler, is addressed at the end of the proof. Fix x > β 0 and ε 1 > 0. Let ∆ > 0 be small enough so that x > β 0 + ∆ and r∆ + V (β 0 ) − V (β 0 − ∆) ≤ ε 1 . Such a ∆ exists since V is continuous, see (27). Let T 1 = V (x) + 3 + r(2 + D − x) h(β 0 + ∆/2) − h(β 0 ) , T 2 = β 0 −∆ 0 1/ y 2 − h(ξ)/s dξ + 1, T = T 1 + T 2 .(40) Above, T 1 is obtained by considerations along these lines. Start with (76), and consider only those policies for which V (x) + 1 > b −2 n log E exp{b 2 n ( T 1 0 h(X n (u))du + rŘ n (T 1 ))}. Obtain a further lower bound on V (x) + 1 by putting on the RHS the indicator of a certain event that assures T 1 0 h(X n (u))du ≥ h(β 0 + ∆/2)T 1 . Moreover, we show thatŘ n (T 1 ) ≥ x − D − 2 + (y + r 2s )T 1 . When combined together with the definition of β 0 in (28), we get the above formula for T 1 . The formula for T 2 is based on the termination time of the game (32). The proof proceeds in several steps. In Step 1 we analyze the process T n . In Step 2, we consider the process ϕ n 1 := y + ψ ♯,1 − ψ ♯,2 −Ř n(41) and the time τ n 1 when this process first hits β 0 + ∆. We show that for any policy that is nearly optimal τ n 1 < T 1 with probability that is significant in the MD scale. Next, in Step 3, we modify the construction of ϕ n 1 on the time interval [τ n 1 , ∞), and in Step 4 show that the process thus constructed hits zero before timeT , with probability 1. In Step 5, we combine these results to obtain a lower bound on the cost. Finally, in Step 6 we we take limits and obtain the result. Step 1 (Limit property of T n ): W.l.o.g. assume that the sequence of policies {U n } satisfies V (x) + 1 > 1 b 2 n log E e b 2 n T 0h ·X n (u)du+r·R n (T ) .(42) Denote ρ(t) = ρt, t ∈ R + .b 2 n log P T n i − ρ i T ≥ b n √ n K ≤ −m.(43) Proof: Fix i ∈ I. By (12), the l.h.s. of (43) equals lim sup 1 b 2 n log P n µ n i Z n i T ≥ K . Using (11) and the fact that µ n /n → µ, it suffices to prove that for every m > 0 there exists K such that lim sup 1 b 2 n log P ( L n T ≥ K) ≤ −m,(44) for L n =X n i ,à n i ,S n i • T n i andR n i . As far asX n i is concerned, the above is immediate because the process is bounded. For L n =à n i , this property follows from Proposition 2.1 and from the fact that for sufficiently large K one has inf J 1 (T , ψ) : ψ ∈ P I and ψ · e i T ≥ K = J 1 (T , (Kt/T )e i ) = 1 2µσ 2 IA · K 2 T .(45) Since the time change T n i (t) ≤ t for all t, a similar conclusion holds forS n i • T n i . Finally, for R n i , note that by (42), V (x) + 1 ≥ 1 b 2 n log E e b 2 n r iR n i (T ) . Hence by the Chebyshev's inequality, P(R n i (T ) ≥ K) ≤ e −b 2 n (K+V (x)+1) . The result follows since, by the monotonicity,R n i (T ) = R n i T . ✷ Step 2 (Estimate on the time τ n 1 ): We introduce some notation that will be needed in the remainder of the proof. Since h is uniformly continuous on [0, D], one can find δ 1 > 0 such that osc D (2δ 1 , h) < ε 1 .(46) One may take δ 1 so that δ 1 < min{∆/4, 1}.(47) Fix m > I(T , ψ * ) + I(T , ψ ♯ ) + 1 + 6ε 1 . Define the event E n = E n (K) = { T n i − ρ i T < b n √ n K, for all i ∈ I}. Using Lemma 4.3, fix K > 0 such that for all sufficiently large n 1 b 2 n log P ((E n ) c ) ≤ −m.(48) From Lemma A.2 it follows that there existsψ ♯ = (ψ ♯,1 ,ψ ♯,2 ) ∈ P 2I such that (θ ·ψ ♯,1 , θ ·ψ ♯,2 • ρ) = (ψ ♯,1 , ψ ♯,2 )(49) and J k (T,ψ ♯,k ) = I k (T, ψ ♯,k ), k = 1, 2.(50) Note that for all large n, osc T ( b n √ n K, ψ ♯,2 i ) < δ 2 , i ∈ I.(51) For ψ = (ψ 1 , ψ 2 ) ∈ P 2I , and 0 < δ, t < ∞, let A δ,t (ψ) = {ψ ∈ D([0,T ], R 2I ) : ψ − ψ t < δ},(52) and Ω n δ,t (ψ) = {(à n ,S n ) ∈ A δ,t (ψ)}.(53) Recall from (41) the definition of ϕ n 1 , and let τ n 1 = inf{t ≥ 0 : ϕ n 1 (t) ≤ β 0 + ∆}. Divide the time interval [0,T ] into (T /ν) ∈ N intervals of size ν where 2 ν ≤ min 4sε 1 r 2 , ε 1 sy 2 , ∆ |r/(2s) + y| .(54) Denote the intervals by N j = N j (ν) = [νj, ν(j + 1)). For every n, we define an index 0 ≤ j n 1 ≤ ⌊T 1 /ν⌋ in such a way that we can estimate, from below, the probability that the time τ n 1 belongs to the interval N j n 1 . Let j n 1 = arg max j∈{0,...,⌊T 1 /ν⌋} P τ n 1 ∈ N j | Ω n δ 3 ,T 1 (ψ ♯ ) , where δ 2 = δ 1 /(8θ max √ I). The proof of this lemma is differed to the end of the section. Step 3 (Constructing a path beyond time τ n 1 ): In Lemma 4.4 we focused on data for which the process (Ǎ n ,Š n • T n ) is near ψ ♯ . Now we will consider data for which this process is near ψ ♯ up to time (j n 1 + 1)ν, and from that time on, near ψ * β 0 −∆ . Thus we focus on the reference path, ψ 0 (t) = ψ n,0 (t) := ψ ♯ (t) 0 ≤ t ≤ (j n 1 + 1)ν, ψ ♯ ((j n 1 + 1)ν) + ψ * β 0 −∆ (t − (j n 1 + 1)ν) t > (j n 1 + 1)ν. Recall that τ n 1 is defined as the first time when ϕ n 1 ≤ β 0 + ∆. Since y + ψ ♯,1 − ψ ♯,2 is continuous, and the jumps ofŘ n are of size (b n √ n) −1 , one has for sufficiently large n, β 0 < ϕ n 1 (τ n 1 ) ≤ β 0 +∆. Moreover, since on the interval (τ n 1 , (j n 1 + 1)ν) one has y +ψ ♯,1 −ψ ♯,2 = y + r/(2s) and we assumed that ν ≤ ∆/|r/(2s) + y|, it follows that ϕ n 1 ((j n 1 + 1)ν) = ϕ n 1 (τ n 1 ) + (y + r/(2s))((j n 1 + 1)ν − τ n 1 ) − (Ř n ((j n 1 + 1)ν) −Ř n (τ n 1 )) > β 0 − ∆ − (Ř n ((j n 1 + 1)ν) −Ř n (τ n 1 )). We now define a new process that starts at time (j n 1 + 1)ν, having the form of the game dynamics with the initial state β 0 − ∆, the path ψ * := ψ * β 0 −∆ , and rejection process ̺ n . For t ≥ (j n 1 + 1)ν set ϕ n 2 (t) = β 0 − ∆ + y(t − (j n 1 + 1)ν) + ψ * ,1 (t − (j n 1 + 1)ν) − ψ * ,2 (t − (j n 1 + 1)ν) − ̺ n (t − (j n 1 + 1)ν), where ̺ n (s) =Ř n (s) −Ř n (τ n 1 ), s ≥ 0. Notice that at time t = (j n 1 + 1)ν there is an initial amount of rejectionsŘ n ((j n 1 + 1)ν) −Ř n (τ n 1 )) ≥ 0, and therefore, ϕ n 1 ((j n 1 + 1)ν) > β 0 − ∆ − (Ř n ((j n 1 + 1)ν) −Ř n (τ n 1 )) = ϕ n 2 ((j n 1 + 1)ν). Therefore ϕ n 3 := y + ψ 0,1 − ψ 0,2 −Ř n ≥ ϕ n 2 on the interval [(j n 1 + 1)ν,T ].(56) Let τ n 2 := inf{t ≥ (j n 1 + 1)ν : ϕ n 2 (t) ≤ 0}. From (34) it follows that it takes β 0 −∆ 0 1/ y 2 − h(ξ)/s dξ time units for the path β 0 − ∆ + y · +ψ * ,1 (·) − ψ * ,2 (·) to reach the level zero. Therefore, τ n 2 < T 2 with probability 1. Define ϕ n (t) = ϕ n 1 (t) 0 ≤ t ≤ (j n 1 + 1)ν, ϕ n 2 (t) (j n 1 + 1)ν < t ≤ (j n 2 + 1)ν. Step 4 (Estimate on the time τ n 2 ): Consider the P 2I -pathψ 0 , which is defined in a similar way toψ ♯ . From Lemma A.2 it follows that there existsψ 0 = (ψ 0,1 ,ψ 0,2 ) ∈ P 2I such that (θ ·ψ 0,1 , θ · (ψ 0,2 • ρ)) = (ψ 0,1 , ψ 0,2 ) and J k (T,ψ 0,k ) = I k (T, ψ 0,k ), k = 1, 2. Let H n k = {τ n k ∈ N j n k }, k = 1, 2, and similarly to Step 2, set j n 2 = arg max j∈{j n 1 +1,...,j n 1 +⌊T 2 /ν⌋+2} P τ n 2 ∈ N j | Ω n δ 3 ,(j n 1 +1)ν+T 2 (ψ 0 ) ∩ H n 1 . Lemma 4.5 One has lim inf n→∞ 1 b 2 n log P H n 2 | Ω n δ 3 ,(j n 2 +1)ν (ψ 0 ) ∩ H n 1 ≥ −2ε 1 . Proof: Recall that τ n 2 ∈ [(j n 1 + 1)ν, (j n 1 + 1)ν + T 2 ) with probability 1. Therefore P H n 2 | Ω n δ 3 ,(j n 1 +1)ν+T 2 (ψ 0 ) ∩ H n 1 ≥ 1 (⌊T 2 /ν⌋ + 1) . The rest of the proof is similar to the proof of Lemma 4.4 and is therefore omitted. ✷ Step 5 (Bounding the cost from below): Let us denote Ω n 2 = Ω n δ 3 ,(j n 2 +1)ν (ψ 0 ). Consider the event Ω n 3 := H n 1 ∩ H n 2 ∩ Ω n 2 ∩ E n . On this event, we bound from below the sum τ n 2 0 h(X n (s))ds + rŘ n (τ n 2 ). By (75),X n ≥ ϕ n 1 − 2δ 1 on [0, (j n 1 + 1)ν], andX n ≥ ϕ n 3 − 2δ 1 on [(j n 1 + 1)ν, τ n 2 ] (thanks to the fact that in Lemma 4.4, ψ is arbitrary on the latter time interval). Therefore, by (77) and the definition of ϕ n , on the time interval [0, τ n 2 ), X n ≥ ϕ n − 2δ 1 . Since we chose δ 1 such that osc D (2δ 1 , h) < ε 1 it follows that τ n 2 0 h(X n (t))dt + rŘ n (τ n 2 ) (60) ≥ τ n 2 0 h(ϕ n (t))dt + rŘ n (τ n 2 ) − ε 1 τ n 2 ≥ τ n 1 0 h(ϕ n 1 (t))dt + rŘ n (τ n 1 ) + τ n 2 (j n 1 +1)ν h(ϕ n 2 (t))dt + r(Ř n (τ n 2 ) −Ř n (τ n 1 )) − ε 1 τ n 2 , where the last inequality follows since h is nonnegative and τ n 1 < (j n 1 +1)ν. We now bound from below the three terms above. From inequality (29), the inequality τ n 1 ≥ j n 1 ν, the definitions of ψ 0 and ψ ♯ , and (54) it follows that τ n 1 0 h(ϕ n 1 (t))dt + rŘ n (τ n 1 ) ≥ r(x − (β 0 + ∆)) + I(τ n 1 , ψ ♯ ) ≥ r(x − (β 0 + ∆)) + I(j n 1 ν, ψ ♯ ) (61) = r(x − (β 0 + ∆)) + I((j n 1 + 1)ν, ψ 0 ) − (j n 1 +1)ν j n 1 ν [s 1 (ψ ♯,1 ) 2 (t) + s 2 (ψ ♯,2 ) 2 (t)]dt = r(x − (β 0 + ∆)) + I((j n 1 + 1)ν, ψ 0 ) − r 2 ν/(4s) ≥ r(x − (β 0 + ∆)) + I((j n 1 + 1)ν, ψ 0 ) − ε 1 . To bound the second term notice that ϕ n 2 (s) := ϕ n 2 (s + (j n 1 + 1)ν) = β 0 − ∆ + ys + ψ * ,1 (s) − ψ * ,2 (s) − ̺ n (s), s ≥ 0. Denoteτ n 2 = τ n 2 − (j n 1 + 1)ν. This is the first time whenφ n 2 hits zero. Therefore, from the definitions of I,τ n 2 , and ψ 0 , the inequality τ n 2 ≥ j n 2 ν, and from inequality (33) it follows that τ n 2 (j n 1 +1)ν h(ϕ n 2 (t))dt + r(Ř n (τ n 2 ) −Ř n (τ n 1 )) = τ n 2 0 h(φ n 2 (u))du + r̺ n (τ n 2 ) = c(β 0 − ∆,τ n 2 , ψ * , ̺ n ) + I(τ n 2 , ψ * ) = c(β 0 − ∆,τ n 2 , ψ * , ̺ n ) + I(τ n 2 , ψ 0 ) − I((j n 1 + 1)ν, ψ 0 ) ≥ c(β 0 − ∆,τ n 2 , ψ * , ̺ n ) + I(j n 2 ν, ψ 0 ) − I((j n 1 + 1)ν, ψ 0 ) ≥ V (β 0 − ∆) + I(j n 2 ν, ψ 0 ) − I((j n 1 + 1)ν, ψ 0 ). Inequality (57) h(X n (t))dt + rŘ n (τ n 2 ) (63) ≥ r(x − (β 0 + ∆)) + V (β 0 − ∆) − ε 1 (T + 1) + I(j n 2 ν, ψ 0 ). Step 6 (Bounding the limit from below): We are now ready to prove (39). First, notice that there are only finitely many possible pairs {(j 1 , j 2 ) ∈ N 2 : 0 ≤ j 1 ≤ j 2 ≤T /ν − 1}. For each such pair define N (j 1 ,j 2 ) = {n ∈ N : (j n 1 , j n 2 ) = (j 1 , j 2 )}. If we show that for each pair (j 1 , j 2 ) lim inf N (j 1 ,j 2 ) 1 b 2 n log E e b 2 n T 0 h(X n (u))du+rŘ n (T ) ≥ V (x) + C 0 ε 1 ,(64) where C 0 is a constant independent of n and ε 1 , then (39) will follow on applying Lemma 4.1 and taking ε 1 → 0. Thus in the rest of the proof we focus on a fixed (j 1 , j 2 ), and prove (64). Hereafter, lim inf denotes the limit inferior along the subset. DenoteΩ n 2 = Ω n δ 3 ,(j 2 +1)ν (ψ 0 ). From (63) and since τ n 2 ≤T it follows that lim inf 1 b 2 n log E e b 2 n T 0 h(X n (u))du+rŘ n (T ) 1 {H n 1 ∩H n 2 ∩Ω n 2 ∩E n } (65) ≥ r(x − (β 0 + ∆)) + V (β 0 − ∆) − ε 1 (T + 1) + I(j 2 ν, ψ 0 ) + lim inf 1 b 2 n log P H n 1 ∩ H n 2 ∩Ω n 2 ∩ E n . We now estimate the last term above. By Lemmas 4.4 and 4.5, for all n sufficiently large, P H n 1 |Ω n 2 P H n 2 | H n 1 ∩Ω n 2 ≥ e −5ε 1 b 2 n . Hence lim inf 1 b 2 n log P H n 1 ∩ H n 2 ∩Ω n 2 ∩ E n (66) ≥ lim inf 1 b 2 n log P H n 1 ∩ H n 2 ∩Ω n 2 − P ((E n ) c ) = lim inf 1 b 2 n log P H n 2 | H n 1 ∩Ω n 2 × P H n 1 |Ω n 2 P Ω n 2 − P ((E n ) c ) ≥ lim inf 1 b 2 n log e −5ε 1 b 2 n P Ω n 2 − P ((E n ) c ) ≥ lim inf 1 b 2 n log e −b 2 n [J((j 2 +1)ν,ψ 0 )+6ε 1 ] − e b 2 n (−m+1) = −I((j 2 + 1)ν), ψ 0 ) − 6ε 1 . Above, the third inequality follows by Proposition 2.1 and (48). The last equality uses (59) and m − 1 > I((j 2 + 1)ν), ψ 0 ) + 6ε 1 . Substituting (66) in (65) yields lim inf 1 b 2 n log E e b 2 n T 0 h(X n (u))du+rŘ n (T ) 1 {H n 2 ∩H n 1 ∩à δ 3 ,(j 2 +1)ν (ψ 0 )∩E n }(67) ≥ r(x − (β 0 + ∆)) + V (β 0 − ∆) − ε 1 (T + 7) + I(j 2 ν, ψ 0 ) − I((j 2 + 1)ν), ψ 0 ). Using (30) and then (31) and Proposition 3.1(iii) gives I(j 2 ν, ψ 0 ) − I((j 2 + 1)ν), ψ 0 ) = − (j 2 +1)ν j 2 ν s 1 (ψ * ,1 β 0 −∆ ) 2 (t) + s 2 (ψ * ,2 β 0 −∆ ) 2 (t) dt (68) = − (j 2 +1)ν j 2 ν s(ω * β 0 −∆ ) 2 (t) dt = − (j 2 +1)ν j 2 ν s −y − y 2 − h(x + yt + ω * β 0 −∆ (t))/s 2 (t) dt. ≥ −νsy 2 ≥ −ε 1 , where the last two inequalities follow from the negativity of y and (54). From (67) and (68) and by recalling that for x > β 0 one has V (x) = r(x − β 0 ) + V (β 0 ), it follows that lim inf 1 b 2 n log E e b 2 n T 0 h(X n (u))du+rŘ n (T ) 1 {H n 1 ∩H n 2 ∩Ω n δ 3 ,(j n 2 +1)ν (ψ 0 )∩E n } ≥ V (x) − ε 1 (T + 8). This proves (64). Hence the result is proved for the case x > β 0 . Finally, consider x ≤ β 0 . The considerations here are simpler than in the previous case. In case that the initial state is exactly β 0 , the decision maker can reject a (small) amount of ∆ at time t = 0. Then the proof that V (x) is a lower bound requires the focusing only on data near ψ * β 0 −∆ , starting at time zero. In case that the initial state is lower than β 0 , one uses the same arguments, with data near ψ * x . ✷ Proof of Lemma 4.4: The proof has two parts. On the first we show that for sufficiently large n one has P τ n 1 ∈ [0, T 1 ) | Ω n δ 3 ,T 1 (ψ) ≥ 1/2.(69) Since the interval [0, T 1 ] is divided into at most ⌊T 1 /ν⌋ + 1 subintervals, there exists an interval N j such that the conditional probability of τ n 1 ∈ N j is at least 1 2(⌊T 1 /ν⌋+1) . Thus, as a result of (69), P τ n 1 ∈ N j n 1 | Ω n δ 3 ,T 1 (ψ) ≥ 1 2 (⌊T 1 /ν⌋ + 1) .(70) On the second part we use this to deduce (55). Part a: Set E n 1 = {τ n 1 ≥ T 1 }. Write Ω n δ 3 ,T 1 (ψ) as Ω n . We analyze the event Ω n 1 := Ω n ∩ E n ∩ E n 1 . On this event, we bound from below the processX n on the time interval [0, T 1 ] and the total number of rejections until time T 1 . By the triangle inequality it follows that for every i ∈ I and every n S n i • T n i − ψ 2 i • ρ i T ≤ S n i • T n i − ψ 2 i • T n i T + ψ 2 i • T n − ψ 2 i • ρ i T ≤ 2δ 2 ,(71) where we have bounded each of the terms on the r.h.s. by δ 2 ; the bound of the first term follows by (52), and the bound of the second follows by the definition of E n and from (51). Similarly, à n i − ψ 1 i T ≤ δ 2 , i ∈ I.(72) Since θ n → θ it follows from (49), (71), and (72) that for sufficiently large n, Š n • T n − ψ 2 • ρ T = θ n ·S n • T n − θ · ψ 2 • ρ T < δ 1 2 ,(73)Ǎ − ψ 1 T = θ n ·Ã n − θ · ψ 1 T < δ 1 4 .(74) Moreover, for sufficiently large n one has, for t ∈ [0,T ], |X n (0) +y n t − y(t)| ≤ δ 1 /4. Using the above inequalities it follows that for every u ∈ [0, T 1 ] one haš X n (u) =X n (0) +y n t +Ǎ n (u) −Š n • T n (u) +Ž n (u) −Ř n (u) ≥ ϕ n 1 (u) − δ 1 − sup t∈[0,T ] X n (0) +ỹ n t − y(t) − Ǎ n − ψ 1 T − Š n • T n − ψ 2 T ≥ ϕ n 1 (u) − 2δ 1 .(75) By the definition of E n 1 and the choice of δ 1 (see (47)), X n (u) ≥ β 0 + ∆ − 2δ 1 ≥ β 0 + ∆/2.(76) By using similar arguments and the inequalityX n ≤ D n one obtainš R n (T 1 ) ≥ y(T 1 ) − D n − δ 1 + ψ 1 (T 1 ) − ψ 2 (T 1 ) = x − D n − δ 1 + (y + r/(2s))T 1 (77) ≥ x − D n − 1 + (y + r/(2s))T 1 ≥ x − D − 2 + (y + r/(2s))T 1 , where the equality follows by the definition of ψ ♯ , the second inequality follows by the choice of δ 1 , and the last inequality follows since lim n→∞ D n = D. From (37), (42), and since T 1 <T , we obtain 3 for sufficiently large n V (x) + 1 > 1 b 2 n log E e b 2 n T 1 0 h(X n (u))du+rŘ n (T 1 ) . Along with (76) and (77) it follows that V (x) + 1 > 1 b 2 n log E e b 2 n T 1 0 h(X n (u))du+rŘ n (T 1 ) 1 Ω n 1 ≥ 1 b 2 n log E e b 2 n ((h(β0+∆/2)+yr+r 2 /(2s))T 1 +r(x−D−2)) 1 Ω n 1 = r(x − D − 2) + (h(β 0 + ∆/2) − h(β 0 ) + r 2 /(4s))T 1 + 1 b 2 n log P(Ω n 1 ). The above equality follows since r 2 /(4s) + ry + h(β 0 ) = 0, which in turn follows since x > β 0 and therefore β 0 < D . Since P(Ω n 1 ) = P(E n 1 ∩ Ω n ∩ E n ) ≥ P(E n 1 |Ω n )P(Ω n ) − P((E n ) c ), and using (48), it follows that P(E n 1 |Ω n ) ≤ e b 2 n (V (x)+1+r(2+D−x)−T 1 (h(β 0 +∆/2)−h(β 0 )−r 2 /(4s))) P(Ω n ) −1 + e −mb 2 n P(Ω n ) −1 . We show that for sufficiently large n, each of the terms on the r.h.s. can be bounded by 1/4. From Proposition 2.1 and (50), it follows that, for all large n, 1 b 2 n log P(Ω n ) ≥ − inf ψ∈A δ 3 ,T 1 (ψ) J(T 1 ,ψ) − 1 ≥ −J(T 1 , ψ) − 1 = −I(T 1 , ψ ♯ ) − 1 = −r 2 T 1 /(4s) − 1. Hence by the definition of T 1 , the first term is bounded by 1/4. Since m > I(T , ψ ♯ ) andT > T 1 , so is the second term. As a result, (69) holds. Part b: By the definition of an admissible control, R n is adapted to the filtration F t := σ{A n i (u), S n i (T n i (u)), i ∈ I, u ≤ t}, hence so isŘ n , and, by (41), so is ϕ n 1 . Since T n i (t) ≤ t, t ≥ 0, and T n i are themselves adapted to F t , it follows that the event {τ n 1 ∈ N j n 1 } is measurable on F n ν(j n 1 +1) , where F n t = σ{A n i (u), S n i (u), i ∈ I, u ≤ t}. Note that F n t = σ{à n i (u),S n i (u), i ∈ I, u ≤ t}. Fix v > 0 and a sequence v n , with v n < v (both deterministic). We will show the following. Given a constant c 1 > 0 and a sequence of events Q n ∈ F n vn , for every ε > 0 there exists δ > 0 and n 1 ∈ N, such that p n 1 := P(Q n |Ω n δ,v (ψ)) ≥ c 1 , n ≥ 1, implies p n 2 := P(Q n |Ω n δ,vn (ψ)) ≥ e −εb 2 n , n ≥ n 1 . Note that this will prove (55), based on (70) that has now been established by part (a). Extending the definition of Ω n δ,t (ψ) (53), we let Ω n δ,a,b (ψ) = { sup s∈[a,b] (à n ,S n )(u) − ψ(u) < δ}, for 0 ≤ a ≤ b. Also, we drop ψ from the notation Ω n δ,a and Ω n δ,a,b . Note that there is no loss of generality in proving the statement forà n (a collection of I independent renewal processes) in place of (à n ,S n ) (a collection of 2I such processes). Thus we will consider only the former. To prove the aforementioned statement, let ε > 0 be given. Consider the quantities p n 1 and p n 2 , depending on δ. Assume that (78) is valid. Then we can write p n 1 = P{Q n ∩ Ω n δ,v } P{Ω n δ,v } = P{Q n ∩ Ω n δ,vn ∩ Ω n δ,vn,v } P{Ω n δ,vn ∩ Ω n δ,vn,v } . A basic independence property for a renewal process, to be used, is the following. Let A be a renewal process of the form A(t) = sup l ≥ 0 : l k=1 U (k) ≤ t , t ≥ 0, where {U (k)} are iid (compare with (3)). Fix t and let π denote the time of the first jump at or after t, namely π = inf{u ≥ t : A(u) > A(t−)} (convention: A(0−) = 0). Consider an event Q measurable on σ{A(u) : 0 ≤ u ≤ t}. Then, for each k ∈ Z + , the event Q ∩ {A t = k} is statistically independent of the sequence {U k+1 , U k+2 , . . .}. Based on this it is not hard to see that, if we let SA denote the shifted version SA(u) = A(π+u)−A(π), u ≥ 0, of A, we have that Q is independent of SA. For a collection of independent renewal processes, a similar statement holds if each of them is shifted according to its own first jump after t. To state this property for the processes (A n i ), if Q is measurable on F n t , then it is independent of (SA n i ). Now, let us apply this to study (à n ). Let π n = (π n i ) be defined by π n i = inf{u ≥ v n :à n i (u) >à n i (v n −)}. If π n := max i |π n i − v n |, then, given any k > 0, P{ π n > δ} ≤ e −kb 2 n for all sufficiently large n, as can be verified using the exponential moment assumption and applying Chebychev's inequality. Hence, given any k, for all large n, p n 1,1 := P{Q n ∩ Ω n δ,vn ∩ Ω n δ,vn,v } ≤ P{Q n ∩ Ω n δ,vn ∩ Ω n δ,vn,v ∩ { π n ≤ δ}} + e −kb 2 n . Let us denote by ω a modulus (by which we mean a function mapping R + to itself with ω(0+) = 0), that dominates the modulus of continuity ofψ m for all m ≤ M . Then, using the definition of Ω n δ,vn,v , adding and subtracting the shifted version Sà n and using the triangle inequality gives p n 1,1 ≤ P{Q n ∩ Ω n δ,vn ∩ { Sà n − (ψ(v n + ·) − ψ(v n )) v−vn −δ < δ ′ }} + e −kb 2 n , where δ ′ = 2δ + 2ω(δ). Using the independence alluded to above, p n 1,1 ≤ P{Q n ∩ Ω n δ,vn }P{ Sà n − (ψ(v n + ·) − ψ(v n )) v−vn −δ < δ ′ } + e −kb 2 n Shifting back gives p n 1,1 ≤ P{Q n ∩ Ω n δ,vn }P{Ω n δ ′′ ,vn,v−δ } + e −kb 2 n for δ ′′ that can be made arbitrarily small by taking δ to be small. A similar argument shows that, for all large n, p n 1,2 := P{Ω n δ,vn ∩ Ω n δ,vn,v } ≥ P{Ω n δ ′′′ ,vn }P{Ω n δ ′′′ ,vn,v } − e −kb 2 n , for suitably chosen δ ′′′ > 0, that again, can be made arbitrarily small by taking small δ. Thus for arbitrary ε ′ > 0, provided that δ is sufficiently small. Hence by selecting k sufficiently large, p n 1 ≤ p n 2 e b 2 n (−I[0,v−δ]+2ε ′ ) + e −kb 2 n 1 2 e b 2 n (−I[0,v]−2ε ′ ) = 2p n 2 e b 2 n (4ε ′ +I[v−δ,v]) + 2e −kb 2 n e b 2 n (I[0,v]+2ε ′ ) . Hence, again by selecting k large, for all large n, p n 2 ≥ 1 2 c 1 e −b 2 n (4ε ′ +I[v−δ,v]) − e b 2 n (−k+I[0,v−δ]−2ε ′ ) ≥ 1 4 c 1 e −b 2 n (4ε ′ +I[v−δ,v]) . Selecting δ > 0 such that 4ε ′ + I[v − δ, v] < ε gives (79) for some n 1 . This completes the proof of part (b), and the lemma. ✷ A nearly optimal policy In this section we show that the policy from [6] is AO for the present setting. While the policy is similar, the proof of AO is quite different, as the paper [6] addresses the diffusion scale, rather than the MD scale. Let the classes be labeled so that h 1 µ 1 ≥ h 2 µ 2 ≥ · · · ≥ h I µ I . Let γ : [0, D] → X be a Borel measurable mapping satisfying γ(w) ∈ arg min ξ {h · ξ : ξ ∈ X , θ · ξ = w}, w ∈ [0, D].(80) We note on passing that, as shown in [2, Theorem A.1], one can equivalently work with onedimensional dynamics, thanks to the fact that the minimum over queue length ξ in the above expression is a function of only of the (one-dimensional) workload w. Since the mappingξ →h ·ξ is linear and the domain X is polyhedral, it can be assumed, without loss of generality, that γ is continuous and takes values on the boundary of X . We have, by definition, that θ · γ(w) = w, andh · γ(w) = h(w) ≤h · ξ for ξ ∈ X for which θ · ξ = w. A particular selection of γ is as follows. Given w ∈ [0, D], set (j, ξ) = (j, ξ)(w) by w ∈ [D j ,D j−1 ) and ξ = ξ(w) := (w −D j )/θ j , wherê D j := I i=j+1 θ i D i , j ∈ {0, . . . , I} and one has 0 =D I <D I−1 < · · · <D 1 <D 0 = θ ·D = D,D := (D 1 , . . . , D I ). Then γ(w) = I i=j+1 D i e i + ξe j .(81) Approximate γ by a curve that is bounded away from the part of the boundary of X that corresponds to the buffer limit, namely ∂X = {x ∈ X : x i = D i for some i}. Fix 0 < ε 0 < min i D i /4. Let a i = D i − 3ε 0 , i ∈ I, and a * = β 0 ∧ (θ · a) < D. Note that if ε 0 is small then a * = β 0 (unless β 0 = D). Define γ a [0, D] → X first on [0, θ · a] as the function obtained upon replacing the parameters (D i ) by (a i ) in (81). That is, for w ∈ [0, θ · a), the variables j = j(w) and ξ = ξ(w) are determined via w = I i=j+1 θ i a i + θ j ξ, j ∈ {1, 2, . . . , I}, ξ ∈ [0, a j ), γ a (w) = I i=j+1 a i e i + ξe j .(82) Given w ∈ [0, θ · a), we will sometimes refer to the unique pair (j, ξ) alluded to above as the representation (j, ξ) of w via (82). Next, on [θ · a, θ ·D] define γ a as the linear interpolation between the points (θ · a, a) and (θ ·D,D). Also setâ j = I i=j+1 θ i a i , j ∈ {0, 1, . . . , I}. Let h a (w) := min{h · ξ : ξ ∈ X , θ · ξ = w, ξ i ≤ γ a i (w), i = 1, . . . , I} =h · γ a (w), w ∈ [0, θ · a]. Note the similarity to the payoff h in (23). Note that the construction depends on the parameter ε 0 , and denote ω 1 (ε 0 ) = sup [0,θ·a] |h a − h|.(83) By the choice of a it is clear that ω 1 (0+) = 0. Before providing the precise construction of the policy, we explain its rationale. The solution to the DG indicates that rejections should occur when the normalized workload in the system is above the threshold β 0 , and that most rejections should be from a specific class, i * , defined in (24). The DG solution also indicates that prioritization should be according to (80) (see the proof of (100) in [2,Theorem A.1]) and that, consequently, the resulting normalized queue length processes should be close to the curve γ. These two goals are contradictory, as parts of the curve γ lie on the part ∂X of the domain where some of the buffers are full, and so even small stochastic fluctuations cause rejections due to the buffer size constraints. Such rejections do not satisfy the requirement to reject only when the workload is above the specified threshold, nor that rejections are from class i * . To address this issue, we have defined the curve γ a , which approximates γ without intersecting the part ∂X of the boundary. The service policy is designed to keep the normalized queue length processes close to this curve. The precise definition of the policy is provided by specifying (B n (t), R n (t)) as a function of X n (t). Rejection policy: As under any policy, in order to meet the buffer size constraint (8), all forced rejections take place. That is, if a class-i arrival occurs at a time t whenX n i (t−)+ 1 bn √ n > D i , then it is rejected. Apart from that, no rejections occur from any class except class i * , which is defined through (24), and no rejections occur (from any class) when θ ·X n < a * . When θ ·X n ≥ a * , all class-i * arrivals are rejected, and these rejections are called overload rejections. Service policy: For eachx ∈ X define the class of low priority L(x) = max{i : x i < a i }, provided x i < a i for some i, and set L(x) = I otherwise. The complement set is the set of high priority classes: H(x) = I \ {L(x)}. When there is at least one class among H(x) having at least one customer in the system, L(x) receives no service, and all classes within H(x), having at least one customer, receive service at a fraction proportional to their traffic intensities. Namely, denote H + (x) = {i ∈ H(x) : x i > 0}, and define ρ ′ (x) ∈ R I as ρ ′ i (x) =          0, ifx = 0, ρ i 1 {i∈H + (x)} k∈H + (x) ρ k , if H + (x) = ∅, e I , if x i = 0 for all i < I and x I > 0.(84) (Note that H + (x) = ∅ can only happen if x i = 0 for all i < I, which is covered by the first and last cases in the above display). Then for each t, B n (t) = ρ ′ (X n (t)).(85) Note that when H + (x) = ∅, ρ ′ i (x) > ρ i for all i ∈ H + (x).(86) That is, all prioritized classes receive a fraction of effort strictly greater than the respective traffic intensity. Also note that i B n i = 1 wheneverX n is nonzero. This is therefore a work conserving policy. Theorem 5.1 Let Assumption 2.1 hold. For every ε 0 > 0 and n ∈ N, denote the policy constructed above by U n (ε 0 ). Then, for all sufficiently large T , lim sup n→∞ J n (T,X n (0), U n (ε 0 )) ≤ V (x) + ω(ε 0 ), where ω : R + → R + is a function satisfying ω(0+) = 0. Proof of Theorem 5.1: Introduce the notation H n t = t 0h ·X n (u)du +r ·R n (t). Fix T > 0 sufficiently large for the identity V (T, x) = V (x) stated in Proposition 3.1(i) to hold. First, notice that E T 0 e b 2 n H n t dt ≤ T E e b 2 n H n T .(87) The argument will be based on a bound on the r.h.s. of (87). Recall the definition (10) of J, and for J > 0, define AC J = {ψ ∈ D([0, T ], R 2I ) : J(T, ψ) ≤ J}. Then AC J is compact in the J 1 topology, and consists of absolutely continuous paths starting at zero. Fix 0 < ε 1 < θ min ε 0 /8. Fix also δ 1 > 0 such that δ 1 < min{ε 0 /12, ε 1 /(11C), ε 0 /(5C), osc D (ε 0 /(5C), h)},(88) where C is the constant from Lemma A.1. For 0 < δ < t ≤ T and ψ ∈ D([0, T ], R 2I ), denote A δ,t (ψ) = {ψ ∈ D([0, T ], R 2I ) : ψ − ψ t < δ} (where we slightly modified the notation (52)). By the compactness of AC J and the continuity of its members, one can find a finite number of membersψ 1 ,ψ 2 , . . . ,ψ M of AC J , and positive constants δ 1 , . . . , δ M with δ m < δ 1 , satisfying AC J ⊂ ∪ M m=1 A m T , and inf{J(T,ψ) :ψ ∈ A m T } ≥ J(T,ψ m ) − ε 0 , m = 1, 2, . . . , M,(89) where, throughout, for 0 ≤ t ≤ T , A m t = A δ m ,t (ψ m ). By the continuity of each of the pathsψ m , one can find ν 1 > 0 such that for m = 1, . . . , M , osc T (ν 1 ,ψ m i ) ≤ δ 1 2(θ max ∨ 1) √ I , i ∈ I.(90) In this proof, C 1 , C 2 , . . . denote positive constants that do not depend on n, ε 0 , δ m or J. Denote Λ n = à n T + S n T . As argued in [6], at the bottom of page 595, R n (T ) ≤ C 1 (1 + Λ n ).(92) SinceX n is bounded, one has H n T ≤ C 2 (1 + Λ n ). Hence, given any J 1 > 0, H n T > J 1 implies Λ n > C −1 2 J 1 − 1 =: G(J 1 ). Therefore E[e b 2 n H n T ] ≤ E[e b 2 n [H n T ∧J 1 ] ] + E[e b 2 n H n T 1 {H n T >J 1 } ] ≤ A n 1 + A n 2 + A n 3 ,(93)where, with B = (∪ M m=1 A m T ) c , A n 1 = M m=1 E[e b 2 n [H n T ∧J 1 ] 1 Ω n,m ], Ω n,m = {(à n ,S n ) ∈ A m T }, A n 2 = E[e b 2 n [H n T ∧J 1 ] 1 {(à n ,S n )∈B} ], A n 3 = E[e b 2 n C 2 (1+Λ n ) 1 {Λ n >G(J 1 )} ]. An argument to be presented shortly will show that there exist t 1 , . . . , t M ∈ [0, T ] such that for large n, A n 1 ≤ M M max m=1 e b 2 n [ t m 0 h(ϕ m (u))du+r(1+ε 0 )̺ m (t m )−I(t m ,ψ m )+ω 2 (ε 0 )] + ε 0 ,(94) where ω 2 (0+) = 0 (this step translates the multidimensional formulation, by which H n T is defined, into a one-dimensional form, given by ϕ m ). As for A n 2 and A n 3 , note, by Proposition 2.1, that for large n, 1 b 2 n log P((à n ,S n ) ∈ B) ≤ − inf ψ∈B I(T, ψ) + ε 0 . Along with the fact that B ⊂ AC c J and the definition of AC J , this shows A n 2 ≤ e b 2 n [J 1 −J+ε 0 ] .(95) Also, A n 3 ≤ E[e b 2 n ((C 2 +1)Λ n +C 2 −G(J 1 )) ]. As shown in the appendix of [1], Assumption 2.1 implies that, for any K < ∞, lim sup n→∞ 1 b 2 n log E[e b 2 n K( Ãn T + Sn T ) ] < ∞. Hence there exists a constant C 3 such that A n 3 ≤ e b 2 n C 2 +C 3 −G(J 1 ) .(96) Combining (93), (94), (95) and (96), lim sup 1 b 2 n log E[e b 2 n H n T ] ≤ max 1≤m≤M t m 0 h(ϕ m (u))du + r(1 + ε 0 )̺ m (t m ) − I(t m , ψ m ) + ω 2 (ε 0 ) ∨ [J 1 − J + ε 0 ] ∨ [C 2 + C 3 − G(J 1 )] ≤ sup ψ∈P,t∈[0,T ] [c ε 0 (x, t, ψ, α β 0 [ψ]) + ω 2 (ε 0 )] ∨ [J 1 − J + ε 0 ] ∨ [C 2 + C 3 − G(J 1 )], where the cost c ε 0 is defined as c with the rejection cost r(1 + ε 0 ) instead of r. Now, let ε 0 → 0 first, then J → ∞, recalling that C 2 , C 3 and G do not depend on J. Finally let J 1 → ∞, so G(J 1 ) → ∞, to obtain lim sup V n (T,X n (0)) ≤ lim sup 1 b 2 n log E[e b 2 n H n T ] ≤ sup ψ∈P 2 ,t∈[0,T ] c(x, t, ψ, α β 0 [ψ]) = V (x), where in the first inequality we used (87), and in the equality we used the optimality of the β 0 -barrier strategy in the game, see Proposition 3.2, as well as Proposition 3.1(i). We thus turn to the proof of (94). We argue in two steps. In step 1, we show the multidimensional processX n lies close to the minimizing curve. Consequently, we also deduce that no forced rejections occur, provided n is sufficiently large. In step 2, we deduce (94) from step 1. Step 1. We show that for large n, max i ∆ n i T ≤ ε 0 ,(97) where we denote the difference process ∆ n i (t) =X n i (t) − γ a i (X n (t)), t ∈ [0, T ].(98) Denote by G = {x ∈ X : θ · x ≤ a * , x = γ a (θ · x)} the set of points lying on the minimizing curve, and recall ∂ + X := {x ∈ X : x i = b i for some i}, the set corresponding to the buffer limit boundary. By the choice of a and ε 0 it follows that G ε 0 and (∂ + X ) ε 0 do not intersect, where for a set A ∈ R I we denote A ε := {x : dist(x, A) ≤ ε}. Forced rejections occur only at times whenX n lies in (∂ + X ) ε 0 (for all n large). As a result, as long as the processX n lies in G ε 0 , no forced rejections occur. Thus by showing (97), one also obtains R n (T ) =R n i * (T )e i * .(99) Denote τ n = inf{t ≥ 0 : max i |∆ n i (t)| ≥ ε 0 }.S n • T n −ψ m,2 • ρ T ≤ 3δ 1 /2, à n −ψ m,1 T ≤ δ 1(100) and Š n • T n − ψ m,2 • ρ T < 2δ 1 , Ǎ − ψ m,1 T < 2δ 1 .(101) Moreover, for all large n and all t, u ∈ [0, τ n ] such that |t − u| < ν 1 , |X n (t) −X n (u)| ≤ ε 1 .(102) Lemma 5.2 For all large n, (97) holds on the event ∪ M m=1 Ω n,m . These two lemmas are proved at the end of the section. Step 2. As mentioned earlier, for sufficiently small ε 0 one has a * = β 0 . Moreover, by Lemma 5.2 and the discussion in the beginning of step 1, no forced rejections occur on the event under consideration. Consider the balance equation (15) and recall thatŽ n andŘ n are nonnegative, nondecreasing processes. RecallY n defined in (13). Also, these processes are flat on the set of times whereX n > 0 andX n < a * , respectively, where we used work conservation and the absence of forced rejections. Recalling from Section A.1 the characterization of the Skorohod map on an interval, it follows that, on the event under consideration, (X n ,Ž n ,Ř n )(t) = Γ [0,β 0 ] (Y n +Ǎ n −Š n • T n ).(103) Compare this relation with (91). Let n be sufficiently large so that Y n − y T ≤ δ 1 . Then by (101), Y n +Ǎ n −Š n • T n − y + ψ m,1 − ψ m,2 T ≤ 5δ 1 . From the above, using (88), (91), and Lemma A.1 it follows that on the event Ω n.m one has Ř n (T ) − ̺ m (T ) ≤ ε 0 and X n − ϕ m T ≤ osc D (ε 0 , h).(104) Now we bound H n T . Let L = i h i . For sufficiently large n, T 0h ·X n (u)du ≤ T 0h · γ a (X n (u))du + 2Lε 0 T = T 0 h a (X n (u))du + 2Lε 0 T ≤ T 0 h(X n (u))du + (2L + 1)ε 0 T + T ω 1 (ε 0 ) ≤ T 0 h(ϕ m (u))du + (2L + 2)T ε 0 + T ω 1 (ε 0 ), where the first inequality follows by Lemma 5.2, the equality follows by the definitions of h a and γ a , the second inequality follows by (83), and the last inequality follows by (104). Also notice that for sufficiently large n r ·R n = r i * R n i * = r µ i * R n i * ≤ r(1 + ε 0 )θ n i * R n i * = r(1 + ε 0 )θ n ·R n = r(1 + ε 0 )Ř n , where the first and third equalities follow from (99), the second equality follows since r = r i * µ i * , the inequality follows since θ n → θ, and the last equality follows by the definition ofŘ n . Denote ω 2 (ε 0 ) = ((2L + 2)T + r(1 + ε 0 ))ε 0 + T ω 1 (ε 0 ). Then from (83) one has ω 2 (0+) = 0. Moreover, E[e b 2 n [H n T ∧J 1 ] 1 Ω n,m ] (105) ≤ E[e b 2 n [ T 0h ·X n (u)du+r·R n (T )] 1 Ω n,m ] ≤ E[e b 2 n [ T 0 h(ϕ m (u))du+r(1+ε 0 )̺ m (T )+ω 2 (ε 0 )] 1 Ω n,m ] ≤ max 0≤t≤T e b 2 n [ t 0 h(ϕ m (u))du+r(1+ε 0 )̺ m (t)+ω 2 (ε 0 )] P[(à n ,S n ) ∈ A m t ]. Let t m be such that This, along with (106) show that the r.h.s. of (105) is bounded by max 0≤t≤T e b 2 n [ t 0 h(ϕ m (u))du+r(1+ε 0 )̺ m (t)+ω 2 (ε 0 )] P[(à n ,S n ) ∈ A m t ] (106) ≤ e b 2 n [ t m 0 h(ϕ m (u))du+r(1+ε 0 )̺ m (t m )+ω 2 (ε 0 )] P[(à n ,S n ) ∈ A m t m ] + ε 0 .e b 2 n [ t m 0 h(ϕ m (u))du+r(1+ε 0 )̺ m (t m )−I(t m ,ψ m )+ω 2 (ε 0 )] + ε 0 . Thus (94) follows. ✷ Proof of Lemma 5.1: Recall relation (11) and the fact thatX n (t) remains bounded. This, along with (92) give Z n ≤ C 4 (1 + Λ n ). Denote Λ J = sup ψ∈AC J ψ 1 T + ψ 2 T < ∞,(107) where the finiteness follows by (45) and the definition of AC J . Thus, on the event ∪ m Ω n,m , one has Z n T ≤ Λ J +2δ 1 . Recalling the expression (12) forZ n , it follows that ρ−T n T < ν 1 for large n. Therefore, for every m ∈ {1, . . . , M }, ψ m,2 • T n −ψ m,2 • ρ T < δ 1 /2. Moreover, on the event Ω n,m , S n • T n −ψ m,2 • T n T < δ 1 , and therefore S n • T n −ψ m,2 • ρ T ≤ 3δ 1 /2. Similarly, à n −ψ m,1 T ≤ δ 1 . By (100), (90), and since θ n → θ, it follows that on the event Ω n,m , (101) holds. It remains to prove (102). Fix 0 ≤ u ≤ t < τ n such that t − u < ν 1 . By the definition of the time τ n , an argument as that leading to (103) shows (X n ,Ž n ,Ř n )(u) = Γ [0,β 0 ] [W n ](u), u ∈ [0, τ n ). whereW n (u) =X n (0) +y n u +Ǎ n (u) −Š n (T n (u)) −Ř n (u). If we show that |W n (t) −W n (u)| < 11δ 1 then the result follows by (88). Using (101) along with (90) and the fact lim n→∞ Y n − y T = 0 shows that, for large n, |W n (t) −W n (u)| ≤ |W n (t) − (x + yt + ψ m,1 (t) − ψ m,2 (t))| + |W n (u) − (x + yu + ψ m,1 (u) − ψ m,2 (u))| + |ψ m,1 (t) − ψ m,1 (u))| + |ψ m,2 (t) − ψ m,2 (u))| < 11δ 1 . This completes the proof. ✷ Proof of Lemma 5.2: The structure of the proof borrows ideas from the proof of Lemma 4.1 of [6] (however, the content is different, as [6] addresses weak convergence). We begin with the case where the initial state lies close to the minimizing curve. That is, max i |∆ n i (0)| ≤ ε 0 .(108) At the last step of the proof we relax this assumption. By reducing to a subsequence of {n}, one then has that there exists an m ∈ {1, . . . , M } such that on the event (à n ,S n ) ∈ A m T one has τ n ≤ T . Let j = j n and ξ n be the corresponding components from the representation (j, ξ) ofX n (τ n ) (with w =X n (τ n )). Fix a positive integer K = K(ε 1 ) = [D/ε 1 ], where ε 1 ≤ θ min ε 0 /8 as defined right before (88). Consider the covering of [0, D] by the K − 1 intervals Ξ k = B(kε 1 , ε 1 ), k = 1, 2, . . . , K − 1, where B(x, a) denotes [x − a, x + a]. LetΞ k = B(kε 1 , 2ε 1 ). From (102) we obtain ifX n (τ n ) ∈ Ξ k thenX n (t) ∈Ξ k for every t ∈ T n := [((τ n − ν 1 ) ∨ 0), τ n ]. By considering a further subsequence, we may assume that there exists a k = k(m) such thať X n (τ n ) ∈ Ξ k for all n. The value assigned by the policy to B n (see (85)) remains fixed asX n varies within any of the intervals (â j ,â j−1 ). Aiming at showing a contradiction for each k, we consider the following four cases. (i)Ξ k ⊂ (0, a * ) and for all j,â j / ∈Ξ k . (ii)Ξ k ⊂ (0, a * ) butâ j ∈Ξ k for some j ∈ {1, 2, . . . , I − 1}. (iii) 0 ∈Ξ k . (iv) a * ∈Ξ k . (i)Ξ k ⊂ (0, a * ) and for all j,â j / ∈Ξ k . Then all points x inΞ k lead to the same j in the representation (j, ξ) of x given by (82). This j = j(k) depends on k only, and in particular does not vary with n. Fix i > j (except when j = I). Note that γ a i (X n (τ n )) = a i (because i > j). We show first that for sufficiently large n one has for every i > j, one has ∆ n i (t) < ε 0 for all t ∈ [0, T ]. This is done as follows. Assume to the contrary that τ n ≤ T and that ∆ n i (τ n ) ≥ ε 0 . Then, since the jumps ofX n i are of size (b n √ n) −1 it follows that there must exist η n ∈ [0, τ n ] with the properties thatX n i (η n ) < a i + ε 0 /2,X n i (t) > a i for all t ∈ [η n , τ n ]. Therefore, during the time interval [η n , τ n ], i is always a member of H + (X n ), and therefore by (85)-(86), B n i (t) = ρ ′ i (X n (t)) > ρ i + C 4 , for some constant C 4 > 0. Thus by (12), d dtZ n i ≤ − µ n i b n √ n C 4 . Moreover, if we defineη n = η n ∨ (τ n − ν 1 ) then by (109) we get that for all t ∈ [η n , τ n ] one haš X n (t) ∈Ξ k ⊂ (0, a * ) and therefore no rejections occur. Using these facts in (12), we havẽ X n i (τ n )−X n i (η n ) ≤ [à n i (τ n )−à n i (η n )]−[S n i (T n i (τ n ))−S n i (T n i (η n ))]− µ n i b n √ n C 4 (τ n −η n ). (112) Fix a sequence r n > 0 with r n → 0 and r n b n √ n → ∞. If τ n − η n < r n and n is sufficiently large thenη n = η n , thus by the definitions of τ n and η n , one hasX n i (τ n ) −X n i (η n ) ≥ ε 0 /2. As a result, [à n i (τ n ) −à n i (η n )] − [S n i (T n i (τ n )) −S n i (T n i (η n ))] ≥ ε 0 /2. where we used γ a i (X n (t)) = 0 for i < j and γ a i (X n (t)) = a i for i > j. These two equations hold from (109). For all large n, this impliesX n i (t) < a i for at least one i > j, by which j ∈ H + (X n ). We can now show that for every t ∈ [0, T ] one has max i≤I |∆ n i (t)| < ε 0 . Indeed, in the case j = I, we have by (115), max i<I |∆ n i (t)| < ε 0 . By (118), |θ · ∆ n (t)| ≤ q n . Since θ ∈ (0, ∞) I and q n → 0, we obtain max i≤I |∆ n i (t)| < ε 0 . In the case j < I, combining (110), (115), (116), we have max i≤I ∆ n i (t) < ε 0 . Using again the fact |θ · ∆ n (τ n )| ≤ q n → 0 shows that (120) is valid in this case as well. (ii)Ξ k ⊂ (0, a * ) butâ j ∈Ξ k for some j ∈ {1, 2, . . . , I − 1}. Let (j n (t), ξ n (t)) denote the representation (82) forX n (t). Note that in the time window T n , j n varies between two values, namely j and j + 1, and so it is no longer true that γ a j+1 (X n (t)) = a j+1 on that time interval. However, it is true that γ a j+1 (X n (t)) ≥ a j+1 − 4ε 1 /θ min ≥ a j+1 − ε 0 /2, t ∈ T n , where the second inequality follows since ε 1 < θ min ε 0 /8. Indeed, we have for any w ∈Ξ k , |w −â j | ≤ 4ε 1 , sinceâ j is also inΞ k . Now, if w ≥â j then γ a j+1 (w) = a j+1 . Otherwise, w =â j+1 + θ j+1 ξ =â j − θ j+1 a j+1 + θ j+1 ξ, hence |a j+1 − ξ| ≤ 4θ −1 j+1 ε 1 and (121) follows. By the same arguments as in case (i) we get that for every i = j + 1, one has ∆ n i (t) < ε 0 for all t ∈ [0, T ]. As for i = j + 1, assume to the contrry that τ n ≤ T and that ∆ n j+1 (τ n ) ≥ ε 0 . Then by (121) we get thatX n j+1 (τ n ) ≥ a j+1 + ε 0 /2 and the same arguments as in case (i) are valid. Combining all the estimates gives, max i≤I ∆ n i (t) < ε 0 , t ∈ [0, T ]. Along with the fact |θ · ∆ n (τ n )| ≤ q n , this gives max i≤I |∆ n i (t)| < ε 0 , t ∈ [0, T ]. (iii) 0 ∈Ξ k . This differs from case (i) in that during T n ,X n may hit zero, and therefore B n might vanish. Note however that the way case (i) is handled, one focuses only on time intervals whereX n = 0, and therefore the proof is valid here as well. We thus have max i≤I |∆ n i (t)| < ε 0 , t ∈ [0, T ]. (iv) a * ∈Ξ k . In this case, θ ·X n may exceed the threshold a * , and rejections may occur. The argument provided in case (i) is then slightly changed. A negative term is added to the r.h.s. of (112), but the consequences of (112) remain valid with this addition. (Note that for small ε 0 ,â i = a * holds for all i, hence assuming ε 0 is small, we do not need to check case (ii) here.) Having shown that max i≤I |∆ n of finding, for a given ω, a triplet (ϕ, η 1 , η 2 ), such that ϕ = ω + η 1 − η 2 , ϕ(t) ∈ [a, b] for all t, η i are nonnegative and nondecreasing, η i (0−) = 0, and By writing η i (0−) = 0 we adopt the convention that η i (0) > 0 is regarded a jump at zero. This convention, in conjunction with [0,∞) 1 (a,b] (ϕ)dη 1 = 0 (resp., [0,∞) 1 [a,b) (ϕ)dη 2 = 0), means that if ω(0) < a (resp., ω(0) > b) then ϕ(0) = a (resp., b). If, however, ω(0) ∈ [a, b] then ϕ(0) = ω(0), and η i have no jump at zero. See [19] for existence and uniqueness of solutions, and continuity and further properties of the map. In particular, we have the following. A.2 On the rate functions I and J For every T ∈ R + and ψ = (ψ 1 , ψ 2 ) ∈ P 2 setĨ(T, ψ) =Ĩ 1 (T, ψ 1 ) +Ĩ 2 (T, ψ 2 ), I 1 (T, ψ 1 ) = inf{J 1 (T,ψ 1 ) :ψ 1 ∈ P I , θ ·ψ 1 = ψ 1 } (122) andĨ 2 (T, ψ 2 ) = inf{J 2 (T,ψ 2 ) :ψ 2 ∈ P I , θ · (ψ 2 • ρ) = ψ 2 }, Recall that I is defined in (17) and J in (10). Lemma A.2 For every ψ = (ψ 1 , ψ 2 ) ∈ P 2 there existsψ = (ψ 1 ,ψ 2 ) ∈ P I × P I such that (θ ·ψ 1 , θ · (ψ 2 • ρ)) = (ψ 1 , ψ 2 ) (124) R k ) and D([0, T ], R k ) the spaces of absolutely continuous functions [resp., continuous functions, functions that are right-continuous with finite left limits (RCLL)] mapping [0, T ] → R k . Write AC 0 ([0, T ], R k ) and C 0 ([0, T ], R k ) for the subsets of the corresponding function spaces, of functions that start at zero. Endow the space D([0, T ], R k ) with the usual Skorohod topology. For a collection x i ∈ R indexed by i ∈ {1, . . . , I} (I being a positive integer), x denotes the vector (x i ). A similar convention holds for R-valued random variables X i and stochastic process {X i (t), t ∈ R + }, i ∈ {1, . . . , I}, where X and {X(t), t ∈ R + } denote the R I -valued random variable and process. Assumption 2. 1 1There exists u 0 > 0 such that for i ∈ I, E[e u 0 IA i ] and E[e u 0 ST i ] are finite. Lemma 4. 3 3For every m > 0 there exists K > 0 such that for every i ∈ I one has lim sup 1 Lemma 4. 4 4Fix ψ ∈ D([0,T ], R I ) such that ψ =ψ ♯ on [ gives a lower bound on the third term. Combining inequalities (60), ≤ P{Q n ∩ Ω n δ,vn }P{Ω n δ ′′ ,vn,v−δ } + e −kb 2 n P{Ω n δ ′′′ ,vn }P{Ω n δ ′′′ ,vn,v } − e −kb 2 n = P{Q n |Ω n δ,vn }P{Ω n δ,vn }P{Ω n δ ′′ ,vn,v−δ } + e −kb 2 n P{Ω n δ ′′′ ,vn }P{Ω n δ ′′′ ,vn,v } − e −kb 2 n. Using the LDP and writing I[a, vn]−ε ′ ) e b 2 n (−I[vn,v]−ε ′ ) − e −kb 2 n = p n 2 e b 2 n (−I[0,v−δ]+2ε ′ ) + e −kb 2 n e b 2 n (−I[0,v]−2ε ′ ) − e −kb 2 n , Writeψ m ∈ P 2I as (ψ m,1 ,ψ m,2 ). For m = 1, . . . , M let ϕ m = y + ψ m,1 − ψ m,2 + ζ m − ̺ m , where, as before, y(t) = x + yt, ψ m,1 = θ ·ψ m,1 , ψ m,2 = θ ·ψ m,2 , and (ϕ n , ζ m , ̺ m ) = Γ [0,β 0 ] [y + ψ m,1 − ψ m,2 ]. Lemma 5. 1 1For all large n, for every m ∈ {1, . . . , M }, one has on the event Ω n,m , P((à n ,S n ) ∈ A m t m ) ≤ − inf ψ∈Am t m J(t m , ψ) + ε 0 ≤ −J(t m ,ψ m ) + 2ε 0 ≤ −I(t m , ψ m ) + 2ε 0 . Lemma A. 1 1Fix b > 0. Then there exists a constant C such that for every T > 0, δ > 0 and ω,ω ∈ D([0, ∞), R),Γ [0,b] (ω) − Γ [0,b] (ω) T ≤ C ω −ω T , and osc T (δ, Γ [0,b] (ω)) ≤ C osc(δ, ω). i andĨ k (T, ψ k ) = I k (T, ψ k ) = J k (T,ψ k ), k = 1, 2, T > 0. (125) Proof: Define K : [0, ∞) × P I → [0, (u)du if all ψ i ∈ AC 0 ([0, T ], R),∞ otherwise, where α 1 , . . . , α I > 0. Then both J 1 and J 2 are of the form K. Set l 1 , . . . , l I ∈ (0, 1]. Define L : [0, ∞) × P → [0, ∞] by L(T, ψ) = inf{K(T,ψ) :ψ ∈ P I , θ ·ψ(l) = ψ}, In case r 2 4s + ry ≥ −h(0) = 0 we get by (i) above that V (x) = ∞ and we do not define β0. We use the convention that ∆/0 = ∞. Recall the convention ε = 0 and generic r. From Lemma 5.1 and inequality (90) it follows that the l.h.s. of the above is smaller than 6δ 1 . Altogether, 6δ 1 > ε 0 /2, which contradicts the choice of δ 1 (see(88)). If, on the other hand, τ n − η n ≥ r n then by (112),The l.h.s. of the above is bounded from above by 2Λ n , which is bounded by Lemma 5.1 and the definition of Λ J , see (107). This contradicts the fact C 4 r n b n √ n → ∞. Therefore, (110) holds.Next, fix i < j (provided j = 1). Then γ a i (X n (τ n )) = 0 and wheneverX n i > 0, i is a member of the high priority set H + (X n ). This is due to the fact that there must exist l > i such thatX n l < a l ; otherwise, the workload would be at least I p=l a p θ n p >X n . Hence, the same argument yields a contradiction. ThereforeConsider now j itself. We will show, for the case j < I, thatSuppose that we show that for every t ∈ [0, τ n ] and every large n,Then, assuming to the contrary that τ n ≤ T andX n j (τ n ) ≥ γ a j (X n (τ n )) + ε 0 , implies that there exists η n ∈ [0, τ n ] with the properties that. From (117) during the time interval [η n , τ n ], j is always a member of H + (X n j (t)). Therefore, we still have inequality (112) valid. Arguing separately for the cases τ n −η n < r n and τ n −η n ≥ r n , leads, in analogy to(113)and(114), to a contradiction. There is a slight difference in the first case. If τ n − η n < r n and n is sufficiently large thenη n = η n , and nowX n j (τ n ) −X n j (η n ) ≥ ε 0 /4 + γ a j (X n (τ n )) − γ a j (X n (η n )). Recall that τ n −η n < ν 1 . Therefore, by taking δ 1 to be sufficiently small, one can verify that |γ a j (X n (τ n )) − γ a j (X n (η n ))| would be significantly smaller than ε 0 /2. This argument follows by the continuity of γ a and by similar arguments to the ones we used in order to prove (102). Now we show that (117) holds (except in the case j = I). Since θ · γ a (θ ·x) = θ ·x for all x ∈ X , θ n → θ, γ a is uniformly continuous, and X is bounded, we have q n := sup x∈X |θ · γ a (θ n ·x) − θ ·x| → 0, as n → ∞.Note by (118) that |θ ·X n (t) − θ · γ a (X n (t))| ≤ q n → 0. Fix t ∈ [0, τ n ]. If ∆ n j (t) ≥ ε 0 /2 thenT ] in all cases completes the proof of the lemma under (108).The relaxation of (108) is performed by showing that within a short time t, max i |∆ n i (t)| ≤ ε 0 . This is sufficient, because on the remaining time interval the argument provided above for the case (108) gives the result.Fix δ > 0. We will show that for each i and all sufficiently large n, there exists t ∈ [0, δ] such that |∆ n i (t)| ≤ ε 0 . Since the proof provided above for the case (108) treats each i separately, this will assure that once |∆ n i | is bounded by ε 0 for some i, it remains so for the remaining time interval.We thus fix i and prove that for all sufficiently large n, there exists t ∈ [0, δ] such that |∆ n i (t)| ≤ ε 0 . Assume to the contrary that |∆ n i | > ε 0 on [0, δ]. Since the jumps of ∆ n are of order n −1/2 , we either have ∆ n i > ε 0 on [0, δ], or ∆ n i < −ε 0 on [0, δ], provided n is large. In the former case, i is always a member of H + (X n (t)), for every t, and therefore by (85)-(86), B n i (t) = ρ ′ i (X n (t)) > ρ i + C 4 , for some constant C 4 > 0. Thus by(12), d dtZTherefore, as was argued in (112), by(12)we obtain thatThe r.h.s. of the above bounded from above, as follows from Lemma 5.1 and the fact that Λ J < ∞ (see(107)). This contradicts the fact that the l.h.s. goes to infinity.In case that ∆ n i < −ε 0 on [0, δ], i is not a member of H + (X n (t)) for any t, and therefore by (85)-(86), B n i (t) = 0. Thus by(12), d dtZThe r.h.s. of the above is bounded, using similar considerations along with (92), whereas again, the l.h.s. tends to infinity with n. It is denoted by Γ[a,b], and is characterized as the solution map ω → (ϕ, η 1 , η 2 ) to the problem Control of the multiclass G/G/1 queue in the moderate deviation regime. Rami Atar, Anup Biswas, Ann. Appl. Probab. 245Rami Atar and Anup Biswas. Control of the multiclass G/G/1 queue in the moderate deviation regime. Ann. Appl. Probab., 24(5):2033-2069, 2014. A differential game for a multiclass queueing model in the moderate-deviation heavy-traffic regime. Rami Atar, Asaf Cohen, Math. Oper. Res. to appearRami Atar and Asaf Cohen. A differential game for a multiclass queueing model in the moderate-deviation heavy-traffic regime. Math. Oper. Res., to appear, 2016. An escape-time criterion for queueing networks: asymptotic risk-sensitive control via differential games. Rami Atar, Paul Dupuis, Adam Shwartz, Math. Oper. Res. 284Rami Atar, Paul Dupuis, and Adam Shwartz. An escape-time criterion for queueing networks: asymptotic risk-sensitive control via differential games. Math. Oper. Res., 28(4):801-835, 2003. Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic. Rami Atar, Avi Mandelbaum, Martin I Reiman, Ann. Appl. Probab. 143Rami Atar, Avi Mandelbaum, and Martin I. Reiman. Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic. Ann. Appl. Probab., 14(3):1084-1134, 2004. On the non-Markovian multiclass queue under risk-sensitive cost. Rami Atar, Gal Mendelson, Queueing Systems Theory Appl. to appearRami Atar and Gal Mendelson. On the non-Markovian multiclass queue under risk-sensitive cost. Queueing Systems Theory Appl., to appear, 2016. An asymptotic optimality result for the multiclass queue with finite buffers in heavy traffic. Rami Atar, Mark Shifrin, Stochastic Systems. 42Rami Atar, Mark Shifrin, et al. An asymptotic optimality result for the multiclass queue with finite buffers in heavy traffic. Stochastic Systems, 4(2):556-603, 2014. Controlled stochastic networks in heavy traffic: convergence of value functions. Amarjit Budhiraja, Arka P Ghosh, Ann. Appl. Probab. 222Amarjit Budhiraja and Arka P. Ghosh. Controlled stochastic networks in heavy traffic: convergence of value functions. Ann. Appl. Probab., 22(2):734-791, 2012. Leontief systems, RBVs and RBMs. Hong Chen, Avi Mandelbaum, Applied stochastic analysis. London; New YorkGordon and Breach5Hong Chen and Avi Mandelbaum. Leontief systems, RBVs and RBMs. In Applied stochastic analysis (London, 1989), volume 5 of Stochastics Monogr., pages 1-43. Gordon and Breach, New York, 1991. Large deviations techniques and applications. Amir Dembo, Ofer Zeitouni, Applications of Mathematics. 38Springer-Verlagsecond editionAmir Dembo and Ofer Zeitouni. Large deviations techniques and applications, volume 38 of Applications of Mathematics (New York). Springer-Verlag, New York, second edition, 1998. Risk-sensitive and robust escape criteria. Paul Dupuis, William M Mceneaney, SIAM J. Control Optim. 356Paul Dupuis and William M. McEneaney. Risk-sensitive and robust escape criteria. SIAM J. Control Optim., 35(6):2021-2049, 1997. Stochastic control for small noise intensities. Wendell H Fleming, SIAM J. Control. 9Wendell H. Fleming. Stochastic control for small noise intensities. SIAM J. Control, 9:473-517, 1971. Risk sensitive stochastic control and differential games. Wendell H Fleming, Commun. Inf. Syst. 63Wendell H. Fleming. Risk sensitive stochastic control and differential games. Commun. Inf. Syst., 6(3):161- 177, 2006. Risk-sensitive control on an infinite time horizon. Wendell H Fleming, William M Mceneaney, SIAM J. Control Optim. 336Wendell H. Fleming and William M. McEneaney. Risk-sensitive control on an infinite time horizon. SIAM J. Control Optim., 33(6):1881-1915, 1995. Controlled Markov processes and viscosity solutions. Wendell H Fleming, H Mete Soner, Stochastic Modelling and Applied Probability. 25Springersecond editionWendell H. Fleming and H. Mete Soner. Controlled Markov processes and viscosity solutions, volume 25 of Stochastic Modelling and Applied Probability. Springer, New York, second edition, 2006. PDE-viscosity solution approach to some problems of large deviations. Wendell H Fleming, Panagiotis E Souganidis, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 134Wendell H. Fleming and Panagiotis E. Souganidis. PDE-viscosity solution approach to some problems of large deviations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13(2):171-192, 1986. Big Queues. Ayalvadi Ganesh, O&apos; Neil, Damon Connell, Wischik, Lecture Notes in Mathematics. 1838Springer-VerlagAyalvadi Ganesh, Neil O'Connell, and Damon Wischik. Big Queues, volume 1838 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2004. Heavy traffic resource pooling in parallel-server systems. J , Michael Harrison, Marcel J López, Queueing Systems Theory Appl. 334J. Michael Harrison and Marcel J. López. Heavy traffic resource pooling in parallel-server systems. Queueing Systems Theory Appl., 33(4):339-368, 1999. Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games. H David, Jacobson, IEEE Trans. Automatic Control. 182ACDavid H. Jacobson. Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games. IEEE Trans. Automatic Control, AC-18(2):124-131, 1973. An explicit formula for the Skorokhod map on. Lukasz Kruk, John Lehoczky, Kavita Ramanan, Steven Shreve, 0, aLukasz Kruk, John Lehoczky, Kavita Ramanan, and Steven Shreve. An explicit formula for the Skorokhod map on [0, a]. . Ann Probab, 35Ann. Probab., 35(5):1740-1768, 2007. Sample path large deviations for multiclass feedforward queueing networks in critical loading. Kurt Majewski, Ann. Appl. Probab. 164Kurt Majewski. Sample path large deviations for multiclass feedforward queueing networks in critical loading. Ann. Appl. Probab., 16(4):1893-1924, 2006. Moderate deviations for queues in critical loading. A Anatolii, Puhalskii, Queueing Systems Theory Appl. 313-4Anatolii A. Puhalskii. Moderate deviations for queues in critical loading. Queueing Systems Theory Appl., 31(3-4):359-392, 1999. Functional large deviation principles for first-passage-time processes. A Anatolii, Ward Puhalskii, Whitt, Ann. Appl. Probab. 72Anatolii A. Puhalskii and Ward Whitt. Functional large deviation principles for first-passage-time processes. Ann. Appl. Probab., 7(2):362-381, 1997. Asymptotically optimal admission control of a queue with impatient customers. Amy R Ward, Sunil Kumar, Math. Oper. Res. 331Amy R. Ward and Sunil Kumar. Asymptotically optimal admission control of a queue with impatient customers. Math. Oper. Res., 33(1):167-202, 2008. A risk-sensitive maximum principle. P Whittle, Systems Control Lett. 153P. Whittle. A risk-sensitive maximum principle. Systems Control Lett., 15(3):183-192, 1990.
[]
[ "Published for SISSA by Springer Diagrammatic expansion of non-perturbative little string free energies", "Published for SISSA by Springer Diagrammatic expansion of non-perturbative little string free energies" ]
[ "Stefan Hohenegger [email protected] \nUniv. Lyon\nUniv Claude Bernard Lyon 1\n\nUMR 5822\nCNRS/IN2P3\nIP2I, F-69622Lyon, VilleurbanneFrance\n" ]
[ "Univ. Lyon\nUniv Claude Bernard Lyon 1", "UMR 5822\nCNRS/IN2P3\nIP2I, F-69622Lyon, VilleurbanneFrance" ]
[ "JHEP04" ]
In[1]we have studied the single-particle free energy of a class of Little String Theories of A-type, which are engineered by N parallel M5-branes on a circle. To leading instanton order (from the perspective of the low energy U(N ) gauge theory) and partially also to higher order, a decomposition was observed, which resembles a Feynman diagrammatic expansion: external states are given by expansion coefficients of the N = 1 BPS free energy and a quasi-Jacobi form that governs the BPS-counting of an M5-brane coupling to two M2-branes. The effective coupling functions were written as infinite series and similarities to modular graph functions were remarked. In the current work we continue and extend this study: working with the full non-perturbative BPS free energy, we analyse in detail the cases N = 2, 3 and 4. We argue that in these cases to leading instanton order all coupling functions can be written as a simple combination of two-point functions of a single free scalar field on the torus. We provide closed form expressions, which we conjecture to hold for generic N . To higher instanton order, we observe that a decomposition of the free energy in terms of higher point functions with the same external states is still possible but a priori not unique. We nevertheless provide evidence that tentative coupling functions are still combinations of scalar Greens functions, which are decorated with derivatives or multiplied with holomorphic Eisenstein series. We interpret these decorations as corrections of the leading order effective couplings and in particular link the latter to dihedral graph functions with bivalent vertices, which suggests an interpretation in terms of disconnected graphs.JHEP04(2021)275very efficient ways of calculating the BPS counting function Z N,1 . A very useful description in this regards is F-theory compactified on a class of toric Calabi-Yau threefolds X N,1[32]. The partition function Z N,1 is captured by the topological string partition function on X N,1[22][23][24][28][29][30]. With the help of the toric diagram of X N,1 (which can directly be inferred from the M-brane web), the latter can be computed in an algorithmic fashion using the (refined) topological vertex[33,34].This intrinsically geometric description, together with the very explicit form in which Z N,1 (or the related free energy F N,1 ) can be presented, have proven very fruitful and have aided in unravelling numerous interesting and surprising structures and symmetries[24][25][26][35][36][37][38][39][40][41][42]. Here we shall not present all the observations made in recent studies (see[42]for a more complete review), but only recount those that are relevant for the current work: the Calabi-Yau manifolds X N,1 are only a subset of a two-parameter class of manifolds, labelled X N,M , which can be used to describe orbifolds of the M5-brane configurations mentioned above[24]. In [36] it was argued that X N,M is dual to X N ,M if N M = N M and gcd(N, M ) = gcd(N , M ), in the sense that the Kähler cones of both these manifolds are part of a larger common extended moduli space. Within this space, X N,M and X N ,M are related through a combination of flop-and other symmetry transformations. The duality map induced by these symmetry transformations was conjectured 2 in [36] to leave the corresponding partition functions invariant, i.e. Z N,M (ω) = Z N ,M (ω ), where ω and ω denote the dependence on the Kähler parameters. Together with the triality [25] of gauge theories engineered from a single X N,M (i.e. the fact that generically the Kähler moduli space of a given X N,M allows for 3 regions that engineer low energy gauge theories), this leads to a large web of dual supersymmetric gauge theories: these theories are expected to all share the same partition function (as has been checked explicitly in numerous examples [26]) and their instanton series correspond to different (but equivalent) series expansions of Z N,M .It was furthermore argued in [39] that this web of dual gauge theories in fact also implies highly non-trivial (and intrinsically non-perturbative) symmetries for individual theories: focusing on the case of the U(N ) gauge theories engineered by X N,1 , invariance of the free energy F N,1 was shown under a particular dihedral symmetry group. 3 The implications of this symmetry were further explored in[40,41]: by studying series expansions of the single-particle 4 free energy F plet N,1 for N = 2, 3 and partially 4, characteristic patterns were observed, which together with the above mentioned symmetries allowed to conjecture a resummation of the former in an intriguing fashion: although being based on a limited (and a priori formal) series expansion, this allowed to write different contributions of F plet N,1 in terms of generating functions of multiple divisor sums introduced in [44] (see appendix A.4 for their definitions). In the case of N = 2 this was further shown to be equivalent to using 2 This conjecture was subsequently proven for gcd(N, M ) = 1 in [38] and for generic (N, M ) (but in a particular limit of the regularisation parameters that are needed to render ZN,1 well defined) in[43].3We refer the reader to [39] for the details. 4 Starting from the partition function ZN,1, the free energy is defined as FN,1 = ln ZN,1. In contrast to that, F plet N,1 = Plog ZN,1 is defined with the help of the plethystic logarithm Plog(f (x)) = ∞ n=1 µ(n) n ln f (nx), where µ is the Möbius function. Physically, F plet N,1 receives contributions only from single-particle states.
10.1007/jhep04(2021)275
null
226,306,756
2011.06323
7f98ad9f21808a2abc3685cda845ca26b71bb7a5
Published for SISSA by Springer Diagrammatic expansion of non-perturbative little string free energies 2021 Stefan Hohenegger [email protected] Univ. Lyon Univ Claude Bernard Lyon 1 UMR 5822 CNRS/IN2P3 IP2I, F-69622Lyon, VilleurbanneFrance Published for SISSA by Springer Diagrammatic expansion of non-perturbative little string free energies JHEP04 275202110.1007/JHEP04(2021)275Received: December 18, 2020 Accepted: March 19, 2021 Published: April 28, 2021M-TheorySupersymmetric Gauge TheoryField Theories in Higher Dimen- sionsTopological Strings In[1]we have studied the single-particle free energy of a class of Little String Theories of A-type, which are engineered by N parallel M5-branes on a circle. To leading instanton order (from the perspective of the low energy U(N ) gauge theory) and partially also to higher order, a decomposition was observed, which resembles a Feynman diagrammatic expansion: external states are given by expansion coefficients of the N = 1 BPS free energy and a quasi-Jacobi form that governs the BPS-counting of an M5-brane coupling to two M2-branes. The effective coupling functions were written as infinite series and similarities to modular graph functions were remarked. In the current work we continue and extend this study: working with the full non-perturbative BPS free energy, we analyse in detail the cases N = 2, 3 and 4. We argue that in these cases to leading instanton order all coupling functions can be written as a simple combination of two-point functions of a single free scalar field on the torus. We provide closed form expressions, which we conjecture to hold for generic N . To higher instanton order, we observe that a decomposition of the free energy in terms of higher point functions with the same external states is still possible but a priori not unique. We nevertheless provide evidence that tentative coupling functions are still combinations of scalar Greens functions, which are decorated with derivatives or multiplied with holomorphic Eisenstein series. We interpret these decorations as corrections of the leading order effective couplings and in particular link the latter to dihedral graph functions with bivalent vertices, which suggests an interpretation in terms of disconnected graphs.JHEP04(2021)275very efficient ways of calculating the BPS counting function Z N,1 . A very useful description in this regards is F-theory compactified on a class of toric Calabi-Yau threefolds X N,1[32]. The partition function Z N,1 is captured by the topological string partition function on X N,1[22][23][24][28][29][30]. With the help of the toric diagram of X N,1 (which can directly be inferred from the M-brane web), the latter can be computed in an algorithmic fashion using the (refined) topological vertex[33,34].This intrinsically geometric description, together with the very explicit form in which Z N,1 (or the related free energy F N,1 ) can be presented, have proven very fruitful and have aided in unravelling numerous interesting and surprising structures and symmetries[24][25][26][35][36][37][38][39][40][41][42]. Here we shall not present all the observations made in recent studies (see[42]for a more complete review), but only recount those that are relevant for the current work: the Calabi-Yau manifolds X N,1 are only a subset of a two-parameter class of manifolds, labelled X N,M , which can be used to describe orbifolds of the M5-brane configurations mentioned above[24]. In [36] it was argued that X N,M is dual to X N ,M if N M = N M and gcd(N, M ) = gcd(N , M ), in the sense that the Kähler cones of both these manifolds are part of a larger common extended moduli space. Within this space, X N,M and X N ,M are related through a combination of flop-and other symmetry transformations. The duality map induced by these symmetry transformations was conjectured 2 in [36] to leave the corresponding partition functions invariant, i.e. Z N,M (ω) = Z N ,M (ω ), where ω and ω denote the dependence on the Kähler parameters. Together with the triality [25] of gauge theories engineered from a single X N,M (i.e. the fact that generically the Kähler moduli space of a given X N,M allows for 3 regions that engineer low energy gauge theories), this leads to a large web of dual supersymmetric gauge theories: these theories are expected to all share the same partition function (as has been checked explicitly in numerous examples [26]) and their instanton series correspond to different (but equivalent) series expansions of Z N,M .It was furthermore argued in [39] that this web of dual gauge theories in fact also implies highly non-trivial (and intrinsically non-perturbative) symmetries for individual theories: focusing on the case of the U(N ) gauge theories engineered by X N,1 , invariance of the free energy F N,1 was shown under a particular dihedral symmetry group. 3 The implications of this symmetry were further explored in[40,41]: by studying series expansions of the single-particle 4 free energy F plet N,1 for N = 2, 3 and partially 4, characteristic patterns were observed, which together with the above mentioned symmetries allowed to conjecture a resummation of the former in an intriguing fashion: although being based on a limited (and a priori formal) series expansion, this allowed to write different contributions of F plet N,1 in terms of generating functions of multiple divisor sums introduced in [44] (see appendix A.4 for their definitions). In the case of N = 2 this was further shown to be equivalent to using 2 This conjecture was subsequently proven for gcd(N, M ) = 1 in [38] and for generic (N, M ) (but in a particular limit of the regularisation parameters that are needed to render ZN,1 well defined) in[43].3We refer the reader to [39] for the details. 4 Starting from the partition function ZN,1, the free energy is defined as FN,1 = ln ZN,1. In contrast to that, F plet N,1 = Plog ZN,1 is defined with the help of the plethystic logarithm Plog(f (x)) = ∞ n=1 µ(n) n ln f (nx), where µ is the Möbius function. Physically, F plet N,1 receives contributions only from single-particle states. JHEP04(2021)275 Introduction Perturbative methods are one of the cornerstones in modern high energy physics. Besides their phenomenological applications, the study of scattering amplitudes and correlation functions in perturbation theory has revealed many interesting structures and symmetries. JHEP04(2021)275 In particular in the context of supersymmetric field theories and string theory, very efficient computational tools have been devised, which have led to very interesting new insights into these theories. Many of these tools have been inspired by a better understanding of underlying mathematical structures, notably number theoretical concepts. Non-perturbative aspects of field theories are more difficult to tackle directly. However, in the case of supersymmetric field theories in various dimensions, their intimate connection to string theory, along with dualities of the latter, have opened up other approaches. This has equally led to very interesting results in recent years, which have revealed further interesting structures and dualities of the underlying theories. In this paper we shall discuss an instance in which perturbative and non-perturbative aspects can be combined in a specific class of supersymmetric field theories in a somewhat unexpected fashion. This leads to a situation in which many of the above mentioned computational tools can be used at the same time to analyse these theories. To make the discussion concrete, we shall focus on a class of supersymmetric gauge theories that arise in the low energy regime of Little String Theories of A-type. Indeed, we shall elaborate on and largely extend a recent observation in [1] on the structure of their non-perturbative BPS partition function. Little String Theories (LSTs) [2][3][4][5][6][7][8] in general are quantum theories in six dimensions, whose spectrum includes extended (string-like) degrees of freedom in the UV. Such theories can be constructed from (type II) string theory via particular limits that decouple the gravitational sector while keeping the string length finite. On the one hand, this connection to full-fledged (type II) string theory (and in particular its various dual descriptions) opens up a very powerful means to explicitly study these theories. On the other hand, since LSTs can be viewed as a 'simplified version' of string theory, in which symmetries and structures may be easier to access, this may also teach us new lessons about string theory (or its dual descriptions) in return. Accordingly, various different constructions of LSTs have been explored, whose low energy field theory descriptions exhibit different gauge-and matter contents: LSTs follow an ADE classification and recent work [9,10] has focused on exploring the landscape of these theories in more depth, using similar methods from classifying superconformal field theories in 6 or less dimensions [11][12][13][14][15][16][17][18][19][20][21]. A class of LSTs of A-type can be constructed in M-theory via so-called BPS M-brane webs: these consist of N parallel M5-branes arranged on a circle 1 S 1 ρ and compactified on S 1 τ while probing a flat transverse space, with M2-branes stretched between them. These theories allow various different low energy limits [25][26][27] that give rise to supersymmetric gauge theories with different gauge structure. In the current work, we shall exclusively be concerned with the region in the parameter space, that describes a U(N ) gauge theory with matter in the adjoint representation. The BPS states of the M-brane system can be counted from the perspective of the one-dimensional intersections of the M2-and M5-branes (called M-strings) [28][29][30]. Indeed, the BPS partition function Z N,1 can be calculated as the equivariant elliptic genus [31] of an N = (0, 2) supersymmetric sigma model. Furthermore, there exist various dual descriptions [30] of these M5-brane configurations which allow for JHEP04(2021)275 generalised Eisenstein series as described in [45]. Despite being, as mentioned, based on a limited, a priori formal series expansion, this form of writing F plet N,1 is fully compatible with all the expected symmetries, notably modular transformations (with modular parameter ρ) as well as the non-perturbative symmetries discovered in [39]. Moreover, as discussed in [42] this result also exhibits the correct pole-structure as a function of the gauge parameters of the U(N ) low energy gauge theory that can abstractly be inferred from the partition function Z N, 1 . Analysing the form of F plet N, 1 proposed in [40,41] in the so-called unrefined limit (i.e. the limit in which Z N,1 captures the unrefined topological string partition function of X N,1 ) it was further observed in [1] that it can be re-written in a fashion that strongly resembles a Feynman diagrammatic decomposition. Focusing again on the examples N = 2, 3 and partially 4, it was argued that to leading instanton order (from the perspective of the low energy U(N ) gauge theory), F plet N,1 can be written in a way resembling N -point functions: the external states were given either by (coefficients of) the BPS counting function of the LST with N = 1 or a (quasi) Jacobi form that governs the BPS-counting of a single M5-brane coupling to two M2-branes. Furthermore, it was remarked that the effective couplings appearing in this decomposition were akin to modular graph functions (or more generally modular graph forms -see appendix A.3 for a very brief review) [46][47][48][49][50][51][52][53][54][55][56]. In particular, it was observed that the first non-trivial such coupling in the case of N = 2 is related to the (second derivative of the) Greens function of a free scalar field on a torus. Higher orders in the instanton expansion of F plet N,1 exhibit a similar pattern, however, new elements appear and no concrete pattern was put forward in [1]. The observation of [1] has linked an intrinsically non-perturbative quantity (the instanton one-particle free energy F plet N,1 of a supersymmetric gauge theory with U(N ) gauge theory) to a very simple scalar two-point function that is a fundamental building block in perturbative (in this case one-loop) scattering amplitudes of string theory. The purpose of this paper is to elaborate on this connection and in particular to give evidence for the complete structure at leading instanton order. Instead 5 of F plet N,1 we shall work with the (unrefined limit of the) full free energy F N,1 . By analysing in more detail the effective couplings (which have for the most part been written as infinite series in [1]) for N = 2, 3 and computing them for N = 4, we find that they can be written as combinations of (derivatives of) N scalar two-point functions. Based on these results, we propose a simple closed form for generic N , that fits all results that are available in the literature. From a physics perspective, we therefore provide strong evidence that the leading instanton contribution to the free energy of a class of U(N ) gauge theories with adjoint matter is entirely determined by two-point correlators of free scalar fields on a torus, as well as the leading instanton contribution of the free energy for N = 1, for which in fact a closed form expansion can be presented. Higher orders in the instanton expansion of F N,1 can be presented in a fashion that resembles higher (i.e. > N ) point functions, whose external states are still the same building blocks as to leading order. Furthermore, for many examples we present JHEP04(2021)275 evidence that the tentative coupling functions are still composed of scalar Greens functions: however, they now appear 'decorated', either through derivative operators or multiplied by holomorphic modular forms. For all the examples we have studied we show that the latter can in fact be re-written as a class of modular graph forms [47,48] with bivalent vertices. While the structures we observe point towards a unified picture, the fact that the fundamental building blocks appearing in the external states form an overcomplete basis (and thus introduce an intrinsic ambiguity in the decomposition of the 'Feynman graphs'), prevents us from conjecturing a closed form for higher instanton orders at this point. The rest of the paper is organised as follows: in section 2 we review the free energy of LSTs of A-type as well as the results of [1]. Due to the technical nature of the explicit computations, we present a detailed summary of our results in section 3. Sections 4, 5 and 6 present a detailed study of the examples N = 2, 3 and 4 respectively. Finally, section 7 contains our conclusions and an interpretation of our results. Details and definitions of modular objects, the precise definitions of the fundamental building blocks (along with their explicit expressions to low orders) that appear as external states, as well as technical details of some computations that have been deemed too long for the main body of the paper have been relegated to three appendices. Review: LST free energies and graphs In this paper we mostly follow the notation of [42] (see also [1,41]). Little String Theories (LSTs) of type A N can be described via F-theory compactified on a class of toric Calabi-Yau threefolds X N,1 , whose web diagram is schematically shown in figure 1. This diagram is doubly periodic, i.e. the horizontal lines labelled a as well as the diagonal lines labelled (1, . . . , N) are pairwise identified. The manifold X N,1 is parametrised by a total of N + 2 Kähler parameters. While various different bases for the latter can be chosen (see [23,24,36]), in this paper we shall follow [25,26,38] and use the parameters ( a 1 , . . . , a N , S, R), which are schematically shown in figure 1 as the (sum of) areas of certain curves of X N,1 represented by lines in the web diagram. 6 Starting from the web diagram in figure 1, the non-perturbative BPS partition function Z N,1 ( a 1,...,N , S, R; 1,2 ) of the LST is captured by the partition function of the topological string on X N,1 . The latter in turn can be computed in an algorithmic fashion using the (refined) topological vertex formalism. From this point of view, the parameters 1,2 appearing in Z N,1 are related to the coupling constant and the refinement of the topological 6 For completeness, we remark that in previous articles (see e.g. [22-25, 38, 39] for more details) we have used a different basis of Kähler parameters, namely (t1, . . . , tN , m, τ ), which is closer to a description of the system in terms of M5-and M2-branes, as explained in the Introduction: t1,...,N denote the distances between the M5-branes that are spread out on a circle of radius ρ = N i=1 ti and compactified on a circle of radius τ , while m corresponds to a mass deformation from the perspective of a certain low energy gauge theory description. As explained in [39,40] (see also [42]) the two bases are related through a linear transformation (with the identification tN+1 = t1) string. 7 From the perspective of the LST, they can be thought of as regularisation parameters that are necessary to render the non-perturbative partition function well defined. Indeed, in the context of the low-energy gauge theory description, they are identified with the deformation parameters of Nekrasov's Ω-background [64][65][66]. R = τ − 2N m + N ρ , S = −m + ρ , ai = ti+1 , ∀ i = 1, · · · , N . (2.1) JHEP04(2021)275 · · · Explicit expressions for Z N,1 for general N in terms of Jacobi theta functions have been given in [22][23][24]38]. In this paper, we will mostly be concerned with the free energy, which is directly related to Z N,1 in the following way Notice, following [37,42] (but unlike [1]) we consider the full free energy which is defined as the logarithm of Z N,1 . Our strategy in analysing the LST in this paper is to consider the series expansions 8 of F N,1 in the unrefined limit 1 = − 2 = . To this end, we define F N,1 ( a 1 , . . . , a N , S, R; , − ) = r Q r R P (r) N ( a 1 , . . . , a N , S, ) , with Q R = e 2πiR . (2.3) From the perspective of the low-energy U(N ) gauge theory, where R is related to the coupling constant, this corresponds to an instanton expansion and P (r) N can be interpreted 7 One way to define the refined A-model of a Calabi-Yau manifold X is via M-theory compactified on X ×S 1 ×TN, where the Taub-NUT space TN is twisted along S 1 thus giving rise to the two parameters 1,2. The topological partition function can then be directly computed by counting BPS-states in M-theory [57][58][59]. See also [60][61][62][63] for a world-sheet definition of the refined topological string and an interpretation of 1,2. 8 Although a prior only a formal expansion, it was pointed out in [42] that each term in this series is made up from (well-defined) quotients of (derivatives of) Jacobi theta functions. JHEP04(2021)275 as the r-instanton free energy. For further convenience, we also define the -expansion 9 (S) , with Q a i = e 2πi a i . (2.6) For ease of writing we shall also use the shorthand notation n = {n 1 , . . . , n N }. In [41] (see also [42]) the following decomposition has been introduced (ρ, S) + n H (r),n (s) (ρ, S) Q n 1 a 1 . . . Q n N a N , with ρ = n i=1 a i ,(2.H (r),{n 1 ,...,n N } (s) (ρ, S) = ∞ =0 Q ρ P (r),{n 1 + ,n 2 + ,...,n N + } N,(s) (S) , with Q ρ = e 2πiρ = N i=1 Q a i . (2.8) Since they will be important building blocks in many computations in this paper, we discuss the functions H In [1] non-trivial evidence has been provided that the P where we have implicitly used ρ = N i=1 a i . 10 Here H (1),{0} (s) and W (1) (0) are respectively the expansion coefficients of the free energy for N = 1 and the function W (ρ, ), which are reviewed in detail in appendix B (where also explicit expressions (for some low values of s) are given). Physically, the latter has appeared in previous works and was shown 9 To make contact with the coefficients P (2.4) 10 As remarked before, the proposition in [1] was strictly speaking for a reduced free energy, in which the logarithm in (2.2) is replaced by the plethystic logarithm where µ(n) is the Möbius function. To order O(QR), however, only the term n = 1 can contribute. Since, µ(1) = 1, this implies that to this order F plet N,1 and FN,1 are in fact identical, such that the results of [1] for r = 1 directly carry over to our current setup. As we shall see below, this is no longer the case for r > 1. to be related to the BPS counting of configurations of M5-branes with single M2-branes ending on them on either side [22,24]. It has also recently appeared in [42], in the context of extracting Hecke structures in the spectrum of the LST free energy in the Nekrasov-Shatashvili limit. Finally, the functions O (N ),α are independent of S (and s) and encode the only dependence of P (r=1) N,(s) on ( a 1 , . . . , a N −1 ). They were schematically written in [1] as JHEP04(2021)275 · · · W (1) (0) W (1) (0) W (1) (0) ( N − 1 − α ) t i m e s H (1),{0} (0) H (1),{0} (0) H (1),{0} (0) α t i m e s · · · H (1),{0} (s) O (N ),αO (N ),α ( a 1,...,N ) = ∞ n 1 ,...,nα=1 p α (n 1,...,nα )Λ α ( a 1,...,N , n 1,...,α ) α a=1 1 − Q t α (n 1,...,α ) ρ . (2.11) We refer the reader to sections 4, 5 and 6 for explicit examples for N = 2, 3, 4 respectively. In (2.11), the p α denote homogeneous polynomials of order α in the summation variables n 1,...,α , t α are linear functions in n 1,...,α and the Λ α are rational functions of the Q a 1,...,N . Graphically, eq. (2.9) was presented in [1] in a way shown in figure 2: the building blocks H and W (1) (0) are interpreted as external states in a (Feynman)diagram, where the O (N ),α play the role of effective couplings. This interpretation was justified by the observation that the first non-trivial such function (appart from O (N ),0 , which (based on several examples) were conjectured to be equal to N ) O (2),1 is related to the Green's function of a free scalar field on the torus and thus indeed represents a 2-point function. The free energy F plet N,1 to order Q r R with r > 1 was also analysed in [1] for N = 2, 3 and it was concluded that at least partially it also allows for a decomposition in the building blocks H and W (r) (s) . However, also new structures emerge, which were difficult to interpret conceptually. In the current work, we repeat the analysis for the full free energy F N,1 to higher orders in Q R . Summary of results The goal of this paper is two-fold: on the one hand side, we analyse in more detail the coupling functions O (N ),α that appear to leading instanton order. On the other hand, we JHEP04(2021)275 • • • • • • b 0 = 0 b 1 = a 1 b 2 = a 1 + a 2 b 3 = a 1 + a 2 + a 3 b −1 b = a 1 + . . . + a b +1 b +2G ( b − b j ; ρ) + 2πi ρ−ρ . consider the P (r) N,(s) for r > 1 which stem from the full free energy F N,1 (rather than F plet N,1 ) and provide evidence in several examples (i.e. N = 2, 3 and 4) that they similarly exhibit a decomposition that resembles effective couplings. Since our arguments are based on series expansions of simple examples, which are rather technical to some extent, we provide in this section a brief summary of our observations. To order O(Q R ) we provide evidence that the observation in [1] (namely that O (2),1 is given by the derivative of the free scalar two-point function on the torus) can be generalised as follows . Therefore, the product over j contains precisely α scalar Greens functions. While the notation in (3.1) is somewhat involved, a term for fixed S can be represented in a very simple graphical form, as shown in figure 3. Further (more concrete) examples for N = 2, 3, 4 can be found in subsequent sections. O (N ),α ( a 1,...,N −1 , ρ) = 1 (2π) 2α N −1 =0 S⊂{0,...,N −1}\{ } |S|=α j∈S G ( b − b j ; ρ) + 2πi ρ −ρ ,(3. To order O(Q r R ) for r > 1, the structure of the free energy P and W (r) (s) are an over-complete basis renders the decomposition ambiguous (in particular for higher values of s). Nevertheless we observe that tentative coupling functions appearing in such decompositions lend themselves to an interpretation as corrected couplings O (N ),1 with additional internal points. We provide structural evidence for this interpretation by the following observations • The coupling functions for r > 1 are still combinations of (derivatives of) scalar twopoint functions multiplied by modular forms. In the cases of low s and r, where the intrinsic ambiguity is limited, we observe that the latter can be arranged in such a way as to render the couplings holomorphic • The modular forms that appear in this process are in fact (holomorphic) graph forms. In the case of N = 2 and (r, s) = (2, 0), in which case the ambiguity of the decomposition is still under control, we can in fact show that the appearing graph form can be related through Cauchy-Riemann differential operators to a graph function, which in turn can be written as the double integral over two scalar two-point functions. Such terms would indeed be interpreted as disconnected contributions in string one-loop amplitudes. However, while very intriguing, the intrinsic ambiguity in the decomposition of the free energy, prevents us from making this observation more precise at the current time. Example N = 2 Decomposition at order O(Q R ) and scalar correlators The simplest example is the case N = 2. As was already argued in [1] to order O(Q R ), the free energy 11 P (r=1) 2,(s) ( a 1 , S) can be written as the sum of two terms 12 P (r=1) 2,(s) ( a 1 , ρ, S) = H (1),{0} (s) W (1) (0) O (2),0 + H (1),{0} (s) H (1),{0} (0) O (2),1 ( a 1 , ρ) , (4.1) where the two coupling functions are given explicitly as O (2),0 = 2 , and O (2),1 ( a 1 , ρ) = − ∞ n=1 2n 1 − Q n ρ Q n a 1 + Q n ρ Q n a 1 . (4.2) Graphically, the two terms appearing in (4.1) can be represented as shown in figure 4, in the form of two-point functions. In this presentation, H (1),{0} (s) , H (1),{0}(0) and W (1) (0) are attached to the external legs, while O (2),0 and O (2),1 are (effective) couplings. In [1] it was furthermore shown that O (2),1 can be written in the following fashion O (2),1 ( a 1 , ρ) = 2 (2π) 2 ℘( a 1 ; ρ) + π 2 3 E 2 (ρ) = 2 (2π) 2 G ( a 1 ; ρ) + 2πi ρ −ρ . (4.3) 11 This is in fact the same term that appears at order O(QR) in F plet 2,1 as defined in eq. (2.10). 12 For ease of writing we shall drop the arguments (ρ, S) appearing in the building blocks H JHEP04(2021)275 W (1) (0) H (1),{0} (s) O (2),0 (a) H (1),{0} (0) H (1),{0} (s) O (2),1 (b) Figure 4. Diagrammatic expansion of P (r=1) 2,(s) in (4.1) in terms of two-point functions: (a) H (1),{0} (s) coupling to W (1) (0) through O (2),0 ; (b) H (1),{0} (s) coupling to H (1),{0} (0) through O (2),1 . b 0 = 0 b 1 = a 1 (a) b 0 = 0 b 1 = a 1 (b)b i , b j ) represent factors of G ( b i − b j ; ρ) + 2πi ρ−ρ . Here ℘(z; ρ) is Weierstrass' elliptic function, E 2 is the second Eisenstein series (see appendix A.1 for the definitions) and G(z; ρ) is the Green's function of a free scalar field on a torus. Another way of arguing for the result (4.3), which shall be relevant for similar computations in subsequent sections, is presented in appendix C. We can combine (4.2) and (4.3) into O (2),α ( a 1 , ρ) =    2 for α = 0 , 1 (2π) 2 1 =0 j = G ( b − b j ; ρ) + 2πi ρ−ρ for α = 1 , = 1 (2π) 2i 1 =0 S⊂{0,1}\{ } |S|=i j∈S G ( b − b j ; ρ) + 2πi ρ −ρ ,(4.4) where b 0 = 0 and b 1 = a 1 , while |S| in the last relation denotes the cardinality of the set S. Here we have also used that G (z; ρ) = G (−z; ρ). Graphically, the combinations of scalar Greens functions appearing in (4.4) can be represented as shown in figure 5, where dashed lines represent factors of G ( b i − b j ; ρ) + 2πi ρ−ρ . Decomposition at order O(Q r R ) for r > 1 To order O(Q r R ) for r > 1, the expansions of F 2,1 and F plet 2,1 are different, such that the results of [1] cannot be directly carried over. We therefore consider below in detail the cases r = 2, 3. JHEP04(2021)275 Contributions at order O(Q 2 R ) We have analysed P (r=2) 2,(s) as a series expansion in Q ρ to order 10 up to s = 4. The results we find are compatible with a decomposition of the form P (2) 2,(s) ( a 1 , ρ, S) = 4 j=0 (φ −2,1 (ρ, S)) j (φ 0,1 (ρ, S)) 4−j f j,(r=2) (s) (ρ, S) (4.5) + ∞ n=1 Q n a 1 + Q n ρ Q n a 1 s+2 k=0 n 2k+1 1 − Q n ρ g j,k,(r=2) (s) (ρ, S) . Here f j,(r=2) (s) are quasi modular forms of weight 2(s + j − 1) and g j,k,(r=2) (s) are modular forms of weight 2(s + j − 2 − k). Explicit expressions for f j,(2) (s) and g j,k,(2) (s) can be given as combinations of Eisenstein series, for example for s = 0 we find f 0,(2) s=0 =0, f 1,(2) s=0 =− 1 4608 , f 2,(2) s=0 =− E 2 1152 , f 3,(2) s=0 =− E 4 1152 , f 3,(2) s=0 = E 6 −E 4 E 2 144 , g 0,0,(2) s=0 =0, g 1,0,(2) s=0 =0, g 2,0,(2) s=0 =− 1 96 , g 3,0,(2) s=0 =0, g 3,0,(2) s=0 =− E 2 12 , g 0,1,(2) s=0 =0, g 1,1,(2) s=0 =0, g 2,1,(2) s=0 =0, g 3,1,(2) s=0 =− 1 12 , g 3,1,(2) s=0 =0, g 0,2,(2) s=0 =0, g 1,2,(2) s=0 =0, g 2,2,(2) s=0 =0, g 3,2,(2) s=0 =0, g 3,2,(2) s=0 =− 1 24 . (4.6) For s = 0, these explicit expressions afford the following decomposition in terms of the basic building blocks H (2),{0} (0) : 13 P (2) 2,(0) ( a 1 , ρ, S) = 2 3 H (2),{0} (0) W (2) (0) O (2),0 + 4 3 H (2),{0} (0) H (2),{0} (0) O (2),1 ( a 1 , ρ) (4.7) − 1 48 H (1),{0} (0) 4 [dE 4 (ρ) + 4E 4 (ρ) I 0 (ρ, a 1 ) + 2I 2 (ρ, a 1 )] + 4 3 H (1),{0} (0) 2 H (2),{0} (0) I 1 (ρ, a 1 ) , where we defined the differential operator d := Q ρ d dQρ and I k is defined in (A.10). For generic k ∈ N ∪ {0}, the latter can be written as derivatives of I 0 (see (A.11)), which in turn is related to the (derivative of the) scalar Greens function G(ρ, a 1 ) (A.16). In view of the discussion of the free energy to order O(Q R ), it is tempting to give the following interpretation of the various terms appearing in (4.7): 1. Up to different numerical prefactors, the terms in the first line of (4.7) correspond schematically to the same two-point functions as in figure 4, except that the external states have been replaced by their counterparts for r = 2. 2. The first term in the second line of (4.7) has the appearance of a 4-point function (since it contains 4 powers of H (1),{0}(0) ). However, the tentative coupling-function (i.e. without the four 'external states' H (1),{0} (0) ) O (2),4-pt r=2,s=0 ( a 1 , ρ) = − 1 48 [dE 4 (ρ) + 4E 4 (ρ) I 0 (ρ, a 1 ) + 2I 2 (ρ, a 1 )] ,(4. 8) 13 This decomposition is unique if we assume only terms involving H (2),{0}(0) , but not W (0) . This tentative interpretation is supported by two further (somewhat circumstantial) pieces of evidence: JHEP04(2021)275 H (1),{0} (0) H (1),{0} (0) H (1),{0} (0) H (1),{0} (0) O (2),1 (a) H (1),{0} (0) H (1),{0} (0) H (1),{0} (0) H (1),{0} (0) (b) (i) The effective coupling in (4.8) naturally decomposes into the sum of two terms O (2),4-pt r=2,s=0 ( a 1 , ρ) = O (2),4-pt,1 r=2,s=0 + O (2),4-pt,2 r=2,s=0 ,(4.9) where each term stems from one of the two diagrams in figure 6 and where we defined O (2),4-pt,1 r=2,s=0 ( a 1 , ρ) = − 1 48 [dE 4 (ρ) + 4E 4 (ρ) I 0 (ρ, a 1 )] = E 4 24 O (2),1 ( a 1 , ρ) − 1 48 dE 4 = 1 24 E 4 O (2),1 ( a 1 , ρ) − quasi-holomorphic , O (2),4-pt,1 r=2,s=0 ( a 1 , ρ) = − 1 24 I 2 (ρ, a 1 ) = 1 48 D 4 a 1 O (2),1 ( a 1 , ρ) . (4.10) Here we used the shorthand notation D a 1 = 1 2πi ∂ ∂ a 1 . We note in particular, that these couplings are still composed of (combinations of) scalar two-point functions G ( b i − b j ; ρ)+ 2πi ρ−ρ , which are 'decorated' either through multiplication with the JHEP04(2021)275 E4 b 0 = 0 b 1 = a 1 (a) b 0 = 0 b 1 = a 1 (b) Figure 7. Diagrammatic presentation of the couplings in (4.10). Dashed lines with end-points . Furthermore, the term 'quasi-holomorphic' in (4.10) denotes the subtraction of all terms containing the quasi-holomorphic form E 2 appearing in an expansion of E 4 O (2),1 (which is implicitly assumed in figure 7 (a)). Indeed, using (A.8) as well as (4.3) along with the expansion (A.13) of the Weierstrass function we find ( b 0 , b 1 ) represent factors of G ( a 1 ; ρ) + 2πi ρ−ρ ,O (2),4-pt,1 r=2,s=0 ( a 1 , ρ) = E 4 (ρ) 48π 2 a 2 1 + E 6 (ρ) 144 + E 4 (ρ) 48π 2 ∞ k=1 2(2k + 1)ζ(2k + 2)E 2k+2 (ρ) a 2k 1 , (4.11) which only depends on the holomorphic Eisenstein series E 2k (with k > 1), but not E 2 . (ii) The holomorphic Eisenstein series E 4 appearing in the bubble in figure 7 (a) (and which tentatively corresponds to the disconnected contribution in figure 6 (a)) is in fact (proportional to) a (holomorphic) graph form [48] that is related to a graph with two bivalent vertices C 4 0 0 0 (ρ) = 2 ζ(4) E 4 (ρ) • • (4, 0) (0, 0) (4.12) We refer to appendix A.3 as well as to [47,48] (see also [46,[49][50][51][52][53][54][55]) for the definition and our conventions and notation. As was explained in [48], graph forms of the type C 2k 0 0 0 in (4.12) can be related to C k 0 k 0 through the action of Cauchy-Riemann differential operators by using the general relation (A.27) C 4 0 0 0 (ρ) = 1 3! (Imρ) 4 ∇ 2 (Im(ρ)) 2 C 2 0 2 0 (ρ) , with ∇ = 2i (Imρ) 2 ∂ ∂ρ . (4.13) Using furthermore (A.28) we have C 2 0 2 0 (ρ) = C 1 1 1 1 (ρ) = Σ d 2 z 1 Imρ Σ d 2 z 2 Imρ π (Imρ) G(z 1 − z 2 ; ρ) 2 . (4.14) JHEP04 (2021)275 Here C 1 1 1 1 (ρ) is a dihedral modular graph function that can be written as the torus integral over two insertion points and whose integrand is proportional to two powers of the scalar two-point function. Therefore indeed, the bubble in figure 6 (a) can be interpreted as (a differential operator acting on) a disconnected diagram with two integrated positions which are contracted through scalar Greens functions. H (1),{0} (0) H (1),{0} (0) H (2),{0} (0) (a) H (1),{0} (0) H (2),{0} (0) H (1),{0} (0) (b); (b) internal state is given by H (1),{0} (0) . b 0 = 0 b 1 = a 1 3. The last term in the second line of (4.7) has the appearance of a 3-point function (since it contains 3 powers of H (1),{0}(0) ). Graphical representations of such terms are shown in figure 8, which differ by a choice of external and internal points. Just as in the discussion of the 4-point functions, the internal states simply modify the tentative coupling functions O (2),3-pt r=2,s=0 ( a 1 , ρ) = 4 3 I 1 (ρ, a 1 ) = − 8 3 D 2 a 1 O (2),1 ( a 1 , ρ) . (4.15) Graphically, both terms in figure 8 contribute to the this coupling, which itself can be presented as in figure 9. As before, also this coupling can be written as a D 2 a 1 derivative (represented by the cross in figure 9) of the scalar 2-point function on the torus. While it is quite interesting from the perspective of the results we have obtained to order O(Q R ) that such a decomposition of the free energy to order O(Q 2 R ) exists, the latter is still quite speculative. In order to add more credibility to this proposed interpretation as corrections to the two-point function it would be important to also analyse P + 4 3 H (1),{0}(0)2 I 2 + H (1),{0}(0)+ 8 45 + 32β H (1),{0} (0) H (1),{0}(1)H (2),{0}(0)I 1 + 52 45 − 32β H (1),{0} (0) 2 H (2),{0}(1)I 1 , where α, β and γ are undetermined constants. As is evident from this expression, these latter constants not only impact the tentative modified coupling functions that appear for the various 3-point and 4-point functions, but also any potential recurring patterns. Furthermore, since the various H now also appear with s = 0 or s = 1, there are many more choices which of these states are assigned to be 'internal' and which are 'external'. These ambiguities make a precise statement difficult at this point. However, we point out the following observations, which to some extent support the tentative interpretation given in this subsection: • We have checked up to s = 4 that P to cover the S-dependence. • We have checked up to order s = 4 that the numerical factors can always be chosen in such a way that the only terms that would enter into potential higher-point coupling functions are of the form dE 2k + 2kE 2k I 0 , ∀k > 1 or I n ∀n > 0 , or E 2k I n ∀n > 0 , ∀k > 1 . (4.17) All the couplings appearing up to s = 4 are holomorphic in the sense that they can be expanded in a power series in a 1 , where the individual coefficients are combinations of the Eisenstein series E 4 and E 6 only (but not E 2 ). • The Eisenstein series E 2k multiplying I n appearing in (4.16) as written in (4.17) can (as in the case r = 1) be interpreted as dihedral modular graph forms C 2k 0 0 0 (ρ) in (A.31) of graphs with two bivalent vertices. It is therefore again tempting to once more interpret them as disconnected contributions. JHEP04(2021)275 Contributions at order O(Q 3 R ) For completeness, we also briefly discuss the decomposition of the free energy to order O(Q 3 R ). Indeed, P 2,(0) can be presented in the form P (3) 2,(0) ( a 1 ,ρ,S)= 3 4 H (3),{0} (0) W (3) (0) O (2),0 + 9 4 H (3),{0} (0) H (3),{0} (0) O (2),1 ( a 1 ,ρ) + H (1),{0}(0)+ H (1),{0} (0) 2 H (2),{0} (0) 2 − 2(dE 4 +4E 4 I 0 ) 27 +βI 2 − H (1),{0} (0) 3 H (3),{0} (0) 1+ 27β 16 I 2 + H (2),{0} (0) 3 αI 1 +H (1),{0} (0) H (2),{0} (0) H (3),{0} (0) 16 3 − 27α 16 E 6 I 1 ,(4.18) where α and β are two undetermined constants. In the same spirit as to order O(Q 2 R ), the terms in the second to fifth line represent corrections to two-point functions with 4, 3, 2 or 1 internal points respectively. Due to the ambiguity in the decomposition in terms of the building blocks H , it is not possible to determine precisely the arising corrected couplings. However, already the fact that P 2,(0) allows a decomposition in terms of only these particular building blocks, can be seen as a further argument for the picture already developed to order O(Q 2 R ). Example N = 3 After the case N = 2, we shall now discuss Little String Theories with N = 3. Decomposition at order O(Q R ) and scalar correlators While it was already argued in [1] that the free energy P α ( a 1 , a 2 , ρ) was (mostly) only given as infinite sums O (3),0 =3, (5.2) O (3),1 ( a 1 , a 2 ,ρ)= ∞ n=1 −2n 1−Q n ρ Q n a 1 + Q n ρ Q n a 1 +Q n a 2 + Q n ρ Q n a 2 +(Q a 1 Q a 2 ) n + Q n ρ (Q a 1 Q a 2 ) n , (5.3) O (3),2 ( a 1 , a 2 ,ρ)= ∞ n=1 n 2 (1−Q n ρ ) 2 Q n ρ Q n a 1 +Q n a 2 + Q n ρ (Q a 1 Q a 2 ) n + (Q a 1 Q a 2 ) n + Q n ρ Q n a 1 + Q n ρ Q n a 2 + ∞ n 1 ,n 2 =1 n 2 (2n 1 +n 2 ) (1−Q n 1 ρ )(1−Q n 2 ρ ) + (n 1 +n 2 )(n 1 −n 2 ) (1−Q n 1 ρ )(1−Q n 1 +n 2 ρ ) Q n 1 +n 2 a 1 Q n 1 a 1 +Q n 1 a 1 Q n 1 +n 2 a 1 + Q n 1 +n 2 ρ Q n 1 +n 2 a 1 Q n 2 a 2 + Q n 1 +n 2 ρ Q n 1 a 1 Q n 1 +n 2 a 2 + Q n 1 ρ Q n 2 a 1 Q n 1 a 2 + Q n 1 ρ Q n 2 a 2 Q n 1 a 1 . (5.4) The coupling O (3),0 can in a trivial manner be represented as in figure 10: it simply corresponds to no correlator connecting any of the three points b 0 = 0 , b 1 = a 1 , b 2 = a 1 + a 2 ,(5.O (3),1 ( a 1,2 , ρ) = 1 (2π) 2 3 =1 j = G ( b − b j ; ρ) + 2πi ρ −ρ . (5.6) Graphically, up to an overall numerical factor 2 (2π) 2 , this coupling can be represented as in figure 11 as a single two-point function connecting any combination of the points { b 0 , b 1 , b 2 }. Finally, the coupling function O (3),2 ( a 1 , a 2 , ρ) is more complicated to analyse than the previous ones due to the presence of the double summation in the second term in (5.4). Following the discussion in [41], however, we can represent it in terms of generating functions T (z 1 , z 2 ; ρ) (see eq. (A.32)) of multiple divisor sums up to length 2, which were introduced in [44]. Since these functions have a well-defined Taylor series expansion 15 (A.33), we can compute the series expansion of O (3),2 ( a 1 , a 2 ; ρ) in powers of a 1,2 . However, in order to 14 Compared to a slightly different expression in [1], here we have used the implicit periodicity of G (z; ρ) under a shift z → z + ρ along with the symmetry G (z; ρ) = G (−z; ρ). 15 In fact, this is the form in which the T have originally been introduced in [44]. Figure 11. Diagrammatical representation of the coupling O (3),1 in (5.6). The dashed lines represent the two-point function JHEP04(2021)275 b 0 b 1 b 2 + b 0 b 1 b 2 + b 0 b 1 b 2G ( b − b j ) + 2πi ρ−ρ . b 0 b 1 b 2 + b 0 b 1 b 2 + b 0 b 1 b 2( b − b j ) + 2πi ρ−ρ . extract the correct pole structure and also due to the complexity of (5.4), we limit ourself to study the leading powers in a 1,2 as a limited power series in Q ρ which (assuming modularity of the various contributions) can be expressed as polynomials in the Eisenstein series. The relevant details can be found in appendix C.2, where we find that the series expansion matches the combination (C.19) of scalar Greens functions which can also be written in the form O (3),2 ( a 1 , a 2 , ρ) = 1 (2π) 4 2 =0 j = G ( b − b j ; ρ) + 2πi ρ −ρ . (5.7) Yet another way to present this is result is O (3),2 ( a 1 , a 2 , ρ) = 1 (2π) 4 2 =0 S∈{0,1,2}\{ } |S|=2 j∈S G ( b − b j ; ρ) + 2πi ρ −ρ ,(5. Decomposition at order O(Q r R ) for r > 1 We can repeat the analysis of the previous section for contributions to the free energy to orders Q r R for r > 1. As in the case for N = 2, however, we shall encounter the problem that a decomposition in the building blocks H and W (r) (s) and certain (modular) coupling functions. In order to present this form despite the above mentioned ambiguities, we use a condensed notation: starting from P (r=2) 3,(s=0) = 3 a=0 k i 1 ,i 2 ,i 3 ,i 4 c (a) k;i 1 ,i 2 ;i 3 ,i 4 (ρ)K k a ( a 1,2 ,ρ) H (1),{0} (0) i 1 H (2),{0} (0) i 2 W (1) (0) i 3 W (2) (0) i 4 , for a = 0, k can only take the value k = 1 with the coupling K k=1 0 ( a 1,2 , ρ) = 1 and the following non-vanishing coefficient functions c (0) 1;6,0;0,0 = (2 − 90α 1 − 81α 19 − 36α 2 + 18α 3 )E 6 5184 , c(0)(72α 1 + 108α 19 + 48α 2 − 24α 3 − 1)E 6 72 , c (0) 1;3,1;1,0 = (−162α 19 − 36α 2 − 1) E 6 54 , c(0) 1;3,0;1,1 = (576α 1 + 3α 16 For a = 1 there are three different types of graphs K k 1 ( a 1,2 , ρ) (i.e. k ∈ {1, 2, 3}) and the corresponding coefficient functions c (1) k;i 1 ,i 2 ;i 3 ,i 4 (ρ) are tabulated in table 1. Here we are using a symbolic notation for the K k 1 ( a 1,2 , ρ): solid points represent ( b 0 , b 1 , b 2 ) and a sum over all cyclic permutations is understood. Furthermore, dashed lines with n crosses between any of these points represent factors of D 2n + 1296α 17 + 432α 18 − 576α 3 + 4) E 4 288 , c (0) 1;2,2;0,0 = α 19 E 6 , c (0) 1;2,1;0,1 = (−1152α 1 + 3α 16 + 1296α 17 − 2304α 2 + 1152α 3 − 16) E 4 864 , c(0)b i G ( b i − b j ; ρ) + 2πi ρ−ρ , such that e.g. 16 I.e. we study limited series expansions in Qρ, which we match to quasi-modular forms, as explained in appendix C. Similarly, for a = 2, we have four different K k 2 ( a 1,2 , ρ) (for which we use the same condensed graphical representation) such that k ∈ {1, 2, 3, 4}. The coefficients c (2) k;i 1 ,i 2 ;i 3 ,i 4 (ρ) are tabulated in table 2. Finally, for a = 3, there are four different K k 3 ( a 1,2 , ρ) (for which we use the same condensed graphical representation) and therefore k ∈ {1, 2, 3, 4}. The coefficients c (3) k;i 1 ,i 2 ;i 3 ,i 4 (ρ) are tabulated in table 3. The parameters α 1,...,19 appearing in eq. (5.9) and tables 1, 2 and 3 are undetermined constants and label the ambiguity in the decomposition of P Determining precise coupling functions, however, is very difficult, due to the many undetermined parameters α i . We remark, however, that the fact that all of the K k=1 a ( a 1,2 , ρ) can again be written entirely as combinations of 'decorated' scalar Greens functions, is in line with the interpretation we have given in the case of N = 2. In particular, just as before, these 'decorations' again consist of differential operators D 2 a or of multiplication with holomor- phic Eisenstein series that can be interpreted as dihedral modular graph forms C 2k 0 0 0 (ρ) in (A.31) and can thus be interpreted as disconnected contributions. K 2 1 ( a 1,2 , ρ) = = G (4) ( b 0 − b 1 ; ρ) + G (4) ( b 1 − b 2 ; ρ) + G (4) ( b 2 − b 0 ; ρ) .(20−576α1−9α10+72α3)E4 864(2π) 4 (1152α1+288α2−144α3−27α9−16)E4 2593(2π) 4 0 0 0 − 12α3 (2π) 4 α10 (2π) 4 0 2(12α3−α10−8α2) (2π) 4 α9 (2π) 4 0 0 2(8+3α10+96α2−72α3−9α9) 9(2π) 4 0 0 2(48α3+9α9−96α2−8) 27(2π) 4 0 0 3 (184−5184α1−9α13−864α14)E 2 4 82944(2π) 2 (11−216α1+36α3)E6 216(2π) 2 (216Eα1+72α2−36α3−5)E6 324(2π) 2 (36−288α1−α12+192α3)E4 96(2π) 2 α14E4 (2π) 2 (384α1+256(α2−α3)−α11−4α13−192α14)E4 96(2π) 2 0 0 (768α3+576α14−1152α2−1536α2−3α11−16)E4 864(2π) 2 0 0 α12 (2π) 2 α13 (2π) 2 0 −2α12 (2π) 2 α11 (2π) 2 0 2(α12−4−3α11−6α13) 3(2π) 2 16+3α11+12α13 9(2π) 245α 1 +9α 3 −2 18(2π) 10 0 0 0 3 (10−288α 1 −9α 5 )E 4 864(2π) 6 0 2(2+48α 2 −24α 3 +3α 5 ) 9(2π) 6 0 − 12α 3 (2π) 6 2(8α 3 −8α 2 +α 5 ) (2π) 6 0 0 0 α 5 (2π) 6 0 0 4 (11−216α 1 )E 6 864(2π) 4 (96α 1 +64α 2 −32α 3 −α 6 −4)E 4 96(2π) 4 0 2(8+3α 6 ) 9(2π) 4 0 0 2(α 7 −3α 6 −2) 3(2π) 6 − 2α 7 (2π) 6 (18−144α 1 +48α 3 −α 7 )E 4 96(2π) 4 0 α 6 (2π) 6 α 7 (2π) 6JHEP04(2021)275 k K k 3 ( a 1,2 , ρ) c(0 0 0 0 0 2 45α 1 −2 9(2π) 10 0 0 0 0 0 3 0 8(α 3 −2α 2 ) (2π) 8 0 − 12α 3 (2π) 8 0 0 4 12−96α 1 −α 4 ) 96(2π) 6 E 4 2(3α 4 −4) 9(2π) 6 0 0 − 2α 4 (2π) 6 α 4 (2π) 6 Example N = 4 Repeating the above discussion for the case N = 4 is much more difficult due to the increased complexity of the free energy even for r = 1. Nevertheless we can report some non-trivial results to order O(Q R ), which are in line with the general picture advocated above: as was argued in [1] to this order the free energy can be decomposed as Implicit expressions for the couplings O (4),α for α = 0, 1, 2, 3 as infinite series can be inferred from the expansions of the free energy presented in [41]. These series representations can be analysed with the same methods outlined in appendix C. For α = 0, 1, 2 we have checked up to order O( a 6 1,2,3 ) that they are compatible with Figure 13. Diagrammatical representation of the coupling O (4),1 in (6.2). The dashed line represent the two-point function O (4),0 = 4 , O (4),1 ( a 1,2,3 , ρ) = 1 (2π) 2 4 =1 j = G ( b − b j ) + 2πi ρ −ρ , (6.2) O (4),2 ( a 1,2,3 , ρ) = 1 (2π) 4 4 =1 j 1 =j 2 j 1 = =j 2 G ( b − b j 1 ) + 2πi ρ −ρ G ( b − b j 2 ) + 2πi ρ −ρ , JHEP04(2021)275 b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3G ( b − b j ) + 2πi ρ−ρ . where we have introduced the points b 0 = 0 , b 1 = a 1 , b 2 = a 1 + a 2 , b 3 = a 1 + a 2 + a 3 . (6.3) For O (4), 3 , it was remarked in [41] that the coefficients for the free energy for the case N = 4 have been matched up to the maximal order O(Q 5 ρ ), but may receive additional corrections beyond that. This order allows us to unambiguously only determine the leading singularity of O (4),3 , for which we find after a lengthy computation 17 O (4),3 ( a 1,2,3 , ρ) = 1 (2π) 6 1 a 2 1 ( a 1 + a 2 ) 2 ( a 1 + a 2 + a 3 ) 2 + 1 a 2 1 a 2 2 ( a 2 + a 3 ) 2 (6.4) + 1 a 2 2 a 2 3 ( a 1 + a 2 ) 2 + 1 a 2 3 ( a 2 + a 3 ) 2 ( a 1 + a 2 + a 3 ) 2 + O( a −4 1,2,3 ) . This pole-structure is compatible with the closed form expression O (4),3 ( a 1,2,3 , ρ) = 1 (2π) 6 4 =1 j 1 <j 2 <j 3 j 1,2,3 = 3 a=1 G ( b − b ja ) + 2πi ρ −ρ . (6.5) The form of the couplings O (4),α in (6.2) and (6.5) is compatible with the general form (3.1). The (non-trivial) couplings O (4),1,2,3 can also be graphically presented: up to numerical factors O (4),1 in figure 13 corresponds to all possible ways to contract two out of the 4 points b 0,1,2,3 with a single two-point function G ( b − b j ) + 2πi ρ−ρ . The coupling O (4),2 in figure 14 corresponds to all possible ways to contract one out of the 4 points b 0,1,2,3 with two distinct other points through one two-point function respectively. Finally, the coupling O (4),3 in figure 15 corresponds to all possible ways to contract one of the 4 points with all other points through two-point functions. Due to the high complexity and the large amount of intrinsic ambiguity, we refrain from discussing a decomposition of the N = 4 free energy to higher orders in O(Q R ). We leave this problem for further work. Conclusions In this paper we have continued and extended the study of [1] to decompose the free energy of LSTs of type A N . In contrast to [1], we have considered the full free energy that counts 17 O (4),3 receives contributions from all coefficients listed in [41], which involve up to 3 infinite series with various powers of Q a 1,2,3 . We refrain from presenting the details of this calculation. Figure 14. Diagrammatical representation of the coupling O (4),2 in (6.2). The dashed lines represent the two-point function JHEP04(2021)275 b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3G ( b − b j ) + 2πi ρ−ρ . b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 + b 0 b 1 b 2 b 3 JHEP04(2021)275 all BPS states (including multi particle states), which does not impact the results of [1] to leading instanton order (from the perspective of the low energy U(N ) gauge theory), but streamlines some of the results to higher order in Q R . We have studied the examples N = 2, 3 and 4 which exhibit clear repeating patterns, that we thus conjecture to hold in general: to leading order O(Q R ), it was already argued in [1] that the free energy can be presented in a way that resembles a Feynman diagrammatic expansion, which is schematically shown in figure 2. The external states are given by the building blocks H (1),{0}(s) and W (1) (0) , which are the expansion coefficients of the free energy for N = 1 as well as the leading term in the expansion of the quasi-Jacobi form W (ρ, S, 1 ) defined in (B.4). In the current paper, we have analysed in detail the effective couplings O (N ),α (for N = 2, 3 and 4) appearing in this decomposition and have provided evidence that they can be entirely written as combinations of (derivatives of the) scalar two-point function of a free scalar field φ on the torus with the general conjectured form given in (3.1). The latter can in fact be also written as a correlation function in the following form O (N ),α ( a 1,...,N −1 , ρ) = (−1) α (2π) 2α α! N −1 =0 S⊂{0,...,N −1}\{ } |S|=α : (∂φ) α ( b ) : : j∈S ∂φ( b j ) : (7.1) = N −1 =0 S⊂{0,...,N −1}\{ } |S|=α : exp − λ (2π) 2 ∂φ ( b ) : : j∈S ∂φ( b j ) : λ α . In the last line we have introduced the counting parameter λ and it is understood to extract the coefficient of λ α . Furthermore : . . . : denotes normal ordering to prevent selfcontractions. We also note that the form (3.1) (or equivalently (7.1)) also implies a recursive structure, that allows to obtain O (N ),α ( a 1,...,N −1 , ρ) from O (N +1),α+1 ( a 1,...,N , ρ) by contour inte- α+1 ( a 1,...,N , ρ) . gration of a N O (N ),α ( a 1,...,N −1 , ρ) = −iπ C d a N N −2 i=0 ( a N − p i ) O (N +1),(7.2) Here it is understood that a 1,...,N and ρ are independent variables. Furthermore, the details of the choice of the contour C are not important, except that it encircles all N − 1 poles of a N p 0 = 0 , and p i = − N −2 j=i a j ∀i = 1, . . . , N − 2 . (7.3) The contour integration in (7.2) is designed to extract the pole of a single scalar two-point function in the decomposition of O (N +1),α+1 ( a 1,...,N , ρ), while the additional factor of 1 2 takes into account that for a given pole p i , there are exactly two diagrams of the type figure 3 that provide such a singularity. Finally, to orders O(Q r R ) for r > 1, we have found that the free energy still affords a decomposition in the basic building blocks H on the torus, which, however, can be decorated in two different ways: either through the action of a differential operator D 2n a i (for n ∈ N ∪ {0}) or through multiplication with combinations of holomorphic Eisenstein series. We have argued that the latter precisely correspond to (dihedral) modular graph forms with bivalent vertices as defined in [47] and thus lend themselves to be interpreted as disconnected contributions from a Feynman diagrammatic point of view. This suggests that also higher instanton orders allow a decomposition in terms of graphs. However, an inherent ambiguity in the decomposition of the free energy (due to the fact that H (r),{0} (s) and W (r) (s) are an overcomplete basis) prevents us from making this statement more precise: in the future it will therefore be important to get a better understanding of the origin of this diagrammatic expansion. In this regard, it will also be important to understand the appearance of the propagators of a chiral scalar field on a torus. While, for example, decompositions of the topological free energy (on the elliptic curve) in terms of cubic graphs (and a formulation of the topological partition function as a path integral with a cubic scalar action) have been discussed in [67,68] (see also [69]), the scalar field propagators O (N ),α in our current formulation only appear after suitably stripping off external states (in the form of H (r),{0} (s) (ρ, S) and W (r) (s) (ρ, S)). Furthermore (see e.g. figure 3) we have not found evidence that the interaction of the scalar field theory is limited to be cubic. This makes the appearance and interpretation of the scalar field less direct and the precise origin behind this mechanism remains to be understood. From a physical perspective, the observations made in this paper suggest that nonperturbative information about the LSTs of A-type can be obtained using simple, purely perturbative ingredients, namely two-point correlation functions of a free scalar field on the torus and the BPS counting function of a single M5-brane on a torus. It would be interesting to analyse if similar statements can also be made for other LSTs and/or supersymmetric gauge theories. Similarly, it would be interesting to extend the current study to quantities other than the free energy. Yet another question is to understand if the decomposition of the free energy introduced in [1] and further elaborated in the current paper reveals new symmetries of the underlying LSTs, which might for example be linked to the conformal symmetry of the scalar two-point function we have encountered in the effective couplings O (N ),α . We leave these questions for future work. Acknowledgments I am deeply indebted to Oliver Schlotterer, for a carefully reading of the draft and many valuable comments and suggestions. I would also like to warmly thank Pierre Vanhove for several interesting exchanges on modular graph functions as well as Amer Iqbal for many inspiring discussions and collaboration on related topics. A Modular toolkit Throughout this paper we are using various different modular objects. This appendix serves to define them and present various different of their properties along with other concepts, which are useful for the discussion in the main body of this article. JHEP04(2021)275 A.1 Jacobi forms and Eisenstein series Many of the objects discussed in this paper are (related to) Jacobi forms: let H be the upper half-plane. Following [70], a weak Jacobi form φ(ρ, z) of index m ∈ Z and weight w of SL(2, Z) is a holomorphic function φ : H × C → C that satisfies the following properties The Jacobi forms we shall encounter in the main body of this article can be written as homogeneous polynomials (both with regards to their index as well as weight) of the following two standard Jacobi forms φ aρ + b cρ + d , z cρ + d = (cρ + d) w e 2πimc z 2 cρ+d φ(ρ, z) , ∀   a b c d   ∈ SL(2, Z) , φ(ρ, z + k 1 ρ + k 2 ) = e −2πim(k 2 1 ρ+2k 1 z) φ(ρ, z) ,∀kφ −2,1 (ρ, z) = θ 2 1 (z, ρ) η 6 (ρ) , and φ 0,1 (ρ, z) = 8 4 i=2 θ i (z, ρ) θ i (0, ρ) 2 , (A.4) of weight and index (−2, 1) and (0, 1) respectively, where θ i (ρ, z) (for i = 1, 2, 3, 4) are the Jacobi theta functions and η is the Dedekind eta-function. Furthermore, the coefficients of these polynomials in φ −2,1 and φ 0,1 themselves are homogeneous polynomials (with regards to their weight) of (holomorphic) Eisenstein series [70][71][72] E 2n (ρ) = 1 − 4n B 2n ∞ k=1 σ 2n−1 (k) Q k ρ , (A.5) where B 2n are the Bernoulli numbers and σ n (k) is the divisor sigma function. In certain cases, we shall also use the notation G 2n (ρ) = 2ζ(2n)E 2n (ρ). The E 2n for n > 1 are holomorphic modular forms of weight 2n, while E 2 transforms in the following fashion under SL(2, Z) E 2 aρ + b cρ + d = (cρ + d) 2 E 2 (ρ) − 3 π c Im(ρ) , ∀   a b c d   ∈ SL(2, Z) . (A.6) Thus E 2 can be completed into the following quasi-modular form of weight 2 E 2 (ρ,ρ) = E 2 (ρ) − 3 π Im(ρ) . (A.7) JHEP04(2021)275 the infinite series I k in (A.10) can be related to derivatives of the two-point function of a free scalar field φ on the torus G( a; ρ) = φ( a) φ(0) = − ln θ 1 ( a; ρ) θ 1 (0, ρ) 2 − π 2Imρ ( a −¯ a) 2 , (A. 15) in particular I 0 (ρ, a) = 1 (2πi) 2 G ( a; ρ) + 2πi ρ −ρ . (A.16) For later use, we also exhibit another way of presenting this scalar two-point function: let Λ = Z ⊕ ρ Z be a two-dimensional lattice and define ·|· : Λ × C −→ C (p, a) = (mρ + n, α 1 ρ + α 2 ) −→ p| a = mα 2 − nα 1 . (A.17) Eq. (A.15) can then be written as G( a; ρ) = Imρ π p∈Λ e 2πi p| a |p| 2 , (A.18) where the summation excludes the origin of Λ. Furthermore, as is evident from (A.12), the generating function I 0 is not holomorphic and modular, due to the presence of G 2 . However, by multiplying I 0 with holomorphic Eisenstein series E 2k (for k > 1) and subtracting a judicious derivative of the same Eisenstein series (as listed in (A.8)) one obtains an object whose Taylor series expansion in a is purely holomorphic and modular: in addition to the example (4.11) discussed in the main body of this paper, we also find the following expressions relevant up to s = 4 dE 6 + 6E 6 I 0 (ρ, a) = − 3E 6 2π 2 a 2 − E 2 4 2 − 6E 6 4π 2 ∞ k=1 (2k + 1)G 2k+2 (ρ) a 2k ,dE 8 + 8E 2 4 I 0 (ρ, a) = − 2E 2 4 π 2 a 2 − 2E 4 E 6 3 − 8E 2 4 4π 2 ∞ k=1 (2k + 1)G 2k+2 (ρ) a 2k , d(E 4 E 6 ) + 10E 4 E 6 I 0 (ρ, a) = − 5E 4 E 6 2π 2 a 2 − E 3 4 2 − E 2 6 3 − 10E 4 E 6 4π 2 ∞ k=1 (2k + 1)G 2k+2 (ρ) a 2k , d(E 12 ) + 12E 12 I 0 (ρ, a) = − 3E 12 π 2 a 2 − E 6 E 2 4 − 3E 12 ∞ k=1 (2k + 1)G 2k+2 (ρ) a 2k . (A.19) A.3 Graph functions and graph forms In this appendix we review so-called modular graph forms. Our conventions and presentation mostly follow [48] as well as [55]. Modular graph functions have first been introduced in [47] (see also [48-52, 55, 56] the graph Γ is then defined as the following integral over the positions z i (with i = 1, . . . , N ) of the N vertices on the torus Σ ρ that is parametrised by the modular parameter ρ C Γ (ρ) = N k=1 Σρ d 2 z k Imρ 1≤i<j≤N G(z i − z j ; ρ) r ij . (A.20) Due to the properties of the scalar Greens functions, C Γ is invariant under SL(2, Z) acting on ρ. Notice, however, that it is a non-holomorphic modular function, which can be made more manifest by using the presentation (A.18) of G(z; ρ), to write 19 C Γ (ρ) = p 1 ,...,pw∈Λ w a=1 1 |p a | 2 N i=1 δ K w b=1 Γ ib p b , (A.21) where we used the shorthand notation δ K w b=1 Γ ib p b = δ K w a=1 Γ ia m a δ K w b=1 Γ ib n b , for p a = m a + ρn a , (A.22) and we have defined This definition of modular graph functions has been generalised in [48] to so-called modular graph forms. To describe the latter, we first generalise the graph Γ by decorating each edge by two integers: instead of the oriented edges in figure 16 between two vertices i and j, we now consider figure 17 with labels a ij,a and b ij,a (for a = 1, . . . , n). We then define 20 δ K (x) =    1 if x = 0 , 0 if x = 0 , and Γ ia =              1 ifC a ij,r b ij,r (ρ) = p 1 ,...,pw∈Λ r 1 p ar rp br r N i=1 δ K s Γ is p s , (A.24) 19 Since it will turn out more convenient for the discussion in the main body of this paper, our normalisation here follows [55] and is missing a factor of Imρ π w relative to [48]. In (A.20) this factor has been accounted for by using the normalisation Σ d 2 z Imρ = 1. 20 As in (A.20), we follow the normalisation of [55], which differs by a factor (Imρ/π) ar +br 2 relative to [48]. where the sums over r and s run over all possible labels (ij, a). Since in the current paper only the modular graph forms with two vertices are important, we shall not make the notation in (A.24) precise for generic N , 21 but focus on graphs with N = 2. Such graphs are called dihedral graphs and for n edges connecting the two vertices, we arrange the decorations Dihedral graph forms with a single edge vanish JHEP04(2021)275 • • (a ij,1 , b ij,1 ) (a ij,2 , b ij,2 ) (a ij,3 , b ij,3 ) (a ij,n , b ij,n ) · · · i jC a 1 b 1 (ρ) = 0 , ∀a 1 , b 1 ∈ N . (A.26) In this paper we shall exclusively encounter dihedral graph forms with two edges, for which numerous identities and relations are known. For example, in [48] it was shown that ∇ n (Imρ) k C k 0 k 0 (ρ) = (Imρ) k+n (k + n − 1)! (k − 1)! C k+n 0 k−n 0 (ρ) , with ∇ = 2i (Imρ) 2 ∂ ∂ρ , (A.27) as well as C a 1 a 2 b 1 b 2 (ρ) = (−1) a 2 +b 2 C a 1 +a 2 0 b 1 +b 2 0 (ρ) . (A.28) Furthermore, the following explicit expressions have been given in [48] C k 0 k 0 (ρ) = π Imρ k E k (ρ) , ∀k > 1 , (A.29) where E k (ρ) is the so-called non-holomorphic Eisenstein series E k (ρ) = (m,n) =(0,0) 1 |m + nρ| 2k . (A.30) Finally, the modular graph forms (along with the dihedral graph with two bi-valent vertices) relevant for the current paper are C 2k 0 0 0 (ρ) =    2ζ(2) E 2 (ρ,ρ) if k = 1 , 2ζ(2k) E 2k (ρ) if k > 1 . (A.31) JHEP04(2021)275 A.4 Generating functions of multiple divisor sums In [44] the following generating function was introduced ; ρ] = n>0 Q n ρ σ s 1 −1,...,s k −1 (n) (s 1 − 1)! . . . (s k − 1)! , s j ∈ N ∀ j ∈ {1, . . . , k} , (A.34) are generating functions of multiple divisor sums σ p 1 ,...,p k (n) = u 1 v 1 +...+u k v k =n u 1 >...>u k >0 v p 1 1 . . . v p k k , p 1 , . . . , p k ∈ N ∪ {0} , k, n ∈ N . (A.35) B Building blocks The decompositions of the LST free energy studied in the main body of this paper use two classes of functions as their basic building blocks: the expansion coefficients of the free energy for the case N = 1 and the expansion coefficients of a specific function that governs the BPS counting of a configuration of a single M5-brane with M2-branes ending on it on either side. In this appendix we review both of these building blocks and also provide basic expansions for the convenience of the reader. B.1 Free energy for N = 1 We first consider the expansion coefficients of the free energy for N = 1 in the unrefined limit. For N = 1, the decomposition (2.7) of the Fourier modes P where H r is the Hecke operator defined in (A.9). H (1),{0} (s) (ρ, S) =          −φ −2,1 (ρ, S) if s = 0 , φ 0,1 (ρ,S) 24 if s = 1 , (−1) s B 2s E 2s (ρ) (2s−3)!!(2s)!! φ −2,1 (ρ, S) if s > 1 ,(B. B.2 The function W (ρ, S) The second class of building blocks that we shall use in the main body of this paper are the expansion coefficients (in powers of ) of the quasi-Jacobi form W that was first introduced in [23, 37] W (ρ, S, ) = θ 2 1 (ρ, S) − θ 1 (ρ, S + )θ 1 (ρ, S − ) θ 2 1 (ρ, ) = ∞ s=0 2s W (1) (s) (ρ, S) . (B.4) The W (1) (s) are quasi Jacobi forms of weight 2s and index 1 and the first few of them are W (1) (0) = 1 24 (φ 0,1 + 2E 2 φ −2,1 ) , W(1)(1) = E 2 2 − E 4 288 φ −2,1 , W(1)(2) = 5E 3 2 + 3E 2 E 4 − 8E 6 51840 φ −2,1 . Following [42] , similar to (B.3), we can define more general building blocks suitable also for r > 1 through Hecke transformations W (r) (s) (ρ, S) = H r W (1) (s) (ρ, S) . (B.5) For further convenience, we present W (r) (s=0) for r = 2 and r = 3 W (2) (0) (ρ, S) = 1 384 φ 2 0,1 + 4E 2 φ 0,1 φ 2,1 + 4E 4 φ −2,1 , (B.6) W (3) (0) (ρ, S) = 1 18 · 24 2 φ 3 0,1 + 6E 2 φ 2 0,1 φ −2,1 + 12E 4 φ 0 φ 2 −2,1 + 8(9E 2 E 4 − 8E 6 )φ 3 −2,1 . C Series expansion of coupling functions In this appendix we present some technical details of the Taylor series expansions of various couplings O (N ),α . C.1 Coupling O (2),1 The simplest non-trivial example is the coupling O (2),1 ( a 1 , ρ). Although eq. (4.3) has already been argued for in [1], we present a somewhat different approach here, which (in principle) is also applicable to some later (more complicated) examples. We start from (4.2) which, as already remarked in [41], can be written in terms of the generating functions (A.32) O (2),1 ( a 1 , ρ) = −2D a 1 [T ( a 1 − ρ; ρ) − T (− a 1 ; ρ)] . (C.1) JHEP04(2021)275 In principle (A.33) affords a full series expansion in powers of a 1 , except that care has to be taken since T ( a 1 − ρ; ρ) has a singularity for a 1 = 0. To this end, we write T ( a 1 − ρ; ρ) = ∞ n=1 Q a 1 1 + Q n ρ 1 − Q n ρ = T ( a 1 ; ρ) + ∞ n=1 Q n a 1 . (C.2) The derivative of the last term can be written as D a 1 ∞ n=1 Q n a 1 = D a 1 Q a 1 1 − Q a 1 = 1 e πi a 1 − e −iπ a 1 2 = − 1 4 sin 2 (π a 1 ) , (C.3) such that we can rewrite (C.1) as O (2),1 = 1 2 sin 2 (π a 1 ) − 2D a 1 ∞ s=1 (1 + (−1) s )(2πi a 1 ) s−1 ∞ n=1 Q n ρ σ s−1 (n) (s − 1)! = 1 2 sin 2 (π a 1 ) − 4 ∞ k=1 (2πi a 1 ) 2k−2 (2k − 2)! B 2k 4k (1 − E 2k (ρ)) . (C.4) Using furthermore the Laurent series expansion For later use, we also remark that a (limited) series expansion in Q ρ to fixed order in a 1 can be obtained in a much simpler fashion. To this end, we recall that it was shown in [41] that O (2),1 can also be written as 23 O (2),1 ( a 1 , ρ) = 1 2π 2 π 2 sin 2 (π a 1 ) − 4π 2 ∞ k=1 ∞ n=1 n Q nk ρ Q n a 1 + Q −n a 1 . (C.7) This expression is equivalent to the following series expansion in Q ρ O (2),1 ( a 1 , ρ) = 1 2π 2   π 2 sin 2 (π a 1 ) − 4π 2 ∞ n=1 Q n ρ |n Q a 1 + Q − a 1   . (C.8) 23 Here it was assumed that |Qρ| < 1 and |Q a 1 | < 1, such that O (2),1 is absolutely convergent. JHEP04(2021)275 Since the sum over is finite (for fixed n), we can extract limited expansions for fixed orders in a 1 . For the first few orders we find O( a −2 1 ): 1 2π 2 , (C.9) O( a 0 1 ): 1 6 1−24Q ρ −72Q 2 ρ −96Q 3 ρ −168Q 4 ρ +O(Q 5 ρ ) ∼ E 2 (ρ) 6 , O( a 2 1 ): π 2 30 1+240Q ρ +2160Q 2 ρ +6720Q 3 ρ +17520Q 4 ρ +O(Q 5 ρ ) ∼ π 2 E 4 (ρ) 30 , O( a 4 1 ): π 4 189 1−504Q ρ −16632Q 2 ρ −122976Q 3 ρ −532728Q 4 ρ +O(Q 5 ρ ) ∼ π 4 E 6 (ρ) 189 , O( a 6 1 ): π 6 1350 1+960Q ρ +354240Q 2 ρ +61543680Q 3 ρ +4858169280Q 4 ρ +O(Q 5 ρ ) ∼ π 6 E 2 4 (ρ) 1350 . Here the identification in terms of Eisenstein series follows if we assume that the coefficient of a 2k 1 in O (2),1 is a quasi-modular form of weight 2k + 2. These terms of low order in a 1 indeed match the precise expansion in (C.6). In fact, we remark, that in the current case this can be made precise by extracting the coefficient of the term a 2s 1 (for s ≥ 0) in a Laurent series expansion of (C.8) (the terms a 2s+1 1 are identically zero) (2s+1)2ζ(2s+2) 2π 2 −2 ∞ n=1 Q n ρ |n (2πi a 1 ) 2s (2s)! = (2s+1)2ζ(2s+2) 2π 2 − 4(2πi) 2s (2s)! ∞ n=1 Q n ρ σ 2s+1 (n) = (2s+1)2ζ(2s+2) 2π 2 − 4(2πi) 2s (2s)! + B 2s+2 4(s+1) (1−E 2s+2 (ρ))= (2s+1)2ζ(2s+2) 2π 2 E 2s+2 . Combining with the a −2 1 term stemming from which is indeed (C.6), thus confirming once more the result we have inferred from analysing the limited expansions (C.9) and assuming modularity of the final result. C.2 Coupling O (3),2 The coupling O (3),2 is defined in (5.4). Due to the complexity of this expression, a purely analytic approach (as discussed in the first part of the previous subsection) is much more difficult and we therefore resort to studying limited expansions in powers of Q ρ . To leading orders in a 1,2 , we can match the latter to combinations of Eisenstein series, which in turn we can compare to combinations of scalar two-point functions. 24 More precisely, we start from the following (schematic) presentation of the coupling 24 While this procedure does not constitute a rigorous proof, the fact that we find agreement for many orders in a1,2 leads us to believe that (5.8) is indeed the complete result. JHEP04(2021)275 where r (2k) i ( a 1 , a 2 ) are rational functions in a 1,2 , which are homogeneous of order 2k and p (2k) i (ρ) are series expansions in Q ρ . We shall then work out the first orders of the latter and match them to combinations of Eisenstein series of weight 4 + 2k. From the ensuing pattern, we will be able to present a closed form expression that fits to all orders that we were able to compute. We start with the terms in the first line of (5.4), which are governed by the quotient where x can stand for various linear combinations of a 1,2 (with |e 2πix | < 1). We can further write this as J (x) = ∞ n=1 n 2 ∞ k=1 k Q n(k+ −1) ρ e 2πinx =          ∞ n=1 n 2 e 2πinx + ∞ k=1 (k + 1) ∞ n=1 n 2 Q nk ρ e 2πinx for = 0 , ∞ k=1 k ∞ n=1 n 2 Q nk ρ e 2πinx for = 1 , ∞ k=2 (k − 1) ∞ n=1 n 2 Q nk ρ e 2πinx for = 2 . Notice that the first term for = 0 diverges for x = 0, thus leading to a pole in the free energy. Furthermore, in order to obtain a (limited) series expansion in powers of Q ρ it is useful to write this expression in the form (C.13) J (x) =          e 2πix (1+e Since (for fixed m) the sum over k is finite, we can derive a limited series expansion in Q ρ for a given power of x. In order to analyse the terms in the second and third line of (5.4), we consider J 1 , 2 (x 1 ,x 2 )= ∞ n 1 ,n 2 =1 n 2 (2n 1 +n 2 ) (1−Q n 1 ρ )(1−Q n 2 ρ ) + (n 1 +n 2 )(n 1 −n 2 ) (1−Q n 1 ρ )(1−Q n 1 +n 2 ρ ) Q 1 n 1 + 2 n 2 ρ e 2πi(n 1 x 1 +n 2 x 2 ) , for 1 , 2 ∈ {0, 1}. We can start by writing (with Q x 1 = e 2πix 1 and Q x 2 = e 2πix 2 ) J 1 , 2 (x 1 , x 2 ) = ∞ n 1 ,n 2 =1 ∞ k 1 ,k 2 =0 n 2 (2n 1 + n 2 ) + (n 2 1 − n 2 2 )Q n 1 k 2 ρ Q n 1 (k 1 + 1 )+n 2 (k 2 + 2 ) ρ Q n 1 x 1 Q n 2 x 2 . The terms in which (k 1 + 1 ) = 0 and/or (k 2 + 2 ) = 0 lead to divergent sums over n 1,2 and need to be considered separately. In order to do so, we need to distinguish the various cases for 1,2 : JHEP04(2021)275 We have checked up to 2k = 10 that this expansion matches the following closed form expression O (3),2 ( a 1 , a 2 , ρ) = 1 (2π) 4 G ( a 1 ; ρ) + 2πi ρ −ρ G ( a 2 ; ρ) + 2πi ρ −ρ + G ( a 1 ; ρ) + 2πi ρ −ρ G ( a 1 + a 2 ; ρ) + 2πi ρ −ρ + G ( a 2 ; ρ) + 2πi ρ −ρ G ( a 1 + a 2 ; ρ) + 2πi ρ −ρ . (C.19) Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 5B Example N = 3 17 5.1 Decomposition at order O(Q R ) and scalar correlators 17 5.2 Decomposition at order O(Q r R ) for r > Building blocks 33 B.1 Free energy for N = 1 33 B.2 The function W (ρ, S) 34 Figure 1 . 1Web diagram of X N,1 with the parameters ( a 1 , . . . , a N , S, R). F N,1 ( a 1 , . . . , a N , S, R; 1,2 ) = ln Z N,1 ( a 1 , . . . , a N , S, R; 1,2 ) . (2.2) N ( a 1 , . . . , a N , S, (s) ( a 1 , . . . , a N , S) . (2.5) Furthermore, we can also define a Fourier series with respect to the parameters a 1,...,N P (r) N,(s) ( a 1,...,N , S) = n 1 ,...,n N Q n 1 a 1 . . . Q n N a N P (r),{n 1 ,...,n N } N,(s) N ,(s) ( a 1,...,N , S) = H (r),{0,...,0} (s) 7) where the prime indicates that the summation is understood over all n = {n 1 , . . . , n N } ∈ (N ∪ {0}) N with at least one of the n i = 0. The H (r),n (s) (ρ, S) are formally defined as[41] , S) in detail in appendix B.1. O (s) ( a 1,...,N , S) introduced in (2.7) afford a decomposition in terms of rather simple building blocks with a very suggestive graphical presentation that resembles in some way an amplitude expansion. Indeed, in [1] the following decomposition was proposed P (r=1) N,(s) ( a 1,...,N , S) = H (N ),α ( a 1,...,N −1 , ρ) , (2.9) (2s 1 ,2s 2 ) defined in[1], we remark thatP (r) N,(s) ( a1, . . . , aN , S) = s 1 +s 2 =s (−1) s 2 P (r) N,(2s 1 ,2s 2 ) ( a1, . . . , aN , S). F plet N,1 ( a1,...,N , S, R; 1,2) = PlogZN,1( a1,...,N , S, R; 1,2) = ∞ n=1 µ(n) n ln ZN,1(n a1,...,N , nS, nR; n 1,2) , (2.10) Figure 2 . 2Diagrammatic expansion of P (r=1) N,(s) . Figure 3 . 3Diagram of a single term in the summation over S (for fixed ) in eq. (3.1): the dashed lines correspond to scalar two-point functions couplings α = 0, . . . , N − 1 and where we have introduced the new arguments b 0 = 0 and b j = j n=1 a n , ∀j = 1, . . . , N − 1 . (3.2) Furthermore, G(z; ρ) is the two-point function of a free boson on the torus and the prime denotes a derivative with respect to the z-argument (see appendix A.3 for more details). Furthermore, the second summation in (3.1) is over all subsets S of {1, . . . , A , . . . , N − 1} of cardinality α (i.e. which have exactly α elements) N ,(s) is more involved. However, nonetheless, in the examples we have studied, it still affords a decomposition similar to the r = 1 counterpart, albeit in a more intricate fashion: analysing the cases N = 2 JHEP04(2021)275 and N = 3 up to orders O(Q 3 R ) and O(Q 2 R ) respectively and up to s = 2, we find that these examples can entirely be decomposed in terms of H Figure 5 . 5Diagrammatic presentation of the combinations of scalar Greens functions appearing in the summation over S (for fixed ) in (4.4): (a) for O (2),0 and (b) for O (2),1 . Dashed lines with end-points ( Figure 6 . 6Diagrammatic presentation of the 4-point functions appearing in the first term of the second line in (4.7) and which contribute to the coupling (4.8). The nodes represented by • correspond to external states (with insertion points b 0 = 0 and b 1 respectively), while the nodes represented by • correspond to internal states. Following the decomposition (4.10), diagram (a) leads to the coupling O (2),4-pt r=2,s=0 ( a 1 , ρ) before the subtraction of the quasi-holomorphic contribution, while diagram (b) leads to O (2),4-pt,1 r=2,s=0 ( a 1 , ρ). depends (apart from ρ) only on a single 'position', namely a 1 . Following the interpretation of the coupling functions appearing at order O(Q R ) as schematically shown in figure 5, this would suggest that of the 4 states, two are external (and are inserted at positions b 0 = 0 and b 1 = a 1 ), while the positions of the remaining two states are internal (and could be integrated over). Two graphs of this type, which could give rise to the various different terms appearing in the coupling O (2),4-pt r=2,s=0 , are schematically shown infigure 6. These are two distinct corrections where the two internal states modify the two-point function, either in the form of a disconnected diagram as infigure 6(a) (from which still the E 2 -dependent contributions need to be subtracted) or through a direct insertion as infigure 6 (b). while crosses indicate the action of the derivative operator D 2 a1 : (a) the coupling O (2),4-pt,1 r=2,s=0 ( a 1 , ρ) (where the subtraction of the quasi-holomorphic contributions is understood); (b) the coupling O E 4 (and a subsequent removal of non-holomorphic contributions) or through the action of the derivative operators D a 1 . Indeed, following the presentation of figure 5 for r = 1, the two objects in (4.10) can graphically be presented as in figure 7: while as before dashed lines represent the scalar Greens function, crosses indicate the action of the operator D 2 a 1 Figure 8 . 8Diagrammatical representation of the apparent 3-point function O (2),3-pt r=2,s=0 in (4.15) as a two-point function (with external states marked with •) corrected by one internal state (marked with •). (a) internal state is given by H Figure 9 . 9Diagrammatic presentation of the coupling in (4.15). The dashed line with end-points ( b 0 , b 1 ) represents a factor of G ( a 1 ; ρ) + 2πi ρ−ρ , while the cross indicates the action of the derivative operator D 2 a1 . higher values of s: while we have worked out the corresponding functions f to s = 4, it turns out that the decomposition in the basic building blocks H (r),{0} (s)is not unique anymore, but leaves certain ambiguities. For example to order s) ( a 1 , ρ, S) can be decomposed using only H (r),{0} (s) Ob 1 = a 1 b 2 = a 1 + a 2 Figure 10 . 1121210s) ( a 1 , a 2 , ρ, S) = H (3),2 ( a 1 , a 2 , ρ) , Trivial diagrammatic presentation of the constant coupling O (3),0 in (5.4). the form of the O (3), a constant. Furthermore, based on the discussion of similar structures already appearing in the case of N = 2 (see e.g. eq. (4.2)), we can write for O (3),2 ( a 1 , a 2 , ρ)14 Figure 12 . 12Diagrammatical representation of the coupling O (3),2 in (5.7). The dashed lines represent the two-point function G the general expression in (3.1). Graphically (up to a numerical factor), this coupling can be represented as in figure 12 as two scalar two-point functions connecting two distinct pairs of points { b 0 , b 1 , b 2 }. We note that all three coupling functions O (3),0 in (5.2), O (3),1 in (5.6) and O (3),2 in (5.7) match the general form (3.1). ) for r ∈ {0, 1} is not unique since the latter form an overcomplete basis. While this ambiguity prevents us from determining decomposes, the general form of the result still gives further credence to the picture already advocated before. Using the same approach as in the case N = 2,16 we have matched the next-to-leading order O(Q 2 R ) of the free energy P 11520α 1 − 3α 16 − 2592α 17 − 864α 18 − α 16 − 864α 17 + 3072α 2 − 1536α 3 1 − α 16 − 864α 17 − 288α 18 + 384α 2 − 192α 3 − 24) structural ambiguity, we see that (similar to the case of N = 2), a decomposition is possible which lends itself to an interpretation in terms of n-point functions (with n ≤ 6) where the external states are given by H for r ∈ {1, 2}. O (4),3 ( a 1,2,3 , ρ) . Figure 15 . 15Diagrammatical representation of the proposed coupling O (4),3 in (6.5). The dashed line represent the two-point function G ( b − b j ) + 2πi ρ−ρ . ) . The tentative coupling functions are again composed of combinations of second derivatives of the scalar Greens function JHEP04(2021)275 n, k) Q n ρ e 2πizk , with c(n, k) = (−1) w c(n, −k) . (A.2) Further symmetries of Jacobi forms follow from the fact that two Fourier coefficients c(n, ) and c(n , ) (for fixed n, n and , ) are identical if 2 − 4mn = ( ) 2 − 4mn , and = (mod 2m) . (A.3) Figure 16 . 16): let Γ be a graph of N vertices (labelled by i, j = 1, . . . , N ) with r ij oriented edges connecting vertex i to vertex j. An example of n edges connecting the vertices i and j is shown infigure 16. The weight w of the graph is defined as the total number of all edges, i.e. w = 1≤i<j≤N r ij . The modular graph function associated with n ij oriented edges connecting vertex i to j. edge a ends on vertex i and points into vertex i , −1 if edge a ends on vertex i and points out of vertex i Figure 17 . 17n oriented and decorated edges connecting vertex i to j. (a α , b α ) (for α = 1, . . . , n) into the matrix A B = a 1 a 2 ... an b 1 b 2 ... bn . We then write for (A.24) NN ,(s) of the free energy F N,1 , as introduced in eq. (2.6), is simply 22 P (r) N =1,(s) (ρ, S) = H ,(s) (ρ, S) are in fact (quasi) Jacobi forms of weight 2s − 2 and index r. For r = 1, they are related to the basic Jacobi forms φ 0,1 and φ −2,1 in (A.4) 2 ℘( a 1 ; ρ) , (C.6) which is indeed (4.3). O ( 3 ), 2 ( a 1 , a 2 3212, ρJ )2 . In order to sum up their contribution, there are three conceptually different terms ( m − k)k e 2πikx for = 2 . Table 1 . 1Coefficient functions c (1) k;i1,i2;i3,i4 (ρ) for a = 1. Table 2 . 2Coefficient functions c(2) k;i1,i2;i3,i4 (ρ) for a = 2. Table 3 . 3Coefficient functions c(3) k;i1,i2;i3,i4 (ρ) for a = 3. T (z 1 , . . . , z k ; ρ) = From a mathematical perspective, these functions can be related to multiple zeta values and are expressable in terms of (reduced) polylogarithms. These, in turn, have made appearances in the physics literature, related to (loop) amplitudes in (supersymmetric) string theories, e.g.[47,50,53,54,[76][77][78][79][80][81][82][83][84][85][86]. For the current work, we remark that (A.32) affords the following Taylor series expansionT (z 1 , . . . , z k ; ρ) = s 1 ,...,s k >0 [s 1 , . . . , s k ; ρ] (2πiz 1 ) s 1 −1 . . . (2πiz k ) s k −1 . (A.33)Here [s 1 , . . . , s k ; ρ] are called brackets of length k ∈ N in[44] [s 1 , . . . , s kn 1 ,...,n k >0 k j=1 e 2πin j z j Q n 1 +...+n j ρ 1 − Q n 1 +...+n j ρ . (A.32) 2 ) 222 Notice, for N = 1 we have ρ = a1.where E 2s (ρ) are the (holomorphic) Eisenstein series and B 2s the Bernoulli numbers. Higher orders in r are related to HJHEP04(2021)275 (1),{0} (s) through the action of the Hecke operator H (r),{0} (s) (ρ, S) = H r H (1),{0} (s) (ρ, S) , ∀s ≥ 0 , (B.3) Here ρ and τ denote the circumferences of the two circles (which subsequently are complexified). Both are measured in units of the radius of the compact time direction, which throughout this paper is implicitly set equal to 1. We refer to[22][23][24] for more details about the precise brane setup. This difference only plays a role to higher instanton orders, since to leading order there is in fact no difference between F plet N,1 and FN,1. For the purpose of this paper, we understand quasi-Jacobi forms as homogeneous polynomials of φ−2,1 and φ0,1, whose coefficients also depend on the Eisenstein series E2. For a more rigorous definition we refer to[73] (see also[41]). For this purpose, we refer the reader to the original literature[48] (see also[55]). JHEP04(2021)275Furthermore, the holomorphic Eisenstein series display a ring structure in the sense that any E 2n for n > 1 can be written as a combination (of powers) of E 4 and E6.We shall sometimes also encounter derivatives d = Q ρ d dQρ of the Eisenstein series E 2n (for n > 1), which can again be expressed in terms of combinations of Eisenstein series with combined weight 2n + 2, which, however, are linear in E 2Finally, we remark that (quasi)Jacobi 18 forms can be related to one-another through Hecke transformations. Let J w,m be the space of Jacobi forms of weight w and index m and let n ∈ N, thenA.2 Weierstrass' elliptic function and scalar two-point functionA class of infinite series[1,41]that are useful in the discussion of the free energy in the case of N = 2 is defined as (with a ∈ C)The series for generic k can be written as derivatives of the generating function I 0Furthermore, it was shown in[41]that the generating function I 0 can be related to Weierstrass' elliptic function ℘where the Eisenstein series G 2 = 2ζ(2) E 2 is defined in (A.5) andA proof of (A.12) (which is slightly complementary to the argument presented in[41]) can be found in appendix C.1. Finally, it was observed in[1]that via the relation[74,75]JHEP04(2021)275• 1 = 0 and 2 = 0:which can be cast into the form• 1 = 1 and 2 = 0:which can be recast into the form• 1 = 0 and 2 = 1:(C.15)• 1 = 1 and 2 = 1:which can be recast in the formFrom these expressions we can extract an expansion in Q ρ for any (fixed) power in the arguments x 1,2 . Consequently, we can work out the leading terms in a Fourier series expansion of the pwith the following matching rational functions in a 1,21 = π 4 a 2 1 + a 1 a 2 + a 2 2 2 a 6 1 +3 a 5 1 a 2 +8 a 4 1 a 2 2 +11 a 3 1 a 3 2 +8 a 2 1 a 4 2 +3 a 1 a 5 2 + a 6 2 5400 a 2 1 a 2 2 ( a 1 + a 2 ) 2 , r (4) 2 = ( a 1 + a 1 a 2 + a 2 ) 2 π 4 1134 , r1 = π 6 2 a 6 1 +6 a 5 1 a 2 +15 a 4 1 a 2 2 +20 a 3 1 a 3 2 +15 a 2 1 a 4 2 +6 a 1 a 5 2 +2 a 6 2 16200 , r (6) 2 = π 6 124740 a 2 1 a 2 2 ( a 1 + a 2 ) 2 3 a 12 1 +18 a 11 1 a 2 +83 a 10 1 a 2 2 +250 a 9 1 a 3 2 +509 a 8 1 a 4 2 +734 a 7 1 a 5 2 +817 a 6 1 a 6 2 +734 a 5 1 a 7 2 +509 a 4 1 a 8 2 +250 a 3 1 a 9 2 +83 a 2 1 a 10 2 +18 a 1 a 11 2 +3 a 12 2 . (C.18) From little string free energies towards modular graph functions. S Hohenegger, 10.1007/JHEP03(2020)077arXiv:1911.08172JHEP. 0377INSPIRES. Hohenegger, From little string free energies towards modular graph functions, JHEP 03 (2020) 077 [arXiv:1911.08172] [INSPIRE]. Some comments on string dynamics. E Witten, hep-th/9507121Strings 95: future perspectives in string theory. INSPIREE. Witten, Some comments on string dynamics, in Strings 95: future perspectives in string theory, (1995) [hep-th/9507121] [INSPIRE]. Point-like instantons and the Spin(32)/Z 2 heterotic string. P S , 10.1016/S0550-3213(97)00232-0hep-th/9612108Nucl. Phys. B. 496149INSPIREP.S. Aspinwall, Point-like instantons and the Spin(32)/Z 2 heterotic string, Nucl. Phys. B 496 (1997) 149 [hep-th/9612108] [INSPIRE]. Point-like instantons on K3 orbifolds. P S Aspinwall, D R Morrison, 10.1016/S0550-3213(97)00516-6hep-th/9705104Nucl. Phys. B. 503533INSPIREP.S. Aspinwall and D.R. Morrison, Point-like instantons on K3 orbifolds, Nucl. Phys. B 503 (1997) 533 [hep-th/9705104] [INSPIRE]. N Seiberg, 10.1016/S0370-2693(97)00805-8hep-th/9705221New theories in six-dimensions and matrix description of M-theory on T 5 and T 5. 298INSPIREN. Seiberg, New theories in six-dimensions and matrix description of M-theory on T 5 and T 5 /Z 2 , Phys. Lett. B 408 (1997) 98 [hep-th/9705221] [INSPIRE]. New string theories in six-dimensions via branes at orbifold singularities. K A Intriligator, 10.4310/ATMP.1997.v1.n2.a5hep-th/9708117Adv. Theor. Math. Phys. 1271INSPIREK.A. Intriligator, New string theories in six-dimensions via branes at orbifold singularities, Adv. Theor. Math. Phys. 1 (1998) 271 [hep-th/9708117] [INSPIRE]. Branes and six-dimensional supersymmetric theories. A Hanany, A Zaffaroni, 10.1016/S0550-3213(98)00355-1hep-th/9712145Nucl. Phys. B. 529180INSPIREA. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys. B 529 (1998) 180 [hep-th/9712145] [INSPIRE]. Branes at orbifolds versus Hanany Witten in six-dimensions. I Brunner, A Karch, 10.1088/1126-6708/1998/03/003hep-th/9712143JHEP. 033INSPIREI. Brunner and A. Karch, Branes at orbifolds versus Hanany Witten in six-dimensions, JHEP 03 (1998) 003 [hep-th/9712143] [INSPIRE]. F-theory and the classification of little strings. L Bhardwaj, M Zotto, J J Heckman, D R Morrison, T Rudelius, C Vafa, 10.1103/PhysRevD.93.086002arXiv:1511.05565Phys. Rev. D. 9386002Erratum ibid. 100 (2019) 029901. INSPIREL. Bhardwaj, M. Del Zotto, J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, F-theory and the classification of little strings, Phys. Rev. D 93 (2016) 086002 [Erratum ibid. 100 (2019) 029901] [arXiv:1511.05565] [INSPIRE]. Revisiting the classifications of 6d SCFTs and LSTs. L Bhardwaj, 10.1007/JHEP03(2020)171arXiv:1903.10503JHEP. 03171INSPIREL. Bhardwaj, Revisiting the classifications of 6d SCFTs and LSTs, JHEP 03 (2020) 171 [arXiv:1903.10503] [INSPIRE]. On the classification of 6D SCFTs and generalized ADE orbifolds. J J Heckman, D R Morrison, C Vafa, 10.1007/JHEP05(2014)028arXiv:1312.5746JHEP. 0528Erratum ibid. 06 (2015) 017. INSPIREJ.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 06 (2015) 017] [arXiv:1312.5746] [INSPIRE]. Atomic classification of 6D SCFTs. J J Heckman, D R Morrison, T Rudelius, C Vafa, 10.1002/prop.201500024arXiv:1502.05405Fortsch. Phys. 63468INSPIREJ.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic classification of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE]. D Xie, S.-T Yau, arXiv:1510.013244d N = 2 SCFT and singularity theory. Part I. Classification. INSPIRED. Xie and S.-T. Yau, 4d N = 2 SCFT and singularity theory. Part I. Classification, arXiv:1510.01324 [INSPIRE]. P Jefferson, H.-C Kim, C Vafa, G Zafrir, arXiv:1705.05836Towards classification of 5d SCFTs: single gauge node. INSPIREP. Jefferson, H.-C. Kim, C. Vafa and G. Zafrir, Towards classification of 5d SCFTs: single gauge node, arXiv:1705.05836 [INSPIRE]. On geometric classification of 5d SCFTs. P Jefferson, S Katz, H.-C Kim, C Vafa, 10.1007/JHEP04(2018)103arXiv:1801.04036JHEP. 04103INSPIREP. Jefferson, S. Katz, H.-C. Kim and C. Vafa, On geometric classification of 5d SCFTs, JHEP 04 (2018) 103 [arXiv:1801.04036] [INSPIRE]. Geometric classification of 4d N = 2 SCFTs. M Caorsi, S Cecotti, 10.1007/JHEP07(2018)138arXiv:1801.04542JHEP. 07138INSPIREM. Caorsi and S. Cecotti, Geometric classification of 4d N = 2 SCFTs, JHEP 07 (2018) 138 [arXiv:1801.04542] [INSPIRE]. Classifying 5d SCFTs via 6d SCFTs: arbitrary rank. L Bhardwaj, P Jefferson, 10.1007/JHEP10(2019)282arXiv:1811.10616JHEP. 10282INSPIREL. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: arbitrary rank, JHEP 10 (2019) 282 [arXiv:1811.10616] [INSPIRE]. Fibers add flavor. Part I. Classification of 5d SCFTs, flavor symmetries and BPS states. F Apruzzi, C Lawrie, L Lin, S Schäfer-Nameki, Y.-N Wang, 10.1007/JHEP11(2019)068arXiv:1907.05404JHEP. 1168INSPIREF. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, Fibers add flavor. Part I. Classification of 5d SCFTs, flavor symmetries and BPS states, JHEP 11 (2019) 068 [arXiv:1907.05404] [INSPIRE]. On the classification of 5d SCFTs. L Bhardwaj, 10.1007/JHEP09(2020)007arXiv:1909.09635JHEP. 097INSPIREL. Bhardwaj, On the classification of 5d SCFTs, JHEP 09 (2020) 007 [arXiv:1909.09635] [INSPIRE]. Towards the classification of rank-r N = 2 SCFTs. Part I. Twisted partition function and central charge formulae. M Martone, 10.1007/JHEP12(2020)021arXiv:2006.16255JHEP. 1221INSPIREM. Martone, Towards the classification of rank-r N = 2 SCFTs. Part I. Twisted partition function and central charge formulae, JHEP 12 (2020) 021 [arXiv:2006.16255] [INSPIRE]. Towards a classification of rank-r N = 2 SCFTs. Part II. Special Kähler stratification of the Coulomb branch. P C Argyres, M Martone, 10.1007/JHEP12(2020)022arXiv:2007.00012JHEP. 1222INSPIREP.C. Argyres and M. Martone, Towards a classification of rank-r N = 2 SCFTs. Part II. Special Kähler stratification of the Coulomb branch, JHEP 12 (2020) 022 [arXiv:2007.00012] [INSPIRE]. M-strings, monopole strings, and modular forms. S Hohenegger, A Iqbal, S.-J Rey, 10.1103/PhysRevD.92.066005arXiv:1503.06983Phys. Rev. D. 9266005INSPIRES. Hohenegger, A. Iqbal and S.-J. Rey, M-strings, monopole strings, and modular forms, Phys. Rev. D 92 (2015) 066005 [arXiv:1503.06983] [INSPIRE]. Instanton-monopole correspondence from M-branes on S 1 and little string theory. S Hohenegger, A Iqbal, S.-J Rey, 10.1103/PhysRevD.93.066016arXiv:1511.02787Phys. Rev. D. 9366016INSPIRES. Hohenegger, A. Iqbal and S.-J. Rey, Instanton-monopole correspondence from M-branes on S 1 and little string theory, Phys. Rev. D 93 (2016) 066016 [arXiv:1511.02787] [INSPIRE]. Self-duality and self-similarity of little string orbifolds. S Hohenegger, A Iqbal, S.-J Rey, 10.1103/PhysRevD.94.046006arXiv:1605.02591Phys. Rev. D. 9446006INSPIRES. Hohenegger, A. Iqbal and S.-J. Rey, Self-duality and self-similarity of little string orbifolds, Phys. Rev. D 94 (2016) 046006 [arXiv:1605.02591] [INSPIRE]. Triality in little string theories. B Bastian, S Hohenegger, A Iqbal, S.-J Rey, 10.1103/PhysRevD.97.046004arXiv:1711.07921Phys. Rev. D. 9746004INSPIREB. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Triality in little string theories, Phys. Rev. D 97 (2018) 046004 [arXiv:1711.07921] [INSPIRE]. Beyond triality: dual quiver gauge theories and little string theories. B Bastian, S Hohenegger, A Iqbal, S.-J Rey, 10.1007/JHEP11(2018)016arXiv:1807.00186JHEP. 1116INSPIREB. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Beyond triality: dual quiver gauge theories and little string theories, JHEP 11 (2018) 016 [arXiv:1807.00186] [INSPIRE]. Five-dimensional gauge theories from shifted web diagrams. B Bastian, S Hohenegger, A Iqbal, S.-J Rey, 10.1103/PhysRevD.99.046012arXiv:1810.05109Phys. Rev. D. 9946012INSPIREB. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Five-dimensional gauge theories from shifted web diagrams, Phys. Rev. D 99 (2019) 046012 [arXiv:1810.05109] [INSPIRE]. M-strings. B Haghighat, A Iqbal, C Kozçaz, G Lockhart, C Vafa, 10.1007/s00220-014-2139-1arXiv:1305.6322Commun. Math. Phys. 334779INSPIREB. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, M-strings, Commun. Math. Phys. 334 (2015) 779 [arXiv:1305.6322] [INSPIRE]. Orbifolds of M-strings. B Haghighat, C Kozcaz, G Lockhart, C Vafa, 10.1103/PhysRevD.89.046003arXiv:1310.1185Phys. Rev. D. 8946003INSPIREB. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, Orbifolds of M-strings, Phys. Rev. D 89 (2014) 046003 [arXiv:1310.1185] [INSPIRE]. M-strings, elliptic genera and N = 4 string amplitudes. S Hohenegger, A , 10.1002/prop.201300035arXiv:1310.1325Fortsch. Phys. 62155INSPIRES. Hohenegger and A. Iqbal, M-strings, elliptic genera and N = 4 string amplitudes, Fortsch. Phys. 62 (2014) 155 [arXiv:1310.1325] [INSPIRE]. V Gritsenko, math.AG/9906191Complex vector bundles and Jacobi forms. INSPIREV. Gritsenko, Complex vector bundles and Jacobi forms, math.AG/9906191 [INSPIRE]. Local Calabi-Yau manifolds of typeà via SYZ mirror symmetry. A Kanazawa, S.-C Lau, 10.1016/j.geomphys.2018.12.015arXiv:1605.00342J. Geom. Phys. 139103INSPIREA. Kanazawa and S.-C. Lau, Local Calabi-Yau manifolds of typeà via SYZ mirror symmetry, J. Geom. Phys. 139 (2019) 103 [arXiv:1605.00342] [INSPIRE]. The topological vertex. M Aganagic, A Klemm, M Mariño, C Vafa, 10.1007/s00220-004-1162-zhep-th/0305132Commun. Math. Phys. 254425INSPIREM. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE]. The refined topological vertex. A Iqbal, C Kozcaz, C Vafa, 10.1088/1126-6708/2009/10/069hep-th/0701156JHEP. 1069INSPIREA. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE]. Five-brane webs and highest weight representations. B Bastian, S Hohenegger, 10.1007/JHEP12(2017)020arXiv:1706.08750JHEP. 1220INSPIREB. Bastian and S. Hohenegger, Five-brane webs and highest weight representations, JHEP 12 (2017) 020 [arXiv:1706.08750] [INSPIRE]. Dual little strings from F-theory and flop transitions. S Hohenegger, A Iqbal, S.-J Rey, 10.1007/JHEP07(2017)112arXiv:1610.07916JHEP. 07112INSPIRES. Hohenegger, A. Iqbal and S.-J. Rey, Dual little strings from F-theory and flop transitions, JHEP 07 (2017) 112 [arXiv:1610.07916] [INSPIRE]. Bound states of little strings and symmetric orbifold conformal field theories. A Ahmed, S Hohenegger, A Iqbal, S.-J Rey, 10.1103/PhysRevD.96.081901arXiv:1706.04425Phys. Rev. D. 9681901INSPIREA. Ahmed, S. Hohenegger, A. Iqbal and S.-J. Rey, Bound states of little strings and symmetric orbifold conformal field theories, Phys. Rev. D 96 (2017) 081901 [arXiv:1706.04425] [INSPIRE]. Dual little strings and their partition functions. B Bastian, S Hohenegger, A Iqbal, S.-J Rey, 10.1103/PhysRevD.97.106004arXiv:1710.02455Phys. Rev. D. 97106004INSPIREB. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Dual little strings and their partition functions, Phys. Rev. D 97 (2018) 106004 [arXiv:1710.02455] [INSPIRE]. Dihedral symmetries of gauge theories from dual Calabi-Yau threefolds. B Bastian, S Hohenegger, 10.1103/PhysRevD.99.066013arXiv:1811.03387Phys. Rev. D. 9966013INSPIREB. Bastian and S. Hohenegger, Dihedral symmetries of gauge theories from dual Calabi-Yau threefolds, Phys. Rev. D 99 (2019) 066013 [arXiv:1811.03387] [INSPIRE]. Symmetries in A-type little string theories. Part I. Reduced free energy and paramodular groups. B Bastian, S Hohenegger, 10.1007/JHEP03(2020)062arXiv:1911.07276JHEP. 0362INSPIREB. Bastian and S. Hohenegger, Symmetries in A-type little string theories. Part I. Reduced free energy and paramodular groups, JHEP 03 (2020) 062 [arXiv:1911.07276] [INSPIRE]. Symmetries in A-type little string theories. Part II. Eisenstein series and generating functions of multiple divisor sums. B Bastian, S Hohenegger, 10.1007/JHEP03(2020)016arXiv:1911.07280JHEP. 0316INSPIREB. Bastian and S. Hohenegger, Symmetries in A-type little string theories. Part II. Eisenstein series and generating functions of multiple divisor sums, JHEP 03 (2020) 016 [arXiv:1911.07280] [INSPIRE]. Symmetric orbifold theories from little string residues. S Hohenegger, A , 10.1103/PhysRevD.103.066004arXiv:2009.00797Phys. Rev. D. 10366004INSPIRES. Hohenegger and A. Iqbal, Symmetric orbifold theories from little string residues, Phys. Rev. D 103 (2021) 066004 [arXiv:2009.00797] [INSPIRE]. B Haghighat, R Sun, 10.1007/JHEP10(2019)192arXiv:1811.04938M5 branes and theta functions. 10192INSPIREB. Haghighat and R. Sun, M5 branes and theta functions, JHEP 10 (2019) 192 [arXiv:1811.04938] [INSPIRE]. The algebra of generating functions for multiple divisor sums and applications to multiple zeta values. H Bachmann, U Kühn, arXiv:1309.3920INSPIREH. Bachmann and U. Kühn, The algebra of generating functions for multiple divisor sums and applications to multiple zeta values, arXiv:1309.3920 [INSPIRE]. Elliptic functions according to Eisenstein and Kronecker. A Weil, 10.1007/978-3-642-66209-6SpringerBerlin, Heidelberg, GermanyA. Weil, Elliptic functions according to Eisenstein and Kronecker, Springer, Berlin, Heidelberg, Germany (1976). Relations between elliptic multiple zeta values and a special derivation algebra. J Broedel, N Matthes, O Schlotterer, 10.1088/1751-8113/49/15/155203arXiv:1507.02254J. Phys. A. 49155203INSPIREJ. Broedel, N. Matthes and O. Schlotterer, Relations between elliptic multiple zeta values and a special derivation algebra, J. Phys. A 49 (2016) 155203 [arXiv:1507.02254] [INSPIRE]. Modular graph functions. E Hoker, M B Green, Ö Gürdogan, P Vanhove, 10.4310/CNTP.2017.v11.n1.a4arXiv:1512.06779Commun. Num. Theor. Phys. 11165INSPIREE. D'Hoker, M.B. Green, Ö. Gürdogan and P. Vanhove, Modular graph functions, Commun. Num. Theor. Phys. 11 (2017) 165 [arXiv:1512.06779] [INSPIRE]. Identities between modular graph forms. E Hoker, M B Green, 10.1016/j.jnt.2017.11.015arXiv:1603.00839J. Number Theor. 18925INSPIREE. D'Hoker and M.B. Green, Identities between modular graph forms, J. Number Theor. 189 (2018) 25 [arXiv:1603.00839] [INSPIRE]. Higher genus modular graph functions, string invariants, and their exact asymptotics. E Hoker, M B Green, B Pioline, 10.1007/s00220-018-3244-3arXiv:1712.06135Commun. Math. Phys. 366927INSPIREE. D'Hoker, M.B. Green and B. Pioline, Higher genus modular graph functions, string invariants, and their exact asymptotics, Commun. Math. Phys. 366 (2019) 927 [arXiv:1712.06135] [INSPIRE]. Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes. F Zerbini, arXiv:1804.07989Bonn, GermanyPh.D. thesis, Bonn U.. INSPIREF. Zerbini, Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes, Ph.D. thesis, Bonn U., Bonn, Germany (2017) [arXiv:1804.07989] [INSPIRE]. . JHEP04. 275JHEP04(2021)275 Modular and holomorphic graph functions from superstring amplitudes. F Zerbini, 10.1007/978-3-030-04480-0_18arXiv:1807.04506KMPB conference: elliptic integrals, elliptic functions and modular forms in quantum field theory. Cham, SwitzerlandSpringerINSPIREF. Zerbini, Modular and holomorphic graph functions from superstring amplitudes, in KMPB conference: elliptic integrals, elliptic functions and modular forms in quantum field theory, Springer, Cham, Switzerland (2018) [arXiv:1807.04506] [INSPIRE]. Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings. J E Gerken, A Kleinschmidt, O Schlotterer, 10.1007/JHEP01(2019)052arXiv:1811.02548JHEP. 0152INSPIREJ.E. Gerken, A. Kleinschmidt and O. Schlotterer, Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings, JHEP 01 (2019) 052 [arXiv:1811.02548] [INSPIRE]. All order α expansion of one-loop open-string integrals. C R Mafra, O Schlotterer, 10.1103/PhysRevLett.124.101603arXiv:1908.09848Phys. Rev. Lett. 124101603INSPIREC.R. Mafra and O. Schlotterer, All order α expansion of one-loop open-string integrals, Phys. Rev. Lett. 124 (2020) 101603 [arXiv:1908.09848] [INSPIRE]. One-loop open-string integrals from differential equations: all-order α -expansions at n points. C R Mafra, O Schlotterer, 10.1007/JHEP03(2020)007arXiv:1908.10830JHEP. 037INSPIREC.R. Mafra and O. Schlotterer, One-loop open-string integrals from differential equations: all-order α -expansions at n points, JHEP 03 (2020) 007 [arXiv:1908.10830] [INSPIRE]. All-order differential equations for one-loop closed-string integrals and modular graph forms. J E Gerken, A Kleinschmidt, O Schlotterer, 10.1007/JHEP01(2020)064arXiv:1911.03476JHEP. 0164INSPIREJ.E. Gerken, A. Kleinschmidt and O. Schlotterer, All-order differential equations for one-loop closed-string integrals and modular graph forms, JHEP 01 (2020) 064 [arXiv:1911.03476] [INSPIRE]. Generating series of all modular graph forms from iterated Eisenstein integrals. J E Gerken, A Kleinschmidt, O Schlotterer, 10.1007/JHEP07(2020)190arXiv:2004.05156JHEP. 07190INSPIREJ.E. Gerken, A. Kleinschmidt and O. Schlotterer, Generating series of all modular graph forms from iterated Eisenstein integrals, JHEP 07 (2020) 190 [arXiv:2004.05156] [INSPIRE]. R Gopakumar, C Vafa, hep-th/9809187M-theory and topological strings. 1INSPIRER. Gopakumar and C. Vafa, M-theory and topological strings. 1, hep-th/9809187 [INSPIRE]. R Gopakumar, C Vafa, hep-th/9812127M-theory and topological strings. 2INSPIRER. Gopakumar and C. Vafa, M-theory and topological strings. 2, hep-th/9812127 [INSPIRE]. Matrix models, geometric engineering and elliptic genera. T J Hollowood, A Iqbal, C Vafa, 10.1088/1126-6708/2008/03/069hep-th/0310272JHEP. 0369INSPIRET.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE]. Deformed topological partition function and Nekrasov backgrounds. I Antoniadis, S Hohenegger, K S Narain, T R Taylor, 10.1016/j.nuclphysb.2010.04.021arXiv:1003.2832Nucl. Phys. B. 838253INSPIREI. Antoniadis, S. Hohenegger, K.S. Narain and T.R. Taylor, Deformed topological partition function and Nekrasov backgrounds, Nucl. Phys. B 838 (2010) 253 [arXiv:1003.2832] [INSPIRE]. Worldsheet realization of the refined topological string. I Antoniadis, I Florakis, S Hohenegger, K S Narain, A Zein Assi, 10.1016/j.nuclphysb.2013.07.004arXiv:1302.6993Nucl. Phys. B. 875101INSPIREI. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Worldsheet realization of the refined topological string, Nucl. Phys. B 875 (2013) 101 [arXiv:1302.6993] [INSPIRE]. Non-perturbative Nekrasov partition function from string theory. I Antoniadis, I Florakis, S Hohenegger, K S Narain, A Zein Assi, 10.1016/j.nuclphysb.2014.01.006arXiv:1309.6688Nucl. Phys. B. 88087INSPIREI. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Non-perturbative Nekrasov partition function from string theory, Nucl. Phys. B 880 (2014) 87 [arXiv:1309.6688] [INSPIRE]. Probing the moduli dependence of refined topological amplitudes. I Antoniadis, I Florakis, S Hohenegger, K S Narain, A Zein Assi, 10.1016/j.nuclphysb.2015.10.016arXiv:1508.01477Nucl. Phys. B. 901252INSPIREI. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Probing the moduli dependence of refined topological amplitudes, Nucl. Phys. B 901 (2015) 252 [arXiv:1508.01477] [INSPIRE]. Integrating over Higgs branches. G W Moore, N Nekrasov, S Shatashvili, 10.1007/PL00005525hep-th/9712241Commun. Math. Phys. 20997INSPIREG.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE]. Testing Seiberg-Witten solution. A Lossev, N Nekrasov, S L Shatashvili, hep-th/9801061NATO Sci. Ser. C. 520359INSPIREA. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, NATO Sci. Ser. C 520 (1999) 359 [hep-th/9801061] [INSPIRE]. Seiberg-Witten prepotential from instanton counting. N A Nekrasov, 10.4310/ATMP.2003.v7.n5.a4hep-th/0206161Adv. Theor. Math. Phys. 7831INSPIREN.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE]. M R Douglas, hep-th/9311130NATO advanced research workshop on new developments in string theory, conformal models and topological field theory. Conformal field theory techniques in large N Yang-Mills theory. INSPIREM.R. Douglas, Conformal field theory techniques in large N Yang-Mills theory, in NATO advanced research workshop on new developments in string theory, conformal models and topological field theory, (1993) [hep-th/9311130] [INSPIRE]. Chiral deformations of conformal field theories. R Dijkgraaf, 10.1016/S0550-3213(97)00153-3hep-th/9609022Nucl. Phys. B. 493588INSPIRER. Dijkgraaf, Chiral deformations of conformal field theories, Nucl. Phys. B 493 (1997) 588 [hep-th/9609022] [INSPIRE]. . JHEP04. 275JHEP04(2021)275 Mirror symmetry and elliptic curves, in The moduli space of curves. R Dijkgraaf, 10.1007/978-1-4612-4264-2_5pg. 149Birkhäuser, Boston, MA, U.S.A.R. Dijkgraaf, Mirror symmetry and elliptic curves, in The moduli space of curves, Birkhäuser, Boston, MA, U.S.A. (1995), pg. 149. The theory of Jacobi forms. M Eichler, D Zagier, 10.1007/978-1-4684-9162-3Birkhäuser, Boston, MA, U.S.A.M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, Boston, MA, U.S.A. (1985). Introduction to modular forms. S Lang, 10.1007/978-3-642-51447-0SpringerBerlin, Heidelberg, GermanyS. Lang, Introduction to modular forms, Springer, Berlin, Heidelberg, Germany (1987). Modular forms, a computational approach. W Stein, American Mathematical SocietyProvidence, RI, U.S.A.W. Stein, Modular forms, a computational approach, American Mathematical Society, Providence, RI, U.S.A. (2007). A Libgober, arXiv:0904.1026Elliptic genera, real algebraic varieties and quasi-Jacobi forms. A. Libgober, Elliptic genera, real algebraic varieties and quasi-Jacobi forms, arXiv:0904.1026. Conformal and current algebras on general Riemann surface. T Eguchi, H Ooguri, 10.1016/0550-3213(87)90686-9Nucl. Phys. B. 282308INSPIRET. Eguchi and H. Ooguri, Conformal and current algebras on general Riemann surface, Nucl. Phys. B 282 (1987) 308 [INSPIRE]. Equations for two point correlation functions on compact Riemann surfaces. S M Kuzenko, O A Solovev, 10.1007/BF01027692Theor. Math. Phys. 88901Teor. Mat. Fiz.. INSPIRES.M. Kuzenko and O.A. Solovev, Equations for two point correlation functions on compact Riemann surfaces, Theor. Math. Phys. 88 (1991) 901 [Teor. Mat. Fiz. 88 (1991) 323] [INSPIRE]. Motivic multiple zeta values and superstring amplitudes. O Schlotterer, S Stieberger, 10.1088/1751-8113/46/47/475401arXiv:1205.1516J. Phys. A. 46475401INSPIREO. Schlotterer and S. Stieberger, Motivic multiple zeta values and superstring amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE]. All order α -expansion of superstring trees from the Drinfeld associator. J Broedel, O Schlotterer, S Stieberger, T Terasoma, 10.1103/PhysRevD.89.066014arXiv:1304.7304Phys. Rev. D. 8966014INSPIREJ. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma, All order α -expansion of superstring trees from the Drinfeld associator, Phys. Rev. D 89 (2014) 066014 [arXiv:1304.7304] [INSPIRE]. Polylogarithms, multiple zeta values and superstring amplitudes. J Broedel, O Schlotterer, S Stieberger, 10.1002/prop.201300019arXiv:1304.7267Fortsch. Phys. 61812INSPIREJ. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, multiple zeta values and superstring amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE]. Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator. S Stieberger, 10.1088/1751-8113/47/15/155401arXiv:1310.3259J. Phys. A. 47155401INSPIRES. Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A 47 (2014) 155401 [arXiv:1310.3259] [INSPIRE]. Elliptic multiple zeta values and one-loop superstring amplitudes. J Broedel, C R Mafra, N Matthes, O Schlotterer, 10.1007/JHEP07(2015)112arXiv:1412.5535JHEP. 07112INSPIREJ. Broedel, C.R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112 [arXiv:1412.5535] [INSPIRE]. Elliptic multiple zeta value. N Matthes, Hamburg, GermanyUniversität HamburgPh.D. thesisN. Matthes, Elliptic multiple zeta value, Ph.D. thesis, Universität Hamburg, Hamburg, Germany (2016). A class of non-holomorphic modular forms I. F Brown, 10.1007/s40687-018-0130-8arXiv:1707.01230Res. Math. Sci. 57INSPIREF. Brown, A class of non-holomorphic modular forms I, Res. Math. Sci. 5 (2018) 7 [arXiv:1707.01230] [INSPIRE]. F Brown, arXiv:1708.03354A class of non-holomorphic modular forms II: equivariant iterated Eisenstein integrals. F. Brown, A class of non-holomorphic modular forms II: equivariant iterated Eisenstein integrals, arXiv:1708.03354. From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop. J Broedel, O Schlotterer, F Zerbini, 10.1007/JHEP01(2019)155arXiv:1803.00527JHEP. 01155INSPIREJ. Broedel, O. Schlotterer and F. Zerbini, From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop, JHEP 01 (2019) 155 [arXiv:1803.00527] [INSPIRE]. Absence of irreducible multiple zeta-values in melon modular graph functions. E Hoker, M B Green, 10.4310/CNTP.2020.v14.n2.a2arXiv:1904.06603Commun. Num. Theor. Phys. 14315INSPIREE. D'Hoker and M.B. Green, Absence of irreducible multiple zeta-values in melon modular graph functions, Commun. Num. Theor. Phys. 14 (2020) 315 [arXiv:1904.06603] [INSPIRE]. Genus-zero and genus-one string amplitudes and special multiple zeta values. D Zagier, F Zerbini, 10.4310/CNTP.2020.v14.n2.a4arXiv:1906.12339Commun. Num. Theor. Phys. 14413INSPIRED. Zagier and F. Zerbini, Genus-zero and genus-one string amplitudes and special multiple zeta values, Commun. Num. Theor. Phys. 14 (2020) 413 [arXiv:1906.12339] [INSPIRE].
[]
[ "YOUR CLASSIFIER IS SECRETLY AN ENERGY BASED MODEL AND YOU SHOULD TREAT IT LIKE ONE", "YOUR CLASSIFIER IS SECRETLY AN ENERGY BASED MODEL AND YOU SHOULD TREAT IT LIKE ONE" ]
[ "Will Grathwohl [email protected] \nGoogle Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n\n", "Kuan-Chieh Wang \nGoogle Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n\n", "Jörn-Henrik Jacobsen [email protected] \nGoogle Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n\n", "David Duvenaud [email protected] \nGoogle Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n\n", "Kevin Swersky [email protected] \nGoogle Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n\n", "Mohammad Norouzi [email protected] \nGoogle Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n\n" ]
[ "Google Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n", "Google Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n", "Google Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n", "Google Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n", "Google Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n", "Google Research\nUniversity of Toronto & Vector Institute Google Research\nUniversity of Toronto & Vector Institute\nUniversity of Toronto & Vector Institute\n" ]
[]
We propose to reinterpret a standard discriminative classifier of p(y|x) as an energy based model for the joint distribution p(x, y). In this setting, the standard class probabilities can be easily computed as well as unnormalized values of p(x) and p(x|y). Within this framework, standard discriminative architectures may be used and the model can also be trained on unlabeled data. We demonstrate that energy based training of the joint distribution improves calibration, robustness, and out-of-distribution detection while also enabling our models to generate samples rivaling the quality of recent GAN approaches. We improve upon recently proposed techniques for scaling up the training of energy based models and present an approach which adds little overhead compared to standard classification training. Our approach is the first to achieve performance rivaling the state-of-the-art in both generative and discriminative learning within one hybrid model.
null
[ "https://arxiv.org/pdf/1912.03263v2.pdf" ]
208,857,409
1912.03263
9e950f0d2a5edf389788379544c84c7df7fbd574
YOUR CLASSIFIER IS SECRETLY AN ENERGY BASED MODEL AND YOU SHOULD TREAT IT LIKE ONE Will Grathwohl [email protected] Google Research University of Toronto & Vector Institute Google Research University of Toronto & Vector Institute University of Toronto & Vector Institute Kuan-Chieh Wang Google Research University of Toronto & Vector Institute Google Research University of Toronto & Vector Institute University of Toronto & Vector Institute Jörn-Henrik Jacobsen [email protected] Google Research University of Toronto & Vector Institute Google Research University of Toronto & Vector Institute University of Toronto & Vector Institute David Duvenaud [email protected] Google Research University of Toronto & Vector Institute Google Research University of Toronto & Vector Institute University of Toronto & Vector Institute Kevin Swersky [email protected] Google Research University of Toronto & Vector Institute Google Research University of Toronto & Vector Institute University of Toronto & Vector Institute Mohammad Norouzi [email protected] Google Research University of Toronto & Vector Institute Google Research University of Toronto & Vector Institute University of Toronto & Vector Institute YOUR CLASSIFIER IS SECRETLY AN ENERGY BASED MODEL AND YOU SHOULD TREAT IT LIKE ONE Under review as a conference paper at ICLR 2020 We propose to reinterpret a standard discriminative classifier of p(y|x) as an energy based model for the joint distribution p(x, y). In this setting, the standard class probabilities can be easily computed as well as unnormalized values of p(x) and p(x|y). Within this framework, standard discriminative architectures may be used and the model can also be trained on unlabeled data. We demonstrate that energy based training of the joint distribution improves calibration, robustness, and out-of-distribution detection while also enabling our models to generate samples rivaling the quality of recent GAN approaches. We improve upon recently proposed techniques for scaling up the training of energy based models and present an approach which adds little overhead compared to standard classification training. Our approach is the first to achieve performance rivaling the state-of-the-art in both generative and discriminative learning within one hybrid model. INTRODUCTION For decades, research on generative models has been motivated by the promise that generative models can benefit downstream problems such as semi-supervised learning, imputation of missing data, and calibration of uncertainty (e.g., Chapelle et al. (2006);Dempster et al. (1977)). Yet, most recent research on deep generative models ignores these problems, and instead focuses on qualitative sample quality and log-likelihood on heldout validation sets. Currently, there is a large performance gap between the strongest generative modeling approach to downstream tasks of interest and hand-tailored solutions for each specific problem. One potential explanation is that most downstream tasks are discriminative in nature and state-of-the-art generative models have diverged quite heavily from state-of-the-art discriminative architectures. Thus, even when trained solely as classifiers, the performance of generative models is far below the performance of the best discriminative models. Hence, the potential benefit from the generative component of the model is far outweighed by the decrease in discriminative performance. Recent work (Behrmann et al., 2018;Chen et al., 2019) attempts to improve the discriminative performance of generative models by leveraging invertible architectures, but these methods still underperform their purely discriminative counterparts jointly trained as generative models. This paper advocates the use of energy based models (EBMs) to help realize the potential of generative models on downstream discriminative problems. While EBMs are currently challenging to work with, they fit more naturally within a discriminative framework than other generative models and facilitate the use of modern classifier architectures. Figure 1 illustrates an overview of the architecture, where the logits of a classifier are re-interpreted to define the joint density of data points and labels and the density of data points alone. The contributions of this paper can be summarized as: 1) We present a novel and intuitive framework for joint modeling of labels and data. 2) Our models considerably outperform previous state-of-the-art hybrid models at both generative and discriminative modeling. 3) We show that the incorporation of generative modeling gives our models improved calibration, out-of-distribution detection, and adversarial robustness, performing on par with or better than hand-tailored methods for multiple tasks. ENERGY BASED MODELS Energy based models (LeCun et al., 2006) hinge on the observation that any probability density p(x) for x ∈ R D can be expressed as p θ (x) = exp(−E θ (x)) Z(θ) ,(1) where E θ (x) : R D → R, known as the energy function, maps each point to a scalar, and Z(θ) = x exp(−E θ (x)) is the normalizing constant (with respect to x) also known as the partition function. Thus, one can parameterize an EBM using any function that takes x as the input and returns a scalar. For most choices of E θ , one cannot compute or even reliably estimate Z(θ), which means estimating the normalized densities is intractable and standard maximum likelihood estimation of the parameters, θ, is not straightforward. Thus, we must rely on other methods to train EBMs. We note that the derivative of the log-likelihood for a single example x with respect to θ can be expressed as ∂ log p θ (x) ∂θ = E p θ (x ) ∂E θ (x ) ∂θ − ∂E θ (x) ∂θ ,(2) where the expectation is over the model distribution. Unfortunately, we cannot easily draw samples from p θ (x), so we must resort to MCMC to use this gradient estimator. This approach was used to train some of the earliest EBMs. For example, Restricted Boltzmann Machines (Hinton, 2002) were trained using a block Gibbs sampler to approximate the expectation in Eq. (2). Despite a long period of little development, there has been recent work using this method to train large-scale EBMs on high-dimensional data, parameterized by deep neural networks (Nijkamp et al., 2019b;a;Du & Mordatch, 2019;Xie et al., 2016). These recent successes have approximated the expectation in Eq. (2) using a sampler based on Stochastic Gradient Langevin Dynamics (SGLD) (Welling & Teh, 2011) which draws samples following x 0 ∼ p 0 (x), x i+1 = x i − α 2 ∂E θ (x i ) ∂x i + , ∼ N (0, α)(3) where p 0 (x) is typically a Uniform distribution over the input domain and the step-size α should be decayed following a polynomial schedule. In practice the step-size, α, and the standard deviation of is often chosen separately leading to a biased sampler which allows for faster training. See Appendix H.1 for further discussion of samplers for EBM training. WHAT YOUR CLASSIFIER IS HIDING In modern machine learning, a classification problem with K classes is typically addressed using a parametric function, f θ : R D → R K , which maps each data point x ∈ R D to K real-valued numbers known as logits. These logits are used to parameterize a categorical distribution using the so-called Softmax transfer function: p θ (y | x) = exp (f θ (x)[y]) y exp (f θ (x)[y ]) ,(4) where f θ (x)[y] indicates the y th index of f θ (x), i.e., the logit corresponding the the y th class label. Our key observation in this work is that one can slightly re-interpret the logits obtained from f θ to define p(x, y) and p(x) as well. Without changing f θ , one can re-use the logits to define an energy based model of the joint distribution of data point x and labels y via: p θ (x, y) = exp (f θ (x)[y]) Z(θ) ,(5) where Z(θ) is the unknown normalizing constant and E θ (x, y) = −f θ (x)[y]. By marginalizing out y, we obtain an unnormalized density model for x as well, p θ (x) = y p θ (x, y) = y exp (f θ (x)[y]) Z(θ) .(6) Notice now that the LogSumExp(·) of the logits of any classifier can be re-used to define the energy function at a data point x as E θ (x) = −LogSumExp y (f θ (x)[y]) = − log y exp(f θ (x)[y]) .(7) Unlike typical classifiers, where shifting the logits f θ (x) by an arbitrary scalar does not affect the model at all, in our framework, shifting the logits for a data point x will affect log p θ (x). Thus, we are making use of the extra degree of freedom hidden within the logits to define the density function over input examples as well as the joint density among examples and labels. Finally, when we compute p θ (y | x) via p θ (x, y)/p θ (x) by dividing Eq. (5) to Eq. (6), the normalizing constant cancels out, yielding the standard Softmax parameterization in Eq. (4). Thus, we have found a generative model hidden within every standard discriminative model! Since our approach proposes to reinterpret a classifier as a Joint Energy based Model we refer to it throughout this work as JEM. OPTIMIZATION We now wish to take advantage of our new interpretation of classifier architectures to gain the benefits of generative models while retaining strong discriminative performance. Since our model's parameterization of p(y|x) is normalized over y, it is simple to maximize its likelihood as in standard classifier training. Since our models for p(x) and p(x, y) are unnormalized, maximizing their likelihood is not as easy. There are many ways we could train f θ to maximize the likelihood of the data under this model. We could apply the gradient estimator of Equation 2 to the likelihood under the joint distribution of Equation 5. Using Equations 6 and 4, we can also factor the likelihood as log p θ (x, y) = log p θ (x) + log p θ (y|x). The estimator of Equation 2 is biased when using a MCMC sampler with a finite number of steps. Given that the goal of our work is to incorporate EBM training into the standard classification setting, the distribution of interest is p(y|x). For this reason we propose to train using the factorization of Equation 8 to ensure this distribution is being optimized with an unbiased objective. We optimize p(y|x) using standard cross-entropy and optimize log p(x) using Equation 2 with SGLD where gradients are taken with respect to LogSumExp y (f θ (x)[y]). We find alternative factorings of the likelihood lead to considerably worse performance as can be seen in Section 5.1. Following Du & Mordatch (2019) we use persistent contrastive divergence (Tieleman, 2008) to estimate the expectation in the right-hand-side of Equation 2 since it gives an order of magnitude savings in computation compared to seeding new chains at each iteration as in Nijkamp et al. (2019b). This comes at the cost of decreased training stability. These trade-offs are discussed in Appendix H.2. APPLICATIONS We completed a thorough empirical investigation to demonstrate the benefits of JEM over standard classifiers. First, we achieved performance rivaling the state of the art in both discriminative and generative modeling. Even more interesting, we observed a number of benefits related to the practical application of discriminative models including improved uncertainty quantification, outof-distribution detection, and robustness to adversarial examples. Generative models have been long-expected to provide these benefits but have never been demonstrated to do so at this scale. All architectures used are based on Wide Residual Networks (Zagoruyko & Komodakis, 2016) where we have removed batch-normalization 1 to ensure that our models' outputs are deterministic functions of the input. This slightly increases classification error of a WRN-28-10 from 4.2% to 6.4% on CIFAR10 and from 2.3 to 3.4% on SVHN. All models were trained in the same way with the same hyper-parameters which were tuned on CIFAR10. Intriguingly, the SGLD sampler parameters found here generalized well across datasets and model architectures. All models are trained on a single GPU in approximately 36 hours. Full experimental details can be found in Appendix A. First, we show that a given classifier architecture can be trained as an EBM to achieve competitive performance as both a classifier and a generative model. We train JEM on CIFAR10, SVHN, and CIFAR100 and compare against other hybrid models as well as standalone generative and discriminative models. We find JEM performs near the state of the art in both tasks simultaneously, outperforming other hybrid models ( Table 1). HYBRID Given that we cannot compute normalized likelihoods, we present inception scores (IS) (Salimans et al., 2016) and Frechet Inception Distance (FID) (Heusel et al., 2017) as a proxy for this quantity. We find that JEM is competitive with SOTA generative models at these metrics. These metrics are not commonly reported on CIFAR100 and SVHN so we present accuracy and qualitative samples on these datasets. Our models achieve 96.7% and 72.2% accuracy on SVHN and CIFAR100, respectively. Samples from JEM can be seen in Figures 2, 3 and in Appendix C. JEM is trained to maximize the likelihood factorization shown in Eq. 8. This was to ensure that no bias is added into our estimate of log p(y|x) which can be computed exactly in our setup. Prior work (Du & Mordatch, 2019;Xie et al., 2016) proposes to factorize the objective as log p(x|y)+log p(y). In these works, each p(x|y) is a separate EBM with a distinct, unknown normalizing constant, meaning that their model cannot be used to compute p(y|x) or p(x). This explains why the model of Du & Mordatch (2019) (we will refer to this model as IGEBM) is not a competitive classifier. As an ablation, we trained JEM to maximize this objective and found a considerable decrease in discriminative performance (see Table 1, row 4). CALIBRATION A classifier is considered calibrated if its predictive confidence, max y p(y|x), aligns with its misclassification rate. Thus, when a calibrated classifier predicts label y with confidence .9 it should have a 90% chance of being correct. This is an important feature for a model to have when deployed in real-world scenarios where outputting an incorrect decision can have catastrophic consequences. The classifier's confidence can be used to decide when to output a prediction or deffer to a human, for example. Here, a well-calibrated, but less accurate classifier can be considerably more useful than a more accurate, but less-calibrated model. Accuracy Confidence Confidence While classifiers have grown more accurate in recent years, they have also grown considerably less calibrated (Guo et al., 2017). Contrary to this behavior, we find that JEM notably improves classification while retaining high accuracy. We focus on CIFAR100 since SOTA classifiers achieve approximately 80% accuracy. We train JEM on this dataset and compare to a baseline of the same architecure without EBM training. Our baseline model achieves 74.2% accuracy and JEM achieves 72.2% (for reference, a ResNet-110 achieves 74.8% accuracy (Zagoruyko & Komodakis, 2016)). We find the baseline model is very poorly calibrated outputting highly over-confident predictions. Conversely, we find JEM produces a nearly perfectly calibrated classifier when measured with Expected Calibration Error (see Appendix E.1). Compared to other calibration methods such as Platt scaling (Guo et al., 2017), JEM requires no additional training data. Results can be seen in Figure 4 and additional results can be found in Appendix E.2. OUT-OF-DISTRIBUTION DETECTION In general, out-of-distribution (OOD) detection is a binary classification problem, where the model is required to produce a score s θ (x) ∈ R, where x is the query, and θ is the set of learnable parameters. We desire that the scores for in-distribution examples are higher than that out-of-distribution examples. Typically for evaluation, threshold-free metrics are used, such as the area under the receiver-operating curve (AU-ROC) (Hendrycks & Gimpel, 2016). There exist a number of distinct OOD detection approaches to which JEM can be applied. We expand on them below. Further results and experimental details can be found in Appendix F.2. INPUT DENSITY A natural approach to OOD detection is to fit a density model on the data and consider examples with low likelihood to be OOD. While intuitive, this approach is currently not competitive on highdimensional data. Nalisnick et al. (2018) showed that tractable deep generative models such as Kingma & Dhariwal (2018) and Salimans et al. (2017) can assign higher densities to OOD examples than in-distribution examples. Further work (Nalisnick et al., 2019) shows examples where the densities of an OOD dataset are completely indistinguishable from the in-distribution set, e.g., see Table 2, column 1. Conversely, Du & Mordatch (2019) have shown that the likelihoods from EBMs can be reliably used as a predictor for OOD inputs. As can be seen in Table 2 column 2, JEM consistently assigns higher likelihoods to in-distribution data than OOD data. One possible explanation for JEM's further improvement over IGEBM is its ability to incorporate labeled information during training while also being able to derive a principled model of p(x). Intriguingly, Glow does not appear to benefit in the same way from this supervision as is demonstrated by the little difference between our unconditional and class-conditional Glow results. Quantitative results can be found in Table 3 (top). PREDICTIVE DISTRIBUTION Many successful approaches have utilized a classifier's predictive distribution for OOD detection (Gal & Ghahramani, 2016;Wang et al., 2018;Liang et al., 2017). A useful OOD score that can be derived from this distribution is the maximum prediction probability: s θ (x) = max y p θ (y|x) (Hendrycks & Gimpel, 2016). It has been demonstrated that OOD performance using this score is highly correlated with a model's classification accuracy. Since JEM is a competitive classifier, we find it performs on par (or beyond) the performance of a strong baseline classifier and considerably outperforms other generative models. Results can be seen in Table 3 (middle). A NEW SCORE: APPROXIMATE MASS It has been recently proposed that likelihood may not be enough for OOD detection in high dimensions (Nalisnick et al., 2019). It is possible for a point to have high likelihood under a distribution yet be nearly impossible to be sampled. Real samples from a distribution lie in what is known as the "typical" set. This is the area of high probability mass. A single point may have high density but if the surrounding areas have very low density, then that point is likely not in the typical set and therefore likely not a sample from the data distribution. For a high-likelihood datapoint outside of the typical set, we expect the density to change rapidly around it, thus the norm of the gradient of the log-density will be large compared to examples in the typical set (otherwise it would be in an area of high mass). We propose an alternative OOD score based on this quantity: s θ (x) = − ∂ log p θ (x) ∂x 2 .(9) For EBMs (JEM and IGEBM), we find this predictor greatly outperforms our own and other generative model's likelihoods -see Table 2 column 3. For tractable likelihood methods we find this predictor is anti-correlated with the model's likelihood and neither is reliable for OOD detection. Results can be seen in Table 3 (bottom). ROBUSTNESS Recent work (Athalye et al., 2017) has demonstrated that classifiers trained to be adversarially robust can be re-purposed to generate convincing images, do in-painting, and translate examples from one class to another. This is done through an iterative refinement procedure, quite similar to the SGLD used to sample from EBMs. We also note that adversarial training (Goodfellow et al., 2014) bears many similarities to SGLD training of EBMs. In both settings, we use a gradient-based optimization procedure to generate examples which activate a specific high-level network activation, then optimize the weights of the network to minimize the generated example's effect on that activation. Further connections have been drawn between adversarial training and regularizing the gradients of the network's activations around the data (Simon-Gabriel et al., 2018). This is similar to the objective of Score Matching (Hyvärinen, 2005) which can also be used to train EBMs (Kingma & Lecun, 2010;Song & Ermon, 2019). Given these connections one may wonder if a classifier derived from an EBM would be more robust to adversarial examples than a standard model. This behavior has been demonstrated in prior work on EBMs (Du & Mordatch, 2019) but their work did not produce a competitive discriminative model and is therefore of limited practical application for this purporse. Similarly, we find JEM achieves considerable robustness without sacrificing discriminative performance. Glow log p(x) JEM log p(x) Approx. Mass JEM SVHN CIFAR100 CelebA IMPROVED ROBUSTNESS THROUGH EBM TRAINING inputsx = x + δ, which change a model's pre- diction subject to ||x−x|| p < . These examples exploit semantically meaningless perturbations to which the model is overly sensitive. However, closeness to real inputs in terms of a given metric does not imply that adversarial examples reside within areas of high density according to the model distribution, hence it is not surprising that the model makes mistakes when asked to classify inputs it has rarely or never encountered during training. This insight has been used to detect and robustly classify adversarial examples with generative models (Song et al., 2017;Fetaya et al., 2019). The state-of-theart method for adversarial robustness on MNIST classifies by comparing an input to samples generated from a classconditional generative model (Schott et al., 2018). This can be thought of as classifying an example similar to the input but from an area of higher density under the model's learned distribution. This refined input resides in areas where the model has already "seen" sufficient data and is thus able to accurately classify. Albeit promising, this family of methods has not been able to scale beyond MNIST due to a lack of sufficiently powerful conditional generative models. We believe JEM can help close this gap. We propose to run a few iterations of our model's sampling procedure seeded at a given input. This should be able to transform low-probability inputs to a nearby point of high probability, "undoing" any adversarial attack and enabling the model to classify robustly. Perturbation Robustness We run a number of powerful adversarial attacks on our CIFAR10 models. We run a white-box PGD attack, giving the attacker access to the gradients through our sampling procedure 2 . Because our sampling procedure is stochastic, we compute the "expectation over transformations" Athalye et al. (2018), the expected gradient over multiple runs of the sampling procedure. We also run gradient-free black-box attacks; the boundary attack and the brute-force pointwise attack . All attacks are run with respect to the L 2 and L ∞ norms and we test JEM with 0, 1, and 10 steps of sampling seeded at the input. Results from the PGD experiments can be seen in Figure 5. Experimental details and remaining results, including gradient-free attacks, can be found in Appendix G. Our model is considerably more robust than a baseline with standard classifier training. With respect to both norms, JEM delivers considerably improved robustness when compared to the baseline but for many epsilons falls below state-of-the-art adversarial training (Madry et al., 2017;Santurkar et al., 2019) and the state-of-the-art certified robutness method of Salman et al. (2019) ("RandAdvSmooth" in Figure 5). We note that each of these baseline methods is trained to be robust to the norm through which it is being attacked and it has been shown that attacking an L ∞ adversarially trained model with an L 2 adversary decreases robustness considerably (Madry et al., 2017). However, we attack the same JEM model with both norms and observe competitive robustness in both cases. JEM with 0 steps refinement is noticeably more robust than the baseline model trained as a standard classifier, thus simply adding EBM training can produce more robust models. We also find that increasing the number of refinement steps further increases robustness to levels at robustness-specific approaches. We expect that increasing the number of refinement steps will lead to more robust models but due to computational constraints we could not run attacks in this setting. Figure 6: Distal Adversarials. Confidently classified images generated from noise, such that: p(y = "car"|x) > .9. Distal Adversarials Another common failure mode of non-robust models is their tendency to classify nonsensical inputs with high confidence. To analyze this property, we follow Schott et al. (2018). Starting from noise we generate images to maximize p(y = "car"|x). Results are shown in figure 6. The baseline confidently classifies unstructured noise images. The L 2 adversarially trained ResNet with = 0.5 (Santurkar et al., 2019) confidently classifies somewhat structured, but unrealistic images. JEM does not confidently classify nonsensical images, so instead, car attributes and natural image properties visibly emerge. LIMITATIONS Energy based models can be very challenging to work with. Since normalized likelihoods cannot be computed, it can be hard to verify that learning is taking place at all. When working in domains such as images, samples can be drawn and checked to assess learning, but this is far from a generalizable strategy. Even so, these samples are only samples from an approximation to the model so they can only be so useful. Furthermore, the gradient estimators we use to train JEM are quite unstable and are prone to diverging if the sampling and optimization parameters are not tuned correctly. Regularizers may be added (Du & Mordatch, 2019) to increase stability but it is not clear what effect they have on the final model. The models used to generate the results in this work regularly diverged throughout training, requiring them to be restarted with lower learning rates or with increased regularization. See Appendix H.3 for a detailed description of how these difficulties were handled. While this may seem prohibitive, we believe the results presented in this work are sufficient to motivate the community to find solutions to these issues as any improvement in the training of energy based models will further improve the results we have presented in this work. RELATED WORK Prior work (Xie et al., 2016) made a similar observation to ours about classifiers and EBMs but define the model differently. They reinterpret the logits to define a class-conditional EBM p(x|y), similar to Du & Mordatch (2019). This setting requires additional parameters to be learned to derive a classifier and an unconditional model. We believe this subtle distinction is responsible for our model's success. The model of (Song & Ou, 2018) is similar as well but is trained using a GAN-like generator and is applied to different applications. Our work builds heavily on Nijkamp et al. (2019b;a); Du & Mordatch (2019) which scales the training of EBMs to high-dimensional data using Contrastive Divergence and SGLD. While these works have pushed the boundaries of the types of data to which we can apply EBMs, many issues still exist. These methods require many steps of SGLD to take place at each training iteration. Each step requires approximately the same amount of computation as one iteration of standard discriminitive model training, therefore training EBMs at this scale is orders of magnitude slower than training a classifier -limiting the size of problems we can attack with these methods. There exist orthogonal approaches to training EBMs which we believe have promise to scale more gracefully. Score matching (Hyvärinen, 2005) attempts to match the derivative of the model's density with the derivative of the data density. This approach saw some development towards high-dimensional data (Kingma & Lecun, 2010) and recently has been successfully applied to large natural images (Song & Ermon, 2019). This approach required a model to output the derivatives of the density function, not the density function itself, so it is unclear what utility this model can provide to the applications we have discussed in this work. Regardless, we believe this is a promising avenue for further research. Noise Contrastive Estimation (Gutmann & Hyvärinen, 2010) rephrases the density estimation problem as a classification problem, attempting to distinguish data from a known noise distribution. If the classifier is properly structured, then once the classification problem is solved, an unnormalized density estimator can be derived from the classifier and noise distribution. While this method has been recently extended (Ceylan & Gutmann, 2018), these methods are challenging to extend to high-dimensional data. CONCLUSION AND FURTHER WORK In this work we have presented JEM, a novel reinterpretation of standard classifier architectures which retains the strong performance of SOTA discriminative models while adding the benefits of generative modeling approaches. Our work is enabled by recent work scaling techniques for training EBMs to high dimensional data. We have demonstrated the utility of incorporating this type of training into discriminative models. While there exist many issues in training EBMs we hope the results presented here will encourage the community to improve upon current approaches. A TRAINING DETAILS We train all models with the Adam optimizer (Kingma & Ba, 2014) for 150 epochs through the dataset using a staircase decay schedule. All network architecutres are based on WideResNet-28-10 with no batch normalization. We generate samples using PCD with hyperparameters in Table 4. We evolve the chains with 20-steps of SGLD per iteration and with probability .05 we reiniatilize the chains with uniform random noise. For preprocessing, we scale images to the range [−1, 1] and add Gaussian noise of stddev = .03. Pseudo-code for our training procedure is in Algorithm 1. When training via contrastive divergence there are a few different ways one could potentially draw samples from p θ (x). We could: 1. Sample y ∼ p(y) then sample x ∼ p θ (x|y) via SGLD with energy E(x|y) = −f θ (x)[y] then throw away y. 2. Sample x ∼ p θ (x) via SGLD with energy E(x) = −LogSumExp y f θ (x)[y]. We experimented with both methods during training and found that while method 1 produced more visually appealing samples (to a human's perspective), method 2 produced slightly stronger discirminative performance -92.9% vs. 91.2% accuracy on CIFAR10. For this reason we use method 2 in all results presented. Algorithm 1 JEM training: Given network f θ , SGLD step-size α, SGLD noise σ, replay buffer B, SGLD steps η, reinitialization frequency ρ 1: while not converged do 2: Sample x and y from dataset 3: L clf (θ) = xent(f θ (x), y) 4: Sample x 0 ∼ B with probability 1 − ρ, else x 0 ∼ U(−1, 1) Initialize SGLD 5: for t ∈ [1, 2, . . . , η] do SGLD 6: In this section we describe the details for reproducing the Inception Score (IS) and FID results reported in the paper. First we note that both IS and FID are scores computed based on a pretrained classifier network, and thus can be very dependent on the exact model/code repository used. For a more detailed discussion on the variability of IS, please refer to Barratt & Sharma (2018). To gauge our model against the other papers, we document our attempt to fairly compare the scores across papers in Table 6. As a direct comparison of IS, we got 8.76 using the code provided by Du & Mordatch (2019), and is better than their best reported score of 8.3. For FID, we used the official implementation from Heusel et al. (2017). Note that FID computed from this repository assigned much worse FID than reported in Chen et al. (2019). x t = x t−1 + α · ∂LogSumExp y (f θ ( xt−1)[y ]) ∂ xt−1 + σ · N ( Conditional vs unconditional samples. Since we are interested in training a Hybrid model, our model, by definition, is a conditional generative model as it has access to label information. In Table 5, unconditional samples mean samples directly obtained from running SGLD using p(x). Conditional samples are obtained by taking the max of our p(y|x) model. The reported scores are obtained by keeping the top 10 percentile samples with the highest p(y|x) values. Scores obtained on a "single" model are computed directly on the training replay buffer of the last checkpoint. "Ensemble" here are obtained by lumping together 5 buffers over the last few epochs of training. As we initialize SGLD with uniform noise, using the training buffer is exactly the same as re-sampling from the model. (2019), Heusel et al. (2017). *denotes numbers copied from Chen et al. (2019), but not the original papers. As unfortunate as the case is with Inception Score and FID (i.e., taking different code repository yields vastly different results), from this table we can still see that our model performs well. Using D&M Inception Score we beat their own model, and using the official repository for FID we beat the Glow 3 model by a big margin. C FURTHER HYBRID MODEL SAMPLES Additional samples from CIFAR10 and SVHN can be seen in Figure 7 and samples from CIFAR100 can be seen in Figure D QUALITATIVE ANALYSIS OF SAMPLES Visual quality is difficult to quantify. Of the known metrics like IS and FID, using samples that have higher p(y|x) values results in higher scores, but not necessary if we use samples with higher log p(x). However, this is likely because of the downfalls of the evaluation metrics themselves rather than reflecting true sample quality. Based on our analysis (below), we find 1. Our log p(x) model assigns values that cluster around different means for different classes. The class automobiles has the highest log p(x). Of all generated samples, all top 100 samples are of this class. 2. Given the class, the samples that have higher log p(x) values all have white background and centered object, and lower log p(x) samples have colorful (e.g., forest-like) background. 3. Of all samples, higher p(y|x) values means clearly centered objects, and lower p(y|x) otherwise. For a perfectly calibrated classifier, this value will be 0 for any choice of M . In our analysis, we choose M = 20 throughout. E.2 FURTHER RESULTS We find that JEM also improves calibration on CIFAR10 as can be seen in Table 13. There we see an improvement in calibration, but both classifiers are well calibrated because their accuracy is so high. In a more interesting experiment, we limit the size of the training set to 4,000 labeled examples. In this setting the accuracy drops to 78.0% and 74.9% in the baseline and JEM, respectively. Given the JEM can be trained on unlabeled data, we treat the remainder of the training set as unlabeled and train in a semi-supervised manner. We find this gives a noticeable boost in the classifier's calibration as seen in Figure 13. Surprisingly this did not improve generalization. We leave exploring this phenomenon for future work. To obtain OOD results for unconditional Glow, we used the pre-trained model and implementation of https://github.com/y0ast/Glow-PyTorch. We trained a Class-Conditional model as well using this codebase which was used to generate the class-conditional OOD results. We obtained the IGEBM of Du & Mordatch (2019) from their open-source implementation at https://github.com/openai/ebm_code_release. For likelihood and likelihoodgradient OOD scores we used their pre-trained cifar10 large model uncond model. We were able to replicate the likelihood based OOD results presented in their work. We implemented our likelihood-gradient approximate-mass score on top of their codebase. For predictive distribution based OOD scores we used their cifar cond model which was the model used in their work to generate their robustness results. Figure 7 contains results on two datasets, Constant and Uniform, which were omitted for space. Most models perform very well at the Uniform dataset. On the Constant dataset (all examples = 0) generative models mainly fail -with JEM being the only one whose likelihoods can be used to derive a predictive score function for OOD detection. Intrestinly, we could not obtain approximate mass scores on this dataset from the Glow models due to numerical stability issues. F.2 FURTHER RESULTS G ATTACK DETAILS AND FURTHER ROBUSTNESS RESULTS We use foolbox for our experiments. PGD uses binary search to determine minimal epsilons for every input and we plot the resulting robustness-distortion curves. PGD runs with 20 random restarts and 40 iterations. For the boundary attack, we run default foolbox settings with one important difference. The random initialization often fails for JEM and thus we initialize the attack with a correclty classified input of another class. This other class is chosen based on the top-2 prediction for the image to be attacked. As all our attacks are expensive to run, we only attacked 300 randomly chosen inputs. The same randomly chosen inputs were used to attack each model. In Figure 14 we see the results of the boundary attack and pointwise attack on JEM and a baseline. The main point to running these attacks was to demonstrate that our model was not able to "cheat" by having vanishing gradients through our gradient-based sampling procedure. Since PGD was more successful than these gradient-free methods, this is clearly not the case and the attacker was able to use the gradients of the sampling procedure to attack our model. Further, we observe the same behavior across all attacks; the EBM with 0 steps sampling is more robust than the baseline and the robustness increases as we add more steps of sampling. We also compare JEM to the IGEBM of Du & Mordatch (2019) with 10 steps of sampling refinement, see Figure 15. We run the same gradient-based attacks on their model and find that despite not having competitive clean accuracy, it is quite robust to large attacks -especially with respect to the L ∞ norm. After = 12 their model is more robust than ours and after = 18 it is more robust than the adversarial training baseline. With respect to the L 2 norm their model is more robust than the adversarial training baseline above = 280 but remains less robust than JEM until = 525. We believe these results demonstrate that EBMs are a compelling class of models to explore for further work on building robust models. G.1 EXPECTATION OVER TRANSFORMATIONS Our SGLD-based refinement procedure is stochastic in nature and it has been shown that stochastic defenses to adversarial attacks can provide a false sense of security (Athalye et al., 2018). To deal with this, when we attack our stochastically refined classifiers, we average the classifier's predictions over multiple samples of this refinement procedure. This makes the defense more deterministic and easier to attack. We redefine the logits of our classifier as: log p k n (y|x) = 1 n n i=1 log p(y|x i ), x i ∼ SGLD(x, k)(11) where we have defined SGLD(x, k) as an SGLD chain run for k steps seeded at x. Intuitively, we draw n different samples {x i } n i=1 from our model seeded at input x, then compute log p(y|x i ) for each of these samples, then average the results. We then attack these averaged logits with PGD to generate the results in Figure 5. We experimented with different numbers of samples and found that 10 samples yields very similar results to 5 samples on JEM with one refinement step (see Figure 16). Because 10 samples took very long to run on the JEM model with ten refinement steps, we settled on using 5 samples in the results reported in the main body of the paper. : Comparing the effect of the number of samples in the EOT attack. We find negligible difference between 5 and 10 for JEM-1 (red and green curves). G.2 TRANSFER ATTACKS We would like to see if JEM's refinement procedure can correct adversarial perturbed inputs -inputs which cause the model to fail. To do this, we generate a series of adversarial examples for JEM-0, with respect to the l ∞ norm, and test the accuracy of JEM-{1,10} on these examples. Ideally, with further refinement the accuracy will increase. The results of this experiment can be seen in Figure 17. We see here that JEM's refinement procedure can correct for adversarial perturbations. H A DISCUSSION ON SAMPLERS H.1 IMPROPER SGLD Recall the transition kernel of SGLD: In the proper formulation of this sampler (Welling & Teh, 2011), the step-size and the variance of the Gaussian noise are related Var( ) = α. If the stepsize is decayed with a polynomial schedule, then samples from SGLD converge to samples from our unnomralized density as the number of steps goes to ∞. x 0 ∼ p 0 (x) x i+1 = x i − α 2 ∂E θ (x i ) ∂θ + , ∼ N (0, α) In practice, we approximate these samples with a sampler that runs for a finite number of steps. When using the proper step-size to noise ratio, the signal from the gradient is overtaken by the noise when step-sizes are large enough to be informative. In practice the sampler is typically "relaxed" in that different values are used for the step-size and the amount of Guassian noise added -typically the amount of noise is significantly reduced. While we are no longer working with a valid MCMC sampler, this approximation has been successfully applied in practice in most recent work scaling EBM training to high dimensional data (Nijkamp et al., 2019b;a;Du & Mordatch, 2019) with the exception of Song & Ermon (2019) (which develops a clever work-around). The model they train is actually an ensemble of models trained on data with different amounts of noise added. They use a proper SGLD sampler decaying the step size as they sample, moving from their high-noise models to their low-noise models. This provides one possible explanation for the compelling results of their model. In our work we have set the step-size α = 2 and draw ∼ N (0, .01 2 ). We have found these parameters to work well across a variety of datasets, domains, architectures, and sampling procedures (persistent vs. short-run). We believe they are a decent "starting place" for energy-functions parameterized by deep neural networks. H.2 PERSISTENT OR SHORT-RUN CHAINS? Both persistent and short-run markov chains have been able to succesfully train EBMs. Nijkamp et al. (2019a) presents a careful study of various samplers which can be used and the tradeoffs one makes when choosing one sampler over another. In our work we have found that if computation allows, short-run MCMC chains are preferable in terms of training stability. Given that each step of SGLD requires approximately the computation of 1 training iteration of a standard classifier we are incentivized to find a sampler which can stably train EBMs requiring as few steps as possible per training iteration. In our experiments we found the smallest number of SGLD steps we could take to stably train an EBM at the scale of this work was 80 steps. Even so, these models eventually would diverge late into training. At 80 steps, we found the cost of training to be prohibitively high compared to a standard classifier. We found that by using persistent markov chains, we could further reduce the number of steps per iteration to 20 and still allow for relatively stable training. This gave a 4x speedup over our fastest short-run MCMC sampler. Still, this PCD sampler was noticebly less stable than the fastest shortrun sampler we could use but we found the multiple factor increase in speed to be a worth-while trade-off. If time allows, we recommend using a short-run MCMC sampler with a large enough number of steps to be stable. Given that is not always possible on problems of scale, PCD can be made to work more efficiently, but at the cost of a greater number of stability-related hyper-parameters. These additional parameters include the buffer size and the re-initialization frequency of the Markov chains. We found both to be important for training stability and found no general recipe for which to set them. We ran most of our experiments with re-initialization frequency at 5%. A particualrly interesting observation we discovered while using PCD is that the model would use the length of the markov chains to encode semantic information. We found that when training models on CIFAR10, when chains were young they almost always could be identified as frogs. When chains were old they could almost always be identified as cars. This behavior is likely some degeneracy of PCD which would not be possible with a short-run MCMC since all chains have the same length. H.3 DEALING WITH INSTABILITY Training a model with the gradient estimator of Eq. (2) can be quite unstable -especially when combined with other objective as was the case with all models presented in this work. There exists a "stable region" of sorts when training these models where the energy values of the true data are in the same range as the energy values of the generated samples. Intuitively, if the generated samples create energies that are not trivially separated from the training data, then real learning has to take place. Nijkamp et al. (2019b;a) provide a careful analysis of this and we refer the reader there for a more in-depth analysis. We find that when using PCD occasionally throughout training a sample will be drawn from the replay buffer that has a considerably higher-than average energy (higher than the energy of a random initialization). This causes the gradients w.r.t this example to be orders of magnitude larger than gradients w.r.t the rest of the examples and causes the model to diverge. We tried a number of heuristic approaches such as gradient clipping, energy clipping, ignoring examples with atypical energy values, and many others but could not find an approach that stabilized training and did not hurt generative and discriminative performance. The only two approaches we found to consistently work to increase stability of a model which has diverged is to 1) decrease the learning rate and 2) increase the number of SGLD steps in each PCD iteration. Unfortunately, both of these approaches slow down learning. We also had some success simply restarting models from a saved checkpoint with a different random seed. This was the main approach taken unless the model was late into training. In this case, random restarts were less effective and we increased the number of SGLD steps from 20 to 40 which stabilized training. While we are very optimistic about the future of large-scale EBMs we believe these are the most important issues that must be addressed in order for these models to be succeful. Figure 1 : 1Visualization of our method, JEM, which defines a joint EBM from classifier architectures. Figure 2 : 2Flow (Chen et al., 2019), Glow (Kingma & Dhariwal, 2018), IGEBM (Du & Mordatch, 2019), SNGAN (Miyato et al., 2018), NCSN (Song & Ermon, 2019) CIFAR10 classconditional samples. Figure 3: Classconditional samples. Figure 4 : 4CIFAR100 calbration results. ECE = Expected Calibration Error(Guo et al., 2017), see Appendix E.1. Figure 5 : 5Adversarial Robustness Results with PGD attacks. JEM adds considerable robustness. A common threat model for adversarial robustness is that of perturbation-based adversarial examples with an L p -norm constraint(Goodfellow et al., 2014). They are defined as perturbed LL gen (θ) = LogSumExp y (f (x)[y ]) − LogSumExp y (f ( x t )[y ]) (θ) = L clf (θ) + L gen (θ) 8 Figure 7 : 7Class-conditional Samples. Left to right: CIFAR10, SVHN. Figure 8 : 8CIFAR100 Class-conditional Samples. Figure 9 : 9Each row corresponds to 1 class, subfigures corresponds to different values of log p(x). left: highest, mid: random, right: lowest. Figure 10 : 10Histograms (oriented horizontally for easier visual alignment) of log p(x) arranged by class. Figure 11 :Figure 12 : 1112left: samples with highest log p(x), right: left: samples with lowest log p(x) left: samples with highest p(y|x), right: left: samples with lowest p(y|x) Error (ECE) is a metric to measure the calibration of a classifier. It works by first computing the confidence, max y p(y|x i ), for each x i in some dataset. We then group the items into equally spaced buckets {B m } M m=1 based on the classifier's output confidence. For example, if M = 20, then B 0 would represent all examples for which the classifier's confidence was between 0.0 and 0.05. (B m ) − conf(B m )| (10) where n is the number of examples in the dataset, acc(B m ) is the averaged accuracy of the classifier of all examples in B m and conf(B m ) is the averaged confidence over all examples in B m . Figure 13 : 13) CIFAR100 Baseline (4k labels) (d) CIFAR100 JEM (4k labels) CIFAR10 Figure 14 : 14Gradient-free adversarial attacks. Figure 15 : 15PGD attacks comparing JEM to the IG EBM of Du & Mordatch (2019). Figure 16 16Figure 16: Comparing the effect of the number of samples in the EOT attack. We find negligible difference between 5 and 10 for JEM-1 (red and green curves). Figure 17 : 17PGD transfer attack L ∞ . We attack JEM-0 and evaluate success of the same adversarial examples under JEM-1 and JEM-10. Whenever an adversarial example is refined back to its correct class, we set the distance to infinity. Note that the adversarial examples do not transfer well from JEM-0 to JEM-1/-10. Table 2 : 2Histograms for OOD detection. All models trained on CIFAR10. Green corresponds to the score on (in-distribution) CIFAR10, and red corresponds to the score on the OOD dataset.CIFAR10 Table 3 : 3OOD Detection Results. Models trained on CIFAR10. Values are AUROC. Table 4 : 4Hyperparameters B SAMPLE QUALITY EVALUTION Table 5 : 5Conditional vs. unconditional Inception Scores.Inception Score FID Method from paper B&S D&M from paper H D&M Residual Flow X 3.6 - 46.4 - - Glow X - 3.9 48.9* 107 - JEM (Ours) X 7.13 8.76 X 38.4 - JEM p(x|y) factored X - 6.36 X 61.8 - EBM (D&M) 8.3 - 8.3 37.9 - 37.9 SNGAN 8.59 - - 25.5 - - NCSN 8.91 - - 25.3 - - Table 6 : 6The headings: B&S, D&M, and H denotes scores computed using code provided byBarratt & Sharma (2018),Du & Mordatch Table 7 : 7OOD Detection Results. Values are AUROC. This was done to remove sources of stochasticity in early experiments. Since then we have been able to successfully train Joint-EBMs with Batch Normalization and other forms of stochastic regularization (such as dropout) without issue. We leave the incorporation of these methods to further work. In Du & Mordatch (2019) the attacker was not given access to the gradients of the refinement procedure. We re-run these stronger attacks on their model as well and provide a comparison in Appendix G. Code taken from https://github.com/y0ast/Glow-PyTorch ACKNOWLEDGEMENTSWe would like to thank Ying Nian Wu and Mitch Hill for providing some EBM training tips and tricks which were crucial in getting this project off the ground. We would also like to thank Jeremy Cohen for his useful feedback which greatly strengthened our adversarial robustness results. We would like to thank Lukas Schott for feedback on the robustness evaluation, Alexander Meinke and Francesco Croce for spotting some typos and suggesting the transfer attack. Anish Athalye, Logan Engstrom, Andrew Ilyas, Kevin Kwok, arXiv:1707.07397Synthesizing robust adversarial examples. arXiv preprintAnish Athalye, Logan Engstrom, Andrew Ilyas, and Kevin Kwok. Synthesizing robust adversarial examples. arXiv preprint arXiv:1707.07397, 2017. Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. Anish Athalye, Nicholas Carlini, David Wagner, arXiv:1802.00420arXiv preprintAnish Athalye, Nicholas Carlini, and David Wagner. Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples. arXiv preprint arXiv:1802.00420, 2018. A note on the inception score. Shane Barratt, Rishi Sharma, arXiv:1801.01973arXiv preprintShane Barratt and Rishi Sharma. A note on the inception score. arXiv preprint arXiv:1801.01973, 2018. Jens Behrmann, Will Grathwohl, T Q Ricky, David Chen, Jörn-Henrik Duvenaud, Jacobsen, arXiv:1811.00995Invertible residual networks. arXiv preprintJens Behrmann, Will Grathwohl, Ricky TQ Chen, David Duvenaud, and Jörn-Henrik Jacobsen. Invertible residual networks. arXiv preprint arXiv:1811.00995, 2018. Decision-based adversarial attacks: Reliable attacks against black-box machine learning models. Wieland Brendel, Jonas Rauber, Matthias Bethge, arXiv:1712.04248arXiv preprintWieland Brendel, Jonas Rauber, and Matthias Bethge. Decision-based adversarial attacks: Reliable attacks against black-box machine learning models. arXiv preprint arXiv:1712.04248, 2017. Ciwan Ceylan, Michael U Gutmann, arXiv:1806.03664Conditional noise-contrastive estimation of unnormalised models. arXiv preprintCiwan Ceylan and Michael U Gutmann. Conditional noise-contrastive estimation of unnormalised models. arXiv preprint arXiv:1806.03664, 2018. Semi-Supervised Learning. Olivier Chapelle, Bernhard Scholkopf, Alexander Zien, MIT PressOlivier Chapelle, Bernhard Scholkopf, and Alexander Zien. Semi-Supervised Learning. MIT Press, 2006. T Q Ricky, Jens Chen, David Behrmann, Jörn-Henrik Duvenaud, Jacobsen, arXiv:1906.02735Residual flows for invertible generative modeling. arXiv preprintRicky TQ Chen, Jens Behrmann, David Duvenaud, and Jörn-Henrik Jacobsen. Residual flows for invertible generative modeling. arXiv preprint arXiv:1906.02735, 2019. Maximum likelihood from incomplete data via the em algorithm. P Arthur, Nan M Dempster, Donald B Laird, Rubin, Journal of the Royal Statistical Society: Series B (Methodological). 391Arthur P Dempster, Nan M Laird, and Donald B Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39(1): 1-22, 1977. Implicit generation and generalization in energy-based models. Yilun Du, Igor Mordatch, arXiv:1903.08689arXiv preprintYilun Du and Igor Mordatch. Implicit generation and generalization in energy-based models. arXiv preprint arXiv:1903.08689, 2019. Ethan Fetaya, Jörn-Henrik Jacobsen, Richard Zemel, arXiv:1906.01171Conditional generative models are not robust. arXiv preprintEthan Fetaya, Jörn-Henrik Jacobsen, and Richard Zemel. Conditional generative models are not robust. arXiv preprint arXiv:1906.01171, 2019. Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. Yarin Gal, Zoubin Ghahramani, International Conference on Machine Learning (ICML). Yarin Gal and Zoubin Ghahramani. Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In International Conference on Machine Learning (ICML), pp. 1050-1059, 2016. J Ian, Goodfellow, arXiv:1412.6572Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. arXiv preprintIan J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. arXiv preprint arXiv:1412.6572, 2014. On calibration of modern neural networks. Chuan Guo, Geoff Pleiss, Yu Sun, Kilian Q Weinberger, Proceedings of the 34th International Conference on Machine Learning. the 34th International Conference on Machine Learning70Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1321-1330. JMLR. org, 2017. Noise-contrastive estimation: A new estimation principle for unnormalized statistical models. Michael Gutmann, Aapo Hyvärinen, Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. the Thirteenth International Conference on Artificial Intelligence and StatisticsMichael Gutmann and Aapo Hyvärinen. Noise-contrastive estimation: A new estimation principle for unnormalized statistical models. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pp. 297-304, 2010. A baseline for detecting misclassified and out-of-distribution examples in neural networks. Dan Hendrycks, Kevin Gimpel, arXiv:1610.02136arXiv preprintDan Hendrycks and Kevin Gimpel. A baseline for detecting misclassified and out-of-distribution examples in neural networks. arXiv preprint arXiv:1610.02136, 2016. Gans trained by a two time-scale update rule converge to a local nash equilibrium. Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, Sepp Hochreiter, Advances in Neural Information Processing Systems. Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. In Advances in Neural Information Processing Systems, pp. 6626-6637, 2017. Training products of experts by minimizing contrastive divergence. E Geoffrey, Hinton, Neural computation. 148Geoffrey E Hinton. Training products of experts by minimizing contrastive divergence. Neural computation, 14(8):1771-1800, 2002. Estimation of non-normalized statistical models by score matching. Aapo Hyvärinen, Journal of Machine Learning Research. 6Aapo Hyvärinen. Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research, 6(Apr):695-709, 2005. Adam: A method for stochastic optimization. P Diederik, Jimmy Kingma, Ba, arXiv:1412.6980arXiv preprintDiederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. Glow: Generative flow with invertible 1x1 convolutions. P Durk, Prafulla Kingma, Dhariwal, Advances in Neural Information Processing Systems. Durk P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1x1 convolutions. In Advances in Neural Information Processing Systems, pp. 10215-10224, 2018. Regularized estimation of image statistics by score matching. P Durk, Yann Kingma, Lecun, Advances in neural information processing systems. Durk P Kingma and Yann Lecun. Regularized estimation of image statistics by score matching. In Advances in neural information processing systems, pp. 1126-1134, 2010. A tutorial on energy-based learning. Yann Lecun, Sumit Chopra, Raia Hadsell, M Ranzato, F Huang, Predicting structured data. 10Yann LeCun, Sumit Chopra, Raia Hadsell, M Ranzato, and F Huang. A tutorial on energy-based learning. Predicting structured data, 1(0), 2006. Are generative classifiers more robust to adversarial attacks?. Yingzhen Li, John Bradshaw, Yash Sharma, arXiv:1802.06552arXiv preprintYingzhen Li, John Bradshaw, and Yash Sharma. Are generative classifiers more robust to adversarial attacks? arXiv preprint arXiv:1802.06552, 2018. Enhancing the reliability of out-of-distribution image detection in neural networks. Shiyu Liang, Yixuan Li, R Srikant, arXiv:1706.02690arXiv preprintShiyu Liang, Yixuan Li, and R Srikant. Enhancing the reliability of out-of-distribution image detec- tion in neural networks. arXiv preprint arXiv:1706.02690, 2017. Towards deep learning models resistant to adversarial attacks. Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, Adrian Vladu, arXiv:1706.06083arXiv preprintAleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. arXiv preprint arXiv:1706.06083, 2017. Takeru Miyato, Toshiki Kataoka, Masanori Koyama, Yuichi Yoshida, arXiv:1802.05957Spectral normalization for generative adversarial networks. arXiv preprintTakeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral normalization for generative adversarial networks. arXiv preprint arXiv:1802.05957, 2018. Do deep generative models know what they don. Eric Nalisnick, Akihiro Matsukawa, Yee Whye Teh, Dilan Gorur, Balaji Lakshminarayanan, arXiv:1810.09136t know? arXiv preprintEric Nalisnick, Akihiro Matsukawa, Yee Whye Teh, Dilan Gorur, and Balaji Lakshminarayanan. Do deep generative models know what they don't know? arXiv preprint arXiv:1810.09136, 2018. Detecting out-of-distribution inputs to deep generative models using a test for typicality. Eric Nalisnick, Akihiro Matsukawa, Yee Whye Teh, Balaji Lakshminarayanan, arXiv:1906.02994arXiv preprintEric Nalisnick, Akihiro Matsukawa, Yee Whye Teh, and Balaji Lakshminarayanan. Detecting out-of-distribution inputs to deep generative models using a test for typicality. arXiv preprint arXiv:1906.02994, 2019. On the anatomy of mcmcbased maximum likelihood learning of energy-based models. Erik Nijkamp, Mitch Hill, Tian Han, Song-Chun Zhu, Ying Nian Wu, arXiv:1903.12370arXiv preprintErik Nijkamp, Mitch Hill, Tian Han, Song-Chun Zhu, and Ying Nian Wu. On the anatomy of mcmc- based maximum likelihood learning of energy-based models. arXiv preprint arXiv:1903.12370, 2019a. On learning non-convergent short-run mcmc toward energy-based model. Erik Nijkamp, Song-Chun, Ying Nian Zhu, Wu, arXiv:1904.09770arXiv preprintErik Nijkamp, Song-Chun Zhu, and Ying Nian Wu. On learning non-convergent short-run mcmc toward energy-based model. arXiv preprint arXiv:1904.09770, 2019b. Foolbox: A python toolbox to benchmark the robustness of machine learning models. Jonas Rauber, Wieland Brendel, Matthias Bethge, arXiv:1707.04131arXiv preprintJonas Rauber, Wieland Brendel, and Matthias Bethge. Foolbox: A python toolbox to benchmark the robustness of machine learning models. arXiv preprint arXiv:1707.04131, 2017. Improved techniques for training gans. Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, Xi Chen, Advances in neural information processing systems. Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In Advances in neural information processing systems, pp. 2234-2242, 2016. Pixelcnn++: Improving the pixelcnn with discretized logistic mixture likelihood and other modifications. Tim Salimans, Andrej Karpathy, Xi Chen, Diederik P Kingma, arXiv:1701.05517arXiv preprintTim Salimans, Andrej Karpathy, Xi Chen, and Diederik P Kingma. Pixelcnn++: Improving the pixelcnn with discretized logistic mixture likelihood and other modifications. arXiv preprint arXiv:1701.05517, 2017. Greg Hadi Salman, Jerry Yang, Pengchuan Li, Huan Zhang, Ilya Zhang, Sebastien Razenshteyn, Bubeck, arXiv:1906.04584Provably robust deep learning via adversarially trained smoothed classifiers. arXiv preprintHadi Salman, Greg Yang, Jerry Li, Pengchuan Zhang, Huan Zhang, Ilya Razenshteyn, and Sebastien Bubeck. Provably robust deep learning via adversarially trained smoothed classifiers. arXiv preprint arXiv:1906.04584, 2019. Computer vision with a single (robust) classifier. CoRR, abs/1906.09453. Shibani Santurkar, Dimitris Tsipras, Brandon Tran, Andrew Ilyas, Logan Engstrom, Aleksander Madry, Shibani Santurkar, Dimitris Tsipras, Brandon Tran, Andrew Ilyas, Logan Engstrom, and Aleksander Madry. Computer vision with a single (robust) classifier. CoRR, abs/1906.09453, 2019. URL http://arxiv.org/abs/1906.09453. Lukas Schott, Jonas Rauber, Matthias Bethge, Wieland Brendel, arXiv:1805.09190Towards the first adversarially robust neural network model on mnist. arXiv preprintLukas Schott, Jonas Rauber, Matthias Bethge, and Wieland Brendel. Towards the first adversarially robust neural network model on mnist. arXiv preprint arXiv:1805.09190, 2018. Carl-Johann Simon-Gabriel, Yann Ollivier, Léon Bottou, Bernhard Schölkopf, David Lopez-Paz, arXiv:1802.01421Adversarial vulnerability of neural networks increases with input dimension. arXiv preprintCarl-Johann Simon-Gabriel, Yann Ollivier, Léon Bottou, Bernhard Schölkopf, and David Lopez- Paz. Adversarial vulnerability of neural networks increases with input dimension. arXiv preprint arXiv:1802.01421, 2018. Generative modeling by estimating gradients of the data distribution. Yang Song, Stefano Ermon, arXiv:1907.05600arXiv preprintYang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. arXiv preprint arXiv:1907.05600, 2019. Pixeldefend: Leveraging generative models to understand and defend against adversarial examples. Yang Song, Taesup Kim, Sebastian Nowozin, Stefano Ermon, Nate Kushman, arXiv:1710.10766arXiv preprintYang Song, Taesup Kim, Sebastian Nowozin, Stefano Ermon, and Nate Kushman. Pixeldefend: Leveraging generative models to understand and defend against adversarial examples. arXiv preprint arXiv:1710.10766, 2017. Yunfu Song, Zhijian Ou, arXiv:1806.00271Learning neural random fields with inclusive auxiliary generators. arXiv preprintYunfu Song and Zhijian Ou. Learning neural random fields with inclusive auxiliary generators. arXiv preprint arXiv:1806.00271, 2018. Training restricted boltzmann machines using approximations to the likelihood gradient. Tijmen Tieleman, Proceedings of the 25th international conference on Machine learning. the 25th international conference on Machine learningACMTijmen Tieleman. Training restricted boltzmann machines using approximations to the likelihood gradient. In Proceedings of the 25th international conference on Machine learning, pp. 1064- 1071. ACM, 2008. Adversarial distillation of bayesian neural network posteriors. Kuan-Chieh Wang, Paul Vicol, James Lucas, Li Gu, Roger Grosse, Richard Zemel, International Conference on Machine Learning (ICML). Kuan-Chieh Wang, Paul Vicol, James Lucas, Li Gu, Roger Grosse, and Richard Zemel. Adversar- ial distillation of bayesian neural network posteriors. In International Conference on Machine Learning (ICML), 2018. Bayesian learning via stochastic gradient langevin dynamics. Max Welling, Yee W Teh, Proceedings of the 28th international conference on machine learning (ICML-11). the 28th international conference on machine learning (ICML-11)Max Welling and Yee W Teh. Bayesian learning via stochastic gradient langevin dynamics. In Proceedings of the 28th international conference on machine learning (ICML-11), pp. 681-688, 2011. A theory of generative convnet. Jianwen Xie, Yang Lu, Song-Chun Zhu, Yingnian Wu, International Conference on Machine Learning. Jianwen Xie, Yang Lu, Song-Chun Zhu, and Yingnian Wu. A theory of generative convnet. In International Conference on Machine Learning, pp. 2635-2644, 2016. . Sergey Zagoruyko, Nikos Komodakis, arXiv:1605.07146Wide residual networks. arXiv preprintSergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016.
[ "https://github.com/y0ast/Glow-PyTorch.", "https://github.com/openai/ebm_code_release.", "https://github.com/y0ast/Glow-PyTorch" ]
[ "Planet Hunters VI: An Independent Characterization of KOI-351 and Several Long Period Planet Candidates from the Kepler Archival Data 1", "Planet Hunters VI: An Independent Characterization of KOI-351 and Several Long Period Planet Candidates from the Kepler Archival Data 1", "Planet Hunters VI: An Independent Characterization of KOI-351 and Several Long Period Planet Candidates from the Kepler Archival Data 1", "Planet Hunters VI: An Independent Characterization of KOI-351 and Several Long Period Planet Candidates from the Kepler Archival Data 1" ]
[ "Joseph R Schmitt [email protected] \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n", "Ji Wang \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n", "Debra A Fischer \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n", "Kian J Jek \nPlanet Hunter\n\n", "John C Moriarty \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n", "Tabetha S Boyajian \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n", "Megan E Schwamb \nInstitute of Astronomy and Astrophysics\n11F of Astronomy-Mathematics Building\nAcademia Sinica\nNational Taiwan University. No\n1, Sec. 4, Roosevelt Rd10617TaipeiTaiwan\n", "Chris Lintott \nOxford Astrophysics\nDenys Wilkinson Building, Keble Road, 1300 S. Lake Shore DriveOX1 3RH 5, 60605Oxford, Adler Planetarium, ChicagoILUSA\n", "Stuart Lynn ", "Arfon M Smith ", "Michael Parrish ", "Kevin Schawinski \nInstitute for Astronomy\nDepartment of Physics\nETH Zurich\nWolfgang-Pauli-Strasse 16CH-8093ZurichSwitzerland\n", "Robert Simpson \nOxford Astrophysics\nDenys Wilkinson Building, Keble Road, 1300 S. Lake Shore DriveOX1 3RH 5, 60605Oxford, Adler Planetarium, ChicagoILUSA\n", "Daryll Lacourse \nPlanet Hunter\n\n", "Mark R Omohundro \nPlanet Hunter\n\n", "Troy Winarski \nPlanet Hunter\n\n", "Samuel Jon Goodman \nPlanet Hunter\n\n", "Tony Jebson \nPlanet Hunter\n\n", "Hans Martin Schwengeler \nPlanet Hunter\n\n", "David A Paterson \nPlanet Hunter\n\n", "Johann Sejpka \nPlanet Hunter\n\n", "Ivan Terentev \nPlanet Hunter\n\n", "Tom Jacobs \nPlanet Hunter\n\n", "Nawar Alsaadi \nPlanet Hunter\n\n", "Robert C Bailey \nPlanet Hunter\n\n", "Tony Ginman \nPlanet Hunter\n\n", "Pete Granado \nPlanet Hunter\n\n", "Vonstad Kristoffer ", "Guttormsen \nPlanet Hunter\n\n", "Franco Mallia \nPlanet Hunter\n\n", "Alfred L Papillon \nPlanet Hunter\n\n", "Franco Rossi \nPlanet Hunter\n\n", "Miguel Socolovsky \nPlanet Hunter\n\n", "Joseph R Schmitt [email protected] \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n", "Ji Wang \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n", "Debra A Fischer \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n", "Kian J Jek \nPlanet Hunter\n\n", "John C Moriarty \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n", "Tabetha S Boyajian \nDepartment of Astronomy\nYale University\n06511New HavenCTUSA\n", "Megan E Schwamb \nInstitute of Astronomy and Astrophysics\n11F of Astronomy-Mathematics Building\nAcademia Sinica\nNational Taiwan University. No\n1, Sec. 4, Roosevelt Rd10617TaipeiTaiwan\n", "Chris Lintott \nOxford Astrophysics\nDenys Wilkinson Building, Keble Road, 1300 S. Lake Shore DriveOX1 3RH 5, 60605Oxford, Adler Planetarium, ChicagoILUSA\n", "Stuart Lynn ", "Arfon M Smith ", "Michael Parrish ", "Kevin Schawinski \nInstitute for Astronomy\nDepartment of Physics\nETH Zurich\nWolfgang-Pauli-Strasse 16CH-8093ZurichSwitzerland\n", "Robert Simpson \nOxford Astrophysics\nDenys Wilkinson Building, Keble Road, 1300 S. Lake Shore DriveOX1 3RH 5, 60605Oxford, Adler Planetarium, ChicagoILUSA\n", "Daryll Lacourse \nPlanet Hunter\n\n", "Mark R Omohundro \nPlanet Hunter\n\n", "Troy Winarski \nPlanet Hunter\n\n", "Samuel Jon Goodman \nPlanet Hunter\n\n", "Tony Jebson \nPlanet Hunter\n\n", "Hans Martin Schwengeler \nPlanet Hunter\n\n", "David A Paterson \nPlanet Hunter\n\n", "Johann Sejpka \nPlanet Hunter\n\n", "Ivan Terentev \nPlanet Hunter\n\n", "Tom Jacobs \nPlanet Hunter\n\n", "Nawar Alsaadi \nPlanet Hunter\n\n", "Robert C Bailey \nPlanet Hunter\n\n", "Tony Ginman \nPlanet Hunter\n\n", "Pete Granado \nPlanet Hunter\n\n", "Vonstad Kristoffer ", "Guttormsen \nPlanet Hunter\n\n", "Franco Mallia \nPlanet Hunter\n\n", "Alfred L Papillon \nPlanet Hunter\n\n", "Franco Rossi \nPlanet Hunter\n\n", "Miguel Socolovsky \nPlanet Hunter\n\n" ]
[ "Department of Astronomy\nYale University\n06511New HavenCTUSA", "Department of Astronomy\nYale University\n06511New HavenCTUSA", "Department of Astronomy\nYale University\n06511New HavenCTUSA", "Planet Hunter\n", "Department of Astronomy\nYale University\n06511New HavenCTUSA", "Department of Astronomy\nYale University\n06511New HavenCTUSA", "Institute of Astronomy and Astrophysics\n11F of Astronomy-Mathematics Building\nAcademia Sinica\nNational Taiwan University. No\n1, Sec. 4, Roosevelt Rd10617TaipeiTaiwan", "Oxford Astrophysics\nDenys Wilkinson Building, Keble Road, 1300 S. Lake Shore DriveOX1 3RH 5, 60605Oxford, Adler Planetarium, ChicagoILUSA", "Institute for Astronomy\nDepartment of Physics\nETH Zurich\nWolfgang-Pauli-Strasse 16CH-8093ZurichSwitzerland", "Oxford Astrophysics\nDenys Wilkinson Building, Keble Road, 1300 S. Lake Shore DriveOX1 3RH 5, 60605Oxford, Adler Planetarium, ChicagoILUSA", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Department of Astronomy\nYale University\n06511New HavenCTUSA", "Department of Astronomy\nYale University\n06511New HavenCTUSA", "Department of Astronomy\nYale University\n06511New HavenCTUSA", "Planet Hunter\n", "Department of Astronomy\nYale University\n06511New HavenCTUSA", "Department of Astronomy\nYale University\n06511New HavenCTUSA", "Institute of Astronomy and Astrophysics\n11F of Astronomy-Mathematics Building\nAcademia Sinica\nNational Taiwan University. No\n1, Sec. 4, Roosevelt Rd10617TaipeiTaiwan", "Oxford Astrophysics\nDenys Wilkinson Building, Keble Road, 1300 S. Lake Shore DriveOX1 3RH 5, 60605Oxford, Adler Planetarium, ChicagoILUSA", "Institute for Astronomy\nDepartment of Physics\nETH Zurich\nWolfgang-Pauli-Strasse 16CH-8093ZurichSwitzerland", "Oxford Astrophysics\nDenys Wilkinson Building, Keble Road, 1300 S. Lake Shore DriveOX1 3RH 5, 60605Oxford, Adler Planetarium, ChicagoILUSA", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n", "Planet Hunter\n" ]
[]
We report the discovery of 14 new transiting planet candidates in the Kepler field from the Planet Hunters citizen science program. None of these candidates overlapped with Kepler Objects of Interest (KOIs) at the time of submission. We report the discovery of one more addition to the six planet candidate system around KOI-351, making it the only seven planet candidate system from Kepler. Additionally, KOI-351 bears some resemblance to our own solar system, with the inner five planets ranging from Earth to mini-Neptune radii and the outer planets being gas giants; however, this system is very compact, with all seven planet candidates orbiting 1 AU from their host star. A Hill stability test and an orbital integration of the system shows that the system is stable. Furthermore, we significantly add to the population of long period 1 This publication has been made possible through the work of more than 280,000 volunteers in the Planet Hunters project, whose contributions are individually acknowledged at http://www.planethunters.org/authors. The authors especially thank the Planet Hunters volunteers who participated in identifying and analyzing the candidates presented in this paper. They are individually recognized at http://www.planethunters.org/PH6.
10.1088/0004-6256/148/2/28
[ "https://arxiv.org/pdf/1310.5912v3.pdf" ]
119,238,163
1310.5912
be5316740e537b3b4380cb02bc84d4618e6d4350
Planet Hunters VI: An Independent Characterization of KOI-351 and Several Long Period Planet Candidates from the Kepler Archival Data 1 2 Jul 2014 Joseph R Schmitt [email protected] Department of Astronomy Yale University 06511New HavenCTUSA Ji Wang Department of Astronomy Yale University 06511New HavenCTUSA Debra A Fischer Department of Astronomy Yale University 06511New HavenCTUSA Kian J Jek Planet Hunter John C Moriarty Department of Astronomy Yale University 06511New HavenCTUSA Tabetha S Boyajian Department of Astronomy Yale University 06511New HavenCTUSA Megan E Schwamb Institute of Astronomy and Astrophysics 11F of Astronomy-Mathematics Building Academia Sinica National Taiwan University. No 1, Sec. 4, Roosevelt Rd10617TaipeiTaiwan Chris Lintott Oxford Astrophysics Denys Wilkinson Building, Keble Road, 1300 S. Lake Shore DriveOX1 3RH 5, 60605Oxford, Adler Planetarium, ChicagoILUSA Stuart Lynn Arfon M Smith Michael Parrish Kevin Schawinski Institute for Astronomy Department of Physics ETH Zurich Wolfgang-Pauli-Strasse 16CH-8093ZurichSwitzerland Robert Simpson Oxford Astrophysics Denys Wilkinson Building, Keble Road, 1300 S. Lake Shore DriveOX1 3RH 5, 60605Oxford, Adler Planetarium, ChicagoILUSA Daryll Lacourse Planet Hunter Mark R Omohundro Planet Hunter Troy Winarski Planet Hunter Samuel Jon Goodman Planet Hunter Tony Jebson Planet Hunter Hans Martin Schwengeler Planet Hunter David A Paterson Planet Hunter Johann Sejpka Planet Hunter Ivan Terentev Planet Hunter Tom Jacobs Planet Hunter Nawar Alsaadi Planet Hunter Robert C Bailey Planet Hunter Tony Ginman Planet Hunter Pete Granado Planet Hunter Vonstad Kristoffer Guttormsen Planet Hunter Franco Mallia Planet Hunter Alfred L Papillon Planet Hunter Franco Rossi Planet Hunter Miguel Socolovsky Planet Hunter Planet Hunters VI: An Independent Characterization of KOI-351 and Several Long Period Planet Candidates from the Kepler Archival Data 1 2 Jul 2014Accepted to AJSubject headings: Planets and satellites: detection -surveys We report the discovery of 14 new transiting planet candidates in the Kepler field from the Planet Hunters citizen science program. None of these candidates overlapped with Kepler Objects of Interest (KOIs) at the time of submission. We report the discovery of one more addition to the six planet candidate system around KOI-351, making it the only seven planet candidate system from Kepler. Additionally, KOI-351 bears some resemblance to our own solar system, with the inner five planets ranging from Earth to mini-Neptune radii and the outer planets being gas giants; however, this system is very compact, with all seven planet candidates orbiting 1 AU from their host star. A Hill stability test and an orbital integration of the system shows that the system is stable. Furthermore, we significantly add to the population of long period 1 This publication has been made possible through the work of more than 280,000 volunteers in the Planet Hunters project, whose contributions are individually acknowledged at http://www.planethunters.org/authors. The authors especially thank the Planet Hunters volunteers who participated in identifying and analyzing the candidates presented in this paper. They are individually recognized at http://www.planethunters.org/PH6. transiting planets; periods range from 124-904 days, eight of them more than one Earth year long. Seven of these 14 candidates reside in their host star's habitable zone. Subject headings: Planets and satellites: detection -surveys Introduction Over the last 20 years, hundreds of exoplanets have been discovered. One powerful method to discover planet candidates is the photometric transit technique, in which a planet crosses in front of its host star as seen from Earth. The Kepler mission (Borucki et al. 2010) has been observing ∼160,000 stars nearly continuously for almost four years searching for these transit signals. In the first 16 quarters, spanning four years, more than 3,800 planet candidates have been discovered (and about 800 more have yet to be dispositioned) via this photometric transit technique 1 (Borucki et al. 2011;Batalha et al. 2013;Burke et al. 2014). The Kepler team searches for transit signals using a matched filter search algorithm that employs wavelets for numerical efficiency, the transit planet search (TPS) (Jenkins et al. 2002(Jenkins et al. , 2010, which requires three transits with a significance of 7.1σ to be placed on the Threshold Crossing Event (TCE) list (Tenenbaum et al. 2013(Tenenbaum et al. , 2014. TPS's Q1-16 run has discovered more than 16,000 transit signals (Tenenbaum et al. 2014). Those TCEs which pass additional tests, including a human review stage (Batalha et al. 2013), become Kepler Objects of Interest (KOIs). It is expected that most KOI candidates are true planets (Morton & Johnson 2011). The false positive rate has been found to depend on the planet radius, with the lowest false positive rate (6.7 − 8.8%) in the range of 1.25 − 6.00R ⊕ , where we find a majority of the new Planet Hunters candidates. Larger planets suffer a false positive rate of 15.9 − 17.7% (Fressin et al. 2013). However, the false positive rate for multiple planet candidates is very low, approximately only two out of all Kepler targets (Lissauer et al. 2012(Lissauer et al. , 2014. The Planet Hunters project 2 (Fischer et al. 2012) is one of the Zooniverse projects 3 (Lintott et al. 2008(Lintott et al. , 2011Fortson et al. 2012) and is designed to have humans visually check Kepler light curves, broken into 30 day increments, to search for undiscovered transit signals. Since December, 2010, approximately 280,000 public volunteers have searched through more than 21 million Kepler light curves hunting for transiting planets, contributing a cumulative total of 200 years of work. While Planet Hunters has identified hundreds of transit signals, we only announce candidates that have not been listed as KOI candidates at the time of submission. Planet Hunters has discovered more than 40 new planet candidates (Fischer et al. 2012;Lintott et al. 2013;Wang et al. 2013;Schwamb et al. 2013), including two confirmed planets. The first confirmed planet from the Planet Hunters project is PH1 b (Kepler-64b), a circumbinary planet in a ∼137 day orbit around an eclipsing binary, and is the first known planet in a quadruple star system (Schwamb et al. 2013). The second confirmed planet from the Planet Hunters project is PH2 b (Kepler-86b), a gas giant planet residing in its host star's habitable zone (Wang et al. 2013). Statistical completeness analysis within the Planet Hunters project is performed by injecting fake transit events into real Kepler light curves (Schwamb et al. 2012). This analysis shows that Planet Hunters are effective at detecting transits of Neptune-sized planets or larger ( 85% completeness for short periods, P < 15 days), although smaller planets can still be recovered. In this paper, we present a total of 14 new candidates from Planet Hunters project, all with period greater than 124 days, eight of them more than 1 Earth year long. Additionally, seven of the new candidates lie within the most recent HZ estimates (Kopparapu et al. 2013). Another candidate discovered around KIC 6436029 makes this a multiple planet candidate system. The new planet candidate we detect in the known KOI six candidate system, KOI-351, makes it the only Kepler star with a seven planet candidate system. While this paper was in the review process, Cabrera et al. (2014) and Lissauer et al. (2014) independently discovered and characterized the seventh candidate in KOI-351. Also while in the review process, six of our new candidates were classified as candidates (KIC 2437209) or "NOT DISPOSITIONED" (KIC 5094412, 6372194, 6436029, 6805414, 11152511) KOIs on the Exoplanet Archive (accessed March 11, 2014), all of which were detected as TCEs in the latest TPS Q1-16 search (Tenenbaum et al. 2014). Of the new candidates with 3+ transits in Q1-16, all except the seventh candidate in KOI-351 have now been detected by TPS. Section 2 explains how these new planet candidates were discovered. Section 3 explains our method to calculate the transit parameters, stellar parameters, whether the planet resides in the HZ, and a discussion of the false positive tests that we have carried out for our new planet candidates. Section 4 discusses characteristics of notable new candidates. We conclude in Section 5. Planet Hunters Candidate Discoveries Candidates are identified through one of two ways. The classic method is through Planet Hunters interface (Fischer et al. 2012;Schwamb et al. 2012), in which users are shown a 30-day light curve and asked to identify transit-like features. The Planet Hunters interface has shown quarters 1, 2, part of 3, 4, 5, 7, 9, 14, and part of 16. However, once a planet candidate is identified, volunteers typically search through all available quarters for additional transits of the same planet candidate or for transits of new candidates within the same system. We have implemented a weighting scheme (Schwamb et al. 2012) in order to rank the quality of user transit classifications. In brief, all users start out with an equal weight, and synthetic light curves are used to seed the user weighting. Users who properly identify synthetic transits are given higher weights. The user weightings continue to evolve depending on whether or not individual rankings agree with the majority rankings. Transits above a threshold score are then sent to the science team to be analyzed. The other way candidates are identified is via the Planet Hunters Talk page 4 (Lintott et al. 2013;Wang et al. 2014). This discussion tool allows users to publicly post and discuss interesting light curves with others. It is through this interface that users are easily able to download all data and collectively scrutinize potentially interesting light curves. The interface provides quick links to the MAST 5 and UKIRT databases for each object. Publicly available web-based tools hosted at the NASA Exoplanet Archive (NEA) and SkyView 6 are used frequently to calculate periodograms, normalize, and phase-fold the data, as well as performing data validation to rule out false positives. Planet Hunters volunteers are instrumental in the success of the Planet Hunters project. The Planet Hunter volunteers organize different types of light curves in collections on the Planet Hunters Talk page. Once a collection has been established, users can compare the light curves they are examining with the light curves in the collections. If the user decides to discuss the light curve on the Talk page, they can suggest that the light curve be added to an existing collection. The Planet Hunters science team searches the collections for interesting candidates. Alternatively, some of the active users compile lists of prospective candidates and pass the spreadsheets to the science team for further vetting. The science team then carries out data validations tests described in Section 3.1. This paper reflects discoveries made through the Planet Hunters Talk interface. Transit Characterization Data Validation Once a system is identified as having a possible transit signal, we perform a full analysis of the system using the Kepler light curve (data validation; Batalha et al. 2010) and any other publicly available archival data. This includes using the PyKE package (Still & Barclay 2012) and screening the surrounding field for background eclipsing binaries (BGEB) by ensuring that the flux-weighted centroid remains stationary during an object's transit (Bryson et al. 2013). By measuring the flux centroid shift between in-and out-of-transit, a limit can be put on the angular separation of a possible contamination source. This method has limitations when the target star is brighter than 11th magnitude or if it is in a crowded field (Wang et al. 2014). The flux-weighted centroid method reported large (> 3σ) flux centroid offsets for the aforementioned cases even for confirmed planet candidates. However, all but one objects in this paper are fainter than 13th magnitude. Our own flux-weighted centroid analysis package is described in Wang et al. (2013) and Wang et al. (2014). We use UKIRT and 2MASS images to search for nearby contaminating sources and asymmetric point spread functions, necessary for probing nearby unresolved sources due to the low-resolution of the Kepler CCD detector. Two of the stars have a visual companion within 4 : KIC 6372194, and 10255705 (see Figure 1 for the UKIRT images and Table 1). If these companions are outside the computed confusion radius, the candidates are unlikely to be orbiting the contaminating sources. If they are orbiting the contaminating sources and outside the confusion radius, an apparent pixel centroid offset should have been detected. However, KIC 10255705 has a 3σ confusion radius of 3.2 and a companion at 1.4 , meaning that no apparent pixel centroid offset would be seen, so we cannot be sure the planet is truly orbiting the brighter star. The neighboring stars for both KIC 6372194 and 10255705 are at least two magnitudes fainter than the central star. The uncertainty caused by the flux contamination is smaller than the one from the stellar radius estimation, so we did not include the dilution factor in our parameter estimates. Afterwards, the light curves are then fully modeled and inspected for variations in even-odd transit depth, the presence of secondary eclipses, and misshaped transit profiles, any of which can indicate a falsely identified planetary candidate with tools from Still & Barclay (2012). Stellar Properties The revised set of Kepler stellar parameters provides estimates of T eff , log g, mass, radius, and [Fe/H] of all stars in the Kepler field (Huber et al. 2014). We adopt these as stellar inputs for the Wang et al. (2013) transit fitting routine from the Kepler stellar data hosted by the NEA Exoplanet Archive 7 . Table 2 lists the primary stellar parameters. We use the values in Table 2 as inputs for our own transit fitting routine (Wang et al. 2013(Wang et al. , 2014 developed to iterate between the light curve solution and the Yonsei-Yale (Y 2 ) isochrones (Demarque et al. 2004) for a range of ages spanning from 0.08 − 15 Gyr, with [α/Fe] = 0. The high T eff of KIC 5522786, T = 8941 K, fell on the upper edge of the temperature range for our stellar models, which caused error bars to be clipped. As such, we retain the Huber et al. (2014) stellar parameters for KIC 5522786. We ran 1000 trials of Monte Carlo simulations using the Y 2 Isochrones to obtain stellar properties such as effective temperature, mass, radius, metallicity, and luminosity. The distributions of the inputs are assumed to follow a Gaussian function. The mean and standard deviation are the reported value and error bar in the Kepler stellar parameter table. The distributions of the outputs are used to constrain the transit light curve fitting, excluding mathematically acceptable fits that are physically unlikely. The formal error bars sometimes result in unrealistically low error bars for log g, radius, and mass, so we adopt a floor on the log g error bar of ±0.10, 20% for the radius error, and 10% for the mass error. The outputs of the transiting light curve fitting (e.g., stellar density) can also be used to constrain the Y 2 isochrone. Thus, the iterative transit fitting routine is able to provide a set of self-consistent stellar and orbital solutions. Transit Fitting All available long-cadence light curves from Kepler quarters 1-16 were flattened, normalized, and phase-folded using the PyKE package (Still & Barclay 2012). We used a custom-made package (Wang et al. 2013) to find the best fit values for the orbital period (P ), the ratio of the planet radius to the stellar radius (R PL /R * ), the ratio of the semi-major axis to the stellar radius (a/R * ), inclination (i), eccentricity (e), longitude of periastron (ω), and midtransit times. Quadratic limb darkening parameters are determined by interpolating a table provided by Claret & Bloemen (2011). The best fit parameters were determined through a Levenberg-Marquardt least square algorithm, while the error bars are estimated with a bootstrapping method in the following way. We repeatedly fit the simulated transiting light curves, which were generated from the observations but perturbed by photon noise. In order to reduce the dependence of the initial guess of orbital parameters, we perturbed the initial guess based on the standard deviation of previous runs. We used a range of five times of standard deviation to explore a large phase space. The reported orbital solutions in Table 3 are the weighted averages based on the goodness of fit. The phase-folded solutions for each star are shown in Figure 2. Odd and even transits are colored blue and red, respectively, and there is no significant odd-even depth variation, which would be indicative of an eclipsing binary star system. The strongest odd-even depth variation is 1.49σ for KIC 5522786, a two transit candidate. Table 3 contains best fit parameters for the new planet candidates. Habitable Zone Planet Hunters has produced a relatively large proportion of the known long period candidates from the Kepler data (see Figure 3). Importantly, it is these longer periods that probe the HZs of G-and K-type stars. The inner edge of the HZ is typically defined as the orbital radius of the water-loss limit, the point at which a terrestrial planet will quickly lose its water, and the outer edge is defined by a maximum greenhouse effect from CO 2 (Kasting et al. 1993). These two boundaries define the "conservative" HZ. "Empirical" HZ estimates expand the HZ by assuming Venus had water for much of its history ("Recent Venus") and that Mars had water early in its history ("Early Mars"). A revised estimate of the HZ by Kopparapu et al. (2013) proposes to define the HZ by its level of incident flux, S, rather than the equilibrium temperature of the planet, removing the dependency on Bond albedo. The HZ also varies with spectral type as the peak wavelength of the stellar emission changes. As such, cooler stars have slightly more distant HZs relative to the incident flux on the planet. In this revised HZ estimate, only the a/R * and T eff are required to determine whether the planet resides in its host's HZ. Seven planet candidates lie directly in the conservative HZ: KIC 2437209, 5010054, 5094412, 5732155, 6372194, 9662267, and 9704149. KIC 11152511 sits in the Recent Venus HZ with error bars into the conservative HZ. KIC 6436029 and KIC 10255705 are outside of the HZ, but are within 1 of the Early Mars and Recent Venus HZs, respectively. See Figure 4 for the location of each planet candidate relative to its host star's HZ. Of these, KIC 11152511 has the lowest radius, R = 1.93 ± 1.50R ⊕ . The best-fit value falls very near or on the transition zone between high density super Earths with thin atmospheres and low density mini-Neptunes with thick atmospheres, which recent studies show is likely between 1.5 and 2.0 R ⊕ (Lopez & Fortney 2013;Weiss & Marcy 2014;Marcy et al. 2014). However, this candidate is likely shrunk due to neglected light dilution from its neighboring star. KIC 5522786's planet candidate also lies in this transition zone (R = 1.86 ± 0.25R ⊕ ), leaving uncertainty as to whether this candidate would be a super Earth or mini-Neptune. However, although it lies outside the T eff range studied by Kopparapu et al. (2013), it certainly orbits too close to the star, with S ≈ 5S 0 (five times the solar incident flux). New Planet Candidates The best-fit parameters for the 14 new planet candidates in this paper are listed in Table 3. Their periods and radii are plotted in Figure 3. Comments on individual candidates are given below. Kopparapu et al. (2013), but at 5S 0 , it is certainly too hot to be habitable. The candidate around KIC 2437209 (T eff = 4842 K, S = 14.39 ± 11.87S 0 ) is excluded for convenience. KIC 2437209 In quarters 1 − 16, the light curve for KIC 2437209 has four large (∼ 14000 ppm) and long transits (duration of 71 hours; see Figure 2, Table 3). Although there were three transits (and a fourth in a data gap), one transit was de-emphasized due to a sudden pixel sensitivity dropout detector in TPS, leaving it without the requisite three detected transits to be labeled a TCE (J. M. Jenkins 2014, private communication). With a fourth transit in Q14, the Q1-16 TPS search (Tenenbaum et al. 2014) has detected this signal, and it is now a KOI candidate. KIC 2437209's transit duration, approximately equivalent to the duration of Neptune transiting the Sun, is extremely long for a 281 day orbit. Our first attempts at modeling this system resulted in a degeneracy in the stellar and planetary radius. This system was consistent with two scenarios: an evolved star on the giant branch with a stellar companion in a circular orbit or with a main sequence dwarf with a planetary companion in a highly eccentric e = 0.974 orbit viewed at apastron, an unfavorable probability of occurrence. In the evolved star interpretation, the stellar parameters were log g = 3.07 +0.60 −0.10 , R * = 4.97 +2.09 −1.14 R , R PL = 62.64 ± 19.61R ⊕ , e = 0.43 +0.23 −0.43 , while the dwarf interpretation had best-fit values of log(g) = 4.56±0.10, This degeneracy led us to obtain one spectrum of the star with Keck/HIRES (Vogt et al. 1994) in an attempt to better determine the stellar parameters. We used Spectroscopy Made Easy (SME) (Valenti & Piskunov 1996;Valenti & Fischer 2005) to model the spectrum; however, due to the faintness of the star (K p = 16.353), the resulting signal to noise ratio was ∼ 15, well below the preferred regime for reliable SME analysis. Nevertheless, our results fitting the spectrum with SME gave T eff = 4965 ± 100 K, log g = 3.75 ± 0.2, and [Fe/H] = 0.564 ± 0.10. This has overlapping error bars with our evolved star interpretation and is inconsistent with input stellar parameters from Huber et al. (2014). R * = 0.80±0.10R , R P L = 9.79±0.62R ⊕ , e = 0.974 +0.013 −0.029 , KIC 2437209 has six quarters of short-cadence data and likely belongs to the open cluster NGC 6791. With g = 17.239 and (g − r ) = 1.001 given by the Kepler Input Catalog, KIC 2437209 is placed on the giant branch of NGC 6791 on the g vs. (g − r ) color-magnitude diagram (Platais et al. 2011). Evolved cluster stars are usually targets for short-cadence Kepler observations to study pulsations. Assuming that the star is a member of NGC 6791, the stellar parameters become T eff = 4664 ± 100 K, R = 3.89 ± 0.22R ⊕ , and log g = 3.34 (D. Huber 2014, private communication), which is consistent with our evolved star fit. Therefore, KIC 2437209 is very likely to be a giant star with a smaller stellar companion. However, should this planet indeed have e = 0.974, then this planet may be one of the super-eccentric planets predicted in Socrates et al. (2012) caught in the act of high eccentricity migration. KIC 5010054 Planet Hunters volunteers discovered an additional transit signal in Q16 at 2496333.91 JD, a signal consistent with a second planet in the system. No accompanying transit is seen. A TAP analysis (Carter & Winn 2009;Gazak et al. 2012;Eastman et al. 2013) suggests a period of P = 504 +170 −160 days, meaning an earlier transit may have been located in a data gap. The TAP fit with the modeled stellar parameters gives this signal a radius of R = 2.9 ± 0.48R ⊕ . KIC 5094412 KIC 5094412 has the shortest transit duration of candidates in this paper (3.77 hours), leading to few in-transit data points. We note that two of the four transits appear V-shaped, a shape that can be produced by a grazing eclipsing binary system, but one is an even numbered transit and the other is an odd numbered transit. The transit morphology of this system is unclear due to the under-sampling induced by the short duration transits. This is the only candidate in our sample that has such a potential V-shaped transit profile. With three transits in Q1-12 (and one in the data gap between Q6 and Q7), one transit was de-emphasized for a reaction wheel zero crossing due to possible false alarms caused by such events. It has since been shown that a reaction wheel zero crossing does not cause false alarms, so they are no longer de-emphasized (J. M. Jenkins 2014, private communication). This change and an extra transit in Q14 led to a detection by TPS in the latest run. As such, it is now an undispositioned KOI. KIC 5522786 KIC 5522786 has only two transits and both exhibit a very sharp ingress and egress. This is the second smallest candidate in this paper, a planet on the super-Earth/mini-Neptune transition zone at R = 1.86 ± 0.25R ⊕ . Despite a depth of only 90 ± 13 ppm, the transits are visually apparent due to the host star's brightness, a Kepler-magnitude = 9.350 star. A third transit of this candidate would be extremely valuable and is expected to take place at about 2457387.7614 JD (December 31, 2015). We also note that the star has the highest effective temperature in our sample with T eff = 8941 +258 −396 K. KIC 6372194 KIC 6372194 was undetected in the Q1-12 TPS search due to a bug in the Robust Statistic code (J. M. Jenkins 2014, private communication). The star was not observed until Q4. However, the bug resulted in the code checking the expected transit in Q3, for which KIC 6372194 was not observed. As such, it was rejected. This bug has since been corrected, and the signal is now detected and is designated an undispositioned KOI. KIC 6436029 KIC 6436029 already has one KOI candidate (KOI 2828.01; P = 59.5d, R = 4.1 ± 2.0R ⊕ ). We have detected three transits of another planet around this star with P = 505.45 days. This new planet candidate possibly lies in the outer HZ; its upper error bar overlaps the Early Mars zone of the optimistic HZ. It has a radius of R = 4.16 ± 4.04R ⊕ . KIC 6805414 There are four transits in Q1-12 (a fifth was in the data gap between Q9 and Q10). Two of these were de-emphasized due to the TPS sudden pixel sensitivity dropout detector (Jon M. Jenkins 2014, private communication). It was subsequently detected in the Q1-16 search and is now an undispositioned KOI. This candidate was also independently discovered in Huang et al. (2013). KIC 11442793 (Kepler-90) KIC 11442793 has six listed KOI candidates (KOI-351) at periods of 7.01, 8.72, 59.74, 91.94, 210.60, and 331.64 days (see Table 4 for transit parameters). Planet Hunters has detected an additional 124.92 day signal with five full transits and two partial transits. However, after the first transit (Q2), the next transit fell within a data gap. The following transit contained only the egress due to a data gap within Q5. In the next transit at the end of Q6, only the ingress is observed. The next transit fell into a data gap. Two other transits were de-emphasized due to the aforementioned reaction wheel zero crossings. Furthermore, the large number of candidates in the system caused the data validation to time out. All these conspired together and led to its non-detection in the Q1-12 TPS search (J. M. Jenkins 2014, private communication). While this paper was being reviewed, Cabrera et al. (2014) and Lissauer et al. (2014) independently discovered the 124.92 day signal and characterized all seven candidates claiming confirmation, with Lissauer et al. (2014) statistically confirming all seven with high confidence. For the 124.92 day signal described in this paper, both of the previous works agree well with each other and with our analysis. To test whether the seventh signal is merely an alias of one or more of the previous six candidates, we assumed a linear ephemeris for all planets and calculated each planet's midtransit points (see Table 5). None of the planets matched the seventh candidate. In fact, somewhat surprisingly, only one midtransit of the previously known six planet candidates comes within 24 hours of a midtransit of the seventh candidate, but unsurprisingly, it's the candidate with the shortest period (7.01 days). However, we recognize the fact that we cannot assume a perfectly We therefore analyzed the system using the IDL program TAP to determine transit midpoints and compared the observed midtransits to the midtransits expected from the KOI epochs and periods. A plot of the observed midtransit times minus calculated midtransit times (O − C) shows deviations from a perfectly linear ephemeris due to TTVs for the outer three planets, indicating significant gravitational interactions between planets. Figure 5 shows the O − C plot for the outer three planet candidates of KOI-351. The last transit of KOI-351.02 has O − C = +14.4 hours, while the second-to-last transit has O − C = −9.4 hours, a change of 24 hours over the course of one orbit. KOI-351.02 has one of the largest TTVs known, excluding circumbinary planets (Mazeh et al. 2013). The presence of TTVs significantly increases the likelihood that our planet candidates are real, as they show mutual gravitational interactions with their neighbors. A more in-depth analysis can likely be used to determine masses and confirm several planets in this system, but this is left for future studies. The TTVs show that we cannot rely solely on comparisons of linear ephemerides. Therefore, we also compared the transit depths and transit durations of all seven candidates. The transit depth for KOI-351.07, 480 ± 87 ppm, is only consistent with two of the KOI candidates, KOI-351.03 and KOI-351.04, but the duration of the seventh candidate is hours longer than either. These two candidates also have significantly different midtransit times and thus cannot be the same object as KOI-351.07. The depths and durations are compared in Table 5, which also shows the phase information of each KOI-351.07 transit relative to the other six candidates. If KOI-351.07 were in fact a secondary eclipse from one of the other six objects, we would see a constant phase offset of KOI-351's transit relative to the primary eclipses. No such phase offset is observed. A combination of looking at the differences in duration, depth, midtransit points, and phases of KOI-351.07 relative to the other six candidates clearly demonstrates that KOI-351.07 is a distinct object. Simply put, the seventh candidate is not an alias of any of the other six candidates. We also performed two stability tests to assess the feasibility of this system. Hill stability is a simple stability diagnostic that can be used to determine whether this somewhat compact system is at least feasible. To test the stability of the system, the masses of the planets must be assumed. For the inner five planets, which all have R < 4.0R ⊕ , we used the recent Weiss & Marcy (2014) mass-radius relationship shown below based on the masses and radii of confirmed planets, which was valid for 1.5R ⊕ ≤ R ≤ 4.0R ⊕ . The middle three planets fall into this range, while the inner two are within 1σ of this range. We also performed two stability tests to assess the feasibility of this system. Hill stability is a simple stability diagnostic that can be used to determine whether this somewhat compact system is at least feasible. To test the stability of the system, the masses of the planets must be assumed. For the inner five planets, which all have R < 4.0R ⊕ , we used the recent Weiss & Marcy (2014) mass-radius relationship shown below based on the masses and radii of confirmed planets, which was valid for 1.5R ⊕ ≤ R ≤ 4.0R ⊕ . The middle three planets fall into this range, while the inner two are within 1σ of this range. We also performed two stability tests An 'x' corresponds to the approximate time a transit was expected to occur, but was missed due to a data gap. Solid lines are used to connect consecutive observed transits, while dashed lines are used to connect non-consecutive observed transits. Error bars are plotted and are smaller than the markers where not seen. The O − C of the last midtransit of KOI-351.02 is ∼ 24 hours larger than the O − C of the second-to-last midtransit of KOI-351.02, making KOI-351.02's TTVs one of the largest known (Mazeh et al. 2013). The presence of TTVs increases the likelihood that the planets are real and within the same system. to assess the feasibility of this system. Hill stability is a simple stability diagnostic that can be used to determine whether this somewhat compact system is at least feasible. To test the stability of the system, the masses of the planets must be assumed. For the inner five planets, which all have R < 4.0R ⊕ , we used the recent Weiss & Marcy (2014) mass-radius relationship shown below based on the masses and radii of confirmed planets, which was valid for 1.5R ⊕ ≤ R ≤ 4.0R ⊕ . The middle three planets fall into this range, while the inner two are within 1σ of this range. We also performed two stability tests to assess the feasibility of this system. Hill stability is a simple stability diagnostic that can be used to determine whether this somewhat compact system is at least feasible. To test the stability of the system, the masses of the planets must be assumed. For the inner five planets, which all have R < 4.0R ⊕ , we used the recent Weiss & Marcy (2014) mass-radius relationship shown below based on the masses and radii of confirmed planets, which was valid for 1.5R ⊕ ≤ R ≤ 4.0R ⊕ . The middle three planets fall into this range, while the inner two are within 1σ of this range. M P M ⊕ = 2.69 R P R ⊕ 0.93 , R < 4R ⊕ (1) The masses of the upper two planets were calculated with the same equation used by Lissauer et al. (2011b), shown below. This equation was determined by fitting for the solar system planets Earth, Uranus, Neptune, and Saturn. However, the upper error bar on the outermost planet is consistent with a Jupiter radius where the mass-radius relationship breaks down, meaning the mass could be significantly larger. M P M ⊕ = R P R ⊕ 2.06 , R > 4R ⊕(2) Again using the same method as Lissauer et al. (2011b), we measured the mutual Hill radii for the six consecutive planet pairs. When the following equation holds true, the pair of planets are Hill stable, meaning that they will never develop crossing orbits, assuming circular, coplanar orbits. Whether the equation holds true depends more heavily on stellar radius (via the R PL /R * and the mass-radius relationship) than stellar mass (via a/R * and Newton's Third Law). The best fit values for these two parameters from Y 2 interpolation are R = 1.04 +0.12 −0.10 R and M = 0.99 +0.10 −0.10 M , consistent within 1σ of the stellar parameters from Huber et al. (2014). The Hill stability metric, where a is the semi-major axis of the planet and R H is the Hill radius, is: ∆ = a outer − a inner R H > 2 √ 3 ≈ 3.46(3) Because of the extremely large TTVs, we might expect that the mutual Hill spheres of the outer two planets are very close to the Hill stability criterion. As expected from the TTVs, the least Hill stable pair of planets are KOI-351.01 (P = 331.6 days) and KOI-351.02 (P = 210.6 days), the outer two planets, with ∆ = 5.60. All planet pairs remain Hill stable across the 3σ range of stellar radius and mass. Lastly, we ran an orbital integration with the Mercury code (Chambers 1999) using the best fit values and for planetary masses corresponding to ±1σ in stellar mass and radius, assuming coplanarity and circular orbits. The systems were stable for > 100 Myr. However, doubling the mass of the outer planet resulted in instability, suggesting that the mass of the outer planet is on the less massive end of the flat part of the mass-radius relationship for gas giants. These orbital integrations are only feasibility tests and are by no means an exhaustive analysis proving dynamic stability. KOI-351 is also interesting as it is close to being a compact analog to the solar system; see Figure 6 for an orbital representation of the system. The nominal radius values of the inner five planet candidates are all in the Earth to mini-Neptune regime ( 3.0R ⊕ ), while the outer two candidates are gas giants. This differs from Kepler-11 (Lissauer et al. 2011a), in that all of Kepler-11's planets are in the super-Earth to Neptune sizes. However, we stress that the error bars on the KOI-351's stellar radius and thus the planetary radii are very large at roughly 50% each. The new candidate has a radius of R = 2.70 ± 2.07R ⊕ . KOI 351 deserves a strong follow-up observation to better constrain the radii, analyze the TTVs, and confirm these planets. This new candidate also exemplifies how complicated signals that may confuse or overload computers can be deciphered by visual checks. We also note that Planet Hunters had independently discovered the 8.72 and the 91.94 day signals. However, during preparation of this paper, both were upgraded to KOI candidate status. Conclusions Planet Hunters is designed to be complementary to the Kepler team's own planet search algorithm, TPS, which has proven itself to be extremely successful with over 18,000 TCEs and more than 3,500 candidates discovered. One of the most important discoveries presented here is the addition of one new planetary candidate around KOI-351, a known six planet candidate system. According to http://exoplanets.org/ (Wright et al. 2011), the two stars with the largest number of confirmed planets, excluding our Sun, contain six planets each: HD 10180 (Lovis et al. 2011) and Kepler-11 (Lissauer et al. 2011a). KOI-435 is the only other Kepler system with six planet candidates Ofir & Dreizler (2013). Furthermore, there are currently only five exoplanetary systems with five confirmed planets (Wright et al. 2011): 55 Cancri (Fischer et al. 2008, Kepler-20 (Borucki et al. 2011;Fressin et al. 2012), Kepler-32 (Borucki et al. 2011;Swift et al. 2013), Kepler-33 (Borucki et al. 2011;Lissauer et al. 2012), and Kepler-62 (Borucki et al. 2013). Although HD 10180 has been claimed to have seven (Lovis et al. 2011) to nine (Tuomi 2012) periodic signals, and GJ 667C is claimed to have up to seven periodic signals (Anglada-Escudé et al. 2013), the planetary nature of those signals is yet to be confirmed. Conversely, we believe that KOI-351 is a true seven planet system with the highest level of certainty short of official confirmation. Analysis shows that this 10 -1 10 0 Semi-major axis (AU) seventh signal is not an alias of the other six and is feasibly stable. Large TTVs for the outer two planet pairs strongly indicate that the outer two planet pairs are likely interacting gravitationally, which helps to validate the system. It is also well known that candidates in multiple candidate systems have much lower false positive probabilities than single candidate systems (Lissauer et al. 2012). KOI-351's system looks somewhat like our own, but much more compact; all seven planet candidates are 1 AU from their host star. While the radii have large error bars, all five of the inner planets have sub-Neptune radii, with KOI-351.07 being the outermost of these. The outer two planets appear to be gas giants. The high multiplicity of this system may merit an increased scrutiny of known planetary systems for additional planet candidates. The most important contribution Planet Hunters provides are the considerable number of new long period candidates. These long period planets can probe the HZ of solar-like stars. Indeed half of the 14 planet candidates reported here are located in their host star's HZ. One such candidate (KIC 11152511) even straddles the transition between super Earth and mini-Neptune radii, making it especially deserving of follow-up analysis. However, many of these new candidates contain only two transits. With the failure of Kepler 's third reaction wheel, one and two transit systems become important in order to study cool planets orbiting far from their host stars. This is especially so for known planetary systems, since the probability that a false positive occurs in a known planetary system is much lower (Lissauer et al. 2012(Lissauer et al. , 2014. Planet Hunters continue to collect one and two transit systems, and these will be further explored in a future paper, Picard et al. (2014, in preparation). Also noteworthy is the new planet candidate around KIC 2437209, which boasts a 74 hour transit every 281 days. Two scenarios fit the transit light curve: an evolved star with a large secondary object transiting the star and a highly eccentric gas giant. If this is indeed a highly eccentric e = 0.974 gas giant, as the long duration might suggest, it may be a planet potentially undergoing high eccentricity migration. However, its position in the g vs. (g − r ) color-magnitude diagram places KIC 2437209 on the giant branch of NGC 6791. This paper brings the total new planetary candidates discovered by Planet Hunters to ∼ 60 plus the two confirmed planets, PH1 b and PH2 b, and new candidates continue to be passed to the science team. Planet Hunters will continue their search for more planetary candidates in the archived Kepler data. The failure of Kepler 's third reaction wheel has led to a proposal for a two wheeled extended mission, K2, which plans to observe new fields of view in the ecliptic for 75 days each (Howell et al. 2014). Should the K2 Kepler mission be approved, Planet Hunters will be analyzing each new campaign as well. and analysis of Kepler data. This open source software project is developed and distributed by the NASA Kepler Guest Observer Office. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. 2437209 · · · · · · · · · · · · 5010054 · · · · · · · · · · · · 5094412 · · · · · · · · · · · · 5522786 · · · · · · · · · · · · 5732155 · · · · · · · · · · · · 6372194 2.0 1.7 · · · 277.6 6436029 · · · · · · · · · · · · 6805414 · · · · · · · · · · · · 9662267 · · · · · · · · · · · · 9704149 · · · · · · · · · · · · 10255705 2.1 1.4 · · · 165.0 11152511 · · · · · · · · · · · · 11442793 · · · · · · · · · · · · 12454613 · · · · · · · · · · · · Note. -Nearest neighbors of the planet candidate hosting stars out to 4 . Only KIC 6805414 has a companion significantly above the background. KIC 10255705 has a neighbor within its 3σ confusion radius. 2.34 ± 0.69 a KIC 5522786 has a T eff above the limits of our stellar modeling, so we used the stellar parameters from (Huber et al. 2014). Note. -Stellar inputs (Huber et al. 2014) and outputs from the iterative light curve and stellar isochrone fitting routine. See Section 3.2 for details. -25 - a Planet candidates with only two transits. Of these, KIC 9704149 actually only has 1.5 transits due to a data gap interrupting the second transit. Note. -Orbital fit to the Kepler light curves from the iterative light curve and stellar isochrone fitting routine. 11,2014), scaled to our model's stellar parameters of M = 0.99 ± 0.10M , R = 1.04 +0.12 −0.10 R , and T eff = 6258 +150 103 K. -27 - Fig. 1 . 1-UKIRT images of the stars with companions within 4 . The images are 20 on a side Only KIC 10255705 has a companion within its 3σ confusion radius. similar to the Huber et al. (2014) stellar parameters. linear ephemeris on account of transit timing variations, or TTVs (Miralda-Escudé 2002; Agol et al. 2005; Holman & Murray 2005; Holman et al. 2010). Fig. 5 . 5-Observed−calculated (O − C) midtransit times for the outer three planet candidates of KOI-351: KOI-351.01 (black circles), KOI-351.02 (blue squares), and KOI-351.07 (red triangles). Fig. 6 . 6-Orbital representation of KOI-351's seven planet candidates with the habitable zone as reference, the black circle being the new candidate. The relative planetary radii are shown as the size of the symbol. Fig. 2.-Phase-folded transit models for new Planet Hunters candidates. Blue circles represent odd transits and are overplotted onto the red diamonds, which represent even transits. There are no significant differences between the depths of the even and odd transits for any candidate. See Section 3.3 for details.Fig. 3.-Radius vs. period for the new Planet Hunters candidates (blue squares), previous Planet Hunters candidates or discoveries (red triangles) (Fischer et al. 2012; Lintott et al. 2013; Wang et al. 2014), and the KOI candidates as of February 6, 2014 (grey circles). Planet Hunters provide a large number of the known, transiting long period candidates. The candidate around KIC 2437209 (P = 281.329 days, R = 62.64 ± 19.61R ⊕ ) is excluded for convenience.Fig. 4.-Locations of the new Planet Hunters candidates relative to their host star's HZ. The green region is the conservative HZ, and the red dashed and dotted lines are the Recent Venus and Early Mars HZ edges, respectively. The size of the symbol is directly proportional to the physical size of the planet. Seven planet candidates lie directly in the conservative HZ: KIC 2437209, 5010054, 5094412, 5732155, 6372194, 9662267, and 9704149. KIC 11152511 sits in the Recent Venus HZ with error bars into the conservative HZ. KIC 6436029 and KIC 10255705 are outside of the HZ, but are within 1 of the Early Mars and Recent Venus HZs, respectively. KIC 5522786 is too hot for the calibration in−40 −20 0 20 40 0.999000 0.999500 1.000000 1.000500 KIC 5010054 −30 −20 −10 0 10 20 30 0.996000 0.997000 0.998000 0.999000 1.000000 1.001000 1.002000 Normalized flux KIC 9704149 −30 −20 −10 0 10 20 30 Phase (hours) 0.998500 0.999000 0.999500 1.000000 1.000500 1.001000 1.001500 KIC 12454613 −60 −40 −20 0 20 40 60 0.990000 0.995000 1.000000 KIC 6805414 Missed by TPS −10 −5 0 5 10 0.997000 0.998000 0.999000 1.000000 1.001000 KIC 5094412 Missed by TPS −40 −30 −20 −10 0 10 20 30 40 0.999850 0.999900 0.999950 1.000000 1.000050 1.000100 Normalized flux KIC 5522786 −30 −20 −10 0 10 20 30 Phase (hours) 0.999000 0.999500 1.000000 1.000500 1.001000 1.001500 KIC 11152511 −15 −10 −5 0 5 10 15 0.988000 0.990000 0.992000 0.994000 0.996000 0.998000 1.000000 1.002000 1.004000 KIC 6372194 Missed by TPS −30 −20 −10 0 10 20 30 0.997000 0.998000 0.999000 1.000000 1.001000 1.002000 1.003000 Normalized flux KIC 6436029 −20 −10 0 10 20 Phase (hours) 0.998500 0.999000 0.999500 1.000000 1.000500 1.001000 Normalized flux KIC 11442793 Missed by TPS −150 −100 −50 0 50 100 150 0.980000 0.985000 0.990000 0.995000 1.000000 1.005000 1.010000 Normalized flux KIC 2437209 Missed by TPS −30 −20 −10 0 10 20 30 0.995000 0.996000 0.997000 0.998000 0.999000 1.000000 1.001000 1.002000 1.003000 KIC 5732155 −20 −10 0 10 20 0.997500 0.998000 0.998500 0.999000 0.999500 1.000000 1.000500 1.001000 1.001500 KIC 9662267 −50 0 50 0.998500 0.999000 0.999500 1.000000 1.000500 1.001000 KIC 10255705 0 200 400 600 800 1000 Period (days) 0 5 10 15 20 25 Radius (R ⊕ ) KOIs Previous PHs Jupiter Neptune Earth New PHs (this paper) 10 -1 10 0 10 1 Flux (F ⊕ ) 4500 5000 5500 6000 6500 T eff (K) Conservative HZ Recent Venus Early Mars R ⊕ R Nep R Jup Table 1 . 1Visual companion detections with UKIRT images.KIC ∆ K P Separation Significance PA (mag) (arcsec) (σ) (deg) Table 2 . 2Stellar ParametersInput Values Table 3 . 3Orbital ParametersStar T 0 Period Impact R PL /R * e ω i a/R * a R PL S Depth Duration (KIC) (MJD) (d) Parameter (radian) (deg) (AU) (R ⊕ ) (S 0 ) (ppm) (hours) 2437209 55329.2790 281.3290 +0.0012 −0.0040 0.48 +0.10 −0.48 0.1154 +0.0006 −0.0017 0.43 +0.23 −0.43 4.72 +0.50 −2.38 89.31 +0.24 −1.26 39.79 +25.10 −7.25 0.88 ± 0.13 62.64 ± 19.61 14.39 ± 11.76 13115 ± 1137 73.56 5010054 a 55188.9590 904.0905 +0.0114 −0.0533 0.03 +0.84 −0.03 0.0257 +0.0006 −0.0010 0.31 +0.68 −0.31 3.72 +1.43 −3.72 90.00 +0.00 −0.31 372.31 +38.58 −8.93 1.86 ± 0.09 2.92 ± 0.48 0.37 ± 0.07 789 ± 136 21.23 5094412 55009.0210 276.8800 +0.0000 −0.0000 0.99 +0.01 −0.09 0.0895 +0.0187 −0.0591 0.04 +0.10 −0.04 5.97 +0.32 −5.97 89.74 +0.21 −0.17 220.60 +12.59 −13.51 0.77 ± 0.03 7.38 ± 3.24 0.77 ± 0.17 2014 ± 280 3.77 5522786 a 55115.4928 757.1570 +0.0089 −0.0299 0.36 +0.40 −0.29 0.0089 +0.0003 −0.0014 0.45 +0.41 −0.45 1.53 +2.79 −1.53 89.91 +0.09 −0.28 236.70 +25.98 −20.35 2.07 ± 0.06 1.86 ± 0.25 4.85 ± 0.98 90 ± 13 14.71 5732155 a 55369.2000 644.1978 +0.0077 −0.0182 0.39 +0.12 −0.39 0.0573 +0.0021 −0.0028 0.52 +0.20 −0.24 4.40 +0.34 −0.52 89.93 +0.03 −0.12 307.37 +22.70 −13.83 1.47 ± 0.09 6.34 ± 0.48 0.59 ± 0.10 3644 ± 398 24.03 6372194 55375.9140 281.5904 +0.0031 −0.0094 0.97 +0.03 −0.11 0.1027 +0.1905 −0.0086 0.27 +0.09 −0.27 2.34 +3.95 −2.34 89.76 +0.01 −0.07 232.59 +12.76 −10.00 0.78 ± 0.02 7.90 ± 7.68 0.59 ± 0.18 10515 ± 344 5.90 6436029 55290.6000 505.4611 +0.0152 −0.0340 0.56 +0.43 −0.56 0.0470 +0.0680 −0.0228 0.63 +0.36 −0.63 4.63 +1.66 −4.63 89.90 +0.08 −0.08 308.28 +11.00 −29.10 1.17 ± 0.04 4.16 ± 4.04 0.26 ± 0.07 1972 ± 389 13.49 6805414 55137.9480 200.2473 +0.0010 −0.0027 0.39 +0.12 −0.02 0.1083 +0.0001 −0.0006 0.74 +0.09 −0.05 4.86 +0.27 −0.57 89.84 +0.02 −0.03 142.17 +12.84 −8.93 0.67 ± 0.04 11.91 ± 1.58 2.67 ± 0.50 12482 ± 296 23.72 9662267 a 55314.3800 466.1710 +0.0112 −0.0370 0.74 +0.19 −0.26 0.0426 +0.0052 −0.0094 0.65 +0.34 −0.65 5.37 +0.92 −5.37 89.83 +0.02 −0.11 257.93 +22.65 −16.43 1.18 ± 0.04 4.50 ± 0.98 0.77 ± 0.15 1511 ± 229 10.55 9704149 a 55252.2220 697.0159 +0.0000 −0.0000 0.38 +0.34 −0.38 0.0524 +0.0039 −0.0018 0.03 +0.11 −0.03 6.29 +0.00 −6.29 89.94 +0.02 −0.05 367.03 +23.65 −10.04 1.49 ± 0.06 5.01 ± 0.39 0.33 ± 0.09 3214 ± 406 13.86 10255705 a 55378.2200 707.8201 +0.0000 −0.0000 0.57 +0.09 −0.57 0.0362 +0.0010 −0.0120 0.52 +0.18 −0.52 4.82 +1.47 −4.82 89.76 +0.07 −0.19 136.78 +55.92 −6.74 1.69 ± 0.13 10.08 ± 3.09 1.67 ± 0.66 1118 ± 102 39.72 11152511 55193.2608 287.3377 +0.0348 −0.0837 0.36 +0.49 −0.36 0.0187 +0.0113 −0.0166 0.00 +0.08 −0.00 0.00 +6.16 −0.00 89.89 +0.11 −4.14 190.60 +4.72 −48.07 0.83 ± 0.05 1.93 ± 1.50 1.06 ± 0.80 463 ± 95 9.38 11442793 55087.1710 124.9199 +0.0236 −0.0540 0.54 +0.05 −0.43 0.0239 +0.0147 −0.0215 0.69 +0.30 −0.69 4.73 +0.25 −1.22 89.69 +0.31 −3.86 99.70 +10.81 −4.08 0.49 ± 0.03 2.70 ± 2.07 5.82 ± 1.70 480 ± 87 10.48 12454613 a 55322.7620 736.3819 +0.0065 −0.0211 0.37 +0.42 −0.37 0.0289 +0.0064 −0.0015 0.16 +0.21 −0.16 0.00 +0.17 −0.00 89.95 +0.03 −0.18 399.19 +22.45 −17.10 1.54 ± 0.04 2.56 ± 0.42 0.23 ± 0.04 1006 ± 142 12.79 Table 4 . 4KOI-351 Candidates Note. -KOI-351.01 through KOI-351.06 values are directly from the public KOI data (http://exoplanetarchive.ipac.caltech.edu, last accessed MarchCandidate T 0 Period Impact R PL /R * i a/R * a R PL S Depth Duration (MJD) (d) Parameter (deg) (AU) (R ⊕ ) (S 0 ) (ppm) (hours) KOI-351.01 54972.98 331.64 0.326 0.0833 89.95 180 0.94 9.4 ± 4.0 1.67 8277 14.54 KOI-351.02 54979.55 210.60 0.313 0.0584 89.95 133 0.69 6.6 ± 2.9 3.09 4073 12.23 KOI-351.03 54991.45 59.74 0.01 0.0217 89.95 57.4 0.30 2.5 ± 1.1 16.3 567 8.12 KOI-351.04 54966.82 91.94 0.29 0.0192 89.95 76.5 0.40 2.2 ± 1.0 9.12 432 8.97 KOI-351.05 54971.96 8.72 0.36 0.0116 88.81 15.9 0.083 1.3 ± 0.51 213 148 3.97 KOI-351.06 54970.17 7.01 0.34 0.0099 88.81 13.8 0.072 1.1 ± 0.48 284 104 3.71 KOI-351.07 55087.22 124.92 0.54 0.0239 89.69 99.8 0.49 2.7 ± 2.07 6.12 480 10.48 Table 5 . 5Nearest linear ephemerides of KOI-351's previously known candidates to T0 Phase T2 Phase T5 Phase T6 Phase T8 Phase T9 Phase Depth Duration 254.64 Difference 504.57 † Difference 879.27 Difference 1004.17 Difference 1253.97 Difference 1378.87 Difference (ppm) (hours) http://exoplanetarchive.ipac.caltech.edu, last accessed March 11, 2014 2 http://www.planethunters.org/ 3 https://www.zooniverse.org/ http://talk.planethunters.org 5 http://archive.stsci.edu/kepler/ 6 http://skyview.gsfc.nasa.gov/ http://exoplanetarchive.ipac.caltech.edu/ AcknowledgementsWe thank all Planet Hunter volunteers, who were indispensable for their work in discovering and analyzing planet candidates in this paper. All Planet Hunters are individually acknowledged at http://www.planethunters.org/authors. The authors thank the Planet Hunters volunteers who participated in identifying and analyzing the candidates presented in this paper. They are individually recognized at http://www.planethunters.org/PH6.We thank the anonymous referee for helpful comments and for suggesting showcasing Planet Hunters considerable contributions to the number of long period candidates.DF acknowledges funding support for PlanetHunters.org from Yale University and support from the NASA Supplemental Outreach Award, 10-OUTRCH.210-0001 and the NASA ADAP12-0172. TSB acknowledges support provided through NASA grant ADAP12-0172. KS gratefully acknowledges support from Swiss National Science Foundation Grant PP00P2 138979/1. Planet Hunters is partially supported by NASA JPL's PlanetQuest program. The Talk system was supported by the National Science Foundation under Grant No. DRL-0941610. The Zooniverse Project is supported by The Leverhulme Trust and by the Alfred P. Sloan foundation. We have used public data from NASA/IPAC/NExScI Star and the Exoplanet Database, which is maintained by JPL at Caltech, under contract with NASA. Our research has utilized NASA's Astrophysics Data System Bibliographic Services. We acknowledge the Kepler science team and others involved for their great and ongoing work. Funding for the Kepler mission is provided by the NASA Science Mission directorate. The publicly released Kepler light curves were downloaded from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. This work made use of PyKE, a software package for the reduction256.81 · · · 509.11 · · · 880.55 · · · 1006.70 · · · 1259.00 · · · 1385.15 · · · · · · · · · Note. - † Transit 5 (T5) is only a partial transit. A data gap within quarter 5 blocks out everything but the egress. Midtransit times of our new KOI-351.07 sandwiched between the nearest two midtransit times of all other six candidates in the system, according to a linear ephemeris as calculated by the NASA Exoplanet Archive's Exoplanet Transit Ephemeris Service (http://exoplanetarchive.ipac.caltech.edu/applications/TransitSearch/). The only midtransit of KOI-351.07 falling within one day of any other transit in the system occurs for KOI-351.06 for Transit 9 (T9), the planet with the shortest period, and thus the one most likely to fall within a day of KOI-351.07 by chance, although its midtransit still falls 17 hours too early. All times are measured in Barycentric Julian Day (BJD) -2454833.00 days. We also show the phase of midtransit of KOI-351.07 relative to the period of the other six candidates and demonstrate that this is not a secondary eclipse of another object. Only the very start of T3's ingress is observed before the end of Q6. Missing transits T1, T4, and T7, fell into data gaps. . E Agol, J Steffen, R Sari, W Clarkson, MNRAS. 359567Agol, E., Steffen, J., Sari, R., & Clarkson, W. 2005, MNRAS, 359, 567 . G Anglada-Escudé, A&A. 556126Anglada-Escudé, G., et al. 2013, A&A, 556, A126 . N M Batalha, ApJ. 713103Batalha, N. M., et al. 2010, ApJ, 713, L103 . ApJS. 20424-. 2013, ApJS, 204, 24 . W J Borucki, Science. 327977Borucki, W. J., et al. 2010, Science, 327, 977 . ApJ. 73619-. 2011, ApJ, 736, 19 . Science. 340587-. 2013, Science, 340, 587 . S T Bryson, PASP. 125889Bryson, S. T., et al. 2013, PASP, 125, 889 . C J Burke, ApJS. 21019Burke, C. J., et al. 2014, ApJS, 210, 19 . J Cabrera, ApJ. 78118Cabrera, J., et al. 2014, ApJ, 781, 18 . J A Carter, J N Winn, ApJ. 70451Carter, J. A., & Winn, J. N. 2009, ApJ, 704, 51 . J E Chambers, MNRAS. 304793Chambers, J. E. 1999, MNRAS, 304, 793 . A Claret, S Bloemen, A&A. 52975Claret, A., & Bloemen, S. 2011, A&A, 529, A75 . P Demarque, J.-H Woo, Y.-C Kim, S K Yi, ApJS. 155667Demarque, P., Woo, J.-H., Kim, Y.-C., & Yi, S. K. 2004, ApJS, 155, 667 . J Eastman, B S Gaudi, E Agol, PASP. 12583Eastman, J., Gaudi, B. S., & Agol, E. 2013, PASP, 125, 83 . D A Fischer, ApJ. 675790Fischer, D. A., et al. 2008, ApJ, 675, 790 . MNRAS. 4192900-. 2012, MNRAS, 419, 2900 L Fortson, Advances in Machine Learning and Data Mining for Astronomy. M. J. Way, J. D. Scargle, K. M. Ali, & A. N. SrivastavaCRC PressFortson, L., et al. 2012, in Advances in Machine Learning and Data Mining for Astronomy, CRC Press, Taylor & Francis Group, Eds.: Michael J. Way, Jeffrey D. Scargle, Kamal M. Ali, Ashok N. Srivastava, p. 213-236, ed. M. J. Way, J. D. Scargle, K. M. Ali, & A. N. Srivastava, 213-236 . F Fressin, Nature. 482195Fressin, F., et al. 2012, Nature, 482, 195 . ApJ. 76681-. 2013, ApJ, 766, 81 . J Z Gazak, J A Johnson, J Tonry, D Dragomir, J Eastman, A W Mann, E Agol, Advances in Astronomy. Gazak, J. Z., Johnson, J. A., Tonry, J., Dragomir, D., Eastman, J., Mann, A. W., & Agol, E. 2012, Advances in Astronomy, 2012 . M J Holman, N W Murray, Science. 3071288Holman, M. J., & Murray, N. W. 2005, Science, 307, 1288 . M J Holman, Science. 33051Holman, M. J., et al. 2010, Science, 330, 51 . S B Howell, Howell, S. B., et al. 2014, ArXiv e-prints . X Huang, G Á Bakos, J D Hartman, MNRAS. 429Huang, X., Bakos, G.Á., & Hartman, J. D. 2013, MNRAS, 429, 2001 . D Huber, ApJS. 2112Huber, D., et al. 2014, ApJS, 211, 2 . J M Jenkins, D A Caldwell, W J Borucki, ApJ. 564495Jenkins, J. M., Caldwell, D. A., & Borucki, W. J. 2002, ApJ, 564, 495 . J M Jenkins, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series. 7740Society of Photo-Optical Instrumentation Engineers (SPIE) Conference SeriesJenkins, J. M., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Confer- ence Series, Vol. 7740, Society of Photo-Optical Instrumentation Engineers (SPIE) Confer- ence Series . J F Kasting, D P Whitmire, R T Reynolds, Icarus. 101108Kasting, J. F., Whitmire, D. P., & Reynolds, R. T. 1993, Icarus, 101, 108 . R K Kopparapu, ApJ. 765131Kopparapu, R. K., et al. 2013, ApJ, 765, 131 . C Lintott, MNRAS. 410166Lintott, C., et al. 2011, MNRAS, 410, 166 . C J Lintott, MNRAS. 3891179Lintott, C. J., et al. 2008, MNRAS, 389, 1179 . AJ. 145151-. 2013, AJ, 145, 151 . J J Lissauer, Nature. 47053Lissauer, J. J., et al. 2011a, Nature, 470, 53 . ApJS. 1978-. 2011b, ApJS, 197, 8 . ApJ. 750112-. 2012, ApJ, 750, 112 . ApJ. 78444-. 2014, ApJ, 784, 44 . E D Lopez, J J Fortney, Lopez, E. D., & Fortney, J. J. 2013, ArXiv e-prints . C Lovis, A&A. 528112Lovis, C., et al. 2011, A&A, 528, A112 . G W Marcy, ApJS. 21020Marcy, G. W., et al. 2014, ApJS, 210, 20 . T Mazeh, ApJS. 20816Mazeh, T., et al. 2013, ApJS, 208, 16 . J Miralda-Escudé, ApJ. 5641019Miralda-Escudé, J. 2002, ApJ, 564, 1019 . T D Morton, J A Johnson, ApJ. 738170Morton, T. D., & Johnson, J. A. 2011, ApJ, 738, 170 . A Ofir, S Dreizler, A&A. 55558Ofir, A., & Dreizler, S. 2013, A&A, 555, A58 . I Platais, K M Cudworth, V Kozhurina-Platais, D E Mclaughlin, S Meibom, C Veillet, ApJ. 7331Platais, I., Cudworth, K. M., Kozhurina-Platais, V., McLaughlin, D. E., Meibom, S., & Veillet, C. 2011, ApJ, 733, L1 . M E Schwamb, ApJ. 754129Schwamb, M. E., et al. 2012, ApJ, 754, 129 . ApJ. 768127-. 2013, ApJ, 768, 127 . A Socrates, B Katz, S Dong, S Tremaine, ApJ. 750106Socrates, A., Katz, B., Dong, S., & Tremaine, S. 2012, ApJ, 750, 106 . M Still, T Barclay, ascl:1208.8004Astrophysics Source Code Library. Still, M., & Barclay, T. 2012, Astrophysics Source Code Library, ascl:1208.8004 . J J Swift, J A Johnson, T D Morton, J R Crepp, B T Montet, D C Fabrycky, P S Muirhead, ApJ. 764105Swift, J. J., Johnson, J. A., Morton, T. D., Crepp, J. R., Montet, B. T., Fabrycky, D. C., & Muirhead, P. S. 2013, ApJ, 764, 105 . P Tenenbaum, ApJS. 2065Tenenbaum, P., et al. 2013, ApJS, 206, 5 . ApJS. 2116-. 2014, ApJS, 211, 6 . M Tuomi, A&A. 54352Tuomi, M. 2012, A&A, 543, A52 . J A Valenti, D A Fischer, ApJS. 159141Valenti, J. A., & Fischer, D. A. 2005, ApJS, 159, 141 . J A Valenti, N Piskunov, A&AS. 118595Valenti, J. A., & Piskunov, N. 1996, A&AS, 118, 595 S S Vogt, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series. 2198362Society of Photo-Optical Instrumentation Engineers (SPIE) Conference SeriesVogt, S. S., et al. 1994, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 2198, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 362-+ . J Wang, J.-W Xie, T Barclay, D A Fischer, ApJ. 7834Wang, J., Xie, J.-W., Barclay, T., & Fischer, D. A. 2014, ApJ, 783, 4 . J Wang, ApJ. 77610Wang, J., et al. 2013, ApJ, 776, 10 . L M Weiss, G W Marcy, ApJ. 7836Weiss, L. M., & Marcy, G. W. 2014, ApJ, 783, L6 . J T Wright, PASP. 123412Wright, J. T., et al. 2011, PASP, 123, 412
[]
[ "A Categorical Treatment of Ornaments *", "A Categorical Treatment of Ornaments *" ]
[ "Pierre-Evariste Dagand ", "Conor Mcbride " ]
[]
[]
Ornaments aim at taming the multiplication of special-purpose datatypes in dependently typed programming languages. In type theory, purpose is logic. By presenting datatypes as the combination of a structure and a logic, ornaments relate these special-purpose datatypes through their common structure. In the original presentation, the concept of ornament was introduced concretely for an example universe of inductive families in type theory, but it was clear that the notion was more general. This paper digs out the abstract notion of ornaments in the form of a categorical model. As a necessary first step, we abstract the universe of datatypes using the theory of polynomial functors. We are then able to characterise ornaments as cartesian morphisms between polynomial functors. We thus gain access to powerful mathematical tools that shall help us understand and develop ornaments. We shall also illustrate the adequacy of our model. Firstly, we rephrase the standard ornamental constructions into our framework. Thanks to its conciseness, this process gives us a deeper understanding of the structures at play. Secondly, we develop new ornamental constructions, by translating categorical structures into type theoretic artefacts.The theory of inductive types is generally understood as the study of initial algebras in some appropriate category. A datatype definition is therefore abstracted away as a signature functor that admits a least fixpoint. This naturally leads to the study of polynomial functors[Gambino and Kock, 2010], a class of functors that all admit an initial algebra. These functors have been discovered and studied under many guises. In type theory, they were introduced by Martin-Löf under the name of well-founded trees[Martin-Löf, 1984, Moerdijk and Palmgren, 2000, Petersson and Synek, 1989, or W-types for short. Containers [Abbott et al., 2005a] and their indexed counterparts[Morris, 2007]generalise these definitions to a fibrational setting. Polynomial functorsHyland, 2004, Gambino andKock, 2010]are the category theorists' take on containers, working in a locally cartesian-closed category.There is a significant gap between this unified theoretical framework and the implementations of inductive types: in systems such as Coq [The Coq Development
10.1109/lics.2013.60
[ "https://arxiv.org/pdf/1212.3806v3.pdf" ]
699,271
1212.3806
e77e2fbd1776b0dcbb99d4fd146a35cdea9a1dd7
A Categorical Treatment of Ornaments * 21 Apr 2013 Pierre-Evariste Dagand Conor Mcbride A Categorical Treatment of Ornaments * 21 Apr 2013Team] or * Revised May 5, 20141 Ornaments aim at taming the multiplication of special-purpose datatypes in dependently typed programming languages. In type theory, purpose is logic. By presenting datatypes as the combination of a structure and a logic, ornaments relate these special-purpose datatypes through their common structure. In the original presentation, the concept of ornament was introduced concretely for an example universe of inductive families in type theory, but it was clear that the notion was more general. This paper digs out the abstract notion of ornaments in the form of a categorical model. As a necessary first step, we abstract the universe of datatypes using the theory of polynomial functors. We are then able to characterise ornaments as cartesian morphisms between polynomial functors. We thus gain access to powerful mathematical tools that shall help us understand and develop ornaments. We shall also illustrate the adequacy of our model. Firstly, we rephrase the standard ornamental constructions into our framework. Thanks to its conciseness, this process gives us a deeper understanding of the structures at play. Secondly, we develop new ornamental constructions, by translating categorical structures into type theoretic artefacts.The theory of inductive types is generally understood as the study of initial algebras in some appropriate category. A datatype definition is therefore abstracted away as a signature functor that admits a least fixpoint. This naturally leads to the study of polynomial functors[Gambino and Kock, 2010], a class of functors that all admit an initial algebra. These functors have been discovered and studied under many guises. In type theory, they were introduced by Martin-Löf under the name of well-founded trees[Martin-Löf, 1984, Moerdijk and Palmgren, 2000, Petersson and Synek, 1989, or W-types for short. Containers [Abbott et al., 2005a] and their indexed counterparts[Morris, 2007]generalise these definitions to a fibrational setting. Polynomial functorsHyland, 2004, Gambino andKock, 2010]are the category theorists' take on containers, working in a locally cartesian-closed category.There is a significant gap between this unified theoretical framework and the implementations of inductive types: in systems such as Coq [The Coq Development Ornaments aim at taming the multiplication of special-purpose datatypes in dependently typed programming languages. In type theory, purpose is logic. By presenting datatypes as the combination of a structure and a logic, ornaments relate these special-purpose datatypes through their common structure. In the original presentation, the concept of ornament was introduced concretely for an example universe of inductive families in type theory, but it was clear that the notion was more general. This paper digs out the abstract notion of ornaments in the form of a categorical model. As a necessary first step, we abstract the universe of datatypes using the theory of polynomial functors. We are then able to characterise ornaments as cartesian morphisms between polynomial functors. We thus gain access to powerful mathematical tools that shall help us understand and develop ornaments. We shall also illustrate the adequacy of our model. Firstly, we rephrase the standard ornamental constructions into our framework. Thanks to its conciseness, this process gives us a deeper understanding of the structures at play. Secondly, we develop new ornamental constructions, by translating categorical structures into type theoretic artefacts. The theory of inductive types is generally understood as the study of initial algebras in some appropriate category. A datatype definition is therefore abstracted away as a signature functor that admits a least fixpoint. This naturally leads to the study of polynomial functors [Gambino and Kock, 2010], a class of functors that all admit an initial algebra. These functors have been discovered and studied under many guises. In type theory, they were introduced by Martin-Löf under the name of well-founded trees [Martin-Löf, 1984, Moerdijk and Palmgren, 2000, Petersson and Synek, 1989, or W-types for short. Containers [Abbott et al., 2005a] and their indexed counterparts [Morris, 2007] generalise these definitions to a fibrational setting. Polynomial functors Hyland, 2004, Gambino andKock, 2010] are the category theorists' take on containers, working in a locally cartesian-closed category. There is a significant gap between this unified theoretical framework and the implementations of inductive types: in systems such as Coq [The Coq Development Team] or Agda [Norell, 2007], datatypes are purely syntactic artefacts. A piece of software, the positivity checker, is responsible for checking that the definition entered by the user is valid, i.e. does not introduce a paradox. The power of the positivity checker depends on the bravery of its implementers: for instance, Coq's positivity checker is allegedly simple, therefore safer, but rather restrictive. On the other hand, Agda's positivity checker is more powerful, hence more complex, but also less trusted. For example, the latter checks the positivity of functions in datatype declarations, while the former conservatively rejects them. The more powerful the positivity checker, the harder it is to relate the datatype definitions to some functorial model. An alternative presentation of inductive types is through a universe construction [Martin-Löf, 1984, Dybjer, 1991, Chapman et al., 2010. The idea is to reflect the grammar of polynomial functors into type theory itself. Having internalised inductive types, we can formally manipulate them and, for example, create new datatypes from old. The notion of ornament [McBride, 2013] is an illustration of this approach. Ornaments arise from the realisation that inductive families can be understood as the integration of a datastructure together with a data-logic. The structure captures the dynamic, operational behavior expected from the datatype. It corresponds to, say, the choice between a list or a binary tree, which is governed by performance considerations. The logic, on the other hand, dictates the static invariants of the datatype. For example, by indexing lists by their length, thus obtaining vectors, we integrate a logic of length with the data. We can then take an m × n matrix to be a plainly rectangular m-vector of n-vectors, rather than a list of lists together with a proof that measuring each length yields the same result. In dependent type theory, logic is purpose: when solving a problem, we want to bake the problem's invariants into the datatype we manipulate. Doing so, our code is correct by construction. The same data-structure will be used for different purposes and will therefore integrate as many logics: we assist to a multiplication of datatypes, each built upon the same structure. This hinders any form of code reuse and makes libraries next to pointless: every task requires us to duplicate entire libraries for our special-purpose datatypes. Ornaments tame this issue by organising datatypes along their structure: given a datatype, an ornament gives an effective recipe to extend -introducing more information -and refine -providing a more precise indexing -the initial datatype. Applying that recipe gives birth to a new datatype that shares the same structure as the original datatype. Hence, ornaments let us evolve datatypes with some special-purpose logic without severing the structural ties between them. In an earlier work [Dagand and McBride, 2012], we have shown how that information can be used to regain code reuse. The initial presentation of ornaments and its subsequent incarnation [McBride, 2013, Dagand andMcBride, 2012] are however very syntactic and tightly coupled with their respective universe of datatypes. We are concerned that their syntactic nature obscures the rather simple intuition governing these definitions. In this paper, we give a semantic account of ornaments, thus exhibiting the underlying structure of the original definitions. To do so, we adopt a categorical approach and study ornaments in the framework of polynomial functors. Our contributions are the following: • In Section 2, we formalise the connection between a universe-based presentation of datatypes and the theory of polynomial functors. In particular, we prove that the functors represented by our universe are equivalent to polynomial functors. This key result lets us move seamlessly from our concrete presentation of datatypes to the more abstract polynomial functors. • In Section 3, we give a categorical presentation of ornaments as cartesian morphisms of polynomial functors. This equivalence sheds some light on the original definition of ornaments. It also connects ornaments to a mathematical object that has been widely studied: we can at last organise our universe of datatypes and ornaments on them into a category -in fact a framed bicategory [Shulman, 2009] -and start looking for categorical structures that would translate into interesting type theoretic objects. • In Section 4, we investigate the categorical structure of ornaments. The contribution here is twofold. On one hand, we translate the original, type theoretic constructions -such as the ornamental algebra and the algebraic ornament -in categorical terms and uncover the building blocks out of which they were carved out. On the other hand, we interpret the mathematical properties of ornaments into type theory -such as the pullback of ornaments and the ornamentation of derivatives -to discover meaningful software artefacts that were previously unknown. Being at the interface between type theory and category theory, this paper targets both communities. To the type theorist, we offer a more semantic account of ornaments and use the intuition thus gained to introduce new type theoretic constructions. Functional programmers of the non-dependent kind will find a wealth of examples that should help them grasp the more abstract concepts, both type theoretic and category theoretic. To the category theorist, we present a type theory, i.e. a programming language, that offers an interesting playground for categorical ideas. Our approach can be summarised as categorically structured programming. For practical reasons, we do not work on categorical objects directly: instead, we materialise these concepts through universes, thus reifying categorical notions through computational objects. Ornaments are merely an instance of that interplay between a categorical concept -cartesian morphism of polynomial functor -and an effective, type theoretic presentation -the universe of ornaments. To help bridge the gap between type theory and category theory, we have striven to provide the type theorist with concrete examples of the categorical notions and the category theorist with the computational intuition behind the type theoretic objects. Categorical Toolkit In this section, we recall a few definitions and results from category theory that will be used throughout this paper. None of these results are new -most of them are folklore -we shall therefore not dwell on the details. However, to help readers not familiar with these tools, we shall give many examples, thus providing an intuition for these concepts. Locally cartesian-closed categories Locally cartesian-closed categories (LCCC) were introduced by Seely [1983] to give a categorical model of (extensional) dependent type theory. A key idea of that presentation is the use of adjunctions to model Π-types and Σ-types. Definition 1 (Locally cartesian-closed category). A locally cartesian-closed category is a category E that is pullback complete and such that, for f : E(X, Y ), each base change functor ∆ f : E /Y → E /X , defined by pullback along f , has a right adjoint Π f . Throughout this paper, we work in a locally cartesian-closed category E with a terminal object 1 E and sums. An object f : E → I in the slice E /I can be thought of as an I-indexed set, which we shall informally denote using a set comprehension notation {E i | i ∈ I}, where E i can be understood as the inverse image of f at i, or equivalently a fibre of the discrete fibration f . By construction, the base change functor has a left adjoint Σ f = f • . We therefore have the adjunctions Σ f ⊣ ∆ f ⊣ Π f Using a set theoretic notation, the base change functor writes as reindexing by f : ∆ f : E /B → E /A ∆ f {E b | b ∈ B} → {E f a | a ∈ A} While the left and right adjoints correspond to the following definitions: Σ f : E /A → E /B Σ f {E a | a ∈ A} → a∈A b E a | b ∈ B Π f : E /A → E /B Π f {E a | a ∈ A} → a∈A b E a | b ∈ B Where and respectively represent the (set theoretic) disjoint union and cartesian product. Details of this construction can be found elsewhere [Mac Lane and Moerdijk, 1992]. The internal language of E corresponds to an extensional type theory denoted Set, up to bureaucracy [Curien, 1993]. This type theory comprises a unit type 1, Σ-types, Π-types, and equality is extensional. Syntactically, this type theory is specified by the following judgments: Formation rules: Introduction rules: Elimination rules: Γ ⊢ 1 : Set Γ ⊢ * : 1 Γ ⊢ A : Set Γ ⊢ B : Set Γ ⊢ A + B : Set Γ ⊢ a : A Γ ⊢ inj l a : A Γ ⊢ b : B Γ ⊢ inj r b : B Γ ⊢ f : A → C Γ ⊢ g : B → C Γ ⊢ x : A + B Γ ⊢ f , g x : C Γ ⊢ S : Set Γ; x : S ⊢ T : Set Γ ⊢ (x : S ) × T : Set Γ ⊢ a : S Γ ⊢ b : T [a/x] Γ ⊢ (a, b) : (x : S ) × T Γ ⊢ p : (x : S) × T Γ ⊢ π 0 p : S Γ ⊢ p : (x : S) × T Γ ⊢ π 1 p : T [π 0 p/x] Γ ⊢ S : Set Γ; x : S ⊢ T : Set Γ ⊢ (x : S) → T : Set Γ ⊢ S : Set Γ; x : S ⊢ t : T Γ ⊢ λ S x. t : (x : S) → T Γ ⊢ f : (x : S) → T Γ ⊢ s : S Γ ⊢ f s : T [s/x] We chose to work in an extensional model for simplicity. However, all the constructions presented in this paper have been modelled in Agda, an intuitionistic type theory. Polynomials and polynomial functors Polynomials Hyland, 2004, Gambino andKock, 2010] provide a categorical model for indexed families [Dybjer, 1991] in a LCCC. Polynomials themselves are small, diagrammatic objects that admit a rich categorical structure. They are then interpreted as strong functors -the polynomial functors -between slices of E. In this section, we shall illustrate the categorical definitions with the corresponding notion on (indexed) container [Petersson and Synek, 1989, Hancock and Hyvernat, 2006, Morris, 2007, an incarnation of polynomials in the internal language Set. I s ←− B f −→ A t −→ J. Application 1 (Container). In type theory, it is more convenient to work with (proof relevant) predicates rather than arrows. Hence, inverting the arrow t : A → J, we obtain a predicate S : J → Set -called the shapes. Similarly, inverting f : B → A, we obtain a predicate P : ∀j. S j → Set -called the positions. The indexing map s remains unchanged but, following conventional notation, we rename it n -the next index function. We obtain the following definition:    S : J → Set P :S j → Set n : P sh → I Note that, to remove clutter, we (implicitly) universally quantify unbound type variables, such as j in the definition of P or sh in the definition of n. The data of S, P , and n is called a container and is denoted S n P . The class of containers indexed by I and J is denoted ICont I J . Remark 1 (Intuition). Polynomials, and more directly containers, can be understood as multi-sorted signatures. The indices specifies the sorts. The shapes at a given index specify the set of symbols at that sort. The positions specify the arities of each symbol. The next index function specifies, for each symbol, the sort of its arguments. Definition 3 (Polynomial functor [Gambino and Kock, 2010, §1.4]). We interpret a polynomial F : I s ←− B f −→ A t −→ J into a functor, conventionally denoted P F , between slices of E with the construction P F E /I ∆s −→ E /B Π f −→ E /A Σt −→ E /J A functor F is called polynomial if it is isomorphic to the interpretation of a polyno- mial, i.e. there exists s, f , and t such that F ∼ = Σ t Π f ∆ s . Application 2 (Interpretation of container). Unfolding this definition in the internal language, we interpret a container as, first, a choice (Σ-type) of shape ; then, for each (Π-type) position, a variable X taken at the next index n for that position: (C : ICont I J) Cont (X : I → Set) : J → Set S n P Cont X → λj. (sh : S j) × ((pos : P sh) → X (n pos)) hence justifying the name polynomial functor : a polynomial interprets into an S-indexed sum of monomials X taken at some exponent pos : P sh, or put informally: S n P Cont {X i | i ∈ I} →    sh∈S j pos∈P sh X n pos | j ∈ J    Example 1 (Container: natural number). Natural numbers are described by the signature functor X → 1 + X. The corresponding container is given in Figure 1a. There are two shapes, one to represent the 0 case, the other to represent the successor case, suc. For the positions, none is offered by the 0 shape, while the suc shape offers one. Note that the signature functor is not indexed: the container is therefore indexed by the unit set and the next index is trivial. Example 2 (Container: list). The signature functor describing a list of parameter A is X → 1 + A × X. The container is presented Figure 1b. Note the similarity with natural numbers. There are 1 + A shapes, i.e. either the empty list nil or the list constructor cons of some a : A. There are no subsequent position for the nil shape, while one position is offered by the cons shapes. Indices are trivial, for lists are not indexed. NatCont                        S Nat ( * : 1) : Set S Nat * → 1 + 1 P Nat (sh : S Nat * ) : Set P Nat (inj l * ) → 0 P Nat (injr * ) → 1 n Nat (pos : P Nat sh) : 1 n Nat pos → * (a) Natural number ListContA                        S List ( * : 1) : Set S List * → 1 + A P List (sh : S List * ) : Set P List (inj l * ) → 0 P List (injr a) → 1 n List (pos : P List sh) : 1 n List pos → * (b) List VecContA                            S Vec (n : Nat) : Set S Vec 0 → 1 S Vec (suc n) → A P Vec (n : Nat) (sh : S Vec n) : Set P Vec 0 * → 0 P Vec (suc n) a → 1 n Vec (n : Nat) (sh : S Vec n) (pos : P Vec pos) : Nat n Vec (suc n) a * → n (c) Vector Figure 1: Examples of containers Example 3 (Container: vector). To give an example of an indexed datatype, we consider vectors, i.e. lists indexed by their length. The signature functor of vectors is given by {X n | n ∈ Nat} → {n = 0 | n ∈ Nat} + {A × X n−1 | n ∈ Nat * } where the empty vector nil requires the length n to be 0, while the vector constructor cons must have a length n of at least one and takes its recursive argument X at index n − 1. The container representing this signature is given Figure 1c. At index 0, only the nil shape is available while index suc n offers a choice of a : A shapes. As for lists, the nil shape has no subsequent position while the cons shapes offer one. It is necessary to compute the next index (i.e. the length of the tail) only when the input index is suc n, in which case the next index is n. We leave it to the reader to verify that the interpretation of NatCont (Example 1), ListCont (Example 2), and VecCont (Example 3) are indeed equivalent to the signature functors we aimed at representing. With this exercise, one gains a better intuition of the respective contribution of shapes, positions, and the next index to the encoding of signature functors. Definition 4 (Polynomial morphism [Gambino and Kock, 2010, §3.8 ]). A morphism from F : I s ′ ←− B f ′ −→ A t ′ −→ J to G : K s ←− D f −→ C t −→ L is uniquely represented -up to the choice of pullback -by the diagram: I B A J D ′ A K D C L s ′ f ′ t ′ s f t u v α ω Example 4 (Container morphism). Let u : I → K and v : J → L. A morphism from a container S ′ n ′ P ′ to a container S n P framed by u and v is given by two functions and a coherence condition:    σ :S ′ j → S (v j) ρ : P (σ sh ′ ) → P ′ sh ′ q : ∀sh ′ : S ′ j. ∀pos : P (σ sh ′ ). u (n ′ (ρ pos)) = n pos Remark that σ and ρ correspond exactly to their diagrammatic counterparts, respectively α and ω, while the coherence condition q captures the commutativity of the left square. Commutativity of the right square is ensured by construction, since we reindex S by v in the definition of σ. A container morphism, i.e. the data σ, ρ, and q, is denoted σ ρ (leaving implicit the coherence condition). The hom-set of morphisms from S ′ n ′ P ′ to S n P framed by u and v is denoted S ′ n ′ P ′ u =⇒ v S n P . In this paper, we are particularly interested in a sub-class of polynomial morphisms: the class of cartesian morphisms. Cartesian morphisms represent only cartesian natural transformationsi.e. for which the naturality square forms a pullback. Definition 5 (Cartesian morphism [Gambino and Kock, 2010, §3.14]). A cartesian mor- phism from F : I s ′ ←− B f ′ −→ A t ′ −→ J to G : K s ←− D f −→ C t −→ L is uniquely represented by the diagram: I B A J K D C L u v α Where the α is pulled back along f , as conventionally indicated by the right angle symbol. Application 3 (Cartesian morphism of containers). In the internal language, a cartesian morphism from S ′ n ′ P ′ to S n P framed by u and v corresponds to the triple:    σ :S ′ j → S (v j) ρ : ∀sh ′ : S ′ j. P (σ sh ′ ) = P ′ sh ′ q : ∀sh ′ : S ′ j. ∀pos : P (σ sh ′ ). u (n ′ pos) = n pos The diagrammatic morphism α translates into an operation on shapes, denoted σ. The pullback condition translates into a proof ρ that the source positions are indeed obtained by pulling back the target positions along σ. As for the indices, the coherence condition q captures the commutativity of the left square. Commutativity of the right square is ensured by construction, since we reindex S by v in the definition of σ. A cartesian morphism is denoted σ c , leaving implicit the proof obligations. The hom-set of cartesian morphisms from S ′ n ′ P ′ to S n P is denoted S ′ n ′ P ′ u =⇒ c v S n P . Because polynomials and containers conventionally use different notations, we sum-up the equivalences in Figure 2. Polynomial Container Obtained by Example 5 (Cartesian morphism). We build a cartesian morphism from ListCont A (Example 2) to NatCont (Example 1) by mapping shapes of ListCont A to shapes of NatCont: t : A → J S : J → Set Inverse image f : B → A P : S j → Set Inverse image s : B → I n : P sh → I Identity α : A → C σ : S ′ j → S (v j) Identityσ (sh l : S List * ) : S Nat * σ (inj l * ) → inj l * -nil to 0 σ (inj r a) → inj r * -cons a to suc We are then left to check that positions are isomorphic: this is indeed true, since, in the nil/0 case, there is no position while, in the cons/suc case, there is only one position. The coherence condition is trivially satisfied, since both containers are indexed by 1. We shall relate this natural transformation to the function computing the length of a list in Example 15. We have seen that polynomials interpret to (polynomial) functors. We therefore expect morphisms of polynomials to interpret to natural transformations between these functors. Definition 6 (Interpretation of polynomial morphism [Gambino and Kock, 2010, §2.1 and §2.7]). For the morphism of polynomials given in Definition 4, we construct the following natural transformation: E /I E /B E /B E /A E /J E /K E /D ′ E /A E /J E /K E /D E /C E /C E /L ∼ = ∼ = ∼ = ∼ = ∼ = ∼ = ⇓ η ⇓ ǫ ∆ s ′ Π f ′ Σ t ′ ∆ s Π f Σ t ∆ s•α † Π f † Σ t Σ u Σ v ∆ α † ∆ α Σ α ∆ ω Π ω The diagrammatic construction of the interpretation on morphism is perhaps intimidating. Its actual simplicity is revealed by containers, in the internal language. Example 6 (Interpretation of container morphism). A morphism from S ′ n ′ P ′ to S n P simply maps shapes S ′ to shapes S covariantly using σ and maps positions P to positions P ′ contravariantly using ρ: (m : S ′ n ′ P ′ u =⇒ v S n P ) Cont (xs : S ′ n ′ P ′ Cont X) : S n P Cont X σ ρ Cont (sh ′ , Xs) → (σ sh ′ , Xs • ρ) Note that, thinking of X as being parametrically quantified, there is not much choice anyway: the shapes are in a covariant position while positions are on the left on an arrow, i.e. contravariant position. Hancock and Hyvernat [2006] present these morphisms as defining a (constructive) simulation relation: having a morphism from C ′ S ′ n ′ P ′ to C S n P gives you an effective recipe to simulate the behavior of C ′ using C: a choice of shape sh ′ in S ′ is translated to a choice of shape in S through σ while the subsequent response pos : P (σsh ′ ) is back-translated through ρ to a response in P ′ . We shall need the following lemma that creates polynomial functors from a cartesian natural transformations to a polynomial functor: Lemma 1 (Lemma 2.2 [Gambino and Kock, 2010]). Let P F : E /I → E /J be a polynomial functor. Let Q a functor from E /I to E /J . If φ : Q . → P F is a cartesian natural transformation, then Q is also a polynomial functor. Finally, we shall need the following algebraic characterization of polynomial functors: Lemma 2 (Corollary 1.14 [Gambino and Kock, 2010]). The class of polynomial functors is the smallest class of functors between slices of E containing the pullback functors and their adjoints, and closed under composition and natural isomorphism. Framed bicategory We have resisted the urge of defining a category of polynomials and polynomial functors. Such a category can be defined for a given pair of indices I and J, with objects being polynomials indexed by I and J (Definition 2) and morphisms (Definition 4) specialised to the case where u = id : I → I and v = id : J → J. From there, we are naturally lead to organise polynomials and their indices in a 2category. However, this fails to capture the fact that indices have a life of their own: it makes sense to have morphisms between differently indexed functors, i.e. between different slices of E. Indeed, morphisms between indices -the objectsinduce 1-morphisms. The 2-categorical presentation does not capture this extra power. Following the steps of Gambino and Kock [2010], we organize polynomials and their functors in a framed bicategory [Shulman, 2009]. We refer the reader to that latter paper for a comprehensive presentation of this concept. A framed bicategory is a double category with some more structure. We therefore recall the definition of a double category and give a few examples. Definition 7 (Double category). A double category D consists of a category of objects D 0 and a category of morphisms D 1 , together with structure functors: D 0 D 1 D 1 × D 0 D 1 L U R ⊙ Satisfying the usual axioms of categories relating the left frame L, right frame R, identity U , and composition ⊙. Example 7 (Double category PolyFun E [Gambino and Kock, 2010, §3.5]). The double category PolyFun E is defined as follow: • Objects (i.e. objects of D 0 ): slices E /I • Vertical arrows (i.e. morphism of D 0 ): colift Σ u : E /I → E /K , for u : I → K in E • Horizontal arrows (i.e. objects of D 1 ) with left frame E /I and right frame E /J : polynomial functor P F : E /I → E /J • Squares (i.e. morphisms of D 1 ) with left frame Σ u and right frame Σ v : strong natural transformation φ: E /I E /J E /K E /L ⇓ φ P F Σ u Σ v P G Example 8 (Double category Poly E ). The double category Poly E is defined as follow: • D 0 = Set, i.e. -Objects: index set I, J, K, L, . . . -Vertical arrows: u : I → K, v : J → L, . . . • Horizontal arrows (i.e. objects of D 1 ) with left frame I and right frame J: polynomial indexed by I and J • Squares (i.e. morphisms of D 1 ) with left frame Σ u and right frame Σ v : polynomial morphism, with u closing the diagram on the left and v closing the diagram on the right. We are naturally tempted to establish a connection between the double category Poly E of polynomials and the double category PolyFun E of polynomial functors. We expect the interpretation of polynomials to play that role, behaving, loosely speaking, as a functor from Poly E to PolyFun E . To formalize that intuition, we need a notion of functor between double categories: Definition 8 (Lax double functor [Shulman, 2009, §6.1]). A lax double functor F : C → D consists of: • Two functors F 0 : C 0 → D 0 and F 1 : C 1 → D 1 such that L • F 1 = F 0 • L and R • F 1 = F 0 • R. • Two natural transformations transporting the ⊙ and U functors in the expected way. Having presented the double-categorical framework, we now move on to framed bicategories. The key intuition here comes from our earlier observation: morphisms between indices, i.e. morphisms in D 0 , induce polynomials, i.e. objects in D 1 . Categorically, this translates into a bifibrational structure on the (L, R) functor: Definition 9 (Framed bicategory [Shulman, 2009]). A framed bicategory is a double category for which the functor (L, R) : D 1 → D 0 × D 0 is a bifibration. Example 9 (Framed bicategory Poly E [Gambino and Kock, 2010, §3.7]). The bifibration is therefore the endpoints functor [Gambino and Kock, 2010, §3.10] for which the cobase change of polynomial F : I s ←− B f −→ A t −→ J along (u, v), denoted (u, v) ! F , is I B A J K B A L s f t u • s f v • t u v While the base change of G along (u, v), denoted (u, v) * G is defined as: I · · J · · · · · · K D C L s f t v † † v † v u ǫ u † s † ∆ f Π f u † Π f u † As before, these definitions straightforwardly translates to operations on containers. Example 10 (Framed bicategory PolyFun E [Gambino and Kock, 2010, §3.6]). The fibrational structure of the framed bicategory gives rise to a transporter lift (the cartesian lifting of the fibration (L, R)) and a cotransporter lift (the op-cartesian lifting of the op-fibration (L, R)). The transporter lift of (u, v) to P G is given by: E /I E /K E /L E /K E /K E /K E /L E /L ∼ = ∼ = ⇓ η Σ u P G ∆ v P G Σ u Σ v The cotransporter lift of (u, v) to P F is given by: E /I E /I E /J E /J E /K E /I E /J E /L ⇓ ǫ ∼ = ∼ = P F ∆ u P F Σ v Σ u Σ v Following Gambino and Kock [2010], we define the base change of P G along (u, v) by (u, v) * P G = ∆ v • P G • Σ u . Dually, we define the cobase change of P F along (u, v) by (u, v) ! P F = Σ v • P F • ∆ u . At this stage, it should be clear that the interpretation of polynomials is more than a mere functor from Poly E to PolyFun E : loosely speaking, it establishes an equivalence of categories. Equivalence of framed bicategory is formally defined as follow: Definition 10 (Framed biequivalence [Shulman, 2009, §7.1]). A framed equivalence between the framed bicategory C and D consists of: • Two lax double functors F : C → D and G : D → C, and • Two framed natural isomorphism η : 1 ∼ = G • F and ǫ : F • G ∼ = 1 We then recall this result of Gambino and Kock [2010]: Theorem 1 (Theorem 3.8 [Gambino and Kock, 2010]). Squares of PolyFun E are uniquely represented (up to a choice of pullback) by a square of Poly E . Consequently, the interpretation of polynomials is a framed biequivalence. The interpretation functor is an equivalence of framed bicategory between Poly E and the framed bicategory PolyFun E . We thus conflate the category of polynomials Poly E and the category of polynomial functors PolyFun E . Polynomials are a "small" presentation of the larger functorial objects. Since both categories are equivalent, we do not lose expressive power by working in the small language. Earlier, we have isolated a class of cartesian morphisms. This defines a sub-category of polynomials, which will be of prime interest in this paper. For clarity, we expound its definition: ExampleI B A J K D C L s t α That is, we take D 0 E and D 1 I,J Poly c E (I, J) for which we define: • The identity functor U that maps an index to the identity polynomial at that index ; • The left frame L that projects the source index I ; • The right frame R that projects the target index J ; • The composition ⊙ of polynomial functors. The frames defined by L and R thus correspond to, respectively, the left-hand side and right-hand side of polynomials and polynomial morphisms. As for the bifibration structure, consider a pair of morphism u, v : K → I, L → J in the base category E × E, we have: • A cobase-change functor reindexing a polynomial P : Poly c E (I, J) to a polynomial (u, v) ! P : Poly c E (K, L) ; • A base-change functor reindexing a polynomial P : Poly c E (K, L) to a polynomial (u, v) * P : Poly c E (I, J). This extra-structure lets us transport polynomials across frames: given a polynomial, we can reindex or op-reindex it to any frame along a pair of index morphisms. The interpretation functor is again an equivalence of framed bicategory between Poly c E and the framed bicategory PolyFun c E which morphisms of functors consists of cartesian natural transformations. We therefore have the following result: Theorem 2 (Theorem 3.13 [Gambino and Kock, 2010] The interpretation functor is an equivalence of framed bicategory between Poly c E and the framed bicategory PolyFun c E [Gambino and Kock, 2010, Theorem 3.13]. We can therefore conflate, once and for all, the category of polynomials Poly c E and the category of polynomial functors PolyFun c E . In a sense, polynomials are a "small" presentation of the larger functorial objects. However, we do not lose expressive power by working in the small language, since both categories are equivalent. Inductive Families in Type Theory In this section, we set out to establish a formal connection between a presentation of inductive families in type theory and the categorical model of polynomial functors. On the type theoretical side, we adopt the universe-based presentation introduced by Chapman et al. [2010]. Working on a universe gives us a syntactic internalisation of inductive families within type theory. Hence, we can manipulate and reason about inductive families from within the type theory itself. The original motivation for this design is generic programming: the programmer can compute over the structure of datatypes, or even compute new datatypes from old. In mathematical term, "generic programming" reads as reflection: we can reflect the meta-theory of inductive types within the type theory. For systems like Agda or Coq, we can imagine reducing their syntactic definition to such a universe by elaboration [Dagand and McBride, 2013]. We recall the definition of the universe in Figure 3. A Desc code is a syntactic object describing a functor from Set I to Set. To obtain this functor, we have to interpret the code using . The reader will gain intuition for the codes by looking at their interpretation, i.e. their semantics. To describe functors from Set I to Set J , we use the isomorphism [Set I , Set] J ∼ = [Set I , Set J ]. Hence, in idesc, we pull the J-index to the front and thus capture functors on slices of Set. The interpretation extends pointwise to idesc. Inhabitants of the idesc type are called descriptions. By construction, the interpretation of a description is a strictly positive functor: for a description D, the initial algebra always exists and is denoted (µD, in : D µD → µD). Definition 11 (Described functor). A functor is described if it is isomorphic to the interpretation of a description. Example 12 (Natural numbers). The signature functor of natural numbers is described by: NatD : idesc 1 1 NatD → λ * . 'Σ (1 + 1) λ inj l * → '1 inj r * → 'var * The reader will check that the interpretation of this code gives a functor isomorphic to the expected X → 1 + X. Descriptions are equivalent to polynomials We can now prove the equivalence between described functors and polynomial functors. The first step is to prove that described functors are polynomial: Lemma 3. The class of described functors is included in the class of polynomial functors. Proof. Let F : Set /I → Set /J a described functor. By definition of the class of described functor, F is naturally isomorphic to the interpretation of a description. That is, for any j : J, there is a D : Desc I such that: F j ∼ = D By induction over D, we show that D is naturally isomorphic to a polynomial: Case D = '1: We have '1 X ∼ = 1 × X 0 , which is clearly polynomial Case D = 'var i: We have 'var i X ∼ = 1 × (X i) 1 , which is clearly polynomial Case D = 'Σ S T : We have 'Σ S T X = (s : S) × T s X. By induction hypothesis, T s X ∼ = (x : S T s ) × (p : P T s x) → X (n T s p). Therefore, we obtain that: 'Σ S T X ∼ = (s : S) × (x : S T s ) × (p : P T s ) → X (n T s p) ∼ = (sx : (s : S) × S T s ) × (p : P T (π 0 sx) (π 1 sx)) → X (n T (π 0 sx) p) This last functor being clearly polynomial. Case D = 'Π S T : We have 'Π S T X = (s : S) → T s X. By induction hypothesis, T s X ∼ = (x : S T s ) × (p : P T s x) → X (n T s p). Therefore, we obtain that: 'Π S T X ∼ = (s : S) → (x : S T s ) × (p : P T s x) → X (n T s p) ∼ = (f : (s : S) → S T s ) × (s : S)(p : P T s (f s)) → X (n T s p) ∼ = (f : (s : S) → S T s ) × (sp : (s : S) × P T s (f s)) → X (n T (π 0 sp) (π 1 sp)) This last functor being clearly polynomial. To prove the other inclusion -that polynomials functors on Set are a subset of described functors -we rely on Lemma 2. To this end, we must prove some algebraic properties of the class of described functors, namely that they are closed under reindexing, its adjoints, and composition. To do so, the methodology is simply to code these operations in idesc. Lemma 4. Described functors are closed under reindexing and its adjoints. Proof. We describe the pullback functors and their adjoints by: D∆ (f:A → B) : idesc B A D∆ f → λa. 'var (f a) DΣ (f:A → B) : idesc A B DΣ f → λb. 'Σ f −1 b λa. 'var a DΠ (f:A → B) : idesc A B DΠ f → λb. 'Π f −1 b λa. 'var a Where the inverse of a function f is represented by the following inductive type: data [f : A → B] −1 (b : B) : Set where f −1 (b = f a) ∋ inv (a : A) It is straightforward to check that these descriptions interpret to the expected operation on slices of Set, i.e. that we have: D∆ f ∼ = ∆ f DΣ f ∼ = Σ f DΠ f ∼ = Π f Lemma 5. Described functors are closed under composition. Proof. We define composition of descriptions by: (D : idesc B C) • D (E : idesc A B) : idesc A C D • D E → λc. compose (D c) E where compose (D : Desc B) (E : idesc B A) : Desc A compose ('var b) E → E b compose '1 E → '1 compose ('Π S T ) E → 'Π S λs. compose (T s) E compose ('Σ S T ) E → 'Σ S λs. compose (T s) E It is then straightforward to check that this is indeed computing the composition of the functors, i.e. that we have: D • D E ∼ = D • E Lemma 6. The class of polynomial functors is included in the class of described functors. Proof. Described functors are closed under reindexing, its left and right adjoint (Lemma 4), are closed under composition (Lemma 5) and are defined up to natural isomorphism. By Lemma 2, the class of polynomial functors is the least such set. Therefore, the class of polynomial functor is included in the class of described functors. We conclude with the desired equivalence: Proposition 1. The class of described functors corresponds exactly to the class of polynomial functors. Proof. By Lemma (3) and Lemma (6), we have both inclusions. The benefit of this algebraic approach is its flexibility with respect to the universe definition: for practical purposes, we are likely to introduce new Desc codes. However, the implementation of reindexing and its adjoints will remain unchanged. Only composition would need to be verified. Besides, these operations are useful in practice, so we are bound to implement them anyway. In the rest of this paper, we shall conflate descriptions, polynomials, and polynomial functors, silently switching from one to another as we see fit. An alternative proof An alternative approach, followed by Morris [2007] for example, consists in reducing these codes to containers. We thus obtain the equivalence to polynomial functors, relying on the fact that containers are an incarnation of polynomial functors in the internal language Gambino and Kock [2010, §2.18]. This less algebraic approach is more constructive. However, to be absolutely formal, it calls for proving some rather painful (extensional) equalities. If the proofs are laborious, the translation itself is not devoid of interest. In particular, it gives an intuition of descriptions in terms of shape, position and indices. This slightly more abstract understanding of our universe will be useful in this paper, and is useful in general when reasoning about datatypes. We formalise the translation in Figure 4, mapping descriptions to containers. The message to take away from that translation is which code contributes to which part of the container, i.e. shape, position, and/or index. Crucially, the '1 and 'Σ codes contribute only to the shapes. The 'var and 'Π codes, on the other hand, contribute to the positions. Finally, the 'var code is singly defining the next index. The inverse translation is otherwise trivial and given here for the sake of completeness: (D : idesc I J) : ICont I J D → λj. Shape (D j) λj. Index (D j) λj. Pos (D j) where Shape (D : Desc I) : Set Shape 'var i → 1 Shape '1 → 1 Shape 'Π S T → (s : S) → Shape (T s) Shape 'Σ S T → (s : S) × Shape (T s) Pos (D : Desc I) (sh : Shape D) : Set Pos 'var i * → 1 Pos '1 * → 0 Pos 'Π S T f → (s : S) × Pos (T s) (f s) Pos 'Σ S T (s, t) → Pos (T s) t(C : ICont I J) −1 : idesc I J S n P −1 → 'Σ S λsh. 'Π (P sh) λpos. 'var (n pos) We are left to prove that these translations are indeed inverse of each other: while this proof is extremely tedious to carry formally, it should be intuitively straightforward. We therefore assume the following lemma: Lemma 7 (Described functors to polynomials, alternatively). is essentially surjective. Discussion Let us reflect on the results obtained in this section. By establishing an equivalence between descriptions -a programming artefact -and polynomial functors -a mathematical object -we connect software to mathematics, and conversely. On the one hand, descriptions are suitable for practical purposes: they are a syntactic object, fairly intensional, and can therefore be conveniently manipulated by a computer. Polynomial functors, on the other hand, are fit for theoretical work: they admit a diagrammatic representation and are defined extensionally, up to natural isomorphism. Better still, we have introduced containers as a middle ground between these two presentations. Containers are an incarnation of polynomials in the internal language. Reasoning extensionally about them is equivalent to reasoning about polynomials. Nonetheless, they are also rather effective type theoretic procedures: we can implement them in Agda 1 . We shall traverse this bridge between software and mathematics in both directions. Going from software to mathematics, we hope to gain a deeper understanding of our constructions. Case in point is generic programming in type theory: we develop many constructions over datatypes, such as ornaments, but the justification for these is often extremely operational, one might even say "ad-hoc". By putting our polynomial glasses on, we can finally see through the syntax and understand the structure behind these definitions. Conversely, going from mathematics to software, we translate mathematical structures to new software constructions. The theory of polynomial functors is indeed well developed. Most programming examples presented in this paper -such as derivatives or ornaments -were first presented in the polynomial functor literature. Besides, by exploring the structure of polynomial functors, we discover new and interesting programming idioms -such as the pullback and composition of ornaments. The categorically minded reader might be tempted to look for an equivalence of category. However, we have not yet introduced any notion of morphism between descriptions. What we have established is a lowly "set theoretic" equivalence between the class of descriptions and the class of polynomial functors. In terms of equivalence of categories, we have established that the object part of a functor, yet to be determined, maps descriptions to polynomial functors in an essentially surjective way. We shall complete this construction in the following section. We will set up descriptions in a double category with ornaments as morphisms. The translation will then functorially map it to the double category PolyFun c E . A Categorical Treatment of Ornaments The motivation for ornaments comes from the frequent need, when using dependent types, to relate datatypes that share the same structure. In this setting, ornaments play the role of an organisation principle. Intuitively, an ornament is the combination of two datatype transformations: we may extend the constructors, and/or refine the indices. Ornaments preserve the underlying data-structure by enforcing that an extension respects the arity of the original constructors. By extending a datatype, we introduce more information, thus enriching its logical content. A typical example of such an ornament is the one taking natural numbers to lists: data Nat : Set where Nat ∋ 0 | suc (n : Nat) List-Orn A ⇒ data List [A : Set] : Set where List A ∋ nil | cons (a : A)(as : List A) By refining the indices of a datatype, we make it logically more discriminating. For example, we can ornament natural numbers to finite sets: Orn '1 u ∋ '1 Orn ('Π S T ) u ∋ 'Π (T + : (s : S) → Orn (T s) u) Orn ('Σ S T ) u ∋ 'Σ (T + : (s : S) → Orn (T s) u) -Delete 'Σ S: | delete (s : S)(T + : Orn (T s) u) (a) Code (O : Orn D u) orn : Desc I insert S D + orn → 'Σ S λs. D + s orn 'var (inv i) orn → 'var i '1 orn → '1 'Π T + orn → 'Π S λs. T + s orn 'Σ T + orn → 'Σ S λs. T + s orn delete s T + orn → T + s orn (b) Interpretation Ornaments We recall the definition of the universe of ornaments in Figure 5. Besides our ability to copy the original description (with the codes '1, 'Σ, and 'Π), we can insert new Σ-types, delete Σ-types by providing a witness, and use a more precise index in the 'var codes. While this universe is defined on Desc K, i.e. functors from Set /K to Set, it readily lifts to endofunctors on slices, i.e. on descriptions idesc K L: orn (re I : J → I) (re O : P → O) (D : O → Desc I) : Set 1 orn re I re O D → (p : P ) → Orn re I (D (re O p)) (o : orn re I re O D) orn (p : P ) : Desc J o orn p → intOrn (D (re O p)) (o p) Example 13 (Ornamenting natural numbers to list). We obtain list from natural numbers with the following ornament: List-Orn (A : Set) : orn NatD id id List-Orn A → λ * . 'Σ λ inj l * → '1 inj r * → insert A λ . 'var * The reader will check that the interpretation ( orn ) of this ornament followed by the interpretation ( ) of the resulting description yields the signature functor of list X → 1 + A × X. Example 14 (Ornamenting natural numbers to finite sets). We obtain finite sets by inserting a number n ′ : Nat, constraining the index n to suc n ′ , and -in the recursive case -indexing at n ′ : Fin-Orn : orn NatD (λn. * ) (λn. * ) Fin-Orn → λn. insert Nat λn ′ . insert (n = suc n ′ ) λ . 'Σ λ inj l * → '1 inj r * → 'var n ′ Again, the reader will verify that this is indeed describing the signature of finite sets. A detailed account of ornaments from a programmer's perspective will be found elsewhere [McBride, 2013, Dagand and McBride, 2012, Ko and Gibbons, 2011. For the purpose of this paper, these definitions are enough. We shall refer to the aforementioned papers when programming concepts reappear in our categorical framework. Ornaments are cartesian morphisms Relating the definition of ornaments with our polynomial reading of descriptions, we make the following remarks. Firstly, the ornament code lets us only insert -with the insert code -or delete -with the delete code -'Σ codes while forbidding deletion or insertion of either 'Π or 'var codes. In terms of container, this translates to: shapes can be extended, while positions must be isomorphic. Secondly, on the 'var code, the ornament code lets us pick any index in the inverse image of u. In terms of container, this corresponds to the coherence condition: the initial indexing must commute with applying the ornamented indexing followed by u. Concretely, for a container S n P , an ornament can be modelled as an extension ext, a refined indexing n + subject to coherence condition q with respect to the original indexing:    ext : S (v l) → Set n + : ext sh → P sh → K q : ∀pos : P sh. u (n + e pos) = n pos Equivalently, the family of set ext can be understood as the inverse image of a function σ : S + l → S (v l). The function n + is then the next index function of a container with shapes S + and positions P • σ. Put otherwise, the morphism on shapes σ together with the coherence condition q form a cartesian morphism from S + n + P • σ to S n P ! To gain some intuition, the reader can revisit the cartesian morphism of Example 5 as an ornament of container -by simply inverting the morphism on shapes -and as an ornament of description -by relating it with the ornament List-Orn (Example 13). We shall now formalise this intuition by proving the following isomorphism: Lemma 8. Ornaments describe cartesian morphisms between polynomial functors, i.e. we have the isomorphism orn D u v ∼ = Poly c E ( , D) u,v In terms of cartesian morphism of polynomials, extending the shape corresponds to the morphism α. Enforcing that the positions, i.e. the structure, of the datatype remain the same corresponds to the pullback along α. The refinement of indices corresponds to the frame morphisms commuting. Proof. We develop the proof on the container presentation: this lets us work in type theory, where is anchored the definition of ornaments. It is a necessary hardship, since no other decent model of ornaments is available to us. After this bootstrapping process, we shall have the abstract tools necessary to lift off type theory. The first half of the isomorphism consists of mapping an ornament o of a description D to a cartesian morphism from the container described by o orn to the container described by D. By definition of cartesian morphisms, we simply have to give a map from the shape of o orn to the shape of D: φ (o : orn D u v) : o orn u =⇒ c v D φ o → (λi. forget o (u i)) c where forget (O : Orn D u) (sh : Shape O orn ) : Shape D forget '1 * → * forget ('Π T + ) f → λa. forget (T + a) (f a) forget ('Σ T + ) (a, sh) → (a, forget (T + a) sh) forget ('var (inv j)) * → * forget (insert a D + ) (a, sh) → forget (D + a) sh forget (delete s O) sh → (s, forget O sh) We are then left to check that (extensionally) the positions are constructed by pullback and the indexing is coherent. This is indeed the case, even though proving it in type theory is cumbersome. On positions, the ornament does not introduce or delete any new 'Π or 'var: hence the positions are left unchanged. On indices, we rely on u −1 k to ensure that the more precise indexing is coherent by construction. In the other direction, we are given a cartesian morphism from F to G. We return an ornament of the description of G. For the isomorphism to hold, this ornament must interpret to the description of F : ψ (m : F u =⇒ c v G) : orn G −1 u v ψ (forget c ) → λj. 'Σ λsh. insert (forget −1 sh) λext. 'Π λps. 'var (inv (n F ps)) Indeed, the description of G is a 'Σ of its shape, followed by a 'Π of its positions, terminated by a 'var at the next index. To ornament G to F , we simply have to insert the inverse image of forget, i.e. the information that extends G to F . As for the next index, we can legitimately use F 's indexing function: the coherence condition of the cartesian morphism ensures that it is indeed in the inverse image of the reindexing function. Having carefully crafted the definition of φ and ψ, it should be obvious that these functions are inverse of each other. It is sadly not that obvious to an (intensional) theorem prover. Hence, we will not attempt to prove it in type theory here. Relation with ornamental algebras [McBride, 2013, §4] To introduce the notion of "ornamental algebra", the second author implemented the erase helper function taking an ornamented type to its unornamented form. This actually corresponds to our transformation φ, followed by the interpretation of the resulting cartesian morphism. The erase function given in the original presentation is indeed natural and cartesian. In the previous section, we have established a connection between descriptions and polynomials. We have now established a connection between ornaments and cartesian morphisms of polynomials. It thus makes sense to organise descriptions in a framed bicategory IDesc c : Definition 12 (Framed bicategory IDesc c ). The framed bicategory IDesc c is defined by: • Objects: sets • Vertical morphisms: set morphisms • Horizontal morphisms: descriptions, framed by I and J • Squares: a square from F to G framed by u and v is an ornament o : orn G u v of G that interprets to (a code isomorphic to) F Where, as for Poly c E (Example 11), the frame structure consists in reindexing a description along a pair of functions. A framed biequivalence We are now ready to establish an equivalence of category between IDesc c and PolyFun c E , thus completing our journey from the type theoretical definition of ornaments to its model as cartesian morphisms. Proposition 2. The double category IDesc c is framed biequivalent to PolyFun c E . Proof. As for the proof of Lemma 8, we work from IDesc c to ICont to prove this theorem. Since ICont is equivalent to PolyFun c E , this gives the desired result. To prove a framed biequivalence, we need a functor on the base category and another on the total category. In this particular case, both base categories are Set: we shall therefore take the identity functor, hence trivialising the natural isomorphisms on composition, identity, and frames. On the total category, we prove the equivalence by exhibiting a full and faithful functor from Desc to ICont that is essentially surjective on objects. Unsurprisingly, this functor is defined on objects by , which is indeed essentially surjective by Lemma 7. The morphism part is defined by φ, which is full and faithful by Lemma 8. We have therefore established the following equivalences of framed bicategories: IDesc c Poly c E PolyFun c E ICont We may now conflate the notions of ornament, cartesian morphism, and cartesian natural transformation. In particular, we shall say that "F ornaments G" when we have a cartesian morphism from F to G. Let us now raid the polynomial toolbox for the purpose of programming with ornaments. The next section shows the beginning of what is possible. Tapping into the categorical structure In the previous section, we have characterised the notion of ornament in terms of cartesian morphism. We now turn to the original ornamental constructions [McBride, 2013] such as the ornamental algebra and the algebraic ornament -and rephrase them in our categorical framework. Doing so, we extract the structure governing their type theoretic definition. Next, we study the categorical structure of cartesian morphisms and uncover novel and interesting ornamental constructions. We shall see how the identity, composition, and frame reindexing translate into ornaments. We shall also be interested in pullbacks in the category PolyFun c E and the functoriality of the derivative in that category. Ornamental algebra Ornamenting a datatype is an effective recipe to augment it with new information. We thus expects that, given an ornamented object, we can forget its extra information and regain a raw object. This projection is actually a generic operation, provided by the ornamental algebra. It is a corollary of the very definition of ornaments as cartesian morphisms. Corollary 1 (Ornamental algebra). From an ornament o : F u =⇒ c v G, we obtain the ornamental algebra o-forgetAlg : F (µG • v) → µG • u. Proof. We apply the natural transformation o at µG and post-compose by in: o-forgetAlg : F (µG • v) o µG −→ (G µG) • u in −→ µG • u Folding the ornamental algebra, we obtain a map from the ornamented type µF to its unornamented version µG. In effect, the ornamental algebra describes how to forget the extra-information introduced by the ornament. Example 15 (Ornamental algebra of the List ornament). The cartesian morphism from list to natural numbers (Example 5) maps the nil constructor to 0, while the cons constructor is mapped to suc. Post-composing by in, we obtain a natural number. This is the algebra computing the length of a list. Algebraic ornaments The notion of algebraic ornament was initially introduced by the second author [McBride, 2013]. A similar categorical construction, defined for any functor, was also presented by Atkey et al. [2012]. In this section, we reconcile these two works and show that, for a polynomial functor, the refinement functor can itself be internalised as a polynomial functor. Definition 13 (Refinement functor [Atkey et al., 2012, §4.3]). Let F an endofunctor on E /I . Let (X : E /I , α : F X → X) an algebra over F . The refinement functor is defined by: F α Σ α •F : (E /I ) /X → (E /I ) /X WhereF -the lifting of F [Hermida andJacobs, 1998, Fumex, 2012] -is taken, in an LCCC, to be the morphism part of the functor F . The idea, drawn from refinement types [Freeman and Pfenning, 1991], is that a function α : µF → X can be thought of as a predicate over µF . By integrating the algebra α into the signature F , we obtain a signature F α indexed by X that describes the F -objects satisfying, by construction, the predicate α . Categorically, this translates to: Theorem 3 (Coherence property of algebraic ornament). The fixpoint of the algebraic ornament of P F by α satisfies the isomorphism µP F α ∼ = Σ α 1 µF where 1 : E /I → [E /I , E /I ], the terminal object functor, maps objects X to id X . Proof. This is an application of Theorem 4.6 [Atkey et al., 2012], specialised to the codomain fibration (i.e. an LCCC). Informally, using a set theoretic notation, this isomorphisms reads as µF α i x ∼ = {t : µF i | α t = x}. That is, the algebraic ornament µF α at index i and x corresponds exactly to the pair of a witness t of µF i and a proof that this witness satisfies the indexing equation α t = x. In effect, from an algebraic predicate over an inductive type, we have an effective procedure reifying this predicate as an inductive family. This theorem also has an interesting computational interpretation. Crossing the isomorphism from left to right, we obtain the Recomputation theorem [McBride, 2013, §8]: from any t + : µF α i x, we can extract a t : µF i together with a proof that α t equals x. From right to left, we obtain the remember function [McBride, 2013, §7]: from any t : µF i, we can lift it to its ornamented form with remember t : µF α i ( α t). When F is a polynomial functor, we show that the refinement functor can be internalised and presented as an ornament of F . In practice, this means that from a description D and an algebra α, we can compute an ornament code that describes the functor D α . This should not come as a surprise: algebraic ornaments were originally presented as ornamentations of the initial description [McBride, 2013, §5]. The following theorem abstracts the original definition: Proposition 3. Let F a polynomial endofunctor on E /I . Let (X, α) an algebra over P F , i.e. α : P F X → X. The refinement functor P F α is polynomial and ornaments F . Proof. To show that P F α is a polymonial ornamenting F , we exhibit a cartesian natural transformation from P F α to P F . Since P F is polynomial, we obtain that P F α is polynomial by Lemma 1. First, there is a cartesian natural transformation from the liftingP F to P F . Indeed, for an LCCC, the lifting consists of the morphism part of P F , denoted P F → [Fumex, 2012]. We therefore have the following isomorphism, hence cartesian natural transformation: (E /I ) /X (E /I ) /P F X E /I ∼ = (E /I ) /1 (E /I ) /1 ∼ = E /I (E /I ) /P F 1 P F → P F → Σ ! P F ½ P F Σ ! X Σ P F ! X Σ ! P F X Indeed, unfolding the definition of Σ f f • , the left square reduces to the functoriality of P F . The right triangle is simply the op-cartesian lifting of P F X P F 1 1 P F ! X ! P F X ! P F 1 . The bottom triangle commutes by the isomorphism relating the slice over the terminal and the total category, i.e. (E /I ) /1 ∼ = E /I . There is also a cartesian natural transformation from Σ α to the identity polynomial indexed by J: E /Σ ! P F X ∼ = (E /I ) /P F X (E /I ) /X ∼ = E /Σ ! X E /I ∼ = (E /I ) /1 (E /I ) /1 ∼ = E /I Σ α Σ ! P F X Σ ! X ∼ = By horizontal composition of these two cartesian natural transformations, we obtain a cartesian natural transformation from P F α to id • P F ∼ = P F . Remark 2. This proof is not entirely satisfactory: it is specialised to the predicate lifting in the codomain fibration. The construction of the cartesian natural transformation from the lifting to the functor is therefore a rather pedantic construction. Hopefully, a more abstract proof could be found. Categorical structures Identity A trivial ornamental construction is the identity ornament. Indeed, for every polynomial, there is a cartesian morphism from and to itself, introducing no extension and no refinement. In terms of Orn code, this construction simply consists in copying the code of the description: this is a generic program, taking a description as input and returning the identity ornament. Vertical composition The next structure of interest is composition. Recall that an ornament corresponds to a (cartesian) natural transformation. There are therefore two notions of composition. First, vertical composition lets us collapse chains of ornaments: E /I E /J E /I E /J F G H F H ⇓ o 1 ⇓ o 2 o 2 • o 1 ⇓ ∼ = Example 16 (Vertical composition of ornaments). We have seen that List ornaments Nat. We also know that Vec ornaments List. By vertical composition, we thus obtain that Vec ornaments Nat. Horizontal composition Turning to horizontal composition, we have the following identity: E /I E /J E /K E /I E /K F 1 G 1 F 2 G 2 F 2 • F 1 G 2 • G 1 ⇓ o 1 ⇓ o 2 o 2 • o 1 ⇓ ∼ = Example 17 (Horizontal composition of ornaments). Let us consider the following polynomials: Square X → X × X : Set /1 → Set /1 Height {X n | n ∈ Nat} → {X n × X n+1 | n ∈ Nat} + {X n × X n | n ∈ Nat} : Set /Nat → Set /Nat It is easy to check that VecCont ornaments ListCont and Height ornaments Square. By horizontal composition of these ornaments, we obtain that VecCont • Height -describing a balanced binary tree -is an ornament of ListCont • Square -describing a binary tree. Thus, we obtain that balanced binary trees ornament binary trees. Frame structure Finally, the frame structure of the bicategory lets us lift morphisms on indices to polynomials. Example 18 (Reindexing ornament). Let twice : Nat → Even, the function that multiplies its input by 2. The Vec polynomial is indexed by Nat: we can therefore reindex it with twice. We automatically obtain an ornament of Vec that is indexed by Even. Needless to say, this construction is not very interesting on its own. However, in a larger development, we can imagine retrofitting an indexed datatype to use another index, making it usable by a library function. The identity, vertical, and horizontal compositions illustrate the algebraic properties of ornaments. The categorical simplicity of cartesian morphisms gives us a finer understanding of datatypes and their relation to each other, as illustrated by Example 17. Pullback of ornaments So far, we have merely exploited the fact that PolyFun c E is a framed bicategory. However, it has a much richer structure. That extra structure can in turn be translated into ornamental constructions. We shall focus on pullbacks, but we expect other categorical notions to be of programming interest. → c H two cartesian natural transformation. They are projected to φ 1 and ψ 1 in the base category. Since E /J is pullback complete, we can construct the pullback of φ 1 and ψ 1 , thus obtaining the following pullback square: · F 1 G1 H1 φ 1 ψ 1 ψ † 1 φ † 1 By reindexing, we thus obtain the following square in the total category: · F X GX HX φ X ψ X ψ † X φ † X By Exercise 1.4.4 [Jacobs, 2001], we have that this square is actually a pullback. In a nutshell, we rely on the unicity of cartesian morphisms in the total category to prove the universal property of pullbacks for that square. Example 19 (Pullback of ornament). Natural numbers can be ornamented to lists (Example 13) as well as finite sets (Example 14). Taking the pullback of these two ornaments, we obtain bounded lists that correspond to lists of bounded length, with the bound given by an index n : Nat. Put explicitly, the object thus computed is the following datatype: data BoundedList [A : Set](n : Nat) : Set where BoundedList A (n = suc n ′ ) ∋ nil (n ′ : Nat) | cons (n ′ : Nat)(a : A)(as : BoundedList A n ′ ) The pullback construction is another algebraic property of ornaments: given two ornaments, both describing an extension of the same datatype (e.g. extending natural numbers to lists and extending natural numbers to finite sets), we can "merge" them into one having both characteristics (i.e. bounded lists). In type theory, Ko and Gibbons [2011] have experimented with a similar construction for composing indexing disciplines. Abbott et al. [2005b] have shown that the Zipper [Huet, 1997] data-structure can be computed from the derivative of signature functors. Interestingly, the derivative is characterised by the existence of a universal arrow in the category Poly c E : Definition 14 (Differentiability [Abbott et al., 2005b]). A polynomial F is differentiable in i if and only if, for any polynomial G, we have the following bijection of morphisms: Derivative of ornament Poly c E (G × π i , F ) Poly c E (G, ∂ i F ) Where π i I k i ←− I id −→ I id −→ I. We denote Poly ∂ i E the class of polynomials differentiable in i. Proposition 5. Let F and G two polynomials in Poly ∂ i E . If F ornaments G, then ∂ i F ornaments ∂ i G. Proof. The proof simply follows from the functoriality of ∂ i over Poly ∂ i E [Abbott, 2003, Section 6.4]. In a nutshell, this follows from the existence of the following cartesian morphism: ∂ i F × π i → c F → c G where the first component is the unit of the universal arrow while the second component is the ornament from F to G. By definition of differentiability, we therefore have the desired cartesian morphism: ∂ i F → c ∂ i G Example 20 (Ornamentation of derivative). Let us consider binary trees, with signature functor 1 + A × X 2 . Balanced binary trees are an ornamentation of binary trees (Example 17). By the theorem above, we have that the derivative of balanced binary trees is an ornament of the derivative of binary trees. The derivative is thus an example of an operation on datatypes that preserves ornamentation. Knowing that the derivative of an ornamented datatype is an ornamentation of the derivative of the original datatype, we get that the order in which we ornament or derive a datatype does not matter. This let us relate datatypes across such transformations, thus preserving the structural link between them. Related work Ornaments were initially introduced by the second author [McBride, 2013] as a programming artefact. They were presented in type theory, with a strong emphasis on their computational contribution. Ornaments were thus introduced through a universe. Constructions on ornaments -such as the ornamental algebra, algebraic ornament, and reornament -were introduced as programs in this type theory, relying crucially on the concreteness of the universe-based presentation. While this approach has many pedagogical benefits, it was also clear that more abstract principles were at play. For example, in a subsequent paper [Dagand and McBride, 2012], the authors successfully adapted the notion of ornaments to another universe of inductive families, whilst Ko and Gibbons [2011] explore datatype engineering with ornaments in yet a third. The present paper gives such an abstract treatment. This focus on the theory behind ornaments thus complements the original, computational treatment. Building upon that original paper, our colleagues Ko and Gibbons [2011] also identify the pullback structure -called "composition" in their paper -as significant, giving a treatment for a concrete universe of ornaments and compelling examples of its effectiveness for combining indexing disciplines. The conceptual simplicity of our approach lets us subsume their type theoretic construction as a mere pullback. The notion of algebraic ornament was also treated categorically by Atkey et al. [2012]: instead of focusing on a restricted class of functors, the authors described the refinement of any functor by any algebra. The constructions are presented in the generic framework of fibrations. The refinement construction described in this paper, once specialised to polynomial functors, corresponds exactly to the notion of algebraic ornament, as we have shown. Hamana and Fiore [2011] also give a model of inductive families in terms of polynomial functors. To do so, they give a translation of inductive definitions down to polynomials. By working on the syntactic representation of datatypes, their semantics is defined by this translation. In our system, we can actually prove that descriptions -our language of datatypes -are equivalent to polynomial functors. Finally, it is an interesting coincidence that cartesian morphisms should play such an important role in structuring ornaments. Indeed, containers stem from the work on shapely types [Jay and Cockett, 1994]. In the shape framework, a few base datatypes were provided (such as natural numbers) and all the other datatypes were grown from these basic blocks by a pullback construction, i.e. an ornament. However, this framework was simply typed, hence no indexing was at play. Conclusion Our study of ornaments began with the equivalence between our universe of descriptions and polynomial functors. This result lets us step away from type theory, and gives access to the abstract machinery provided by polynomials. For practical reasons, the type theoretic definition of our universe is very likely to change. However, whichever concrete definition we choose will always be a syntax for polynomial functors. We thus get access to a stable source of mathematical results that informs our software constructions. We then gave a categorical presentation of ornaments. Doing so, we get to the essence of ornaments: ornamenting a datatype consists in extending it with new information, and refining its indices. Formally, this characterisation turns into a presentation of ornaments as cartesian morphisms of polynomials. Finally, we reported some initial results based on our explorations of this categorical structure. We have translated the type theoretic ornamental toolkit to the categorical framework. Doing so, we have gained a deeper understanding of the original definitions. Then, we have expressed the categorical definition of Poly c E in terms of ornaments, discovering new constructions -identity, vertical, and horizontal composition -in the process. Also, we have studied the structure of Poly c E , obtaining the notion of pullback of ornaments. Future work We have barely scratched the surface of Poly c E : a lot remain unexplored. Pursuing this exploration might lead to novel and interesting ornamental constructions. Also, our definition of ornaments in terms of polynomials might be limiting. One can wonder if a more abstract criterion could be found for a larger class of functors. For instance, the functor 1 C : [C, D] → D is a fibration for D pullback complete and C equipped with a terminal object 1 C . Specialised to the categories of slices of E, the cartesian morphisms are exactly our ornaments. What about the general case? Definition 2 ( 2Polynomial[Gambino and Kock, 2010, §1.1]). A polynomial is the data of 3 morphisms f : B → A, s : B → I, and t : A → J in E. Conventionally, a polynomial is diagrammatically represented by Figure 2 : 2Translation polynomial/container data:Figure 3 : 3Desc [I : Set] : Set 1 where Desc I ∋ 'var (i : I) | '1 | 'Π (S : Set) (T : S → Desc I) | 'Σ (S : Set) (T : S → Desc I) (D : Desc I) (X : I → Set) Set) (J : Set) : Set 1 idesc I J → J → Desc I (D : idesc I J ) (X : I → Set) : J → Set D X → λj. D j X Universe of inductive families Index ( D :Figure 4 : D4Desc I) (pos : Pos D sh) : I Index 'var i * → i Index 'Π S T (s, pos) → Index (T s) pos Index 'Σ S T pos → Index (T (π0 sh)) pos From descriptions to containers data Nat : Set where Nat ∋ 0 | suc (n : Nat) Fin-Orn ⇒ data Fin (n : Nat) : Set where Fin (n = suc n ′ ) ∋ f0 (n ′ : Nat) | fsuc (n ′ : Nat)(k : Fin n ′ ) data Orn (D : Desc K)[u : I → K] : Set 1 where -Extend with S: Orn D u ∋ insert (S : Set)(D + : S → Orn D u) -Refine index: Orn ('var k) u ∋ 'var (i : u −1 k) -Copy the original: Figure 5 : 5Universe of ornaments 11 (Framed bicategory Poly c E [Gambino and Kock, 2010, §3.13]). The framed bicategory Poly c E is defined by: • Objects: indices, i.e. objects of E • Vertical arrows: index morphisms, i.e. morphisms of E • Horizontal arrows: polynomial indexed by I and J, respectively left and right frames• Squares: cartesian morphism of polynomial reindexed by u and v, respectively left and right frames: Proposition 4. The category PolyFun cE has all pullbacks.Proof. First, let us recall that the notion of cartesian morphism arises from the fact that the following functor is a fibration:[E /I , E /J ] E /JWhere cartesian natural transformation corresponds to the cartesian morphisms of that fibration. Let φ : F . → c H and ψ : G1 . Such an implementation is available on the the first author's website. Acknowledgements We would like to thank Gabor Greif and José Pedro Magalhães for their input on this paper. We also thank our colleagues Clément Fumex and Lorenzo Malatesta for their feedback on our proofs. The authors are supported by the Engineering and Physical Sciences Research Council, Grant EP/G034699/1. Awodey, Finally, there has been much work recently on homotopy inductive types. Finally, there has been much work recently on homotopy inductive types [Awodey et al., Coincidentally, the formalism used in these works is based on W-types, i.e. the type theoretic incarnation of polynomial functors. It would be there be interesting to study what ornaments could express in this frameworkCoincidentally, the formalism used in these works is based on W-types, i.e. the type theoretic incarnation of polynomial functors. It would be there be interesting to study what ornaments could express in this framework. Categories of Containers. M Abbott, University of LeicesterPhD thesisM. Abbott. Categories of Containers. PhD thesis, University of Leicester, 2003. M Abbott, T Altenkirch, N Ghani, Containers: Constructing strictly positive types. TCS. 342M. Abbott, T. Altenkirch, and N. Ghani. Containers: Constructing strictly positive types. TCS, 342(1):3-27, 2005a. ∂ for data: Differentiating data structures. M Abbott, T Altenkirch, C Mcbride, N Ghani, Fundam. Inform. 651-2M. Abbott, T. Altenkirch, C. McBride, and N. Ghani. ∂ for data: Differentiating data structures. Fundam. Inform., 65(1-2):1-28, 2005b. Refining inductive types. R Atkey, P Johann, N Ghani, Logical Methods in Computer Science. R. Atkey, P. Johann, and N. Ghani. Refining inductive types. Logical Methods in Computer Science, 2012. Inductive types in homotopy type theory. S Awodey, N Gambino, K Sojakova, S. Awodey, N. Gambino, and K. Sojakova. Inductive types in homotopy type theory. Jan. 2012. The gentle art of levitation. J Chapman, P.-E Dagand, C Mcbride, P Morris, ICFP. J. Chapman, P.-E. Dagand, C. McBride, and P. Morris. The gentle art of levitation. In ICFP, pages 3-14, 2010. Substitution up to isomorphism. P.-L Curien, Fundam. Inf. 191-2P.-L. Curien. Substitution up to isomorphism. Fundam. Inf., 19(1-2):51-85, Sept. 1993. ISSN 0169-2968. Transporting functions across ornaments. P.-E Dagand, C Mcbride, ICFP. P.-E. Dagand and C. McBride. Transporting functions across ornaments. In ICFP, pages 103-114, 2012. Elaborating inductive definitions. P.-E Dagand, C Mcbride, JFLA. P.-E. Dagand and C. McBride. Elaborating inductive definitions. In JFLA, February 2013. Inductive sets and families in Martin-Löf's type theory. P Dybjer, Logical Frameworks. P. Dybjer. Inductive sets and families in Martin-Löf's type theory. In Logical Frame- works. 1991. Refinement types for ML. T Freeman, F Pfenning, SIGPLAN Not. 26T. Freeman and F. Pfenning. Refinement types for ML. SIGPLAN Not., 26:268-277, May 1991. Induction and coinduction schemes in category theory. C Fumex, University of StrathclydePhD thesisC. Fumex. Induction and coinduction schemes in category theory. PhD thesis, University of Strathclyde, 2012. Wellfounded trees and dependent polynomial functors. N Gambino, M Hyland, TYPES. 3085N. Gambino and M. Hyland. Wellfounded trees and dependent polynomial functors. In TYPES, volume 3085 of LNCS, pages 210-225. 2004. Polynomial functors and polynomial monads. CoRR, abs/0906. N Gambino, J Kock, 4931N. Gambino and J. Kock. Polynomial functors and polynomial monads. CoRR, abs/0906.4931, Mar. 2010. A foundation for GADTs and inductive families: dependent polynomial functor approach. M Hamana, M Fiore, WGP'11. ACMM. Hamana and M. Fiore. A foundation for GADTs and inductive families: dependent polynomial functor approach. In WGP'11, pages 59-70. ACM, 2011. Programming interfaces and basic topology. P Hancock, P Hyvernat, Annals of Pure and Applied Logic. 1371-3P. Hancock and P. Hyvernat. Programming interfaces and basic topology. Annals of Pure and Applied Logic, 137(1-3):189-239, Jan. 2006. Structural induction and coinduction in a fibrational setting. C Hermida, B Jacobs, Inf. Comput. 1452C. Hermida and B. Jacobs. Structural induction and coinduction in a fibrational setting. Inf. Comput., 145(2):107-152, 1998. The zipper. G Huet, Journal of Functional Programming. 705G. Huet. The zipper. Journal of Functional Programming, 7(05):549-554, Sept. 1997. Categorical Logic and Type Theory. B Jacobs, Studies in Logic and the Foundations of Mathematics). Elsevier Science141B. Jacobs. Categorical Logic and Type Theory, Volume 141 (Studies in Logic and the Foundations of Mathematics). Elsevier Science, May 2001. Shapely types and shape polymorphism. C Jay, J Cockett, ESOP '94. 788C. Jay and J. Cockett. Shapely types and shape polymorphism. In ESOP '94, volume 788 of LNCS, pages 302-316. 1994. Modularising inductive families. H.-S Ko, J Gibbons, Workshop on Generic Programming. H.-S. Ko and J. Gibbons. Modularising inductive families. In Workshop on Generic Programming, pages 13-24, 2011. Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext). S , Mac Lane, I Moerdijk, Springercorrected editionS. Mac Lane and I. Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext). Springer, corrected edition, May 1992. . P Martin-Löf, Intuitionistic Type Theory. Bibliopolis·NapoliP. Martin-Löf. Intuitionistic Type Theory. Bibliopolis·Napoli, 1984. Ornamental algebras, algebraic ornaments. C Mcbride, Journal of Functional Programming. to appearC. McBride. Ornamental algebras, algebraic ornaments. Journal of Functional Program- ming, to appear, 2013. Wellfounded trees in categories. I Moerdijk, E Palmgren, Annals of Pure and Applied Logic. 104I. Moerdijk and E. Palmgren. Wellfounded trees in categories. Annals of Pure and Applied Logic, 104:189-218, July 2000. Constructing Universes for Generic Programming. P Morris, University of NottinghamPhD thesisP. Morris. Constructing Universes for Generic Programming. PhD thesis, University of Nottingham, 2007. Towards a practical programming language based on dependent type theory. U , Chalmers University of TechnologyPhD thesisU. Norell. Towards a practical programming language based on dependent type theory. PhD thesis, Chalmers University of Technology, 2007. A set constructor for inductive sets in martin-löf's type theory. K Petersson, D Synek, CTCS. K. Petersson and D. Synek. A set constructor for inductive sets in martin-löf's type theory. In CTCS, pages 128-140, 1989. Locally cartesian closed categories and type theory. R A G Seely, Mathematical Proceedings of the Cambridge Philosophical Society. 95R. A. G. Seely. Locally cartesian closed categories and type theory. Mathematical Proceedings of the Cambridge Philosophical Society, 95, 1983. Framed bicategories and monoidal fibrations. CoRR, abs/0706.1286. M A Shulman, M. A. Shulman. Framed bicategories and monoidal fibrations. CoRR, abs/0706.1286, Jan. 2009. The Coq Development Team. The Coq Proof Assistant Reference Manual. The Coq Development Team. The Coq Proof Assistant Reference Manual.
[]
[ "POMDPs Make Better Hackers: Accounting for Uncertainty in Penetration Testing", "POMDPs Make Better Hackers: Accounting for Uncertainty in Penetration Testing" ]
[ "Carlos Sarraute [email protected] \nCore Security & ITBA\nBuenos AiresArgentina\n", "Olivier Buffet [email protected] \nINRIA Nancy\nFrance\n", "Jörg Hoffmann [email protected] \nSaarland University\nSaarbrückenGermany\n" ]
[ "Core Security & ITBA\nBuenos AiresArgentina", "INRIA Nancy\nFrance", "Saarland University\nSaarbrückenGermany" ]
[]
Penetration Testing is a methodology for assessing network security, by generating and executing possible hacking attacks. Doing so automatically allows for regular and systematic testing. A key question is how to generate the attacks. This is naturally formulated as planning under uncertainty, i.e., under incomplete knowledge about the network configuration. Previous work uses classical planning, and requires costly pre-processes reducing this uncertainty by extensive application of scanning methods. By contrast, we herein model the attack planning problem in terms of partially observable Markov decision processes (POMDP). This allows to reason about the knowledge available, and to intelligently employ scanning actions as part of the attack. As one would expect, this accurate solution does not scale. We devise a method that relies on POMDPs to find good attacks on individual machines, which are then composed into an attack on the network as a whole. This decomposition exploits network structure to the extent possible, making targeted approximations (only) where needed. Evaluating this method on a suitably adapted industrial test suite, we demonstrate its effectiveness in both runtime and solution quality.
10.1609/aaai.v26i1.8363
[ "https://arxiv.org/pdf/1307.8182v1.pdf" ]
281,555
1307.8182
48d2e8cea51a1c9fa4a92e135aab35544191e626
POMDPs Make Better Hackers: Accounting for Uncertainty in Penetration Testing Carlos Sarraute [email protected] Core Security & ITBA Buenos AiresArgentina Olivier Buffet [email protected] INRIA Nancy France Jörg Hoffmann [email protected] Saarland University SaarbrückenGermany POMDPs Make Better Hackers: Accounting for Uncertainty in Penetration Testing Penetration Testing is a methodology for assessing network security, by generating and executing possible hacking attacks. Doing so automatically allows for regular and systematic testing. A key question is how to generate the attacks. This is naturally formulated as planning under uncertainty, i.e., under incomplete knowledge about the network configuration. Previous work uses classical planning, and requires costly pre-processes reducing this uncertainty by extensive application of scanning methods. By contrast, we herein model the attack planning problem in terms of partially observable Markov decision processes (POMDP). This allows to reason about the knowledge available, and to intelligently employ scanning actions as part of the attack. As one would expect, this accurate solution does not scale. We devise a method that relies on POMDPs to find good attacks on individual machines, which are then composed into an attack on the network as a whole. This decomposition exploits network structure to the extent possible, making targeted approximations (only) where needed. Evaluating this method on a suitably adapted industrial test suite, we demonstrate its effectiveness in both runtime and solution quality. Introduction Penetration Testing (short pentesting) is a methodology for assessing network security, by generating and executing possible attacks exploiting known vulnerabilities of operating systems and applications (e.g., (Arce and McGraw 2004)). Doing so automatically allows for regular and systematic testing without a prohibitive amount of human labor, and makes pentesting more accessible to non-experts. A key question is how to automatically generate the attacks. A natural way to address this issue is as an attack planning problem. This is known in the AI Planning community as the "Cyber Security" domain (Boddy et al. 2005). Independently (though considerably later), the approach was put forward also by Core Security (Lucangeli, Sarraute, and Richarte 2010), a company from the pentesting industry. In that form, attack planning is very technical, addressing the low-level system configuration details that are relevant to vulnerabilities. Herein, we are concerned exclusively with this setting. We consider regular automatic pentesting as done in Core Security's "Core Insight Enterprise" tool. We will use the term "attack planning" in that sense. Lucangeli et al. (2010) encode attack planning into PDDL, and use off-the-shelf planners. This already is useful-in fact, it is currently employed commercially in Core Insight Enterprise, using a variant of Metric-FF (Hoffmann 2003). However, the approach is limited by its inability to handle uncertainty. The pentesting tool cannot be upto-date regarding all the details of the configuration of every machine in the network, maintained by individual users. Core Insight Enterprise currently addresses this by extensive use of scanning methods as a pre-process to planning, which then considers only exploits, i.e., hacking actions modifying the system state. The drawbacks of this are that (a) this pre-process incurs significant costs in terms of running time and network traffic, and (b) even so, since scans are not perfect, a residual uncertainty remains (Metric-FF is run based on the configuration that appears to be most likely). Prior work (Sarraute, Richarte, and Lucangeli 2011) has addressed (b) by associating each exploit with a success probability. This is unable to model dependencies between the exploits, and it still requires extensive scanning (to obtain realistic success probabilities) so does not solve (a). Herein, we provide the first solution able to address both (a) and (b), intelligently mixing scans with exploits like a real hacker would. The basic insight is that penetration testing can be naturally modeled in terms of solving a POMDP. We encode the incomplete knowledge as an uncertainty of state, thus modeling the possible network configurations in terms of a probability distribution. Scans and exploits are deterministic in that their outcome depends only on the state they are executed in. Negative rewards encode the cost (the duration) of scans and exploits; positive rewards encode the value of targets attained. The model incorporates firewalls, detrimental side-effects of exploits (crashing programs or entire machines), and dependencies between exploits relying on similar vulnerabilities. POMDP solvers fail to scale to large networks. This is not surprising-even the input model grows exponentially in the number of machines. We show how to address this based on exploiting network structure. We view networks as graphs whose vertices are fully-connected subnetworks, and whose arcs encode the connections between these, filtered by firewalls. We decompose this graph into biconnected components. We approximate the attacks on these components by combining attacks on individual subnetworks. We approxi-mate the latter by combining attacks on individual machines. The approximations are conservative, i.e., they never overestimate the value of the policy returned. Attacks on individual machines are modeled and solved as POMDPs, and the solutions are propagated back up. We evaluate this approach based on the test suite of Core Insight Enterprise, showing that, compared to a global POMDP model, it vastly improves runtime at a small cost in attack quality. We next discuss some preliminaries. We then describe our POMDP model, our decomposition algorithm, and our experimental findings, before concluding the paper. Preliminaries We fill in some details on network structure and penetration testing. We give a brief background on POMDPs. Network Structure Networks can be viewed as directed graphs whose vertices are given by the set M of machines, and whose arcs are connections between pairs of m ∈ M . However, in practice, these network graphs have a particular structure. They tend to consist of subnetworks, i.e., clusters N of machines where every m ∈ N is directly connected to every m ∈ N . By contrast, not every subnetwork N is connected to every other subnetwork N , and typically, if such a connection does exist, then it is filtered by a firewall. From the perspective of an attacker, the firewalls filter the connections and thus limit the attacks that can be executed when trying to hack into a subnetwork N from another subnetwork N . On the other hand, once the hacker managed to get into a subnetwork N , access to all machines within N is easy. Thus a natural representation of the network, from an attack planning point of view, is that of a graph whose vertices are subnetworks, and whose arcs are annotated with firewalls F . We herein refer to this graph as the logical network LN , and we denote its arcs with N F − → N . We formalize firewalls as sets of rules describing which kinds of communication (e.g., ports) are disallowed. Thus smaller sets correspond to "weaker" firewalls, and the empty firewall blocks no communication at all. We remark that, in our POMDP model, we do not provide for privilege escalation, or obtaining passwords. This can instead be modeled at the level of LN . Different privilege levels on the same machine m can be encoded via different copies of m. If controlling m allows the retrieval of passwords, then m can be connected via empty firewalls to the machines m who can be accessed by using these passwords, more precisely to high-privilege copies of these m . Penetration Testing Uncertainty in pentesting arises because it is impossible to keep track of all the configuration details of individual machines, i.e., exactly which versions of which programs are installed etc. However, it is safe to assume that the pentesting tool knows the structure of the network, i.e., the graph LN and the filtering done by each firewall: changes to this are infrequent and can easily be registered. The objective of pentesting is to gain control over certain machines (with critical content) in the network. At any point in time, each machine has a unique status. A controlled machine m has already been hacked into. A reached machine m is connected to a controlled machine, i.e., either m is in a subnetwork N one of whose machines is controlled, or m is in a subnetwork N with a LN arc N F − → N where one of the machines in N is controlled. All other machines are not reached. The algorithm starts with one controlled machine, denoted here by * . 1 We will use the following (small but real-life) situation as a running example: Example 1 The attacker has already hacked into a machine m , and now wishes to attack a machine m within the same subnetwork. The attacker knows two exploits: SA, the "Symantec Rtvscan buffer overflow exploit"; and CAU, the "CA Unicenter message queuing exploit". SA targets a particular version of "Symantec Antivirus", that usually listens on port 2967. CAU targets a particular version of "CA Unicenter", that usually listens on port 6668. Both work only if a protection mechanism called DEP ("Data Execution Prevention") is disabled. If SA fails, then it is likely that CAU will fail as well (because DEP is enabled). The attacker is then better off trying something else. Achieving such behavior requires the attack plan to observe the outcomes of actions, and to react accordingly. Classical planning (which assumes perfect world knowledge at planning time) cannot accomplish this. Furthermore, port scans-observation actions testing whether or not a particular port is open-should be used only if one actually intends to execute a relevant exploit. Here, if we start with SA, we should scan only port 2967. We accomplish such behavior through the use of POMDPs. By contrast, to reduce uncertainty, classical planning requires a pre-process executing all possible scans. In this example, there are only two-ports 2967 and 6668-however in general there are many, causing significant network traffic and waiting time. POMDPs POMDPs are usually defined (e.g., (Monahan 1982;Kaelbling, Littman, and Cassandra 1998)) by a tuple S, A, O, T, O, r, b 0 . If the system is in state s ∈ S (the state space), and the agent performs an action a ∈ A (the action space), then that results in (1) a transition to a state s according to the transition function T (s, a, s ) = P r(s |s, a), (2) an observation o ∈ O (the observation space) according to the observation function O(s , a, o) = P r(o|s , a) and (3) a scalar reward r(s, a, s ). b 0 , the initial belief, is a probability distribution over S. The agent must find a decision policy π choosing, at each step, the best action based on its past observations and actions so as to maximize its future gain, which we measure here through the total accumulated reward. The expected value of an optimal policy is denoted with V * . The agent typically reasons about the hidden state of the system using a belief state b, a probability distribution over S. For our experiments we use SARSOP (Kurniawati, Hsu, and Lee 2008), a state of the art point-based algorithm, i.e., an algorithm approximating the value function as the upper envelope of a set of hyperplanes, corresponding to a selection of particular belief states (referred to as "points"). POMDP Model A preliminary version of our POMDP model appeared at the SecArt'11 workshop (Sarraute, Buffet, and Hoffmann 2011). The reader may refer to that paper for a more detailed example listing complete transition and observation models for some actions, and exemplifying the evolution of belief states when applying these actions. In what follows, we keep the description brief in the interest of space. States Several aspects of the problem-notably the network structure and the firewall filtering rules-are known and static. POMDP variables encoding these aspects can be compiled out in a pre-process, and are not included in our model. The states capture the status of each machine (controlled/reached/not reached). For non-controlled machines, they also specify the software configuration (operating system, servers, open ports, . . . ). We specify the vulnerable programs, as well as programs that can provide information about these (e.g., the protection mechanism "DEP" in our running example is relevant to both exploits). The states also indicate whether a given machine or program has crashed. Finally, we introduce one special terminal state into the POMDP model (of the entire network, not of individual machines). That state corresponds to giving up the attack, when for every available action (if any) the potential benefit is not worth the action's cost. Example 2 The states describe the attacked machine m. For simplicity, we assume that the exploits here do not risk crashing the machine (see also next sub-section). Apart from the terminal state and the state representing that m is controlled, the states specify which programs ("SA" or "CAU") are present, whether they are vulnerable, and whether "DEP" is enabled. Each application is listening on a different port, so a port is open iff the respective application is present, and we do not need to model ports separately. Thus we have a total of 20 states: In short, the states for each machine m essentially are tuples of status values for each relevant program. Global system states then are tuples of these machines-states, with one entry for each m ∈ M . The state space enumerates these tuples. In other words, the state space is factored in a natural way, by programs and machines. An obvious option is, thus, to model and solve the problem using factored POMDPs (e.g., (Hansen and Feng 2000)). We did not try this yet; our POMDP model generator internally enumerates the states, and feeds the ground model to SARSOP. 2 The factored nature of our problem also implies that the state space is huge. In a realistic setting, the set C of possible configuration tuples for each machine m ∈ M is very large, yielding an enormous state space |S| = O(|C| |M | ). In practice, we will run POMDPs only on single machines, i.e., |M | = 1. Actions To reach the terminal state, we need a terminate action indicating that one gives up on the attack. There are two main types of actions, scans and exploits, which both have to be targeted at reachable machines. Scans can be OS detection actions or port scans. In most cases, they have no effect on the state of the target machine. Their purpose is to gain knowledge about a machine's configuration, by an observation that typically allows to prune some states from the belief (e.g., observing that the OS must be some Windows XP version). Exploits make use of a vulnerability-if present-to gain control over a machine. The outcome of the exploit is observed by the attacker, so a failed exploit may, like a scan, yield information about the configuration (e.g., that a protection mechanism is likely to be running). For a minority of exploits, a failed attempt crashes the machine. For all actions, the outcome is deterministic: which observation is returned, and whether an exploit succeeds/fails/crashes, is uniquely determined by the target machine's configuration. Example 3 In our example, there are five possible actions: m_exploit_SA m_exploit_CAU m_scan_port_2967 m_scan_port_6668 terminate The POMDP model specifies, for each state in Example 2, the outcome of each action. For example, m_exploit_SA succeeds if and only if SA is present and vulnerable, and DEP is disabled. Hence, when applied to either of the states 9, 10, or 11, m_exploit_SA results in state 2, and returns the observation succeeded. Applied to any other state, m_exploit_SA leaves the state unchanged, and the observation is failed. The outcomes of actions also depend on what firewall (if any) stands between the pentester and the target. If the firewall filters out the relevant port, then the action is unusable: its transition model leaves the state unchanged, and no observation is returned. For example, if a firewall F filters out port 2967, then m_scan_port_2967 and m_exploit_SA are unusable through F , but can be employed as soon as a machine behind F is under control. Rewards No reward is obtained when using the terminate action or when in the terminal state. The instant reward of any scan/exploit action depends on the transition it induces in the present state. Our simple model is to additively decompose the instant reward r(s, a, s ) into r(s, a, s ) = r e (s, a, s ) + r t (a) + r d (a). Here, (i) r e (s, a, s ) is the value of the attacked machine in case the transition (s, a, s ) corresponds to a successful exploit, and is 0 for all other transitions; (ii) r t (a) is a cost that depends on the action's duration; and (iii) r d (a) is a cost that reflects the risk of detection when using this action. (iii) is orthogonal to the risk of crashing a program/machine, which as described we model as a possible outcome of exploits. Note that (ii) and (iii) may be correlated; however, there is no 1-to-1 correspondence between the duration and detection risk of an exploit, so it makes sense to be able to distinguish these two. Finally, note that (i) results in summing up rewards for successful exploits on different machines. That is not a limiting assumption: one can reward breaking into [m 1 OR m 2 ] by introducing a new virtual machine, accessible at no cost from each of m 1 and m 2 . Example 4 In our example, we set r e = 100 in case of success, 0 otherwise; r t = −10 for all actions; and r d = 0 (no risk of detection). We will see below what effect these settings have on an optimal policy. Since all actions are deterministic, there is no point in repeating them on the same target through the same firewallthis will not produce new effects or bring any new information. In particular, positive rewards cannot be received multiple times. Thus cyclic behaviors incur infinite negative costs. This implies that the expected reward of an optimal policy is finite even without discounting. 3 Designing the Initial Belief Penetration testing is done at regular time intervals. The initial belief-our knowledge of the network when we start the pentesting-depends on (a) what was known at the end of the previous pentest, and on (b) what may have changed since then. We assume for simplicity that knowledge (a) is perfect, i.e., each machine m at time 0 (the last pentest) is assigned one concrete configuration I(m). We then compute the initial belief as a function b 0 (I, T ) where T is the number of days elapsed since the last pentest. The uncertainty in this belief arises from not knowing which software updates were applied. We assume that the updates are made independently on each machine (simplifying, but reasonable given that updates are controlled by individual users). A simple model of updates (Sarraute, Buffet, and Hoffmann 2011) encodes the uncertain evolution of each program independently, in terms of a Markov chain. The states in each chain correspond to the different versions of the program, and the transitions model the possible program updates (with estimated probabilities that these updates will be made). The initial belief then is the distribution resulting from this chain after T steps. .998 Figure 1: The three independent Markov chains used to model the update mechanism in our example network. Example 5 In our running example, the three components in the single machine are DEP, CAU and SA. They are updated via three independent Markov chains, each with two states, as illustrated in Figure 1. The probabilities indicate how likely the machine is to transition from one state to another during one day. Say we set T = 30, and run the Markov chains on the configuration I in which m has DEP disabled, and both SA and CAU are vulnerable to the attacker's exploit. In the resulting initial belief b 0 (I, T ), DEP is likely to be enabled; the weight of states 12-20 in Example 2 is high in b 0 (> 70%). Here, we use this simple model as the basic building block in a method taking into account that version x of program A may need version y or z of program B. We assume that programs are organized in a hierarchical manner, the operating system being at the root of a directed acyclic graph, and a program having as its parents the programs it directly depends on. This yields a Dynamic Bayesian Network, where each conditional probability distribution is derived from a Markov chain P r(X t = x |X t−1 = x) filtered by a compatibility function δ(X = x, parent 1 (X) = x 1 , . . . , parent k (X) = x k ), that returns 1 iff the value of X is compatible with the parent versions, 0 otherwise. This model of updates is reasonable, but of course still not realistic; future work needs to investigate such models in detail. We now illustrate how reasoning with the probabilities of the initial belief results in the desired intelligent behavior. Example 6 Say we compute the initial belief b 0 (I, T ) as in Example 5. Since the weight of states 12-20 is high in b 0 , if m_exploit_SA fails, then the success probability of m_exploit_CAU is reduced to the point of not being worth the effort anymore, and the attacker (the optimal policy) gives up, i.e., would try a different attack not prevented by DEP. Namely, consider P r(CAU + |2967 + ), i.e., the probability of m_exploit_CAU succeeding, after observing that port 2967 is open. This corresponds to the weight of (A) states 8 and 11 in Example 2, within the states (B) 6-11 plus 15-20. That weight (A/B) is about 20%. Thus the expected value of m_exploit_CAU in this situation is about 100 * 0.2 [success reward] −10 [action cost] = 10, cf. Example 4, so the action is worthwhile. By contrast, say that m_exploit_SA has been tried and failed. Then (A) is reduced to state 8 only, while (B) still contains (in particular) all the DEP states 15-20. The latter states have a lot of weight, and thus P r(CAU + |2967 + ,SA − ) is only about 5%. Given this, the expected value of m_exploit_CAU is negative, and it is better to apply terminate instead. 4AL Decomposition Algorithm As hinted, POMDPs do not scale to large networks (cf. the experiments in the next section). We now present an approach using decomposition and approximation to overcome C5 C2 C3 C4 C6 C7 * C1 * C2 C3 N2 N1 N3 C 1 F 1 3 F * 1 F * 3 F 1 2 F 3 2 C 3 N 1 m m k . . . m 1 F 1 3 F 1 3 F 1 3 ∅ ∅ N 3 (a) LN as tree of components C. (b) Paths for attacking C 1 . (c) Attacking N 3 from N 1 , using m first. this problem, relying on POMDPs only to attack individual machines. The approach is called 4AL since it addresses network attack at 4 different levels of abstraction. 4AL is a POMDP solver specialized to attack planning as addressed here. Its input are the logical network LN and POMDP models encoding attacks on individual machines. Its output is a policy (an attack) for the global POMDP encoding LN , as well as an approximation of the value of the global value function. We next overview the algorithm, then fill in some technical details. To simplify the presentation, we will focus on the approximation of the value function, and outline only briefly how to construct the policy. 4AL Overview and Basic Properties The four levels of 4AL are: (1) Decomposing the Network, (2) Attacking Components, (3) Attacking Subnetworks, and (4) Attacking Individual Machines. We outline these levels in turn before providing technical details. Figure 2 provides illustrations. • Level 1: Decompose the logical network LN into a tree of biconnected components, rooted at * . In reverse topological order, call Level 2 on each component; propagate the outcomes upwards in the tree. Every graph decomposes into a unique tree of biconnected components (Hopcroft and Tarjan 1973). A biconnected component is a sub-graph that remains connected when removing any one vertex. In pentesting, intuitively this means that there is more than one possibility (more than one path) to attack the subnetworks within the component, requiring to reason about the component as a whole (which is the job of Level 2). By contrast, if removing subnetwork X (e.g., N 2 in Figure 2 (b)) makes the graph fall apart into two separate sub-graphs (C 2 vs. the rest of LN , compare also Figure 2 (a)), then all attacks from * to one of these subgraphs (C 2 here) must first traverse X (N 2 here). Thus the overall expected value of the attack can be computed by (1) computing the value of attacking that sub-graph (C 2 ) alone, and (2) adding the result as a pivoting reward to the reward of breaking into X (N 2 ). In other words, we "propagate the outcomes upwards" in the tree displayed in Figure 2 (a). It is important to note that this tree decomposition will typically result in a huge reduction of complexity. Bicon-nected components in LN arise only from clusters of more than 2 subnetworks sharing a common (physical) firewall machine. Such clusters tend to be small. In the real-world test scenario used by Core Security and in our experiment here, there is only one cluster, of size 3. In case there are no clusters at all, LN is a tree and 4AL Level 2 trivializes completely. • Level 2: Given component C, consider, for each rewarded subnetwork N ∈ C, all paths P in C that reach N . Backwards along each P , call Level 3 on each subnetwork and associated firewall. Choose the best path for each N . Aggregate these path values over all N , by summing up but disregarding rewards that were already accounted for by a previous path in the sum. In case a biconnected component C contains more than one subnetwork, to obtain the best attack on C, in general we have no choice but to encode the entire component as a POMDP. Since that is not feasible, Level 2 considers individual "attack paths" within C. Any single path P is equivalent to a sequence of attacks on individual subnetworks; these attacks are evaluated using Level 3. We consider the rewarded vertices N in separation, enumerating the attack paths and choosing a best one. The values of the best paths are aggregated over all N in a conservative (pessimistic) manner, by accounting for each reward at most once. A strict under-estimation occurs in case the best paths for some rewarded vertices are not disjoint: then these attacks share some of their cost, so a combined attack has a higher expected reward than the sum of independent attacks. In Figure 2 (b), N 2 and N 3 have a pivoting reward because they allow to reach the components C 2 and C 3 respectively. If the best paths for both N 2 and N 3 go via N 1 (because the firewall F * 3 is very strict), then these paths are not disjoint, duplicating the effort for breaking into N 1 . Obviously, enumerating attack paths within C is exponential in the size of C. This is the only point in 4AL-apart of course from calls to the POMDP solver-that has worst-case exponential runtime. In practice, biconnected components are typically small, cf. the above. Consider Figure 2 (c). When attacking N (here, N 3 ) from some machine behind the firewall F (here, F 1 3 ), we have to choose which machine inside N to attack. Given we commit to one such choice m, the attack problem becomes that of breaking into m and afterwards exploiting the direct connection to any m = m ∈ N , and any descendant network (here, C 3 ) we can now pivot to. As described, that can be dealt with by combining attacks on individual machines with modified rewards. (The pivoting reward for descendant networks is computed beforehand by Levels 1 and 2.) Like Level 2, Level 3 makes a conservative approximation. It fixes a choice of which m ∈ N to attack. By contrast, the best strategy may be to switch between different m ∈ N depending on the success of the attack so far. For example, if one exploit is very likely to succeed, then it may pay off to try this on all m first, before trying anything else. • Level 4: Given a machine m and a firewall F , model the single-machine attack planning problem as a POMDP, and run an off-the-shelf POMDP solver. Cache known results to avoid duplicate effort. Summing up, 4AL has low-order polynomial runtime except for the enumeration of paths within biconnected components (Level 2), and solving single-machine POMDPs (Level 4). The decomposition at Level 1 incurs no information loss. Levels 2 and 3 make conservative approximations, so, if the POMDP solutions are conservative (e.g., optimal), then the overall outcome of 4AL is conservative as well. Technicalities To provide a more detailed understanding of 4AL, we now discuss pseudo-code for the algorithm, provided in Figure 3. Consider first Algorithm 1. It should be clear how the overall structure of the algorithm corresponds to our previous discussion. It calls the linear-time algorithm by Hopcroft and Tarjan (1973) (hereafter, HT) to find the decomposition. The loop i = k, . . . , 1 processes the components in reverse topological order. The pivoting reward function pr encodes the propagation of rewards upwards in the tree; this should be self-explanatory apart for the expression "the parent" of C i in LN . The latter relies on the fact that, after "clean-up" (line 2), each component has exactly one such parent. To explain the clean-up, note first that HT works on undirected graphs; when applying it, we ignore the direction of the arcs in LN . The outcome is an undirected tree of biconnected components, where the cut vertices-those vertices removing which makes the graph break apart-are shared between several components. In Figure 2 (b), e.g., N 2 prior to the clean-up belongs to both, C 1 and C 2 . The clean-up sets the root of the tree to * , and assigns each cut-vertex to the component closest to * (e.g., N 2 is assigned to C 1 ); * itself is turned into a separate component. Re-introducing the direction of arcs in LN , we then prune vertices not reachable from * . Next, we remove arcs that cannot participate in any non-redundant attack path starting in * . Since moving towards * in the decomposition tree necessarily leads any attack back to a vertex it has visited (broken into) already, after such removal the arcs between components form a directed tree as in Figure 2 (a). Each non-root component C i (e.g., C 3 ) has exactly one parent component C in the cleaned-up tree (e.g., C 1 ). The respective subnetwork N ∈ C (e.g., N 3 ) is a cut vertex in LN . Thus, as claimed above, N is the only vertex, in LN , that connects into C i . Obviously, all attacks on C i must pass through its parent N . Further, the vertices and arcs removed by clean-up cannot be part of an optimal attack. Thus Level 1 is loss-free. To state this-and the other properties of 4AL-formally, we need some notations. We will use V * to denote the real (optimal) expected value of an attack, and V to denote the 4AL approximation. The attacked object is given as the argument. For example, V * (LN ) is the expected value of attacking LN ; V (C, pr) is the outcome of running 4AL Level 2 on component C and pivoting reward function pr. Proposition 1 Let LN be a logical network. Say that, for all calls to 4AL Level 2 made by 4AL Level 1 when run on LN , we have V (C, pr) = V * (C, pr). Then V (LN ) = V * (LN ). If V (C, pr) ≤ V * (C, pr) for all calls to 4AL Level 2, then V (LN ) ≤ V * (LN ). Consider now Algorithm 2. Our previous description was imprecise in omitting the additional algorithm argument pr. This integrates with the algorithm by being passed on, for every subnetwork on the paths we consider (line 7), to Algorithm 3 which adds it to the reward obtained for hacking into that subnetwork (Algorithm 3 line 4). R aggregates the values (lines 1, 9), over all rewarded subnetworks N . This aggregation is made conservative by removing all rewards-pivoting rewards as well as the own rewards of the individual machines involved-that have already been accounted for (line 10). Regarding the individual machines, Algorithm 2 uses the shorthands (a) r(N ) > 0 (line 2) and (b) r(N ) ← 0 (line 10); (a) means that there exists m ∈ N so that r(m) > 0; (b) means that r(m) ← 0 for all m ∈ N . Regarding pivoting rewards, note that line 10 of Algorithm 2 modifies the function pr maintained by Algorithm 1. This does not lead to conflicts because, at the time when Algorithm 1 calls Algorithm 2 on component C, all descendants of C in LN have already been processed, and thus in particular Algorithm 1 will make no further updates to the value of pr(N ), for any N ∈ C. By C * (line 4) we denote the set {N ∈ C | ∃N ∈ LN, N ∈ C : (N , N ) ∈ LN } of subnetworks that serve as an entry into C (e.g., N 1 and N 3 for C 1 in Figure 2 (b)). Note in line 4 that the path P starts with a firewall F 0 . To understand this, consider the situation addressed. The algorithm assumes that the parent N of C ( * , for component C 1 in Figure 2 (b)) is under control. But then, to break into C, we still need to traverse an arc from N into C. F 0 is the firewall on the arc chosen by P (F * 1 or F * 3 in Figure 2 (b)). The calls to Level 3 (line 7) comprise the network N i to be hacked into, the firewall F i−1 that must be traversed for doing so, the pivoting reward of N i , as well as the ongoing path reward R(P ) which gets propagated backwards along the path. Clearly, this is equivalent to the sequence of attacks required to execute P , and harvesting all pivoting rewards associated with such an attack. Thus, with the conservativeness of the aggregation across the subnetworks N , we get: Proposition 2 Let C be a biconnected component, and let pr be a pivoting reward function. Say that, for all calls to 4AL Level 3 made by 4AL Level 2 when run on (C, pr), we have V (F, N, pR, pathR) ≤ V * (F, N, pR, pathR). Then V (C, pr) ≤ V * (C, pr). Algorithms 3 and 4 should be self-explanatory, given our previous discussion. Just note that the pivoting reward pR is represented by the arc from m to C 3 in Figure 2 (c), which is accounted for by simply adding it to the value of m (Algorithm 3 line 4). The path reward pathR (not illustrated in Figure 2 (c)) is also added to the value of m (Algorithm 3 line 4). Max'ing over attacks on the individual machines m is, obviously, a conservative approximation because attack strategies are free to choose m. Thus: Proposition 3 Let F be a firewall, let N be a subnetwork, let pR be a pivoting reward, and let pathR be a path reward. Say that, for all calls to 4AL Level 4 made by 4AL Level 3 when run on (F, N, pR, pathR), we have V (F, m, R) ≤ V * (F, m, R). Then V (F, N, pR, pathR) ≤ V * (F, N, pR, pathR). Policy Construction At Level 1, the global policy is constructed from the Level 2 policies simply by following the tree decomposition: starting at the tree root, we execute the Level 2 policies for all reached components (in any order); once a hack into a component succeeds, the respective children components become reached. At Level 2, i.e., within a bi-connected component C, the policy corresponds to the set of paths P considered by Algorithm 2. Each P is processed in turn. For each node N in P (until failure to enter that subnetwork), we call the corresponding Level 3 policy. At Level 3, i.e., considering a single subnetwork N , our policy simply attacks the machine m ∈ N that yielded the maximum in Algorithm 3. The policy first attacks m through the firewall, using the respective Level 4 policy. In case the attack succeeds, the policy attacks the remaining machines m ∈ N in any order (i.e., for each m , we perform the associated Level 4 policy until termination). At Level 4, the policy is the POMDP policy returned by our POMDP solver. Experiments We evaluated 4AL against the "global" POMDP model, encoding the entire attack problem into a single POMDP. The experiments are run on a machine with an Intel Core2 Duo CPU at 2.2 GHz and 3 GB of RAM. The 4AL algorithm is implemented in Python. To solve and evaluate the POMDPs generated by Level 4, we use the APPL toolkit. 4 Test Scenario Our test scenario is based on the network structure shown in Figure 5. The attack begins from the Internet ( * is the cloud in the top left corner). The network consists of three areas-Exposed, Sensitive, User-separated by firewalls. Internally, each of Exposed and Sensitive is fully connected (i.e., these areas are subnetworks), whereas User consists of a tree of subnetworks separated by empty firewalls. Only two machines are rewarded, one in Sensitive (reward 9000) and one in a leaf subnetwork of User (reward 5000). The cost of port scans and exploits is 10, the cost of OS detection is 50. We allow to scale the number of machines |M | by distributing, of every 40 machines, the first one to Exposed, the second one to Sensitive, and the remaining 38 to User. The exploits are taken from Core Security's database. The number of exploits |E| is scaled by distributing these over 13 templates, and assigning to each machine m one such template as I(m) (the known configuration at the time of the last pentest). The initial belief b 0 (I, T ), where T is the time elapsed since the last pentest, is then generated as outlined. The fixed parameters here (rewards, action costs, distribution of machines over areas, number of templates) are estimated based on practical experiences at Core Security. The network structure and exploits are realistic, and are used for industrial testing in that company. The main weakness of the scenario lies in the approximation of software updates underlying b 0 (I, T ). Altogether, the scenario is still simplified, but is natural and does approach the complexity of real-world penetration testing. For lack of space, in what follows we scale only |M | and |E|, fixing |T | = 50. The latter is realistic but challenging: pentesting is typically performed about once a month; smaller T are easier to solve as there is less uncertainty. Approximation Loss Figure 4 (a) shows the relative loss of quality when running 4AL instead of a global POMDP solution, for values of |E| and |M | where the latter is feasible. We show quality(global -POMDP ) − quality(4AL) in percent of quality(global -POMDP ). Policy quality here is estimated by running 2000 simulations. That measurement incurs a variance, which is almost stronger than the very small quality advantage of the global POMDP solution. The maximal loss for any combination of |E| and |M | is 14.1% (at |E| = 7, |M | = 6), the average loss over all combinations is 1.96%. The average loss grows monotonically over |M |, from −1.14% for |M | = 1 to 4.37% for |M | = 6. Over |E|, the behavior is less regular; the maximum average loss, 5.4%, is obtained when fixing |E| = 5. Scaling Up Figure 4 (b) shows the runtime of 4AL when scaling up to much larger values of |E| and |M |. The scaling behavior over |M | clearly reflects the fact that 4AL is polynomial in that parameter, except for the size of biconnected components (which is 3 here). Scaling E yields more challenging single-machine POMDPs, resulting in a sometimes steep growth of runtime. However, even with |M | and |E| both around 100, which is a realistic size in practice, the runtime is always below 37 seconds. Conclusion We have devised a POMDP model of penetration testing that allows to naturally represent many of the features of this application, in particular incomplete knowledge about the network configuration, as well as dependencies between different attack possibilities, and firewalls. Unlike any previous methods, the approach is able to intelligently mix scans with exploits. While this accurate solution does not scale, large networks can be tackled by a decomposition algorithm. Our present empirical results suggest that this is accomplished at a small loss in quality relative to a global POMDP solution. An important open question is to what extent our POMDP + decomposition approach is more cost-effective than the classical planning solution currently employed by Core Security. Our next step will be to answer this question experimentally, comparing the attack quality of 4AL against that of the policy that runs extensive scans and then attaches FF's plan for the most probable configuration. 4AL is a domain-specific algorithm and, as such, does not contribute to the solution of POMDPs in general. At a high level of abstraction, its idea can be understood as imposing a template on the policy constructed, thus restricting the space of policies explored (and employing special-purpose algorithms within each part of the template). In this, the approach is somewhat similar to known POMDP decomposition approaches (e.g., (Pineau, Gordon, and Thrun 2003;Müller and Biundo 2011)). It remains to be seen whether this connection can turn out fruitful for either future work on attack planning, or POMDP solving more generally. The main directions for future work are to devise more accurate models of software updates (hence obtaining more realistic designs of the initial belief); to tailor POMDP solvers to this particular kind of problem, which has certain special features, in particular the absence of non-deterministic actions and that some of the uncertain parts of the state (e.g. the operating systems) are static; and to drive the industrial application of this technology. We hope that these will inspire other researchers as well. Figure 2 : 2Illustration of Levels 1, 2, and 3 (from left to right) of the 4AL algorithm. • Level 3 :1N 3Given a subnetwork N and a firewall F through which to attack N , for each machine m ∈ N approximate the reward obtained when attacking m first. For this, Algorithm 1: Level 1 (Decomposing the Network) Input: LN : Logical Network. Output: Approximation V of expected value V * of attacking LN from controlled machine * . / * Decompose LN into tree DLN of biconnected components, rooted at * ; DLN ←HopcroftTarjan(LN ); 2 Set tree root to * and clean-up LN and DLN ; 3 C 1 , . . . , C k ← a topological ordering of DLN ; 4 Intitialize pivoting reward pr(N ) for all N ∈ LN to 0; 5 for i = k, ..., 1 do / * Call Level 2 to attack each component. ← the parent of C i in LN ; 8 pr(N ) ← pr(N ) + V (C i ); Biconnected component C, reward function pr. Output: Approximation V of expected value V * of attacking C, given its parent is controlled and its pivoting rewards are pr. ∃N ∈ C s.t. r(N ) > 0 or pr(N ) > 0 do 3 P ← ; R(P ) ← 0; P (N ) ← P ; / * Maximize over all simple paths (no repeated vertices) from an entry vertex to N . F k− 1 −Figure 3 : 13−− → N k = N where N 1 , . . . , N k ∈ C and N 1 ∈ C * do / * Propagate rewards along P P ) ← Level3(N i , F i−1 , pr(N i ), R(P )); 8 P (N ) ← arg max(R(P (N )), R(P )); 9 R ← R + R(P (N ));10 r(N i ), pr(N i ) ← 0 for all vertices N i on P (N ); 11 return R Algorithm 3: Level 3 (Attacking Subnetworks) Input: Firewall F , subnetwork N , rewards pR, pathR. Output: Approximation V of expected value V * of attacking N through F , given F is reached, N 's pivoting reward is pR, and the path reward behind N is pathR. 1 R ← 0; / * Maximize over reward obtained when hacking first into a particular machine m ∈ N . * / 2 foreach m ∈ N do 3 R(m) ← r(m); / * After breaking m, we can pivot behind N , and reach all m = m ∈ N without F . * / 4 R(m) ← R(m) + pR + pathR; 5 foreach m = m ∈ N do 6 R(m) ← R(m)+Level4(m , ∅, r(m )); 7 R ← max(R, Level4(m, F, R(m))); 8 return R Algorithm 4: Level 4 (Attacking Individual Machines) Input: Firewall F , machine m, reward R. Output: Approximation V of expected value V * of attacking m through F , given m is reached and the current reward of breaking it is R. 1 if (m, F, R) is cached then 2 return V (m, F, R) 3 M ←createPOMDP(m, F, R); 4 V ←solvePOMDP(M ); 5 Cache (m, F, R) with V ; 6 return V 4AL algorithm, pseudo-code.modify m's reward to take into account that, after breaking m, we are behind F : call Level 4 to obtain the values of all m = m with an empty firewall; then add these values, plus any pivoting reward, to the reward of m and call Level 4 on this modified m with firewall F . Maximize the resulting value over all m ∈ N . 4Figure 5 : 5APPL 0.93 at http://bigbird.comp.nus.edu.sg/pmwiki/farm/appl/ Network structure in our test suite. Figure 4 : 4Empirical results for 4AL compared to a global POMDP model. For simplicity, we will notate * as a separate vertex in LN . If * is part of a subnetwork N , this means to turn N \ { * } into a separate vertex in LN , connected to * via the empty firewall. Note that this approach enables certain non-trivial optimizations: some of the states in Example 2 could be merged. If DEP is enabled, then it does not matter whether or not CAU/SA are vulnerable. For brevity, we do not discuss this in detail here. In fact, the problem falls into the class of Stochastic Shortest Path Problems(Bertsekas and Tsitsiklis 1996). This last step should be self-explanatory. The POMDP model is created as described earlier. Note that Level 3 may, during the execution of 4AL, call the same machine with the same firewall more than once. For example, inFigure 2(c), when we switch to attacking m 1 instead of m, the call of Level 4 with m k and an empty firewall is repeated. Acknowledgments. Work performed while Jörg Hoffmann was employed by INRIA, Nancy, France. Why attacking systems is a good idea. I Arce, G Mcgraw, IEEE Computer Society -Security & Privacy Magazine. 24Arce, I., and McGraw, G. 2004. Why attacking systems is a good idea. IEEE Computer Society -Security & Privacy Magazine 2(4). Neurodynamic Programming. D Bertsekas, J Tsitsiklis, Athena ScientificBertsekas, D., and Tsitsiklis, J. 1996. Neurodynamic Pro- gramming. Athena Scientific. Course of action generation for cyber security using classical planning. M S Boddy, J Gohde, T Haigh, S A Harp, Proc. of ICAPS'05. of ICAPS'05Boddy, M. S.; Gohde, J.; Haigh, T.; and Harp, S. A. 2005. Course of action generation for cyber security using classi- cal planning. In Proc. of ICAPS'05. Dynamic programming for POMDPs using a factored state representation. E Hansen, Z Feng, Proceedings of the International Conference on AI Planning and Scheduling (AIPS'00). the International Conference on AI Planning and Scheduling (AIPS'00)Hansen, E., and Feng, Z. 2000. Dynamic programming for POMDPs using a factored state representation. In Proceed- ings of the International Conference on AI Planning and Scheduling (AIPS'00). The Metric-FF planning system: Translating "ignoring delete lists" to numeric state variables. J Hoffmann, Journal of Artificial Intelligence Research. 20Hoffmann, J. 2003. The Metric-FF planning system: Translating "ignoring delete lists" to numeric state variables. Journal of Artificial Intelligence Research 20:291-341. Algorithm 447: efficient algorithms for graph manipulation. J Hopcroft, R Tarjan, Communications of the ACM. 16Hopcroft, J., and Tarjan, R. 1973. Algorithm 447: efficient algorithms for graph manipulation. Communications of the ACM 16:372-378. Planning and acting in partially observable stochastic domains. L Kaelbling, M Littman, Cassandra , A , Artificial Intelligence. 1011-2Kaelbling, L.; Littman, M.; and Cassandra, A. 1998. Plan- ning and acting in partially observable stochastic domains. Artificial Intelligence 101(1-2):99-134. SARSOP: Efficient point-based POMDP planning by approximating optimally reachable belief spaces. H Kurniawati, D Hsu, W Lee, Robotics: Science and Systems IV. Kurniawati, H.; Hsu, D.; and Lee, W. 2008. SARSOP: Ef- ficient point-based POMDP planning by approximating op- timally reachable belief spaces. In Robotics: Science and Systems IV. Attack planning in the real world. J Lucangeli, C Sarraute, G Richarte, Workshop on Intelligent Security. Lucangeli, J.; Sarraute, C.; and Richarte, G. 2010. Attack planning in the real world. In Workshop on Intelligent Secu- rity (SecArt 2010). A survey of partially observable Markov decision processes. G Monahan, Management Science. 28Monahan, G. 1982. A survey of partially observable Markov decision processes. Management Science 28:1-16. HTN-style planning in relational POMDPs using first-order FSCs. F Müller, S Biundo, Proceedings of the 34th German Conference on AI (KI'11). the 34th German Conference on AI (KI'11)Müller, F., and Biundo, S. 2011. HTN-style planning in relational POMDPs using first-order FSCs. In Proceedings of the 34th German Conference on AI (KI'11), 216-227. Policycontingent abstraction for robust robot control. J Pineau, G Gordon, S Thrun, Proceedings of the 19th Conference on Uncertainty in Articifical Intelligence (UAI'03). the 19th Conference on Uncertainty in Articifical Intelligence (UAI'03)Pineau, J.; Gordon, G.; and Thrun, S. 2003. Policy- contingent abstraction for robust robot control. In Proceed- ings of the 19th Conference on Uncertainty in Articifical In- telligence (UAI'03), 477-484. Penetration testing == POMDP solving?. C Sarraute, O Buffet, J Hoffmann, Proceedings of the 3rd Workshop on Intelligent Security (SecArt'11). the 3rd Workshop on Intelligent Security (SecArt'11)Sarraute, C.; Buffet, O.; and Hoffmann, J. 2011. Penetra- tion testing == POMDP solving? In Proceedings of the 3rd Workshop on Intelligent Security (SecArt'11). An algorithm to find optimal attack paths in nondeterministic scenarios. C Sarraute, G Richarte, J Lucangeli, In ACM Workshop on Artificial Intelligence and Security. 11Sarraute, C.; Richarte, G.; and Lucangeli, J. 2011. An algorithm to find optimal attack paths in nondeterministic scenarios. In ACM Workshop on Artificial Intelligence and Security (AISec'11).
[]
[ "Probing Leptogenesis and Pre-BBN Universe with Gravitational Waves Spectral Shapes", "Probing Leptogenesis and Pre-BBN Universe with Gravitational Waves Spectral Shapes" ]
[ "Rome Samanta [email protected] \nInstitute of Physics\nCEICO\nCzech Academy of Sciences\nNa Slovance\n1999/2182 21Prague 8Czech Republic\n", "Satyabrata Datta [email protected] \nSaha Institute of Nuclear Physics\nHBNI\n1/AF Bidhannagar700064KolkataIndia\n" ]
[ "Institute of Physics\nCEICO\nCzech Academy of Sciences\nNa Slovance\n1999/2182 21Prague 8Czech Republic", "Saha Institute of Nuclear Physics\nHBNI\n1/AF Bidhannagar700064KolkataIndia" ]
[]
On the frequency-amplitude plane, Gravitational Waves (GWs) from cosmic strings show a flat plateau at higher frequencies due to the string loop dynamics in standard radiation dominated post-inflationary epoch. The spectrum may show an abrupt upward or a downward trend beyond a turning point frequency f * , if the primordial dark age prior to the Big Bang Nucleosynthesis (BBN), exhibits non-standard cosmic histories. We argue that such a spectral break followed by a rising GW amplitude which is a consequence of a postinflationary equation of state (ω > 1/3) stiffer than the radiation (ω = 1/3), could also be a strong hint of a leptogenesis in the seesaw model of neutrino masses. Dynamical generation of the right handed (RH) neutrino masses by a gauged U (1) symmetry breaking leads to the formation of a network of cosmic strings which emits stochastic GWs. A gravitational interaction of the lepton current by an operator of the form ∂ µ Rj µ -which can be generated in the seesaw model at the two-loop level through RH neutrino mediation, naturally seeks a stiffer equation of state to efficiently produce baryon asymmetry proportional to 1 − 3ω.We discuss how GWs with reasonably strong amplitudes complemented by a neutrino-less double beta decay signal could probe the onset of the most recent radiation domination and lightest RH neutrino mass at the intermediate scales.a
10.1007/jhep11(2021)017
[ "https://arxiv.org/pdf/2108.08359v2.pdf" ]
237,213,666
2108.08359
1f37e0a519aacacd1447bd65d80b3972af039b56
Probing Leptogenesis and Pre-BBN Universe with Gravitational Waves Spectral Shapes 20 Oct 2021 Rome Samanta [email protected] Institute of Physics CEICO Czech Academy of Sciences Na Slovance 1999/2182 21Prague 8Czech Republic Satyabrata Datta [email protected] Saha Institute of Nuclear Physics HBNI 1/AF Bidhannagar700064KolkataIndia Probing Leptogenesis and Pre-BBN Universe with Gravitational Waves Spectral Shapes 20 Oct 20212 On the frequency-amplitude plane, Gravitational Waves (GWs) from cosmic strings show a flat plateau at higher frequencies due to the string loop dynamics in standard radiation dominated post-inflationary epoch. The spectrum may show an abrupt upward or a downward trend beyond a turning point frequency f * , if the primordial dark age prior to the Big Bang Nucleosynthesis (BBN), exhibits non-standard cosmic histories. We argue that such a spectral break followed by a rising GW amplitude which is a consequence of a postinflationary equation of state (ω > 1/3) stiffer than the radiation (ω = 1/3), could also be a strong hint of a leptogenesis in the seesaw model of neutrino masses. Dynamical generation of the right handed (RH) neutrino masses by a gauged U (1) symmetry breaking leads to the formation of a network of cosmic strings which emits stochastic GWs. A gravitational interaction of the lepton current by an operator of the form ∂ µ Rj µ -which can be generated in the seesaw model at the two-loop level through RH neutrino mediation, naturally seeks a stiffer equation of state to efficiently produce baryon asymmetry proportional to 1 − 3ω.We discuss how GWs with reasonably strong amplitudes complemented by a neutrino-less double beta decay signal could probe the onset of the most recent radiation domination and lightest RH neutrino mass at the intermediate scales.a On the frequency-amplitude plane, Gravitational Waves (GWs) from cosmic strings show a flat plateau at higher frequencies due to the string loop dynamics in standard radiation dominated post-inflationary epoch. The spectrum may show an abrupt upward or a downward trend beyond a turning point frequency f * , if the primordial dark age prior to the Big Bang Nucleosynthesis (BBN), exhibits non-standard cosmic histories. We argue that such a spectral break followed by a rising GW amplitude which is a consequence of a postinflationary equation of state (ω > 1/3) stiffer than the radiation (ω = 1/3), could also be a strong hint of a leptogenesis in the seesaw model of neutrino masses. Dynamical generation of the right handed (RH) neutrino masses by a gauged U (1) symmetry breaking leads to the formation of a network of cosmic strings which emits stochastic GWs. A gravitational interaction of the lepton current by an operator of the form ∂ µ Rj µ -which can be generated in the seesaw model at the two-loop level through RH neutrino mediation, naturally seeks a stiffer equation of state to efficiently produce baryon asymmetry proportional to 1 − 3ω. We discuss how GWs with reasonably strong amplitudes complemented by a neutrino-less double beta decay signal could probe the onset of the most recent radiation domination and lightest RH neutrino mass at the intermediate scales. Leptogenesis [1][2][3][4][5][6][7] is a simple mechanism to explain the observed baryon asymmetry of the universe [8]. The right handed (RH) heavy neutrinos which are introduced in the Standard Model (SM) to generate light neutrino masses (Type-I seesaw), decay CP asymmetrically to create lepton asymmetry which is then converted to baryon asymmetry via Sphaleron transition [9]. When it comes to the testability of leptogenesis, there are subtleties. If the heavy neutrino masses are not protected by any symmetry [10], it is quite natural to assume that they are hierarchical in nature like any other family of SM fermions. In that case, the lightest RH mass scale is bounded from below M ≳ 10 9 GeV [11]which is beyond the reach of the present collider experiments. Nonetheless, still the colliders and other low energy neutrino experiments can probe leptogenesis mechanisms that do not constitute hierarchical RH neutrinos-starting from O(TeV) to O(MeV) scale heavy neutrinos [12][13][14][15]. A shift of attention from the collider experiments to the Gravitational Waves (GWs) physics is not less interesting in terms of testing leptogenesis. Particularly, this new cosmic frontier, in which after the discovery of GWs from black hole mergers by LIGO and Virgo collaboration [16,17], plenty of efforts are being made to detect primordial GWs from the Early Universe (EU) within a wide range of frequencies-starting from the Pulsar Timing Arrays (PTAs, ∼ nHz) to the LIGO(∼ 25Hz). A network of cosmic strings [18][19][20] which is a generic consequence of breaking symmetries such as U (1), is one of the prominent sources of strong stochastic primordial gravitational waves which can be tested in a complementary way in most of the planned GW detectors. Numerical simulations based on the Nambu-Goto action [21,22] indicate that cosmic string loops loose energy dominantly via GW radiation, if the underlying broken symmetry corresponds to a local gauge symmetry. In the context of seesaw, this sounds music to the ears, since such a gauge symmetry is U (1) B−L [23][24][25], breaking of which could be responsible for the dynamical generation of the heavy RH masses and hence the lepton number violation as well as creation of a network of cosmic strings. Having this set-up, there could be two categories to look for the GWs as a probe of leptogenesis. Category A: A scale separation between the RH masses and the typical Grand Unified Theory (GUT) scale (∼ 10 16 GeV), imposed by seesaw perturbativity condition and the neutrino oscillation data [26] implies that residual symmetries like U (1) B−L protects the RH neutrinos to get mass at the GUT scale. Therefore, breaking of that symmetry at a later stage and consequent emission of GWs from cosmic strings are natural probes of the scale of leptogenesis. In this case, it is the amplitude (GW energy density normalised by the critical energy density) of the GWs that matters as a probe and this approach has been taken in Refs. [27,28]. Category B: To make the testability more robust, along with the amplitudes, one can associate leptogenesis also to the spectral shapes of the GWs [29,30]. Cosmic string loops that originate and decay in the radiation domination, exhibit a flat plateau on the amplitude-frequency plane at the higher frequencies. This spectral shape may show an upward or a downward trend if something other than radiation dominates the energy density of the EU before the onset (T * ) of most recent radiation domination prior to the BBN (T ∼ 5 MeV) [31][32][33]. Such a non-standard cosmic history that is responsible for this spectral break which along with the GW amplitude, one aims to claim also as a probe, should therefore be a natural/well-motivated call from the perspective of leptogenesis. Two well-known scenarios in this context can be opted for. Category B1: A matter domination (ω = 0 < 1/3) [34,35]. Category B2: Scenarios such as kination (ω = 1 > 1/3) [36,37]. For the former (latter), one finds a spectral break followed by a downward (upward) going GW amplitude [38][39][40][41]. Two leptogenesis mechanisms in the Category B1-a low-scale leptogenesis and a leptogenesis from ultralight primordial black holes (M P BH ≲ 13g) have been studied in Ref. [29] and Ref. [30] respectively. In this article, we discuss a scenario that falls in the Category B2, i.e., interpreting a flat then a spectral break followed by a rising GW amplitude as a signature of leptogenesis. Note that, two crucial ingredients for this typical signal are of course cosmic string network itself and then a non-standard equation of state (ω = 1 in our discussion). In the context of leptogenesis from decays [1], though the former is a natural consequence in the sense of Category A [27], a stiffer equation of state is not an indispensable criterion. However, in seesaw models, even when the Lagrangian is minimally coupled to gravity, through massive RH neutrino mediation one can generate an operator of the form ∂ µ Rj µ /M 2 at two-loop level [42][43][44] (see also Ref. [28,45] for a flavour generalisation and Ref. [46] for a recent review), where R is the Ricci scalar and j µ is the lepton current. This operator is a well-studied operator [47][48][49][50] with the corresponding mechanism dubbed as "gravitational lepto/baryogenesis" and produces final baryon asymmetry proportional toṘ ∝ (1 − 3ω). Interestingly, note now that two primary ingredients of the GW signal are also natural requirements to obtain non-zero lepton asymmetry, i.e., the symmetry breaking which gives rise to massive RH neutrinos (mediate in the loops [44]) as well as cosmic strings and then an equation of state ω ≠ 1/3 [51]. We shall discuss later on, that indeed a stiffer equation of state is needed to efficiently produce lepton asymmetry. Plateau amplitudes corresponding to Gµ ≲ 10 −12 with G being the Newton constant and µ being the string tension, with a post LISA spectral break supplemented by a potential test in neutrino-less double beta decay experiments, make the scenario generally robust. The above introduction summarises the basic idea and the main results of this paper. The next sections are dedicated to a more detailed description and technicalities. II. GRAVITATIONAL WAVES FROM COSMIC STRINGS Cosmic strings may originate as the fundamental or composite objects in string theory [52,53] as well as topological defects from spontaneous symmetry breaking (SSB) when the vacuum manifold M has a non-trivial first homotopy group π 1 (M). A theory with spontaneous breaking of a U (1) symmetry exhibits string solution [19,20], since π 1 (M) = Z. An example of a field theory containing string like solution is a theory of U (1)-charged complex scalar field φ that in the context of seesaw could be a SM scalar singlet φ B−L which is responsible for the dynamical generation of RH neutrino masses. After the formation, strings get randomly distributed in space and form a network of horizon-size long strings [54,55] characterised by a correlation length L = µ/ρ ∞ , where µ-the string tension or energy per unit length is in general constant (however, e.g., in case of global strings [56] and recently introduced melting strings[57] µ ∼ f (T )) and typically taken to be the square of the symmetry breaking scale Λ CS and ρ ∞ is the long string energy density. When two segments of long strings cross each other they inter-commute and form loops with a probability P = 1 [58] (exceptions [59]). A string network may interact strongly with thermal plasma and thereby its motion gets damped [60]. After the damping stops, the strings oscillate and enter a phase of scaling evolution that constitute two competing dynamics namely the stretching of the correlation length due to the cosmic expansion and fragmentation of the long strings into loops which oscillate independently and produce particle radiation or gravitational waves [61][62][63]. Out of these two competing dynamics, there is an attractor solution called the scaling regime [64][65][66] in which the characteristic length scales as L ∼ t. This implies, for constant string tension, ρ ∞ ∝ t −2 . Therefore, the network tracks any cosmological background energy density ρ bg ∝ a −3(1+ω) ∝ t −2 with the same equation of state and hence cosmic strings do not dominate the energy density of the universe like any other defects. The loops radiate GWs at a constant rate which sets up the time evolution of a loop of initial size [61,63] and the initial loops size parameter α ≃ 0.1-a value preferred by numerical simulations [67,68]. The total energy loss from a loop is decomposed into a set of normal-mode oscillations with frequencies f k = 2k/l = a(t 0 )/a(t)f , where k = 1, 2, 3...k max (k max is for numerical purpose, otherwise ∞) and f is the frequency observed today. Given the loop number density n t , l k , the present time gravitational wave density parameter is given by l i = αt i as l(t) = αt i − ΓGµ(t − t i ), where Γ ≃ 50Ω GW (t 0 , f ) ≡ f ρ −1 c dρ GW /df = ∑ k Ω (k) GW (t 0 , f ), with the kth mode amplitude Ω (k) GW (t 0 , f ) as[67] Ω (k) GW (f ) = 2kGµ 2 Γ k f ρ c t 0 t osc a(t) a(t 0 ) 5 n t , l k dt. (II.1) The quantity Γ k depends on the small scale structures of the loop and is given by Γ (k) = Γk −δ ζ(δ) , e.g., δ = 4/3 and 5/3 for cusps and kinks [69]. The integration in Eq.II.1 is subjected to a Heaviside func- tion Θ ≡ Θ(t i −t osc )Θ(t i − l cric α ), with t osc = Max [network formation time(t F ) , end of damping(t fric )] and l cric is the critical length below which massive particle radiation dominates over GWs [70,71]. Both these Θ functions set a high-frequency cut-off in the spectrum (a systematic analysis can be found in Ref. [40]). The most important aspect to obtain the GW spectrum is the computation of the loop number density n t , l k which we calculate from the Velocity-dependent-One-Scale (VOS) model [72][73][74] which assumes the loop production function to be a delta function, i.e. all the loops are created with the same fraction of the horizon size with a fixed value of α. Given a general equation of state parameter ω, the number density n ω t , l k is computed as n ω (t, l k (t)) = A β α (α + ΓGµ) 3(1−β) l k (t) + ΓGµt 4−3βt 3β , (II.2) where β = 2/3(1 + ω) and we assume A β = 29.6 (ω = 1), 5.4 (w = 1/3) and 0.39 (ω = 0) [74] is a step-function while changing the cosmological epochs. The most interesting feature of GWs from cosmic string is that the amplitude increases with the symmetry breaking scale Λ CS . This can be seen by computing the Ω GW , considering loop production as well as decay in the radiation domination which gives an expression for a flat plateau at higher frequencies (see AUX A for an exact formula) Ω (1) GW (f ) = 128πGµ 9ζ(δ) A r r Ω r (1 + r ) 3/2 − 1 , (II.3) where r = α/ΓGµ and Ω r ≃ 9 × 10 −5 . Such strong GWs as a consequence of a very high scale symmetry breaking thus serves as an outstanding probe of particle physics models [75][76][77][78][79][80][81]. Possibly the most important recent development is the finding of a stochastic common spectrum process across 45 pulsars by NANOGrav PTA [82], which if interpreted as GWs, corresponds to a strong amplitude and is better fitted with cosmic strings [28,83,84] than the single value power spectral density as predicted by supermassive black hole models. Let's also mention that a very recent analysis by PPTA [85] is in agreement with the NANOGrav result. In presence of an additional early kination era, the entire GW spectrum is determined by four dynamics. I) A peak at a lower frequency-caused by the loops which are produced in the radiation era and decay in the standard matter era. II) The flat plateau, Ω plt GW , as mention while describing Eq.II.3. III) A spectral break at f * = 8 αΓGµ t −1/2 * t −2/3 0 t 1/6 eq -so called the turning point frequency [30,39,40], followed by a rising GW amplitude Ω (1) GW (f > f * ) ≃ Ω plt GW (f /f * ) , caused by modified redshifting of the GWs during kination era V) a second turning point frequency f ∆ after which the GWs amplitude falls, e.g, due to particle productions below l < l cric = β m µ −1/2 (ΓGµ) m , with β m ∼ O(1) and m = 1, 2 for loops with kinks or cusps [70,71]. If the falling is caused due to thermal friction, then one needs to consider the damping of the smaller loops along with the long-string network for t < t f ric , discarding any GWs production by the smaller loops, i.e, the entire dynamics is completely frozen until t f ric [60]. In fact, in our computation we do not take into account any GWs produced from smaller loops prior to t f ric and consider that the falling is due to particle production which sets the high-frequency cut-off that is much more stronger (appears at lower frequencies) than the friction cut-off [40]. Note also that if the two turning-point frequencies are close to each other, potentially the GW detectors could see a small bump after the flat plateau with a peak amplitude ≃ Ω plt GW (f ∆ /f * ). Nevertheless, as we show in the next section that given a successful leptogenesis, the second turning point frequency as well as small bumps are most likely to be outside the frequency range of the GW detectors. Before concluding the section, we note two important points. Firstly, the VOS model overestimates the number density of the loops by an order of magnitude compared to the numerical simulations [67]. This is due to the fact that VOS model considers all the loops are of same size at the production. However, there could be a distribution of α. Numerical simulation finds that only 10% of the energy of the long-string network goes to the large loops (α ≃ 0.1) while the rest 90% goes to the highly boosted smaller loops that do not contribute to the GWs. This fact is taken into account by including a normalisation factor F α ∼ 0.1 in Eq.II.2 [74]. Secondly, the amplitude beyond f * goes as f 1 even after taking into account high-k modes (see AUX A) unlike the case of an early matter domination where the same changes from f −1 → f −1/3 for cusps like structures [29,30,40]. III. GRAVITATIONAL LEPTOGENESIS, RESULTS AND DISCUSSION The idea behind gravitational leptogenesis [48] is, a C and CP-violating operator L CP V ∼ b∂ µ Rj µ ∼ b∂ µ R¯ γ µ with b as a real effective coupling, corresponds to a chemical potential µ = bṘ for the lepton number in the theory. Therefore, the normalised (by photon density n γ ∼ T 3 ) equilibrium lepton asymmetry (using standard Fermi-Dirac statistics with energies E ± = E ±µ) is given by N eq B−L ∼ bṘ T . Interestingly, L CP V can be generated in a UV framework using the seesaw Lagrangian even when it is minimally coupled to gravity (see e.g., Ref. [43] for an in-depth discussion, sec.II of Ref. [28] for a brief summary). As a computational insight, one calculates an effective h vertex corresponding to the operator L CP V using a conformally flat metric g µν = (1 + h)η µν with R = −3∂ 2 h, capitalising the fact that the coupling 'b' is independent of the choice of background. In seesaw model, a similar h vertex that manifests the L CP V operator, can be constructed at two-loop level, where the Higgs and the RH masses mediate the loops. Then simply comparing the coefficients of both the vertices up to linear order in h, the coupling b can be calculated in terms of the Yukawa coupling f (where, f αi¯ LαH N Ri is the Yukawa interaction in seesaw, with Lα , H and N R being the lepton doublet, Higgs and RH fields respectively) and RH neutrino masses M i . The expression for the equilibrium asymmetry then reads N eq B−L = π 2Ṙ 36(4π) 4 j>i Im k 2 ij ζ(3)T M i M j ln M 2 j M 2 i , (III.1) where k ij = (f † f ) ij . The above expression could be modulated by a factor (M 2 j /M 2 i ) γ , where γ = 0, 1. However, γ = 0 appears to be the most natural solution which can be calculated exactly [43,44]. In any case, even if one considers γ = 1 or the 'hierarchical enhancement', tuning the complex part in k 2 ij , correct baryon asymmetry can always be reproduced. The most important part is, N B−L ∝Ṙ ∝ 1 − 3ω which is still vanishing in radiation domination at high temperatures with SM-QCD thermodynamic potential [51]. Therefore, a general cosmological background other than radiation that is quite a natural call now, always stems a non-vanishing equilibrium asymmetry unless the Yukawa couplings are real or purely imaginary. In the EU, any dynamically produced lepton asymmetry tracks the N eq B−L if the interaction that causes the asymmetry production is strong enough. When the interaction rate becomes weaker (compared to the Hubble expansion), the asymmetry freezes out with the potential to reproduce correct baryon asymmetry N B−L ∼ 6 × 10 −8 [8]. In seesaw model, ∆L = 2 interactions [86] play this role. The general evolution equation that governs the entire dynamics is given by dN B−L dz = − κ z p + W ID N B−L − β z q , (III.2) where z = M 1 /T , W ∆L=2 (z) = κ z p with p = 5−3ω 2 , N eq B−L = β z q with q = 7+9ω 2 and W ID represents the inverse decay H → N 1 rate. The parameters κ ∼ f κ (m i , M 1 )z 1 2 (1−3ω) * and N eq B−L ∝ β ∼ f β (m i , M 1 , Im[f ij ])(1 − 3ω)z 3 2 (3ω−1) * , where z * = M 1 /T * and m i is the i-th light neutrino mass with i = 1, 2, 3. All the exact expressions can be found in AUX B. Before proceeding further, let us mention that we do not include the charged lepton flavour effects in this analysis for simplicity. Nonetheless, a systematic description with flavour issues can be found in Ref. [45] along with a more finer description in Ref. [28]. To proceed further, the process consists of two distinct temperature regimes. At a higher temperature T in ∼ Λ CS , as soon as the symmetry breaks, the RH neutrinos become massive and Eq.III.2 starts acting without W ID which is negligible at this regime. In this gravitational leptogenesis scenario, typically, z in (= M 1 /T in ) can be constrained with so called weak field condition as z in ≥ M 1 /M P l , whereM P l is the reduced Planck constant. Once the asymmetry freezes out, at the lower temperatures, it faces a washout by the inverse decays which are strongly active at T ∼ M 1 . The final asymmetry is therefore of the form N where the dimensionless washout/decay parameter K 1 is a function of Yukawa couplings. Eq.III.3 that matches with the numerical solutions of the Eq.III.2 with quite a high accuracy, is the master equation which we use to present all the results. Prior to the explanation of Fig.1, let's introduce a parametrisation of the Yukawa matrix as m D = U mΩ M , where m D = f v with v = 174 GeV, U is the leptonic mixing matrix and Ω is a 3 × 3 complex orthogonal matrix with a standard parametrisation in terms of three complex rotation matrices [28] with complex angles θ ij = x ij +iy ij . In general, Ω is a completely 'free' matrix unless one invokes additional symmetries to fix the flavour structure of the theory. A plethora of works is dedicated in this direction [10]. With this orthogonal parametrisation it is easy to show that the equilibrium asymmetry is independent of U . Therefore, as far as the seesaw parameters are concerned, the light, heavy neutrino masses and the orthogonal matrix take part in the process. The decay parameter can also be expressed in terms of these parameters as K 1 = m −1 * ∑ k m k |Ω k1 | 2 with m * ≃ 10 −3 being the equilibrium neutrino mass [4]. In Fig.1, we show the variation of the produced asymmetry with the lightest neutrino mass for three benchmark values; M 1 = 10 6,7,8 GeV with a fixed orthogonal matrix and different values of z * . The basic nature of the curves is quite interesting. Let's focus on the z * = 10 3 curve (yellow) for M 1 = 10 8 GeV. It shows a plateau until m 1 ≃ 10 −2 eV, then an increase followed by a downfall at large m 1 values. First of all, for w = 1, the parameter κ ∼ z −1 * and therefore for large values of z * , the strength of the ∆L = 2 process becomes so weak that the asymmetry instantly freezes out without tracking the equilibrium number density. The coefficient f κ does not change much until m 1 ∼ 10 −2 eV and then increases for m 1 ≳ 10 −2 eV [28]. This increase in f κ pushes the asymmetry more towards the equilibrium and hence the overall magnitude of N B−L increases for m 1 ≳ 10 −2 eV. A downfall at large m 1 is caused by the exponential term in Eq.III.3. The washout is in fact modulated by two parameters, K 1 and z * . However, for large values of m 1 , the parameter K 1 becomes huge and therefore, even if one has a large z * , the frozen out asymmetry is completely washed out. On the other hand, when z * is small, e.g., z * = 10 2 , the overall magnitude of N B−L decreases since β ∼ z 3 * . In this case however, z −1 * suppression in κ is not that significant compared to the previous one. Until m 1 ∼ 10 −2 eV, it shows the constant behaviour due to the mentioned nature of f κ , however, at large m 1 values, it becomes strong enough to maintain the asymmetry in equilibrium for a period of time. The downfall is mostly dominated due to this equilibrium asymmetry tracking and not due to the late time washout. Note that for ω < 1/3, for a fixed value of z * , κ increases (causes delayed freeze out and hence dilution of the asymmetry N G0 B−L ) and β decreases (causes a decrease in N eq B−L ). A concrete example is a matter domination, i.e., ω = 0, where κ ∼ z * and β ∼ z −3/2 * . Moreover, these kind of scenarios are inclusive of a late time entropy production which dilutes the produced asymmetry significantly [30,35]. Therefore, ω < 1/3 scenarios are utterly inefficient. This possibly strengthens the claim that in the future, should the GW detectors find a flat and then a rising signal, RH neutrino induced gravitational leptogenesis with a stiffer equation of state is a natural mechanism to associate with, since both of them, successful leptogenesis and the GW signal, are triggered by common theoretical ingredients. In Fig.2, we show the future sensitivities of the GW detectors such as LISA [87], BBO [88], CE [89], ET [90] on the Gµ − T * plane. In the case of strong GW amplitudes, the most stringent constraint comes from the effective number of neutrino species which reads ∫ df f −1 Ω GW (f )h 2 < 5.6 × 10 −6 ∆N ef f . Considering ∆N ef f ≤ 0.2, the peak of the spectrum at f ∆ , and taking into account contributions from the infinite number of modes that give a factor of ζ(7/3) amplification , where we consider particle production from cusps [71]. The corresponding region has been shaded in red. We have ignored the variation of the effective relativistic degrees of freedom even when T * is below the QCD phase transition. Proper temperature dependence of the same, would include a factor of 1.5-3 modification. Since we are entirely onto the gravitational leptogenesis (to motivate ω ≠ 1/3), we take M max 1 ∼ 10 8 GeV so that the contribution from the decays are negligible. This gives an upper bound on the T in (Λ CS ) that corresponds to Gµ ≲ 10 −12 . Therefore, the mechanism can be tested with reasonably strong GW amplitudes even for the flat part (Eq.II.3). For strong amplitudes, the spectral breaks are likely to happen at high-frequency GW detectors like CE and ET plus the bump like signals (f * and f ∆ are close to each other) in general lie outside those detectors. In Fig.2, the black point represented by ♠ (♣), should (not) be a signal (see a supplementary Fig.3). We shall end the discussion with a 'Neutrino-Gravitational Waves Complementarity (NGWC)' or more generally, how this type of GW signal could be supplemented by low energy neutrino experiments. NGWC depends on the z * and flavour structure of the theory or more precisely, on the orthogonal matrix. From Fig.1, it can be seen that, depending on the RH neutrino mass (hence Gµ), various z * values are sensitive to the neutrinoless double beta decay experiments (the N B−L curves intersect with the N Obs B−L at the same time falls within the vertical green region). For the parameter set in Fig.1, the NGWC points fall unfortunately in the red region as well as they are well outside the GW detectors (showed by the ♡ and ♢ points in Fig.2). However, if one decreases the y ij , to produce correct N B−L , for a fixed value of Gµ one needs larger values of z * , meaning the NGWC points would move towards the left side, i.e., towards the smaller values of T * . The entire picture can be encapsulated within the triangle drawn on the test parameter space shaded in green in Fig.2. The red horizontal arm represents the constant Gµ line along which the entries of Ω decrease as one goes from larger to smaller T * . The yellow arm represents the constant z * line, as one goes along the line towards smaller Gµ values, entries of Ω increase and the . A fall at a high frequency is due to the particle production from cusps for l < l cric = µ −1/2 (ΓGµ) 2 [40,71]. We have shown the spectrum only for the fundamental mode. blue arm represents the constant (already predicted) orthogonal matrix and as one goes towards the higher values of Gµ, z * decreases or in other words, T * increases. The blue arm is of great interest. If one has a completely determined orthogonal matrix, from Fig.1 the NGWC points can be determined with the sets of M 1 and T * . This means the blue arm is a line of predictions from the GW experiments, i.e., we can predict at which amplitude and at which frequency the spectral break would occur. The triangle as a whole can be pushed towards the larger T * values increasing y ij . This implies, seesaw models which exhibit an orthogonal matrix with large imaginary part entries, would likely to show the spectral break at higher frequencies and therefore may not be tested with the planned detectors. These models are dubbed as 'boosted' seesaw models where the light neutrino basis vectors and heavy neutrino basis vectors are strongly misaligned [91]. On the other hand, models with flavour structures close to 'form dominance' [92] that typically predicts a real orthogonal matrix (Ω = P , where P is a permutation matrix), would show a spectral break within the frequency range of the current or planned GW detectors. Ω GW (t 0 , f ) = f ρ c dρ GW df = k Ω (k) GW (t 0 , f ). (III.4) The frequency derivative of ρ GW is given by dρ (k) GW df = t 0 t F a(t) a(t 0 ) 4 P GW (t, f k ) dF df dt, (III.5) where dF df = f a(t 0 ) a(t) the quantity P GW (t, f k ) represents the power emitted by the loops and is given by (see e.g., Ref. [68]) P GW (t, f k ) = Gµ 2 Γ k n(l,t)δ f k − 2k l dl. (III.6) Integrating Eq.III.6 over the loop lengths gives P GW (t, f k ) = 2kGµ 2 Γ k f 2 k n(t, f k ) = 2kGµ 2 Γ k f 2 a(t 0 ) a(t) 2 n t , 2k f a(t) a(t 0 ) . (III.7) From Eq.III.7 and Eq.III.5 one gets dρ (k) GW df = 2kGµ 2 Γ k f 2 t 0 t osc a(t) a(t 0 ) 5 n t , 2k f a(t) a(t 0 ) dt (III.8) and therefore the energy density corresponding to the mode 'k' is given by Ω (k) GW (t 0 , f ) = 2kGµ 2 Γ k f ρ c t 0 t osc a(t) a(t 0 ) 5 n t , 2k f a(t) a(t 0 ) dt. (III.9) Using the VOS equations and considering the loop production function as a delta function (see, e.g., Ref. [74]), it is easy to obtain the most general formula for the number density in an expanding background that scales as a ∼ t β . The expression is given by n(t, l k (t)) = A β α (α + ΓGµ) 3(1−β) l k (t) + ΓGµt 4−3βt 3β . (III.10) The Eq.III.9 can be expressed in the conventional form that are used in many papers (e.g., Ref. [39,40]) using the time dependence of the loop length which gives initial time t Now using Eq.III.11, the number density in Eq.III.10 can be re-expressed as n(t, l k (t)) = A β (t (k) i ) α(α + ΓGµ)t (k)4 i a(t k i ) a(t) 3 . (III. 12) time t * at which the most recent radiation domination begins, an approximate frequency up to which the spectrum shows a flat plateau is given by f * = 8 αΓGµ t −1/2 * t −2/3 0 t 1/6 eq ≃ 8z eq αΓGµ t eq t * 1/2 t −1 0 . (III.20) Similarly, using the critical length l cric = µ −1/2 (ΓGµ) 2 for cusp like structures, the second turning point frequency can be computed as f ∆ ≃ 9 α (Gµ) 5/4 M pl T * f * , (III.21) where we have assumed that f ∆ /f * ≃ t * /t ∆ (cf. Eq.III.21), and for simplicity we consider g * (T ) = g * ≃ 106 throughout. Therefore, for post QCD phase transition T ≲ 200 MeV, the formula is bit errorful. To observe both the frequencies distinctively, one should have f ∆ > f * . This gives the following restriction on the parameter space Gµ > T 4/5 * 2.88 × 10 −20 4/5 (III.22) which is shown by the red region in Fig.2. A4. The BBN limit: To be consistent with the the number of effective neutrino species, the GW energy density has to comply with f max f BBN df f Ω GW h 2 < 5.6 × 10 −6 ∆N ef f , (III.23) with ∆N ef f < 0.2. Considering the dominant contribution from the non-flat part after the first turning point frequency f * , the following constraint on the parameter space can be obtained On the other hand considering the analytic approach, i.e., using Velocity dependent One Scale (VOS) the same is obtained as n(t, l k (t)) = A r α (α + ΓGµ) 3/2 l k (t) + ΓGµt 5/2t 3/2 ≡ A r N α l k (t) + ΓGµt 5/2t 3/2 ,(III.26) where A r = 5.4. As mentioned before, the VOS model assumes all the loops are of same length at creation. However, at the moment of creation, the loops may follow a distribution depending on α. If so, the above formula should be modified as n(t, l k (t)) = A r ∫ w(α)N α dα l k (t) + ΓGµt 5/2t 3/2 . (III.27) Therefore, to the make VOS formula in Eq.III.26 consistent with the numerical result, one has to normalise Eq.III.26, i.e., n(t, l k (t)) = F α A r N α l k (t) + ΓGµt . (III.38) The most general Boltzmann equations (BEs) for leptogenesis with seesaw Lagrangian minimally coupled to gravity are dN N 1 dz = −D N N 1 − N eq N 1 , (III.39) dN B−L dz = −Dε 1 N N 1 − N eq N 1 − (W ∆L=2 + W ID ) N B−L − N eq B−L ,(III.40) where the first equation governs the production of RH neutrinos and the first term in the second equation represents the contribution to the lepton asymmetry from RH neutrino decays. Since we are neglecting the contribution from decays, only the second equation with the terms in 'bold' is relevant. Note that recently in Ref. [44], another curvature-induced evolution term that modulates of the asymmetry production dynamics at ultra-high temperatures has been introduced. We neglect that term in our computation. However, that will not change the qualitative features of our final results. To obtain a simpler form of the Boltzmann equation it is convenient to simplify the expression of the equilibrium asymmetry and the lepton number violating processes. Using the orthogonal parametrisation of the Dirac neutrino mass matrix m D = U mΩ M , the equilibrium asymmetry can be expressed as a power law in z as a [email protected] b [email protected] arXiv:2108.08359v2 [hep-ph] 20 Oct 2021 I. INTRODUCTION FIG. 1 . 1Eos: ω = 1. The yellow, red and green lines correspond to the lightest RH mass M 1 = 10 8,7,6GeV. For M 1 = 10 7,6 we do not show the lines corresponding to z * = 10 1 . We take M 3 = M 1 /z in GeV,M 2 = 10 −1 M 3GeV, x ij = π/4, y ij = 10 −1 and two mass-squared differences are at their best-fit values. ID (z)dz , where N G0B−L is the frozen out asymmetry after the system is done with ∆L = 2 interaction, and the exponential term represents a late-time washout by the inverse decays. A general solution of Eq.III.2 is complicated and depends on the properties of incomplete Gamma functions. However, for ω = 1, that corresponds to p = 1 and q = 8, a simpler solution can be obtained. FIG. 2 . 2Gµ vs. T * plot against the sensitivities of various GW detectors.compared to the fundamental mode, the BBN constraint translates to Gµ < T been shown by the blue exclusion region. On the other hand, to observe two spectral breaks (at f * and f ∆ ) distinctly, one should have f ∆ > f * which translates to the constraint Gµ > FIG. 3 . 3EOS: ω = 1. The curve in blue (red) is a valid (invalid) signal of leptogenesis. The curves are generated with Gµ = 10 −12 and T * = 10 −1 GeV (red) and T * = 10 2 GeV (blue) plateau, loop number density normalisation and the turning point frequencies A1. The standard expression: The normalised energy density parameter of gravitational waves at present time is expressed as k (t) + ΓGµt α + ΓGµ , (III.11) shown in the blue region inFig.2. The constraints in Eq.III.22 and Eq.III.24 are derived for α = 0.1.A5. Numerical simulation vs. VOS model loop number density and the normalisation:The number density obtained from numerical simulation is given by (see, Ref.[68]) (considering the loops created during radiation domination) zz = M 1 /T N B-L eq N B-L Obs z *~1 0 2 z *~1 0 3 m 1 = 6ev m 1 = 6ev m 1 = 200mev m 1 = 200mev m 1 = 20mev m 1 = M 1 /T FIG. 4. Top: EOS: ω = 1. Evolution of the gravitationally produced asymmetry for different values of m 1 . We have taken M 1 = 10 8 GeV, z * = 1.3 × 10 3 (red), 10 4 (green) as benchmark values. Bottom: EOS: ω = 1. Evolution of the gravitationally produced asymmetry for different values of m 1 . We have taken M 1 = 10 8 GeV, z * = 1.3 × 10 3 (red), 10 2 (green) as benchmark values. For both the plots we use M 3 = 10 14 GeV, M 2 = 10 12 GeV, x ij = π/4, y ij = 10 2 and z in = M 1 /M pl . The thick blue dashed line shows a match bewteen numerical solutions and the master equation obtained in Eq.III.53. The thin blue dashed lines show a match bewteen numerical solutions and solution obtained (without the late time N 1 -washout) in Eq.III.50. Acknowledgements: RS is supported by the MSCA-IF IV FZU -CZ.02.2.69/0.0/0.0/20 079/0017754 project and acknowledges European Structural and Investment Fund and the Czech Ministry of Education, Youth and Sports. RS acknowledges Graham M. Shore and Pasquale Di Bari for an useful discussion on gravitational leptogenesis and boosted seesaw models respectively, Kai Schmitz for a helpful chat on Ref.[29]and Sabir Ramazanov for discussions on cosmic strings in general.Putting the value of n(t, l k (t)) from Eq.III.12 into Eq.III.9, one gets the standard expression Ω (k)3dt, (III.13)where we have renamed A β as C eff .A2. The flat plateau: To obtain the GW spectrum from the loops that are produced and decay during the radiation domination, it is convenient to do the integration in Eq.III.9 with respect to the scale factor which reads Ω (k)wherein Eq.III.10 (in radiation domination) can also be expressed in terms of the scale factor asPutting Eq.III.16 in Eq.III.14 and after performing the integration one gets Ω (1)is the minimum frequency emitted by a given loop. Given the scaling solution of the loop production rate, which decreases with the fourth power in time, f ≃ f min is a reasonable assumption. Then, with t i ≪ t eq one has Ω (1)The expression for the flat plateau matches with Ref.[74]barring the factor ζ(δ) in the denominator. This is due to the fact that definition of the Ω (k)A3. The turning point frequencies: In the above, it is assumed that the dominant emission comes from the very earliest epoch of loop creation. Nonetheless, a precise value of the time can be calculated by maximizing the integral in Eq.III.14 with respect tot which giveswhere f is the frequency observed today which was emitted at timet M when the a given initial loop l i = αt i reached to the half of its size l i /2, i.e.,t M is eventually the half-life of the loop. IfAs one can see that for α = 0.1, Eq.III.25 and Eq.III.26 is consistent forA6. The spectral shape beyond the first turning point frequency: As mentioned previously in the main text, when the number of modes increases in the sum, the spectral behaviour beyond the turning point deviates from that of the fundamental mode (see e.g., Ref.[29,40]). The reason being the following:From Eq.III.13 it is evident thatNow to perform the sum one can expand the RHS of Eq.III.29 for some first few benchmark modes, i.e.,where the integers obey 1 < m < n < r. This suggests, if one keeps on increasing the mode numbers, there should be a critical value k ≡ k * for which the amplitude ΩGW (f * = f /k * ) contributes to the frequency f . Therefore, the sum can be split into two parts. The first one is from k = 1 up to k * for which the the amplitude at f receives contributions from the non-flat part and the second one is from k * to k max for which the test point receives contribution from the flat part, i.e.,The first term in Eq.III.32 gives the dominant contribution. In the large k * limit, the sum is thereforeTherefore, for an equation of states like kination, the spectral shape is quite similar to the k = 1 mode even after adding the contributions from the larger number of modes.AUX B: Evolution of the lepton asymmetry and derivation of the master equationThe energy density in a general equation of state red-shifts as ρ ω ∝ a −3(1+ω) . We assume there is no further entropy production after the instantaneous reheating. Therefore the scale factor is inversely proportional to the temperature, i.e., ρ ω ∝ T3(1+ω). Since the energy density of radiation and field φ ω should be equal at the critical temperature T * , the proportionality constant σ ω can then be obtained aswhere the energy density in radiation domination is given by ρ rad = σ rad T 4 ≡ (π 2 g * /30)T 4 . The total energy at an arbitrary temperature T is then given bywhere we define z * (z) = M 1 /T * (M 1 /T ). The modified Hubble parameter and theṘ in the general equation of state are then given by.(III.37)Given the Hubble parameter in Eq.III.36, the expression for the lepton number violating interactions W (z) ≡ Γ ∆L=2 /Hz can be generalised asAs mentioned earlier, at very high temperature the inverse decays are negligible. Therefore one can simply solve the BEto obtain the an expression for the frozen out asymmetry N (III.50)The washout by the inverse decays can be obtained using Eq.III.46 and integral properties of the Bessel function K n (z)(III.51)The washout factor comes out asTherefore the master formula for the final asymmetry that can be used for a numerical scan is given by which very accurately reproduces the numerical result as shown inFig.4with the phrase "FULL ANALYTICAL". . M Fukugita, T Yanagida, Phys. Lett. B. 17445M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986). . A Riotto, M Trodden, Ann. Rev. Nucl. Part. Sci. 4935A. Riotto and M. Trodden, Ann. Rev. Nucl. Part. Sci. 49, 35 (1999). . A Pilaftsis, T E J Underwood, Nucl. Phys. B. 692303A. Pilaftsis and T. E. J. Underwood, Nucl. Phys. B 692, 303 (2004). . W Buchmuller, P Di Bari, M Plumacher, Annals Phys. 315305W. Buchmuller, P. Di Bari and M. Plumacher, Annals Phys. 315, 305 (2005). . S Davidson, E Nardi, Y Nir, Phys. Rept. 466105S. Davidson, E. Nardi and Y. Nir, Phys. Rept. 466, 105 (2008). . D Bodeker, W Buchmuller, arXiv:2009.07294hep-phD. Bodeker and W. Buchmuller, arXiv:2009.07294 [hep-ph]. . P , Di Bari, arXiv:2107.13750hep-phP. Di Bari, [arXiv:2107.13750 [hep-ph]]. . Y Akrami, Planck CollaborationAstron. Astrophys. 64110Y. Akrami et al. [Planck Collaboration], Astron. Astrophys. 641, A10 (2020). . V A Kuzmin, V A Rubakov, M E Shaposhnikov, Phys. Lett. B. 15536V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 155, 36 (1985). . G Altarelli, F Feruglio, Rev. Mod. Phys. 82G. Altarelli and F. Feruglio, Rev. Mod. Phys. 82, 2701-2729 (2010). . S Davidson, A Ibarra, Phys. Lett. B. 535S. Davidson and A. Ibarra, Phys. Lett. B 535, 25-32 (2002). . E K Akhmedov, V A Rubakov, A Y Smirnov, Phys. Rev. Lett. 811359E. K. Akhmedov, V. A. Rubakov and A. Y. Smirnov, Phys. Rev. Lett. 81, 1359 (1998). . A Pilaftsis, T E J Underwood, Nucl. Phys. B. 692A. Pilaftsis and T. E. J. Underwood, Nucl. Phys. B 692, 303-345 (2004). . T Hambye, D Teresi, Phys. Rev. Lett. 117991801T. Hambye and D. Teresi, Phys. Rev. Lett. 117, no. 9, 091801 (2016). . P S Dev, P Millington, A Pilaftsis, D Teresi, Nucl. Phys. B. 886569P. S. Bhupal Dev, P. Millington, A. Pilaftsis and D. Teresi, Nucl. Phys. B 886 (2014) 569. . B P Abbott, LIGO Scientific and Virgo CollaborationsPhys. Rev. Lett. 116661102B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. Lett. 116, no. 6, 061102 (2016). . B P Abbott, LIGO Scientific and Virgo CollaborationsPhys. Rev. Lett. 11624241103B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. Lett. 116, no. 24, 241103 (2016). . T W B Kibble, J. Phys. A. 91387T. W. B. Kibble, J. Phys. A 9, 1387 (1976). . R Jeannerot, J Rocher, M Sakellariadou, Phys. Rev. D. 68103514R. Jeannerot, J. Rocher and M. Sakellariadou, Phys. Rev. D 68, 103514 (2003). . H B Nielsen, P Olesen, Nucl. Phys. B. 6145H. B. Nielsen and P. Olesen, Nucl. Phys. B 61, 45 (1973). . C Ringeval, M Sakellariadou, F Bouchet, JCAP. 070223C. Ringeval, M. Sakellariadou and F. Bouchet, JCAP 0702, 023 (2007). . J J Blanco-Pillado, K D Olum, B Shlaer, Phys. Rev. D. 8383514J. J. Blanco-Pillado, K. D. Olum and B. Shlaer, Phys. Rev. D 83, 083514 (2011). . A Davidson, Phys. Rev. D. 20776A. Davidson, Phys. Rev. D 20, 776 (1979). . R E Marshak, R N Mohapatra, Phys. Lett. 91222R. E. Marshak and R. N. Mohapatra, Phys. Lett. 91B, 222 (1980). . R N Mohapatra, R E Marshak, Phys. Rev. Lett. 441643Phys. Rev. Lett.R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett. 44, 1316 (1980) Erratum: [Phys. Rev. Lett. 44, 1643 (1980)]. . I Esteban, M C Gonzalez-Garcia, M Maltoni, T Schwetz, A Zhou, JHEP. 09178I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, JHEP 09, 178 (2020). . J A Dror, T Hiramatsu, K Kohri, H Murayama, G White, Phys. Rev. Lett. 124441804J. A. Dror, T. Hiramatsu, K. Kohri, H. Murayama and G. White, Phys. Rev. Lett. 124, no. 4, 041804 (2020). . R Samanta, S Datta, JHEP. 05211R. Samanta and S. Datta, JHEP 05, 211 (2021). . S Blasi, V Brdar, K Schmitz, Phys. Rev. Res. 2443321S. Blasi, V. Brdar and K. Schmitz, Phys. Rev. Res. 2, no.4, 043321 (2020). . S Datta, A Ghosal, R Samanta, arXiv:2012.14981hep-phS. Datta, A. Ghosal and R. Samanta, [arXiv:2012.14981 [hep-ph]]. . R H Cyburt, B D Fields, K A Olive, T H Yeh, Rev. Mod. Phys. 8815004R. H. Cyburt, B. D. Fields, K. A. Olive and T. H. Yeh, Rev. Mod. Phys. 88, 015004 (2016) . M Kawasaki, K Kohri, N Sugiyama, Phys. Rev. Lett. 824168M. Kawasaki, K. Kohri and N. Sugiyama, Phys. Rev. Lett. 82, 4168 (1999). . T Hasegawa, N Hiroshima, K Kohri, R S L Hansen, T Tram, S Hannestad, JCAP. 1212T. Hasegawa, N. Hiroshima, K. Kohri, R. S. L. Hansen, T. Tram and S. Hannestad, JCAP 12, 012 (2019). . M S Turner, Phys. Rev. D. 281243M. S. Turner, Phys. Rev. D 28, 1243 (1983) . K Hamaguchi, H Murayama, T Yanagida, Phys. Rev. D. 6543512K. Hamaguchi, H. Murayama and T. Yanagida, Phys. Rev. D 65, 043512 (2002) . L H Ford, Phys. Rev. D. 352955L. H. Ford, Phys. Rev. D 35, 2955 (1987). . P J E Peebles, A Vilenkin, Phys. Rev. D. 5963505P. J. E. Peebles and A. Vilenkin, Phys. Rev. D 59, 063505 (1999). . Y Cui, M Lewicki, D E Morrissey, J D Wells, Phys. Rev. D. 9712123505Y. Cui, M. Lewicki, D. E. Morrissey and J. D. Wells, Phys. Rev. D 97, no.12, 123505 (2018). . Y Cui, M Lewicki, D E Morrissey, J D Wells, JHEP. 190181Y. Cui, M. Lewicki, D. E. Morrissey and J. D. Wells, JHEP 1901, 081 (2019). . Y Gouttenoire, G Servant, P Simakachorn, JCAP. 0732Y. Gouttenoire, G. Servant and P. Simakachorn, JCAP 07, 032 (2020). . Y Cui, M Lewicki, D E Morrissey, Phys. Rev. Lett. 12521211302Y. Cui, M. Lewicki and D. E. Morrissey, Phys. Rev. Lett. 125, no.21, 211302 (2020). . J I Mcdonald, G M Shore, Phys. Lett. B. 751J. I. McDonald and G. M. Shore, Phys. Lett. B 751, 469-473 (2015) . J I Mcdonald, G M Shore, JHEP. 0430J. I. McDonald and G. M. Shore, JHEP 04, 030 (2016). . J I Mcdonald, G M Shore, JHEP. 1025J. I. McDonald and G. M. Shore, JHEP 10, 025 (2020). . R Samanta, S Datta, JHEP. 1267R. Samanta and S. Datta, JHEP 12, 067 (2020) . G M Shore, arXiv:2106.09562hep-phG. M. Shore, [arXiv:2106.09562 [hep-ph]]. . A G Cohen, D B Kaplan, Phys. Lett. B. 199A. G. Cohen and D. B. Kaplan, Phys. Lett. B 199, 251-258 (1987). . H Davoudiasl, R Kitano, G D Kribs, H Murayama, P J Steinhardt, Phys. Rev. Lett. 93201301H. Davoudiasl, R. Kitano, G. D. Kribs, H. Murayama and P. J. Steinhardt, Phys. Rev. Lett. 93, 201301 (2004). . S Mohanty, A R Prasanna, G Lambiase, Phys. Rev. Lett. 9671302S. Mohanty, A. R. Prasanna and G. Lambiase, Phys. Rev. Lett. 96, 071302 (2006) . G Lambiase, S Mohanty, A R Prasanna, Int. J. Mod. Phys. D. 221330030G. Lambiase, S. Mohanty and A. R. Prasanna, Int. J. Mod. Phys. D 22, 1330030 (2013). . K Saikawa, S Shirai, JCAP. 0535K. Saikawa and S. Shirai, JCAP 05, 035 (2018). . E Witten, Phys. Lett. B. 153E. Witten, Phys. Lett. B 153, 243-246 (1985). . G Dvali, A Vilenkin, JCAP. 0310G. Dvali and A. Vilenkin, JCAP 03, 010 (2004). . M B Hindmarsh, T W B Kibble, Rept. Prog. Phys. 58M. B. Hindmarsh and T. W. B. Kibble, Rept. Prog. Phys. 58, 477-562 (1995). A Vilenkin, E P S Shellard, Cosmic Strings and Other Topological Defects. Cambridge University PressA. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press,2000). . C F Chang, Y Cui, arXiv:2106.09746hep-phC. F. Chang and Y. Cui, [arXiv:2106.09746 [hep-ph]]. . W T Emond, S Ramazanov, R Samanta, arXiv:2108.05377hep-phW. T. Emond, S. Ramazanov and R. Samanta, [arXiv:2108.05377 [hep-ph]]. . E P S Shellard, Nucl. Phys. B. 283E. P. S. Shellard, Nucl. Phys. B 283, 624-656 (1987) . E J Copeland, T W B Kibble, D A Steer, Phys. Rev. D. 7565024E. J. Copeland, T. W. B. Kibble and D. A. Steer, Phys. Rev. D 75, 065024 (2007). . A Vilenkin, Phys. Rev. D. 43A. Vilenkin, Phys. Rev. D 43, 1060-1062 (1991). . A Vilenkin, Phys. Lett. B. 107A. Vilenkin, Phys. Lett. B 107, 47-50 (1981). . N Turok, Nucl. Phys. B. 242N. Turok, Nucl. Phys. B 242, 520-541 (1984) . T Vachaspati, A Vilenkin, Phys. Rev. D. 313052T. Vachaspati and A. Vilenkin, Phys. Rev. D 31, 3052 (1985) . D P Bennett, F R Bouchet, Phys. Rev. Lett. 60257D. P. Bennett and F. R. Bouchet, Phys. Rev. Lett. 60, 257 (1988). . D P Bennett, F R Bouchet, Phys. Rev. Lett. 632776D. P. Bennett and F. R. Bouchet, Phys. Rev. Lett. 63, 2776 (1989). . A Albrecht, N Turok, Phys. Rev. D. 40A. Albrecht and N. Turok, Phys. Rev. D 40, 973-1001 (1989). . J J Blanco-Pillado, K D Olum, B Shlaer, Phys. Rev. D. 89223512J. J. Blanco-Pillado, K. D. Olum and B. Shlaer, Phys. Rev. D 89, no.2, 023512 (2014). . J J Blanco-Pillado, K D Olum, Phys. Rev. D. 9610104046J. J. Blanco-Pillado and K. D. Olum, Phys. Rev. D 96, no.10, 104046 (2017). . T Damour, A Vilenkin, Phys. Rev. D. 6464008T. Damour and A. Vilenkin, Phys. Rev. D 64, 064008 (2001) . D Matsunami, L Pogosian, A Saurabh, T Vachaspati, Phys. Rev. Lett. 12220201301D. Matsunami, L. Pogosian, A. Saurabh and T. Vachaspati, Phys. Rev. Lett. 122, no.20, 201301 (2019). . P Auclair, D A Steer, T Vachaspati, Phys. Rev. D. 101883511P. Auclair, D. A. Steer and T. Vachaspati, Phys. Rev. D 101, no.8, 083511 (2020). . C J A P Martins, E P S Shellard, Phys. Rev. D. 54C. J. A. P. Martins and E. P. S. Shellard, Phys. Rev. D 54, 2535-2556 (1996). . C J A P Martins, E P S Shellard, Phys. Rev. D. 6543514C. J. A. P. Martins and E. P. S. Shellard, Phys. Rev. D 65, 043514 (2002). . P Auclair, J J Blanco-Pillado, D G Figueroa, A C Jenkins, M Lewicki, M Sakellariadou, S Sanidas, L Sousa, D A Steer, J M Wachter, JCAP. 0434P. Auclair, J. J. Blanco-Pillado, D. G. Figueroa, A. C. Jenkins, M. Lewicki, M. Sakellariadou, S. Sanidas, L. Sousa, D. A. Steer and J. M. Wachter, et al. JCAP 04, 034 (2020). . W Buchmuller, V Domcke, H Murayama, K Schmitz, Phys. Lett. B. 809135764W. Buchmuller, V. Domcke, H. Murayama and K. Schmitz, Phys. Lett. B 809, 135764 (2020). . S F King, S Pascoli, J Turner, Y L Zhou, Phys. Rev. Lett. 126221802S. F. King, S. Pascoli, J. Turner and Y. L. Zhou, Phys. Rev. Lett. 126, no.2, 021802 (2021). . B Fornal, B Shams Es, Haghi, Phys. Rev. D. 10211115037B. Fornal and B. Shams Es Haghi, Phys. Rev. D 102, no.11, 115037 (2020). . W Buchmuller, JHEP. 04168W. Buchmuller, JHEP 04, 168 (2021). . W Buchmuller, V Domcke, K Schmitz, arXiv:2107.04578hep-phW. Buchmuller, V. Domcke and K. Schmitz, [arXiv:2107.04578 [hep-ph]]. . M A Masoud, M U Rehman, Q Shafi, arXiv:2107.09689hep-phM. A. Masoud, M. U. Rehman and Q. Shafi, [arXiv:2107.09689 [hep-ph]]. . L Bian, X Liu, K P Xie, arXiv:2107.13112hep-phL. Bian, X. Liu and K. P. Xie, [arXiv:2107.13112 [hep-ph]]. . Z Arzoumanian, NANOGravAstrophys. J. Lett. 905234Z. Arzoumanian et al. [NANOGrav], Astrophys. J. Lett. 905, no.2, L34 (2020). . J Ellis, M Lewicki, Phys. Rev. Lett. 126441304J. Ellis and M. Lewicki, Phys. Rev. Lett. 126, no.4, 041304 (2021). . S Blasi, V Brdar, K Schmitz, Phys. Rev. Lett. 126441305S. Blasi, V. Brdar and K. Schmitz, Phys. Rev. Lett. 126, no.4, 041305 (2021). . B Goncharov, R M Shannon, D J Reardon, G Hobbs, A Zic, M Bailes, M Curylo, S Dai, M Kerr, M E Lower, arXiv:2107.12112astro-ph.HEB. Goncharov, R. M. Shannon, D. J. Reardon, G. Hobbs, A. Zic, M. Bailes, M. Curylo, S. Dai, M. Kerr and M. E. Lower, et al. [arXiv:2107.12112 [astro-ph.HE]]. . W Buchmuller, P Di Bari, M Plumacher, Nucl. Phys. B. 643362Nucl. Phys. BW. Buchmuller, P. Di Bari and M. Plumacher, Nucl. Phys. B 643, 367-390 (2002) [erratum: Nucl. Phys. B 793, 362 (2008)]. . P Amaro-Seoane, LISAarXiv:1702.00786[astro-ph.IMP. Amaro-Seoane et al. [LISA], [arXiv:1702.00786 [astro-ph.IM]]. . V Corbin, N J Cornish, Class. Quant. Grav. 23V. Corbin and N. J. Cornish, Class. Quant. Grav. 23, 2435-2446 (2006). . B P Abbott, LIGO ScientificClass. Quant. Grav. 34444001B. P. Abbott et al. [LIGO Scientific], Class. Quant. Grav. 34, no.4, 044001 (2017) . B Sathyaprakash, M Abernathy, F Acernese, P Ajith, B Allen, P Amaro-Seoane, N Andersson, S Aoudia, K Arun, P Astone, Class. Quant. Grav. 2979501Class. Quant. Grav.B. Sathyaprakash, M. Abernathy, F. Acernese, P. Ajith, B. Allen, P. Amaro-Seoane, N. Andersson, S. Aoudia, K. Arun and P. Astone, et al. Class. Quant. Grav. 29, 124013 (2012) [erratum: Class. Quant. Grav. 30, 079501 (2013)]. . P Di Bari, M Fiorentin, R Samanta, JHEP. 0511P. Di Bari, M. Re Fiorentin and R. Samanta, JHEP 05, 011 (2019). . M C Chen, S F King, JHEP. 0672M. C. Chen and S. F. King, JHEP 06, 072 (2009).
[]
[ "Green Vehicle Routing Problem: State of the Art and Future Directions", "Green Vehicle Routing Problem: State of the Art and Future Directions" ]
[ "Saba Sabet \nLaboratory of Innovations in Transportation (LiTrans)\nRyerson University\n\n", "Bilal Farooq \nLaboratory of Innovations in Transportation (LiTrans)\nRyerson University\n\n" ]
[ "Laboratory of Innovations in Transportation (LiTrans)\nRyerson University\n", "Laboratory of Innovations in Transportation (LiTrans)\nRyerson University\n" ]
[]
Green vehicle routing problem (GVRP) aims to consider greenhouse gas emissions reduction, while routing the vehicles. It can be either through adopting Alternative Fuel Vehicles (AFVs) or with existing conventional fossil fuel vehicles in fleets. GVRP also takes into account environmental sustainability in transportation and logistics. We critically review several variations and specializations of GVRP to address issues related to charging, pickup, delivery, and energy consumption. Starting with the concepts and definitions of GVRP, we summarize the key elements and contributors to GVRP publications. Afterward, the issues regarding each category of green vehicle routing are reviewed, based on which key future research directions and challenges are suggested. It was observed that the main focus of previous publications is on the operational level routing decision and not the supply chain issues. The majority of publications used metaheuristic methods, while overlooking the emerging machine learning methods. We envision that in addition to machine learning, reinforcement learning, distributed systems, internet of vehicles (IoV), and new fuel technologies have a strong role in developing the GVRP research further.
10.1109/access.2022.3208899
[ "https://arxiv.org/pdf/2202.01695v1.pdf" ]
246,485,384
2202.01695
a8cf575c9d50077113ce09fc5b0ecf1fd264e60c
Green Vehicle Routing Problem: State of the Art and Future Directions Saba Sabet Laboratory of Innovations in Transportation (LiTrans) Ryerson University Bilal Farooq Laboratory of Innovations in Transportation (LiTrans) Ryerson University Green Vehicle Routing Problem: State of the Art and Future Directions Green vehicle routing problemvehicle routing problemalternative fuel VRPliterature review Green vehicle routing problem (GVRP) aims to consider greenhouse gas emissions reduction, while routing the vehicles. It can be either through adopting Alternative Fuel Vehicles (AFVs) or with existing conventional fossil fuel vehicles in fleets. GVRP also takes into account environmental sustainability in transportation and logistics. We critically review several variations and specializations of GVRP to address issues related to charging, pickup, delivery, and energy consumption. Starting with the concepts and definitions of GVRP, we summarize the key elements and contributors to GVRP publications. Afterward, the issues regarding each category of green vehicle routing are reviewed, based on which key future research directions and challenges are suggested. It was observed that the main focus of previous publications is on the operational level routing decision and not the supply chain issues. The majority of publications used metaheuristic methods, while overlooking the emerging machine learning methods. We envision that in addition to machine learning, reinforcement learning, distributed systems, internet of vehicles (IoV), and new fuel technologies have a strong role in developing the GVRP research further. Introduction The Vehicle Routing Problem (VRP) is used to design optimal routes for a fleet of vehicles to service a set of customers, considering a certain set of constraints. VRP was first introduced by Dantzig and Ramser [1] in their seminal work on truck dispatching. In that study, the first algorithmic approach was proposed and applied to optimize fuel deliveries. According to the study, VRP aims to find optimal routes for a fleet of delivery trucks, each with limited capacity, so as to minimize the total distance travelled. There can be one or more depots and customer nodes in a vehicle routing network. Green Vehicle Routing Problem (GVRP) is a branch of green logistics, which refers to vehicle routing problems where externalities of using vehicles, such as carbon dioxideequivalents emissions, are to be reduced. In this way, although research in this field has a long history, Erdogan and Miller-Hooks [2] formally introduced the term GVRP for the first time. GVRP incorporates the environmental aspects of transportation into VRP, which is one of the most interesting problems in the field of logistics and transportation. The goal of this problem is to earn economic benefits, while also taking into account environmental considerations. In its most general form, GVRP aims to minimize GHG emissions with solely conventional gasoline and diesel vehicles (GDVs) or with Alternative Fuel Vehicles (AFVs) in the fleets. It also takes into account the environmental sustainability in freight transportation. The aim of this study is to provide a systematic state of the art and outline new insights and perspectives into GVRP, based on a wide range of relevant searches by answering the main review questions below: 1. What are the main variants of GVRP developed to incorporate the environmental goals in the VRP field? 2. What are the strategies to address environmental issues in GVRP? 3. What multiple objectives associated with the GVRP variants should be considered? 4. What are the emerging issues and future trends in GVRP? For the existing literature on GVRP, the availability of extensive resources from reputed journals, books, technical reports, surveys and conference proceedings helped the present study. Specifically, we collected the relevant reserach articles from academic databases, including Google Scholar, Scopus, Springer, Taylor & Francis Elsevier, ScienceDirect, Wiley, and IEEEXplore. To conduct the search, we identified keywords such as green vehicle routing, green logistics, electric vehicle routing,alternative fuel vehicle routing, and chargingdischarging scheduling of electric vehicles. The overall structure of the study is as follows: next section presents GVRP research background, Section 3 presents GVRP Algorithms, Section 4 presents GVRP classification, Section 5 presents new insights into GVRP, and the final section presents a detailed conclusion and future research direction. Analysis of Existing Literature The aim of this study is to provide a systematic literature review based on a wide range of relevant searches. The flowchart in Figure 1 refers to the review methodology applied in this article. The structure in the figure illustrates the current study in several steps. First, we determined the basis for the review work; retrieving and refining the selected papers and organizing the outline of the review paper. In the next step, we provided descriptive analysis using quantitative figures. Then, we used software and apply scientometric analysis on the refined papers to visualize the clustering of keywords used in the literature. In the next step, we focused on the fundamental aspects of GVRP, extracting various types of trends and the gap existing in the current GVRP research. In the end, the conclusion is described, as well as providing opportunities for future research on GVRP. Scientometric Analysis of GVPR Literature Considering the importance of mitigating GHG emission, many studies have looked into green vehicle routing during the last decades using different terms such as Pollution Routing Problem (PRP), Eco-Routing and GVRP. Thus, it would be complicated to pinpoint a particular period in time when publications of green vehicle routing officially started. The earliest publications of this domain, as recorded by the web of science, date back to 2012 [2]. Since then, publications of GVRP have constantly been the center of attention with an overall number of more than 450 publications over the years. During the last five years leading to the time of the current publication, more than 400 publications on GVRP have been indexed each year, 100 items of which associated with 2019. In other words, the size of this literature has risen sharply within the last five years. See Figure 2 for a visualisation of this trend in GVRP literature. Contributing authors in GVRP publications were mainly from the countries shown in Figure 4. The U.S started investigating GVRP earlier than any other country as most early publications are from USA. China, as the first rank, has improved in GVRP research in recent years. Furthermore, five major European countries (Germany, England, Spain, Italy and France) have engaged considerably in the improvement of GVRP field. Iran, Canada and India have a high rate of publication, playing an influential role in developing GVRP solutions in the recent years. Probably, the high rate of publication in China, Europe and the U.S is associated with their share of the AFV market. As of December 2020, China had the largest stock of EVS, with 42% of the global plug-in passenger EV fleet in use. China also dominates the plug-in light commercial electric vehicle and electric bus deployment, with 65% of the global commercial EV fleet [3]. Europe had about 3 million plug-in passenger EVs by the end of 2020, accounting for 30% of the global stock [4]. It also has the secondlargest electric light commercial vehicle fleet, with about 31% of the global stock in 2019 [3]. As of November 2021, the U.S has the third-largest share of the EV market, after China and Europe [5]. The terms used in the title, abstract and keyword lists of the green vehicle routing problem could be useful semantic lenses through the structure and composition of this literature and theme and nature of its studies. They could reflect the methods used in GVRP literature, the range of applications to which GVRP have been extended over various periods of time. Here, networks of keyword co-occurrence are analysed for GVRP publications as well as bursts of citations to keywords (or more specifically, to the articles where such keywords have been used). Two terms (which could be keywords or terms identified in the title or abstract) are considered co-occurred when they appear in the (keyword list or title/abstract of) the same publication. A map of term co-occurrence identifies those that have occurred most frequently and visualises those that have frequently co-occurred in closer spatial proximity [6]. Groups of terms that are highly related (i.e. those with strong cooccurrence relationship) form clusters of terms that could represent various sectors of the literature. A map of co-occurrence could be visualised in various formats including Figure 5, where the frequency of occurrence is presented with the years they have been appealing to scholars the most. This map indicates that the terms green logistics, vehicle routing problem and fuel consumption are amongst the recently occurred keywords in the publications. GA, ALNS, and TS were the most frequent algorithms in the GVRP literature. GVRP Algorithms In the literature, algorithms have been developed to resolve variants of Vehicle routing problems. The objective function of a VRP is defined based on the particular purpose of the study. Depending on applications, various types of GVRP are introduced, formulated and solved by exact or approximate methodologies. The main limitation of GVRP's proposed algorithms is generality. Specific heuristic and metaheuristic methods presented in one paper and one case study do not guarantee the effectiveness of solving other types of GVRP problems ; Therefore, in order to solve other variant problems more efficiently, more general methodologies are to be adopted. Exact methods are currently applied in order to find an optimal solution for a few customers with limited capacity and fixed time windows [7,8]. Popular exact procedures include direct tree search, dynamic programming, integer linear programming, etc. Although simple variants of real problems can be modelled with graphs naturally, even some simple variants of vehicle routing problems are considered to be NP-hard due to the size and frequency of real world VRPs in networks. So, the size of problems that can be solved, optimally, using mathematical programming or combinatorial optimization may be limited. Therefore, several recent studies have turned to approximate algorithms, evaluating their efficiencies in solving vehicle routing problems. Approximate algorithms can be heuristic, meta-heuristic, hybrid heuristic or machine learning methods. Heuristic methods are used to resolve the vehicle routing problem according to the specific knowledge of the problem, which is in most cases suboptimal or close enough to a reliable solution [9]. In GVRP literature, heuristic methods can be split into constructive and improvement heuristics. Constructive heuristics seek to propose an initial solution by providing either serial or parallel route construction [10]. Such solutions are constructed in a greedy way, which usually generates solutions slightly far from an optimal solution of the VRP. In this regard, the modified savings method is used to provide an initial solution of several types of GVRP, and especially Electric-VRP with the insertion of charging stations. Traditional local search algorithms usually evaluate the whole neighborhood, but only perform one single move at each step. However, there are often many neighborhood moves in the current neighborhood that are independent of each other and can be simultaneously performed without interference [11]. The local search stops when no improvement in the solution can be noticed in the neighborhood of the incumbent solution, also named as local optima. Metaheuristic methods can be defined as heuristics guiding other heuristics. These methods are either neighborhood-oriented (local) metaheuristics or population metaheuristics. Neighborhood-oriented heuristics keep exploring the neighborhood of the optimal solution. Simulated Annealing (SA), Tabu Search (TS), Variable Neighborhood Search (VNS), and Adaptive Large Neighborhood Search (ALNS) are among popular local metaheuristic methods [12]. Population metaheuristic methods are based on the natural selection procedure to evolve a population and let the fittest survive. Among them, Genetic Algorithms (GA), Ant Colony (AC), Bee Colony (BC), and Particle Swarm Optimization (PSO) are often used [13]. Hybrid metaheuristic methods take advantage of the meta-heuristic procedures to keep searching even after reaching the first local optima. In some cases, several metaheuristic and heuristic methods are combined and applied to a vehicle routing problem since using a specific approach leads to difficulties, such as a low-quality solution, trapping in local optima in search space, or high computation time. Therefore, several studies hybridize two or more algorithms to simultaneously employ strengths. Hybrid methods include exact-metaheuristic [14], metaheuristic-metaheuristic [15] and metaheuristic-heuristic [16] algorithms in order to obtain better results and add to the robustness of the solution. Moreover, several studies have attempted to solve GVRP with general exact solvers, such as CPLEX, Lingo and GAMS [17]. It should be noted that despite the development of exact methods, very few exact methods have been proposed for the EVRP and its extensions, which is a branch of GVRP. Exact methods are found to be inadequate to solve a large-scale optimization problem [18]. While researchers have used a population metaheuristic to solve the problem, only a few studies were able to generate high-quality solutions in a reasonable computational time. Most importantly, the use of emerging machine learning and data mining tools has been overlooked in the literature of GVRP algorithms. Figure 6 refers to the overall percentage usage of each type of method in GVRP literature. It can be inferred that metaheuristic algorithms are increasingly recognized as a significant option, the most applicable methodology, and are currently receiving global attention from scholars and practitioners. Additionally, the small contribution of heuristic methods (about 7%) shows that most scholars tend to combine them into other methods due to local optima deficiency. This usually relates to these methodologies by concentrating on solving a specific problem. GVRP Classification We can divide GVRP literature into three main categories: (1) GVRP with conventional vehicles (2) GVRP with alternative fuel vehicles, and (3) GVRP with mixed fleet of vehicles. We also identified several variants as subcategories for the GVRP with Conventional Vehicles (CV) and Alternative Fuel Vehicles (AFV). Conventional Fossil Fuel-Powered Vehicles Routing The CO2 and NOx emissions problem has negative impact on the environment as well as on human health. In CVRP, any type of emission is considered in the objective function, with the main focus on minimizing the routing cost and polluting emissions. Several variants of GVRP with conventional vehicles have been explored by scholars in recent years. Figure 8 is associated with different GVRP with CV variants and their origination year in the literature. The figure reveals that while issues such as time dependency, refueling location and eco-driving have been investigated since the beginning of last decade, emerging issues like multi-objective, connectivity and especially automation has been overlooked and only in the last 2-3 years have the researchers found interest in examining GVRP considering connected and automated vehicles. Multi-Objective Conventional GVRP In this type of CVRP, more than 1 objective is taken into account. First introduced by Demir et al. [9], this variant aims to minimize both route cost and fuel consumption or vehicle kilometers travelled (VKT), or travel time or speed. In their study, they solved a bi-objective pollution routing problem (PRP), minimizing two conflicting factors: fuel consumption and driver time. Several scholars have recently developed Demir's methodology for solving PRP such as [19,20]. Poonthalir and Nadarajan [19] resolved a multi-objective problem with varying speed constraints. Their model was able to minimize both route cost and fuel consumption, using particle swarm optimization with a new mutation operator called greedy mutation operator. Rauniyar et al. [20] developed a reliable solution methodology based on genetic algorithm to solve a bi-objective C-GVRP defined by Demir et al. Some studies have investigated multi-objectives like minimizing marginal cost and fuel consumption, VKT and travel time, combined with other variants. Alfaseeh et al. [21] and Djavadian et al. [22] incorporated connectivity and automation to their multi-objective Eco-routing study considering two different routing strategies as myopic and anticipatory routing. They showed that the anticipatory routing strategies outperform myopic ones due to the consideration of future traffic state in their routing calculations. Refueling Locations In several papers such as Yi et al. [23], constraints like road conditions, congestion, topography, vehicle load and their impact on route cost and fuel consumption have been included. Rezaei et al. [24] examined green vehicle routing problem with time windows constraint considering a heterogeneous fleet of vehicles and fuel stations. However, their methodology can be only applied to products with hard time windows such as like milk, meat, and newspaper. Heterogeneous Fleet This variant is associated with a fleet of vehicles with different capacities and costs, available for distribution activities. The problem is also known as the Mixed Fleet VRP or as the Heterogeneous Fleet VRP. Koc et al. [25] first modelled and solved a GVRP variant with a heterogeneous fleet. They applied a hybrid evolutionary metaheuristic method to solve the problem and concluded that in an urban area, the advantages of having a heterogeneous fleet outweigh those of a homogeneous one. Time Dependency Sometimes, GHG emission varies over time. Thus, emission can be a function of time because of travel speed variability creating time dependency in GVRP. This variant deals with the difficulties brought by the various factors of travel speed variability, such as the travel speed, congestion, and land use affecting CO2 emissions [26]. Also, VRP with Time Windows (VRPTW) is to deliver the goods with time constraints and limited capacity of the vehicle fleet. This variant includes both soft/hard Time Windows and a combination of both. The complexity is that there are various uncertainties in this variant due to unexpected occurrences. Also, the algorithms used in the previous studies, were mostly applied to products that hard time windows are suitable for them such as meat, and newspaper. Franceschetti et al. [27] directly considered the effect of traffic congestion into conventional GVRP with hard time windows. Jabali et al. [28] investigated the two phases of free-flow traffic and congestion. They minimized the emissions per kilometer as a function of speed, developing a relationship between the reduction of emissions and marginal costs. Pickup and Delivery This is a more general issue in green logistics. The aim of this variant is to minimize operational delivery costs and the environmental impacts at the same time. In such cases, there are several issues to be considered. Some of them are: locating depots where vehicle with limited capacity is sent to deliver orders, minimizing emissions by scheduling customers. Tajik et al. [29] showed that considering fixed cost, the vehicle routes would better be composed of large vehicles, despite their higher emissions. Automation and Connectivity These variants are quite novel in the literature of GVRP. Issues regarding connectivity and Automation were only discussed by very few scholars such as [21,22]. They aimed to minimize travel time, GHG and NOx emissions with different costing approaches and routing strategies. They asserted that as vehicular ad hoc network (VANET) impact on ecorouting is under the direct influence of ITS application, its sensitivity to the availability, robustness, accuracy, and temporal and spatial distribution of the network data should be further investigated. Furthermore, the authors believed that routing based on GHG as the objective offers a considerable reduction in average TT, average VKT, total GHG, and total NOx compared to the alternative where TT is the main objective. Routing Strategy Routing strategy can be an influential factor in optimization purposes. Inventory Routing allows suppliers to deliver their products to a given set of customers while optimizing inventory management, vehicle routing, and delivery schedule all at the same time. This is mostly applicable when delivering goods are subject to various constraints. It is assumed that emissions are associated with routing decisions, whereas waste is more linked to inventory decisions. As for future research direction, the multiproduct distribution, and non-deterministic consumption rates are among the features that have to be explored in order to minimize both the total cost of distribution and emission. Eco-Driving Eco-driving is defined as driving in such a way that minimizes emission by considering the dynamics of the traffic flow and safety measures. Zhou et al. [30] investigated fuel consumption models to evaluate eco-driving and eco-routing. They proposed that drivers often have more difficulty in applying eco-driving techniques on roads with high congestion. When traffic conditions are stable, eco-driving is more successful compared with facilities with lower speed limits and several roundabouts and ramps. Also, on higher speed limit roads, free-flow conditions can cause an increase in cruising speed, which lead to higher instant fuel consumption and consequently higher emission. Alternative Fuel Vehicle Routing The main goal of Alternative Fuel Vehicle Routing Problem (AFVRP) is to provide optimal routes with minimum energy consumption, time or cost for a fleet of alternative fuel vehicles, while considering their operation limitations such as limited driving distance and capacity. The number of publications on AFVRP is illustrated in Figure 9. There are more than 300 publications on AFVRP according to Scopus. Alternative Fuel Vehicle Routing Problem (AFVRP) can be divided into six categories based on their fuel type, as illustrated by Figure 10. Figure 11 displays the frequency publications on AFVRP per year divided into 3 main categories of EVs, Hybrid EVs, and other alternative fuel types. Figure 12 illustrates the share of each type of fuel in AFVRP literature on the whole. Although there is a wide range of fuels associated with this type of GVRP, only a small fraction of the literature is focused on non-electric AFVRP. Also, from Figure 11, it can be inferred that 23 (18%), 90 (72%), and 12 (10%) studies belong to the AFVRP, EVRP, and Hybrid VRP (HVRP), respectively. Previous studies have mostly been focused on EVRP variants, and there exists a research gap on the other two variants of the problem, and specifically the HVRP variant. Figure 13 shows the frequency of occurrence for various keywords, their co-occurrence, and the year they were most attractive to the researchers. This map indicates that the terms optimization, liquefied natural gas, refueling strategies, and time windows are amongst the most occurred keywords in the AFVRP publications. The term adaptive behavior is departed from the whole graph being used only once in the literature back in 2014. Time Windows This variant was considered earlier than other EVRPs, while it is more complex than the classical VRP with time windows. In this variant, a set of EVs deliver goods to customers, considering a given time window. Not visiting in pre-specified intervals decreases customers' satisfaction or may lead to infeasible solutions in practice. In one of the most recent papers associated with this variant, Keskin and Catay [31] considered stochastic waiting times at recharging stations with time windows. In another study, Schneider et al. [32] proposed a hybrid heuristic, which is a combination of a variable neighborhood search algorithm with a tabu search heuristic, considering limited vehicle freight capacities as well as customer time windows. Also, in a similar case study, Ding et al. [33] developed a heuristic method, according to variable neighborhood search and tabu search, by applying simple charging time adjustment processes to provide a more efficient solution. Moreover, Mao et al. [34] used an improved ant colony optimization (ACO) algorithm and hybridized it with insertion heuristic and enhanced local search to provide a solution for this problem, with respect to partial recharging and battery swapping. In addition, they proposed a new probabilistic selection model in ACO by considering the impact of both distances and time windows. In addition, Keskin and Catay [35] conducted more practical research by applying the full recharge restriction and allowing partial recharging (EVRPTW-PR). They defined this problem as a 0-1 mixed-integer linear program and proposed an Adaptive Large Neighborhood Search (ALNS) algorithm to solve it efficiently. Replenishment In this variant, it is usually assumed that a vehicle gets fully recharged every time it is at the refueling station-thus, continuing its service as long as its battery can support it. While EV battery packs are capable of supporting travel in the 100-mile range on a single charge, in order to replenish their batteries, they need to have access to charging stations. Meng and Ma [36] explored electric vehicle routing with soft time windows and offered two ways of power replenishment. The design of mobile charging stations have been investigated by Huang et al. [37]. In another study, Wen et al. [38] considered the problem of locating electronic replenishment stations for electric vehicles on a traffic network with flow-based demand to optimize the network performance. Baouche et al. [39] presented a new approach for the EV routing problem with recharging stage(s) along the way on the available charging stations to solve the autonomy limitation. Yet, issues like compatibility of battery EVs with chargers in their Charging Station(CS)s, and the impact of recharging at public or private CSs are still overlooked in the literature. Moreover, the time spent for recharging or battery swapping is a critical factor in this variant. A straightforward assumption usually used in vehicle routing studies is that the recharging time is constant across the whole network. Service time and charging time are also traditionally considered fixed. Charging Type In most of the existing EVRP papers, the battery charge level is a linear function of charging time, while in reality, this function is nonlinear. A practical linear charging estimation may lead to infeasible, unrealistic or expensive solutions. Therefore, Froger et al. [40] considered a realistic nonlinear relationship between the time spent on refueling and the amount of fuel consumed by the vehicle. Moreover, Xiao et al. [41] proposed a new model of the electric vehicle routing problem with respect to a general energy/electricity consumption function for EVs that factor in energy losses, nonlinear charging function with the piecewise linearization technique, efficient visits to charge stations, and continuous decision variables for speed, payload, travel time, recharging, etc. Karakatic [42] developed a Two-Layer Genetic Algorithm (TLGA) to overcome the capacitated Multi-Depot Vehicle Routing Problem by considering Time Windows (MDVRPTW) and Electric Vehicles (EV) with partial nonlinear recharging times. Zuo et al. [43] developed a new technical formulation for vehicle route selection and charging station visits, which reduces the formulation complexity. In addition, they proposed a new linearization method that applies a set of secant lines to surrogate the concave nonlinear charging function with linear constraints. Partial Recharging Conrad and Figliozzi [44] first considered the possibility of partial and overnight recharging for EVs at customer sites, which brought about one of the operational variants of the E-VRP. Subsequently, Felipe et al. [45] modelled and solved an EVRP with partial recharging for the first time. Since then, different aspects such as limited number of chargers in charging stations, various kinds of charging, fuzzy optimization models, etc. have been looked into. A method of charging is to establish some stations for swapping their batteries along the route either via stations or the road infrastructure. However, the impact of the different elements of this variant such as vehicle size, geographical configuration of site, recharge stations, autonomy, and recharging technologies have to be further checked. As one of the recent studies in this regard, Kancharla and Ramadurai [46] investigated capacitated and load-dependent discharging in a partial recharging EVRP. Also, several other combinations of this variant with time windows [47] and congestion tolls [48] have been studied to date. Hiermann et al. [49] proposed an electric vehicle routing problem combining conventional, plug-in hybrid, and electric vehicles. Location of Charging Station The driving range of AFVs is typically limited. Fuel cell electric vehicles are fueled with pure hydrogen gas stored in a tank on the vehicle and have a driving range of over 500 kilometres. Thus, the choice of routing with the focus on the driving range and long duration of the recharging process as well as the location of charging stations to utilize the necessary charging infrastructure is of significance. This variant accounts for the limited driving range of BEVs, which directly leads to the more frequent recharging needs at CS. However, only a few studies have considered the problem of CS capacity, due to the limited number of BEVs which can be charged at a CS, simultaneously. Only in recent years, have scholars started to combine a nonlinear charging process and CS location problem with distance constraint into the EVRP models [50]. They have proposed that waiting time at the charging stations can increase the total cost. Thus, considering waiting times, battery reservations, and adaptive routing with uncertainties in CS availability is an integral part of future EVRP research direction. Zhang et al. [51] also developed an Improved Whale Optimization Algorithm (IWOA), presented the locating problem of Electric Vehicle (EV) charging stations model with service risk constraints and applied IWOA to solve it. In another study, Bilal and Rizwan [52] investigated different approaches, objective functions, constraints and range of optimization techniques that were addressed by researchers for optimal placement of CS during recent years. Pickup and Delivery BEVs have a shorter driving range (160-240 km) compared to the driving range of conventional vehicles (480-650km). Therefore, they are more likely to be used on short distances or in urban areas where they are more effective than conventional vehicles due to their lower driving speed, lower noise production, and cost [53]. When the average route length is short, BEVs can be easily used as recharging can happen upon their return to the depot. BEVs are already being used by companies like DHL, and FedEx mostly for last-mile delivery of light goods as distances are short enough. Several studies have been conducted to investigate pickup and delivery effects in AFVs. For instance, Madankumar and Rajendran [54] presented two Mixed Integer Linear Programming (MILP) models for solving the Green Vehicle Routing Problems with Pickups and Deliveries in a Semiconductor Supply Chain. In another research, Ahmadi et al. [55] also investigated the importance of vehicle dynamics parameters in energy models for EV routing, especially in the Pickup-and-Delivery Problem (PDP). Grandinetti et al [56] developed a multi-objective mixed integer linear model for minimising the total travel distance, the total cost for the EVs used and the total penalty cost for the unsatisfied time windows. Energy Consumption Rate/Model Several energy consumption models were used by the literature, but only a few can predict realistic energy consumption at the road segment level, such as [57]. Additionally, Qi et al. [58] proposed a model to obtain an accurate link-level energy consumption estimation for EVs, considering the energy consumption under real-world traffic congestion on two proposed impact factors i.e. positive kinetic energy (PKE) and negative kinetic energy (NKE). Yi and Bauer [59] developed an adaptive multiresolution framework for electric vehicle (EV) energy consumption estimation with real-time capability. Accordingly, three key parameters, namely powertrain efficiency, wind speed, and rolling resistance, are adaptively estimated using a two-step nonlinear iterative algorithm. Basso et al. [60] introduced the Two-stage Electric Vehicle Routing Problem that incorporates improved energy consumption estimation with respect to detailed topography and speed profiles. Multi-Depot This variant arises as a practical and functional issue in VRP, where vehicles are dispatched from and returned to one of the multiple depot locations. Therefore, apart from the routing choice, it is necessary to decide from which depot the goods are going to be delivered. Only 4% of the publications have incorporated this variant, including [61]. In this regard, Zhang et al. [62] used an Ant Colony System-based metaheuristic to find a solution to A Multi-depot Green Vehicle Routing Problem (MDGVRP). Schneider et al. [63] considered the Multi-Depot Electric Vehicle Location Routing Problem with Time Windows (MDVLRP). Fuel Swapping This variant is defined to establish some stations for swapping batteries and other fuel types which correspond to increased recharging speed and reduced time loss. A battery swapping operation is faster than recharging operation, by taking only 10 minutes. Also, the used-up batteries can be recharged at night when electricity is charged at a discount. This variant accounts for over 15% of the publications on AFVRP including [64]. Some scholars have proposed (recharging) or battery swapping services can be available at all or some of the customers' sites as well [65]. In other studies, such as [66], the authors developed robust optimization models that aid the planning process for deploying battery-swapping infrastructure. Also, Wu [67] reviewed the state-of-the-art battery swapping station (BSS) literature and business models, where the BSS offers a recharged battery to an incoming EV with a low state-of-charge. First, four operation modes are presented i.e., a single BSS, multiple BSSs, an integrated BSS and battery charging station (BCS), and multiple BSSs and BCSs. BSS problems in routing are surveyed in different operational areas including charging schedule, construction and planning, dispatching and routing optimization, and power management. Upstream Effect Both the EU and U.S.A boasted about electric vehicles producing zero emission. However, unlike conventional vehicles, a significant proportion of the emissions produced by electric vehicles occurs 'upstream', i.e., when the electricity is produced at the source. Thus, current regulations, which only account for exhaust emissions, do not fully capture the GHG emissions from EVs. If retaining the 0g/km rate is an actual goal, it is required that manufacturers buy carbon credits to compensate for the 0g/km rate or switch to a full life cycle analysis. Lutsey and Sperling [68] found that if upstream emission's effect are considered, an EV powered from the American electricity grid produced an average of 56% less CO2 emission than a similar brand new petrol car (62g/km compared with 142g/km). Sen et al. [69] uses a hybrid life-cycle assessment method to analyze and compare alternative fuelpowered heavy-duty trucks (HDTs) applying a Monte Carlo simulation to account for the uncertainty in the data. New Insights and Perspectives into GVRP Even considering a pervasive application of alternative fuel vehicles and their role in emission reduction, a large number of vehicles remain on the roads, producing congestion and polluting emissions. Hence, to ultimately reduce the emission, we must consider emerging solutions that can also take into account the increasing customer demand for vehicles, while considering environmental aspects. Technological Consideration Several emerging technologies are opening up new research directions in the context of GVRP. Here we discuss the most promising ones. Drones Unmanned Aerial/Ground Vehicles (UAV/UGV) or drones are an emerging technology solution to the last-mile delivery problem. Drones are either controlled by a remote controller or an on-board intelligence. They have the potential to cut the pollution caused by trucks on a congested road network by utilizing the unused airspace. A mixed fleet of drones and trucks can also be used for delivery as proposed by Wang el al. [70]. Drone technology is considered more reliable and faster since neither are drones affected by road congestion nor traffic accidents on the networks. Several research directions can be considered. First, the environment that the UAV/UGV is operating in should be specified. That is, within which obstacles and urban air mobility limitations the vehicle is making its way. To do so, vehicle flight planning and optimization are required and considered as a path planning problem. As a result, yet another version of GVRP can be formulated where the drone must optimize the path, while conserving the fuel and minimizing emissions. Sarath et al. [71] discussed several techniques to achieve UAV path detection, planning and obstacle avoidance for realtime communicative environments. Alongside obstacle avoidance, there are some aerial restrictions in the path planning of drones, such as no-fly zone areas. Feng [72] proposed an improved method to achieve path planning for UAVs in complex surroundings. However, the fuel consumption and emissions dimension is rarely considered in the path planning problem. Secondly, it is noted that the wind and weather play a critical role in the flight planning of the drones, especially if they are small and light. Cheng et al. [73] introduced a distributionally robust optimization model to solve a two-period drone scheduling problem with uncertain flight times, which can be implemented in a data-driven framework using historical weather information. The operation of drones for the last-mile delivery, where the mobile base-station is a truck, creates another emerging GVRP, where the integrated truck and drone operations need to be optimized in a dynamic environment. As the drone traffic is expected to increase in the near future, we expect that cities will regulate the airspace more and may consider extending the existing 2D on-land road network to 3D land and air road network. This will result in further potential research directions in GVRP. Distributed Systems In recent years, distributed ledger technologies, for instance, blockchain have been used in transportation and logistics to manage information. Lopez and Farooq [74] proposed a blockchain based smart mobility data market (BSMD) that provides the underlying framework for the use of distributed ledgers in transportation applications. Eckert et al. [75] developed a carbon credit (C2) market for multimodal passenger mobility using BSMD, where individuals could track and trade C2s based on their mode, trips, and availability of credits. Such a market has the potential to be used in GVRP where the emissions are minimized not only based on the cost objective, but also with cooperation/competition among the individuals in the market by exchanging carbon credits with dynamic pricing. Due to the distributed and dynamic characteristics of the Internet of Vehicles (IoV), contentcentric decentralized vehicular named data networking (VNDN) has become more suitable for content-oriented applications in IoV [76]. The existing centralized architecture is prone to the failure of single points, which results in trust problems in key verification between crossdomain nodes, consuming more power and reducing the lifetime. Focusing on secure key management and power-efficient routing, [76] proposed a blockchain-based key management and green routing scheme for VNDN. Due to advancements in information and communication technologies, there is a strong focus on developing highly intelligent intersections in urban areas that can control and route traffic. Farooq and Djavadian [77] proposed a distributed traffic management system where the intersections actively cooperate and exchange information among each other via Infrastructure to Infrastructure (I2I) communication technology. These intelligent intersections (I2s) use the information to predict traffic conditions and proactively route vehicles from origin to destination via vehicle to infrastructure (V2I) communication technology. We are of the view that such distributed and intelligent systems can be used to develop new solutions to GVRP. Fuel Technologies Advanced research on lithium-ion batteries done over several decades has resulted in high energy density, high cycle life, and high-efficiency batteries. However, the research is still ongoing on new electrode materials to enhance the performance of energy density, power density, cycle life, safety, and cost. The current generation of anode and cathode materials are suffering from several issues, including, slow Li-ion transport, high volume expansion, limited electrical conductivity, low thermal stability, dissolution or other unfavorable interactions with electrolyte, and mechanical brittleness [78]. Several approaches have been developed to solve these problems. A variety of intercalation cathodes have been available on the market, and conversion material technology is going to become more common. In terms of GVRP, lithium-ion battery electrode materials' technological advances would result in solutions with less stringent constraints in terms of trip lengths and better utilization of the capacity of the vehicle, especially when the demand is highly stochastic. In the view of the Fraunhofer Institute, synthetic fuels and drive technologies such as hydrogen combined with the fuel cell has the potential to play a crucial role in the future of transportation. It is expected that such a role might be negligible in private vehicles, but significant in long-distance and heavy-duty vehicles used for goods movement. However, the drawbacks of hydrogen-based fuel cell technology should not be overlooked as it is very costly in terms of efficiency and operating costs. Horváth et al. [79] study, compares two types of EVs from the customer's point of view. In their study they had a detailed investigation carried out into whether battery-or hydrogen-powered electric vehicles will become ubiquitous in the future. The question of which energy storage system has the best efficiency and is the most cost-effective one for powering electric vehicles is still unanswered. With BEVs, only eight percent of the energy is lost upstream, and another 18 percent is wasted to convert the electrical energy to drive power. Depending on the model, BEVs' efficiency is about 70 to 80 percent. In the case of the hydrogen-powered EVs, 45 percent of the energy is already lost during the production of hydrogen through electrolysis. Of this remaining original energy, another 55 percent is lost to convert hydrogen to electricity. This means that the hydrogen-powered EVs' efficiency is about 25 to 35 percent, depending on the model. The efficiency is even worse with alternative fuels. The efficiency in this case is only 10 to 20 percent, which can convey the meaning that the use of Hydrogen would therefore be a mistake for passenger cars [79]. Therefore, Horváth et al. implied that investments should rather focus on long-distance and heavy duty transportation where ecological and economic constraints play an important role. Although alternative fuels may become prevalent in the long term, yet it is unlikely that they would completely substitute the fossil fuels in the near future. Given the scarcity of such resources, alternative fuels should be prioritized for different transportation sectors to which it is cost-effective. This is not only because alternative fuels are competitive in those sectors but also because it is hard to decarbonize them. On the other hand, handling hydrogen from storage to transportation is difficult as it requires additional infrastructure such as hydrogen grid, and additional transformation on the demand side like fuel cells for heavy-duty transportation. All in all, considering the complexity of the adoption of these fuel technologies and the scarcity of vehicles that are powered from such technologies, green vehicle routing problem is less likely to be affected by studies in AFV areas rather than EVs and HEVs in the near future. Given that to inspect different aspects of GVRP with fuel cell vehicles, a sufficient rate of them is required on transportation network. To the best of our knowledge, very few studies have looked into this area of AFVRP. In addition, since the use of alternative fuel in heavy duty transportation is preferred to passenger vehicles, it is observed that scholars tend to investigate AFVRP in urban transit, logistics and air and rail transportation. Therefore, AFVRP studies will grow vastly based on the demand and penetration rates of alternative fuel vehicles. Methodological Considerations The application of machine learning methods in green vehicle routing and optimization has attracted scholars' attention in the last few years. While demonstrations using AI in GVRP are rare, in a recent publication, Guiladi and Eriksson [80] referred to the dynamic routing of electric commercial vehicles with a large amount of data considering random customer requests when predicting the optimal route. This work introduced artificial intelligence applied to routing and energy prediction of electric vehicles with a Deep Q-Learning method proposed to solve the problem. In a similar study, Chen et al. [81] proposed a Deep Q-Learning method to assign customers to vehicles and drones for same-day delivery. Although reinforcement learning in GVRP is not commonly used, it is considered a powerful tool for considering generalized GVRP studies considering different variants and applications of it where specific heuristic and metaheuristic methods can't be generalized to another problem. Also, it is a powerful method of dealing with uncertainties in the real world GVRP problems. In a study by Basso [82], it has been shown that the reinforcement learning (RL) method could save on average 4.8% (up to 12%) energy by planning the route and charging anticipatively, rather than the deterministic online reoptimization method. The research addresses the dynamic Stochastic Electric Vehicle Routing Problem (DS-EVRP) with a safe reinforcement learning method to solve the problem. In similar studies, a deep reinforcement learning method is proposed to minimize the total cost of a fleet of electric-autonomous vehicles for ride hailing services [83], where the complete system state is approximated using neural networks. Recently, approaches like chaos theory, quantum computation, and fuzzy logic have been introduced in GVRP, which can help deal with uncertainties. From the fuzzy theory perspective, most studies in this area consider the fuzzy chance-constrained mixed integer non-linear programming model ( [84], [85]). In terms of stochastic optimizations, however, only a few studies have shown an application of this approach in the green freight transportation context [86]. Diversification is another crucial factor to the performance of the population-based algorithm, but the initial population in the many methods such as bee colony is generated using a greedy heuristic, which has insufficient diversification. Therefore the ways in which the sequential optimization for the initial population drives the population toward improved solutions are examined [87]. Emerging Services and their Requirements Cars travelling on the roads are not completely exploited and can be used to deliver goods as well as passengers. Crowd shipping is replacing conventional delivery companies with occasional drivers using their personal vehicles to deliver goods. Archetti et al. [88] put forward the concept of VRP with occasional drivers (VRPOD). Macrina et al. [89] added time windows, multiple deliveries for origin-destination pairs, and split and delivery policy in several publications [90,91]. It also introduced a VRP with a mixed fleet of CVs and EVs [89]. This strategy has proven to positively impact emission and routing costs reduction as well as offering a higher reliability and customer satisfaction level. Although crowd shipping companies have expanded their business in the last several years. However, those companies mainly have businesses within some metropolitan areas. A system may work for the last-mile delivery, but how it will perform for inter-city delivery is still to be investigated [92]. Another concern is that a business model may have differential performance in different contexts, possibly due to cultural differences and infrastructure networks that support CS markets [93]. In that a case, the promising application areas challenge stakeholders on both supply and demand sides (e.g., market segments, network issues), operations and management (e.g., reverse logistics) to implement CS systems that function collaboratively, dynamically, and sustainably. Conclusion To the best of our knowledge, no existing study has comprehensively considered all GVRP variants and its future direction. Review studies are crucial for understanding the existing body of literature and, in this regard, a clear gap exists. Other operational constraints driven from GVRP variants such as site-dependent GVRP, and periodic GVRP are to be investigated in the future. Also, if retaining the zero emission rate is a goal in reality, it is required that manufacturers buy carbon credits to make up for their emission, or switch to a full life-cycle analysis. However, this requires more data collection on GHG emission from the electricity grid, which may lessen the popularity of electric vehicles among manufacturers, leading to a sharp fall in their investment in EV market. Also, it can be concluded that the main focus of previous literature is on the operational level routing decision, overlooking other aspects associated with the supply chain management such as network design, road tolls, reliability index, etc. While there is a large body of literature developed on EVRP in recent years, AFVRP is yet to be further explored -especially in terms of mixed fleet, connectivity and autonomous vehicles. The other limitation of AFVRP is that some alternative fuels are not cost-effective for private vehicles. Thus, the demand side of AFVRP is yet to be developed based on fuel technologies. The forecasting methods such as data-driven machine learning and reinforcement learning methods were rarely investigated in GVRP. However, metaheuristic methods are dominant for all variants of GVRPs due to their less time consumption and reasonable precision, indicating the significance of this approach compared with others. Nevertheless, there is no consensus among scholars about the reliability of such methodology. Also, exact methods are rarely applied in time dependent VRP. Opportunities in solution methodologies such as the application of Deep Q-Learning method, quantum computing, chaos theory, reinforcement learning, etc. are still not popular. Further GVRP research using such methods is highly recommended due to their strength and robustness in forecasting different parameters. In addition, the advancement in information and communication technologies has given rise to distributed systems related issues which still need further development in terms of Internet of Vehicles (IoV) and transportation infrastructures. Opportunities for considering uncertainties are provided as previous studies considered time-dependent concepts to deal with uncertainties, while demand, speed, and travel time are still the most important parameters remained as uncertainty in this field. In this regard it can be noticed that other non-deterministic parameters such as uncertain travel time have been neglected. Sustainability related indices such as social concerns, customer willingness, driver pattern, and operational risks are among the uncertainties of the supply chain network, which are totally overlooked by the former research. In addition, to develop a more promising routing system, especially on a macroscopic scale with plenty of complex and hazardous areas, a road network should be designed to enhance network reliability and overcome vehicle routing issues. There are various issues, such as weather conditions, restricted zones areas, and several moving obstacles, that must be taken into account. In other words, the more real-world challenges to be considered, the more operational routing network will be achieved. Opportunities for multiple-objective approaches still exist as in reality, there is rarely a single-objective problem, while only 20% of the publications consider multi-objective problems, which confirms the need for multi-objective optimization problems in the green transportation context. Last but not least, new opportunities in urban logistics have paved the way for UAV/UGVs to influence GVRP in the last mile delivery. Crowd shipping in all the transportation modes has also been another strategic element of green logistics in recent years, saving the delivery time and cost, while producing less GHG emission. It is expected that in the near future, GVRP will not be limited to the land roads networks, but will also involve virtual air networks. Figure 1 : 1Procedure of applied literature analysis Figure 2 : 2The number of GVRP publications by year A significant number of studies are associated with scientific articles published in transportation research journals such as IEEE annual conferences, Sustainability, Sensors, Transportation Research (Parts: B, C, and E), Journal of Advanced Transportation, Transportation Research Procedia, Transportation Science; European Journal of Operational Research, Cities, etc. Others include Expert Systems with Applications, Journal of Cleaner Production, Applied Energy, etc. Figure 3 refers to the network of journal bibliographic coupling. Figure 3 : 3The network of journal bibliographic coupling 5 Figure 4 : 4GVRP literature contribution by country Figure 5 : 5Map of keyword co-occurrence for GVRP Figure 6 : 6Usage of different solution methodologies in the existing literature Figure 7 7refers to the number of publications per year using each type of above-mentioned methodologies. This figure demonstrates the increasing trend in using metaheuristic algorithms and confirms the significance of this type of methodologies. Overall, the exact methods and metaheuristic algorithms are the methods most preferred by researchers. Furthermore, software applications and exact solvers have hit the bottom as the least popular methods due to their high computational time and complexity in the first years, while heuristic methods have become less popular, taking the place of exact solvers in the diagram. Figure 7 : 7The number of publications per year based on methodology Figure 8 : 8GVRP with CV variants origination Figure 9 : 9Alternative fuel vehicle routing publications per year Figure 10 :Figure 11 : 1011Alternative fuel vehicle classification Number of EVs, Hybrid EVs, and other alternative fuel vehicles' publications per year Figure 12 : 12Percentage of AFVRP publications based on fuel type Figure 13 : 13Map of keyword co-occurrence for GVRPFigure 14is associated with different GVRP with AFV variants and their origination year in literature. The figure reveals that while issues related to refueling location have been investigated since the beginning of the last decade, emerging issues like partial recharging have only gained interest in recent years with the advancement of new charging technologies and battery swapping. Figure 14 : 14GVRP with AFVRP variants origination The truck dispatching problem. G B Dantzig, J H Ramser, Management science. 61G. B. Dantzig, J. H. Ramser, The truck dispatching problem, Management science 6 (1) (1959) 80-91. S Erdogan, E Miller-Hooks, A green vehicle routing problem, Transportation research part E: logistics and transportation review. 48S. Erdogan, E. Miller-Hooks, A green vehicle routing problem, Transportation research part E: logistics and transportation review 48 (1) (2012) 100-114. Global ev outlook 2020: Entering the decade of electric drive?. I E Agency, I. E. Agency, Global ev outlook 2020: Entering the decade of electric drive? (2020). New passenger car registrations by fuel type in the european union-quarter 1. E A M Association, ACEA. Rep 195E. A. M. Association, et al., New passenger car registrations by fuel type in the european union-quarter 1 2019, ACEA, Brussels, Tech. Rep 195. Assessment of light-duty plug-in electric vehicles in the united states. D Gohlke, Y Zhou, Argonne National Lab.(ANL). Argonne, IL (United StatesTech. repD. Gohlke, Y. Zhou, Assessment of light-duty plug-in electric vehicles in the united states, 2010-2020, Tech. rep., Argonne National Lab.(ANL), Argonne, IL (United States) (2021). Software survey: Vosviewer, a computer program for bibliometric mapping. N J Van Eck, L Waltman, scientometrics. 842N. J. Van Eck, L. Waltman, Software survey: Vosviewer, a computer program for bibliometric mapping, scientometrics 84 (2) (2010) 523-538. Improved branch-cut-and-price for capacitated vehicle routing. D Pecin, A Pessoa, M Poggi, E Uchoa, Mathematical Programming Computation. 91D. Pecin, A. Pessoa, M. Poggi, E. Uchoa, Improved branch-cut-and-price for capacitated vehicle routing, Mathematical Programming Computation 9 (1) (2017) 61-100. Recent exact algorithms for solving the vehicle routing problem under capacity and time window constraints. R Baldacci, A Mingozzi, R Roberti, European Journal of Operational Research. 2181R. Baldacci, A. Mingozzi, R. Roberti, Recent exact algorithms for solving the vehicle routing problem under capacity and time window constraints, European Journal of Operational Research 218 (1) (2012) 1-6. An adaptive large neighborhood search heuristic for the pollutionrouting problem. E Demir, T Bektaş, G Laporte, European Journal of Operational Research. 2232E. Demir, T. Bektaş, G. Laporte, An adaptive large neighborhood search heuristic for the pollution- routing problem, European Journal of Operational Research 223 (2) (2012) 346-359. Recent advances in selection hyper-heuristics. J H Drake, A Kheiri, E Zcan, E K Burke, European Journal of Operational Research. 2852J. H. Drake, A. Kheiri, E. Ö zcan, E. K. Burke, Recent advances in selection hyper-heuristics, European Journal of Operational Research 285 (2) (2020) 405-428. Learning improvement heuristics for solving routing problems. Y Wu, W Song, Z Cao, J Zhang, A Lim, IEEE Transactions on Neural Networks and Learning Systems. Y. Wu, W. Song, Z. Cao, J. Zhang, A. Lim, Learning improvement heuristics for solving routing problems.., IEEE Transactions on Neural Networks and Learning Systems. A survey on metaheuristics for stochastic combinatorial optimization. L Bianchi, M Dorigo, L M Gambardella, W J Gutjahr, Natural Computing. 82L. Bianchi, M. Dorigo, L. M. Gambardella, W. J. Gutjahr, A survey on metaheuristics for stochastic combinatorial optimization, Natural Computing 8 (2) (2009) 239-287. Evolutionary algorithms, swarm intelligence methods, and their applications in water resources engineering: a state-of-the-art review. M Reddy, D Nagesh Kumar, H2Open Journal. 31M. Janga Reddy, D. Nagesh Kumar, Evolutionary algorithms, swarm intelligence methods, and their applications in water resources engineering: a state-of-the-art review, H2Open Journal 3 (1) (2021) 135-188. Fuel emissions optimization in vehicle routing problems with time-varying speeds. J Qian, R Eglese, European Journal of Operational Research. 2483J. Qian, R. Eglese, Fuel emissions optimization in vehicle routing problems with time-varying speeds, European Journal of Operational Research 248 (3) (2016) 840-848. Modelling and analysis of a green vehicle routing problem. E Jabir, V V Panicker, R Sridharan, Proceedings of the 12th AIMS International Conference on Management. the 12th AIMS International Conference on ManagementE. Jabir, V. V. Panicker, R. Sridharan, Modelling and analysis of a green vehicle routing problem, in: Proceedings of the 12th AIMS International Conference on Management, 2015, pp. 1310-1318. Vehicle routing and scheduling with time-varying data: A case study. W Maden, R Eglese, D Black, Journal of the Operational Research Society. 613W. Maden, R. Eglese, D. Black, Vehicle routing and scheduling with time-varying data: A case study, Journal of the Operational Research Society 61 (3) (2010) 515-522. The green vehicle routing problem: A systematic literature review. R Moghdani, K Salimifard, E Demir, A Benyettou, Journal of Cleaner Production. 279123691R. Moghdani, K. Salimifard, E. Demir, A. Benyettou, The green vehicle routing problem: A systematic literature review, Journal of Cleaner Production 279 (2021) 123691. Decomposition for large-scale optimization problems with overlapping components. Y Sun, X Li, A Ernst, M N Omidvar, IEEE congress on evolutionary computation (CEC). IEEEY. Sun, X. Li, A. Ernst, M. N. Omidvar, Decomposition for large-scale optimization problems with overlapping components, in: 2019 IEEE congress on evolutionary computation (CEC), IEEE, 2019, pp. 326-333. A fuel efficient green vehicle routing problem with varying speed constraint (f-gvrp). G Poonthalir, R Nadarajan, Expert Systems with Applications. 100G. Poonthalir, R. Nadarajan, A fuel efficient green vehicle routing problem with varying speed con- straint (f-gvrp), Expert Systems with Applications 100 (2018) 131-144. Multi-factorial evolutionary algorithm based novel solution approach for multi-objective pollution-routing problem. A Rauniyar, R Nath, P K Muhuri, Computers and Industrial Engineering. 130A. Rauniyar, R. Nath, P. K. Muhuri, Multi-factorial evolutionary algorithm based novel solution ap- proach for multi-objective pollution-routing problem, Computers and Industrial Engineering 130 (2019) 757-771. Multi-objective eco-routing in a distributed routing framework. L Alfaseeh, S Djavadian, R Tu, B Farooq, M Hatzopoulou, 2019 IEEE International Smart Cities Conference (ISC2). IEEEL. Alfaseeh, S. Djavadian, R. Tu, B. Farooq, M. Hatzopoulou, Multi-objective eco-routing in a dis- tributed routing framework, in: 2019 IEEE International Smart Cities Conference (ISC2), IEEE, 2019, pp. 747-752. Multi-objective eco-routing for dynamic control of connected and automated vehicles. S Djavadian, R Tu, B Farooq, M Hatzopoulou, Transportation Research Part D: Transport and Environment. 87102513S. Djavadian, R. Tu, B. Farooq, M. Hatzopoulou, Multi-objective eco-routing for dynamic control of connected and automated vehicles, Transportation Research Part D: Transport and Environment 87 (2020) 102513. Energy impact evaluation for eco-routing and charging of autonomous electric vehicle fleet: Ambient temperature consideration. Z Yi, J Smart, M Shirk, Transportation Research Part C: Emerging Technologies. 89Z. Yi, J. Smart, M. Shirk, Energy impact evaluation for eco-routing and charging of autonomous electric vehicle fleet: Ambient temperature consideration, Transportation Research Part C: Emerging Technologies 89 (2018) 344-363. A green vehicle routing problem with time windows considering the heterogeneous fleet of vehicles: two metaheuristic algorithms. N Rezaei, S Ebrahimnejad, A Moosavi, A Nikfarjam, European Journal of Industrial Engineering. 134N. Rezaei, S. Ebrahimnejad, A. Moosavi, A. Nikfarjam, A green vehicle routing problem with time win- dows considering the heterogeneous fleet of vehicles: two metaheuristic algorithms, European Journal of Industrial Engineering 13 (4) (2019) 507-535. The fleet size and mix pollution-routing problem. Ç Koç, T Bektaş, O Jabali, G Laporte, Transportation Research Part B: Methodological. 70Ç . Koç, T. Bektaş, O. Jabali, G. Laporte, The fleet size and mix pollution-routing problem, Trans- portation Research Part B: Methodological 70 (2014) 239-254. A cost-effective methodology to compare travel time and speed: a tale of 11 cities. S Sabet, F Namdarpour, M Mesbah, Proceedings of the Institution of Civil Engineers-Municipal Engineer. the Institution of Civil Engineers-Municipal EngineerThomas Telford LtdS. Sabet, F. Namdarpour, M. Mesbah, A cost-effective methodology to compare travel time and speed: a tale of 11 cities, in: Proceedings of the Institution of Civil Engineers-Municipal Engineer, Thomas Telford Ltd, 2021, pp. 1-11. The time-dependent pollutionrouting problem. A Franceschetti, D Honhon, T Van Woensel, T Bektaş, G Laporte, Transportation Research Part B: Methodological. 56A. Franceschetti, D. Honhon, T. Van Woensel, T. Bektaş, G. Laporte, The time-dependent pollution- routing problem, Transportation Research Part B: Methodological 56 (2013) 265-293. Analysis of travel times and co2 emissions in time-dependent vehicle routing. O Jabali, T Van Woensel, A De Kok, Production and Operations Management21O. Jabali, T. Van Woensel, A. De Kok, Analysis of travel times and co2 emissions in time-dependent vehicle routing, Production and Operations Management 21 (6) (2012) 1060-1074. A robust optimization approach for pollution routing problem with pickup and delivery under uncertainty. N Tajik, R Tavakkoli-Moghaddam, B Vahdani, S M Mousavi, Journal of Manufacturing Systems. 332N. Tajik, R. Tavakkoli-Moghaddam, B. Vahdani, S. M. Mousavi, A robust optimization approach for pollution routing problem with pickup and delivery under uncertainty, Journal of Manufacturing Systems 33 (2) (2014) 277-286. A review of vehicle fuel consumption models to evaluate eco-driving and eco-routing. M Zhou, H Jin, W Wang, Transportation Research Part D: Transport and Environment. 49M. Zhou, H. Jin, W. Wang, A review of vehicle fuel consumption models to evaluate eco-driving and eco-routing, Transportation Research Part D: Transport and Environment 49 (2016) 203-218. A simulation-based heuristic for the electric vehicle routing problem with time windows and stochastic waiting times at recharging stations. M Keskin, B Atay, G Laporte, Computers and Operations Research. 125105060M. Keskin, B. Ç atay, G. Laporte, A simulation-based heuristic for the electric vehicle routing problem with time windows and stochastic waiting times at recharging stations, Computers and Operations Research 125 (2021) 105060. The electric vehicle-routing problem with time windows and recharging stations. M Schneider, A Stenger, D Goeke, Transportation science. 484M. Schneider, A. Stenger, D. Goeke, The electric vehicle-routing problem with time windows and recharging stations, Transportation science 48 (4) (2014) 500-520. Conflict-free electric vehicle routing problem with capacitated charging stations and partial recharge. N Ding, R Batta, C Kwon, N. Ding, R. Batta, C. Kwon, Conflict-free electric vehicle routing problem with capacitated charging stations and partial recharge, 2015. The electric vehicle routing problem with time windows and multiple recharging options. H Mao, J Shi, Y Zhou, G Zhang, 10.1109/ACCESS.2020.3003000IEEE Access. 8H. Mao, J. Shi, Y. Zhou, G. Zhang, The electric vehicle routing problem with time windows and multiple recharging options, IEEE Access 8 (2020) 114864-114875. doi:10.1109/ACCESS.2020.3003000. Partial recharge strategies for the electric vehicle routing problem with time windows. M Keskin, B Atay, 10.1016/j.trc.2016.01.013.URLhttps:/www.sciencedirect.com/science/article/pii/S0968090X16000322Transportation Research Part C: Emerging Technologies. 65M. Keskin, B. Ç atay, Partial recharge strategies for the electric vehicle routing problem with time windows, Transportation Research Part C: Emerging Technologies 65 (2016) 111-127. doi:https: //doi.org/10.1016/j.trc.2016.01.013. URL https://www.sciencedirect.com/science/article/pii/S0968090X16000322 Route optimization of electric vehicle considering soft time windows and two ways of power replenishment. M Meng, Y Ma, Advances in Operations Research. M. Meng, Y. Ma, Route optimization of electric vehicle considering soft time windows and two ways of power replenishment, Advances in Operations Research 2020. Design of a mobile charging service for electric vehicles in an urban environment. S Huang, L He, Y Gu, K Wood, S Benjaafar, IEEE Transactions on Intelligent Transportation Systems. 162S. Huang, L. He, Y. Gu, K. Wood, S. Benjaafar, Design of a mobile charging service for electric vehicles in an urban environment, IEEE Transactions on Intelligent Transportation Systems 16 (2) (2014) 787-798. Locating replenishment stations for electric vehicles: application to danish traffic data. M Wen, G Laporte, O B G Madsen, A V Nørrelund, A Olsen, 10.1057/jors.2013.100Journal of the Operational Research Society. 6510M. Wen, G. Laporte, O. B. G. Madsen, A. V. Nørrelund, A. Olsen, Locating replenishment stations for electric vehicles: application to danish traffic data, Journal of the Operational Research Society 65 (10) (2014) 1555-1561. doi:10.1057/jors.2013.100. Electric vehicle shortest path problem with replenishment constraint. F Baouche, R Trigui, R Billot, N E El Faouzi, 10.1109/ICCVE.2014.72976002014 International Conference on Connected Vehicles and Expo (ICCVE). F. Baouche, R. Trigui, R. Billot, N. E. El Faouzi, Electric vehicle shortest path problem with replenish- ment constraint, in: 2014 International Conference on Connected Vehicles and Expo (ICCVE), 2014, pp. 514-515. doi:10.1109/ICCVE.2014.7297600. Improved formulations and algorithmic components for the electric vehicle routing problem with nonlinear charging functions. A Froger, J E Mendoza, O Jabali, G Laporte, Computers and Operations Research. 104A. Froger, J. E. Mendoza, O. Jabali, G. Laporte, Improved formulations and algorithmic components for the electric vehicle routing problem with nonlinear charging functions, Computers and Operations Research 104 (2019) 256-294. Electric vehicle routing problem: A systematic review and a new comprehensive model with nonlinear energy recharging and consumption, Renewable and Sustainable. Y Xiao, Y Zhang, I Kaku, R Kang, X Pan, 10.1016/j.rser.2021.111567.URLhttps:/www.sciencedirect.com/science/article/pii/S1364032121008455Energy Reviews. 151111567Y. Xiao, Y. Zhang, I. Kaku, R. Kang, X. Pan, Electric vehicle routing problem: A systematic review and a new comprehensive model with nonlinear energy recharging and consumption, Renewable and Sus- tainable Energy Reviews 151 (2021) 111567. doi:https://doi.org/10.1016/j.rser.2021.111567. URL https://www.sciencedirect.com/science/article/pii/S1364032121008455 Optimizing nonlinear charging times of electric vehicle routing with genetic algorithm. S Karakatič, 10.1016/j.eswa.2020.114039.URLhttps:/www.sciencedirect.com/science/article/pii/S0957417420308083Expert Systems with Applications. 164114039S. Karakatič, Optimizing nonlinear charging times of electric vehicle routing with genetic algorithm, Expert Systems with Applications 164 (2021) 114039. doi:https://doi.org/10.1016/j.eswa.2020. 114039. URL https://www.sciencedirect.com/science/article/pii/S0957417420308083 A new formulation of the electric vehicle routing problem with time windows considering concave nonlinear charging function. X Zuo, Y Xiao, M You, I Kaku, Y Xu, 10.1016/j.jclepro.2019.117687.URLhttps:/www.sciencedirect.com/science/article/pii/S0959652619325375Journal of Cleaner Production. 236117687X. Zuo, Y. Xiao, M. You, I. Kaku, Y. Xu, A new formulation of the electric vehicle routing problem with time windows considering concave nonlinear charging function, Journal of Cleaner Production 236 (2019) 117687. doi:https://doi.org/10.1016/j.jclepro.2019.117687. URL https://www.sciencedirect.com/science/article/pii/S0959652619325375 The recharging vehicle routing problem. R G Conrad, M A Figliozzi, Proceedings of the 2011 industrial engineering research conference. the 2011 industrial engineering research conferenceNorcross, GA8R. G. Conrad, M. A. Figliozzi, The recharging vehicle routing problem, in: Proceedings of the 2011 industrial engineering research conference, IISE Norcross, GA, 2011, p. 8. A heuristic approach for the green vehicle routing problem with multiple technologies and partial recharges. Á Felipe, M T Ortuño, G Righini, G Tirado, Transportation Research Part E: Logistics and Transportation Review. 71Á . Felipe, M. T. Ortuño, G. Righini, G. Tirado, A heuristic approach for the green vehicle routing problem with multiple technologies and partial recharges, Transportation Research Part E: Logistics and Transportation Review 71 (2014) 111-128. Electric vehicle routing problem with non-linear charging and loaddependent discharging. S R Kancharla, G Ramadurai, Expert Systems with Applications. 160113714S. R. Kancharla, G. Ramadurai, Electric vehicle routing problem with non-linear charging and load- dependent discharging, Expert Systems with Applications 160 (2020) 113714. A time-dependent electric vehicle routing problem with congestion tolls. R Zhang, J Guo, J Wang, IEEE Transactions on Engineering Management. R. Zhang, J. Guo, J. Wang, A time-dependent electric vehicle routing problem with congestion tolls, IEEE Transactions on Engineering Management. Fuzzy optimization model for electric vehicle routing problem with time windows and recharging stations. S Zhang, M Chen, W Zhang, X Zhuang, Expert systems with applications. 145113123S. Zhang, M. Chen, W. Zhang, X. Zhuang, Fuzzy optimization model for electric vehicle routing problem with time windows and recharging stations, Expert systems with applications 145 (2020) 113123. Routing a mix of conventional, plug-in hybrid, and electric vehicles. G Hiermann, R F Hartl, J Puchinger, T Vidal, 10.1016/j.ejor.2018.06.025.URLhttps:/www.sciencedirect.com/science/article/pii/S0377221718305605European Journal of Operational Research. 2721G. Hiermann, R. F. Hartl, J. Puchinger, T. Vidal, Routing a mix of conventional, plug-in hybrid, and electric vehicles, European Journal of Operational Research 272 (1) (2019) 235-248. doi:https: //doi.org/10.1016/j.ejor.2018.06.025. URL https://www.sciencedirect.com/science/article/pii/S0377221718305605 The location routing problem using electric vehicles with constrained distance. A Almouhanna, C L Quintero-Araujo, J Panadero, A A Juan, B Khosravi, D Ouelhadj, Computers and Operations Research. 115104864A. Almouhanna, C. L. Quintero-Araujo, J. Panadero, A. A. Juan, B. Khosravi, D. Ouelhadj, The location routing problem using electric vehicles with constrained distance, Computers and Operations Research 115 (2020) 104864. Locating electric vehicle charging stations with service capacity using the improved whale optimization algorithm. H Zhang, L Tang, C Yang, S Lan, 10.1016/j.aei.2019.02.006.URLhttps:/www.sciencedirect.com/science/article/pii/S1474034618302647Advanced Engineering Informatics. 41100901H. Zhang, L. Tang, C. Yang, S. Lan, Locating electric vehicle charging stations with service capacity using the improved whale optimization algorithm, Advanced Engineering Informatics 41 (2019) 100901. doi:https://doi.org/10.1016/j.aei.2019.02.006. URL https://www.sciencedirect.com/science/article/pii/S1474034618302647 Electric vehicles in a smart grid: a comprehensive survey on optimal location of charging station. M Bilal, M Rizwan, M. Bilal, M. Rizwan, Electric vehicles in a smart grid: a comprehensive survey on optimal location of charging station, 2020. Granular tabu search for the pickup and delivery problem with time windows and electric vehicles. D Goeke, European Journal of Operational Research. 2783D. Goeke, Granular tabu search for the pickup and delivery problem with time windows and electric vehicles, European Journal of Operational Research 278 (3) (2019) 821-836. Mathematical models for green vehicle routing problems with pickup and delivery: A case of semiconductor supply chain. S Madankumar, C Rajendran, Computers and Operations Research. 89S. Madankumar, C. Rajendran, Mathematical models for green vehicle routing problems with pickup and delivery: A case of semiconductor supply chain, Computers and Operations Research 89 (2018) 183- 192. S Ahmadi, G Tack, D Harabor, P Kilby, 10.4230/LIPIcs.CP.2021.11of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl -Leibniz-Zentrum für Informatik. L. D. Michel21017DagstuhlS. Ahmadi, G. Tack, D. Harabor, P. Kilby, Vehicle Dynamics in Pickup-And-Delivery Problems Using Electric Vehicles, in: L. D. Michel (Ed.), 27th International Conference on Principles and Practice of Constraint Programming (CP 2021), Vol. 210 of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl -Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2021, pp. 11:1-11:17. doi:10.4230/LIPIcs.CP.2021.11. URL https://drops.dagstuhl.de/opus/volltexte/2021/15302 A pick-up and delivery problem with time windows by electric vehicles. L Grandinetti, F Guerriero, F Pezzella, O Pisacane, International Journal of Productivity and Quality Management. 18L. Grandinetti, F. Guerriero, F. Pezzella, O. Pisacane, A pick-up and delivery problem with time windows by electric vehicles, International Journal of Productivity and Quality Management 18 (2016) 403-423. The electric vehicle routing problem with energy consumption uncertainty. S Pelletier, O Jabali, G Laporte, Transportation Research Part B: Methodological. 126S. Pelletier, O. Jabali, G. Laporte, The electric vehicle routing problem with energy consumption uncertainty, Transportation Research Part B: Methodological 126 (2019) 225-255. Data-driven decomposition analysis and estimation of link-level electric vehicle energy consumption under real-world traffic conditions. X Qi, G Wu, K Boriboonsomsin, M J Barth, 10.1016/j.trd.2017.08.008.URLhttps:/www.sciencedirect.com/science/article/pii/S1361920916307714Transportation Research Part D: Transport and Environment. 64the contribution of electric vehicles to environmental challenges in transport. WCTRS conference in summerX. Qi, G. Wu, K. Boriboonsomsin, M. J. Barth, Data-driven decomposition analysis and estimation of link-level electric vehicle energy consumption under real-world traffic conditions, Transportation Research Part D: Transport and Environment 64 (2018) 36-52, the contribution of electric vehicles to environmental challenges in transport. WCTRS conference in summer. doi:https://doi.org/10. 1016/j.trd.2017.08.008. URL https://www.sciencedirect.com/science/article/pii/S1361920916307714 Adaptive multiresolution energy consumption prediction for electric vehicles. Z Yi, P H Bauer, IEEE Transactions on Vehicular Technology. 6611Z. Yi, P. H. Bauer, Adaptive multiresolution energy consumption prediction for electric vehicles, IEEE Transactions on Vehicular Technology 66 (11) (2017) 10515-10525. Energy consumption estimation integrated into the electric vehicle routing problem. R Basso, B Kulcsár, B Egardt, P Lindroth, I Sánchez-Díaz, 10.1016/j.trd.2019.01.006Transportation Research Part D: Transport and Environment. 69R. Basso, B. Kulcsár, B. Egardt, P. Lindroth, I. Sánchez-Díaz, Energy consumption estimation in- tegrated into the electric vehicle routing problem, Transportation Research Part D: Transport and Environment 69 (2019) 141-167. doi:10.1016/j.trd.2019.01.006. Bi objective hybrid vehicle routing problem with alternative paths and reliability. M Nosrati, A Khamseh, Decision Science Letters. 92M. Nosrati, A. Khamseh, Bi objective hybrid vehicle routing problem with alternative paths and reliability, Decision Science Letters 9 (2) (2020) 145-162. Ant Colony Algorithm for Routing Alternate Fuel Vehicles in Multi-depot Vehicle Routing Problem. S Zhang, W Zhang, Y Gajpal, S S Appadoo, SpringerSingapore; SingaporeS. Zhang, W. Zhang, Y. Gajpal, S. S. Appadoo, Ant Colony Algorithm for Routing Alternate Fuel Vehicles in Multi-depot Vehicle Routing Problem, Springer Singapore, Singapore, 2019, pp. 251-260. The multi-depot electric vehicle location routing problem with time windows. J Paz, M Granada-Echeverri, J Escobar, doi:10.5267/j. ijiec.2017.4.001International Journal of Industrial Engineering Computations. 9J. Paz, M. Granada-Echeverri, J. Escobar, The multi-depot electric vehicle location routing problem with time windows, International Journal of Industrial Engineering Computations 9. doi:10.5267/j. ijiec.2017.4.001. The two-echelon capacitated electric vehicle routing problem with battery swapping stations: Formulation and efficient methodology. W Jie, J Yang, M Zhang, Y Huang, European Journal of Operational Research. 2723W. Jie, J. Yang, M. Zhang, Y. Huang, The two-echelon capacitated electric vehicle routing problem with battery swapping stations: Formulation and efficient methodology, European Journal of Operational Research 272 (3) (2019) 879-904. Charging strategies for economic operations of electric vehicles in commercial applications. M Schücking, P Jochem, W Fichtner, O Wollersheim, K Stella, Transportation Research Part D: Transport and Environment. 51M. Schücking, P. Jochem, W. Fichtner, O. Wollersheim, K. Stella, Charging strategies for economic operations of electric vehicles in commercial applications, Transportation Research Part D: Transport and Environment 51 (2017) 173-189. Infrastructure planning for electric vehicles with battery swapping. H.-Y Mak, Y Rong, Z.-J M Shen, Management Science. 597H.-Y. Mak, Y. Rong, Z.-J. M. Shen, Infrastructure planning for electric vehicles with battery swapping, Management Science 59 (7) (2013) 1557-1575. A survey of battery swapping stations for electric vehicles: Operation modes and decision scenarios. H Wu, 10.1109/TITS.2021.3125861IEEE Transactions on Intelligent Transportation Systems. 1H. Wu, A survey of battery swapping stations for electric vehicles: Operation modes and decision scenarios, IEEE Transactions on Intelligent Transportation Systems (2021) 1-23doi:10.1109/TITS. 2021.3125861. Regulatory adaptation: Accommodating electric vehicles in a petroleum world. N Lutsey, D Sperling, Energy Policy. 45N. Lutsey, D. Sperling, Regulatory adaptation: Accommodating electric vehicles in a petroleum world, Energy Policy 45 (2012) 308-316. Does a battery-electric truck make a difference?-life cycle emissions, costs, and externality analysis of alternative fuel-powered class 8 heavy-duty trucks in the united states. B Sen, T Ercan, O Tatari, Journal of cleaner production. 141B. Sen, T. Ercan, O. Tatari, Does a battery-electric truck make a difference?-life cycle emissions, costs, and externality analysis of alternative fuel-powered class 8 heavy-duty trucks in the united states, Journal of cleaner production 141 (2017) 110-121. Vehicle routing problem with drones, Transportation research part B: methodological. Z Wang, J.-B Sheu, 122Z. Wang, J.-B. Sheu, Vehicle routing problem with drones, Transportation research part B: method- ological 122 (2019) 350-364. Prototype survey of path planning and obstacle avoidance in uav systems. S Chandra, A Sastry, Int. J. Eng. Technol.(UAE). 7S. Sarath Chandra, A. Sastry, Prototype survey of path planning and obstacle avoidance in uav systems, Int. J. Eng. Technol.(UAE) 7 (2018) 316-322. L Feng, arXiv:2004.09864Reinforcement learning to optimize the logistics distribution routes of unmanned aerial vehicle. arXiv preprintL. Feng, Reinforcement learning to optimize the logistics distribution routes of unmanned aerial vehicle, arXiv preprint arXiv:2004.09864. Robust drone delivery with weather information. C Cheng, Y Adulyasak, L.-M Rousseau, M Sim, History. C. Cheng, Y. Adulyasak, L.-M. Rousseau, M. Sim, Robust drone delivery with weather information, History. A multi-layered blockchain framework for smart mobility data-markets. D Lopez, B Farooq, Transportation Research Part C: Emerging Technologies. 111D. Lopez, B. Farooq, A multi-layered blockchain framework for smart mobility data-markets, Trans- portation Research Part C: Emerging Technologies 111 (2020) 588-615. A blockchain-based user-centric emission monitoring and trading system for multi-modal mobility. J Eckert, D López, C L Azevedo, B Farooq, 2020 Forum on Integrated and Sustainable Transportation Systems (FISTS). IEEEJ. Eckert, D. López, C. L. Azevedo, B. Farooq, A blockchain-based user-centric emission monitoring and trading system for multi-modal mobility, in: 2020 Forum on Integrated and Sustainable Transportation Systems (FISTS), IEEE, 2020, pp. 328-334. Blockchain-based key management and green routing scheme for vehicular named data networking, Security and Communication Networks. H Liu, R Zhu, J Wang, W Xu, H. Liu, R. Zhu, J. Wang, W. Xu, Blockchain-based key management and green routing scheme for vehicular named data networking, Security and Communication Networks 2021. Distributed Traffic Management System with Dynamic End-to-End Routing. B Farooq, S Djavadian, Provisional Pat. Ser. 86562B. Farooq, S. Djavadian, Distributed Traffic Management System with Dynamic End-to-End Routing (2019, U.S. Provisional Pat. Ser. No. 62/865,725). Li-ion battery materials: present and future. N Nitta, F Wu, J T Lee, G Yushin, Materials today. 185N. Nitta, F. Wu, J. T. Lee, G. Yushin, Li-ion battery materials: present and future, Materials today 18 (5) (2015) 252-264. Techno-economic analysis of a decarbonized shipping sector: Technology suggestions for a fleet in 2030 and 2040. S Horvath, M Fasihi, C Breyer, Energy Conversion and Management. 164S. Horvath, M. Fasihi, C. Breyer, Techno-economic analysis of a decarbonized shipping sector: Technol- ogy suggestions for a fleet in 2030 and 2040, Energy Conversion and Management 164 (2018) 230-241. Artificial intelligence applied to routing and energy prediction of electric vehicles. E Guiladi, J Eriksson, E. Guiladi, J. Eriksson, Artificial intelligence applied to routing and energy prediction of electric vehi- cles. Deep q-learning for same-day delivery with vehicles and drones. X Chen, M W Ulmer, B W Thomas, European Journal of Operational Research. X. Chen, M. W. Ulmer, B. W. Thomas, Deep q-learning for same-day delivery with vehicles and drones, European Journal of Operational Research. Dynamic stochastic electric vehicle routing with safe reinforcement learning. R Basso, B Kulcsár, I Sanchez-Diaz, X Qu, Transportation Research Part E: Logistics and Transportation Review. 157102496R. Basso, B. Kulcsár, I. Sanchez-Diaz, X. Qu, Dynamic stochastic electric vehicle routing with safe reinforcement learning, Transportation Research Part E: Logistics and Transportation Review 157 (2022) 102496. Dynamic pricing and fleet management for electric autonomous mobility on demand systems. B Turan, R Pedarsani, M Alizadeh, Transportation Research Part C: Emerging Technologies. 121102829B. Turan, R. Pedarsani, M. Alizadeh, Dynamic pricing and fleet management for electric autonomous mobility on demand systems, Transportation Research Part C: Emerging Technologies 121 (2020) 102829. A time-dependent fuzzy programming approach for the green multimodal routing problem with rail service capacity uncertainty and road traffic congestion. Y Sun, M Hrušovskỳ, C Zhang, M Lang, Complexity. Y. Sun, M. Hrušovskỳ, C. Zhang, M. Lang, A time-dependent fuzzy programming approach for the green multimodal routing problem with rail service capacity uncertainty and road traffic congestion, Complexity 2018. Solving the green-fuzzy vehicle routing problem using a revised hybrid intelligent algorithm. R Wang, J Zhou, X Yi, A A Pantelous, Journal of Ambient Intelligence and Humanized Computing. 101R. Wang, J. Zhou, X. Yi, A. A. Pantelous, Solving the green-fuzzy vehicle routing problem using a revised hybrid intelligent algorithm, Journal of Ambient Intelligence and Humanized Computing 10 (1) (2019) 321-332. Multi-objective inventory routing problem: A stochastic model to consider profit, service level and green criteria. M Rahimi, A Baboli, Y Rekik, Transportation Research Part E: Logistics and Transportation Review. 101M. Rahimi, A. Baboli, Y. Rekik, Multi-objective inventory routing problem: A stochastic model to consider profit, service level and green criteria, Transportation Research Part E: Logistics and Trans- portation Review 101 (2017) 59-83. Sequential insertion heuristic with adaptive bee colony optimisation algorithm for vehicle routing problem with time windows. S Jawarneh, S Abdullah, PloS one. 107130224S. Jawarneh, S. Abdullah, Sequential insertion heuristic with adaptive bee colony optimisation algo- rithm for vehicle routing problem with time windows, PloS one 10 (7) (2015) e0130224. The vehicle routing problem with occasional drivers. C Archetti, M Savelsbergh, M G Speranza, European Journal of Operational Research. 2542C. Archetti, M. Savelsbergh, M. G. Speranza, The vehicle routing problem with occasional drivers, European Journal of Operational Research 254 (2) (2016) 472-480. Green logistics and crowd-shipping: challenges and opportunities. G Macrina, N Leone, F Guerriero, G Laporte, Ph.D. thesisG. Macrina, N. Leone, F. Guerriero, G. Laporte, Green logistics and crowd-shipping: challenges and opportunities, Ph.D. thesis (2018). Crowd-shipping with time windows and transshipment nodes. G Macrina, L D P Pugliese, F Guerriero, G Laporte, Computers and Operations Research. 113104806G. Macrina, L. D. P. Pugliese, F. Guerriero, G. Laporte, Crowd-shipping with time windows and transshipment nodes, Computers and Operations Research 113 (2020) 104806. Crowd-shipping: a new efficient and eco-friendly delivery strategy. G Macrina, L D P Pugliese, F Guerriero, Procedia Manufacturing. 42G. Macrina, L. D. P. Pugliese, F. Guerriero, Crowd-shipping: a new efficient and eco-friendly delivery strategy, Procedia Manufacturing 42 (2020) 483-487. Supply, demand, operations, and management of crowd-shipping services: A review and empirical evidence. T V Le, A Stathopoulos, T Van Woensel, S V Ukkusuri, Transportation Research Part C: Emerging Technologies. 103T. V. Le, A. Stathopoulos, T. Van Woensel, S. V. Ukkusuri, Supply, demand, operations, and man- agement of crowd-shipping services: A review and empirical evidence, Transportation Research Part C: Emerging Technologies 103 (2019) 83-103. Demand and supply modeling of crowd-shipping markets. T V Le, Purdue University Graduate SchoolPh.D. thesisT. V. Le, Demand and supply modeling of crowd-shipping markets, Ph.D. thesis, Purdue University Graduate School (2019).
[]
[ "Magnetic and topological transitions in three-dimensional topological Kondo insulator *", "Magnetic and topological transitions in three-dimensional topological Kondo insulator *" ]
[ "Huan Li [email protected]‡correspondingauthor \nCollege of Science\nGuilin University of Technology\n541004GuilinChina\n", "† ", "Zhi-Yong Wang \nCollege of Science\nGuilin University of Technology\n541004GuilinChina\n", "Xiao-Jun Zheng [email protected] \nCollege of Science\nGuilin University of Technology\n541004GuilinChina\n", "Yu Liu \nInstitute of Applied Physics and Computational Mathematics\n100088BeijingChina\n\nSoftware Center for High Performance Numerical Simulation\nAcademy of Engineering Physics\n100088BeijingChina, China\n", "Yin Zhong \nCenter for Interdisciplinary Studies\nKey Laboratory for Magnetism and Magnetic Materials of the MoE\nLanzhou University\n730000LanzhouChina\n" ]
[ "College of Science\nGuilin University of Technology\n541004GuilinChina", "College of Science\nGuilin University of Technology\n541004GuilinChina", "College of Science\nGuilin University of Technology\n541004GuilinChina", "Institute of Applied Physics and Computational Mathematics\n100088BeijingChina", "Software Center for High Performance Numerical Simulation\nAcademy of Engineering Physics\n100088BeijingChina, China", "Center for Interdisciplinary Studies\nKey Laboratory for Magnetism and Magnetic Materials of the MoE\nLanzhou University\n730000LanzhouChina" ]
[]
By using an extended slave-boson method, we draw a global phase diagram summarizing both magnetic phases and paramagnetic (PM) topological insulating phases (TIs) in three-dimensional topological Kondo insulator (TKI). By including electron hopping (EH) up to third neighbor, we identify four strong topological insulating (STI) phases and two weak topological insulating (WTI) phases, then the PM phase diagrams characterizing topological transitions between these TIs are depicted as functions of EH, f -electron energy level and hybridization constant. We also find an insulator-metal transition from a STI phase which has surface Fermi rings and spin textures in qualitative agreement to TKI candidate SmB6. In weak hybridization regime, antiferromagnetic (AF) order naturally arises in the phase diagrams, and depending on how the magnetic boundary crosses the PM topological transition lines, AF phases are classified into AF topological insulator (AFTI) and non-topological AF insulator (nAFI), according to their Z2 indices. In two small regions of parameter space, two distinct topological transition processes between AF phases occur, leading to two types of AFTI, showing distinguishable surface dispersions around their Dirac points.
10.1088/0256-307x/35/12/127501
[ "https://arxiv.org/pdf/1809.09867v1.pdf" ]
59,939,328
1809.09867
e9bdd6269a18f40019116410eadc879d7557a3b1
Magnetic and topological transitions in three-dimensional topological Kondo insulator * 26 Sep 2018 Huan Li [email protected]‡correspondingauthor College of Science Guilin University of Technology 541004GuilinChina † Zhi-Yong Wang College of Science Guilin University of Technology 541004GuilinChina Xiao-Jun Zheng [email protected] College of Science Guilin University of Technology 541004GuilinChina Yu Liu Institute of Applied Physics and Computational Mathematics 100088BeijingChina Software Center for High Performance Numerical Simulation Academy of Engineering Physics 100088BeijingChina, China Yin Zhong Center for Interdisciplinary Studies Key Laboratory for Magnetism and Magnetic Materials of the MoE Lanzhou University 730000LanzhouChina Magnetic and topological transitions in three-dimensional topological Kondo insulator * 26 Sep 2018 By using an extended slave-boson method, we draw a global phase diagram summarizing both magnetic phases and paramagnetic (PM) topological insulating phases (TIs) in three-dimensional topological Kondo insulator (TKI). By including electron hopping (EH) up to third neighbor, we identify four strong topological insulating (STI) phases and two weak topological insulating (WTI) phases, then the PM phase diagrams characterizing topological transitions between these TIs are depicted as functions of EH, f -electron energy level and hybridization constant. We also find an insulator-metal transition from a STI phase which has surface Fermi rings and spin textures in qualitative agreement to TKI candidate SmB6. In weak hybridization regime, antiferromagnetic (AF) order naturally arises in the phase diagrams, and depending on how the magnetic boundary crosses the PM topological transition lines, AF phases are classified into AF topological insulator (AFTI) and non-topological AF insulator (nAFI), according to their Z2 indices. In two small regions of parameter space, two distinct topological transition processes between AF phases occur, leading to two types of AFTI, showing distinguishable surface dispersions around their Dirac points. Topological Kondo insulator (TKI), 6 a heavy-fermion system with strong Coulomb interaction and d-f hybridization governing by spin-orbit coupling, preserves time-reversal symmetry (TRS), therefore can generate topological insulating phases (TI s ) with Kondo screening effect. As revealed by previous works, variation of electron hopping (EH) strength, f -electron energy level ǫ f and hybridization constant V can drive topological transitions among phases of strong topological insulator (STI), weak topological insulator (WTI) and normal Kondo insulator (nKI). 11 However, existing works in literature are restricted to their interested parameter regime, hence the studied TI s are still confined to a limited number of STI and WTI, and the full STI and WTI phases in TKI have not been explored adequately, particularly at the presence of strong electron-electron correlation. 11 (PAM), we uncover all possible TI s in three-dimensional (3D) TKI: four STI and two WTI, each possessing distinct surface states and Dirac cones. We also present the paramagnetic (PM) phase diagrams characterizing topological transitions between these TI s , as functions of EH, ǫ f and V . By proper fitting of EH, we verify a STI phase with Fermi surfaces and spin textures which can qualitatively simulate the TKI material SmB 6 , 12 confirming the applicability of PAM to TKI, and it is find that this STI phase is in vicinity to an insulator-metal transition driving by enhancement of V . In heavy-fermion systems, the interplay and competition between Kondo screening and magnetic correlation can motivate magnetic transitions when the Kondo interaction is reduced. 13 Similarly, in half-filled TKI, theoretical calculations have verified a transition to AF phase when the hybridization interaction V is weakened, 10,14 reminiscent of the induced magnetism in pressurized SmB 6 . 15-17 Besides, our earlier work has proved that due to the combined S symmetry of time reversal and translation operations, the AF states in TKI remain topological distinguishable, regardless of the breaking of TRS by magnetic order. We has developed a Z 2 topological classification to the AF states in TKI and proposed a novel AFTI phase under unique setting of model parameters, together with an AFTI-nontopological AF insulator (nAFI) topological transition while EH was shifted in some way. 10 Unfortunately, why AFTI should appear in such parameter region is still not clear, and it remains confused whether new AF phases exist in other parameter regions. We have shown that at least near the magnetic boundary (MB), the Z 2 index for AF directly relies on that of TI phase from which the AF order develops, 10 therefore, in order to investigate all possible AF phases with distinct topologies, the magnetic transition and classification of AF phases should be discussed on the basis of the PM phase diagrams summarizing all TI s , i.e., the four STI and two WTI should be included properly to study the AF transition as well as the topological transitions between AF phases. We use the spin-1/2 PAM to character the 3D TKI in cubic lattice: 18 H = H d + H f + H df + H U ,(1) where H d = k,α (ǫ d k − µ)d † kα d kα , H f = k,α (ǫ f + ǫ f k − µ)f † kα f kα . H df = V k,α,β S k · σ αβ d † kα f kβ + h.c. is the Kondo hybridization with spin-orbit coupling, in which S k = (sin k · a 1 , sin k · a 2 , sin k · a 3 ), 18 with the element vectors a 1 , a 2 , a 3 for cubic lattice. H U = U i n f i↑ n f i↓ is the on-site coulomb repulsion between f electrons, and we consider infinite U in this work. We includes EH up to third neighbor, with t d(f ) , t ′ d(f ) , t ′′ d(f ) denote nearest-neighbor (NN) , next-nearest-neighbor (NNN), and next-next-nearestneighbor (NNNN) hopping amplitudes, respectively, which determine the tight-binding dispersions ǫ d(f ) k . The chemical potential µ is used to fix the total electron number to half filling n t = 2, and variable EH, hybridization interaction V and f energy level ǫ f are considered. In what follows, t d = 1 is set as energy unit, and we choose t f = −0.2 and keep t ′ d /t d = t ′ f /t f to get a medium gapped insulating phase (unless when the insulator-metal transition is discussed). We employ the Kotliar-Ruckenstein (K-R) slave-boson method 10,19,20 to solve PAM. Similar to Coleman's slave boson theory, 18 the mean-field approximation of PAM Eq. 1 in large-U limit reads 10 H MF = N (−ηn f ) + k,α,β (d † kα , f † kα ) (ǫ d k − µ)δ αβṼ S k · σ αβ V S k · σ αβ (ǫ f k − µ)δ αβ d † kβ f † kβ ,(2) where the effective hybridizationṼ = V Z is renormalized by factor Z = 2(1 − n f )/(2 − n f ), and the effective f dispersionǫ f k = ǫ f +η+Z 2 ǫ f k , in which η shifts the f level. n f is the density of f electron per site, N is the number of lattice sites. The PM mean field parameters n f , η, µ are solvable through saddle-point solution for H MF , then the quasi-particle dispersions which are the eigenvalues of the Hamiltonian matrix in Eq. 2 (in a modified form) are used to identify the Z 2 index. In last two rows of Fig. 1, we show six types of distinct quasi-particle spectrums of PM TI s , each with different model parameters listed in Tab. I, comparing with d and f dispersions. At the eight high symmetric points (HSP s ) k m in 3D Brillouin zone (BZ) (i.e.,Γ=(0,0,0); X=(π,0,0), (0,π,0), (0,0,π); M =(π,π,0), (π,0,π), (0,π,π); and R=(π,π,π)), the hybridization vanishes (due to its odd parity), consequently the quasi-particle energy equals either ǫ d k orǫ f k , leading to the parity of occupied states δ m = 1 or -1 at k m , respectively. Therefore, the strong topological index ν 0 can be easily obtained by observing the bulk dispersions in Fig. 1 via (−1) ν0 = m∈HSPs δ m , and the weak topological indices ν j (j = x, y, z) are calculated from the HSP s on k j plane P j t f = −0.2, t ′ f /t f = t ′ d /t d , (a) V = 1, t ′′ d = t ′′ f = 0; (b) ǫ f = −2, t ′′ d = t ′ d , t ′′ f = t ′ f . Second and third rows: slab dispersions of TIs. The red lines denote the surface states. Last two rows: quasi-particle dispersions (black solid lines) in six TIs, the red and green lines are dand renormalized f -dispersions, respectively. Parameters and Z2 invariants for each TIs are listed in Tab. I. through (−1) νj = m∈Pj δ m . 6 The quantities δ m and Z 2 indices for six TI s are listed in Tab. I. By diagonalizing 40 slabs to simulate the 3D lattice with opened (001) surface, the surface states of the six TI s are computed and displayed by second and third rows in Fig. 1. On (001) surface, there are four HSP s :Γ=(0,0);X=(π,0), (0,π); andM =(π,π), each of the six TI s in Fig. 1 has Dirac points locating at different HSP s . The requirement of odd number of Dirac points on surface of STI leads to four inequivalent STI s : STIΓ, STIM , STIΓX , and STIMX , in which the subscripts denote the locations of Dirac points. For WTI s , there are even number of Dirac points, resulting in two WTI s : WTIΓM and WTIX . For nKI, generally no Dirac point exists, however, there is a special nKI with Dirac points at all four HSP s , since Fermi level crosses its surface states even times between two arbitrary HSP s , this phase is actually non-topological phase rather than a topological one. In the first row of Fig. 1, with varying t ′ d , ǫ f and V , we have located the topological boundaries among all possible TI s , determining by the change of Z 2 index. The topological transitions between TI s are generated by closing and reopening of the insulating gap at certain HSP, leading to an inversion of parity and consequently the shifting of Z 2 index. 10 In above, we have set t ′ f /t f = t ′ d /t d , under which the Dirac points in TI s all cross the Fermi energy, leading to the vanishment of Fermi surface. For TKI candidate SmB 6 , medium- sized surface Fermi rings aroundΓ andX were detected through ARPES, verifying it in a STIΓX phase. 12 Though our study of TKI is based on the simplified PAM, it can still produce a STIΓX with similar surface states to SmB 6 . To do this, we chose EH departing from t ′ f /t f = t ′ d /t d , and found a STIΓX phase with Fermi surfaces and helical spin textures quite similar to SmB 6 , see Fig. 2(c) and (d). Furthermore, this phase is in vicinity to an insulator-metal transition generated by shifting of V or ǫ f (see Fig. 2(a) and (b)), which may be account for the metallic phase observed in pressurized SmB 6 . 15,21 Based on the PM phase diagrams of TI s , we now study the AF transitions in TKI. In our previous work, the original K-R method of symmetric PAM 19,20 has been generalized to treat AF phases in non-symmetric case, 10 which can be applied to TKI. The resulting mean-field Hamiltonian is rather complicated in that in addition to n f , η and µ, two AF order parameters m f and h should be determined, besides, two renormalization factors Z 1 and Z 2 arise. Due to the S symmetry combined by TRS and lattice translation, the AF phases in TKI fall into Z 2 topological class, and the Z 2 index ν is calculated from the parities of the occupied spectrums at four Kramers degenerate momenta (KDM) p m (Γ and three X points) via (−1) ν = pm∈KDM δ m , in which δ m = i ξ i (p m ), with Parameters: the parity ξ i (p m ) of i-th occupied state at p m equals either 1 or -1, when quasi-particle energy equals that of d or f at p m , respectively. 10 Particularly, the strong topological index ν 0 on the PM side of the MB directly determines ν of the AF phase near MB, namely, ν 0 = 1 (STI) leads to ν = 1 (AFTI), while ν 0 = 0 (WTI or nKI) leads to ν = 0 (nAFI), 7,10 giving a straightforward verification of the AF phases near MB. Around t ′ d = −0.35, the MB is divided by the STIΓX -WTIX transition line into two parts, leading to AFTI and nAFI just below the two parts of MB, respectively. While V is lowered further from MB, ν of AF phases should be computed from the 3D spectrums (e.g., Fig. 4 (b) and (d)) to determine the AFTI-nAFI topological boundary, which is demonstrated by the green line near t ′ d = −0.35 in Fig. 3. The AFTI-nAFI transition is realized via parity inversion during gap closing and reopening at Γ, and it converges with STIΓX -WTIX boundary at the MB, see the lower inset in Fig. 3. ǫ f = −1.5, t ′′ d = t ′ d , t ′′ f = t ′ f , t f = −0.2 and t ′ f /t f = t ′ d /t d . Near t ′ d = 0.25, the MB is separated by WTIX -STIM and STIM -nKI lines into three parts. Below the middle part of MB (which touches STIM ), an AFTI arises, while below the other two parts of MB, nAFI emerges. The AFTI-nAFI transition forms a narrow water-drop-shaped area in which AFTI survives (green solid line in the upper inset in Fig. 3). Besides, though nAFI s above and below t ′ d = 0.25 have quite different dispersions (compare Fig. 5(b) with (f)), they still have equal ν = 0, since their magnetic orders grow from nKI and WTI, respectively. Though band gap is closed at the boundary between two nAFI s , no parity inversion occurs, consequently no topological transition takes place (see the green dashed line in upper inset of Fig. 3). The surface states of AF phases are shown in Fig. 4(a), (c) near t ′ d = −0.35, and in Fig. 5(a), (c), (e) around t ′ d = 0.25, respectively. In AFTI s , the Dirac points atΓ andM are protected by topology hence are robust, see Fig. 4(a) and Fig. 5(c). Furthermore, the Dirac surface states of two AFTI s (one near STIM and the other near STIΓX ) disperse quite differently, in which the former constructs a valley shape (Fig. 5(c)). In contrast, the gapless surface states atX in both AFTI and nAFI ( Fig. 4(a), (c) and Fig. 5(a)) are not robust, since they can be gapped by additional factor such as gate voltage. 10 In summary, we have performed a slave-boson mean-field analysis of the 3D TKI using spin-orbit coupled PAM, and presented the phase diagrams including all possible PM TI s in TKI: four STI and two WTI, each with distinguishable locations of Dirac points. We also obtained a STIΓX phase with similar surface states to SmB 6 , and found it can be driven to conducting state through an insulator-metal transition by enhanced hybridization. We also investigated the magnetic boundary of AF phases in TKI, and found the topological transitions between AFTI and nAFI in two narrow regions in parameter space. Besides, we found two types of AFTI with distinct dispersions at the Dirac points. Though our work is based on an uniform mean-field approximation, any site-dependent treatment will not break the application of Z 2 classification of both PM and AF states. 14, 17 We hope our work can help to reach a comprehensive understanding of novel AFTI phases in strongly correlated electrons systems. FIG. 1 : 1First row: PM phase diagrams of 3D TKI. Parameters: FIG. 2 : 2(a) Insulator-metal transition from STIΓX to conducting phase. (b) Closing of bulk gap during this transition. (c) Slab dispersions of STIΓX . (d) Surface Fermi rings and spin textures of STIΓX . t ′ d = t ′′ d = −0.375, t f = −0.23, t ′ f = t ′′ f = 0.2. ǫ f = −4 for (b) to (d), and V = 1 for (c) and (d). FIG. 3 : 3(a) Magnetic boundary in 3D TKI (blue lines). Near t ′ d = 0.25 and −0.35, topological transitions between AF phases take place (green solid lines). FIG. 4 : 4Surface dispersions (left column) and quasi-particle dispersions (right column) of AFTI and nAFI near the AF topological boundary in lower inset ofFig. 3. V = 1.4, t ′ d = −0.37 for AFTI, and t ′ d = −0.35 for nAFI. FIG. 5 : 5Surface spectrums (left column) and quasi-particle dispersions (right column) of the three AF phases near the AF topological boundary in upper inset of Fig. 3. V = 1.19 for all. From up, middle to down rows, t ′ d = 0.235, 0.253, and 0.264, respectively.The magnetic critical hybridization V c is calculated as a function of t ′ d , then the MB plotted by V c is added to the phase diagram, seeFig. 3. The MB crosses the topological boundaries of TI s in two parts, one near t ′ d = 0.25, the other around t ′ d = −0.35, see insets ofFig. 3for details. In this work, by considering adequate parameter space of periodic Anderson model * Supported by the National Natural Science Foundation of China under Grant Nos 11764010, 11504061, 11564008, 11704084, 11704166, Guangxi NSF under Grant Nos 2017GXNSFAA198169, 2017GXNSFBA198115, and SPC-Lab Research Fund (No. XKFZ201605). † Corresponding author, Email: [email protected] ‡ Corresponding author, Email: [email protected] TABLE I : IParameters and Z2 invariants of the TI phases shown inFig. 1. In all phases, t f = −0.2.phase t ′ d t ′′ d t ′ f t ′′ f ǫ f V δΓ δX δM δR ν0 νj Dirac points a STIΓ 0.26 0.26 -0.052 -0.052 -2 0.7 1 -1 -1 -1 1 -Γ STIΓX -0.35 -0.35 0.07 0.07 -2 1 -1 1 -1 -1 1 -Γ,X STIM 0.252 0.252 -0.0504 -0.0504 -2 1.5 1 1 1 -1 1 -M STIMX 0.4 0 -0.08 0 -1 1 1 1 -1 1 1 -M ,X WTIΓM -0.6 0 0.12 0 -1 1 -1 1 1 -1 0 1Γ,M WTIX -0.2 0 0.04 0 -2 1 1 1 -1 -1 0 1X a The surface dispersions are calculated on (001) surface. . X L Qi, S C Zhang, Rev. Mod. Phys. 831057Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83, 1057 . A Go, Phys. Rev. Lett. 109664012 Go A et al 2012 Phys. Rev. Lett. 109, 066401 . S L Yu, X Xie, J X Li, Phys. Rev. Lett. 10710401Yu S L, Xie X C and Li J X 2011 Phys. Rev. Lett. 107,010401 . Wang Z Zhang, S C , Phys. Rev. B. 86165116Wang Z and Zhang S C 2012 Phys. Rev. B 86, 165116 . X Wan, Phys. Rev. B. 83205101Wan X G et al 2011 Phys. Rev. B 83, 205101 . Phys. Rev. B. 8545130Dzero M el al 2012 Phys. Rev. B 85, 045130 . A Mong R S K, Essin, J E Moore, Phys. Rev. B. 81245209Mong R S K, Essin A M and Moore J E 2010 Phys. Rev. B 81, 245209 . C Fang, M J Gilbert, B A Bernevig, Phys. Rev. B. 8885406Fang C, Gilbert M J and Bernevig B A 2013 Phys. Rev. B 88, 085406 . Z Li, Phys. Rev. B. 91235128Li Z et al 2015 Phys. Rev. B 91, 235128 . H Li, 10.1088/1361-648X/aae17bJ. Phys.: Condens. Matter. Li H et al 2018 J. Phys.: Condens. Matter https://doi.org/10.1088/1361-648X/aae17b . M Legner, A Rüegg, M Sigrist, Phys. Rev. B. 8985110Legner M, Rüegg A and Sigrist M 2014 Phys. Rev. B 89, 085110 . N Xu, H Ding, M Shi, J. Phys.: Condens. Matter. 28363001Xu N, Ding H and Shi M 2016 J. Phys.: Condens. Matter 28, 363001 . M Vekic, Phys. Rev. Lett. 742367Vekic M et al 1995 Phys. Rev. Lett. 74, 2367 . R Peters, T Yoshida, N Kawakami, Phys. Rev. B. 9875104Peters R, Yoshida T and Kawakami N 2018 Phys. Rev. B 98, 075104 . Y Zhou, Science Bulletin. 621439Zhou Y Z el al 2017 Science Bulletin 62, 1439 . N Butch, Phys. Rev. Lett. 116156401Butch N P el al 2016 Phys. Rev. Lett. 116, 156401 . K W Chang, P J Chen, Phys. Rev. B. 97195145Chang K W and Chen P J 2018 Phys. Rev. B 97, 195145 . V Alexandrov, P Coleman, O Erten, Phys. Rev. Lett. 114177202Alexandrov V, Coleman P, and Erten O 2015 Phys. Rev. Lett. 114, 177202 . G Kotliar, A E Ruckenstein, Phys. Rev. Lett. 571362Kotliar G and Ruckenstein A E 1986 Phys. Rev. Lett. 57, 1362 . S J Sun, M F Yang, T M Hongt, Phys. Rev. B. 4816127Sun S J, Yang M F and Hongt T M 1993 Phys. Rev. B 48, 16127 . P Paraskevas, Martin B , Mohamed M , Europhys. Letts. 11066002Paraskevas P, Martin B and Mohamed M 2015 Europhys. Letts. 110, 66002
[]
[ "Accumulation of elastic strain toward crustal fracture in magnetized neutron stars", "Accumulation of elastic strain toward crustal fracture in magnetized neutron stars" ]
[ "Yasufumi Kojima \nDepartment of Physics\nHiroshima University Higashi-Hiroshima\n739-8526Japan\n" ]
[ "Department of Physics\nHiroshima University Higashi-Hiroshima\n739-8526Japan" ]
[]
This study investigates elastic deformation driven by the Hall drift in a magnetized neutron-star crust. Although the dynamic equilibrium initially holds without elastic displacement, the magneticfield evolution changes the Lorentz force over a secular timescale, which inevitably causes the elastic deformation to settle in a new force balance. Accordingly, elastic energy is accumulated, and the crust is eventually fractured beyond a particular threshold. We assume that the magnetic field is axially symmetric, and we explicitly calculate the breakup time, maximum elastic energy stored in the crust, and spatial shear-stress distribution. For the barotropic equilibrium of a poloidal dipole field expelled from the interior core without a toroidal field, the breakup time corresponds to a few years for the magnetars with a magnetic field strength of ∼ 10 15 G; however, it exceeds 1 Myr for normal radio pulsars. The elastic energy stored in the crust before the fracture ranges from 10 41 to 10 45 erg, depending on the spatial-energy distribution. Generally, a large amount of energy is deposited in a deep crust. The energy released at fracture is typically ∼ 10 41 erg when the rearrangement of elastic displacements occurs only in the fragile shallow crust. The amount of energy is comparable to the outburst energy on the magnetars.
10.3847/1538-4357/ac9184
[ "https://export.arxiv.org/pdf/2209.04136v1.pdf" ]
252,185,162
2209.04136
17483ee8633d6712aec61631930dde509f9b8789
Accumulation of elastic strain toward crustal fracture in magnetized neutron stars September 12, 2022 Yasufumi Kojima Department of Physics Hiroshima University Higashi-Hiroshima 739-8526Japan Accumulation of elastic strain toward crustal fracture in magnetized neutron stars September 12, 2022arXiv:2209.04136v1 [astro-ph.HE] 9 Sep 2022 Draft version Typeset using L A T E X default style in AASTeX631Neutron starsCompact objectsMagnetarsHigh-energy astrophysics This study investigates elastic deformation driven by the Hall drift in a magnetized neutron-star crust. Although the dynamic equilibrium initially holds without elastic displacement, the magneticfield evolution changes the Lorentz force over a secular timescale, which inevitably causes the elastic deformation to settle in a new force balance. Accordingly, elastic energy is accumulated, and the crust is eventually fractured beyond a particular threshold. We assume that the magnetic field is axially symmetric, and we explicitly calculate the breakup time, maximum elastic energy stored in the crust, and spatial shear-stress distribution. For the barotropic equilibrium of a poloidal dipole field expelled from the interior core without a toroidal field, the breakup time corresponds to a few years for the magnetars with a magnetic field strength of ∼ 10 15 G; however, it exceeds 1 Myr for normal radio pulsars. The elastic energy stored in the crust before the fracture ranges from 10 41 to 10 45 erg, depending on the spatial-energy distribution. Generally, a large amount of energy is deposited in a deep crust. The energy released at fracture is typically ∼ 10 41 erg when the rearrangement of elastic displacements occurs only in the fragile shallow crust. The amount of energy is comparable to the outburst energy on the magnetars. INTRODUCTION Neutron-star crust is considered a key aspect for understanding several astrophysical phenomena. The solid layer near the stellar surface can support non-spherical deformations, called mountains, with a height of less than 1 cm. Such asymmetries on a spinning star cause the continuous emission of gravitational waves. Therefore, calculation of the maximum size of these mountains is key to the detection of gravitational waves and has been discussed in several theoretical studies (Ushomirsky et al. 2000;Payne & Melatos 2004;Haskell et al. 2006;Gittins et al. 2021). Gravitational-wave observation provides valuable information (Abbott et al. 2021a(Abbott et al. ,b, 2022, for recent upper limit); thus, by continuously improving the sensitivity of the LIGO-Virgo-KAGRA detectors, the physics relevant to the phenomenon may be explored. Pulsar glitches are sudden spin-up events that are observed in radio pulsars (Espinoza et al. 2011;Basu et al. 2022, for glitch catalogue). Similar spin-up and peculiar spin-down events are observed in anomalous X-ray pulsars (Dib et al. 2008;Kaspi & Beloborodov 2017). A sudden spin-up in a radio pulsar is produced by the transfer of angular momentum from the superfluid components of the core to the normal crust (Anderson & Itoh 1975;Alpar et al. 1984). Crust quakes were also discussed in other models (Franco et al. 2000;Giliberti et al. 2020;Rencoret et al. 2021, for recent studies). The elastic deformation is caused by a decrease in centrifugal force, owing to a secular spin-down, and the crust eventually fractures when the strain exceeds a critical threshold. However, this simple model does not explain the observation; the loading of the solid crust between glitches is too insignificant to trigger a quake. Giant flares in magnetars are rare, albeit highly energetic. They are typically ∼ 10 44 − 10 46 erg released within a second (Turolla et al. 2015;Kaspi & Beloborodov 2017;Esposito et al. 2021, for a review). Quasi-periodic oscillations (QPOs) with some discrete frequencies in the range of 20 Hz -2 kHz were observed in the tails of these flares. Per the order-of-magnitude estimate, these frequencies correspond to the torsional shear or Alfvén modes with magnetic field strength ∼ 10 15 G. Outbursts, which are less energetic, are also observed in magnetars. These activities are considered to be powered by internal strong magnetic fields of ∼ 10 15 G. The crustal fracture of the magnetar is proposed as a model for fast radio bursts (FRBs) (Suvorov & Kokkotas 2019;Wadiasingh & Chirenti 2020), and it may be supported by QPOs (Li et al. 2022) in the radio burst from SGR J1935+2154 in the galaxy (Mereghetti et al. 2020;CHIME/FRB Collaboration et al. 2020;Bochenek et al. 2020). Most FRBs are located at a cosmological distance, and further observation sheds light on whether FRBs originate from magnetars or a subclass. A recent observation of the magnetar SGR 1830-0645 revealed pulse-peak migration during the first 37 days of outburst decay (Younes et al. 2022). This provides important information concerning the crust motion coupled with the exterior magnetosphere. Most theoretical studies have been focused on the crustal-deformation limit. Elastic stresses gradually accumulate until a particular threshold. Beyond this threshold, the elastic behavior of the lattice abruptly ceases, and the transition is exhibited as a star-quake or burst. An evolutionary calculation of the deformation is necessary to understand the related astrophysical phenomena. In this study, we consider the crust in a magnetized neutron star. The static magneto-elastic force balance was studied for various magnetic-field configurations (Kojima et al. 2021a(Kojima et al. , 2022. A variety of magneto-elastic equilibria was demonstrated, and is considerably different from the barotropic equilibrium without a solid crust. Herein, we explore the accumulation of shear stress induced by the Hall evolution, which is an important process in the strong field-strength regime. Suppose that the magneto-hydro-dynamical (MHD) equilibrium in the crust holds at a particular time without the elastic force. The equilibrium is not that for electrons (Gourgouliatos et al. 2013;Gourgouliatos & Cumming 2014); thus, the magnetic field tends to the Hall equilibrium in a secular timescale. According to the magnetic-field evolution, the Lorentz force also changes. The deviation is assumed to be balanced with the elastic force. Thus, the shear stress in the crust gradually accumulates and reaches a critical limit. We cannot follow the post-failure evolution because some uncertainties are involved in the discontinuous transition. Therefore, our study provides the recurrent time and magnitude of the bursts. The models and equations used in the study are discussed in Section 2. For MHD equilibrium in a barotropic star, the evolution of the magnetic field is driven by a spatial gradient of electron density. In Section 3, the critical configuration at the elastic limit is evaluated, and the accumulating elastic energy is calculated. In Section 4, we also consider non-barotropic effects using simple models. The non-barotropicity results in another driving process of the magnetic-field evolution, and consequently elastic deformation. The numerical results of these models are given. Finally, our conclusions are presented in Section 5. FORMULATION AND MODEL Magnetic Equilibrium We consider the dynamical force balance between pressure, gravity, and the Lorentz force. The MHD equilibrium is described as follows: − 1 ρ ∇P − ∇Φ g + 1 cρ j × B = 0,(1) where Φ g is the gravitational potential including the centrifugal terms. The third term has a magnitude ∼ 10 −7 (B/10 14 G) 2 times smaller than those of the first and second terms. Deviation owing to the Lorentz force is small enough to be treated as a perturbation on a background equilibrium. We limit our consideration to an axially symmetric configuration for the magnetic-field configuration. The poloidal and toroidal components of the magnetic field are expressed by two functions Ψ and S, respectively, as follows: B = ∇ × Ψ ̟ e φ + S ̟ e φ ,(2) where ̟ = r sin θ is the cylindrical radius, and e φ is the azimuthal unit-vector in (r, θ, φ) coordinates. When the equilibrium is barotropic, i.e., the constant surfaces of ρ and P are parallel, the azimuthal current j φ is described in the form 4πj φ c = ρ̟ dK dΨ + S ̟ dS dΨ ,(3) where the current function S should be a function of Ψ, and K is another function of Ψ. For the axially symmetric barotropic equilibrium, acceleration due to the Lorentz force, which is abbreviated as a ≡ (cρ) −1 j × B, is given by a = 1 4π ∇K.(4) The force balance (1) is described by gradient terms of scalar functions. The magnetic function Ψ is obtained by solving the Ampére-Biot-Savart law with the source term (Equation (3)) after the functional forms S(Ψ) and K(Ψ) are specified. For simplicity, we assume that K is a linear function of Ψ, K = K 0 Ψ, and S = 0 in Equation (3). The poloidal magnetic field is purely dipolar and is given by Ψ = g(r) sin 2 θ. The radial function g is solved with appropriate boundary conditions: in a vacuum at the surface R, and g = 0 at the core-crust interface r c . The latter refers to the magnetic field expelled from the core. For the case where the field penetrates into the core, g smoothly connects to the interior solution at r c . We normalize the radial function by the dipole field strength B 0 at the surface, g(R) = B 0 R 2 /2. The magnetic geometry discussed above is a simple initial model to examine the magnetic field evolution. However, purely poloidal magnetic field configurations are unstable according to an energy principle (Tayler 1973;Markey & Tayler 1973;Wright 1973) and numerical MHD simulation (Braithwaite & Nordlund 2006;Braithwaite 2009;Lander & Jones 2011a,b;Mitchell et al. 2015). Dynamical simulations revealed that the final state after a few Alfvén-wave crossing times is a twisted-torus configuration, in which the poloidal and toroidal components of comparable field strengths are tangled. Moreover, recent three-dimensional simulation shows very asymmetric equilibrium (Becerra et al. 2022). These studies are concerned with the configuration in an entire star. The information relevant to our study corresponds to the magnetic field in a thin outer layer; therefore, the present understanding is quite incomplete. For example, the ratio of toroidal to poloidal components decreases near the surface, because the toroidal field should vanish outside the exterior. However, the ratio in the crust located near the surface is uncertain, almost zero or in the order of unity, although both components are comparable in magnitude inside the star. Initially, a simple magnetic field configuration is used in this paper, but it is necessary to improve the configuration. We discuss the non-barotropic equilibrium of the magnetic field. From Equation (1), the acceleration owing to the Lorentz force satisfies ∇ × a = ∇ × 1 ρ ∇P = 0.(5) The acceleration a is no longer described by the gradient of a scalar, but may be generalized as a = 1 4π ∇K + α∇β,(6) where α and β are functions of r and θ, and ∇ × a = ∇α × ∇β = 0 is assumed. Owing to almost arbitrary functions α and β, the constraint on the electric current, and hence to the magnetic-field configuration, is relaxed in the non-barotropic case. Non-barotropicity has been studied in magnetic deformation (Mastrano et al. 2011(Mastrano et al. , 2013(Mastrano et al. , 2015. Barotropic (Haskell et al. 2008;Kojima et al. 2021b) and non-barotropic models are significantly different. (Mastrano et al. 2011(Mastrano et al. , 2013(Mastrano et al. , 2015 up to approximately one order of magnitude in the resulting ellipticity. The effect is important; however, the treatment remains unclear. Therefore, we introduce the models of ∇ × a to study nonbarotropicity in Section 4. Magnetic-field Evolution Our consideration is limited to the inner crust of a neutron star, where the mass density ranges from ρ c = 1.4 × 10 14 g cm −3 at the core-crust boundary r c to the neutron-drip density ρ 1 = 4 × 10 11 g cm −3 at R = 12 km. We ignore the outer crust compared to "ocean" and treat the exterior region of r > R as the vacuum. The crust thickness ∆r ≡ R − r c is assumed to be ∆r/R = 0.05, ∆r = 0.6 km. The Lorentz force j × B due to the magnetic-field evolution is not fixed in a secular timescale. The evolution in the crust is governed by the induction equation ∂ ∂t B = −∇ × 1 en e j × B + c σ j ,(7) where n e is the electron-number density, and σ is the electric conductivity. The first term in Equation (7) represents the Hall drift, and the Hall timescale τ H is estimated as follows: τ H = 4πen ec (∆r) 2 cB 0 = 7.9 × 10 5 yr B 0 10 14 G −1 ,(8) where the electron-number density n ec at the core-crust boundary, crust thickness ∆r, and dipole field strength B 0 at the surface are used. This timescale is shorter than that of the Ohmic decay, which is the second term in Equation (7) in the strong-magnetic-field regime. The Hall-Ohmic evolution was numerically simulated in the Hall timescale (Pons & Geppert 2007;Kojima & Kisaka 2012;Viganò et al. 2013;Gourgouliatos et al. 2013;Gourgouliatos & Cumming 2014;Viganò et al. 2021) for axially symmetric models. Recently, the calculation has been extended to 3D models (Wood & Hollerbach 2015;Viganò et al. 2019;De Grandis et al. 2020;Gourgouliatos et al. 2020;Igoshev et al. 2021), revealing some of the effects ignored in the 2D models. Here, our calculation is limited to the early phase of the evolution in a simpler axial-symmetric model. We consider only the Hall drift term in Equation (7) and rewrite the equation as follows: ∂ ∂t B = −∇ × 1 en e j × B = −∇χ × a − χ∇ × a,(9) where χ ≡ cρ en e = 4πρ c (∆r) 2 τ H B 0χ .(10) In Equation (10),χ is a dimensionless function, which represents an inverse of the electron fraction. The electronnumber density is obtained from the proton fraction of the equilibrium nucleus in "cold catalyzed matter," i.e., it is determined in the ground state at T = 0 K. The data given by Douchin & Haensel (2001) is approximately fitted by a smooth function:χ =ρ 1/2 0.32 + 0.66ρ ,(11) whereρ ≡ ρ/ρ c . The spatial-density profile of a neutron star in r c ≤ r ≤ R is approximated as follows (Lander & Gourgouliatos 2019): ρ = ρ ρ c = 1 − 1 − ρ 1 ρ c 1/2 r − r c ∆r 2 .(12) The radial derivative dχ/dr = (dχ/dρ)(dρ/dr) sharply changes, owing to an abrupt decrease in the density near the surface; however,χ is a smoothly varying function of O(1). This functional behavior, which originates from the stellardensity profile that is inherent in neutron stars, is crucial in our numerical calculation. Different fitting formulae are discussed for different equation of state in Pearson et al. (2018); however, the difference inχ is not significant in our analysis. We consider the early phase of the evolution in the axially symmetric equilibrium model, in which a φ = 0. From Equation (9) at t = 0, we obtain that ∂B φ /∂t = 0, but ∂B r /∂t = ∂B θ /∂t = 0. The azimuthal component B φ changes linearly with time t, whereas the poloidal components change with t 2 . We limit our consideration to the lowest order of t only and ignore the change in the poloidal magnetic field. The early phase of the toroidal magnetic field may be approximated as δB φ = δS ̟ t τ H ,(13) where δS is a function of r and θ. Because it is associated with δB φ , the poloidal current changes; thus, the Lorentz force δf = c −1 (δj × B + j × δB) also changes. We observe that the non-zero component is δf φ because j p = B φ = 0 at t = 0, and we explicitly write δf φ = 1 4π̟ 2 [∇δS × ∇Ψ] φ t τ H .(14) Quasi-stationary Elastic Response We assume that the solid crust elastically acts against the force δf φ . The change is so slow that the elastic evolution is quasi-stationary. The acceleration ∂ 2 ξ i /∂t 2 of the elastic displacement vector ξ i is dismissed; thus, the elastic force is balanced with the change in the Lorentz force, i.e., when the gravity and pressure in Equation (1) is assumed to be fixed. The elastic force is expressed by the trace-free strain tensor σ ij and a shear modulus µ; therefore, the force balance is ∇ i 2µσ i φ + δf φ = 0,(15)σ ij = 1 2 (∇ i ξ j + ∇ j ξ i ),(16) where incompressible motion ∇ k ξ k = 0 is assumed. Alternatively, the equivalent form is ∇ i 2µσ iφ + 1 4π B i δB φ = 0.(17) The relevant component induced by δf φ is the azimuthal displacement ξ φ only, and the shear tensors that are determined by it are σ rφ = r 2 ξ φ r ,r , σ θφ = sin θ 2r ξ φ sin θ ,θ .(18) The shear modulus µ increases with density, and it may be approximated as a linear function of ρ, which is overall fitted to the results of a detailed calculation reported in a previous study (see Figure 43 in Chamel & Haensel 2008). µ = µ c ρ ρ c ,(19) where µ c = 10 30 erg cm −3 at the core-crust interface. The shear speed v s in Equation (19) is constant through the crust: v s = µ ρ 1/2 = 8.5 × 10 7 cm s −1 .(20) To solve Equation (15), we use an expansion method with the Legendre polynomials P l (cos θ) and radial functions k l (r), a l (r) as follows: ξ φ = − l≥1 rk l P l,θ t τ H ,(21)δf φ = − l≥1 r −3 a l P l,θ t τ H .(22) The displacement ξ φ is decoupled with respect to the index l, owing to spherical symmetry, i.e., µ(r). Equation (15) is reduced to a set of ordinary differential equations for k l (Kojima et al. 2022): (µr 4 k ′ l ) ′ − (l − 1)(l + 2)µr 2 k l = −a l ,(23) where a prime ′ denotes a derivative with respect to r. The boundary conditions for the radial functions k l are given by the force-balance across the surfaces at r c and R. That is, the shear-stress tensor σ rφ vanishes because other stresses for the fluid and magnetic field are assumed to be continuous. Explicitly, we have k ′ l = 0 both at r c and R. Note that a mode with l = 1 is special in Equation (23), and k 1 is simply obtained by integrating a 1 with respect to r. ELASTIC DEFORMATION IN BAROTROPIC MODEL The magnetic-field evolution in Equation (9) is driven by two terms, which are separately examined. We first consider the barotropic case, in which ∇ × a = 0. The evolution is driven by the first term in Equation (9), i.e., the distribution of the electron fraction. The linear growth term δS in Equation (13) is obtained using Equation (4) as δS = 2K 0 ρ c (∆r) 2 3B 0χ ′ g sin θP 2,θ .(24) In the case where the poloidal magnetic field (Ψ = g sin 2 θ) is confined in the crust, the constant K 0 is numerically obtained as K 0 = 6.0 × B 0 /(ρ c (∆r) 2 ). Alternatively, we express K 0 = 8.6 × 10 1 v 2 a /(B 0 R 2 ), where v a is the Alfvén speed in terms of B 0 and ρ 1 : v a = B 0 √ 4πρ 1 = 4.5 × 10 7 B 0 10 14 G cm s −1 .(25) The Lorentz force δf φ in Equation (14) is calculated using Equation (24). For later convenience, we consider a general form δS = y l (r) sin θP l,θ . By using an identity for the Legendre polynomials, we reduce Equation (14) to δf φ = 1 4πr 3 (lg ′ y l + 2gy ′ l ) l + 1 2l + 1 P l−1,θ + (−(l + 1)g ′ y l + 2gy ′ l ) l 2l + 1 P l+1,θ t τ H .(26) Thus, the radial functions a l−1 and a l+1 in Equation (22) are induced by y l . By numerically solving Equation (23) for k l 's, we obtain ξ φ in Equation (21) and the shear stress tensors, σ rφ and σ θφ , in Equation (18). For Equation (24), the results are expressed using a combination of k 1 and k 3 . Results The shear stress increases homologously with time, i.e., the spatial profile of the shear force is unchanged, but the magnitude increases with time. The numerical calculation provides the maximum shear stress σ max with respect to (r, θ) in the crust as follows: σ max = 6.5 × 10 1 v a v s 2 t τ H .(27) The maximum is determined using a ratio of the shear speed v s in Equation (20) to the Alfvén speed v a in Equation (25). Elastic equilibrium is possible, only when the shear strain satisfies a particular criterion. We adopt the following (the Mises criterion) to determine the elastic limit: 1 2 σ ij σ ij ≤ (σ max ) 2 ≤ (σ c ) 2 ,(28) where σ c is a number σ c ≈ 10 −2 − 10 −1 (Horowitz & Kadau 2009;Caplan et al. 2018;Baiko & Chugunov 2018). Thus, the period of the elastic response is limited by the constraint σ c : t ≤ t * ≡ 1.5 × 10 −3 σ c 0.1 v s v a 2 τ H ,(29) The breakup time t * becomes short, i.e., a few years for magnetars with B 0 = 10 15 G. This is in agreement with the recurrent time of the activity in magnetars. However, the timescale exceeds 1 Myr for most neutron stars with B < 10 13 G. Moreover, other evolution effects are important, and present results are not applicable. Energy The stored elastic energy is obtained by numerically integrating over the entire crust: ∆E ela = 2π R rc r 2 dr π 0 sin θdθ µσ ij σ ij = 5.8 × 10 −7 µ c R 3 σ c 0.1 2 t t * 2 ,(31) where we have used R 3 instead of R 2 ∆r to normalize the crustal volume because ∆r/R = 0.05 is fixed. The elastic energy E ela increases up to ≈ 10 41 erg at the breakup time t * . The magnetic energy ∆E mag is also obtained by ∆E mag = 2π R rc r 2 dr π 0 sin θdθ (δB φ ) 2 8π = 2.0 × 10 −4 B 2 0 R 3 σ c 0.1 2 v s v a 4 t t * 2 .(32) Here, the shear speed appears in ∆E mag because the breakup time t * in Equation (29) is used instead of τ H . The magnetic energy ∆E mag at t * is ∆E mag ≈ 2 × 10 43 (B 0 /10 14 G) −2 erg. However, it is considerably smaller than the (r-r c )/∆r ε mag (r) ε ela (r) Figure 1. Energy distribution in crust as a function of the radius. Normalized energy density ε(r) is plotted for magnetic energy, which is denoted by a dotted curve, and elastic energy, denoted by a solid curve. poloidal magnetic energy E mag,p , which is numerically calculated as E mag,p = 3.8B 2 0 R 3 ≈ 4 × 10 46 (B 0 /10 14 G) 2 erg. Note that total magnetic energy is conserved by the Hall evolution. Therefore, the same amount of polar magnetic energy decreases. However, we ignored the change in the poloidal component and its energy, which are proportional to t 2 . The ratio of Equations (31) and (32) is d∆E ela /dt d∆E mag /dt = ∆E ela ∆E mag = 8.1 × 10 −2 v a v s 2 = 2.3 × 10 −2 B 0 10 14 G 2 ,(33) where µ c and B 2 0 are eliminated using v s and v a . The ratio is proportional to B 2 0 ; thus, ∆E mag decreases in more strongly magnetized neutron stars. From the viewpoint of the energy flow from the poloidal component, the breakup energy ∆E ela ≈ 10 41 at the terminal is fixed, but the ∆E mag stored in the middle depends on the Hall drift speed. The elastic energy is efficiently accumulated through toroidal magnetic energy with an increase in B 0 ; ∆E ela > ∆E mag for B 0 > 6.7 × 10 14 G. Figure 1 shows spatial-energy densities ε ela (r) and ε mag (r), with regard to ∆E ela and ∆E mag in the crust, respectively. They are normalized as ε ela dr = ε ela dr = 1. Evidently, both energies are highly concentrated near the surface r ≈ R. This property comes from the radial derivative ofχ in Equation (10). dχ/dr = (dχ/dρ)(dρ/dr) is steep there even thoughχ is O(1). The large value comes from |dρ/dr|, i.e., a sharp decrease in density near the stellar surface, and it results in a smaller evolution timescale ≪ τ H in Equation (29). Spatial Distribution Shear stresses σ θφ and σ rφ are induced by the axial displacement ξ φ . A numerical calculation shows that the component σ rφ is considerably larger than σ θφ ; (σ rφ ) max ∼ 200(σ θφ ) max . Figure 2 shows their spatial distribution using a contour map of 2µσ θφ and 2µσ rφ in the r − θ plain. The angular dependence of σ θφ is σ θφ ∝ sin 2 θ cos θ, and it is anti-symmetric with respect to the equator (θ = π/2). Moreover, σ rφ is the sum of P 1,θ and P 3,θ , and it is symmetric with respect to θ = π/2. The magnitude σ = (σ ij σ ij /2) 1/2 is also shown in the right panel, and σ is sharp near the surface, as expected from the sharp energy-density-distribution in Figure 1 We discuss the modification of the poloidal magnetic field at the core-crust boundary. Thus far, the magnetic field is expelled there. When the field is penetrated to the core, the function g near the boundary and the constant K 0 in Equation (24) are changed. The former is unimportant because the functionχ ′ is sharp near the surface, and this fact determines the result, as shown in Figures 1 and 2. The constant K 0 for the penetrated field is 4.1 × 10 −2 times smaller than that for the expelled one. Consequently, the profile is almost unchanged, but the break time t * increases by a factor of 24 with the same dipole field strength. We consider the evolution driven by the second term, ∇ × a = 0 in Equation (9), which originates from the nonbarotropic material distribution. However, ∇ × a and the corresponding magnetic field cannot be easily estimated, unless the non-barotropic property is specified. A large freedom of choice hinders our analysis. Therefore, we simply model the term ∇ × a and understand the non-barotropic effect in its magnitude and property. For this purpose, we assume a φ = 0 and (∇ × a) φ = N 2 (∆r) 2 F n (r)P 2,θ ,(34) where N is a constant that characterizes the non-barotropic strength, and it has the dimension of velocity. Additionally, F n is a non-dimensional radial function. We consider a small deviation from the barotropic case, for which the second term in Equation (6) is smaller than the first term. Therefore, the magnetic field is approximated using the barotropic case, i.e., the poloidal magnetic function Ψ and S = 0. This treatment constrains the normalization N in Equation (34) with respect to the magnitude. By the dimensional argument, we have |N | ≪ R/t ff ∼ 10 9 cm s −1 , where t ff is a freefall timescale. Moreover, |N | < v a and |N | < v s ∼ 10 8 cm s −1 are also likely because the crust is in magneto-elastic equilibrium. The angular dependence of Equation (34) is chosen for δS to be the same as in Equation (24). The radial function F n has a maximum that is normalized as unity, and it vanishes at inner and outer boundaries; the function is modeled as follows: F n = 256 27 r − r c ∆r n R − r ∆r 4−n ,(35) where n = 1 or 3. The model with n = 1 is referred to as the "in" model because the maximum is located at r = r c + ∆r/4, whereas that with n = 3 is referred to as the "out" model because the maximum is located at r = R − ∆r/4. Results of a Simple Model We neglect the first term in Equation (9), and consider the magnetic-field evolution driven by the term ∇ × a (Equation (34)) only. Similar to the calculations in Section 3, a linearly growing shear-stress is obtained, owing to ξ φ . The period of the elastic response is limited by t ≤ t * ≡ n 1 × σ c 0.1 v s N 2 τ H ,(36) where n 1 is a number of the order of 10 −2 , depending on the models, as listed in Table 1. Owing to our simple modeling, the comparison between the barotropic and non-barotropic models is uncomplicated; the Alfvén speed v a in Equation (29) is formally substituted by N in Equation (36). model n1 (t * ) n2 (∆E ela ) n3 (∆Emag) in 8.0 × 10 −3 2.1 × 10 −3 7.2 × 10 −7 out 1.2 × 10 −2 2.4 × 10 −4 1.0 × 10 −6 ave 3.6 × 10 −3 2.9 × 10 −4 2.9 × 10 −7 min 1.9 × 10 −3 0.49 × 10 −4 0.73 × 10 −7 max 5.5 × 10 −3 7.0 × 10 −4 6.4 × 10 −7 The elastic energy ∆E ela and toroidal magnetic energy ∆E mag stored inside the crust are also summarized as follows: ∆E ela = n 2 × µ c R 3 σ c 0.1 2 t t * 2 ,(37)∆E mag = n 3 × B 2 0 R 3 σ c 0.1 2 v s N 4 t t * 2 ,(38) where n 2 ≈ 6×10 −4 , and n 3 ≈ 10 −6 , as listed in Table 1. The elastic energy ∆E ela does not depend on N ; however, the timescale (36) does. The amount of elastic energy is unrelated to the detailed process, which affects the accumulation speed in the crust. At the breakup time, the elastic energy is ∆E ela = 2 × 10 44 − 2 × 10 45 erg. This energy is more than three orders of magnitude larger than the energy (31) considered in the previous section. The difference is made clear when considering the energy-density distribution. Figure 3 shows the energy-density distribution in the crust. The difference in the toroidal magnetic energy clearly originates in the model choice; the energy density spreads more towards the interior for the "in" model, whereas it spreads more towards the exterior for the "out" model. Most of the elastic energy is localized near the inner core-crust boundary; however, the distribution in the "out" model is shifted outwardly with the second peak (∼ r c + 0.8∆r) produced by the input model. The integrated elastic energy in the "out" model becomes one order of magnitude smaller than that in the "in" model. The amount of elastic energy at the breakup clearly depends on the spatial distribution of the energy density because the shear modulus µ is a strongly decreasing function toward the surface. The elastic limit of the entire crust is typically determined using a condition to the shear σ ij near the surface. The total elastic energy ∼ µσ ij σ ij d 3 x thereby decreases as σ ij σ ij is localized towards the exterior. The breakup elastic energy ∆E ela ∼ 10 41 erg at t * in the previous section is an extreme case because the energy density is concentrated near the surface. Figure 4 shows the magnitude of shear stress σ inside the crust. The contours of σ in the two models are different. We identified that the dominant component in the "in" model (left panel) is σ θφ , which has an angular dependence that is described by σ θφ ∝ sin 2 θ cos θ. The maximum of σ is given along a line cos θ = ±1/ √ 3 (θ ∼ 55 • , 125 • ). The component σ rφ is dominant in the "out" model (right panel). Sharp peaks are localized near the surface, similar to the right panel in Figure 2; however, the localization is not as pronounced as in Figure 2. The magnitude σ, which is important to determine the critical limit, is large near the surface. Figure 4. Crust contour map of magnitude of stress tensor normalized using the maximum. The left panel for the "in" model shows σ ≈ σ θφ , whereas the right panel for the "out" model shows σ ≈ σ rφ . Results of a Model Including Higher Multipoles In a more realistic situation, the solenoidal acceleration may fluctuate spatially. We consider the sum of multi-pole components P l,θ : (∇ × a) φ = N 2 (∆r) 2 lmax l=2 l 2 −4/3 λ l F n (r)P l,θ ,(39) where λ l F n is a radial function that is randomly selected from ±F 1 or ±F 3 depending on l. As discussed for Equation (26), the radial function k l in the azimuthal displacement ξ φ is solved for the source term that originates from λ l−1 F n and λ l+1 F n ; thus, the amplitude |k l | fluctuates according to the randomness. We fix the overall constant N . Equation (39) is reduced to Equation (34) when l max = 2. Moreover, higher l-modes, up to l max = 30 with the power-law weight, are included. The power-law index is considerably steep; therefore, the dominant component is still described by l = 2. We calculated 20 models by randomly mixing λ l F n . The numerical results are summarized in the same forms as eqs. (36)-(38), and the numerical values n i (i = 1, 2, 3) in the breakup time and energies are listed in Table 1 according to the average, minimum, and maximum for the 20 models. These numerical values are of the same order as those by a single mode with l = 2 because we include higher l-modes as the correction. Interestingly, the breakup time t * generally becomes shorter than that for a single mode with l = 2 because the higher modes l > 3 are cooperative. Figure 5 demonstrates the spatial distribution of the shear stress tensor. Two models are shown using contours of the magnitude of σ. In the left panel, the sub-critical regions are on a constant θ line with a sharp peak near the surface. In the other model (right panel), a peak is observed at θ = 0 near the surface. The angular position of the peak is different between the two models. As shown in Figure 4, the spatial pattern along a constant θ comes from the component σ θφ , whereas the sharp peak near the surface is due to σ rφ . The mixing of the two types of radial functions, F 1 and F 3 , and angular functions P l with random phases and weights only complicates the spatial distribution of σ. A sharp peak is likely to be located near the surface. The outer part of the crust is always fragile; thus, the breakup time becomes shorter. SUMMARY AND DISCUSSION We have considered the evolution of elastic deformation over a secular timescale (> 1 yr) starting from zero displacement. The initial state is related to the dynamic force balance that is determined within a second. When a neutron star cools below the melting temperature T ∼ 10 9 K, its crust is crystallized. Meanwhile, the pressure, gravity, and Lorentz force are balanced without the elastic force. In another situation, the elastic energy settles to the ground state, and zero displacement occurs after the energy is completely released at crustal fracture. Therefore, the initial condition is simple and natural. When the MHD equilibrium is axisymmetric, the azimuthal component of the magnetic field increases linearly according to the Hall evolution. Consequently, the elastic deformation in the azimuthal direction is induced to cancel the change in the Lorentz force, and the shear strain gradually increases. We estimate the range of the elastic response. Beyond the critical limit, the crust responds plastically or fractures. Our calculations provide the breakup time and shear distribution at the threshold. For the barotropic case, the breakup time until fracture is proportional to the cube of the magnetic-field strength. The time becomes a few years for a magnetar with a surface dipole of B 0 ∼ 10 15 G, when the field is located outside the core. However, it exceeds 1 Myr for most radio pulsars (B 0 < 10 13 G), and the process is irrelevant to them. In addition to the field strength, the timescale is typically shortened by a factor of 10 −3 smaller than the Hall timescale because the elastic displacement is highly concentrated near the surface. The driven mechanism is related to the instability associated with electron-density gradients (Wood et al. 2014). The distribution in any realistic model of neutron-star crusts is considerably sharp; therefore, the evolution is general. Another type of Hall-drift instability occurs in the presence of a non-uniform magnetic field (Rheinhardt & Geppert 2002), which is not considered here, and its energy would be smaller, owing to the size of the irregularity. In our calculation, we do not follow the instability; instead, we estimate the energy transferred to the elastic deformation. The elastic energy at the critical limit in the model driven by the electron-number-density gradient is ∼ 10 41 erg. The amount of energy is of the same order as that of short bursts in magnetars. The breaktime of ∼ 10 years also reconciles with the observed recurrent-time of the bursts. However, the energy ∼ 10 41 erg is smaller than that of giant flares ∼ 10 44 − 10 46 erg (Turolla et al. 2015;Kaspi & Beloborodov 2017;Esposito et al. 2021, for a review). The total elastic energy derived in Section 3 is based on the electron-number density in cold catalyzed matter, i.e., the ground state at T = 0 K. If the assumption was relaxed, the non-barotropic effects might increase the total elastic energy, as considered in Section 4. When the pressure distribution is no longer expressed solely by the density ρ, the magnetic evolution is affected by the solenoidal acceleration a = ∇× (ρ −1 ∇P ) = 0. We have also considered this effect by creating the model in terms of a spatial function and an overall strength parameter, which are assumed to be constant in time in our non-barotropic model. Using the simplified model, we calculated the breakup time of the crustal failure and the energies stored in the crust. The results were comparable to those for the barotropic case. The strength parameter significantly affects the breakup time; the larger the magnitude, the shorter the breakup time. However, the amount of elastic energy at the breakup does not depend on the strength parameter, but only on the spatial function. The maximum elastic energy considerably increases up to ∼ 10 45 erg. However, the model is still primitive, and thermal evolution should also be incorporated to investigate a more realistic situation. The maximum energy has been explored thus far; however, a natural question arises. What is the fraction of energy that is released at the crustal fracture when the strain exceeds the threshold? This question is important, but at present unclear, owing to our lack of understanding of the fracture dynamics. Therefore, we present the following discussion. As depicted in Figure 5, in the realistic mixture model, a peak of shear strain σ is probably located near the surface, where the crust is fragile. Therefore, the fracture should not include the whole crust, but only the shallow crust. For this case, the released energy is not the whole elastic energy ∼ 10 45 erg, but the energy stored in the restricted region, i.e., a small fraction of the total, probably ∼ 10 41 erg. The elastic deformation driven by the Hall evolution is simulated for the first time. The critical structure at the breakup time is crucial for subsequent evolution, irrespective of plastic evolution or fracturing. The transition may appear as a burst on a magnetar. The magnetic-field rearrangement due to a mimic burst was incorporated in the Hall evolution (Pons & Perna 2011;Viganò et al. 2013;Dehman et al. 2020), without solving elastic deformation. These studies estimated the critical state based on the magnetic stress M ij . In numerical simulations, M ij changed, and the critical state was assumed when a condition among M ij reached the threshold value. Similar approximations for the elastic limit, which were derived solely from M ij , were used in a previous study (Lander et al. 2015;Lander & Gourgouliatos 2019;Suvorov & Kokkotas 2019). Mathematically, the shear stress σ ij cannot be derived from M ij without solving the appropriate differential equation ∇ j (2µσ j i + M j i ) = 0 (see Equation (17)). Therefore, previous results with the criterion M ij are questionable. Our calculation shows that the period of elastic evolution is typically 10 −3 times the Hall timescale; however, this value depends on the strength and geometry of the magnetic field. The timescale is shorter than the Ohmic timescale for B ≥ 10 13 G. The magnetic-field evolution beyond the period may be described by including the viscous bulk flow when the crust responds plastically. The effect of the plastic flow on the Hall-Ohmic evolution was considered by assuming a plastic flow everywhere in the crust (Kojima & Suzuki 2020) or by using an approximated criterion (Lander & Gourgouliatos 2019;Gourgouliatos & Lander 2021;Gourgouliatos 2022). The effect may be regarded as additional energy lost to the Ohmic decay. However, the post-failure evolution significantly depends on the modeling in the numerical simulation (Gourgouliatos & Lander 2021;Gourgouliatos 2022). That is, the region of plastic flow is either local or global when the failure criterion is satisfied. Therefore, the manner of incorporation of crust failure in the numerical simulation must be explored. Finally, further investigation is required before the elastic deformation toward the crust failure can be considered a viable model. The effect of magnetic field configuration should be considered because there are many degrees of freedom concerning it. Moreover, the outer boundary, i.e., inner-outer crust boundary or exterior magnetosphere is crucial as the crust becomes more fragile with increasing radius. Meanwhile, the electric conductivity decreases and the Ohmic loss becomes more important. By considering coupling to an exterior magnetosphere, twisting of the magnetosphere as well as crustal motion will be calculated in a secular timescale to match astrophysical observations, e.g., to describe the pre-stage of outbursts, like SGR 1830-0645 (Younes et al. 2022). Figure 2 . 2Contour map of stress tensor in crust. The left panel shows 2µσ θφ , the middle 2µσ rφ , and the right the magnitude σ. They are normalized according to the maximum. Negative values are plotted using a dotted curve in the left panel. The magnitude σ in the right panel is shown in the outer small region near the surface, R − 0.1∆r ≤ r ≤ R. Figure 3 . 3Spatial distribution of magnetic energy (dotted curve) and elastic energy (solid curve) in the crust. Normalized energy densities ε(r) are plotted for two models. Figure 5 . 5Color contour map of crust for the magnitude of the stress tensor σ normalized using the maximum; two models are compared. Table 1 . 1Numericalvalues in eqs. (36), (37), and (38). . ELASTIC DEFORMATION IN NON-BAROTROPIC MODEL ACKNOWLEDGEMENTSThis work was supported by JSPS KAKENHI Grant Number JP17H06361,JP19K03850. . R Abbott, H Abe, F Acernese, 10.3847/1538-4357/ac17eaApJ. 92180Abbott, R., Abe, H., Acernese, F., et al. 2021a, ApJ, 921, 80, doi: 10.3847/1538-4357/ac17ea . 10.3847/2041-8213/abffcdarXiv:2204.04523ApJL. 91327arXiv e-prints-. 2021b, ApJL, 913, L27, doi: 10.3847/2041-8213/abffcd -. 2022, arXiv e-prints, arXiv:2204.04523. https://arxiv.org/abs/2204.04523 . M A Alpar, D Pines, P W Anderson, J Shaham, 10.1086/161616ApJ. 276325Alpar, M. A., Pines, D., Anderson, P. W., & Shaham, J. 1984, ApJ, 276, 325, doi: 10.1086/161616 . P W Anderson, N Itoh, 10.1038/256025a0Nature. 25625Anderson, P. W., & Itoh, N. 1975, Nature, 256, 25, doi: 10.1038/256025a0 . D A Baiko, A Chugunov, 10.1093/mnras/sty2259MNRAS. 4805511Baiko, D. A., & Chugunov, A. I. 2018, MNRAS, 480, 5511, doi: 10.1093/mnras/sty2259 . A Basu, B Shaw, D Antonopoulou, 10.1093/mnras/stab3336MNRAS. 5104049Basu, A., Shaw, B., Antonopoulou, D., et al. 2022, MNRAS, 510, 4049, doi: 10.1093/mnras/stab3336 . L Becerra, A Reisenegger, J A Valdivia, M E Gusakov, 10.1093/mnras/stac102MNRAS. 511732Becerra, L., Reisenegger, A., Valdivia, J. A., & Gusakov, M. E. 2022, MNRAS, 511, 732, doi: 10.1093/mnras/stac102 . C D Bochenek, V Ravi, K V Belov, 10.1038/s41586-020-2872-xNature. 58759Bochenek, C. D., Ravi, V., Belov, K. V., et al. 2020, Nature, 587, 59, doi: 10.1038/s41586-020-2872-x . J Braithwaite, 10.1111/j.1365-2966.2008.14034.xMNRAS. 397763Braithwaite, J. 2009, MNRAS, 397, 763, doi: 10.1111/j.1365-2966.2008.14034.x . J Braithwaite, Å Nordlund, 10.1051/0004-6361:20041980A&A. 4501077Braithwaite, J., & Nordlund,Å. 2006, A&A, 450, 1077, doi: 10.1051/0004-6361:20041980 . M E Caplan, A S Schneider, C J Horowitz, 10.1103/PhysRevLett.121.132701PhRvL. 121132701Caplan, M. E., Schneider, A. S., & Horowitz, C. J. 2018, PhRvL, 121, 132701, doi: 10.1103/PhysRevLett.121.132701 . N Chamel, P Haensel, 10.12942/lrr-2008-10Living Reviews in Relativity. 1110Chamel, N., & Haensel, P. 2008, Living Reviews in Relativity, 11, 10, doi: 10.12942/lrr-2008-10 . B C Andersen, CHIME/FRB CollaborationK M Bandura, CHIME/FRB Collaboration10.1038/s41586-020-2863-yNature. 587CHIME/FRB Collaboration, Andersen, B. C., Bandura, K. M., et al. 2020, Nature, 587, 54, doi: 10.1038/s41586-020-2863-y . D De Grandis, R Turolla, T S Wood, 10.3847/1538-4357/abb6f9ApJ. 90340De Grandis, D., Turolla, R., Wood, T. S., et al. 2020, ApJ, 903, 40, doi: 10.3847/1538-4357/abb6f9 . C Dehman, D Viganò, N Rea, 10.3847/2041-8213/abbda9ApJL. 90232Dehman, C., Viganò, D., Rea, N., et al. 2020, ApJL, 902, L32, doi: 10.3847/2041-8213/abbda9 . R Dib, V M Kaspi, F P Gavriil, 10.1086/524653ApJ. 6731044Dib, R., Kaspi, V. M., & Gavriil, F. P. 2008, ApJ, 673, 1044, doi: 10.1086/524653 . F Douchin, P Haensel, 10.1051/0004-6361:20011402A&A. 380151Douchin, F., & Haensel, P. 2001, A&A, 380, 151, doi: 10.1051/0004-6361:20011402 . C M Espinoza, A G Lyne, B W Stappers, M Kramer, 10.1111/j.1365-2966.2011.18503.xMNRAS. 4141679Espinoza, C. M., Lyne, A. G., Stappers, B. W., & Kramer, M. 2011, MNRAS, 414, 1679, doi: 10.1111/j.1365-2966.2011.18503.x . P Esposito, N Rea, G L Israel, 10.1007/978-3-662-62110-3_3Astrophysics and Space Science Library. T. M. Belloni, M. Méndez, & C. Zhang461Astrophysics and Space Science LibraryEsposito, P., Rea, N., & Israel, G. L. 2021, in Astrophysics and Space Science Library, Vol. 461, Astrophysics and Space Science Library, ed. T. M. Belloni, M. Méndez, & C. Zhang, 97-142, doi: 10.1007/978-3-662-62110-3 3 . L M Franco, B Link, R I Epstein, 10.1086/317121ApJ. 543987Franco, L. M., Link, B., & Epstein, R. I. 2000, ApJ, 543, 987, doi: 10.1086/317121 . E Giliberti, G Cambiotti, M Antonelli, P M Pizzochero, 10.1093/mnras/stz3099MNRAS. 4911064Giliberti, E., Cambiotti, G., Antonelli, M., & Pizzochero, P. M. 2020, MNRAS, 491, 1064, doi: 10.1093/mnras/stz3099 . F Gittins, N Andersson, D I Jones, 10.1093/mnras/staa3635MNRAS. 5005570Gittins, F., Andersson, N., & Jones, D. I. 2021, MNRAS, 500, 5570, doi: 10.1093/mnras/staa3635 . K N Gourgouliatos, arXiv:2202.06662arXiv e-printsGourgouliatos, K. N. 2022, arXiv e-prints, arXiv:2202.06662. https://arxiv.org/abs/2202.06662 . K N Gourgouliatos, A Cumming, 10.1093/mnras/stt2300MNRAS. 4381618Gourgouliatos, K. N., & Cumming, A. 2014, MNRAS, 438, 1618, doi: 10.1093/mnras/stt2300 . K N Gourgouliatos, A Cumming, A Reisenegger, 10.1093/mnras/stt1195MNRAS. 4342480Gourgouliatos, K. N., Cumming, A., Reisenegger, A., et al. 2013, MNRAS, 434, 2480, doi: 10.1093/mnras/stt1195 . K N Gourgouliatos, R Hollerbach, A P Igoshev, 10.1093/mnras/staa1295MNRAS. 4951692Gourgouliatos, K. N., Hollerbach, R., & Igoshev, A. P. 2020, MNRAS, 495, 1692, doi: 10.1093/mnras/staa1295 . K N Gourgouliatos, S K Lander, 10.1093/mnras/stab1869MNRAS. 5063578Gourgouliatos, K. N., & Lander, S. K. 2021, MNRAS, 506, 3578, doi: 10.1093/mnras/stab1869 . B Haskell, D I Jones, N Andersson, 10.1111/j.1365-2966.2006.10998.xMNRAS. 3731423Haskell, B., Jones, D. I., & Andersson, N. 2006, MNRAS, 373, 1423, doi: 10.1111/j.1365-2966.2006.10998.x . B Haskell, L Samuelsson, K Glampedakis, N Andersson, 10.1111/j.1365-2966.2008.12861.xMNRAS. 385531Haskell, B., Samuelsson, L., Glampedakis, K., & Andersson, N. 2008, MNRAS, 385, 531, doi: 10.1111/j.1365-2966.2008.12861.x . C J Horowitz, K Kadau, 10.1103/PhysRevLett.102.191102PhRvL. 102Horowitz, C. J., & Kadau, K. 2009, PhRvL, 102, 191102, doi: 10.1103/PhysRevLett.102.191102 . A P Igoshev, K N Gourgouliatos, R Hollerbach, T S Wood, 10.3847/1538-4357/abde3eApJ. 909101Igoshev, A. P., Gourgouliatos, K. N., Hollerbach, R., & Wood, T. S. 2021, ApJ, 909, 101, doi: 10.3847/1538-4357/abde3e . V M Kaspi, A M Beloborodov, 10.1146/annurev-astro-081915-023329ARA&A. 55261Kaspi, V. M., & Beloborodov, A. M. 2017, ARA&A, 55, 261, doi: 10.1146/annurev-astro-081915-023329 . Y Kojima, S Kisaka, 10.1111/j.1365-2966.2012.20509.xMNRAS. 4212722Kojima, Y., & Kisaka, S. 2012, MNRAS, 421, 2722, doi: 10.1111/j.1365-2966.2012.20509.x . Y Kojima, S Kisaka, K Fujisawa, 10.1093/mnras/staa3489doi: 10.1093/mnras/staa3489MNRAS. 5062097MNRASKojima, Y., Kisaka, S., & Fujisawa, K. 2021a, MNRAS, 506, 3936, doi: 10.1093/mnras/stab1848 -. 2021b, MNRAS, 502, 2097, doi: 10.1093/mnras/staa3489 . 10.1093/mnras/stac036MNRAS. 511480-. 2022, MNRAS, 511, 480, doi: 10.1093/mnras/stac036 . Y Kojima, K Suzuki, 10.1093/mnras/staa1045MNRAS. 4943790Kojima, Y., & Suzuki, K. 2020, MNRAS, 494, 3790, doi: 10.1093/mnras/staa1045 . S K Lander, N Andersson, D Antonopoulou, A L Watts, 10.1093/mnras/stv432MNRAS. 4492047Lander, S. K., Andersson, N., Antonopoulou, D., & Watts, A. L. 2015, MNRAS, 449, 2047, doi: 10.1093/mnras/stv432 . S K Lander, K N Gourgouliatos, 10.1093/mnras/stz1042MNRAS. 4864130Lander, S. K., & Gourgouliatos, K. N. 2019, MNRAS, 486, 4130, doi: 10.1093/mnras/stz1042 . S K Lander, D I Jones, 10.1111/j.1365-2966.2010.18009.xdoi: 10.1111/j.1365-2966.2010.18009.xMNRAS. 4121730MNRASLander, S. K., & Jones, D. I. 2011a, MNRAS, 412, 1394, doi: 10.1111/j.1365-2966.2010.17998.x -. 2011b, MNRAS, 412, 1730, doi: 10.1111/j.1365-2966.2010.18009.x . X Li, M Ge, L Lin, 10.3847/1538-4357/ac6587ApJ. 93156Li, X., Ge, M., Lin, L., et al. 2022, ApJ, 931, 56, doi: 10.3847/1538-4357/ac6587 . P Markey, R J Tayler, 10.1093/mnras/163.1.77MNRAS. 16377Markey, P., & Tayler, R. J. 1973, MNRAS, 163, 77, doi: 10.1093/mnras/163.1.77 . A Mastrano, P D Lasky, A Melatos, 10.1093/mnras/stt1131MNRAS. 4341658Mastrano, A., Lasky, P. D., & Melatos, A. 2013, MNRAS, 434, 1658, doi: 10.1093/mnras/stt1131 . A Mastrano, A Melatos, A Reisenegger, T Akgün, 10.1111/j.1365-2966.2011.19410.xMNRAS. 4172288Mastrano, A., Melatos, A., Reisenegger, A., & Akgün, T. 2011, MNRAS, 417, 2288, doi: 10.1111/j.1365-2966.2011.19410.x . A Mastrano, A G Suvorov, A Melatos, 10.1093/mnras/stu2671MNRAS. 4473475Mastrano, A., Suvorov, A. G., & Melatos, A. 2015, MNRAS, 447, 3475, doi: 10.1093/mnras/stu2671 . S Mereghetti, V Savchenko, C Ferrigno, 10.3847/2041-8213/aba2cfApJL. 89829Mereghetti, S., Savchenko, V., Ferrigno, C., et al. 2020, ApJL, 898, L29, doi: 10.3847/2041-8213/aba2cf . J P Mitchell, J Braithwaite, A Reisenegger, 10.1093/mnras/stu2514MNRAS. 4471213Mitchell, J. P., Braithwaite, J., Reisenegger, A., et al. 2015, MNRAS, 447, 1213, doi: 10.1093/mnras/stu2514 . D J B Payne, A Melatos, 10.1111/j.1365-2966.2004.07798.xMNRAS. 351569Payne, D. J. B., & Melatos, A. 2004, MNRAS, 351, 569, doi: 10.1111/j.1365-2966.2004.07798.x . J M Pearson, N Chamel, A Y Potekhin, 10.1093/mnras/sty2413MNRAS. 4812994Pearson, J. M., Chamel, N., Potekhin, A. Y., et al. 2018, MNRAS, 481, 2994, doi: 10.1093/mnras/sty2413 . J A Pons, U Geppert, 10.1051/0004-6361:20077456A&A. 470303Pons, J. A., & Geppert, U. 2007, A&A, 470, 303, doi: 10.1051/0004-6361:20077456 . J A Pons, R Perna, 10.1088/0004-637X/741/2/123ApJ. 741123Pons, J. A., & Perna, R. 2011, ApJ, 741, 123, doi: 10.1088/0004-637X/741/2/123 . J A Rencoret, C Aguilera-Gómez, A Reisenegger, 10.1051/0004-6361/202141499A&A. 65447Rencoret, J. A., Aguilera-Gómez, C., & Reisenegger, A. 2021, A&A, 654, A47, doi: 10.1051/0004-6361/202141499 . M Rheinhardt, U Geppert, 10.1103/PhysRevLett.88.101103PhRvL. 88101103Rheinhardt, M., & Geppert, U. 2002, PhRvL, 88, 101103, doi: 10.1103/PhysRevLett.88.101103 . A G Suvorov, K D Kokkotas, 10.1093/mnras/stz2052MNRAS. 4885887Suvorov, A. G., & Kokkotas, K. D. 2019, MNRAS, 488, 5887, doi: 10.1093/mnras/stz2052 . R J Tayler, 10.1093/mnras/161.4.365MNRAS. 161365Tayler, R. J. 1973, MNRAS, 161, 365, doi: 10.1093/mnras/161.4.365 . R Turolla, S Zane, A L Watts, 10.1088/0034-4885/78/11/116901Reports on Progress in Physics. 78116901Turolla, R., Zane, S., & Watts, A. L. 2015, Reports on Progress in Physics, 78, 116901, doi: 10.1088/0034-4885/78/11/116901 . G Ushomirsky, C Cutler, L Bildsten, 10.1046/j.1365-8711.2000.03938.xMNRAS. 319902Ushomirsky, G., Cutler, C., & Bildsten, L. 2000, MNRAS, 319, 902, doi: 10.1046/j.1365-8711.2000.03938.x . D Viganò, A Garcia-Garcia, J A Pons, C Dehman, V Graber, 10.1016/j.cpc.2021.108001Computer Physics Communications. 265108001Viganò, D., Garcia-Garcia, A., Pons, J. A., Dehman, C., & Graber, V. 2021, Computer Physics Communications, 265, 108001, doi: 10.1016/j.cpc.2021.108001 . D Viganò, N Rea, J A Pons, 10.1093/mnras/stt1008MNRAS. 434123Viganò, D., Rea, N., Pons, J. A., et al. 2013, MNRAS, 434, 123, doi: 10.1093/mnras/stt1008 . D Viganò, D Martínez-Gómez, J A Pons, 10.1016/j.cpc.2018.11.022Computer Physics Communications. 237168Viganò, D., Martínez-Gómez, D., Pons, J. A., et al. 2019, Computer Physics Communications, 237, 168, doi: 10.1016/j.cpc.2018.11.022 . Z Wadiasingh, C Chirenti, 10.3847/2041-8213/abc562ApJL. 90338Wadiasingh, Z., & Chirenti, C. 2020, ApJL, 903, L38, doi: 10.3847/2041-8213/abc562 . T S Wood, R Hollerbach, 10.1103/PhysRevLett.114.191101PhRvL. 114Wood, T. S., & Hollerbach, R. 2015, PhRvL, 114, 191101, doi: 10.1103/PhysRevLett.114.191101 . T S Wood, R Hollerbach, M Lyutikov, 10.1063/1.4879810Physics of Plasmas. 2152110Wood, T. S., Hollerbach, R., & Lyutikov, M. 2014, Physics of Plasmas, 21, 052110, doi: 10.1063/1.4879810 . G A Wright, 10.1093/mnras/162.4.339MNRAS. 162339Wright, G. A. E. 1973, MNRAS, 162, 339, doi: 10.1093/mnras/162.4.339 . G Younes, S K Lander, M G Baring, 10.3847/2041-8213/ac4700ApJL. 92427Younes, G., Lander, S. K., Baring, M. G., et al. 2022, ApJL, 924, L27, doi: 10.3847/2041-8213/ac4700
[]
[ "Prototype Helps Federated Learning: Towards Faster Convergence", "Prototype Helps Federated Learning: Towards Faster Convergence" ]
[ "Yu Qiao [email protected] \nDepartment of Artificial Intelligence\nKyung Hee University\nYongin-si 17104Republic of Korea\n", "Seong-Bae Park [email protected] \nDepartment of Computer Science and Engineering\nKyung Hee University\nYongin-si 17104Republic of Korea\n", "Sun Moo Kang \nDepartment of Computer Science and Engineering\nKyung Hee University\nYongin-si 17104Republic of Korea\n", "Choong Seon \nDepartment of Computer Science and Engineering\nKyung Hee University\nYongin-si 17104Republic of Korea\n", "Hong [email protected] " ]
[ "Department of Artificial Intelligence\nKyung Hee University\nYongin-si 17104Republic of Korea", "Department of Computer Science and Engineering\nKyung Hee University\nYongin-si 17104Republic of Korea", "Department of Computer Science and Engineering\nKyung Hee University\nYongin-si 17104Republic of Korea", "Department of Computer Science and Engineering\nKyung Hee University\nYongin-si 17104Republic of Korea" ]
[]
Federated learning (FL) is a distributed machine learning technique in which multiple clients cooperate to train a shared model without exchanging their raw data. However, heterogeneity of data distribution among clients usually leads to poor model inference. In this paper, a prototype-based federated learning framework is proposed, which can achieve better inference performance with only a few changes to the last global iteration of the typical federated learning process. In the last iteration, the server aggregates the prototypes transmitted from distributed clients and then sends them back to local clients for their respective model inferences. Experiments on two baseline datasets show that our proposal can achieve higher accuracy (at least 1%) and relatively efficient communication than two popular baselines under different heterogeneous settings.
10.48550/arxiv.2303.12296
[ "https://export.arxiv.org/pdf/2303.12296v1.pdf" ]
257,663,990
2303.12296
c75dff121c385a8af155550aed924137510967e0
Prototype Helps Federated Learning: Towards Faster Convergence Yu Qiao [email protected] Department of Artificial Intelligence Kyung Hee University Yongin-si 17104Republic of Korea Seong-Bae Park [email protected] Department of Computer Science and Engineering Kyung Hee University Yongin-si 17104Republic of Korea Sun Moo Kang Department of Computer Science and Engineering Kyung Hee University Yongin-si 17104Republic of Korea Choong Seon Department of Computer Science and Engineering Kyung Hee University Yongin-si 17104Republic of Korea Hong [email protected] Prototype Helps Federated Learning: Towards Faster Convergence Federated learning (FL) is a distributed machine learning technique in which multiple clients cooperate to train a shared model without exchanging their raw data. However, heterogeneity of data distribution among clients usually leads to poor model inference. In this paper, a prototype-based federated learning framework is proposed, which can achieve better inference performance with only a few changes to the last global iteration of the typical federated learning process. In the last iteration, the server aggregates the prototypes transmitted from distributed clients and then sends them back to local clients for their respective model inferences. Experiments on two baseline datasets show that our proposal can achieve higher accuracy (at least 1%) and relatively efficient communication than two popular baselines under different heterogeneous settings. I. INTRODUCTION Federated Learning (FL) was first proposed by Google in 2016 and was originally used to address the update issues of Android terminals. Its original motivation is to carry out privacy-preserving Machine Learning (ML) based on datasets distributed on multiple computing nodes [1]. Essentially, FL is a distributed ML paradigm. With the further development of Artificial Intelligence (AI), the concept of FL has been further refined and developed. The essential feature of federated training is that the data of all parties are kept locally without revealing privacy issues. In addition, the data distribution of clients in federated scenarios is usually heterogeneous. Therefore, the key challenge is revealed: the data distribution among clients is usually not Independent and Identically Distributed (non-IID), which can lead to poor performance in model inferences [2], [11]. There are existing various studies trying to tackle the non-IID issue. FedAvg [1] is the first optimization algorithm for federated scenarios. MOON [3] employs a contrastive loss to minimize the difference between the representations learned by global and local models. FedNova [4] focuses on the aggregation stage. It lets different clients execute different numbers of local epochs in each global iteration, and then normalizes and scales their local updates before aggregating all local updates to ensure that the aggregated global updates have little biases. Inspired by [5], the classifier of the model in the federated settings has a greater bias than other layers. In other words, this means that predictions made through the classifier tend to be highly biased. Therefore, in this paper, we creatively propose to make predictions through the previous layer of the classifier. The output of the previous layer of the classifier is the feature space of the class, and the average of the feature spaces in the same class space can be defined as the "prototype" of this class. Their work [6] has shown that the information of prototypes can be effectively exploited to resolve heterogeneity in FL. The main contributions are summarized as follows: • We design a novel prototype-based federated learning framework where clients communicate with the server as typical federated training does, but model inferences are made based on the aggregated prototypes instead of the classifier. • We propose a prototype aggregation method, in which clients can upload their prototypes to the server for aggregation before finishing the last federation iteration. • Experiments over two popular benchmark datasets: MNIST [7] and Fashion-MNIST [8], show that our proposal has a higher test accuracy (at least 1% higher) and is relatively efficient in communication than the two popular baselines. II. PROPOSED FRAMEWORK A. Problem Statement Consider a distributed clients set φ with private sensitive dataset D i = {(x i , y i )} of size D i in the distributed edge network. Following the typical training process of FL, clients and the edge server cooperate to train a shared model F(ω; x i ), where ω is the model parameters of the global model and x i denotes the feature vector of one client i. Our objective is to minimize the loss function across heterogeneous clients as follows: arg min ω L(ω) = i∈φ D i i∈φ D i L i (F(ω; x i ), y i ),(1) where y i denotes the label of a sample, and L i is the empirical risk (e.g. cross-entropy loss) of client i, respectively. B. Proposed Federated Learning Framework The typical federated training process can be summarized as: (1). The server distributes the model to each client for local training; (2). Clients update the model parameters individually and send them back to the server for aggregation; (3). The server aggregates these uploaded latest model parameters, and then sends them back again to local clients for the next global iteration, repeating the above process until convergence. Our proposal is similar to this typical process, but in the last global round, clients not only need to update their own local model parameters as usual, but also calculate their own prototypes for each class. The calculated prototypes along with model parameters are sent to the server for aggregation. Finally, the aggregated prototypes and the latest model parameters are then sent back to local clients for model inferences. The overview of our proposed framework is presented in Fig. 1. C. Prototype-based Model Inference Strategy Generally, a deep learning model consists of two parts: feature extractor layers and decision-making layers. The former is denoted as f e (ω e ; x), and the latter is denoted as f d (ω d ; y). Therefore, the shared model can be written as F(ω; x) = f d (f e (ω e ; x), y), which means that the output of feature extractor layers acts as the input of the decisionmaking layers. 1) Prototype Calculation: Following the definition for prototypes given by [6], the prototype representing j-th class of client i-th can be formulated as: y i,j = 1 D i,j (x,y)∈Di,j f e (ω e ; x),(2) where D i,j is the distribution of client i-th belong to the j-th class, and D i,j is the size for D i,j . 2) Global Prototype Aggregation: As calculated by Eq. 2, these computed prototypes can be sent to the server for aggregation, which can be defined as follows: for i = 0, 1,..., N in parallel do 5: if t = T then 6: y j = 1 |φ| i∈φ y i,j ,(3) Local model updates. where |φ| is the number of clients in the network. This Equation means to average the prototypes from those clients with j-th class as global prototypes. 3) Prototype-based Model Inference: The results of Eq.3 can be viewed as a global representation of each class for model inference. Therefore, the prediction can be made by measuring the L2 distance between the local prototype f e (ω e ; x) and the global representation prototypes y j , which can be expressed as [10]: y = arg min j f e (ω e ; x) − y j 2 ,(4) where the final predicted label is denoted asŷ. Further details about prototype-based federated learning are shown in Algorithm.1. III. EXPERIMENTS A. Dataset and Local Model We use MNIST [7] and Fashion-MNIST [8] as benchmarks for comparison. MNIST is a handwritten digit recognition dataset with 10 digits and an image size of 28x28x1. Fashion-MNIST is more complex than MNIST, covering a total of 70,000 images of different items in 10 categories and an image size of 28x28x1. We use a 4-layer CNN network with 2 convolutional layers and 2 fully connected layers for MNIST and Fashion-MNIST, similar networks are also adopted in their work [5], [6]. B. Implementation Details We use 20 clients for all experiments, and the default training parameters for the MNIST and Fashion-MNIST datasets are set to B = 8, E = 1, η = 0.01, representing the local batch size, local epochs, and learning rate, respectively. We sample 5000 samples from the training dataset and distribute them to all clients to mimic the situation where training samples are limited in the real-world [6]. Further, we adopt the Dirichlet distribution [9] to simulate the heterogeneous setting among clients, which can be expressed as Dir(α), where the smaller α, the more unbalanced the data distribution among clients is. C. Accuracy and Communication Efficiency Comparison We compare our proposal with two popular baselines Local and FedAvg under different heterogeneities, where Local indicates that clients train their own model individually without any communication with the server or other clients, and FedAvg as the first federated algorithm is the most popular baseline. Note that we perform prototype-based model inference in each round during training in order to compare their test accuracies after each global iteration. The accuracy and communication efficiency for comparison of all methods on MNIST and Fashion-MNIST are shown in Fig.2 and Fig.3, respectively. It appears that our prototype-based model inference strategy achieves higher test accuracy and a relatively faster convergence rate in each global communication round than other methods under different heterogeneous settings on both these two datasets. To be more specific, our proposal on MNIST outperforms Local and FedAvg by 18.9% and 2.4% in terms of accuracy when α = 0.05, and also outperforms them by 31.5% and 1.0% when α = 0.05, respectively. Further, our proposal on Fashion-MNIST outperforms Local and FedAvg by 1.4% and 14.3% in terms of accuracy when α = 0.05, and also outperforms them by 18.4% and 7.4% when α = 0.1, respectively. IV. CONCLUSION In this paper, a prototype-based federated learning framework is proposed, which can help boost the performance of model inference. We first present the prototype calculation method, and then we introduce the prototype aggregation approach. Finally, we propose the prototype-based model inference strategy. Experiments show that our strategy can improve the accuracy by at least 1% on MNIST and Fashion-MNIST compared to two popular baselines Local and FedAvg, and can achieve relatively efficient communication. For future works, our proposal has the potential to be combined with other state-of-the-art methods and tests on more datasets. Figure 1 : 1The overview of the proposed prototype-based FL framework. Note that in previous T -1 global rounds, clients only transmit model parameters with the server to train a shared model, which does not illustrate in the figure. We only illustrate the training process in T -th global iteration (i.e. the final global iteration.). In the T -th round, clients not only transmit their model parameters, but also their prototypes to the server for model aggregation (step. 1 in this figure) and prototype aggregation (step. 2). Finally, the prediction can be made based on the distance of each query to aggregated prototypes (step. 3). Figure 2 : 212: end for 13: Model Testing: 14: for each sample i in testing dataset do 15: for each class j in {y} do 16:Measure L2 distance between f e (ω e ; x i ) and y The top-1 average test accuracy of all methods on MNIST in different communication rounds with the degree of skewness α = 0.05 and α = 0.1. Figure 3 : 3The top-1 average test accuracy of all methods on Fashion-MNIST in different communication rounds with the degree of skewness α = 0.05 and α = 0.1. Initialize ω 0 , {y}. 3: for t = 1, 2, ..., T doAlgorithm 1: Prototype-based FL (ProtoFed) Input: Dataset D i , ω i 1: Model Training: 2: 4: Communication-efficient learning of deep networks from decentralized data. B Mcmahan, E Moore, D Ramage, Artificial intelligence and statisticsMcMahan B, Moore E, Ramage D, et al. "Communication-efficient learning of deep networks from decentralized data," Artificial intelli- gence and statistics, pp. 1273-1282, 2017. The non-iid data quagmire of decentralized machine learning. Kevin Hsieh, Amar Phanishayee, Onur Mutlu, Phillip Gibbons, International Conference on Machine Learning. Kevin Hsieh, Amar Phanishayee, Onur Mutlu, and Phillip Gibbons. "The non-iid data quagmire of decentralized machine learning," Inter- national Conference on Machine Learning, pp. 4387-4398, 2020. Model-contrastive federated learning. Q Li, B He, D Song, IEEE/CVF Conference on Computer Vision and Pattern Recognition. Li, Q., B. He, D. Song. "Model-contrastive federated learning," IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021. Tackling the objective inconsistency problem in heterogeneous federated optimization. J Wang, Q Liu, H Liang, G Joshi, H V Poor, Advances in Neural Information Processing Systems. 33J. Wang, Q. Liu, H. Liang, G. Joshi, and H. V. Poor. "Tackling the ob- jective inconsistency problem in heterogeneous federated optimization," Advances in Neural Information Processing Systems, vol. 33, 2020. No fear of heterogeneity: Classifier calibration for federated learning with non-iid data. M Luo, F Chen, D Hu, Advances in Neural Information Processing Systems. 34Luo M, Chen F, Hu D, et al. "No fear of heterogeneity: Classifier calibration for federated learning with non-iid data," Advances in Neural Information Processing Systems, vol. 34, pp. 5972-5984, 2021. Fedproto: Federated prototype learning across heterogeneous clients. Y Tan, G Long, L Liu, AAAI Conference on Artificial Intelligence. 13Tan Y, Long G, Liu L, et al. "Fedproto: Federated prototype learning across heterogeneous clients," AAAI Conference on Artificial Intelli- gence, vol. 1. no. 3, 2022. Gradient-based learning applied to document recognition. Yann Lecun, Proceedings of the IEEE. 8611LeCun, Yann, et al. "Gradient-based learning applied to document recognition," Proceedings of the IEEE, vol. 86, no.11, pp. 2278-2324, 1998. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. Han Xiao, Kashif Rasul, Roland Vollgraf, arXiv:1708.07747arXiv preprintXiao, Han, Kashif Rasul, and Roland Vollgraf. "Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms," arXiv preprint arXiv:1708.07747, 2017. Bayesian nonparametric federated learning of neural networks. M Yurochkin, M Agarwal, S Ghosh, International Conference on Machine Learning. Yurochkin M, Agarwal M, Ghosh S, et al. "Bayesian nonparametric federated learning of neural networks," International Conference on Machine Learning, pp: 7252-7261, 2019. A Framework for Multi-Prototype Based Federated Learning: Towards the Edge Intelligence. Yu Qiao, IEEE2023 International Conference on Information NetworkingQiao, Yu, et al. "A Framework for Multi-Prototype Based Federated Learning: Towards the Edge Intelligence." 2023 International Confer- ence on Information Networking. IEEE, 2023. CDFed: Contributionbased Dynamic Federated Learning for Managing System and Statistical Heterogeneity. Yu Qiao, Md Munir, Shirajum, IEEE/IFIP Network Operations and Management Symposium. Qiao, Yu and Munir, Md Shirajum, et al. "CDFed: Contribution- based Dynamic Federated Learning for Managing System and Statistical Heterogeneity." IEEE/IFIP Network Operations and Management Sym- posium, 2023.
[]
[ "Linking discrete and continuum diffusion models: Well-posedness and stable finite element discretizations", "Linking discrete and continuum diffusion models: Well-posedness and stable finite element discretizations" ]
[ "Christina Schenk \nIMDEA Materials Institute\nEric Kandel 228906Tecnogetafe, MadridSpain\n", "David Portillo \nIMDEA Materials Institute\nEric Kandel 228906Tecnogetafe, MadridSpain\n\nUniversidad Politécnica de Madrid\nJosé Gutiérrez Abascal\n\n\n28006MadridSpain\n", "Ignacio Romero \nIMDEA Materials Institute\nEric Kandel 228906Tecnogetafe, MadridSpain\n\nUniversidad Politécnica de Madrid\nJosé Gutiérrez Abascal\n\n\n28006MadridSpain\n" ]
[ "IMDEA Materials Institute\nEric Kandel 228906Tecnogetafe, MadridSpain", "IMDEA Materials Institute\nEric Kandel 228906Tecnogetafe, MadridSpain", "Universidad Politécnica de Madrid\nJosé Gutiérrez Abascal\n", "28006MadridSpain", "IMDEA Materials Institute\nEric Kandel 228906Tecnogetafe, MadridSpain", "Universidad Politécnica de Madrid\nJosé Gutiérrez Abascal\n", "28006MadridSpain" ]
[]
In the context of mathematical modeling, it is sometimes convenient to integrate models of different nature. These types of combinations, however, might entail difficulties even when individual models are wellunderstood, particularly in relation to the well-posedness of the ensemble. In this article, we focus on combining two classes of dissimilar diffusive models: the first one defined over a continuum and the second one based on discrete equations that connect average values of the solution over disjoint subdomains. For stationary problems, we show unconditional stability of the linked problems and then the stability and convergence of its discretized counterpart when mixed finite elements are used to approximate the model on the continuum. The theoretical results are highlighted with numerical examples illustrating the effects of linking diffusive models. As a side result, we show that the methods introduced in this article can be used to infer the solution of diffusive problems with incomplete data.
10.1002/nme.7204
[ "https://export.arxiv.org/pdf/2208.07600v2.pdf" ]
255,643,666
2208.07600
c31921b1c706154ba18e11f63f4cfe0e9000fd21
Linking discrete and continuum diffusion models: Well-posedness and stable finite element discretizations Christina Schenk IMDEA Materials Institute Eric Kandel 228906Tecnogetafe, MadridSpain David Portillo IMDEA Materials Institute Eric Kandel 228906Tecnogetafe, MadridSpain Universidad Politécnica de Madrid José Gutiérrez Abascal 28006MadridSpain Ignacio Romero IMDEA Materials Institute Eric Kandel 228906Tecnogetafe, MadridSpain Universidad Politécnica de Madrid José Gutiérrez Abascal 28006MadridSpain Linking discrete and continuum diffusion models: Well-posedness and stable finite element discretizations Mixed finite elementsinf-sup conditionmulti-scalenetwork modeldiffusion problemsstability In the context of mathematical modeling, it is sometimes convenient to integrate models of different nature. These types of combinations, however, might entail difficulties even when individual models are wellunderstood, particularly in relation to the well-posedness of the ensemble. In this article, we focus on combining two classes of dissimilar diffusive models: the first one defined over a continuum and the second one based on discrete equations that connect average values of the solution over disjoint subdomains. For stationary problems, we show unconditional stability of the linked problems and then the stability and convergence of its discretized counterpart when mixed finite elements are used to approximate the model on the continuum. The theoretical results are highlighted with numerical examples illustrating the effects of linking diffusive models. As a side result, we show that the methods introduced in this article can be used to infer the solution of diffusive problems with incomplete data. Introduction Diffusion problems of interest in Applied Math and Engineering can be studied with discrete models as well as partial differential equations (PDEs). The first approach is naturally simpler than the second one since it sidesteps the difficulties that result from the spatial description of the solution fields. In addition, the solution of these simple models can be approximated very efficiently at the expense of all spatial details, in contrast with the approximation of PDEs. The latter, in fact, invariably requires working with (large) systems of equations that arise from the spatial approximation of the problem. Not surprisingly, the coexistence of a hierarchy of models for a single physical phenomenon is a common trait of most, if not all, scientific endeavors. Precisely because several models of different complexity often exist for one single physical problem, sometimes it proves convenient to combine several of them to exploit their relative advantages. For example, when studying complex deformable bodies, it has been proven effective to combine models for beams and solids [1,2,3], since the economy of the beam equations can be exploited to analyze whole structures, whereas the (more complex) equations of solid mechanics are used to describe with detail the mechanics of regions with intricate stress distributions. This same strategy can be found in the analysis of complex -typically multiscale -problems in, e.g., the study of ground and subsurface hydraulic flow [4], arteries and the heart [5], capillary network and major blood vessels [6,7], etc. In these situations, always motivated by a reduction of complexity or computational cost, each of the connected models might be wellknown, but linking them poses difficulties. In particular, the well-posedness of the joint problem is a delicate matter: even when each individual model is described with well-posed equations, one still needs to prove that the connection does not spoil this property. Hence, links between models of different nature are of theoretical as well as practical interest. In this article, we analyze the coupled solution of two diffusive models, the first one defined over a continuum -and thus described by a PDE -and the second one defined over discrete network elements -and described with an ordinary differential equation (ODE). A prototypical example of the problem of interest in this article is thermal equilibrium. Its most general description in a three-dimensional body employs Poisson's equation, the paradigmatic elliptic model. In addition, when a body is slender, its thermal equilibrium might be described by a second-order ODE. In practical applications, however, we might be interested in modeling the temperature of a part that is best described as a conductive body with a wire that connects some regions, a wire that need not be inside the body. While one could model the ensemble as a continuum, it proves more convenient -especially when using numerical discretizationsto use different models for the bulk and the wire. This is a relevant problem, for example, in the thermal design of printed circuit boards (PCBs) where the thermal conductivity of the electronic components and their (thin) connections are very different, as also their geometries. A second relevant example that involves coupled models of different nature appears when modeling the diffusion of infectious diseases. While multi-compartmental network models have been used for centuries [8] and many results have been obtained (e.g., [9]), they lack spatial resolution. More recent efforts -especially related to COVID-19attempt to use diffusive boundary value problems instead [10,11,12,13]. Given its practical and theoretical interests, the numerical approximation of problems that mix models of different dimensionality has been studied before (e.g., [14,3,15,16,6]). In particular, some works have studied the approximation of mixed-dimensional problems where the low-dimension model is embedded within the high-dimensional one, with coupling fluxes between them [14,6]. One of the main difficulties for coupling 3D-1D models is that the trace operator from the 3D domain to the 1D domain is not well-posed if the lower dimensional problem is a 1D manifold with a co-dimension larger than one. Here, however, we are only interested in coupling elements of a discrete (net-work) model with a continuum model. In contrast with the references before, the network is just a mathematical abstraction, where edges represent connections (i.e., carriers of flux) and are not, strictly speaking, 1D submanifolds of the continuum. More specifically, the networks that we will study in this work model diffusive phenomena between disjoint regions of the continuum domain, connecting the average values of the linked variables through a discrete diffusive law. Note that our solution approach differs from the so-called network diffusion models aiming to solve PDEs on discrete graphs [17,18]. Given the difficulties in formulating and analyzing general coupled continuumdiscrete models, we restrict our presentation to the study of the problems that link two dissimilar models, each of them being the solution of a minimization problem. The motivation for this choice is double: first, many diffusive problems of interest have this form and, second, they are easily amenable to analysis. In particular, if two different models are of this type, their link might be conveniently tackled using the classical method of Lagrange multipliers that results in a saddle point problem. The mathematical aspects of saddle point problems and their approximations with Galerkin-type methods [19,20,21] are well-understood both in R n [22] as well as in Hilbert spaces. The current interest remains, thus, in formulating new links for different models and proving that the resulting coupled problem as well as its discretization are stable. In the current work, we study linked formulations of bulk and one-dimensional diffusive network models, aiming to provide a rigorous footing for the family of continuum/discrete problems mentioned above. The results presented herein address first the continuum model, i. e., the well-posedness of the mixed problem that appears when the bulk and one-dimensional discrete type problems are linked. Once this problem is studied using standard tools, we formulate mixed finite elements for discrete/continuum diffusion problems and analyze also their well-posedness. In contrast to elliptic problems, finite element discretizations of mixed problems do not inherit the well-posedness from the continuum counterpart and a different stability analysis has to be performed. In this work, we show the unconditional stability of the continuum/discrete problems of interest by proving a discrete inf-sup condition. These theoretical findings are then illustrated with numerical examples. The main result of this work is thus that linked discrete/continuum finite element formulations of the coupled diffusive problems considered are unconditionally stable and convergent. Also, as we will show, the ideas of the proposed coupling can be used for the solution of diffusive problems with missing data or partial information. An elegant solution to this seemingly unrelated problem -closer to data science than diffusion -can be obtained in a straightforward manner employing the formulation presented in this work. The article is structured as follows. In Section 2 we describe the mathematical formulation of the continuum/discrete problems. The theoretical investigations that prove the well-posedness of the joint problem are presented in Section 3. Then, in Section 4, we introduce the finite element discretization of the coupled problem and prove that the well-posedness of the original problem is inherited by the fully discrete problem. In Section 5, we demonstrate the applicability of the previously introduced concepts and methods for several problems that are of practical interest. Finally, Section 6 summarizes the main results of the article. Mixed-dimensional Poisson problems In this section, we define the diffusive problems whose analysis and approximated solution are the central topic in this article. The first part of this boundary-value problem describes the stationary solution of a transport problem in a continuum and is ubiquitous in Mathematical Physics. For concreteness, in the following, we use the language of thermal analysis assuming that the unknown field is the temperature in a body. Throughout the article, all the equations could be reinterpreted in terms, for example, of mass concentration or electrical charge. The second type of problem refers to the temperature distribution on a one-dimensional wire whose solution is given by a second-order ordinary differential equation. In certain situations, moreover, a closed-form solution to this problem can be found, yielding a discrete diffusion relation between the temperature at the ends of the wire. Finally, we describe a joint solution of these two problems when they are solved simultaneously, that is when we consider a body in thermal equilibrium with two or more disjoint subsets connected by a thermal wire. The Poisson problem in a continuum domain The continuum body where we would like to study its temperature distribution is a bounded open set Ω ⊂ R d with boundary denoted as ∂Ω. For simplicity, in what follows, we will restrict to d = 2, but no fundamental problem arises in the three-dimensional case. The temperature in this body is a field φ ∈ H 1 0 (Ω), the Hilbert space of (Lebesgue) square-integrable functions with square-integrable weak first derivatives and vanishing trace at the boundary. If f ∈ H −1 (Ω) is a known field of heat supply, the stationary Poisson problem can be written in its standard strong form −κ ∆φ = f in Ω , φ = 0 on ∂Ω .(1) Here κ is the thermal conductivity of the medium and will be assumed to be constant, for simplicity, and ∆ is the Laplacian operator. See, e.g., [23] for a detailed description of this canonical elliptic boundary-value problem. With a view to the analysis and discretization of Eq. (1), we rewrite the Poisson problem in weak form. For that we define U ≡ H 1 0 (Ω) and recall that its norm is given, for any φ ∈ U by the standard expression φ U := φ 2 L 2 (Ω) + 2 ∇φ 2 L 2 (Ω) 1/2 ,(2) where is the characteristic length of the problem (for example, the diameter of Ω) and ∇ denotes the gradient operator. Then, the weak form of Poisson's problem consists in finding φ ∈ U such that a Ω (φ, ψ) = f Ω (ψ) (3) for all ψ ∈ U, where a Ω (φ, ψ) := Ω ∇φ · κ∇ψ dV, f Ω (ψ) := Ω f · ψ dA,(4) are, respectively, a bilinear and a linear form on U. The Poisson problem on a segment We describe next the stationary heat problem for a one-dimensional body, which is referred to throughout as a wire. The governing equations of this problem can be derived from the statement (1) of the Poisson problem, simply by assuming that the body Ω has a prismatic shape and the temperature field is constant in all the points of a cross section. Details of this projection are omitted and the final form of the equation is given. Consider a wire of length L with temperature θ : [0, L] → R. When the wire is in thermal equilibrium, the temperature must satisfȳ κ d 2 θ dx 2 = r,(5) where r is the heat supply per unit length andκ is the cross-sectional conductivity. The problem, however, is not well-posed unless we append suitable boundary conditions. Rather, the temperature is only defined modulo an affine function. As before, we are interested in the weak formulation of this problem. Postponing for the moment the issue of the uniqueness in the solution, we can introduce the function space W = H 1 (0, L), analogous to the solution space for the body but without the trace constraint, and look for solutions θ ∈ W such that a L (θ, η) = f L (η),(6)for all η ∈ W, where a L (θ, η) := L 0κ θ η dx, f L (η) := L 0 r η dx.(7) For future reference, we recall that for every function θ ∈ W, its norm is θ W := θ 2 L 2 (0,L) + L 2 θ 2 L 2 (0,L)) 1/2 .(8) Let us note that the actual geometry of the wire plays no role in the equations above, except for its length. As explained in Sec. 1, Eq. (6) merely defines the behavior of a network edge, an abstract entity that connects temperature of two points through a diffusive equation. Moreover, if the heat supply r is identically zero and the diffusivity is constant, the solution to Eq. (5) is an affine function. In this case, for all practical purposes, the wire will just establish a discrete diffusive relation between the temperature at its two end points. Linked formulation We study next a joint problem consisting of a body Ω with a temperature field φ satisfying the problem (3), where additionally we identify two disjoint non-empty regions B 1 , B 2 Ω whose temperatures are connected by means of a thermally conductive wire. This wire is such that the temperature at each of its ends coincides with the mean temperature of the regions B 1 , B 2 , respectively. See Fig. 1 for an illustration of the linked problem. We note that, alternatively, we could have considered two disjoint conductive bodies Ω 1 and Ω 2 that are in thermal equilibrium while a wire connects regions B α ⊂ Ω α , with α = 1, 2. The analysis of both of the problems described is analogous and, for conciseness, we focus on the former. Let us insist, once again, that the purpose of this coupled model is not the actual representation of a true wire that connects two points in the body (say, the centers of B 1 and B 2 in Fig. 1). Rather, the wire connects regions B 1 and B 2 through a discrete diffusive equation. More precisely, let θ 1 , θ 2 denote the mean temperature in the body regions B 1 , B 2 , that is, θ α := 1 |B α | Bα φ dV,(9) with α = 1, 2, and |B α | denoting the non-zero measure of the set. Then, the problem governing the temperature field on the wire can be fully described by the boundary value problem κ d 2 θ dx 2 = 0 on (0, L),(10a)θ(0) = θ 1 , (10b) θ(L) = θ 2 .(10c) If the scalars θ 1 , θ 2 were known, the problem (10) would be standard and its wellposedness would require no further analysis. However, here we are interested in the situation where the values θ 1 , θ 2 are not known a priori but rather, obtained through the averages (9) of the solution to problem (3). It remains, thus, to formulate the coupled problem that includes the thermal equilibrium of the body and wire, as well as the link conditions. We will use Lagrange multipliers (λ 1 , λ 2 ) =: λ ∈ Q ≡ R 2 to enforce the two constraints (10b) and (10c), and we will use the notation · Q to indicate the Euclidean norm in R 2 . For convenience, we introduce the product space V = U × W with norm (φ, θ) V := φ 2 U + d−1 θ 2 W 1 2 ,(11) for all (φ, θ) ∈ V. On this space we can define the bilinear form a(·, ·) : V×V → R and linear form f : V → R as a(φ, θ; ψ, δ) := a Ω (φ, ψ) + a L (θ, δ) , f (ψ, δ) := f Ω (ψ) + f L (δ) ∈ V.(12) for all (φ, θ) and (ψ, δ) in V. The joint equilibrium of the solid and the wire then results from the saddle point of the Lagrangian L : V × Q → R: L(φ, θ, λ 1 , λ 2 ) := 1 2 a(φ, θ; φ, θ) − f (φ, θ) + λ 1 θ(0) − 1 |B 1 | B1 φ dV + λ 2 θ(L) − 1 |B 2 | B2 φ dV .(13) Hence, we aim to solve the following problem (φ, θ, λ 1 , λ 2 ) = arg inf φ,θ sup λ L(φ, θ, λ 1 , λ 2 )(14) where the Lagrangian as in Eq. (13). The optimality conditions of the functional L are satisfied by the functions (φ, θ, λ 1 , λ 2 ) ∈ V × Q such that a(φ, θ; ψ, δ) + b(ψ, δ; λ 1 , λ 2 ) = f (ψ, δ) ,(15)b(φ, θ; Γ 1 , Γ 2 ) = 0,(16) for all (ψ, δ, Γ 1 , Γ 2 ) ∈ V × Q with b(φ, θ; Γ 1 , Γ 2 ) := Γ 1 θ(0) − 1 |B 1 | B1 φ dV + Γ 2 θ(L) − 1 |B 2 | B2 φ dV .(17) The bilinear form a(·, ·) defines a linear continuum operator A : V → V by the relation Au, v V ×V = a(u, v),(18)for all u = (φ, θ) ∈ V, v = (ψ, δ) ∈ V. Also, the bilinear form b(·, ·) on V × Q defines a linear operator B : V → Q with transpose B T : Q → V by Bv, Γ Q ×Q = v, B T Γ V×V = b(v, Γ ),(19) for all v ∈ V, Γ ∈ Q. Employing these definitions, Eq. (16) can be alternatively rewritten as: Au + B T λ = f in V Bu = 0 in Q .(20) This last expression is the standard form of a mixed problem [20]. The solvability of this problem depends on conditions over the bilinear forms a(·, ·) and b(·, ·) as well as properties of the spaces where they are defined on. In particular, often the properties of the linear operator B are delicate to ascertain. Remarks 1. Two modifications of problem (16) are interesting in their own right: 1. One could consider that the one-dimensional diffusive mechanism connects, instead of disjoint subsets B 1 , B 2 ⊂ Ω, the boundary of two different bodies, or parts of them. The description of this modified problem and the functional setting are slightly different than the one presented up to here, since the new problem will be formulated in terms of the traces of functions. 2. Second, we could consider a situation where there is no connecting onedimensional diffusive wire between regions of the body but only that the average temperature on a measurable set S ⊂ Ω is known to have a fixed valueθ. In this simple case we will be left with Poisson's problem with a constraint. The problem will still be of the form (16), more precisely, a Ω (φ, ψ) +b(ψ, γ) = f Ω (ψ) , b(φ, η) = 0,(21)withb(·, ·) on V × R defined as b(φ, η) := η θ − 1 |S| S φ dV .(22) Analysis In this section, we study the well-posedness of problem (16). According to the standard theory of mixed problems [19,20,21], we need to show that the bilinear forms a(·, ·) and b(·, ·) are continuum, that a(·, ·) is elliptic on the kernel of the operator B, and that a certain inf-sup condition, to be defined later, also holds for b(·, ·). In the following, for simplicity, let us assume that the conductivities κ andκ are constant and positive. The first step of the analysis is to study the continuity of all the linear and bilinear forms appearing in the problem statement (16). This is trivial and we summarize all the results, without proof, in the following theorem: Theorem 3.1. The bilinear forms a Ω (·, ·), a L (·, ·), and a(·, ·) are continuous in their corresponding spaces of definition, i.e., |a Ω (φ, ψ)| ≤ c Ω φ U ψ U , |a L (θ, δ)| ≤ c L θ W δ W , |a(φ, θ; ψ, δ)| ≤ c (ψ, δ) V ,(23) for all φ, ψ ∈ U, θ, δ ∈ W and some generic positive constants c Ω , c L and c. Next, we show the continuity of b(·, ·). Since this bilinear form is nonstandard, we provide the full proof of the result. Theorem 3.2. The bilinear form b is continuous on (V × Q), i.e. |b(φ, θ; λ)| ≤ c (φ, θ) V λ Q(24) for all (φ, θ; λ) ∈ V × Q and some constant c > 0. Proof. Since H 1 (0, L) → C 0 [0, L], we can use the mean value theorem to determine that there exists m ∈ [0, L] such that θ(m) = 1 L L 0 θ(x) dx .(25) Then, by the fundamental theorem of calculus θ(L) = θ(m) + L m θ (x) dx.(26) Hence, |θ(L)| ≤ 1 L L 0 |θ(x)| dx + L m |θ (x)| dx ≤ L −1/2 θ L 2 (0,L) + L 1/2 θ L 2 (0,L) = L −1/2 θ H 1 (0,L) .(27) Similarly, the bound |θ(0)| ≤ L −1/2 θ H 1 (0,L) also holds. Using these results, we proceed to bound from above the bilinear form b(·, ·): |b(φ, θ; λ)| = λ 1 1 |B 1 | B1 φ dV − θ(0) + λ 2 1 |B 2 | B2 φ dV − θ(L) ≤ |λ 1 | 1 |B 1 | B1 φ dV − θ(0) + |λ 2 | 1 |B 2 | B2 φ dV − θ(L) ≤ |λ 1 | 1 |B 1 | B1 |φ| dV + |θ(0)| + |λ 2 | 1 |B 2 | B2 |φ| dV + |θ(L)| ≤ C λ 2 1 |Ω| Ω |φ| dV + L −1/2 θ H 1 (0,L) ≤ C λ 2 −d/2 φ H 1 0 (Ω) + L −1/2 θ H 1 (0,L) ≤ C −d/2 λ Q (φ, θ) V , where, throughout the proof, C denotes a constant whose value might change from one step to another and d denotes the dimension of the characteristic length corresponding to Ω. Next, we have to ensure ellipticity on ker B, the kernel of the operator B. By definition, this set is ker B := {(φ, θ) ∈ V : b(φ, θ; λ) = 0, ∀λ ∈ Q}.(28) Elements (φ, θ) ∈ V in ker B verify λ 1 1 |B 1 | B1 φ dV − θ(0) + λ 2 1 |B 2 | B2 φ dV − θ(L) = 0.(29) for all pairs (λ 1 , λ 2 ) ∈ Q. Since the two Lagrange multipliers are independent, it must hold that ker B = (φ, θ) ∈ V : 1 |B 1 | B1 φ dV = θ(0) and 1 |B 2 | B2 φ dV = θ(L) . (30) As advanced above, for the well-posedness of the mixed problem, it suffices that the bilinear form a(·, ·) be elliptic on ker B ⊂ V, as shown in the following theorem: Theorem 3.3. The bilinear form a(·, ·) is elliptic on ker B, i.e., there exists a constantᾱ > 0 such that a(φ, θ; φ, θ) ≥ᾱ (φ, θ) 2 V ,(31) for all (φ, θ) ∈ ker B. Proof. To start the proof, we need two preliminary results. First, using the weighted Young inequality we note that θ−θ(0) 2 L 2 (0,L) + θ − θ(L) 2 L 2 (0,L) =2 θ 2 L 2 (0,L) + L |θ(0)| 2 + L |θ(L)| 2 − 2 L 0 θ(0) θ dx − 2 L 0 θ(L) θ dx ≥2 θ 2 L 2 (0,L) + L |θ(0)| 2 + L |θ(L)| 2 − 2 0 θ 2 L 2 (0,L) − L 2 0 |θ(0)| 2 − 2 L θ 2 L 2 (0,L) − L 2 L |θ(L)| 2 ≥ 2 − 2 0 − 2 L θ 2 L 2 (0,L) + L 1 − 1 2 0 |θ(0)| 2 + L 1 − 1 2 L |θ(L)| 2 ,(32) for arbitrary scalars 0 , L > 0. In the kernel of B, moreover, we have that |θ(0)| 2 = 1 |B 1 | B1 φ dV 2 ≤ C −d φ 2 H 1 0 (Ω) , |θ(L)| 2 = 1 |B 2 | B2 φ dV 2 ≤ C −d φ 2 H 1 0 (Ω) .(33) Assuming that the following conditions hold 1 − 1 2 0 ≤ 0 , 1 − 1 2 L ≤ 0 ,(34) then, combining Eqs. (32) and (33), the following bound is obtained θ−θ(0) 2 L 2 (0,L) + θ − θ(L) 2 L 2 (0,L) ≥ 2 − 2 0 − 2 L θ 2 L 2 (0,L) + C 1−d 2 − 1 2 0 − 1 2 L φ 2 H 1 0 (Ω) .(35) Also, as in the proof of Theorem 3.2, we use the fundamental theorem of calculus to bound θ 2 L 2 (0,L) ≥ CL −2 θ − θ(0) 2 L 2 (0,L) , θ 2 L 2 (0,L) ≥ CL −2 θ − θ(L) 2 L 2 (0,L) .(36) To finally prove the ellipticity of the bilinear form a(·, ·) we use the definition (12), the bounds (35), (36) and the Poincaré inequality from where it follows that a(φ, θ; φ, θ) = Ω κ |∇φ| 2 dV + L 0κ |θ | 2 dx ≥C κ −2 φ 2 H 1 0 (Ω) +κ 2 θ 2 L 2 (0,L) +κ 2 θ 2 L 2 (0,L) ≥C κ −2 φ 2 H 1 0 (Ω) + C κ d−1 θ 2 L 2 (0,L) + C κ d−3 θ − θ(0) 2 L 2 (0,L) + θ − θ(L) 2 L 2 (0,L) ≥C κ −2 φ 2 H 1 0 (Ω) + C κ d−1 θ 2 L 2 (0,L) + C κ d−3 2 − 2 0 − 2 L θ 2 L 2 (0,L) + C κ −2 2 − 1 2 0 − 1 2 L φ 2 H 1 0 (Ω) , where, as usual, C denotes a generic constant whose value might change in each inequality. Finally, we rewrite this bound as a(φ, θ; φ, θ) ≥C κ −2 3 − 1 2 0 − 1 2 L φ 2 H 1 0 (Ω) + C κ d−3 (2 − 2 0 − 2 L ) θ 2 L 2 (0,L) + L 2 θ 2 L 2 (0,L) ≥C κ −2 φ 2 U + d−1 θ 2 W =C κ −2 (φ, θ) 2 V , as long as there exist 2 0 , 2 L ≥ 0 that verify conditions (34) as well as 3 − 1 2 0 − 1 2 L > 0 , 2 − 2 0 − 2 L > 0 . These three conditions are satisfied, for example, by 2 0 = 2 L = 4/5, completing the proof. Let us note that the bilinear form a(·, ·) is not elliptic in the whole space V, because the bilinear form of the one-dimensional conductor, namely a L (·, ·) is not elliptic in the space W, precisely due to the lack of Dirichlet boundary conditions in problem (5). Finally, to ensure that the mixed problem is well-posed, we also have to show that the following inf-sup condition holds for b(·, ·). Theorem 3.4. Inf-sup condition. There exists a constant β > 0 such that sup (φ,θ)∈V b(φ, θ; λ) (φ, θ) V ≥ β λ Q\ ker B T .(37) Proof. Since B is an operator from V to Q , a finite-dimensional space, its range is closed (Theorem 1.1. in [20]). Hence, the inf-sup condition holds. With Theorems 3.1, 3.2, 3.3 and 3.4, it follows that problem (16) is wellposed. Let us remark that, typically, in other constrained minimization problems, the proof of the inf-sup condition might be quite involved. In the problem discussed in this article, however, this proof is trivial. Approximation of the problem To solve the saddle-point problem (16) we use a mixed finite element discretization. We now briefly recall the details of this method as it applies to the project at hand. The first step in the finite element approximation of problem (16) is the definition of a finite-dimensional subspace, V h ⊂ V of the infinite-dimensional solution space. Moreover, we assume that all the finite element functions in V h are linear combinations of piecewise polynomials defined in their corresponding domains, namely, Ω or [0, L]. Since the space of the Lagrange multipliers is already a space of dimension 2 we do not need to introduce a subspace for it and we simply define Q h ≡ Q. The (mixed) Galerkin finite element method consists in finding u h = (φ h , θ h ) ∈ V h , λ h ∈ Q h such that a(u h , v h ) + b(v h , λ h ) = f (v h ), b(u h , Γ h ) = 0,(38) holds for all v h = (ψ h , δ h ) ∈ V h and Γ h ∈ Q h . In contrast with the finite element approximations of elliptic boundary value problems, the well-posedness of mixed methods such as (38) does not follow from the well-posedness of the corresponding continuous problem. Instead, a new analysis has to be carried out and, in particular, a discrete inf-sup condition needs to be proven. This is typically the most difficult ingredient of this analysis but, as we will show below, it is not the case for the problem at hand given the finite dimension of the space Q. Analysis The well-posedness of the discrete problem (38) can be established by following similar steps as in the analysis of the continuous problem and presented in Section 3. For that, we start by introducing B h , the discrete counterpart of the operator B used in Section 3, and now defined by the relation B h v h , Γ h Q h ×Q h = v h , B T h Γ h V×V = b(v h , Γ h ),(39)for all v h ∈ V h and Γ h ∈ Q h . Since B h is surjective and ker B T h ⊂ ker B T , ker B T h = {0} and B h = B V h . So, we end up in the special case of ker B h ⊂ ker B. Hence, the ellipticity of a(·, ·) on ker B h follows from the ellipticity of a(·, ·) on ker B. As advanced, the key condition for the well-posedness of (38) is thus the discrete inf-sup condition. However, given that the range of B h is finite dimensional, it is closed and thus the discrete inf-sup condition holds. From Theorem 2.1. in [20] we obtain the following estimate u−u h V + λ−λ h Q\ ker B T ≤ c inf v h ∈V h u − v h V + inf q h ∈Q h λ − Γ h Q , (40) where c is a constant that depends on a(·, ·), b(·, ·),ᾱ as in Theorem 3.3 and β as in Theorem 3.4. Details can be found in [20]. Numerical examples In this section, we use first the mixed method (38) for two examples chosen to illustrate its ability to link regions of thermally conductive solids. Then, and to complete the section, we show that the ideas of the mixed network-continuum formulations can be exploited to solve one particular type of inference problems in diffusive situations. Heat flux between dissimilar regions In this example, we choose a square solid with dimensions 2 × 2 where two subregions, the first with the shape of a circle, and the second an ellipse, are linked through a conductive wire. In a Cartesian coordinate system located at the center of the square and with axes parallel to its edges, the circular region has center (x, y) = (−0.625, −0.5625) and radius 0.5. The elliptical region is centered at (x, y) = (0.375, 0.4375), and has horizontal and vertical semi-axes of lengths 0.175 and 0.3, respectively (see Fig. 2). The setting is very similar to the one we employed in Section 2.3 to describe the mixed-dimensional boundary value problem. See Fig. 1 for comparison. The block has conductivity κ = 100 and the wire has a high conductivity of valueκ = 10, 000. The temperature is constrained at the top and bottom edges to values of 500 and 300, respectively. If the problem had no thermal link, the thermal field would be linear with constant vertical heat flux. However, due to the presence of the linking wire, which is selected with a high conductivity, the thermal field is distorted. The elliptical region -closer to the hotter top-serves as a heat source for the circular region -this one closer to the cooler bottom edge. As a result, the temperature fields in the two connected regions are close to 400. In principle, the region and its center can lay anywhere in the domain. The centers of these regions determine the amount of temperature going into and coming out of the wire calculated as the average temperature in the corresponding region. To study the convergence of the formulation, we obtain six finite element solutions for this problem, with increasing resolution (see Fig. 2). In each of the solutions, the linked regions B 1 , B 2 consist of those elements of the mesh whose centers fall inside the circle and ellipse, respectively. As can be observed in Fig. 2, coarse meshes do not represent accurately the linked regions (see the top two figures in Fig. 2). However, when the mesh is refined, both the circle as well as the ellipse are accurately approximated (see, again, Fig. 2). We note that, alternatively, we could have chosen to employ a mesh that was adapted to the linked regions, but the converged solutions would not differ. Convergence of increasingly finer connected regions This second example examines the classical Fourier problem and its relation with diffusive networks, of the type employed in this work, with the continuum notion of flux. For that, we will consider a square domain of unit side, a material with conductivity κ = 100, and temperature boundary conditions described by parabolas at the four edges, all with their maximum values at the center of the edges and zero at the vertices. The maximum temperatures at each edge are 100 (bottom), 200 (right), 300 (top), and 400 (left). This problem can be solved with a classical numerical method, such as the finite element or finite volume method (see Fig. 4a for the finite element solution with a mesh of 512 2 bilinear elements). However, here we solve the problem by discretizing the domain into disjoint and independent square regions. These regions are connected with their neighbors through wires with conductivity κ·∆y (horizontal wires) and κ · ∆x (vertical wires), where ∆x and ∆y are the lengths of the corresponding region in x and y direction, respectively. Then, this linked problem is solved using the formulation described in Section 4. Since we use only one finite element for each region, the method explained above is equivalent to a finite volume method where the fluxes are obtained by solving the boundary value problem of the wires. Fig. 3 shows the solutions obtained with an increasing number of independent regions. In addition to the thermal field, the edges of the diffusive network are also drawn as well as their temperature field, using the same color scale as in the continuum regions. In the coarser solutions, one can clearly see the discontinuity of the thermal field across the boundaries of the regions since, as explained, they are independent meshes and do not share any node. Also, it is apparent in these coarser solutions that the thermal field in the wires differs from the continuum field at corresponding points. Remarkably, as the size of the regions is reduced -and hence the length and conductivity of the wires-the connected solution (Fig. 3) resembles more closely the finite element solution (Fig. 4a) used as reference. This apparent convergence is verified by the results depicted in Fig. 4b. This figure shows the root-mean-square error of the temperature field of each linked solution compared with the reference finite element solution and obtained as e = 1 N N i (φ i − φ F E i ) 2 ,(41) where N is the number of regions and φ i refers to the temperature at the center of the i−th region, or element, respectively. Using linked models to infer diffusive solutions This final example studies the possibility of using the methods introduced in Section 4 to infer the optimal thermal field in a domain when only partial information is available. More precisely, we are interested in determining the most likely temperature field in a domain when there is information about the temperature on part of the boundary and the average temperature in some regions of the domain. When studying a Fourier-type diffusive problem, one needs to know the Dirichlet and Neumann conditions in all of the boundaries as well as the heat applied in the interior of the domain, see Eq. (1). The lack of either of these data renders the problem ill-posed and no solution can be analytically (nor numerically) obtained. Here we are interested in finding the optimal temperature field of a continuum domain, Ω, where the Dirichlet boundary conditions are partially known but also the average temperature in some regions, {B i } Nregions i=1 with B i ⊆ Ω. In particular, the heat supplied through the Neumann boundary and the interior of the domain is completely unknown. This is an optimization problem that searches for the temperature field φ ∈ U that verifies min φ∈U E = Ω κ 2 ∇φ 2 dV, s.t. 1 |B i | Bi φ dV = θ i , i ∈ {1, ..., N regions } φ =φ, ∅ = ∂Ω j ⊆ ∂Ω(42) The solution to this optimization problem is the thermal field on the solid Ω with imposed mean-temperature constraints using degenerated wires that link the assumed temperature with the selected regions using the constraint given in Eq. (22). To analyze the ability of problem (42) to infer an unknown thermal field, we consider a 2 × 1 rectangle with κ = 100. We compute a reference solution with boundary conditions and thermal loading depicted in Fig. 5a. In this figure, the temperature on the left and top edges corresponds to φ = 400 and φ = 200, respectively. Also, a wavy heat supply is imposed in the whole domain of the form h(x, y) = 0, if (x, y) ∈ C sin 4π(x−xx) 2 +(y−yc) 2 r 2 , otherwise ,(43) where C is the circle with center (x, y) = (1.5, 0.25) and radius r = 0.25, where the coordinates refer to a Cartesian system located at the bottom left corner of the rectangle, with the x, y axes parallel to the horizontal and vertical directions, respectively. Fig. 5b shows the solution of this boundary value problem (in what follows, the reference problem) obtained with a finite element mesh of size 128 × 64. For the current example, we define the potentially linked regions as the ones obtained with a simple grid of the whole domain of 8 × 8 equal rectangles. We only employ as known boundary data the temperature on the left edge. Then, we analyze a sequence of discrete problems that employ an increasingly large number of data concerning the average temperature in random regions. Denoting these regions as B i (see Eq. (42)), we proceed as follows: first, we solve the problem with known Dirichlet data, and a known average temperature θ i on a random region B 1 which we can obtain from the exact solution; then, we select a second random region B 2 , obtain its average temperature θ 2 from the exact solution and solve the minimization problem with the two constraints. Proceeding in this fashion, we obtain the solutions illustrated in Fig. 6. This figure shows that when the only available data is the average temperature of one region and the Dirichlet boundary condition, the solution of minimal energy is far from the reference (see top figures in Fig. 6. As expected, the more data available, the better the obtained solution. When the average temperatures are available in all the rectangular regions, the error is minimal and the solution of the optimization problem is close to the reference solution. Naturally, the small oscillations in the solution are not captured by the approximation because the available data do not have enough resolution. One might expect that in the limit when the regions of known average become very small and cover the whole domain, the solution of the optimization problem converges to the exact solution. Finally, in Fig. 7 the energy error ε -relative to the energy of the reference solution -, i.e., ε = |E Nregions − E F E | E F E(44) for the calculated sequence is illustrated. The energy error depends on the random sequence of known data, but it should be always monotonically decreasing. Note that for this sequence, there is almost no improvement in the solution for the last 33 added data instances. The average temperature of the region added in the 32nd iteration makes the difference, reducing the energy error to approximately 20%. From that iteration, the solutions of the constrained optimization problems are very similar. (See Figs. 6h and 6j). Conclusions and main results In this paper, we studied linked continuum/discrete models of diffusion from the theoretical and numerical points of view. More specifically, we consider the solution of diffusion boundary value problems where (discrete) diffusion pathways are introduced between subdomains, more precisely, between the mean values of the diffusing quantity. Although we have described all the results and examples in the language of thermal problems, the scope of the article is more general and applies to any type of linear Fourier-type boundary value problem. After introducing the setting for the linked formulations of bulk and the one-dimensional diffusive models, we proved the well-posedness of the continuum/discrete linked problem in the corresponding functional spaces by resorting to the theory of constrained boundary value problems. Next, we proved that the well-posedness continues to hold for the approximate problem, i.e. the problem discretized via mixed finite elements. A direct consequence of the two previous results is the convergence of the fully discretized equations of the linked model to its exact solution. The previous theoretical findings have been illustrated with three numerical examples. They show that the finite element method is always stable and that the links take care of the diffusive fluxes in a different scale than the continuum, but that in the limit both discrete and continuum mechanisms are equivalent. Finally, we showed that the ideas of the method can be used to carry out inference in the solution of diffusive problems with only partial information available about the average solution in subsets of the whole domain. The results of this article can be employed to link the two different classes of diffusive models, namely, the PDE-based descriptions of the continuum and diffusive network-based formulations when the network connects average values of the diffusive field computed over bounded disjoint regions. Figure 1 : 1Sketch of linked solid-wire problem setting. (c) 32 2 mesh.(d) 64 2 mesh.(e) 128 2 mesh. (f) 256 2 mesh. Figure 2 : 2Thermal link connecting dissimilar regions of circle and ellipse inside a block for six meshes with 8 2 , 16 2 , 32 2 , 64 2 , 128 2 , 256 2 elements (top left to bottom right). Figure 3 : 3Temperature field in thermal problem solved with independent regions linked with wires. The linking wires are depicted as lines.(a) Finite element solution of the several regions and wires problem with a quadratic mesh of 512 × 512 elements. Root-mean-square error of the regions' center temperature compared to the finite element solution. Orange triangle represents linear convergence. Figure 4 : 4Finite element solution of the several regions and wires problem and root-mean-square error of the regions' center temperature compared to the finite element solution. ( b ) bReference solution with a fe mesh of size 128 × 64. Figure 5 : 5Reference boundary value problem solution of the inference example. Figure 6 :Figure 7 : 67Inference problem. A sequence of solutions with a different number of available region data. Left: gray regions are the ones where the average temperature is available. Right: solution with the corresponding average temperatures and only left edge boundary condition known. Energy error for the inference problem for different numbers of available region data. Acknowledgements C.S. and I.R. acknowledge the financial support from Madrid's regional government through a grant with IMDEA Materials addressing research activities on SARS-COV 2 and COVID-19, and financed with REACT-EU resources from the European regional development fund. Transition finite elements for three-dimensional stress analysis. K S Surana, International Journal for Numerical Methods in Engineering. 15K S. Surana. Transition finite elements for three-dimensional stress analy- sis. International Journal for Numerical Methods in Engineering, 15:991- 1020, 1980. Coupling nonlinear beams and continua: Variational principles and finite element approximations. I Romero, International Journal for Numerical Methods in Engineering. 114I. Romero. Coupling nonlinear beams and continua: Variational principles and finite element approximations. International Journal for Numerical Methods in Engineering, 114:1192-1212, 2018. Consistent coupling of positions and rotations for embedding 1D cosserat beams into 3d solid volumes. I Steinbrecher, A Popp, C Meier, Computational Mechanics. 693I. Steinbrecher, A. Popp, and C. Meier. Consistent coupling of positions and rotations for embedding 1D cosserat beams into 3d solid volumes. Computational Mechanics, 69(3):701-732, 2021. Modeling coupled surface-subsurface flow processes: A review. A Furman, Vadose Zone Journal. 72A. Furman. Modeling coupled surface-subsurface flow processes: A review. Vadose Zone Journal, 7(2):741-756, 2008. A modular numerical method for implicit 0D/3D coupling in cardiovascular finite element simulations. M E Moghadam, I E Vignon-Clementel, R Figliola, A L Marsden, Journal of Computational Physics. 244M. E. Moghadam, I. E. Vignon-Clementel, R. Figliola, and A. L. Marsden. A modular numerical method for implicit 0D/3D coupling in cardiovascular finite element simulations. Journal of Computational Physics, 244:63-79, 2013. On the coupling of 1d and 3d diffusionreaction equations: Application to tissue perfusion problems. C Angelo, A Quarteroni, Mathematical Models & Methods in Applied Sciences. 188C. D'Angelo and A. Quarteroni. On the coupling of 1d and 3d diffusion- reaction equations: Application to tissue perfusion problems. Mathematical Models & Methods in Applied Sciences, 18(8):1481-1504, 2008. Geometric multiscale modeling of the cardiovascular system, between theory and practice. A Quarteroni, A Veneziani, C Vergara, Computer Methods in Applied Mechanics and Engineering. 302A. Quarteroni, A. Veneziani, and C. Vergara. Geometric multiscale mod- eling of the cardiovascular system, between theory and practice. Computer Methods in Applied Mechanics and Engineering, 302:193-252, Apr 2016. Essay d'une nouvelle analyse de la mortalité causée par la petite vérole et des avantages de l'inoculation pour la prévenir. Mémoires de Mathématiques et de Physique. D Bernoulli, Académie Royale des Sciences. D. Bernoulli. Essay d'une nouvelle analyse de la mortalité causée par la petite vérole et des avantages de l'inoculation pour la prévenir. Mémoires de Mathématiques et de Physique, Académie Royale des Sciences, Paris, pages 1-45, 1760. An Introduction to Mathematical Epidemiology. M Martcheva, Springer Science & Business MediaNew YorkM. Martcheva. An Introduction to Mathematical Epidemiology. Springer Science & Business Media New York, 2015. Assessing the spatio-temporal spread of covid-19 via compartmental models with diffusion in italy, usa, and brazil. Archives of. Malú Grave, Alex Viguerie, Gabriel F Barros, Alessandro Reali, Alvaro L G A Coutinho, Computational Methods in Engineering. 286Malú Grave, Alex Viguerie, Gabriel F. Barros, Alessandro Reali, and Al- varo L. G. A. Coutinho. Assessing the spatio-temporal spread of covid-19 via compartmental models with diffusion in italy, usa, and brazil. Archives of Computational Methods in Engineering, 28(6):4205-4223, 2021. Modeling nonlocal behavior in epidemics via a reaction-diffusion system incorporating population movement along a network. Malú Grave, Alex Viguerie, Gabriel F Barros, Alessandro Reali, F S Roberto, Alvaro L G A Andrade, Coutinho, 115541, 2022. A Special Issue on Comp. Mod. and Simulation of Infectious Diseases. 401Malú Grave, Alex Viguerie, Gabriel F. Barros, Alessandro Reali, Roberto F.S. Andrade, and Alvaro L.G.A. Coutinho. Modeling nonlocal behavior in epidemics via a reaction-diffusion system incorporating popula- tion movement along a network. Computer Methods in Applied Mechanics and Engineering, 401:115541, 2022. A Special Issue on Comp. Mod. and Simulation of Infectious Diseases. Diffusion-reaction compartmental models formulated in a continuum mechanics framework: Application to covid-19, mathematical analysis, and numerical study. Alex Viguerie, Alessandro Veneziani, Guillermo Lorenzo, Davide Baroli, Nicole Aretz-Nellesen, Alessia Patton, Thomas E Yankeelov, Alessandro Reali, J R Thomas, Ferdinando Hughes, Auricchio, Comput. Mech. 665Alex Viguerie, Alessandro Veneziani, Guillermo Lorenzo, Davide Baroli, Nicole Aretz-Nellesen, Alessia Patton, Thomas E. Yankeelov, Alessandro Reali, Thomas J. R. Hughes, and Ferdinando Auricchio. Diffusion-reaction compartmental models formulated in a continuum mechanics framework: Application to covid-19, mathematical analysis, and numerical study. Com- put. Mech., 66(5):1131-1152, nov 2020. Delay differential equations for the spatially resolved simulation of epidemics with specific application to covid-19. Nicola Guglielmi, Elisa Iacomini, Alex Viguerie, Mathematical Methods in the Applied Sciences. 458Nicola Guglielmi, Elisa Iacomini, and Alex Viguerie. Delay differential equations for the spatially resolved simulation of epidemics with specific application to covid-19. Mathematical Methods in the Applied Sciences, 45(8):4752-4771, 2022. Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction. F Laurino, P Zunino, ESAIM: Mathematical Modelling and Numerical Analysis. 536F. Laurino and P. Zunino. Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction. ESAIM: Mathematical Modelling and Numerical Analysis, 53(6):2047-2080, 2019. Analysis and approximation of mixed-dimensional pdes on 3D-1D domains coupled with lagrange multipliers. M Kuchta, F Laurino, K.-A Mardal, P Zunino, SIAM Journal on Numerical Analysis. 591M. Kuchta, F. Laurino, K.-A. Mardal, and P. Zunino. Analysis and ap- proximation of mixed-dimensional pdes on 3D-1D domains coupled with lagrange multipliers. SIAM Journal on Numerical Analysis, 59(1):558-582, 2021. One-way coupled fluid-beam interaction: capturing the effect of embedded slender bodies on global fluid flow and vice versa. N Hagmeyer, M Mayr, I Steinbrecher, A Popp, Advanced Modeling and Simulation in Engineering Sciences. 912022N. Hagmeyer, M. Mayr, I. Steinbrecher, and A. Popp. One-way coupled fluid-beam interaction: capturing the effect of embedded slender bodies on global fluid flow and vice versa. Advanced Modeling and Simulation in Engineering Sciences, 9(1), 2022. Networks: An Introduction. M Newman, Oxford University PressM. Newman. Networks: An Introduction. Oxford University Press, 2010. Computational Epidemiology-Data-Driven Modeling of COVID-19. E , Springer International PublishingE. Kuhl. Computational Epidemiology-Data-Driven Modeling of COVID- 19. Springer International Publishing, 2021. Mixed and hybrid methods. J E Roberts, J M Thomas, Handbook of Numerical Analysis. P. G. Ciarlet and J. L. LionsAmsterdamNorth-HollandJ. E. Roberts and J. M. Thomas. Mixed and hybrid methods. In P. G. Ciarlet and J. L. Lions, editors, Handbook of Numerical Analysis, pages 523-639. North-Holland, Amsterdam, 1989. Mixed and hybrid finite element methods. F Brezzi, M Fortin, SpringerBerlinF. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer, Berlin, 1991. Mixed finite element methods. F Auricchio, F Brezzi, C Lovadina, Encyclopedia of Computational Mechanics. E. Stein, R. de Borst, and T. J. R. HughesWiley & SonsF. Auricchio, F. Brezzi, and C. Lovadina. Mixed finite element methods. In E. Stein, R. de Borst, and T. J. R. Hughes, editors, Encyclopedia of Computational Mechanics. Wiley & Sons, 2005. R T Rockafellar, Convex analysis. Princeton, N.J.Princeton University Press28R. T. Rockafellar. Convex analysis, volume 28. Princeton University Press, Princeton, N.J., 1970. L C Evans, Partial differential equations. AMS PressL. C. Evans. Partial differential equations. AMS Press, 1999.
[]
[ "Hyperbolic Metamaterial Resonator-Antenna Scheme for Large, Broadband Emission Enhancement and Single Photon Collection", "Hyperbolic Metamaterial Resonator-Antenna Scheme for Large, Broadband Emission Enhancement and Single Photon Collection" ]
[ "Faraz A Inam \nDept. of Physics\nAligarh Muslim University\n202002AligarhU.PIndia\n\nDept. of Condensed Matter Physics and Material Science\nTata Institute of Fundamental Research\n400005MumbaiIndia\n", "Nadeem Ahmed \nDept. of Physics\nAligarh Muslim University\n202002AligarhU.PIndia\n", "Michael J Steel \nMQ Photonics Research Centre\nDept. of Physics and Astronomy\nMacquarie University\n2109SydneyNSWAustralia\n", "Stefania Castelletto \nSchool of Engineering\nRMIT University\n3001MelbourneVictoriaAustralia\n" ]
[ "Dept. of Physics\nAligarh Muslim University\n202002AligarhU.PIndia", "Dept. of Condensed Matter Physics and Material Science\nTata Institute of Fundamental Research\n400005MumbaiIndia", "Dept. of Physics\nAligarh Muslim University\n202002AligarhU.PIndia", "MQ Photonics Research Centre\nDept. of Physics and Astronomy\nMacquarie University\n2109SydneyNSWAustralia", "School of Engineering\nRMIT University\n3001MelbourneVictoriaAustralia" ]
[]
We model the broadband enhancement of single-photon emission from color centres in silicon carbide nanocrystals coupled to a planar hyperbolic metamaterial (HMM) resonator.The design is based on positioning the single photon emitters within the HMM resonator, made of a dielectric index-matched with silicon-carbide material. The broadband response results from the successive resonance peaks of the lossy Fabry-Perot structure modes arising within the high-index HMM cavity. To capture this broadband enhancement in the single photon emitter's spontaneous emission, we placed a simple gold based cylindrical antenna on top of the HMM resonator. We analyzed the performance of this HMM coupled antenna structure in terms of the Purcell enhancement, quantum efficiency, collection efficiency and overall collected photon rate. For perpendicular dipole orientation relative to the interface, the HMM coupled antenna resonator leads to a significantly large spontaneous emission enhancement with Purcell factor of the order of 250 along with a very high average total collected photon rate (CPR) of about 30 over a broad emission spectrum (700 nm -1000 nm). The peak CPR increases to about 80 at 900 nm, corresponding to the emission of silicon-carbide quantum emitters. This is a state-of-the art improvement considering the previous computational designs have reported a maximum average CPR of 25 across the nitrogen-vacancy centre emission spectrum, 600 nm to 800 nm with the highest value being about 40 at 650 nm.
10.1364/josab.35.002153
[ "https://export.arxiv.org/pdf/1801.03277v3.pdf" ]
51,686,243
1801.03277
b74c770b9eccc9b5c9875e915b1614d47f402bc1
Hyperbolic Metamaterial Resonator-Antenna Scheme for Large, Broadband Emission Enhancement and Single Photon Collection Faraz A Inam Dept. of Physics Aligarh Muslim University 202002AligarhU.PIndia Dept. of Condensed Matter Physics and Material Science Tata Institute of Fundamental Research 400005MumbaiIndia Nadeem Ahmed Dept. of Physics Aligarh Muslim University 202002AligarhU.PIndia Michael J Steel MQ Photonics Research Centre Dept. of Physics and Astronomy Macquarie University 2109SydneyNSWAustralia Stefania Castelletto School of Engineering RMIT University 3001MelbourneVictoriaAustralia Hyperbolic Metamaterial Resonator-Antenna Scheme for Large, Broadband Emission Enhancement and Single Photon Collection 1Keyworks: Hyperbolic metamaterial resonatorSingle-photon sourcesSilicon Carbide emittersnitrogen vacancy centresbroadband enhancementsingle photon antenna 2 We model the broadband enhancement of single-photon emission from color centres in silicon carbide nanocrystals coupled to a planar hyperbolic metamaterial (HMM) resonator.The design is based on positioning the single photon emitters within the HMM resonator, made of a dielectric index-matched with silicon-carbide material. The broadband response results from the successive resonance peaks of the lossy Fabry-Perot structure modes arising within the high-index HMM cavity. To capture this broadband enhancement in the single photon emitter's spontaneous emission, we placed a simple gold based cylindrical antenna on top of the HMM resonator. We analyzed the performance of this HMM coupled antenna structure in terms of the Purcell enhancement, quantum efficiency, collection efficiency and overall collected photon rate. For perpendicular dipole orientation relative to the interface, the HMM coupled antenna resonator leads to a significantly large spontaneous emission enhancement with Purcell factor of the order of 250 along with a very high average total collected photon rate (CPR) of about 30 over a broad emission spectrum (700 nm -1000 nm). The peak CPR increases to about 80 at 900 nm, corresponding to the emission of silicon-carbide quantum emitters. This is a state-of-the art improvement considering the previous computational designs have reported a maximum average CPR of 25 across the nitrogen-vacancy centre emission spectrum, 600 nm to 800 nm with the highest value being about 40 at 650 nm. 2 I. INTRODUCTION. Solid-state quantum emitters operating at room temperature [1,2], and their integration in photonic networks are central in the development of quantum information processing. A variety of materials have emerged as an alternative platform to host color centres that could operate as room-temperature single photon sources (SPSs) [3,4] and at the same time, as coherent spin quantum bits with optical read-out [5,6]. This category of quantum emitters includes diamond color centres [7], zinc-oxide based (ZnO) color centres [8], emitters in silicon-carbide (SiC) [3], GaN [9], and 2D materials such as boron nitride [4]. While all these platforms are of great interest for future integrated on-chip quantum photonics at room temperature, as they all have their own set of interesting properties, only diamond and SiC have been proved to host SPSs and spin qubits with optical read-out at the same time [5,6]. In particular SiC can host bright SPSs in the visible [10] that may be efficiently electrically driven using off-the-shelf fabrication methods [11]. Further, SiC possesses many color centres with high spin state [3], that can be used as quantum bits with the longest coherence time in solids, due to the absence of spin-orbit coupling and nuclear spin bath decoupling [12]. Additionally SiC is a CMOS compatible material [13] with a large variety of nanofabrication methods that includes laser micromachining/deposition and ablation [14], making it ideally placed for device fabrication and on-chip integration of quantum systems [15]. These quantum emitters' spectra range from the visible region to the near infrared (up to ~1100nm), the latter emitters being much dimmer [6] and thus most in need of radiative spontaneous emission (SE) enhancement for their application in magnetic sensing [16] and spin nanoscopy [17]. As color centres are point objects emitting radiation as a point dipole with a slow decay rate (nanosecond scale), one of the main aims is to increase their decay rate and direct their emission modes. The typical approach followed is to couple them with microcavities or nanocavities with high quality factors or small mode volume to increase the enhancement on a specific emission mode and couple them to waveguides to extract the photons [2,18]. For optical micro and nanocavities, large Purcell enhancement (Fp) is limited to narrow linewidths. The other approach is to build a single photon antenna [19] based on plasmonic resonators [20]. Here the quality factor is low, thus enhancement can occur over the entire color center spectrum (100 nm), while high confinement can be achieved. A major disadvantage of plasmonic antennas is in relying on materials with plasmonic resonance which entails plasmonic losses [21]. More recently a class of Hyperbolic Metamaterials 3 (HMM) [22], metal-dieletric multilayers, have been proposed to enhance the SE of SPSs due to their in-principle indefinitely large photonic density of states and have opened new avenues in their use for quantum optics applications, including SPSs. HMMs have a highly anisotropic dielectric tensor, with dielectric properties in some directions and metallic properties in another direction. This provides the emitting dipole with asymptotically directed large momentum LDOS modes (high-k modes with unbounded magnitude of Kx and Kz in the lossless limit). This enables high emission directionality, a large increase of LDOS and of SE rate, with overall high quantum yield due to a larger ratio of radiated to dissipated power as compared to plasmonic antennas [23]. All these properties are maintained over a 100 to 200 nm spectral region. The collection enhancement of quantum dots SE has been achieved when positioned within HMM resonators [24], while for the nitrogen vacancy centre (NV), the nanodiamonds were positioned on top of the HMM resonators [25]. This last case has achieved an enhancement of SE of a factor of 3, relative to the emission on a cover glass. Further by introducing a metalloid as metamaterial rather than gold, such as TiN, with lower dissipative losses, a SE enhancement close to 10 for a perpendicular dipole was predicted (5.5 for the average dipole orientation) with an experimental average enhancement of a factor of 4 [26]. The effectiveness of the various out-coupling schemes can be quantified in terms of the overall enhancement in the collected photon rate (CPR) [27]. The CPR is governed by both the Purcell enhancement (Fp) and collection efficiency (CE), defined as the spatial overlap of the emission with the numerical aperture (NA) of the collection objective, the CPR being the product of the above two factors ( ∝ • ) [27]. This means that even if a perfect dielectric antenna is able to achieve perfect collection with a CE of unity but with negligible Purcell enhancement, (Fp ~ 1), it results in a CPR of 1. Present state of the art collection schemes for NV centre emission using anti-reflection coated solid-immersion lens reports a CE ~ 0.4 with Fp ~ 1 [28,29] and dielectric grating achieving a CE ~ 0.7 and Fp ~1 [30,31]. Simulated designs of metal antennas [32] have been shown to achieve a Fp ~ 50 and CE ~ 0.3 in a bandwidth of about 50 nm with an average CPR of just about 2-3 over the full NV spectrum. A recent simulated study of metal-dielectric nano-antenna have reported the highest CPR of about 25 across the NV spectrum [27]. In this work, we study and design a HMM resonator based on only five alternating layers of gold and ZnS (previously 16 layers were used [26]) for non-resonant broad-band 4 enhancement of SE of SiC emitters [33]. This structure would be much simpler to fabricate experimentally and very much preserves the hyperbolicity condition ( ⊥ ⋅ ∥ < 0) in the studied frequency range of 650 nm to 1000 nm. The emitter is positioned in the centre of the resonator rather than on the top. It is thus the first broadband design specifically targeting emitters, such as the carbon antisite vacancy pair [10] and other visible SiC surface quantum emitters [11], the silicon vacancy in SiC [6], and it could be as well applied to NV centres in diamond. The choice of ZnS as dielectric is based on its facile deposition methods with its wide use in thin-film growth [34] and most importantly its refractive index (2.4) being similar to SiC (2.5) and diamond (2.4). Emitters in high index dielectric materials like diamond/SiC generally suffer from low emission rates due to poor index matching at the diamond-air interface [35]. Various methods have been explored to efficiently out-couple the emission from these centers by placing them in an environment of higher refractive index materials [36]. A ZnS resonator with its index matched with both diamond and SiC will be an ideal candidate for the out-coupling of the emission from these host centres. This resonator will be useful in coupling of nanocrystals of SiC containing the red or near infrared emitters [6] or nanodiamonds containing NV centres. The use of SiC thin layers achieved by pulsed laser deposition [37] as a dielectric in the HMM resonators, being the ultimate goal to apply this design to emitters directly embedded in the dielectric constituent [38] II. THEORITICAL BACKGROUND: The spontaneous emission rate  of a quantum emitter can be accelerated by tuning the electromagnetic mode environment in its vicinity through a factor known as the local density of optical states (LDOS), ( , ). The LDOS in terms of the Green function is defined as [19,39] ( , ) = (6 2 ⁄ )[̂T • Im ( ; , ) •̂](1) 5 for the emitter source located at and oriented along the unit vector ̂. Here  is the transition frequency and G is the dyadic Green function signifying the interaction of the radiated electric field with the emitter at the source point. The enhancement of the spontaneous emission by an environment with high LDOS is known as the Purcell effect, which is quantified in terms of the Purcell factor (Fp), i.e. the spontaneous emission decay rate in such an environment compared to that in vacuum. A resonant structure provides an emitter with enhanced LDOS for its emitted photon to couple to when its emission frequency matches with the frequency of the optical modes supported by the structure. In plasmonics, large electric field in the vicinity of the metallic nano-structures significantly enhances the LDOS over the broad scattering spectrum of the metallic nanostructures. Plasmonic structures however provide enhanced LDOS mostly in the visible region of the spectrum as the resonant plasma frequency of the metals lies in this range. A hyperbolic meta-material (HMM) comprises of sub-wavelength scale periodic metaldielectric layers in a specific direction [22]. This provides anisotropy in the dispersion profile resulting in anisotropy in the LDOS for different transition dipole orientations as well as divergence of the LDOS [22]. In HMMs the modes have higher momentum than conventional surface plasmon polaritons, even out of resonance. From the layers' permittivity and the exact fill fraction of metal in the structure the components of the effective anisotropic permittivity can be calculated using effective medium theory [40]. The average index for the multilayered HMM can be calculated using the relation [22]: ⊥ = + + , 1 ∥ = ⁄ + ⁄ +(2) where ⊥ , ∥ are the dielectric components in perpendicular and parallel directions respectively and , md dd are the thickness of metal and dielectric layers respectively. . In HMM the effective permittivity components ∥ parallel and perpendicular ⊥ to the interface have opposite signs ( ⊥ ⋅ ∥ < 0), which lends a hyperboloid shape to the iso-frequency surface (IFC), according to the dispersion relation equation ZnS/Au HMM structure (described in the next section), the above hyperbolicity condition is well displayed within our studied frequency range of 650 -1000 nm (Fig.1). 6 There are two types of resonant modes arising in a HMM structure. The first is that of the plasmonic modes resulting from the alternate sub-wavelength metal layers. The second type are the structure modes as the HMM structure acts as a nanoscale lossy cavity. An interplay of these two type of modes results in the broadband enhancement of the LDOS seen by the transition dipole when placed inside the HMM [41]. III. RESULTS AND DISCUSSION: We carried out calculations using the Radio Frequency (RF) module of the COMSOL Multiphysics suite. To test the validity of our calculations we compared our results with the analytical results of a dipole emission near a metal film (Drexhage experiment) [42]. Figure 2 shows the good agreement of our calculations with the analytical results. In this model we have a dipole emitter positioned in air and 10 nm away from the gold (Au) surface (Fig. 2a). For this HMM calculation, we have considered quantum emitters based on silicon-carbide (SiC) nano-particles. Purcell Factor/Emission Rate Calculations: We start the design of HMM structure by encapsulating the nano-spheres of silicon-carbide The mixing of the plasmonic modes with these structure modes results in the overall broadband response from the structure. The resonance of these structure modes can be tuned across the infrared spectrum by controlling the metal and dielectric layer thicknesses. Here we have chosen a thickness for the metal/dielectric layers for which the HMM structure could be easily fabricated. The main highlight of this study is to present a simple 5-layered HMM structure based on a high index dielectric layer having its index closely matched to the SiC and diamond based single photon emitters. This relatively simple strategy while preserving the hyperbolicity condition ( ⊥ ⋅ ∥ < 0) in our studied frequency range of 650 nm to 1000 nm (Fig. 1 Purcell/emission enhancement is therefore restricted to only the plasmonic resonance in the visible spectrum without any structure modes arising in the infrared region ( Fig. 4 (b)). Here only the perpendicular dipole orientation gave an enhanced broadband response with successive resonance peaks in the infrared spectrum where the emission from SiC as well as being the local electric field profile of the modes with wave-vectors k and polarizations [40]. The Purcell enhancement is therefore expected to go as a function Collected Photon Rate Calculations: We have achieved broadband enhancement from visible to infra-red frequencies using a multilayered higher index dielectric HMM structure. We will now focus on how to capture the maximum light out of these structures. Figure 6 displays the far-field coupling of the dipole emission at 900 nm for a vertically oriented dipole radiating on a silica glass substrate (n = 1.45) and in our HMM structure. Dipole emission on a glass substrate has been well studied with the glass itself being known to act as a dielectric antenna directing most of the emission to the higher index substrate medium [44,45]. For perpendicular dipole emission in the HMM, from Figure 6 it can be seen that most of the dipole emission is being lost in the HMM layer and the bottom glass substrate. We therefore need to design a robust antenna which would be able to direct this broadband Purcell enhancement into the numerical aperture (NA) of the collection objective on top of the HMM structure. This would result in a large photon collection rate (CPR) throughout the broadband spectral range of the HMM. For these photon collection rate calculations, we will consider the objective lens to have a numerical aperture of (NA) of 0.95. This corresponds to a collection solid angle of 71.8 o . 14 The Au cylindrical antenna based HMM structure seems to shift the Purcell enhancement curve giving a resonance peak at 900 nm since we optimized the antenna parameters, its height and radius for dipole emission at 900 nm. However, the order of enhancement is well maintained throughout the broadband spectrum range of the various color centres in SiC and diamond. Though this simple design we are able to achieve an average collected photon rate IV. CONCLUSIONS: We have demonstrated a simple HMM structure based on a high refarctive index dielectric, [27]. In this calculation, the NV dipole is considered to be located at the centre of the bowtie with its orientation along the bowtie axis. As the refractive indices of ZnS (n = 2.30 at 900 nm), SiC (2.59) and diamond (2.39) are matched, our HMM scheme is universal for all SiC and diamond based emitters. It will lead to similar enhancements for the case of SiC nanocrystals fabricated as in [33,47] and embedded in the ZnS middle layer; or for nanodiamond embedded in the ZnS middle layer. Additionally, this model is particularly interesting as it can be applied to enhance emission of single photon sources not only in nanoparticles emitters but also in thin layer. This design can be extended to the case of emitters embedded in diamond [48] or SiC thin-films by replacing the middle ZnS layer. By replacing the dielectric with SiC itself and the metamaterial with TiN (which has similar plasmonic properties of gold) a fully CMOS compatible HMM resonator can be realised. Due to TiN being refractory material, its fabrication is compatible with thin film SiC pulsed laser deposition [38] or SiC chemical vapor deposition, where surface color centres with dipole aligned to the main crystallographic axis can be created [49] in addition to the above mentioned color centres. V. METHODS: All electromagnetic calculations of the dipole emission rate and collected power are performed using the commercial finite-element method (FEM) based COMSOL Multiphysics RF module package 5.2. The single vacancy color center is modelled as an oscillating point dipole [50] located within the SiC nanoparticle of size 40 nm. Spontaneous emission is a purely quantum process. However since the influence of the environment is expressed through the classical LDOS, the emission rate relative to a reference system can be found by classical electromagnetic calculations also [50]. Here in this study we are accounting for only the radiative emission process considering our emitters to have high quantum efficiency. Treating the quantum emitter as a classical point dipole, the electromagnetic fields are excited by a point current source driven at frequency . c   The total power radiated by the dipole is calculated over the surface of the SiC sphere as highlighted in Figure 9. The radiative spontaneous emission rate enhancement is then calculated as R = P/Pr, where Pr is the power corresponding to the reference system. For Purcell factor calculations the reference system is a SiC nanoparticle in vacuum. Within the whole computational domain the minimum mesh size used is 0.5 nm and the maximum mesh size used is 45 nm. Within the SiC sphere where the radiated power is calculated the minimum mesh size is kept at 0.5 nm with the maximum mesh size being 2 nm. of the HMM resonator. This design provides large Purcell factor enhancements ( HMM vacuum ) of about 300 at both visible (680 nm, corresponding to the peak of diamond NV centre emission) and near infrared emission (900 nm, corresponding to the emission of SiC nanoparticles) for perpendicular dipole orientation. Further by using a simple Au based cylindrical antenna on top of the HMM we are able to achieve an average collection efficiency of about 0.15. This gives us a very high collected photon rate ( ∝ • ) of about 30 in the broad spectrum range of 700 nm to 1000 nm. Figure 1 . 1The effective permittivity ⊥ (red) and ∥ (blue) of our ZnS/Au HMM structure as a function of wavelength. Figure 2 . 2(a) Schematic of a dipole emitter placed on top of the gold (Au) surface. The separation between the dipole and Au surface is considered as 10 nm. (b) Calculated and analytical Purcell factor ( vacuum  ) for the above considered case of dipole emission above the Au surface. of diameter 40 nm inside a Poly Vinyl Alcohol (PVA) matrix of thickness 50 nm. The PVA layer is encapsulated on both sides by gold (Au) metal layers of 30 nm thickness. To provide the hyperbolicity to this structure we place a low index dielectric layer of PVA of the same thickness 30 nm above and below the Au layers. This arrangement gives a simple five-layer structure of an HMM. The quantum emitter is placed at the centre of the SiC sphere. The HMM structure is placed on a glass substrate. From this simple low-index PVA/Au HMM, we build our HMM structure by sequentially replacing the PVA layer with the high indexed layer of ZnS(Fig. 3). The ZnS layer has its index matched with the SiC nanoparticle and acts as an effective resonator for enhanced emission from SiC centres. Figure 3 . 3Schematic images showing the design for Au/ZnS HMM starting from a simple Au/PVA counterpart. (a) Au/PVA HMM, (b) Au/PVA+ZnS HMM and (c) Au/ZnS HMM. Figure 4 . 4Purcell factor ( HMM vacuum ) and relative emission rates ( cov HMM erslip  ) for dipole emission in the above three HMM structures corresponding to (a), (c) perpendicular dipole orientation relative to the Au layers and (b), (d) parallel dipole orientation relative to the Au layer. Figure 4 4displays how the Purcell factor (Fp = HMM vacuum ) and relative emission rates ( cov HMM erslip  ) for dipole emission in the HMM structure, gets significantly enhanced by the use of index-matched ZnS layers. On replacing the PVA layer (n = 1.47 at λ = 900 nm) with 9 the high index ZnS layer (n = 2.30 at λ = 900 nm) the enhancement in the Purcell factor/emission rate significantly increases. More prominent resonance peaks emerge in the emission spectrum.These successive peaks results from the lossy Fabry-Perot structure modes[43]. The high index HMM structure itself acting a lossy Fabry-Perot cavity due to large reflections at the interfaces. Since the effective index of the medium decreases with increase in wavelength, the successive resonance peaks appears broader due to reduced reflectivity at the HMM interface. The broadband response is restricted to the case of dipole emission perpendicular to the Au interface. Perpendicular dipole emission experiences reflections at the top and bottom HMM layers. This results in the excitation of lossy Fabry-Perot modes in the HMM structure. ), results in a significantly large broadband enhancement over the full emission range of the various point emitters embedded inside the SiC and diamond nanoparticles. The chosen thicknesses of the metal/dielectric layers provide successive Fabry-Perot modes in the studied emission band of 650 nm -1000 nm. With reduced thicknesses for metal/dielectric layers, more metal/dielectric layers would be required to achieve similar resonance modes in the studied emission range. This would have increased the complexity of the structure. For the parallel dipole emission, the reflections at the top and bottom HMM layers are minimal due to the continuity of the field across the interface boundaries. The of cos 2 2, being about half for dipoles oriented at 45 o to the interface. The Purcell enhancement factor being still more than 100 over the broad spectrum range of 650 nm to 1000 nm (refer to Figure 8a.). To test the sensitivity in the Purcell/emission enhancement with the variations in the dipole positions within the SiC nanoparticles we varied the dipole position along the z (vertical) and the x (horizontal) directions. The x and y directions which are both parallel to the metaldielectric interface are symmetric to each other. The HMM structure is considered to be continuous along the x, y (horizontal) directions and we have modelled these by using scattering boundary conditions along these directions. Figure 5 shows the variation in the Purcell factor with dipole displacement along the three orthogonal coordinate axes for the case of dipole emission at 900 nm, corresponding to vacancy-Si, VSi in SiC. Along the zdirection, as the dipole separation from the centre increases the dipole gets closer to the metal layers leading to significantly enhanced HMM structure modes coupled to the plasmonic resonance. Along the x and y directions, the dipole experiences the same electromagnetic environment due to the top and bottom HMM layer. Slight changes in the Purcell factor arise due to small variations in the dipole emission rates corresponding to the varying dipole separation from the SiC sphere surface for the vacuum case. Our result therefore shows that the enhancement is not specific to the dipole central location and the same order of enhancement exists for all dipole positions within the middle ZnS layer. Figure 5 . 5Variation of Purcell factor SiC based dipole emitter encapsulated in the multilayered Au/ZnS HMM structure with its displacement along the coordinate axes directions. The emission wavelength considered here is 900 nm corresponding to vacancy-Si, VSi in SiC. Figure 6 . 6The far-field distribution of the radiated power for dipole emission in SiC nanoparticle when placed on a glass substrate and in the HMM structure. The emission wavelength considered is 900 nm.To collect light from the top of HMM structures, complex periodic-gratings are commonly used[24,31]. Here we use a simple antenna structure consisting of a single metal cylinder on top of the HMM structure. Traditionally for radio frequencies, antenna parameters are prescribed only in terms of external wavelength. However, at optical frequencies electrons in the metal offer substantial inertia and do not respond instantaneously to the driving fields.Metal electrons therefore have to be treated as a strong coupled plasma at optical frequencies.This leads to a reduced effective wavelength within the antenna[46]. Figure 7a 7ashows the schematic of the Au cylindrical antenna on top of the HMM structure.To optimize the antenna parameters for maximum collection efficiency we measured the collection efficiency (CE) from the structure by varying both the antenna height and antenna diameter at dipole emission wavelength equal to 900 nm. The calculated optimum antenna height and diameter for 900 nm emission wavelength are 110 nm and 820 nm respectively. Figure 7b 7bdisplays the far-field distribution of the radiated power for the dipole emission in SiC nanoparticle at 900 nm with Au cylindrical antenna on the HMM surface. The antenna is clearly able to direct a significant portion of the radiated power within the collection angle of the objective. Figure 8 shows the performance of this antenna for perpendicular and 45 o to Au/ZnS interface dipole orientations in terms of the Purcell enhancement, the quantum efficiency (QE), the collection efficiency (CE) and the photon collected rate (CPR) of the 13 HMM antenna structure. The quantum efficiency being defined as, = far-field radiated power/total emitted power by the dipole. Figure 7 . 7(a) Schematic of the Au cylindrical antenna on top of the HMM structure. (b) The far-field distribution of the radiated power for dipole emission in SiC nanoparticle when placed in the HMM structure with the cylindrical antenna on top. The emission wavelength considered is 900 nm. Figure 8 . 8The total performance of the Au cylindrical antenna based HMM structure for perpendicular and 45 o dipole orientation. (a) the modified Purcell enhancement (Fp), (b) the quantum efficiency (QE), (c) the collection efficiency and (d) the collected photon count rate (CPR). ( CPR) of about 30 throughout the broad spectral range of 700 nm to 1000 nm for perpendicular dipole orientation. Also since our Au antenna diameter is quite large, about 820 nm, the antenna performance will not be critical to the dipole location below the antenna and gives the same order of enhancement for few nanometers variation in the dipole positions within the HMM structure.Since ∝ • with ∝ cos 2 and collection efficiency being mostly a function of the antenna only, the CPR is also expected to scale as ∝ cos 2 . From Figure 8, it can be seen that both Purcell enhancement and collected photon rates are nearly being halved for 45 o dipole orientation relative to the Au/ZnS interface. The collection efficiency is similar for both the dipole orientations. The 45 o dipole orientation also gives a high average CPR value of about 15 over the broad spectral range of 700 nm to 1000 nm.The antenna design here is optimized for dipole emission around 900 nm corresponding toSiC nanoparticles based color centres. However, this scheme can be applied to optimize the antenna performance across any of the color centres in diamond and SiC, including the wellknown NV or Si vacancy centres by tuning the antenna parameters (its height and radius). Figure 9 . 9Computational geometry of our HMM structure. The highlighted bottom domains corresponds to the coverglass surface on which our HMM structure is place. The inner sphere corresponds to the SiC nanoparticle enclosing the point dipole. Figure 10 . 10Computational geometry of our HMM antenna structure for far-field and collected photon rate calculations. Figure 10 10shows the computational geometry of our HMM structure with the radiated power being collected on the top highlighted surface. The diameter of the top surface is chosen such that it acts as a collection lens with a numerical aperture of 0.95. The far-field power distribution is calculated on the surface of the enclosed sphere whose radius is about 3 times the dipole emission wavelength. The Au cylindrical antenna is represented by the highlighted cylindrical surface on top of the HMM structure. Acknowledgement FAI would like to thank Venu Gopal Achanta, TIFR, India, for useful discussions during the course of this study. FAI acknowledge University Grant Commission, India for funding through the Faculty Start-up grant. diamond centres lies. However, as the spontaneous emission rate  of a dipole in the weak10 coupling regime from the Fermi's Golden rule is proportional to     2 2 ,, ,ˆ.         kk k d E r , with  being the transition frequency, μ = d  the dipole moment and   , k Er  ZnS, and Au layers which shows a large broadband Purcell enhancement for dipole emission in both visible and near infra-red regions. The HMM resonator is composed of gold and ZnS layers to achieve a Purcell enhacement of 400 at 850 nm and 300 at 680nm with a similar order of enhancement throughout the broad spectral range of 650 nm to 1000 nm.By employing simple gold (Au) cylindrical antenna on top of the HMM structure we are able to achieve a large average collected photon rate (CPR) of about 30 throughout the broad spectral range of 700 nm to 1000 nm. The peak CPR value of about 80 at 900 nm corresponding to the emission of silicon-carbide quantum emitters. This is therefore a stateof-the art improvement considering that the previous computational design using a hybrid bowtie metal-dielectric antenna has reported a maximum average CPR of 25 across the NV emission spectrum of 600 nm to 800 nm with the highest value being about 40 at 65015 nm . A Sipahigil, R E Evans, D D Sukachev, M J Burek, J Borregaard, M K Bhaskar, C T , A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. . J L Nguyen, H A Pacheco, C Atikian, R M Meuwly, F Camacho, E Jelezko, H Bielejec, M Park, M D Lončar, Lukin, Science. 354847Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, Science 354, 847 (2016). . P Lodahl, S Mahmoodian, S Stobbe, Rev. Mod. Phys. 87347P. Lodahl, S. Mahmoodian, and S. Stobbe, Rev. Mod. Phys. 87, 347 (2015). . A Lohrmann, B C Johnson, J C Mccallum, S Castelletto, Reports Prog. Phys. 8034502A. Lohrmann, B. C. Johnson, J. C. McCallum, and S. Castelletto, Reports Prog. Phys. 80, 034502 (2017). . T T Tran, K Bray, M J Ford, M Toth, I Aharonovich, Nat. Nanotechnol. 1137T. T. Tran, K. Bray, M. J. Ford, M. Toth, and I. Aharonovich, Nat. Nanotechnol. 11, 37 (2015). F J Heremans, C G Yale, D D Awschalom, Proc. IEEE. IEEE104F. J. Heremans, C. G. Yale, and D. D. Awschalom, Proc. IEEE 104, 2009 (2016). . F Fuchs, B Stender, M Trupke, D Simin, J Pflaum, V Dyakonov, G V Astakhov, Nat. Commun. 67578F. Fuchs, B. Stender, M. Trupke, D. Simin, J. Pflaum, V. Dyakonov, and G. V. Astakhov, Nat. Commun. 6, 7578 (2015). . I Aharonovich, S Castelletto, D A Simpson, C.-H Su, A D Greentree, S Prawer, Reports Prog. Phys. 7476501I. Aharonovich, S. Castelletto, D. A. Simpson, C.-H. Su, A. D. Greentree, and S. Prawer, Reports Prog. Phys. 74, 076501 (2011). . A J Morfa, B C Gibson, M Karg, T J Karle, A D Greentree, P Mulvaney, S Tomljenovic-Hanic, Nano Lett. 12949A. J. Morfa, B. C. Gibson, M. Karg, T. J. Karle, A. D. Greentree, P. Mulvaney, and S. Tomljenovic-Hanic, Nano Lett. 12, 949 (2012). . A M Berhane, K.-Y Jeong, Z Bodrog, S Fiedler, T Schröder, N V Triviño, T Palacios, A Gali, M Toth, D Englund, I Aharonovich, Adv. Mater. 291605092A. M. Berhane, K.-Y. Jeong, Z. Bodrog, S. Fiedler, T. Schröder, N. V. Triviño, T. Palacios, A. Gali, M. Toth, D. Englund, and I. Aharonovich, Adv. Mater. 29, 1605092 (2017). . S Castelletto, B C Johnson, V Ivády, N Stavrias, T Umeda, A Gali, T Ohshima, Nat. Mater. 13151S. Castelletto, B. C. Johnson, V. Ivády, N. Stavrias, T. Umeda, A. Gali, and T. Ohshima, Nat. Mater. 13, 151 (2014). . A Lohrmann, N Iwamoto, Z Bodrog, S Castelletto, T Ohshima, T J Karle, A Gali, S Prawer, J C Mccallum, B C Johnson, Nat. Commun. 67783A. Lohrmann, N. Iwamoto, Z. Bodrog, S. Castelletto, T. Ohshima, T. J. Karle, A. Gali, S. Prawer, J. C. McCallum, and B. C. Johnson, Nat. Commun. 6, 7783 (2015). . H Seo, A L Falk, P Klimov, K C Miao, G Galli, D D Awschalom, Nat. Commun. 712935H. Seo, A. L. Falk, P. V Klimov, K. C. Miao, G. Galli, and D. D. Awschalom, Nat. Commun. 7, 12935 (2016). . M H Weng, D T Clark, S N Wright, D L Gordon, M A Duncan, S J Kirkham, M I Idris, H , M. H. Weng, D. T. Clark, S. N. Wright, D. L. Gordon, M. A. Duncan, S. J. Kirkham, M. I. Idris, H. . K Chan, R A R Young, E P Ramsay, N G Wright, A B Horsfall, Semicond. Sci. Technol. 3254003K. Chan, R. A. R. Young, E. P. Ramsay, N. G. Wright, and A. B. Horsfall, Semicond. Sci. Technol. 32, 054003 (2017). . Y Huang, X Wu, H Liu, H Jiang, J. Micromechanics Microengineering. 2765005Y. Huang, X. Wu, H. Liu, and H. Jiang, J. Micromechanics Microengineering 27, 065005 (2017). . S E Economou, P Dev, Nanotechnology. 27504001S. E. Economou and P. Dev, Nanotechnology 27, 504001 (2016). . H Kraus, V A Soltamov, F Fuchs, D Simin, A Sperlich, P G Baranov, G V Astakhov, V Dyakonov, Sci, Rep, 45303H. Kraus, V. A. Soltamov, F. Fuchs, D. Simin, A. Sperlich, P. G. Baranov, G. V. Astakhov, and V. Dyakonov, Sci. Rep. 4, 5303 (2015). . M Pfender, N Aslam, G Waldherr, P Neumann, J Wrachtrup, Proc. Natl. Acad. Sci. U. S. A. 11114669M. Pfender, N. Aslam, G. Waldherr, P. Neumann, and J. Wrachtrup, Proc. Natl. Acad. Sci. U. S. A. 111, 14669 (2014). . M Pelton, Nat. Photonics. 9427M. Pelton, Nat. Photonics 9, 427 (2015). . A F Koenderink, ACS Photonics. 4710A. F. Koenderink, ACS Photonics 4, 710 (2017). . K J Russell, T.-L Liu, S Cui, E L Hu, Nat. Photonics. 6459K. J. Russell, T.-L. Liu, S. Cui, and E. L. Hu, Nat. Photonics 6, 459 (2012). . S Schietinger, M Barth, T Aichele, O Benson, Nano Lett. 91694S. Schietinger, M. Barth, T. Aichele, and O. Benson, Nano Lett. 9, 1694 (2009). . A Poddubny, I Iorsh, P Belov, Y Kivshar, Nat. Photonics. 7948A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, Nat. Photonics 7, 948 (2013). . W D Newman, C L Cortes, Z Jacob, J. Opt. Soc. Am. B. 30766W. D. Newman, C. L. Cortes, and Z. Jacob, J. Opt. Soc. Am. B 30, 766 (2013). . T Galfsky, H N S Krishnamoorthy, W Newman, E E Narimanov, Z Jacob, V M Menon, Optica. 262T. Galfsky, H. N. S. Krishnamoorthy, W. Newman, E. E. Narimanov, Z. Jacob, and V. M. Menon, Optica 2, 62 (2015). . M Y Shalaginov, S Ishii, J Liu, J Liu, J Irudayaraj, A Lagutchev, A V Kildishev, V M Shalaev, Appl. Phys. Lett. 102173114M. Y. Shalaginov, S. Ishii, J. Liu, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, Appl. Phys. Lett. 102, 173114 (2013). . M Y Shalaginov, V V Vorobyov, J Liu, M Ferrera, A V Akimov, A Lagutchev, A , M. Y. Shalaginov, V. V. Vorobyov, J. Liu, M. Ferrera, A. V. Akimov, A. Lagutchev, A. N. . V V Smolyaninov, J Klimov, A V Irudayaraj, A Kildishev, V M Boltasseva, Shalaev, Laser Photon. Rev. 9120Smolyaninov, V. V. Klimov, J. Irudayaraj, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, Laser Photon. Rev. 9, 120 (2015). . A Karamlou, M E Trusheim, D Englund, Opt. Express. 263341A. Karamlou, M. E. Trusheim, and D. Englund, Opt. Express 26, 3341 (2018). . J P Hadden, J P Harrison, A C Stanley-Clarke, L Marseglia, Y.-L D Ho, B R Patton, J L O&apos;brien, J G Rarity, Appl. Phys. Lett. 97241901J. P. Hadden, J. P. Harrison, A. C. Stanley-Clarke, L. Marseglia, Y.-L. D. Ho, B. R. Patton, J. L. O'Brien, and J. G. Rarity, Appl. Phys. Lett. 97, 241901 (2010). . W Pfaff, B J Hensen, H Bernien, S B Van Dam, M S Blok, T H Taminiau, M J Tiggelman, R N Schouten, M Markham, D J Twitchen, R Hanson, W. Pfaff, B. J. Hensen, H. Bernien, S. B. Van Dam, M. S. Blok, T. H. Taminiau, M. J. Tiggelman, R. N. Schouten, M. Markham, D. J. Twitchen, and R. Hanson, (n.d.). . P.-B Li, Y.-C Liu, S.-Y Gao, Z.-L Xiang, P Rabl, Y.-F Xiao, F.-L Li, Phys. Rev. Appl. 444003P.-B. Li, Y.-C. Liu, S.-Y. Gao, Z.-L. Xiang, P. Rabl, Y.-F. Xiao, and F.-L. Li, Phys. Rev. Appl. 4, 044003 (2015). . L Li, E H Chen, J Zheng, S L Mouradian, F Dolde, T Schröder, S Karaveli, M L Markham, D J Twitchen, D Englund, Nano Lett. 151493L. Li, E. H. Chen, J. Zheng, S. L. Mouradian, F. Dolde, T. Schröder, S. Karaveli, M. L. Markham, D. J. Twitchen, and D. Englund, Nano Lett. 15, 1493 (2015). . I Bulu, T Babinec, B Hausmann, J T Choy, M Loncar, Opt. Express. 195268I. Bulu, T. Babinec, B. Hausmann, J. T. Choy, and M. Loncar, Opt. Express 19, 5268 (2011). . S Castelletto, A F M Almutairi, G Thalassinos, A Lohrmann, R Buividas, D W M Lau, P Reineck, S Juodkazis, T Ohshima, B C Gibson, B C Johnson, Opt. Lett. 421297S. Castelletto, A. F. M. Almutairi, G. Thalassinos, A. Lohrmann, R. Buividas, D. W. M. Lau, P. Reineck, S. Juodkazis, T. Ohshima, B. C. Gibson, and B. C. Johnson, Opt. Lett. 42, 1297 (2017). . D H Hwang, J H Ahn, K N Hui, K S Hui, Y G Son, Nanoscale Res. Lett. 726D. H. Hwang, J. H. Ahn, K. N. Hui, K. S. Hui, and Y. G. Son, Nanoscale Res. Lett. 7, 26 (2012). . S Castelletto, J P Harrison, L Marseglia, A C Stanley-Clarke, B C Gibson, B A Fairchild, J P Hadden, Y.-L D Ho, M P Hiscocks, K Ganesan, S T Huntington, F Ladouceur, A D , S. Castelletto, J. P. Harrison, L. Marseglia, A. C. Stanley-Clarke, B. C. Gibson, B. A. Fairchild, J. P. Hadden, Y.-L. D. Ho, M. P. Hiscocks, K. Ganesan, S. T. Huntington, F. Ladouceur, A. D. . S Greentree, J L Prawer, J G O&apos;brien, Rarity, New J. Phys. 1325020Greentree, S. Prawer, J. L. O'Brien, and J. G. Rarity, New J. Phys. 13, 025020 (2011). . A Khalid, K Chung, R Rajasekharan, D W M Lau, T J Karle, B C Gibson, S Tomljenovic-Hanic, Sci. Rep. 511179A. Khalid, K. Chung, R. Rajasekharan, D. W. M. Lau, T. J. Karle, B. C. Gibson, and S. Tomljenovic-Hanic, Sci. Rep. 5, 11179 (2015). . M A Capano, S D Walck, P T Murray, D Dempsey, J T Grant, Appl. Phys. Lett. 643413M. A. Capano, S. D. Walck, P. T. Murray, D. Dempsey, and J. T. Grant, Appl. Phys. Lett. 64, 3413 (1994). . F A Inam, R Kumar, N Rameshwari, D Lau, S Ghosh, B C Gibson, S A Ramakrishna, S , F. A. Inam, R. Kumar, N. Rameshwari, D. Lau, S. Ghosh, B. C. Gibson, S. A. Ramakrishna, and S. Castelletto, 13th Int. Conf. Fiber Opt. Photonics. OSA, Washington, D.C.Castelletto, in 13th Int. Conf. Fiber Opt. Photonics (OSA, Washington, D.C., 2016), p. P1A.11. . G M Akselrod, C Argyropoulos, T B Hoang, C Ciracì, C Fang, J Huang, D R Smith, M H Mikkelsen, Nat. Photonics. 8835G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, Nat. Photonics 8, 835 (2014). . R J Glauber, M Lewenstein, Phys. Rev. A. 43467R. J. Glauber and M. Lewenstein, Phys. Rev. A 43, 467 (1991). . H Yuan, X Jiang, F Huang, X Sun, Sci. Rep. 630818H. Yuan, X. Jiang, F. Huang, and X. Sun, Sci. Rep. 6, 30818 (2016). . J R Lakowicz, Anal. Biochem. 337171J. R. Lakowicz, Anal. Biochem. 337, 171 (2005). . E Hecht, Optics. Addison-WesleyE. Hecht, Optics (Addison-Wesley, 2002). . W Lukosz, R E Kunz, J. Opt. Soc. Am. 671615W. Lukosz and R. E. Kunz, J. Opt. Soc. Am. 67, 1615 (1977). . K G Lee, X W Chen, H Eghlidi, P Kukura, R Lettow, A Renn, V Sandoghdar, S Götzinger, Nat. Photonics. 5166K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, Nat. Photonics 5, 166 (2011). . L Novotny, Phys. Rev. Lett. 98266802L. Novotny, Phys. Rev. Lett. 98, 266802 (2007). . A Muzha, F Fuchs, N V Tarakina, D Simin, M Trupke, V A Soltamov, E N Mokhov, P G Baranov, V Dyakonov, A Krueger, G V Astakhov, Appl. Phys. Lett. 105243112A. Muzha, F. Fuchs, N. V. Tarakina, D. Simin, M. Trupke, V. A. Soltamov, E. N. Mokhov, P. G. Baranov, V. Dyakonov, A. Krueger, and G. V. Astakhov, Appl. Phys. Lett. 105, 243112 (2014). . S Stehlik, M Varga, P Stenclova, L Ondic, M Ledinsky, J Pangrac, O Vanek, J Lipov, A Kromka, B Rezek, ACS Appl. Mater. Interfaces. 938842S. Stehlik, M. Varga, P. Stenclova, L. Ondic, M. Ledinsky, J. Pangrac, O. Vanek, J. Lipov, A. Kromka, and B. Rezek, ACS Appl. Mater. Interfaces 9, 38842 (2017). . A Lohrmann, S Castelletto, J R Klein, T Ohshima, M Bosi, M Negri, D W M Lau, B C , A. Lohrmann, S. Castelletto, J. R. Klein, T. Ohshima, M. Bosi, M. Negri, D. W. M. Lau, B. C. . S Gibson, J C Prawer, B C Mccallum, Johnson, Appl. Phys. Lett. 10821107Gibson, S. Prawer, J. C. McCallum, and B. C. Johnson, Appl. Phys. Lett. 108, 021107 (2016). . Y Xu, J S Vučković, R K Lee, O J Painter, A Scherer, A Yariv, J. Opt. Soc. Am. B. 16465Y. Xu, J. S. Vučković, R. K. Lee, O. J. Painter, A. Scherer, and A. Yariv, J. Opt. Soc. Am. B 16, 465 (1999).
[]
[ "LEARNING TO SELECT CONTEXT IN A HIERARCHICAL AND GLOBAL PERSPECTIVE FOR OPEN-DOMAIN DIALOGUE GENERATION", "LEARNING TO SELECT CONTEXT IN A HIERARCHICAL AND GLOBAL PERSPECTIVE FOR OPEN-DOMAIN DIALOGUE GENERATION" ]
[ "Lei Shen \nInstitute of Computing Technology\nIIP\nChinese Academy of Sciences\n\n\nUniversity of Chinese Academy of Sciences\n\n", "Haolan Zhan \nUniversity of Chinese Academy of Sciences\n\n", "Xin Shen \nAustralian National University\n\n", "Yang Feng \nInstitute of Computing Technology\nIIP\nChinese Academy of Sciences\n\n\nUniversity of Chinese Academy of Sciences\n\n" ]
[ "Institute of Computing Technology\nIIP\nChinese Academy of Sciences\n", "University of Chinese Academy of Sciences\n", "University of Chinese Academy of Sciences\n", "Australian National University\n", "Institute of Computing Technology\nIIP\nChinese Academy of Sciences\n", "University of Chinese Academy of Sciences\n" ]
[]
Open-domain multi-turn conversations mainly have three features, which are hierarchical semantic structure, redundant information, and long-term dependency. Grounded on these, selecting relevant context becomes a challenge step for multiturn dialogue generation. However, existing methods cannot differentiate both useful words and utterances in long distances from a response. Besides, previous work just performs context selection based on a state in the decoder, which lacks a global guidance and could lead some focuses on irrelevant or unnecessary information. In this paper, we propose a novel model with hierarchical self-attention mechanism and distant supervision to not only detect relevant words and utterances in short and long distances, but also discern related information globally when decoding. Experimental results on two public datasets of both automatic and human evaluations show that our model significantly outperforms other baselines in terms of fluency, coherence, and informativeness.
10.1109/icassp39728.2021.9414730
[ "https://arxiv.org/pdf/2102.09282v1.pdf" ]
231,951,585
2102.09282
596bd78a61e79a79d41d6d3eb17ddd71bc5ef91f
LEARNING TO SELECT CONTEXT IN A HIERARCHICAL AND GLOBAL PERSPECTIVE FOR OPEN-DOMAIN DIALOGUE GENERATION Lei Shen Institute of Computing Technology IIP Chinese Academy of Sciences University of Chinese Academy of Sciences Haolan Zhan University of Chinese Academy of Sciences Xin Shen Australian National University Yang Feng Institute of Computing Technology IIP Chinese Academy of Sciences University of Chinese Academy of Sciences LEARNING TO SELECT CONTEXT IN A HIERARCHICAL AND GLOBAL PERSPECTIVE FOR OPEN-DOMAIN DIALOGUE GENERATION Index Terms-Open-domain Dialogue GenerationCon- text SelectionHierarchical and Global Perspective Open-domain multi-turn conversations mainly have three features, which are hierarchical semantic structure, redundant information, and long-term dependency. Grounded on these, selecting relevant context becomes a challenge step for multiturn dialogue generation. However, existing methods cannot differentiate both useful words and utterances in long distances from a response. Besides, previous work just performs context selection based on a state in the decoder, which lacks a global guidance and could lead some focuses on irrelevant or unnecessary information. In this paper, we propose a novel model with hierarchical self-attention mechanism and distant supervision to not only detect relevant words and utterances in short and long distances, but also discern related information globally when decoding. Experimental results on two public datasets of both automatic and human evaluations show that our model significantly outperforms other baselines in terms of fluency, coherence, and informativeness. INTRODUCTION Open-domain multi-turn dialogue generation has gained increasing attentions in recent years, as it is more accordant with real scenarios and aims to produce customized responses. In general, an open-domain multi-turn conversation has following features: (1) The context (including the query and previous utterances in our paper) is in a hierarchical structure, which means it consists of some utterances, and each utterance contains several words. (2) At most cases, many contents of the context are redundant and irrelevant to the response. (3) Some related information (utterances or words) and the response are in a long-term dependency relation. Therefore, Context Selection, detecting the relevant context based on which to generate a more coherent and informative response, is a key point in multi-turn dialogue generation. Based on feature (1), the hierarchical recurrent encoderdecoder network (HRED) [1] has been proposed. It encodes each utterance and the whole context at two levels, and is widely applied to other methods for multi-turn dialogue generation. Then, hierarchical recurrent attention [2] and explicit weighting [3,4], memory networks [5] and self-attention mechanism [6] have been introduced to match feature (2) and (3), respectively. However, few work could cover all these features simultaneously to fulfill context selection and response generation tasks. When it comes to Context Selection, existing methods can be categorised into two ways: (1) Detecting related utterances measured by the similarity between query and each previous utterance [3,4]. (2) Applying the attention mechanism from a local perspective, i.e., based solely on the current state in decoder with the Maximum Likelihood Estimation (MLE) loss [4,6]. The similarity measurement in the former cannot select word-level context, while the guidance from the local perspective in the latter would make the model choose some deviated context and produce an inappropriate response [7,8,9]. To tackle the above mentioned problems, we propose HiSA-GDS, a modified Transformer model with Hierarchical Self-Attention and Globally Distant Supervision. To the best of our knowledge, it is the first time to design these two modules for open-domain dialogue generation. Specifically, we use Transformer encoder to encode each utterance in the context. During training, the response is firstly processed by a masked self-attention layer, and then a word-word attention aggregates related word information in each utterance individually. After that, we conduct utterance-level self-attention to get context-sensitive representations of aggregated information from last layer. Then, we calculate the attention weights between utterance-level outputs of the previous layer and the masked response representation. Finally, we generate the corresponding response based on the fusion of selected information at both word and utterance levels. Besides, to provide a global guidance of decoding, we import a distant supervision module which utilizes the similarity score between the response and each contextual utterance measured by a pre-trained sentence-embedding model. All parameters are learned based on the global Distant Supervision and local MLE in an end-to-end framework. Experimental results on two public datasets along with further discussions show that HiSA-GDS significantly outper- forms other baselines and is capable to generate more fluent, coherent, and informative responses. APPROACH The input is a context containing n utterances {X i } n i=1 , and each utterance is defined as X i = {x i,1 , ..., x i,|Xi| }, where |X i | is the length of the i-th utterance and x i,m is the m-th word of X i . Our goal is to select relevant context consisting of utterances and words, and then generate a response Y = {y 1 , y 2 , ..., y |Y | } by utilizing the related information, where |Y | is the length of response Y . Encoder We consider each utterance independently, and given an utterance X i , the input representation of word x i,j is the sum of its word embedding and position encoding: I(x i,j ) = WE(x i,j ) + WPE(x i,j ), where WE(x i,j ) and WPE(x i,j ) represent word and word position embedding, respectively. The input embedding is then fed into Transformer encoder with N layers. The final encoding of X i is the output from the N -th layer, E (N ) i . Please refer to [10] for more details. Hierarchical Self-Attention based Decoder The decoder also contains N layers, and each layer is composed of five sub-layers. The first sub-layer is a masked selfattention, which is defined as: M (l) t = MHA(D (l−1) t , D (l−1) t , D (l−1) t ),(1) where MHA is the multi-head attention function, D (l−1) t denotes the input representation of the l-th layer, and M (l) t denotes the output of masked self-attention at the l-th layer. D (0) t is the concatenated result of all words before time step t in the response and each word is also represented as the sum of its word embedding and position encoding. The second sub-layer is a word-word attention that summarizes word-level response-related information from each utterance X i into a vector at a specific decoding time: U (l) t,i = MHA(f w (M (l) t ), E (N ) i , E (N ) i ),(2) where f w is a linear transformation. The third sub-layer is an utterance-level self-attention. Inspired by Zhang et al. [6], we also utilize the self-attention mechanism to capture the long-term dependency of utterancelevel information. Similar to word position encoding, we add utterance position encoding (UPE) to U (l) t,i , and denote the sum result asŨ (l) t,i . The output of this sub-layer is calculated as: H (l) D (l) t = λ t * F (l) t + (1 − λ t ) * U (l) t,n ,(7) where W g is parameter metric, σ is the sigmoid activation function, and * means the point-wise product. Globally Distant Supervision Previous attention-based models achieve context selection from a local perspective, i.e., they try to generate one token at a time based solely on the current decoding state, which would detect deviated context and mislead the further generation. Besides, we do not have manual annotations to provide direct signals for selection. To address these problems, we design a globally distant supervision module to help determine relevant information, which provides a global guidance for the response generation process. Firstly, we apply a high quality pre-trained sentence-embedding model to encode contextual utterance X i and response Y into vectors, denoted as x i and y. Then, we use the dot product to measure the semantic relevance between x i and y [11], and compute the selection probability as follows: P (x = x i |y) = exp(x i · y) n j=1 exp (x j · y) .(8) Training Objective We utilize three loss functions in our training process. The first one is MLE loss which is defined as: L M LE (θ) = − 1 |Y | |Y | t=1 logp(y t |y <t , {X i } n i=1 ; θ),(9) where θ represents the model parameters, and y <t denotes the previously generated words. Since MLE loss only provides local (token-wise) supervision, inspired by Ren et al. [12] and Zhan et al. [13], we apply the Kullback-Leibler divergence (KL) loss and the Maximum Causal Entropy (MCE) loss for globally distant supervision. KL loss measures the distance between two distributions: P (x|y), which is the distant ground-truth supervision described in Equation 8, and Q(x|y) = 1 |Y | |Y | t=1 C (N ) t , which is the average sum of estimated probabilities at all steps from the output of wordutterance attention sub-layer in the last decoder layer. We denote the KL loss as: L KL (θ) = KL(P (x|y)||Q(x|y); θ). Then, we use MCE loss to alleviate the negative effects of noises caused by imprecise Q(x|y): L M CE (θ) = 1 |Y | |Y | t=1 w∈V P (y t = w)logP (y t = w),(11) where V denotes the vocabulary. Finally, our overall loss is a linear combination of these three loss functions: L(θ) = L M LE (θ) + η 1 L KL (θ) + η 2 L M CE (θ),(12) where hyper-parameters η 1 and η 2 govern the relative importance of different loss terms. EXPERIMENT SETTINGS Datasets: We evaluate the performance on two public datasets: Ubuntu Dialogue Corpus [14] (Ubuntu) and JD Customer Service Corpus [15] (JDDC). Baselines: (1) Seq2Seq with Attention Mechanism (S2SA) [16], and we concatenate all context utterances as a long sequence; (2) Hierarchical Recurrent Encoder-Decoder (HRED) [1]; (3) Variational HRED (VHRED) [17] with word drop and KL annealing, and the word drop ratio equals to 0.25; (4) Static Attention based Decoding Network (Static) [4]; (5) Hierarchical Recurrent Attention Network (HRAN) [18]; (6) Transformer [10], and we concatenate all context utterances into a long sequence; (7) Relevant Contexts Detection with Self-Attention Model (ReCoSa) [6]. They all focus on multiturn conversations, and ReCoSa is a state-of-the-art model on both Ubuntu and JDDC. For ablation study, HiSA is our model without the globally distant supervision. Hyper-parameters: The utterance padding length is set to 30, and the maximum conversation length is 10. The hidden size of encoder and decoder is 512, and the number of layers is 4 for encoder and 2 for decoder. The head number of multi-head attention is set to 8. The high-quality pre-trained sentence-embedding model we used is Infersent [19]/Familia [20] for Ubuntu/JDDC. These models are both pre-trained on large-scale datasets in either English or Chinese, and perform well on our datasets. For optimization, we use Adam [21] with a learning rate of 0.0001 with gradient clipping. Hyperparameters in Equation 12 are set to 1. Performance Measures: For automatic evaluation, we use 4 groups of metrics: (1) BLEU-2 [22]; (2) Embedding-based Metrics (Average, Greedy, and Extrema) [17]; (3) Coherence [23] that evaluates the semantic coherence between the context and response; (4) Distinct-2 [24]. For human evaluation, we utilize the side-by-side human comparison. We invite 7 postgraduate students as annotators. To each annotator, we show a context with two generated responses, one from HiSA-GDS and the other from a baseline model, but the annotators do not know the order. Then we ask annotators to judge which one wins based on fluency, coherence, and informativeness. Please refer to [18] for more details. Agreements among the annotators are calculated using Fleiss' kappa. RESULTS AND DISCUSSION Automatic Evaluation Results: As shown in Table 1, our model outperforms all baselines significantly on both Ubuntu and JDDC (significance tests, p-value < 0.01) by achieving the highest scores in almost all automatic metrics. Compared with existing baseline models, our model demonstrates its ability of generating relevant and appropriate responses. This is supported by the fact that results of our proposed model have gained improvements on BLEU-2, Embedding-based Metrics, and Coherence. Besides, we also achieve higher Distinct-2 score, which indicates that HiSA-GDS can generate more informative responses. Human Evaluation Results: These results are shown in Table 2. We observe that HiSA-GDS outperforms all baseline models on both Ubuntu and JDDC. Specifically, the percentage of "win" is always larger than that of "loss". Take Ubuntu dataset as an example. Compared with VHRED and Transformer, HiSA-GDS achieves preference gains with 48%, Table 2. Human evaluation between HiSA-GDS and other baselines on Ubuntu and JDDC. 51%, and 44%, respectively. We check responses generated by our model with "win" and find that they are more relevant to contextual utterances. The kappa scores indicate that annotators come to a "Moderate agreement" on judgement. Discussion of Hierarchical Self-Attention: To validate the effectiveness of hierarchical self-attention mechanism, we present the heatmap of an example in Figure 2. In this example, there are seven contextual utterances, and for each utterance, importance of each word is indicated with the depth of blue color on the right part. Besides, we also show an utterance-level attention visualization on the left part. An utterance is more important when the red color is lighter. For example, the third and seventh utterances, i.e., X 3 and X 7 , are more important than the others. The importance of a word (horizontal heatmap on the right of X 1 to X 7 ) or an utterance (vertical heatmap on the left of X 1 to X 7 ) is calculated as the average value of different heads. From the word-level visualization, we find that words including "订 单(order)", "今天(today)", and "送货(deliver)" are selected to be more relevant. Overall, the results are in accordance with humans' judgement and have achieved the goal of our proposed model. Discussion of GDS: Since GDS is only utilized during the training process, we calculate the relevance score between each contextual utterance and the ground-truth response. Af- ter applying Familia [20] over the entire conversation, the relevance scores are 0.1502, 0.1388, 0.1602, 0.1548, 0.0979, 0.1343, and 0.1638 for X 1 to X 7 , which is consistent with humans' intuition. Besides, inspired by Zhang et al. [6], we randomly sample 300 context-response pairs from JDDC. Three annotators who are postgraduate students are invited to label each context. If a contextual utterance is related to the response, then it is labeled as 1. The kappa value is 0.568, which indicates the moderate consistency among different annotators. We then pick out samples that is labeled the same by at least two annotators, and then calculate the kappa value between humans' judgement and the outputs from Familia [20] on these cases. The value 0.863 reflects "Substantial agreement" between them. CONCLUSION In this paper, we propose a novel model for open-domain dialogue generation, HiSA-GDS, which conducts context selection in a hierarchical and global perspective. The hierarchical self-attention is introduced to capture relevant context at both word and utterance levels. We also design a globally distant supervision module to guide the response generation at decoding. Experiments show that HiSA-GDS can generate more fluent, coherent, and informative responses. Fig. 1 . 1Architecture of HiSA-GDS. The white dashed box is Transformer encoder, while the gray one is the modified Transformer decoder. The residual connection and layer normalization are omitted for brevity. "WPE" and "UPE" represent word position encoding and utterance position encoding. The upper right corner shows the globally distant supervision that is only introduced to the N -th layer of decoder. Fig. 2 . 2Left: Utterance-level multi-head attention visualization of HiSA-GDS in the word-utterance attention layer. 0 to 7 are the index of each head. Right: Word-level attention visualization in the word-word attention layer. The importance of a word (horizontal blue heatmap) or an utterance (vertical red heatmap) is calculated as the average value of all heads. IIP stands for Key Laboratory of Intelligent Information Processing. * Equal Contribution. Corresponding to: [email protected] t = MHA(Ũ (l) t ,Ũ (l) t ,Ũ (l) t ),(3)whereŨ(l) t = [Ũ (l) t,1 ,Ũ (l) t,2 , ...,Ũ (l) t,n ].Then, the fourth sublayer is a word-utterance attention layer to find out utterancelevel relevant information which is defined as:C (l) t = f l (MHA(f u (M (l) t ), H (l) t , H (l) t )),(4)where f l and f u are linear transformations, and f l is used for changing the output dimension. The last sub-layer is a feedforward neural network (FFN):F (l) t = FFN(C (l) t ).(5)Each of above mentioned sub-layer is followed by a normalization layer and a residual connection. Finally, we use a fusion gate to regulate the relevant information at word level (U (l) t,n ) and utterance level (F (l) t ):λ t = σ(W g [U (l) t,n , F (l) t ]),(6) Building end-toend dialogue systems using generative hierarchical neural network models. Alessandro Iulian V Serban, Yoshua Sordoni, Aaron Bengio, Joelle Courville, Pineau, AAAI. Iulian V Serban, Alessandro Sordoni, Yoshua Bengio, Aaron Courville, and Joelle Pineau, "Building end-to- end dialogue systems using generative hierarchical neu- ral network models," in AAAI, 2016, pp. 3776-3783. HKA: A hierarchical knowledge attention mechanism for multi-turn dialogue system. Jian Song, Kailai Zhang, Xuesi Zhou, Ji Wu, ICASSP. Jian Song, Kailai Zhang, Xuesi Zhou, and Ji Wu, "HKA: A hierarchical knowledge attention mechanism for multi-turn dialogue system," in ICASSP, 2020, pp. 3512-3516. How to make context more useful? an empirical study on context-aware neural conversational models. Zhiliang Tian, Rui Yan, Lili Mou, Yiping Song, Yansong Feng, Dongyan Zhao, Zhiliang Tian, Rui Yan, Lili Mou, Yiping Song, Yan- song Feng, and Dongyan Zhao, "How to make context more useful? an empirical study on context-aware neu- ral conversational models," in ACL, 2017, pp. 231-236. Contextsensitive generation of open-domain conversational responses. Weinan Zhang, Yiming Cui, Yifa Wang, Qingfu Zhu, Lingzhi Li, Lianqiang Zhou, Ting Liu, in COLING. Weinan Zhang, Yiming Cui, Yifa Wang, Qingfu Zhu, Lingzhi Li, Lianqiang Zhou, and Ting Liu, "Context- sensitive generation of open-domain conversational re- sponses," in COLING, 2018, pp. 2437-2447. Hierarchical variational memory network for dialogue generation. Hongshen Chen, Zhaochun Ren, Jiliang Tang, Yihong Eric Zhao, Dawei Yin, WWWHongshen Chen, Zhaochun Ren, Jiliang Tang, Yi- hong Eric Zhao, and Dawei Yin, "Hierarchical vari- ational memory network for dialogue generation," in WWW, 2018, pp. 1653-1662. Recosa: Detecting the relevant contexts with self-attention for multi-turn dialogue generation. Hainan Zhang, Yanyan Lan, Liang Pang, Jiafeng Guo, Xueqi Cheng, ACL. Hainan Zhang, Yanyan Lan, Liang Pang, Jiafeng Guo, and Xueqi Cheng, "Recosa: Detecting the relevant con- texts with self-attention for multi-turn dialogue genera- tion," in ACL, 2019, pp. 3721-3730. Modeling semantic relationship in multi-turn conversations with hierarchical latent variables. Lei Shen, Yang Feng, Haolan Zhan, ACL. Lei Shen, Yang Feng, and Haolan Zhan, "Modeling se- mantic relationship in multi-turn conversations with hi- erarchical latent variables," in ACL, 2019, pp. 5497- 5502. CDL: Curriculum dual learning for emotion-controllable response generation. Lei Shen, Yang Feng, ACL. Lei Shen and Yang Feng, "CDL: Curriculum dual learn- ing for emotion-controllable response generation," in ACL, 2020, pp. 556-566. Compose like humans: Jointly improving the coherence and novelty for modern chinese poetry generation. Lei Shen, Xiaoyu Guo, Meng Chen, IJCNN. Lei Shen, Xiaoyu Guo, and Meng Chen, "Compose like humans: Jointly improving the coherence and novelty for modern chinese poetry generation," in IJCNN, 2020, pp. 1-8. Attention is all you need. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, Illia Polosukhin, Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin, "Attention is all you need," in NIPS, 2017, pp. 5998-6008. Learning to select knowledge for response generation in dialog systems. Rongzhong Lian, Min Xie, Fan Wang, Jinhua Peng, Hua Wu, IJCAI. Rongzhong Lian, Min Xie, Fan Wang, Jinhua Peng, and Hua Wu, "Learning to select knowledge for response generation in dialog systems," in IJCAI, 2019, pp. 5081- 5087. Thinking globally, acting locally: Distantly supervised global-to-local knowledge selection for background based conversation. Pengjie Ren, Zhumin Chen, Christof Monz, Jun Ma, Maarten De Rijke, AAAI. Pengjie Ren, Zhumin Chen, Christof Monz, Jun Ma, and Maarten de Rijke, "Thinking globally, acting lo- cally: Distantly supervised global-to-local knowledge selection for background based conversation," in AAAI, 2020, pp. 8697-8704. Userinspired posterior network for recommendation reason generation. Hainan Haolan Zhan, Hongshen Zhang, Lei Chen, Yanyan Shen, Zhuoye Lan, Dawei Ding, Yin, SIGIR. Haolan Zhan, Hainan Zhang, Hongshen Chen, Lei Shen, Yanyan Lan, Zhuoye Ding, and Dawei Yin, "User- inspired posterior network for recommendation reason generation," in SIGIR, 2020, pp. 1937-1940. The ubuntu dialogue corpus: A large dataset for research in unstructured multi-turn dialogue systems. Ryan Lowe, Nissan Pow, V Iulian, Joelle Serban, Pineau, SIGDIAL. Ryan Lowe, Nissan Pow, Iulian V Serban, and Joelle Pineau, "The ubuntu dialogue corpus: A large dataset for research in unstructured multi-turn dialogue sys- tems," in SIGDIAL, 2015, pp. 285-294. The JDDC corpus: A large-scale multi-turn chinese dialogue dataset for e-commerce customer service. Meng Chen, Ruixue Liu, Lei Shen, Shaozu Yuan, Jingyan Zhou, Youzheng Wu, Xiaodong He, Bowen Zhou, LREC. Meng Chen, Ruixue Liu, Lei Shen, Shaozu Yuan, Jingyan Zhou, Youzheng Wu, Xiaodong He, and Bowen Zhou, "The JDDC corpus: A large-scale multi-turn chinese dialogue dataset for e-commerce customer ser- vice," in LREC, 2020, pp. 459-466. Sequence to sequence learning with neural networks. Ilya Sutskever, Oriol Vinyals, Quoc V Le, NIPS. Ilya Sutskever, Oriol Vinyals, and Quoc V Le, "Se- quence to sequence learning with neural networks," in NIPS, 2014, pp. 3104-3112. A hierarchical latent variable encoderdecoder model for generating dialogues. Iulian Vlad Serban, Alessandro Sordoni, Ryan Lowe, Laurent Charlin, Joelle Pineau, Aaron Courville, Yoshua Bengio, AAAI. Iulian Vlad Serban, Alessandro Sordoni, Ryan Lowe, Laurent Charlin, Joelle Pineau, Aaron Courville, and Yoshua Bengio, "A hierarchical latent variable encoder- decoder model for generating dialogues," in AAAI, 2017, pp. 3295-3301. Hierarchical recurrent attention network for response generation. Chen Xing, Yu Wu, Wei Wu, Yalou Huang, Ming Zhou, AAAI. Chen Xing, Yu Wu, Wei Wu, Yalou Huang, and Ming Zhou, "Hierarchical recurrent attention network for re- sponse generation," in AAAI, 2018, pp. 5610-5617. Supervised learning of universal sentence representations from natural language inference data. Alexis Conneau, Douwe Kiela, Holger Schwenk, Loïc Barrault, Antoine Bordes, EMNLP. Alexis Conneau, Douwe Kiela, Holger Schwenk, Loïc Barrault, and Antoine Bordes, "Supervised learning of universal sentence representations from natural lan- guage inference data," in EMNLP, 2017, pp. 670-680. Di Jiang, Yuanfeng Song, Rongzhong Lian, Siqi Bao, Jinhua Peng, Huang He, Hua Wu, arXiv:1808.03733Familia: A Configurable Topic Modeling Framework for Industrial Text Engineering. arXiv preprintDi Jiang, Yuanfeng Song, Rongzhong Lian, Siqi Bao, Jinhua Peng, Huang He, and Hua Wu, "Familia: A Con- figurable Topic Modeling Framework for Industrial Text Engineering," arXiv preprint arXiv:1808.03733, 2018. Adam: A method for stochastic optimization. P Diederick, Jimmy Kingma, Ba, ICLR. Diederick P Kingma and Jimmy Ba, "Adam: A method for stochastic optimization," in ICLR, 2015. BLEU: a method for automatic evaluation of machine translation. Kishore Papineni, Salim Roukos, Todd Ward, Wei-Jing Zhu, ACL. Kishore Papineni, Salim Roukos, Todd Ward, and Wei- Jing Zhu, "BLEU: a method for automatic evaluation of machine translation," in ACL, 2002, pp. 311-318. Better conversations by modeling, filtering, and optimizing for coherence and diversity. Xinnuo Xu, Ondřej Dušek, Ioannis Konstas, Verena Rieser, in EMNLP. Xinnuo Xu, Ondřej Dušek, Ioannis Konstas, and Ver- ena Rieser, "Better conversations by modeling, filtering, and optimizing for coherence and diversity," in EMNLP, 2018, pp. 3981-3991. A diversity-promoting objective function for neural conversation models. Jiwei Li, Michel Galley, Chris Brockett, Jianfeng Gao, Bill Dolan, NAACL-HLT. Jiwei Li, Michel Galley, Chris Brockett, Jianfeng Gao, and Bill Dolan, "A diversity-promoting objective func- tion for neural conversation models," in NAACL-HLT, 2016, pp. 110-119.
[]
[ "Path Planning for Air-Ground Robot Considering Modal Switching Point Optimization", "Path Planning for Air-Ground Robot Considering Modal Switching Point Optimization" ]
[ "Xiaoyu Wang ", "Xinyu Zhang [email protected] ", "Jun Li ", "Wenzhuo Liu [email protected] ", "Huaping Liu [email protected] ", "\nThe School of Vehicle and Mobility\nTsinghua University\nBeijingP.R.China\n", "\nDepartment of Computer Science and Technology\nThe School of Vehicle and Mobility\nnd Kangyao Huang The\nTsinghua University\nBeijingP.R.China\n", "\nTsinghua University\nBeijingP.R.China\n", "\nThe School of Vehicle and Mobility\nThe School of Vehicle and Mobility\nth Honglin Sun The School of Vehicle and Mobility Tsinghua University\nTsinghua University\nBeijing, BeijingP.R.China, P.R.China\n", "\nTsinghua University\nBeijingP.R.China\n", "\nth Pingping\nThe Department of Computer Science and Technology\nLu University of Michigan\nAnn ArborUSA\n", "\nTsinghua University\nBeijingP.R.China\n" ]
[ "The School of Vehicle and Mobility\nTsinghua University\nBeijingP.R.China", "Department of Computer Science and Technology\nThe School of Vehicle and Mobility\nnd Kangyao Huang The\nTsinghua University\nBeijingP.R.China", "Tsinghua University\nBeijingP.R.China", "The School of Vehicle and Mobility\nThe School of Vehicle and Mobility\nth Honglin Sun The School of Vehicle and Mobility Tsinghua University\nTsinghua University\nBeijing, BeijingP.R.China, P.R.China", "Tsinghua University\nBeijingP.R.China", "th Pingping\nThe Department of Computer Science and Technology\nLu University of Michigan\nAnn ArborUSA", "Tsinghua University\nBeijingP.R.China" ]
[]
An innovative sort of mobility platform that can both drive and fly is the air-ground robot. The need for an agile flight cannot be satisfied by traditional path planning techniques for air-ground robots. Prior studies had mostly focused on improving the energy efficiency of paths, seldom taking the seeking speed and optimizing take-off and landing places into account. A robot for the field application environment was proposed, and a lightweight global spatial planning technique for the robot based on the graph-search algorithm taking mode switching point optimization into account, with an emphasis on energy efficiency, searching speed, and the viability of real deployment. The fundamental concept is to lower the computational burden by employing an interchangeable search approach that combines planar and spatial search. Furthermore, to safeguard the health of the power battery and the integrity of the mission execution, a trap escape approach was also provided. Simulations are run to test the effectiveness of the suggested model based on the field DEM map. The simulation results show that our technology is capable of producing finished, plausible 3D paths with a high degree of believability. Additionally, the mode-switching point optimization method efficiently identifies additional acceptable places for mode switching, and the improved paths use less time and energy.
10.48550/arxiv.2305.08178
[ "https://export.arxiv.org/pdf/2305.08178v1.pdf" ]
258,685,818
2305.08178
7b1dc2d94352c2922238351924436348bb9d05be
Path Planning for Air-Ground Robot Considering Modal Switching Point Optimization Xiaoyu Wang Xinyu Zhang [email protected] Jun Li Wenzhuo Liu [email protected] Huaping Liu [email protected] The School of Vehicle and Mobility Tsinghua University BeijingP.R.China Department of Computer Science and Technology The School of Vehicle and Mobility nd Kangyao Huang The Tsinghua University BeijingP.R.China Tsinghua University BeijingP.R.China The School of Vehicle and Mobility The School of Vehicle and Mobility th Honglin Sun The School of Vehicle and Mobility Tsinghua University Tsinghua University Beijing, BeijingP.R.China, P.R.China Tsinghua University BeijingP.R.China th Pingping The Department of Computer Science and Technology Lu University of Michigan Ann ArborUSA Tsinghua University BeijingP.R.China Path Planning for Air-Ground Robot Considering Modal Switching Point Optimization path planninghybrid A*BASair-ground robotfield rescue mission An innovative sort of mobility platform that can both drive and fly is the air-ground robot. The need for an agile flight cannot be satisfied by traditional path planning techniques for air-ground robots. Prior studies had mostly focused on improving the energy efficiency of paths, seldom taking the seeking speed and optimizing take-off and landing places into account. A robot for the field application environment was proposed, and a lightweight global spatial planning technique for the robot based on the graph-search algorithm taking mode switching point optimization into account, with an emphasis on energy efficiency, searching speed, and the viability of real deployment. The fundamental concept is to lower the computational burden by employing an interchangeable search approach that combines planar and spatial search. Furthermore, to safeguard the health of the power battery and the integrity of the mission execution, a trap escape approach was also provided. Simulations are run to test the effectiveness of the suggested model based on the field DEM map. The simulation results show that our technology is capable of producing finished, plausible 3D paths with a high degree of believability. Additionally, the mode-switching point optimization method efficiently identifies additional acceptable places for mode switching, and the improved paths use less time and energy. I. INTRODUCTION In recent years, new kinds of flying and driving robots have evolved as the theory and technology of electric ver-tical take-off and landing have advanced. For more flexible transportation, scientists and engineers combine electrically propelled chassis technologies with aviation applications to create hybrid locomotion air-ground vehicles. Great achievements have been made and attracted widespread attention, such as TF-X from Terrafugia [1], AAV from EHang [2], the EPFL jumpglider from Carnegie Mellon University and Swiss Federal Institute of Technology in Lausanne [3,4]. The combined flying and driving skills, as well as the ability to shift power from propellers to wheels for environmental adaptation, are common elements of these breakthroughs. The focus of robotics is on this capacity for multimodal movement. However, when energy consumption is taken into consideration, cruising in the sky requires high-power density to overcome gravity which significantly limits the endurance and task scope [5]. To balance driving and flying, the right path planner for 3D searching and navigating algorithms become essential. There are mainly sampling-based algorithms, graphsearch algorithms, curve interpolation methods, etc. [6][7][8] for 3D path planning. Due to the increasing search volume in 3D scenarios and the requirement for fast planning of UAV [9], sampling-based algorithms, especially RRT [10] and its variants (such as RRT-Connect [11] and RRT* [12]), are still the mainstream. However, when it comes to airground vehicles, modal switching should be considered. In the category of graph-search algorithms, A* and its derivation algorithms are also widely used in 3D scenarios [13]. Incremental searching algorithms such as LPA* [14] and D* Lite [15] are able to dynamically search a path in a partially known environment. ARA* [16] can rapidly generate a sub-optimal path and iteratively optimize by adjusting the weight of the heuristic function. Theta* [17] implements an any-angle path planning through the process of removing parent nodes. HPATheta* [18] implements the hierarchical path planning method, which can complete the large-scale virtual terrain pathfinding, and smooth the path within a reasonable calculation time. The path planning methods for ground robots and aerial robots are different. The hybrid robots are able to fly so that planar planning methods are no longer suitable [19]. While employing the planning method for aerial robots will be unable to make use of the high energy efficiency feature from ground driving. Despite graph-search based 3D path planning algorithms walks through fair development, the planning method for the air-ground robot still stays in a primary stage [20]. Nevertheless the multi-modal movement feature of the air-ground robot brings new challenges [21] [22]. In this field, Brandon Araki et al. presented SIPP and ILP algorithms for hybrid swarm robots [23] which can be used on multiple robots in the continuous-time domain. Meanwhile, this algorithm also focuses on time and energy consumption for future 3D traffic networks. Amir Sharif et al. presented an energy-efficient A* based method implemented by modifying the calculation of heuristic cost for different actions [24]. H.J. Terry Suh et al. presented the first motion planning method for the multi-modal locomotion system [25], which considered energy efficiency and dynamic constraints. All these works presented viable planning methods for the air-ground robot with high energy efficiency but barely mentioned the computation time, realtime obstacle avoidance when the transition between flight and ground driving is more barely mentioned. II. BACKGROUND AND RELATED WORK A path planning method for air-ground robot based on a hybrid A* and BAS (Beetle Antennae Search) algorithm to achieve a balance between flying and driving was proposed, it can switch modes shrewdly in challenging field conditions, and produce task execution routes with minimal overall energy consumption. There are mainly four contributions of our works: • Introducing the novel variable-structure air-ground robot which is capable of operating in challenging field environment. The folding arm of the robot can make the robot structure compact and flexible, and enhance the maneuvering performance in the field environment. • An iterative framework for path optimization is proposed to achieve the balance between computing time and accuracy, 2D searching and 3D searching operate alternatively until the global optimum is reached. Figure 1. The overall structure of the proposed robot • Improved BAS optimization algorithm is proposed inserting the modification algorithm into the real-time computational logic of path planning, to optimize the takeoff and landing node. Not only to save energy consumption, but also increase the feasibility of the calculated route, and enhance the safety of the path for the robot field rescue mission. • The proposed path planning model can deal with the development of various huge slopes and emergency threats through the node information judgment logic in the field environment. It can also take into account the health and mobility of the robot's power system. The remainder of the paper will proceed as follows: Section II introduces the robot platform our team designed for this experiment. In Section III, the structure and detail of the proposed path planning method is described. Section IV presents the simulation experiment designed to verify the effectiveness of the proposed algorithm framework, and includes simulation results. Finally, Section V reviews the conclusions made from simulation results and future work. III. ROBOT PLATFORM In our previous works, several types of air-ground robots are designed to carry workload in rescue and searching [26] [27]. These robots have achieved good results in testing under urban scenarios, even in some complex fields such as urban rescue and logistics. To get a better performance of driving and passability, the innovative robot in [26] is upgraded by folding propellers. Our platform are designed with two modals: the open-wing state for flying in the air, and the wing-folded state for driving on the ground. The two modals and main hardware compositions of the robot are shown in Fig. 1. Our robot is expected to be fully self-organized and intelligent to perform high-risk works replacing human. IV. METHOD The proposed planning algorithm is described in detail in this section. This approach of planning is typically based on the characteristics of air-ground robots: when driving on the ground, a robot has good energy efficiency, limited mobility, low energy consumption, and low computational load; when flying in the air, the robot experiences the opposite outcome. Therefore, we assume that the robot typically travels on land and only flips to flying mode when it becomes difficult to identify a path or when the terrain is insufficient for the airground robot's mobility. The challenge is to enable the robot to complete the rescue operation while using less time and energy, and to do it while simultaneously deciding when to take off and land. By choosing a single mode switching point that is inconvenient for the platform's takeoff and landing, the A* algorithm can result in hazardous scenarios like roll and sideslip. These problems can be avoided by using the BAS algorithm. The planner will alternately execute the 2D and 3D path finding algorithms of the BAS mode switching point optimization method given a start and target position. Following several iterations, the planner will produce a raw path with a greater energy efficiency. The route is then refined using Bessel curve smoothing to gauge the robot's mobility. The robot begins its field rescue mission after reaching the target point. In summary, the overall flow of the proposed algorithm to complete the task is shown in Fig. 2. The flowchart in Fig. 3 shows the basic logic of the proposed path planner. A. Path Planning 1) Ground Planning: 2D A*: The planner searches in a plane according to the original A* cost function when the robot stays on the ground. Meanwhile, the function to judge the feasibility function of air-ground robot driving in terrain (called takeoff decision function) runs together with A*. Takeoff decision function can determine whether the maneuverability index(M-index) of the robot is greater than the slope, flip angle and turning radius(DOF) from the current node to the next node. Based on the maneuverability of air-ground Robot, the experimental tests to obtain empirical values for the maneuverability index was designed. The function will count a number (count) from 0 if the DOF between the current node and adjacent nodes is greater than or equal to M-index until the count exceeds the threshold (thre).And move the node from the open list to the close list which do not match M-index. Otherwise, a new node will be recorded and the count will be reset to 0.When count = 7, the takeoff decision function returns the flag with true value, the planner stops searching and picks current node as the initial mode switching point. In addition to the Manhattan distance in the conventional A* algorithm, the suggested method's cost function addi-tionally accounts for the energy consumption E in Energy Consumption Model between nodes in the field environment. According to the mobility test experiment of the airground Robot, between the feasible node and the previous node shall meet the following conditions:      gx min ≤ gx ≤ gx max gy min ≤ gy ≤ gy max gz min ≤ gz ≤ gz max(1) 2) Flight Planning: 3D A*: According to the distance to the goal, the 3D pathfinding method was categorized into two categories : flying to the goal and flying to the ground. We expect that when the target is close, the robot would fly straight there, ignoring the goal's altitude, as often switching modes might lead to instability and energy waste. The 3D A* was utilized to look for a path that leads to the desired outcome. To preserve energy, the robot was expected to fly as long as it can if the destination is far away. In this scenario, a modified 3D A* was employed to enable the robot to fly away from the present seeking trap, called a trap escape algorithm in short. The trap escaping algorithm is based on the assumption that the robot can across over current terrain obstacles. In this algorithm, a variable H 2D and an SOC (State of Charge) judement rule was introduced. H 2D : The horizontal Manhattan distance from the current position to the goal, same as the heuristic cost in the 2D pathfinding phase. H 2D is used as an indicator of whether the robot is approaching the goal. At the beginning, an initial value named H 2D0 was recorded, and which should also be equal to the final value of H min in the 2D pathfinding phase. In each execution loop, calculate ∆H 2D , which represents the moved distance toward the goal. Next, depending on the value of ∆H 2D , 3 stages of the 3D pathfinding process were defined : 1. takeoff stage; 2. escape stage; 3. landing stage. In the trap escaping algorithm, a dummy altitude variable z dummy we introduce, and the values according to the current stage to let the searching runs as desired were assigned. Concretely, the takeoff stage is to demand the robot fly upward until it can horizontally get closer to the goal. When ∆H 2D < C escape , the dummy altitude z dummy equals to the current robot height z curr added with a small positive value . The dummy altitude is applied to calculate the heuristic value instead of the actual altitude of the goal. Since the original heuristic function would lead to the searching direction attracted to the same altitude with the goal, while z curr is used to cancel this effect, and the positive drives the algorithm searching upwards. Otherwise, when the goal is also on the ground, the attraction effect of the heuristic function would result in meaningless searches near the ground. When C escape ≤ ∆H 2D < C landing , the planner will get into the escape stage when the robot is flying over the obstacle. The value of dummy altitude are required to be set the same value as the current robot height to eliminate the effect of height, which would lead the algorithm to search forward. Finally, as the H 2D gets lower, when ∆H 2D ≥ C landing , the dummy altitude equals were set to the ground height for attracting the searching direction to the ground when considering the robot has gotten rid of the obstacle. Once a ground node is searched, planner will record it as the landing point, and reconstruct the path then terminate the loop. SOC judement rule: When the platform departs from the closest mode changeover point, the SOC record begins. The mode switch condition will be entered if the overall soc consumption of the battery during the flight exceeds the pre-set safety limit (SOC ref ). The regulation is intended to keep the battery soc in an effective and healthy range while in flight and to guarantee that field rescue reconnaissance missions may be carried out without endangering the battery's long-term health. B. Modal Switching Point Optimization The principle of BAS algorithm is shown in Fig. 4. According to the field application scenario of the robot, the main steps of BAS modal switching point optimization algorithm are as follows: Algorithm 2 H Calculation for the Trap Escaping Algorithm Step 1: The updating step direction of individual longicorn is random, and the random vector is expressed as follows: − → b = Rands(k, 1) Rands(k, 1) (2) Where, k is the spatial dimension, which is taken as 3 in this paper. The three dimensions respectively represent the longitude, latitude and height of nodes. Step 2: Set the initial mode switching point determined by 2D and 3D A* as the initial position of Longhorn. Step 3: Longhorn updates the position of the left and right two tentacles: X R = X + D − → b 2 X L = X − D − → b 2 (3) Where, X R and X L are the coordinates of the left and right tentacles of the longhorn respectively. b is the coordinates of the centroid of the longhorn. D is the distance between two tentacles. Step 4: Based on the current position of the left and right tentacles, the size of fitness function F (X R ) and F (X L ) is compared to update the position of the centroid: X = X − δ − → b sign(F (X R ) − F ((X L ))(4)F = E + αR (5) R = a · gx + b · gy + c · gz (6) Where, δ is the Euclidean distance of the step, E is energy consumption, α, a, b and c are hyperparameter, gx, gy and gz are the gradient values of x, y, and z with the current node. The α values in different situations to select different weight relations between E and R can be adjust. For example, to enable the BAS algorithm to search for mode switching points in a longer range, when the SOC of robot's power battery is sufficient the α value can appropriately increased. Step 5: The modal switching point data of each iteration is retained, and the node with the smallest fitness function is selected as the starting point of the next stage of planning. C. Energy Consumption Model To evaluate the energy efficiency of a path, equations below are used to estimate energy consumption. E = E hover (mode) + E move (mode) + E transform (mode) (7) The total energy consumption was assumed consists of hovering energy [24], moving energy, and transform energy. mode represents the operating mode of the robot. E move (mode) = mg∆h + ρAC d v 2 ∆d 2 mode = fly µmg∆d + ρAC d v 2 ∆d 2 mode = drive (9) E transform (mode) = E expand/fold + E Bodeneffekt(10) When hovering, the weight of the air blown downward by propellers is equivalent to its self-weight [28]. In our estimation model, the moving energy consumption includes two parts: to overcome gravity or ground friction; and to overcome air drag. Hovering consumes the most energy. Hence the calculation of air drag was simplified by using fixed C d and A for any direction and apply a fixed µ for ground moving. The E transform (mode) denotes the energy SOC calculation formula of power battery of air-ground robot: SOC t = Q 0 − t 0 E t Q(11) Values of parameters used in the equations are shown in the table. I. V. SIMULATION RESULTS In order to verify the feasibility of the algorithm, the DEM (Digital Elevation Model) data map with 12m resolution is selected as the simulation scene. Mountainous and hilly landscapes with a wide range of altitudes are present in the selected map region, as seen in Fig.5. This simulation is based on the scene setting of a land and air robot mission execution, and it is separated into low altitude area, high altitude area, and composite region for experimental verification. The set starting and goal points in the three areas are shown in Fig. 6. A. Analysis Fig. 7 displays the experimental results of path planning in three distinct altitude situations. As can be observed, the algorithm is capable of adapting to a variety of field conditions and producing robot-friendly routes. The path planning impact without modal switching points is also extremely reasonable, and path finding can be carried out along the edges of contour lines. Since the height span is not very wide, we may discover the area with gradual altitude shift as the alternative area for the path, as shown in the low altitude and high altitude trials, (a) and (b) of Fig. 7, when paired with the 2D and 3D perspective data pictures. While the high altitude lines generally go through the bright yellow high altitude region, the low altitude paths mostly travel through the dark blue low altitude region. Since the two regions' colors are essentially the same, the landscape Since the terrain is more challenging to drive on the ground in high altitude regions, the planner uses flight mode more frequently to advance, and the soc consumption in this case is also the highest, as can be seen from the figure. The difference in elevation between the routes in high and low altitude regions is also greater. The path and direction in the two-dimensional image are unaffected by the optimization of the modal switching point of the BAS algorithm; only the robot's takeoff time is altered. It is also quite flexible when switching between flight and ground forms; the suggested algorithm's recommended flow is followed by the switching operations of the 2d, 3d, and BAS algorithms. The initial modal coordinate points are chosen when the BAS algorithm is applied based on the A* algorithm, and the information of the local real-time environment may be gathered and modeled using Lidar, cameras, etc. Different modifications are made to the routes both before and after the modal conversion. For robot takeoff and landing, the optimal sites have gradient values that are more advantageous. The optimized task paths not only have shorter task execution time, but also have less power consumption, with 2.1%, 2.5% and 2.2% of power consumption saved respectively in the three experiments. Three traditional path planning search algorithms are provided as a comparison in order to evaluate the performance of the modal switching point method suggested in this article. The comparative performance of the four algorithms with the same number of search points is displayed in Fig. 9. Due to its simple iterative logic and cost function, the BAS algorithm has the quickest search time among the four algorithms. The parameter R is employed in the aforementioned research to fully quantify the gradient strength of the modal switching sites, and the modal switching points chosen by the BAS algorithm had lower average R values. Additionally, the entire path with the improved modal switching point uses least energy. In this study, an air-ground robot was developed that can withstand the challenging field conditions. Additionally, we primarily proposed a quick and lightweight spatial path planning method for the robot that incorporated a 2D/3D searching approach and a BAS modal switching point optimization algorithm. The method was tested successfully in simulated flight. To confirm the efficiency and viability of the proposed BAS algorithm, the proposed path planning framework can efficiently minimize the overall energy consumption compared to three different search algorithms.Therefore, Our technology may be used on air-ground robots, particularly those small-sized, highly mobile robots with rescue and reconnaissance capabilities. For future work, we will carry out with two aspects: we generally intend to conduct more real-world trials and advance the air-ground robot in the application. On the other hand, we will utilize our algorithm in completely unpredictable and dynamic contexts because that is more applicable to real-world situations. Figure 2 . 2The operation diagram of the proposed algorithm. (The equipment used are the primary hardware that the robot employs in each of the several operating modes shown inFig. 1. The keywork is the main task in the process of corresponding algorithm iteration) Figure 3 . 3Flowchart of the planner execution DOF, M − index, count, f lag 13: end function Figure 4 . 4Schematic diagram of the BAS modal switching point optimization Figure 5 . 5Map: (a) is the original satellite map of the experimental area and (b) is the map of the digital elevation data after matlab processing consumption during the transform process. In this paper, when switching modes, the energy used to open or fold the wings is the E expand/fold . And the energy consumed by the ground effect is the E Bodeneffekt . The values are obtained by testing the actual working process of the robot. Figure 6 . 6Distribution of start-goal points of the experimental setup and different angle views of the map created by the produced path matches the robot's movements more closely. Figure 7 .Figure 8 . 78Path planning results of three elevation experiment Comparison of SOC trajectories VI. CONCLUSION AND FUTURE WORK Figure 9 . 9Performance of different methods 1 : 1if H 2D0 < H 2D then ∆H 2D ← H 2D0 − H 2D 5: if ∆H 2D < C escape then else if C escape ≤ ∆H 2D < C landing then else if ∆H 2D > C landing then H ← H 2D + |z dummy − z curr | 13: if ∆SOC f lying > SOC ref then2: H 2D0 ← H 2D // update H 2D0 3: end if 4: 6: z dummy ← z curr + // takeoff stage 7: 8: z dummy ← z curr // escape stage 9: 10: z dummy ← z ground // landing stage 11: end if 12: 14: z dummy ← z ground // landing stage 15: end if Table I IPARAMETERS SETTINGS(SUPPLEMENT) Symbol Meaning Value ρ Air density 1.2 kg/m 3 m robot mass 39.5 kg r Propeller radius 0.4191 m x Number of propellers 6 g Gravitational acceleration 9.81 m/s 2 η Motor efficiency 0.58 µ Coefficient of friction 0.06 C d Coefficient of air drag 1.5 v(fly) Flying velocity 2 m/s v(drive) Driving velocity 1 m/s A(fly) Flying windward area 0.6 m 2 A(drive) Drving windward area 0.05 m 2 Constant Standby energy 100 J Roadable aircraft with folding wings and integrated bumpers and lighting. C C Dietrich, S A Schweighart, A M Mracek, 358uS Patent 7,938C. C. Dietrich, S. A. Schweighart, and A. M. Mracek, "Roadable aircraft with folding wings and integrated bumpers and lighting," May 10 2011, uS Patent 7,938,358. Multi-rotor passengercarrying aircraft. H Du, H Hu, Z Li, App. 15/016548uS PatentH. Du, H. Hu, and Z. Li, "Multi-rotor passenger- carrying aircraft," Jun. 29 2017, uS Patent App. 15/016,548. Towards a self-deploying and gliding robot. M Kovač, J C Zufferey, D Floreano, SpringerBerlin HeidelbergM. Kovač, J. C. Zufferey, and D. Floreano, "Towards a self-deploying and gliding robot," Springer Berlin Heidelberg, 2009. The epfl jumpglider: A hybrid jumping and gliding robot with rigid or folding wings. M Kovac, O Wassim-Hraiz, J C Fauria, D Zufferey, Floreano, IEEEM. Kovac, Wassim-Hraiz, O. Fauria, J. C. Zufferey, and D. Floreano, "The epfl jumpglider: A hybrid jumping and gliding robot with rigid or folding wings," IEEE, 2011. Flight control of a rotary wing uav -a practical approach. B Ahmed, H R Pota, M Garratt, IEEE Conference on Decision and Control. B. Ahmed, H. R. Pota, and M. Garratt, "Flight control of a rotary wing uav -a practical approach," in IEEE Conference on Decision and Control, 2008. Comparison between a* and rrt algorithms for uav path planning. C Zammit, E.-J Van Kampen, AIAA guidance, navigation, and control conference. 1846C. Zammit and E.-J. Van Kampen, "Comparison be- tween a* and rrt algorithms for uav path planning," in 2018 AIAA guidance, navigation, and control confer- ence, 2018, p. 1846. Advanced Graph Search Algorithms for Path Planning of Flight Vehicles. Recent Advances in Aircraft Technology. L D Filippis, G Guglieri, L. D. Filippis and G. Guglieri, Advanced Graph Search Algorithms for Path Planning of Flight Vehicles. Re- cent Advances in Aircraft Technology, 2012. Optimal path planning in complex cost spaces with sampling-based algorithms. D Devaurs, T Siméon, J Cortés, IEEE Transactions on Automation Science and Engineering. 132D. Devaurs, T. Siméon, and J. Cortés, "Optimal path planning in complex cost spaces with sampling-based algorithms," IEEE Transactions on Automation Science and Engineering, vol. 13, no. 2, pp. 415-424, 2016. Fast and efficient visible trajectories planning for the dubins uav model in 3d built-up environments. O Gal, Y Doytsher, Robotica. 321O. Gal and Y. Doytsher, "Fast and efficient visible trajectories planning for the dubins uav model in 3d built-up environments," Robotica, vol. 32, no. pt.1, pp. 143-163, 2014. Rapidly-exploring random trees : A new tool for path planning. S M Lavalle, Computer ence Dept. Oct. 98S. M. Lavalle, "Rapidly-exploring random trees : A new tool for path planning," Computer ence Dept. Oct, vol. 98, 1998. Rrt-connect: An efficient approach to single-query path planning. J J K Jr, S M Lavalle, Proceedings of the 2000 IEEE International Conference on Robotics and Automation, ICRA 2000. the 2000 IEEE International Conference on Robotics and Automation, ICRA 2000San Francisco, CA, USAJ. J. K. Jr and S. M. Lavalle, "Rrt-connect: An efficient approach to single-query path planning," in Proceed- ings of the 2000 IEEE International Conference on Robotics and Automation, ICRA 2000, April 24-28, 2000, San Francisco, CA, USA, 2000. Sampling-based algorithms for optimal motion planning. S Karaman, E Frazzoli, The International Journal of Robotics Research. 307S. Karaman and E. Frazzoli, "Sampling-based algo- rithms for optimal motion planning," The International Journal of Robotics Research, vol. 30, no. 7, pp. 846- 894, 2011. Path planning using an improved a-star algorithm. C Ju, Q Luo, X Yan, 2020 11th International Conference on Prognostics and System Health Management. PHM-2020 Jinan2020C. Ju, Q. Luo, and X. Yan, "Path planning using an improved a-star algorithm," in 2020 11th International Conference on Prognostics and System Health Man- agement (PHM-2020 Jinan), 2020. Lifelong planning a. S Koenig, M Likhachev, D Furcy, Artificial Intelligence. 1551-2S. Koenig, M. Likhachev, and D. Furcy, "Lifelong planning a," Artificial Intelligence, vol. 155, no. 1-2, pp. 93-146, 2004. D* lite. S Koenig, M Likhachev, S. Koenig and M. Likhachev, "D* lite," 2002. Anytime dynamic a*: An anytime, replanning algorithm. M Likhachev, D Ferguson, G Gordon, A Stentz, S Thrun, in In ICAPS, 2005M. Likhachev, D. Ferguson, G. Gordon, A. Stentz, and S. Thrun, "Anytime dynamic a*: An anytime, replanning algorithm," in In ICAPS, 2005, pp. 262- 271. Theta*: Any-angle path planning on grids. A Nash, K Daniel, S Koenig, A Felner, A. Nash, K. Daniel, S. Koenig, and A. Felner, "Theta*: Any-angle path planning on grids." 2007. Hierarchical and smoothed topographic path planning for large-scale virtual simulation environments. C Chagas, E Zacarias, L A De Lima, E Silva, Pignaton De Freitas, Expert Systems with Applications. 189116061C. Chagas, E. Zacarias, L. A. de Lima Silva, and E. Pignaton de Freitas, "Hierarchical and smoothed topographic path planning for large-scale virtual simulation environments," Expert Systems with Applications, vol. 189, p. 116061, 2022. [Online]. Available: https://www.sciencedirect.com/ science/article/pii/S0957417421014019 Probabilistic path planning for cooperative target tracking using aerial and ground vehicles. H Yu, R W Beard, M Argyle, C Chamberlain, American Control Conference (ACC). H. Yu, R. W. Beard, M. Argyle, and C. Chamber- lain, "Probabilistic path planning for cooperative target tracking using aerial and ground vehicles," in American Control Conference (ACC), 2011, 2011. A hybrid path planning method in unmanned air/ground vehicle (uav/ugv) cooperative systems. J Li, G Deng, C Luo, Q Lin, Q Yan, Z Ming, IEEE Transactions on Vehicular Technology. J. Li, G. Deng, C. Luo, Q. Lin, Q. Yan, and Z. Ming, "A hybrid path planning method in unmanned air/ground vehicle (uav/ugv) cooperative systems," IEEE Transac- tions on Vehicular Technology, pp. 1-1, 2016. Comparison of parallel genetic algorithm and particle swarm optimization for real-time uav path planning. Vincent Roberge, Tarbouchi, Labonté Mahommed, Gilles , Industrial Informatics. IEEE Transactions onRoberge, Vincent, Tarbouchi, Mahommed, Labonté, and Gilles, "Comparison of parallel genetic algorithm and particle swarm optimization for real-time uav path planning," Industrial Informatics, IEEE Transactions on, 2013. Intelligent amphibious ground-aerial vehicles: State of the art technology for future transportation. X Zhang, J Huang, Y Huang, K Huang, L Yang, Y Han, L Wang, H Liu, J Luo, J Li, IEEE Transactions on Intelligent Vehicles. 81X. Zhang, J. Huang, Y. Huang, K. Huang, L. Yang, Y. Han, L. Wang, H. Liu, J. Luo, and J. Li, "Intelligent amphibious ground-aerial vehicles: State of the art technology for future transportation," IEEE Transac- tions on Intelligent Vehicles, vol. 8, no. 1, pp. 970-987, 2023. Multi-robot path planning for a swarm of robots that can both fly and drive. B Araki, J Strang, S Pohorecky, C Qiu, T Naegeli, D Rus, 2017 IEEE International Conference on Robotics and Automation (ICRA. B. Araki, J. Strang, S. Pohorecky, C. Qiu, T. Naegeli, and D. Rus, "Multi-robot path planning for a swarm of robots that can both fly and drive," in 2017 IEEE International Conference on Robotics and Automation (ICRA), 2017, pp. 5575-5582. Energy efficient path planning of hybrid fly-drive robot (hyfdr) using a* algorithm. A Sharif, H M Lahiru, S Herath, H Roth, 15th International Conference on Informatics in Control, Automation and Robotics. A. Sharif, H. M. Lahiru, S. Herath, and H. Roth, "Energy efficient path planning of hybrid fly-drive robot (hyfdr) using a* algorithm," in 15th International Conference on Informatics in Control, Automation and Robotics, 2018. Energy-Efficient Motion Planning for Multi-Modal Hybrid Locomotion. H J T Suh, X Xiong, A Singletary, A D Ames, J W Burdick, arXiv:1909.10209arXiv e-printsH. J. T. Suh, X. Xiong, A. Singletary, A. D. Ames, and J. W. Burdick, "Energy-Efficient Motion Planning for Multi-Modal Hybrid Locomotion," arXiv e-prints, p. arXiv:1909.10209, Sep. 2019. Multimodal dynamics analysis and control for amphibious fly-drive vehicle. Q Tan, X Zhang, H Liu, S Jiao, M Zhou, J Li, IEEE/ASME Transactions on Mechatronics. 262Q. Tan, X. Zhang, H. Liu, S. Jiao, M. Zhou, and J. Li, "Multimodal dynamics analysis and control for am- phibious fly-drive vehicle," IEEE/ASME Transactions on Mechatronics, vol. 26, no. 2, pp. 621-632, 2021. A multi-modal deformable landair robot for complex environments. X Zhang, Y Huang, K Huang, X Wang, D Jin, H Liu, J Li, arXiv:2210.16875arXiv preprintX. Zhang, Y. Huang, K. Huang, X. Wang, D. Jin, H. Liu, and J. Li, "A multi-modal deformable land- air robot for complex environments," arXiv preprint arXiv:2210.16875, 2022. J G Leishman, Principles of Helicopter Aerodynamics. Cambridge University Press12J. G. Leishman, Principles of Helicopter Aerodynam- ics. Cambridge University Press, 2002, vol. 12.
[]
[ "Inversion Relations, Reciprocity and Polyominoes", "Inversion Relations, Reciprocity and Polyominoes" ]
[ "M Bousquet-Mélou \nDepartment of Mathematics and Statistics\n) LaBRI, CNRS Université Bordeaux I 351 cours de la Libération 33405 Talence Cedex France (2)\nThe University of Melbourne Parkville\n3052VicAustralia\n", "A J Guttmann ", "W P Orrick ", "A Rechnitzer ", "\nDepartment of Mathematics and Statistics\nLaboratoire Bordelais de Recherche en Informatique\nUniversité Bordeaux I\nThe University of Melbourne\n\n" ]
[ "Department of Mathematics and Statistics\n) LaBRI, CNRS Université Bordeaux I 351 cours de la Libération 33405 Talence Cedex France (2)\nThe University of Melbourne Parkville\n3052VicAustralia", "Department of Mathematics and Statistics\nLaboratoire Bordelais de Recherche en Informatique\nUniversité Bordeaux I\nThe University of Melbourne\n" ]
[]
We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons and staircase polygons with a staircase hole. In so doing, we establish a connection between the reciprocity results known to combinatorialists and the inversion relations used by physicists to solve models in statistical mechanics. For several classes of convex polygons, the inversion (reciprocity) relation, augmented by certain symmetry and analyticity properties, completely determines the anisotropic perimeter generating function.
10.1007/bf01608785
[ "https://export.arxiv.org/pdf/math/9908123v1.pdf" ]
15,235,165
math/9908123
dc48b8af31a05620f7b30b923f1ce500aac42212
Inversion Relations, Reciprocity and Polyominoes 23 Aug 1999 June 1999 M Bousquet-Mélou Department of Mathematics and Statistics ) LaBRI, CNRS Université Bordeaux I 351 cours de la Libération 33405 Talence Cedex France (2) The University of Melbourne Parkville 3052VicAustralia A J Guttmann W P Orrick A Rechnitzer Department of Mathematics and Statistics Laboratoire Bordelais de Recherche en Informatique Université Bordeaux I The University of Melbourne Inversion Relations, Reciprocity and Polyominoes 23 Aug 1999 June 1999arXiv:math/9908123v1 [math.CO]Inversion relationscombinatorial reciprocity theoremspolyominoesself- avoiding polygonsconvex polygonsstatistical mechanics AMS Subject Classification: 05A15 (05B5082B2082B23) We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons and staircase polygons with a staircase hole. In so doing, we establish a connection between the reciprocity results known to combinatorialists and the inversion relations used by physicists to solve models in statistical mechanics. For several classes of convex polygons, the inversion (reciprocity) relation, augmented by certain symmetry and analyticity properties, completely determines the anisotropic perimeter generating function. Introduction Symmetries are among the most important guiding principles in all of physics and mathematics. It often happens that a problem may be solved by symmetry considerations alone, and even if not, understanding the symmetries of the solution can greatly reduce the amount of work needed to find it. We study here a symmetry of functions which is known as "selfreciprocity" to combinatorialists and which is referred to as "inversion relations" in lattice statistical mechanics. Our focus will be on polyomino enumeration problems which are of interest in both combinatorics and physics. We shall demonstrate that one can find examples of functional symmetry in the resulting generating functions. The inversion relation rose to prominence in statistical mechanics in the early 1980s as the most direct path to the solution of many integrable models [23,2,3] and was soon realized to be commonplace in both solved and unsolved models [2,3,15]. Let G(x) be a thermodynamic quantity which depends on a collection of parameters, x. An inversion relation is a functional equation G(x) ± x α G(φ(x)) = ψ(x)(1) where φ and ψ are known functions of x. Typically φ involves taking reciprocals of one or more components of x. The inversion relation tightly constrains the function G. For some two-dimensional models a pair of additional conditions holds: that G is symmetric under exchange of horizontal and vertical, and that G is an analytic function of its arguments. Very often, the three constraints taken together uniquely determine the function G. In 1974 Stanley presented a general framework for reciprocity results. He established several powerful general conditions under which a generating function will be self-reciprocal [20]. The language and notation of Stanley [20,22] will be used throughout this paper. Definition 1.1. Let H(y 1 , . . . , y n ) be a rational function in the variables y i , with coefficients in R. We say that H is self-reciprocal if there exists an n-tuple of integers (β 1 , . . . , β n ) such that H(1/y 1 , . . . , 1/y n ) = ±y β 1 1 . . . y βn n H(y 1 , . . . , y n ). In what follows, we write y β ≡ y β 1 1 . . . y βn n and 1/y ≡ (1/y 1 , . . . , 1/y n ). Thus eqn. (2) may be concisely expressed as H(1/y) = ±y β H(y). Note that a rational function is self-reciprocal if and only if both its numerator and denominator are so, and that the self-reciprocity of a polynomial amounts to a certain symmetry in its coefficients. Some explicit examples are given in Subsection 3.2. Let us now demonstrate the relationship between self-reciprocity and inversion relations. Consider the multivariable generating function where m = (m 1 , . . . , m j ), x = (x 1 , . . . , x j ) and similarly for n and y. The summation is over (j +k)-tuples of nonnegative integers representing the objects being enumerated. Performing the summation over n, we reexpress eqn. (3) in terms of partial generating functions, H m (y), G(x, y) = m H m (y)x m .(4) Now suppose that the partial generating functions are self-reciprocal, H m (1/y) = ±ǫ m y β(m) H m (y),(5) where ǫ is a j-tuple of elements in the set {−1, 1} which characterizes the dependence of the sign on m, and where β(m) depends linearly on m: β(m) = Am + α. Here, A = (a ℓ,i ) ℓ,i is a k × j matrix of integers and α is a k-tuple of integers. We can then write G(x, y) ∓ y −α G(ǫxy −A , 1/y) = 0 (7) where ǫxy −A is the j-tuple whose i th entry is ǫ i x i k ℓ=1 y −a ℓ,i ℓ . This is clearly a special case of the inversion relation (1). A few comments are in order: • The right hand side of (7) is zero, but in the more general situation some of the partial generating functions, H m (y) will fail to be self-reciprocal for certain choices of m. If we are fortunate, this will be a small, finite or otherwise controllable set of cases, and we will be able to compute the correction term we need to add to the right hand side explicitly. For many examples in statistical mechanics, this correction term depends on x but not on y. • In all of the cases we shall see below, the denominators of our rational functions will be a product of terms (1 − y α j ), which are self-reciprocal. Stanley has proved that this denominator form always holds for certain classes of problems (see Theorem 4.6.11 of ref. [22]). • It might be asked which of the concepts, inversion or self-reciprocity, is the more general. On one hand, in the derivation of (7) the dependence of the exponent β on m was assumed to be linear, which may not always hold, implying that reciprocity is more fundamental. On the other hand, the function φ occurring in (1) may in principle be more complicated than x → ǫxy −A , y → 1/y. In this case, the partial generating functions might not be self-reciprocal. An example is provided in Section 2 by the Potts model, but in the polyomino examples considered in this paper, this situation does not arise. We now present a nonexhaustive list of recipes for finding and proving reciprocity results and inversion relations. 1. If the generating function (or thermodynamic quantity) is known in closed form, an inversion relation can be demonstrated directly. As an example, we treat the anisotropic perimeter generating function for directed convex polygons in this manner in Section 3. 2. For statistical mechanics models which admit a formulation in terms of a family of commuting transfer matrices, a transformation of parameters can often be found which inverts the transfer matrix. The commutativity property then allows the inversion relation to be derived. We review this in detail in Section 2, with the two-dimensional, zero-field Ising model as primary example. 3. In non-integrable models, the transfer matrix will still be invertible and may suggest a possible inversion relation, but the required analyticity property is lacking. Nevertheless, the suggested inversion relation can often be verified by inspection of the partial generating functions up to some finite order in the low-temperature expansion (4). The q > 2 Potts model inversion relation discussed in Section 2 was derived this way in ref. [13]. Some of the new results reported in the present paper were initially discovered by this method before being rederived by one of the other methods. 4. The "Temperley methodology" [7] can be used to obtain very general reciprocity results for many classes of column-convex polygons. The first step is to derive a functional equation for the generating function which can be interpreted as the gluing of an additional column onto the graph. Step two is to show by induction that appending an additional column preserves self-reciprocity. This is detailed in Section 4. 5. If the problem can be posed as a system of linear diophantine equations, whose solutions are subject to certain types of constraints, we may apply self-reciprocity theorems due to Stanley [20]. We have so far succeeded in applying this method only to families of directed polyominoes (Section 5), but it enables us to treat problems which are impossible, or at least extremely cumbersome, by the method of functional equations. 6. For combinatorial objects with a rational generating function of denominator j (1 − y α j ), one can try to explain self-reciprocity -i.e., the symmetry of the numerator -by interpreting the numerator combinatorially. This has been done by Fédou for a family of objects related to (but distinct from) staircase polygons [10]. In Section 2 we review the motivation for looking at inversion relations in statistical mechanics and describe the methods used to obtain them. This will be useful for making comparisons with the results obtained later, and for suggesting applications and generalizations of the inversion relations. In Section 3, we present examples of reciprocity results and inversion relations for polyominoes, and summarize our main new results. The technical heart of the paper consists of Section 4 on the Temperley methodology, and Section 5 on the application of Stanley's results to polyominoes. Inversion relations in statistical mechanics The first use of the inversion relation in statistical mechanics was the solution by Stroganov of certain two dimensional vertex models on the square lattice [23]. Generalizations of Stroganov's models were later solved by the same means by Schultz [19]. Shortly after Stroganov, Baxter used a similar method to solve the hard hexagon model [1] and recognized its broad applicability, giving the eight-vertex and Ising models as examples [2]. Subsequently, a number of authors pointed out that many known solutions to problems in two-dimensional statistical mechanics can be derived easily using the inversion relation method. Among these were Shankar [18], Baxter [5] and Pokrovsky and Bashilov [17]. It is noteworthy that inversion relations hold also for models that have not been solved. Prominent among such models are the two-dimensional Ising model in a magnetic field whose inversion relation was found by Baxter [2], and the three-dimensional Ising model and noncritical q-state Potts model, both of whose inversion relations were found by Jaekel and Maillard [12,13]. What generally distinguishes solved and unsolved models is the growth rate in the number of poles arising in the partial generating functions in the expansion (4), as a function of order. Roughly speaking, a more complicated pole structure implies that the number of parameters needed to specify a given partial generating function is greater, and makes it less likely that an inversion relation can completely determine all of them. Nevertheless, inversion relations are still invaluable in the study of such problems, not least because they provide an independent check on series data. The fundamental problem of statistical mechanics is to calculate the partition function. Here we consider vertex models defined on a square lattice with each bond colored with one of r possible colors. Each lattice site makes a contribution to the energy of the system which depends on the colors of the adjacent bonds. This defines an r 4 -vertex model if all possible colorings are permitted. Stroganov computed the partition function per site in the thermodynamic limit of several 16-and 81-vertex models. Consider first a finite lattice (on the torus) of N rows and M columns. The partition function can be expressed in terms of the transfer matrix T M as: Z M,N = Tr (T M ) N(8) (see [4,22]). Here, T M is the r M ×r M matrix whose i, jth entry is the contribution to Z M,N of a single row of M sites connected to the row below by a set of vertical bonds in configuration i and to the row above by a set of vertical bonds in configuration j. It depends on the temperature, T , and on r 4 parameters specifying the vertex energies. In the thermodynamic limit, the partition function per site is given by κ = lim M,N →∞ (Z M,N ) 1/M N = lim M →∞ (λ M ) 1/M(9) where λ M is the largest eigenvalue of T M , assumed to be nondegenerate. For simplicity let us consider a family of models whose vertex energies are functions of a single parameter, b. The models solved by Stroganov are integrable by virtue of the commutativity of the transfer matrices at different values of this parameter. This implies that the transfer matrix eigenvectors are common to all members of the family, and that the b dependence is only in the eigenvalues. For this reason b is often called the spectral parameter. The key observation is that the inverse of the transfer matrix in these models is itself a member of the commuting family, up to a scale factor [T M (b)] −1 = ψ(b) −M T M (φ(b)) .(10) Acting on the eigenvector corresponding to λ M (b) with both sides of eqn. (10) yields the functional equation κ(b)κ (φ(b)) = ψ(b).(11) It is the commutativity of the transfer matrices for all values of b that allows the analytical continuation of the function κ from b to φ(b). With knowledge of the functions ψ(b) and φ(b) and using the analyticity of κ(b), Stroganov finds a unique solution, thereby reproducing Baxter's results for the symmetric eight vertex and homogeneous ferroelectric models, and obtaining the result for a certain 81-vertex model [23]. As an illustrative example, we review here the derivation by Baxter [2] of Onsager's expression for the partition function of the two-dimensional zero-field Ising model [16]. Let the square lattice be drawn at 45 • to the horizontal and let the couplings between nearest neighbors along the two lattice directions be J and J ′ . Define low temperature variables x = e −2K , y = e −2K ′ with K = J/k B T, K ′ = J ′ /k B T.(12) Transfer matrices for different choices of parameters will commute provided they have the same value of k = (sinh 2K sinh 2K ′ ) −1 . The transformation which inverts the transfer matrix is K → K + iπ 2 , K ′ → −K ′ ,(13) which does not modify the value of k. Define the reduced partition function per site by Λ(x, y) = exp(−K − K ′ )κ(K, K ′ ).(14) Then Λ(x, y) obeys the inversion relation Λ(x, y)Λ(−x, 1/y) = 1 − x 2 .(15) Note that log Λ(x, y) − 1 2 log(1 − x 2 ) has an inversion relation of precisely the form (7). By the symmetry of the model, we have Λ(x, y) = Λ(y, x).(16) Inspection of the low temperature expansion leads us to conjecture the form Λ(x, y) = 1 + m≥1 P m (y 2 ) (1 − y 2 ) 2m−1 x 2m .(17) That the coefficient of x 2m is a rational function of y 2 is apparent from the nature of the low temperature expansion, but that the denominator has such a simple form is not expected on general grounds. Presumably it is a consequence of the condition of commuting transfer matrices. Here we take it as a hypothesis. Then Baxter has shown that the inversion relation (15), symmetry (16) and the denominator form (17) determine Λ(x, y) completely. We present his argument in Section 3.4 where we use it in the context of polygon enumeration. Up till now we have been assuming integrability and in particular we have relied on the property that the transfer matrix and its inverse are both members of some one-parameter commuting family. What about models for which this property doesn't hold? Since analyticity of κ(b) breaks down, the step (11) in the above derivation is no longer valid. However, it is still possible to obtain an inversion relation by direct analysis of the low-temperature expansion of the partition function to some finite order. As an example, it was shown in ref. [13] that the logarithm of the reduced partition function per site, G(x, y) = ln Λ(x, y), of the q-state Potts model satisfies the inversion relation G(x, y) + G − x 1 + (q − 2)x , 1 y = ln (1 − x)(1 + (q − 1)x) 1 + (q − 2)x .(18) When q = 2 this reduces to the Ising model inversion relation (15). The inversion relations we will be considering in the remainder of the paper are derived by analysis of the generating function (analogous to the low temperature expansion) and do not depend on the models being integrable. An additional new feature is seen in this Potts model example. Neglecting for the moment the nonzero right-hand-side of (18), which can be eliminated by a suitable redefinition of G(x, y), we notice that when q > 2 there is no longer an order-by-order cancellation of the partial generating functions as defined in (4), but rather cancellation of combinations of partial generating functions of different orders. However, we may convert to self-reciprocal form by defining G ′ (x, y) = G x 1 − (q − 2)x/2 , y(19) under which the inversion relation becomes G ′ (x, y) + G ′ (−x, 1/y) = ln 1 − q 2 x 2 /4 1 − (q − 2) 2 x 2 /4 .(20) In the cases we will look at in this paper, the partial generating functions turn out to be self-reciprocal in the natural variables of the problem. We have not investigated the existence of inversion relations involving more complicated changes of variables. 3 Polyomino enumeration and self-reciprocity Definitions The constructions we will consider are defined on the square lattice. All are defined only up to translation on the lattice. Starting at a lattice site and moving to one of the four nearest neighbors constitutes a step which we may identify with the edge connecting the sites. A connected sequence of steps is a path or walk. If no lattice site in the path occurs more than once, the path is self-avoiding. If a path returns to its starting site in the final step, and otherwise does not intersect itself, the result is a self-avoiding polygon. The number of steps taken is the perimeter of the polygon; the number of steps taken in the vertical direction is the vertical perimeter. The horizontal perimeter is defined similarly. The area is the number of cells of the lattice enclosed by the polygon. Enumerating self-avoiding polygons according to perimeter or area is an unsolved problem. However, progress has been made in enumerating certain subclasses of self-avoiding polygons. Rectangles coincide with the rectangles of ordinary geometry whose vertices are lattice points and whose edges lie along lattice directions. A rectangle which contains a given polygon, i.e., all steps of the polygon lie inside or on the rectangle, is a bounding rectangle for that polygon. The smallest such rectangle is the minimal bounding rectangle. A polygon whose perimeter equals that of its minimal bounding rectangle is convex. If a convex polygon contains at least one of the corners of its minimal bounding rectangle (for concreteness say the south-west corner) then it is a directed convex polygon. If it contains also the north-east corner, it is a staircase polygon, so called because it is bounded above and below by two staircase-like or directed paths. On the other hand, if it contains two adjacent corners, say the southwest and southeast (northeast and southeast) then it is a stack polygon with horizontal (vertical) orientation. If it contains three corners, then it is a Ferrers graph. Representative examples of different classes of convex polygons are shown in Figure 1. One way to obtain non-convex polygons is to relax the convexity condition along one direction only. A self-avoiding polygon is column-convex if the intersection of any vertical line with the polygon has at most two connected components. Row-convex polygons are similarly walk is a self-avoiding walk whose steps are taken in accordance with the three-choice rule which allows a step either to the left or the right or straight ahead after any vertical step, but forbids a right turn after any horizontal step. A polygon formed from such a walk is a three-choice polygon. When the walk returns to its starting point, we don't specify whether the next step, i.e., the first step, is a valid continuation of the walk. If it is, the result is a staircase polygon; if not, it is an imperfect staircase polygon (see Figure 3(a)). When we refer to three-choice polygons below, we include only the imperfect ones. A polyomino is a union of connected (sharing an edge) cells of the lattice. We shall consider one class of nonpolygon polyominoes -the staircase polygons with a staircase hole. The outer boundary and the hole are both staircase polygons and must not touch at any point. An example is shown in Figure 3 Self-reciprocity in polyomino enumeration For each of the above classes of column-convex polygons, the anisotropic perimeter and area generating function, G(x, y, q) = m≥1 n≥1 a≥1 C(m, n, a)x m y n q a(21) has been computed exactly (see ref. [7] and references therein). Here C(m, n, a) is the number of polygons of the class with 2m horizontal bonds, 2n vertical bonds and area a. For the classes of convex polygons, the anisotropic perimeter generating function, G(x, y, 1) is an algebraic function of the fugacities, x and y, whereas the area generating function, G(1, 1, q) is a q-series. For classes of polygons that are only column-convex, both G(x, y, 1) and G(1, 1, q) [24] are algebraic, but G(x, y, q) involves q-series. A closed-form expression for the three-choice polygon anisotropic perimeter-area generating function is not yet known, but by means of a transfer matrix technique it can be evaluated in polynomial time [9]. The isotropic perimeter generating function, G(x, x, 1) is known to have a logarithmic singularity [9], and is therefore not algebraic, but is known to be D-finite. The generating function for staircase polygons with a staircase hole is also not known in closed form. Its properties are expected to be similar in many respects to the generating function for three-choice polygons [11]. We shall be concerned with self-reciprocity properties of the generating functions H m (y, q) that count polygons of width m. We first give two examples. 1. The area generating function for staircase polygons of width 4 is the following rational function [6]: H 4 (q) = q 4 (1 + 2q + 4q 2 + 6q 3 + 7q 4 + 6q 5 + 4q 6 + 2q 7 + q 8 ) (1 − q) 2 (1 − q 2 ) 2 (1 − q 3 ) 2 (1 − q 4 ) . It satisfies H 4 (1/q) = −H 4 (q), and is thus self-reciprocal. Observe that the numerator is not only symmetric (due to self-reciprocity), but also unimodal. 2. The (half-)vertical perimeter and area generating function for column-convex polygons of width 3 is the following rational function, which can be derived from the general formula of ref. [7]: H 3 (y, q) = yq 3 (1 − yq) 4 (1 − yq 2 ) 2 (1 − yq 3 ) · (y 6 q 8 + 4y 5 q 7 + 2y 5 q 6 + y 4 q 6 − y 4 q 4 − 4y 3 q 5 − 6y 3 q 4 − 4y 3 q 3 − y 2 q 4 + y 2 q 2 + 2yq 2 + 4yq + 1). It satisfies H 3 (1/y, 1/q) = − 1 yq 3 H 3 (y, q) and hence is self-reciprocal. Again, observe the symmetry of the coefficients in the numerator. We shall generalize these results to polygons of any width. Table 1 summarizes the selfreciprocity properties we have established. Most of them can be proved in various ways. One can for instance use a closed form expression of the generating function (Section 3.3), or a functional equation that defines it (Section 4); one can also encode the polygons by a sequence of numbers constrained by linear diophantine equations and apply Stanley's general results (Section 5). We shall see that the last two methods allow us to introduce many additional parameters and obtain self-reciprocity results that significantly generalize those of Table 1. Self-reciprocity via generating functions When a closed form expression for the generating function of some class of polygons is known, it seems natural to use it to demonstrate an inversion relation. Let us take the example of Class Picture Self Reciprocity Inversion Relation Ferrers H m (1/y, 1/q) = (−1) m y m−2 q m 2 −3m 2 H m (y, q) G(x, y) − y 2 G(−x/y, 1/y) = 0 stack H m (1/y, 1/q) = −y 2m−3 q m 2 −2m H m (y, q) G(x, y) + y 3 G(x/y 2 , 1/y) = 0 staircase H m (1/y, 1/q) = −y m−1 H m (y, q), m ≥ 2 G(x, y, q) + yG(x/y, 1/y, 1/q) = −x directed convex H m (1/y) = −y m−2 H m (y) G(x, y) + y 2 G(x/y, 1/y) = 0 convex Not simple G(x, y) + y 3 G(x/y, 1/y) = xy − x 3 y ∂ ∂x 1−x+y ∆(x,y) bargraph H m (1/y, 1/q) = (−1) m yq m H m (y, q) G(x, y, q) − yG(−xq, 1/y, 1/q) = 0 dir. col.-conv. H m (1/q) = − 1 q H m (q) G(x, q) + qG(x, 1/q) = 0 column-convex H m (1/y, 1/q) = − 1 yq m H m (y, q) G(x, y, q) + yG(xq, 1/y, 1/q) = 0 three-choice Not simple G(x, y, q) + y 2 G(x/y, 1/y, 1/q) = known SC with SC hole Not simple G(x, y, q) + y 2 G(x/y, 1/y, 1/q) = known the anisotropic perimeter generating function for directed convex polygons, which is known to be [14]: G(x, y) = xy ∆(x, y)(22) with ∆(x, y) = 1−2x−2y −2xy +x 2 +y 2 = (1−y) 2 [1 − x(2 + 2y − x)/(1 − y) 2 ]. Expanding this expression in x gives G(x, y) = m≥1 H m (y)x m = y 1 − y x + y(1 + y) (1 − y) 3 x 2 + y(1 + 4y + y 2 ) (1 − y) 5 x 3 + y(1 + 9y + 9y 2 + y 3 ) (1 − y) 7 x 4 + O(x 5 ) which suggests that the partial generating functions, H m (y) are self-reciprocal, and more precisely, that H m (1/y) = −y m−2 H m (y). This is equivalent to the inversion relation G(x, y) + y 2 G(x/y, 1/y) = 0,(23) which is easily checked from the closed form of the generating function. Note that an explicit expression for H m (y) is given in [6]. The inversion relations for convex polygons and directed column-convex polygons may also be obtained from the expression of their generating function. The partial generating functions for directed convex polygons, counted by the area, are not self-reciprocal: for instance, the generating function for width 3 is q 3 (1 + 3q + 3q 2 + 2q 3 + q 4 )/(1 − q) 2 (1 − q 2 ) 2 (1 − q 3 ). However, many other classes of column-convex polygons have an inversion relation for the full anisotropic perimeter and area generating function. Since these generating functions are also known in closed form they could be derived as above. However more can be shown, namely that there is a self-reciprocity for any parameter which is a linear function of the vertical heights in the graph. This very general result will be derived in Section 4. Likewise, the inversion relations for three-choice polygons and staircase polygons with a staircase hole, given in Table 1, are also special cases of more general formulae which will be derived in Section 5. Using inversion relations to compute generating functions As in statistical mechanics, the inversion relation and symmetry, and some general assumptions on analyticity of the generating function, are sometimes sufficient to determine the solution completely. In order to have an algorithm for computing a generating function term by term, it is necessary, but not sufficient, to have some property relating terms of different orders. For our purposes this property will always be x-y symmetry. Thus we restrict our attention to classes of graphs with x-y symmetry, i.e., Ferrers, staircase, directed convex, convex and three-choice polygons, and staircase polygons with a staircase hole. Moreover, we shall only consider the anisotropic perimeter generating function (without area). For the former four classes we will show that the inversion relation provides sufficient additional information to compute the generating function, whereas for the latter two it does not. The general form of the generating function is G(x, y) = H 1 (y)x + H 2 (y)x 2 + H 3 (y)x 3 + · · ·(24) where the partial generating functions, H m (y) are rational functions We assume that in general we know the denominator form either empirically or by rigorous proof, and that D m (y) is of degree d m . Now we proceed inductively, following Baxter [2]. If we have already computed the coefficient functions H 1 (y), . . . , H m−1 (y) in the expansion (24) and if x-y symmetry holds, we also know the coefficients of y, y 2 , . . . , y m−1 in the expansion of G(x, y). In particular, we can compute the coefficients of y, y 2 , . . . , y m−1 in the numerator polynomial P m (y). In order to obtain the unknown coefficients of P m (y), we must be able to express them in terms of the known ones by means of the inversion relation. Writing P m (y) = k a k y k , and using D m (1/y) = ±y −dm D m (y), the inversion relation fixes the value of the combinations of coefficients, a k ± a ℓ , with k + ℓ = α + d m − m. Hence the determination of all the coefficients a k is possible if and only if the arithmetic condition holds: H m (y) = P m (y) D m (y) ,(25)d m < 3m − α.(28) This condition is seen to hold for all the classes of convex polygons we have looked at, since d m ≤ 2m − 1, but not for three-choice polygons or staircase polygons with a staircase hole, since d m ∼ 4m. Self-reciprocity via Temperley methodology We consider column-convex polygons as pairs of partially directed paths having the same endpoints, as indicated in Figure 4. Let P be a column-convex polygon of width m. For 0 ≤ i ≤ m, we denote by N i (resp. S i ) the number of north (resp. south) steps in the top path γ at abscissa i. For 0 ≤ i ≤ m, we denote by N i (resp. S i ) the number of north (resp. south) steps of the bottom path γ at abscissa i. We choose the end points of the paths in such a way that N 0 = S 0 = N m = S m = S 0 = S m = 0. Note that m k=0 (N k + S k − S k − N k ) = 0. We notice that all standard statistics are linear functions of the N i , S i , N i and S i . For instance, the vertical perimeter of the polygon is 2n = m k=0 (N k + S k + S k + N k ) = 2 m k=0 (N k + S k ).(29) The height of the i th column of the polygon is, for 1 ≤ i ≤ m, h i = i−1 k=0 (N k + S k − S k − N k ), and the area of the polygon is a = m k=0 (m − k)(N k + S k − S k − N k ).(30) Theorem 4.1. Let P be one of the following sets: Ferrers diagrams, stacks (drawn as in Figure 1(c)), staircase polygons, bar-graphs, column-convex polygons. Let P m be the subset of P containing all polygons of width m. Let F m be the generating function for polygons in the set P m : F m (y, z, y, z) = P ∈Pm y N z S y N z S . Then F m is a rational function, and it is self-reciprocal: F m (1/y, 1/z, 1/y, 1/z) = C m F m (y, z, y, z),(31) with C m =                                                          (−1) m y m−2 m y 0 m−1 i=1 y i for Ferrers graphs, − y 2m−3 m y 0 m−1 i=1 y i z i for stacks, − m−1 i=1 y i y i for staircase polygons (m ≥ 2), (−1) n y 0 y m for bar-graphs, − 1 y 0 y m for column-convex polygons. The proof of the theorem is based on the so-called Temperley approach for counting column-convex polygons [24], combined with the systematic use of formal power series [7]. Here we provide only the proof for column-convex polygons, since the others are very similar. We commence by showing that the partial generating functions for column-convex polygons, V m (y, z, y, z), can be computed inductively. Proposition 4.2. Let V m (y, z, y, z) be the generating function for column-convex polygons of width m. Let us denote it, for the sake of simplicity, V m (y m ). Then the series V m (y m ) can be defined inductively by: V 1 (y 1 ) = y 0 y 1 1 − y 0 y 1 and V m+1 (y m+1 ) = (1 − y m z m )(1 − y m z m )V m (y m+1 ) (1 − y m+1 y m )(1 − y m+1 z m )(1 − y −1 m+1 y m )(1 − y −1 m+1 z m ) + (z m − y m+1 y m z m )V m (z m ) (1 − y m+1 z m )(1 − y −1 m+1 z m )(y m − z m ) + (y m − y m+1 y m z m )V m (y m ) (1 − y m+1 y m )(1 − y −1 m+1 y m )(z m − y m ) . Proof. The basic idea is build a polygon of width m + 1 by adding a new column to a polygon of width m [7]. It is convenient to use Hadamard products to establish the functional equation. frag replacements Let F (t) = f h t h and R(t) = r h t h be two formal power series in t with coefficients in a ring A. We denote by F (t) ⊙ R(t) the Hadamard product of F (t) and R(t), evaluated at t = 1: > 0 > 0 > 0 > 0 > 0 ≥ 0 ≥ 0 ≥ 0 + + + ⊙ hF (t) ⊙ R(t) = f h r h . In what follows, f h (resp. r h ) will be the generating function for some column-convex polygons whose rightmost (resp. leftmost) column has height h, so that F (t)⊙R(t) will count polygons obtained by matching the rightmost column of a polygon of type F with the leftmost column of a polygon of type R. Also, R(t) will be a rational function of t. We shall use the following simple identity: F (t) ⊙ 1 1 − at = F (a). The expression for V 1 (y 1 ) is obvious. We build a column-convex polygon of width m + 1 as follows: we take a polygon of width m and match its rightmost column with the leftmost column of a column-convex polygon of width 2. This is illustrated by Figure 5, which shows that V m+1 (y m+1 ) = V m (t) ⊙ R(t),(32) where R(t) = ty m+1 1 − ty m+1 · 1 1 − y m+1 y m · 1 1 − y m+1 z m + ty m+1 1 − ty m+1 · 1 1 − y m+1 y m · ty m 1 − ty m + ty m+1 1 − ty m+1 · tz m 1 − tz m · 1 1 − y m+1 z m + ty m+1 1 − ty m+1 · tz m 1 − tz m · ty m 1 − ty m is the generating function for column-convex polygons of width 2. In order to determine the coefficient r h of t h in R(t), we expand R(t) in partial fractions of t: R(t) = z m y m y 2 m+1 (1 − y m+1 y m )(1 − y m+1 z m ) + (1 − y m z m )(1 − y m z m ) (1 − y m+1 y m )(1 − y m+1 z m )(1 − y −1 m+1 y m )(1 − y −1 m+1 z m ) · 1 1 − ty m+1 + (z m − y m+1 y m z m ) (1 − y m+1 z m )(1 − y −1 m+1 z m )(y m − z m ) · 1 1 − tz m + (y m − y m+1 y m z m ) (1 − y m+1 y m )(1 − y −1 m+1 y m )(z m − y m ) · 1 1 − ty m . Note that V m (0) = 0. We now combine eqn. (32) with the above expression for R(t) to obtain the announced expression for V m+1 (y m+1 ). Proof of Theorem 4.1. Induction on m using the functional equation of Proposition 4.2 shows that the partial generating functions for column-convex polygons satisfy: V m (1/y, 1/z, 1/y, 1/z) = − 1 y 0 y m V m (y, z, y, z). We proceed similarly for the other families: the functional equation is obtained by setting some of the variables y i , y i , z i and z i to 0. Then, an inductive argument yields the selfreciprocity result. It would be tempting to write that the self-reciprocity of V m implies the self-reciprocity of, say, the generating function for staircase polygons, obtained by setting z i and z i to 0 in V m . But replacing a variable by 0 in a self-reciprocal rational function might break the self-reciprocity: for instance, take P (y 1 , y 2 ) = 1 + y 1 + 2y 2 1 + 2y 2 + y 1 y 2 + y 2 1 y 2 . Then P (1/y 1 , 1/y 2 ) = 1 y 2 1 y 2 P (y 1 , y 2 ), but P (y 1 , 0) = 1 + y 1 + 2y 2 1 is not self-reciprocal. However, the following simple lemma gives a useful stability property of self-reciprocal rational functions. Lemma 4.3. Let F (y 1 , . . . , y n ) be a self-reciprocal rational function. Let A be an m × n integer matrix. Let u = (u 1 , . . . , u m ), and define u A to be the n-tuple whose i th coordinate is k u a ki k . Then the series G(u) = F (u A ), if defined, is self-reciprocal in the variables u i . More precisely, if F (1/y) = ±y β F (y) then G(1/u) = ±u Aβ G(u). From Theorem 4.1 and Lemma 4.3 we immediately deduce: Corollary 4.4. For any of the sets P listed in Theorem 4.1, and any statistics on columnconvex polygons that can be expressed as linear functions of the quantities N, S, N, S, the generating function for polygons in the set P m according to these statistics is a self-reciprocal rational function. This corollary allows us to complete the top part of Table 1. Let us, for instance, derive the inversion relation satisfied by the tri-variate generating function G(x, y, q) for columnconvex polygons, taking into account the usual parameters of interest: horizontal and vertical half-perimeters (variables x and y), and area (variable q). Eqns. (29) G(x, y, q) + yG(xq, 1/y, 1/q) = 0. (33) Note that in the first two self-reciprocity relations of Table 1, the exponent of q depends quadratically on the width. For this reason, they only yield an inversion relation for q = 1. Self-reciprocity via Stanley's general results Linear homogeneous diophantine systems Stanley has analyzed the situation where the objects to be counted correspond to integer solutions of a system of linear equations with integer coefficients (linear diophantine system) subject to a set of constraints. He has established certain conditions under which reciprocity relations will hold between two combinatorics problems defined by the same linear diophantine system but by different sets of constraints, and also conditions under which the solution to a given problem will be self-reciprocal [20,22]. Consider the linear homogeneous diophantine system (LHD-system), Φα = 0(34) in the unknowns α = (α 1 , . . . , α s ) with Φ a matrix of integers having p rows and s columns and 0 a p-tuple of zeros. The corank κ of the system is defined to be s − rank(Φ). For a linearly independent system, κ = s − p. Let S be a set of integer solutions to eqn. (34). We define the generating function, S(y), as the formal power series S(y) = α∈S y α(35) where y = (y 1 , . . . , y s ) is a vector of fugacities associated with the unknowns in eqn. (34). In our applications, we find two types of constraints on the unknowns, α j . Certain of the unknowns, α j , are required to be strictly positive while the rest are required to be nonnegative. Conveniently, precisely these kinds of constraints have been treated by Stanley. Let the unknowns be α = (γ, δ) ≡ γ ⊕ δ where γ is an n-tuple and δ is an (s − n)-tuple. Likewise let y = (u, v). In what follows, the notation, δ > 0, means that all coordinates of δ are positive. Proposition 5.1. Let E be the set of integer solutions, (γ, δ), to a linear homogeneous diophantine system of corank κ, such that γ ≥ 0 and δ > 0. Let E be the set of solutions to the same system with γ > 0 and δ ≥ 0. If the system has an integer solution, (γ, δ) such that γ > 0 and δ < 0, then E(u, v) and E(u, v) are rational functions obeying the reciprocity relation E(u, v) = (−1) κ E(1/u, 1/v).(36) Proof. This is Proposition 8.3 of ref. [20] and the proof is given there. Proposition 5.1 can be specialized to obtain a self-reciprocity condition, which will be our main tool in the derivations to follow. Corollary 5.2. A sufficient condition for the function E(u, v) to be self-reciprocal is that the linear homogeneous diophantine system has the solution (γ, δ) = (1, −1). In this case E(1/u, 1/v) = (−1) κ u 1 v 1 E(u, v).(37) PSfrag replacements Figure 6: Staircase polygon of width three N 1 N 2 M 0 M 1 M 2 M 3 N 1 N 2 Proof. Since the solution (1, −1) satisfies the conditions of Proposition 5.1, the reciprocity result (36) holds. The result follows immediately from the shift (γ, δ) → (γ + 1, δ −1) which establishes a bijection between the sets E and E. Since the conditions of the corollary are sufficient but not necessary, it is often possible to find a perfectly valid LHD-system describing a given self-reciprocal generating function, E(u, v), which does not admit the solution (1, −1). Hence we are faced with the problem of finding a suitable LHD-system which satisfies the corollary. A useful heuristic is to start with an LHD-system in many unknowns, and selectively eliminate those unknowns whose constraints are not independent of the constraints on the other unknowns. In all the cases we will consider, the resulting system will satisfy the conditions of Corollary 5.2. We do not justify this heuristic here. In a paper subsequent to ref. [20], Stanley [21] develops a more comprehensive theory which overcomes these difficulties, and which additionally gives "correction" terms for systems in which self-reciprocity fails to hold. We have not yet explored the ramifications of this theory. Before applying the above result to staircase polygons with a staircase hole or to threechoice polygons, we use it to derive the reciprocity relation for ordinary staircase polygons of width three. This will serve to illustrate all the basic ingredients of the method. Figure 6. Decomposing the polygon into three columns, and imposing the condition that each column be as high on the left as it is on the right, we obtain the linear homogeneous diophantine system M 0 − M 1 − N 1 = 0 (38a) M 1 + N 1 − M 2 − N 2 = 0 (38b) M 2 + N 2 − M 3 = 0.(38c) All heights must be nonnegative, but the self-avoidance condition additionally requires that the M j be positive. The constraints M 0 > 0 and M 3 > 0 are actually redundant, since they follow from eqns. (38a,38c) and the constraints on the remaining unknowns, namely N 1 , N 2 , N 1 , N 2 ≥ 0 M 1 , M 2 > 0.(39) Since the constraints on M 0 and M 3 play no role in the solution, we are free to eliminate these unknowns, and it turns out to be necessary to do so in order to apply Corollary 5.2. We are left with the single equation (38b) in the six independent unknowns γ = (N 1 , N 2 , N 1 , N 2 ) and δ = (M 1 , M 2 ). Let us associate to the unknown N i (resp. N i , M i ) the fugacity y i (resp. y i , z i ). Let E ′ be the set of solutions to eqn. (38b) subject to the the constraints γ ≥ 0 and δ > 0. Since γ = 1, δ = −1 is a solution to eqn. (38b), Corollary 5.2 tells us that E ′ (y, y, z) is self-reciprocal, E ′ (1/y, 1/y, 1/z) = − y 1 y 2 y 1 y 2 z 1 z 2 E ′ (y, y, z).(40) Equations (38a,38c) imply that to account for the dependent parameters M 0 and M 3 , we make the substitutions z 1 → z 0 z 1 , y 1 → z 0 y 1 , z 2 → z 2 z 3 and y 2 → z 3 y 2 . Applying Lemma 4.3, we obtain for the set E of nonnegative solutions to (38a,38b,38c) such that M > 0: E(1/y, 1/y, 1/z) = − y 1 y 2 y 1 y 2 z 1 z 2 E(y, y, z).(41) Notice that reintroducing the dependent unknowns has not changed the constant factor. This feature holds as well in the more complicated models we will look at. The result (41) may be verified by inspection of the explicit expression for the generating function E(y, y, z) = z 0 z 1 z 2 z 3 (1 − y 1 z 0 z 1 z 2 z 3 y 2 ) (1 − z 0 y 1 )(1 − z 0 z 1 y 2 )(1 − y 1 y 2 )(1 − z 0 z 1 z 2 z 3 )(1 − y 1 z 2 z 3 )(1 − y 2 z 3 ) . Applications We now apply the methods of Section 5.1 to staircase polygons with a staircase hole and to three-choice polygons. All the essential steps have already been seen in the derivation of the reciprocity result for staircase polygons of width three. They are 1. Set up a linear homogeneous diophantine system by decomposing the polyomino into width one rectangles and imposing the condition that the left and right sides of each rectangle have equal height. 2. Sort the unknowns into three classes, γ, δ and τ , according to whether they are constrained to be nonnegative, constrained to be positive or constrained by conditions on the other unknowns. 3. Use Gaussian elimination to remove the unknowns in τ . 4. Verify that the resulting system is solved by setting all members of γ equal to one and all members of δ equal to minus one. Apply Corollary 5.2 to obtain the self-reciprocity result for the reduced system. 5. Reintroduce the unknowns in the set τ by means of Lemma 4.3. We can define three widths for a staircase polygon with a staircase hole: the distance from the left edge of the figure to the left edge of the hole, k, the distance from the left edge of the figure to the right edge of the hole, ℓ, and the width of the entire figure, m. Note that 0 < k < ℓ < m. Recall that for staircase polygons the figures of width one were an exceptional case which did not obey the same reciprocity result as the general case. The staircase polygons with a hole of width one are also an exceptional case, which we must exclude. We thus impose the additional condition ℓ − k > 1. A figure with given k, ℓ and m is specified by the following dimensions, as shown in Figure 7 4. interior heights M j and M j below and above the hole, k ≤ j ≤ ℓ. N 1 N 2 N 3 N 4 N 5 N 6 M 0 M 1 M 2 M 6 M 7 N 1 N 2 N 3 N 4 N 5 N 6 H 3 H 4 H 5 M 2 M 3 M 4 M 5 H 3 H 4 M 2 M 3 M 4 M 5 M 6 M 2 M 3 M 4 M 5 (a) Staircase polygon with staircase hole, (k, ℓ, m) = (3, 5, 7) Three-choice polygons can be regarded as staircase polygons with a hole which doesn't close. The width k has the same meaning as above, ℓ denotes the ultimate horizontal extent of the branch of the figure above the hole, and m denotes the ultimate horizontal extent of the branch below the hole. Note that ℓ ≥ k and m > k. Again an exceptional case, m = k + 1, must be excluded. Hence we impose the restriction m > k + 1. The labeling of the vertical dimensions follows, with a few obvious modifications, the pattern of staircase polygons with a staircase hole and is shown in Figure 7(b). In particular, the heights M j within the hole are defined only for j ≤ min(ℓ, m). When ℓ = k the unknowns M j and H j do not appear. This special case is treated separately. N 1 N 2 N 3 N 4 N 5 N 6 M 0 M 1 M 2 M 6 M 7 N 1 N 2 N 3 N 4 N 5 N 6 H 3 H 4 H 5 M 2 M 3 M 4 M 5 H 3 H 4 M 2 M 3 M 4 M 5 M 6 M 2 M 3 As in the case of column-convex polygons, the standard statistics are linear in these heights. The (half-)vertical perimeter for staircase polygons with a staircase hole is given by n = M 0 + m−1 j=1 N j + M k + ℓ−1 j=k+1 H j(42) and the area is given by a = M 0 + k−1 j=1 (M j + N j ) + ℓ j=k+1 (N j + M j ) + ℓ−1 j=k (M j + N j ) + m−1 j=ℓ+1 (N j + M j ) + M m .(43) In what follows, we associate to the unknowns N i (resp. N i , H i , H i , M i , M i , M i ) the fugacities y i (resp. y i , w i , w i , z i , z i , z i ). Proposition 5.4. Let E k,ℓ,m (y, y, w, w, z, z, z) be the generating function for staircase polygons with a staircase hole where k, ℓ and m are the widths defined above. Then if ℓ − k > 1, the generating function E k,ℓ,m (y, y, w, w, z, z, z) is self-reciprocal, E k,ℓ,m (1/y, 1/y, 1/w, 1/w, 1/z, 1/z, 1/z) = − z k z ℓ m−1 j=1 (y j y j ) ℓ−1 j=k+1 (w j w j ) m−1 j=1 z j ℓ j=k+1 z j ℓ−1 j=k z j E k,ℓ,m (y, y, w, w, z, z, z).(44) Proof. The linear homogeneous diophantine system is the union of five sets of equations which we label L 1 -L 5 . The regions to the left and right of the hole give L 1 and L 2 , the regions below and above the hole give L 3 and L 4 and the inside of the hole gives L 5 : L 1 =          M 0 − M 1 − N 1 = 0 M j + N j − M j+1 − N j+1 = 0 for 1 ≤ j ≤ k − 2 M k−1 + N k−1 − M k − M k − M k − N k = 0 L 2 =      subject to the positivity constraints on the heights. Making appropriate substitutions to restore the unknowns in set τ , and using Lemma 4.3 we obtain eqn. (44). We now treat three-choice polygons. Proposition 5.5. Let E k,ℓ,m (y, y, w, w, z, z, z) be the generating function for three-choice polygons where k, ℓ and m are the widths defined above. Then if m − k > 1, the generating function E k,ℓ,m (y, y, w, w, z, z, z) satisfies a self-reciprocity condition which, when ℓ = k, takes the form E k,k,m (1/y, 1/y, 1/w, 1/w, 1/z, 1/z, 1/z) = Proof. It is simpler to treat the two cases ℓ = k and ℓ > k separately. The proofs follow very closely that of Proposition 5.4. As for column-convex polygons, the two propositions above may be extended to other statistics. Corollary 5.6. Let P be either of the sets staircase polygons with a staircase hole or threechoice polygons. Let P k,ℓ,m be the subset of figures in P with the widths k, ℓ and m defined as above. Then the generating function in P k,ℓ,m according to any statistics linear in the quantities N, N, H, H, M, M, M, is a self-reciprocal rational function (assuming ℓ − k > 1 for staircase polygons with a staircase hole and m − k > 1 for three-choice polygons). The half-horizontal perimeter for either of the sets P is given by m + ℓ − k. Using this in combination with Corollary 5.6, (42) and (43), we obtain the inversion relations specialized to horizontal and vertical perimeter, and area, which are listed in Table 1. The exceptional cases (ℓ − k = 1 and m − k = 1 respectively) can be computed explicitly by the methods of [7]. Discussion Each of the methods we have discussed for obtaining reciprocity or inversion relations has its own particular uses. For example, the method of Stroganov is suitable for lattice models in statistical mechanics which are characterized by a family of commuting transfer matrices. The Temperley methodology is mainly applicable to families of polygons that are columnconvex or nearly so. Stanley's method for obtaining reciprocity results apply to any problem defined by a system of linear homogeneous diophantine (LHD) equations, but the solutions to this system must be constrained by a system of simple inequalities of a certain form. It is probable that for many lattice models in statistical mechanics the low temperature expansion can be framed as an LHD-system. However, most are likely to require more general types of constraints than the simple inequalities of the directed polyomino problems we have considered. Likewise, the non-directed polygon problems that we have successfully treated using the Temperley methodology can be recast as LHD-systems with more complex constraints. How to handle such constraints is a worthy problem for future investigation. In recent work [8] this statistical mechanical language has been adapted for the enumeration of lattice paths, and may apply to polyomino problems as well. It is intriguing to speculate that the inversion relations found here may be connected with this approach. We have not searched for inversion relations for any polyomino problem in variables other than the natural variables for the problem. Yet the example of the Potts model demonstrates that such inversion relations may exist. It is also possible that symmetries in addition to the ones presented here can be found for some problems. It is our hope that such additional symmetries might lead to the solution of currently intractable problems. For the moment, we remark that the search for inversion and symmetry relations appears to provide a new method to tackle certain combinatorial problems. The degree of applicability of this method is still unclear. Figure 1 :Figure 2 : 12Classes Classes of column-convex polygons defined. The set of convex polygons is the intersection of the sets of row-and column-convex polygons. Subclasses of column-convex polygons include the bar-graphs which contain the bottom edge of the minimal bounding rectangle, and directed column-convex polygons whose bottom edge is a directed path. Some examples are shown inFigure 2.A second class of non-convex polygons is made up of four directed paths. A three-choice Figure 3 : 3Non-convex polyominoes with D m ( 0 ) = 1 .Figure 4 : 014The general form of the inversion relation is G(x, y) ± y α G(ǫx/y, 1/y) = RHS(26) where α is an integer and RHS is zero or some simple function. It is equivalent to a selfreciprocity relation of the formH m (1/y) ± ǫ m y m−α H m (y) = RHS. Whether the inversion relation is sufficient to compute the generating function depends on the value of the exponent α and on the degree of the denominator, D m (y). Direct proof of the denominator form can often be obtained. For Ferrers graphs, it is easily shown that D m (y) = (1 − y) m . For staircase polygons, one finds D m (y) = (1 − y) 2m−1 . The same denominator form holds for directed convex and convex polygons also. For the three-choice polygons and staircase polygons with a staircase hole it can be shown that the denominators are D m (y) =    (1 − y) 2m−1 (1 + y) 2m−7 m even (1 − y) 2m−1 (1 + y) 2m−8 m odd. A column-convex polygon Figure 5 : 5Construction of column-convex polygon by Hadamard products. and (30) express the vertical perimeter and the area in terms of the quantities N, N, S and S. They imply that the (half) vertical perimeter and area generating function H m (y, q) for column-convex polygons of width m is H m (y, q) = V m (N, S, N, S) where y k = z k = yq m−k and y k = z k = q −(m−k) . Theorem 4.1 then gives H m (1/y, 1/q) = − 1 yq m H m (y, q), which implies Example 5. 3 . 3Staircase polygons of width three can be characterized by the heights N 1 , N 2 , N 1 , N 2 , M 0 , M 1 , M 2 and M 3 , as shown in heights N j and N j of the lower and upper perimeter segments of the polygon, 1 ≤ j ≤ m − 1, 2. interior heights M j to the left and right of, and within, the hole, 0 ≤ j ≤ m, 3. heights H j and H j of the lower and upper perimeter segments of the hole, k + 1 ≤ j ≤ ℓ − 1, Three-choice polygon, (k, ℓ, m) =(2,6,5) Figure 7 : 7Labels for polyomino vertical heights ,ℓ,m (y, y, w, w, z, z, z). Table 1 : 1Summary of polyomino inversion relations 13 AcknowledgmentsWe have benefited from conversations with George Andrews, Richard Brak, Jean-Marie Maillard, Paul Pearce and Markus Vöge and from correspondence with Jean-Marc Fédou and Richard Stanley. We thank Iwan Jensen for providing us with his series data for staircase polygons with a staircase hole. AJG and WPO acknowledge support from the Australian Research Council. Hard hexagons: exact solution. R J Baxter, J. Phys. A: Math. Gen. 13R.J. Baxter, Hard hexagons: exact solution, J. Phys. A: Math. Gen., 13 (1980) L61- L70. Exactly Solved Models. R J Baxter, Fundamental Problems in Statistical Mechanics V; Proceedings of the 1980 Enschede Summer School. E.G.D. Cohen ed.AmsterdamNorth HollandR.J. Baxter, Exactly Solved Models. In: Fundamental Problems in Statistical Mechanics V; Proceedings of the 1980 Enschede Summer School, E.G.D. Cohen ed. (North Holland, Amsterdam, 1980) 109-141. Two-dimensional models in statistical mechanics. R J Baxter, Statistical Mechanics and Field Theory; Proceedings of the Seventh Physics Summer School. C.J. Burden eds.SingaporeWorld ScientificR.J. Baxter, Two-dimensional models in statistical mechanics. In: Statistical Mechanics and Field Theory; Proceedings of the Seventh Physics Summer School, ANU, 1994; V.V. Bazhanov and C.J. Burden eds. (World Scientific, Singapore, 1995) 129-167. Exactly Solved Models in Statistical Mechanics. R J Baxter, Academic PressLondonR.J.Baxter, Exactly Solved Models in Statistical Mechanics, (Academic Press, London, 1982). The inversion relation method for some two-dimensional exactly solved models in lattice statistics. R J Baxter, J. Stat. Phys. 28R.J. Baxter, The inversion relation method for some two-dimensional exactly solved models in lattice statistics, J. Stat. Phys., 28 (1982) 1-41. Convex polyominoes and heaps of segments. M Bousquet-Mélou, J. Phys. A: Math. Gen. 25M. Bousquet-Mélou, Convex polyominoes and heaps of segments, J. Phys. A: Math. Gen., 25 (1992) 1925-1934. A method for the enumeration of various classes of column-convex polygons. M Bousquet-Mélou, Discrete Math. 154M. Bousquet-Mélou, A method for the enumeration of various classes of column-convex polygons, Discrete Math., 154 (1996) 1-25. R Brak, J W Essam, A L Owczarek, From the Bethe Ansatz to the Gessel-Viennot Theorem. this issueR. Brak, J.W. Essam and A.L. Owczarek, From the Bethe Ansatz to the Gessel-Viennot Theorem, this issue. On the number of three-choice polygons. A Conway, A J Guttmann, M Delest, Math. Comput. Model. 26A. Conway, A.J. Guttmann and M. Delest, On the number of three-choice polygons, Math. Comput. Model., 26 (1997) 51-58. J M Fédou, Proceedings of the 4th Conference on Formal Power Series and Algebraic Combinatorics. P. Leroux and C. Reutenauerthe 4th Conference on Formal Power Series and Algebraic Combinatorics11Université du Québecà MontréalPublications du LACIMJ.M. Fédou, Fonctions de Bessel, empilements et tresses. In: Proceedings of the 4th Conference on Formal Power Series and Algebraic Combinatorics, P. Leroux and C. Reutenauer, eds., Publications du LACIM, Université du Québecà Montréal, 11 (1992) 189-202. Punctured polygons and polyominoes on the square lattice, submitted to. A J Guttmann, I Jensen, I G Enting, J. Phys. A. A.J. Guttmann, I. Jensen and I.G. Enting, Punctured polygons and polyominoes on the square lattice, submitted to J. Phys. A. Symmetry relations in exactly soluble models. M T Jaekel, J.-M Maillard, J. Phys. A: Math. Gen. 15M.T. Jaekel and J.-M. Maillard, Symmetry relations in exactly soluble models, J. Phys. A: Math. Gen., 15 (1982) 1309-1325. Inverse functional relation on the Potts model. M T Jaekel, J.-M Maillard, J. Phys. A: Math. Gen. 15M.T. Jaekel and J.-M. Maillard, Inverse functional relation on the Potts model, J. Phys. A: Math. Gen., 15 (1982) 2241-2257. Rigorous results for the number of convex polygons on the square and honeycomb lattices. K Y Lin, S J Chang, J. Phys. A: Math. Gen. 21K. Y. Lin and S. J. Chang, Rigorous results for the number of convex polygons on the square and honeycomb lattices, J. Phys. A: Math. Gen., 21 (1988) 2635-2642. The inversion relation. J.-M Maillard, J. Physique. 46J.-M. Maillard, The inversion relation, J. Physique, 46 (1985) 329-341. Crystal statistics. I. A two-dimensional model with an order-disorder transition. L Onsager, Phys. Rev. 65L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder tran- sition, Phys. Rev., 65 (1944) 117-149. Star-triangle relations in the exactly solvable statistical models. S V Pokrovsky, Yu A Bashilov, Comm. Math. Phys. 84S.V. Pokrovsky and Yu.A. Bashilov, Star-triangle relations in the exactly solvable sta- tistical models, Comm. Math. Phys., 84 (1982) 103-132. Simple derivation of the Baxter-model free energy. R Shankar, Phys. Rev. Lett. 47R. Shankar, Simple derivation of the Baxter-model free energy, Phys. Rev. Lett., 47 (1981) 1177-1180. Solvable q-state models in lattice statistics and quantum field theory. C L Schultz, Phys. Rev. Lett. 46C.L. Schultz, Solvable q-state models in lattice statistics and quantum field theory, Phys. Rev. Lett., 46 (1981) 629-632. Combinatorial Reciprocity Theorems. R P Stanley, Adv. Math. 14R.P. Stanley, Combinatorial Reciprocity Theorems, Adv. Math., 14 (1974) 194-253. Linear diophantine equations and local cohomology. R P Stanley, Inventiones Math. 68R.P. Stanley, Linear diophantine equations and local cohomology, Inventiones Math., 68 (1982) 175-193. . R P Stanley, Enumerative Combinatorics. IR.P. Stanley, Enumerative Combinatorics, Vol. I, (Wadsworth and Brooks/Cole, Mon- terey Calif. 1986). A new calculation method for partition functions in some lattice models. Yu G Stroganov, Phys. Lett. 74Yu.G. Stroganov, A new calculation method for partition functions in some lattice models, Phys. Lett., A74 (1979) 116-118. Combinatorial problems suggested by the statistical mechanics of domains and of rubber-like molecules. H N V Temperley, Phys. Rev. 103H.N.V. Temperley, Combinatorial problems suggested by the statistical mechanics of domains and of rubber-like molecules, Phys. Rev., 103 (1956) 1-16.
[]
[ "Model-Free Continuation of Periodic Orbits in Certain Nonlinear Systems Using Continuous-Time Adaptive Control", "Model-Free Continuation of Periodic Orbits in Certain Nonlinear Systems Using Continuous-Time Adaptive Control" ]
[ "Yang Li \nDepartment of Mechanical Science and Engineering\nUniversity of Illinois at Urbana-Champaign\n\n", "· Harry Dankowicz \nDepartment of Mechanical Science and Engineering\nUniversity of Illinois at Urbana-Champaign\n\n", "Yang Li \nIntroduction\n\n", "· Harry Dankowicz \nIntroduction\n\n" ]
[ "Department of Mechanical Science and Engineering\nUniversity of Illinois at Urbana-Champaign\n", "Department of Mechanical Science and Engineering\nUniversity of Illinois at Urbana-Champaign\n", "Introduction\n", "Introduction\n" ]
[]
This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package COCO demonstrate successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design.
10.1007/s11071-022-08059-1
[ "https://export.arxiv.org/pdf/2203.10306v4.pdf" ]
247,593,965
2203.10306
91bdff3cc30cf2eccaa13da60910891b1b567892
Model-Free Continuation of Periodic Orbits in Certain Nonlinear Systems Using Continuous-Time Adaptive Control Yang Li Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign · Harry Dankowicz Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Yang Li Introduction · Harry Dankowicz Introduction Model-Free Continuation of Periodic Orbits in Certain Nonlinear Systems Using Continuous-Time Adaptive Control Received: date / Accepted: datemanuscript No. (will be inserted by the editor)Control-based continuation · Model reference adaptive control · Persistent excitation This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package COCO demonstrate successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design. Introduction Control-based continuation provides a model-free approach for tracking periodic orbits of periodically excited nonlinear dynamical systems, independently of their orbital stability, under variations in experimentally accessible parameters p [3,4,6,8,13,15,19,23,25,28,32,33]. More advanced implementations also support tracking of special classes of periodic responses, e.g., those with vanishing phase lag relative to the excitation or corresponding to fold points in oneparameter bifurcation diagrams [1,21,22,24,29]. The approach embeds the experiment in a feedback control loop with control input u, parameterized by an experimentally accessible reference signal r and designed such that the response to zero control input (u = 0) is that of the original system. Provided that the closed-loop dynamics (including in the control input) exhibit asymptotic convergence to limit cycle dynamics for initial conditions in some region C , periodic reference signals in some set R, and parameter values in some region P, control-based continuation seeks to determine r ∈ R and p ∈ P such that lim t→∞ u(t) = 0 given an initial condition in C . If such can be found, then the corresponding steady-state dynamics must coincide with a periodic orbit of the original system, albeit with orbital stability properties determined by the feedback control design. A feedback control design that supports such a determination is said to be non-invasive, since it leaves the family of periodic orbits unchanged [5,9,30]. In an abstract sense, setting aside any concerns about accuracy and precision, the control-based continuation approach is thus straightforward to implement (but see [7,26,27]). Firstly, formulate a non-invasive feedback control design, preferably with some a priori understanding of how to interpret the relationship between the sought reference input and the desired, but unknown, limit cycle dynamics. Secondly, while maintaining dynamics in C , perform iter-arXiv:2203.10306v4 [math.OC] 30 Sep 2022 ative updates on r and p until the control input approaches 0 asymptotically. Of course, in order to result in a finitedimensional problem, the latter must be preceded by discretization of the periodic reference input and a suitably chosen finite-time approximation of the periodic, steady-state control input. In practice, the construction of a non-invasive feedback control design may necessitate some a priori knowledge of the dynamics near the sought periodic orbits, particularly in order to ensure exponentially asymptotically stable limit cycle dynamics for the closed-loop system. This is the case for non-adaptive linear feedback control, for which the gains must be chosen to ensure that all associated Floquet multipliers lie inside the unit circle [1,9]. Additionally, particular feedback designs may fail to maintain dynamics in C or even guarantee bounded response of the closed-loop system with potentially disastrous consequences. The latter is true for linear feedback control. Finally, even as a non-invasive design may have been found for a particular parameter region P, it may need to be retuned repeatedly to accommodate larger variations in p. Once one moves beyond non-adaptive linear feedback control, a theoretical analysis is often restricted to particular classes of problems. In two recent papers [17,18], the present authors investigated the use of adaptive feedback control to overcome the challenges outlined in the previous paragraph, specifically for tracking of fixed points in a class of single-input-single-output discrete-time dynamical systems and periodic orbits in a class of linear systems with matched uncertainties. As shown there, provable performance bounds were accompanied by a significant reduction in tuning effort. This came at the expense, however, of non-exponential rates of convergence, as well as a requirement that the frequency content of the reference input be sufficiently rich to result in persistent excitation of the closedloop dynamics. In this paper, we consider tracking of periodic orbits using adaptive feedback control for a larger class of dynamical systems than in our previous work, assuming nonlinearities of known structure in the case of problems of arbitrary dimension, and restricting attention to scalar problems in the case of uniformly bounded, unmodeled nonlinearities with uniformly bounded first-order partial derivatives. In both cases, we assume matched uncertainty, i.e., that an appropriately chosen control input could cancel the influence of parameter uncertainty. We rely on versions of the model-reference adaptive control approach and, in the case of nonlinearities of known structure, derive guaranteed performance bounds and demonstrate robustness to additive uniformly bounded disturbances. Integration with the COCO continuation package shows successful tracking of stable and unstable periodic orbits with a minimum of manual tuning. In the case of unmodeled nonlinearities, it also highlights practical challenges associated with slow rates of convergence of the closed-loop dynamics. The remainder of this paper is organized as follows. The class of nonlinear systems of initial interest is defined in Section 2, which also includes a discussion of a corresponding non-invasive, non-adaptive, linear feedback control design. Section 3 describes a proposed non-invasive modelreference adaptive control algorithm and associated performance bounds. Numerical simulations in Section 4 illustrate the performance of the controller at a fixed parameter value, while its use for control-based continuation is explored in Section 5 using an implementation in the COCO software package [10]. Robustness of the control design under unmodeled, uniformly bounded, additive disturbances is considered in Section 6. For a class of scalar systems, Section 7 relaxes the assumption that the structure of the nonlinearity be known to the control design and demonstrates the application of a proposed model-reference adaptive control design for parameter continuation. A brief concluding discussion follows in Section 8. Problem formulation Model class Following the discussion in [18] for a class of linear systems, consider the dynamical system, q = Aq + b θ T Q(t, q) + σ ,(1) where A ∈ R n×n is a known constant Hurwitz matrix, b is a known constant vector, θ ∈ R m is an unknown constant vector, σ is a known periodic function of period T , and Q(t, q) represents a known nonlinear function of t and q that is periodic in t with period T . The model form (1) reduces to that in [18] when Q(t, q) ≡ q. For other choices of Q(t, q), (1) captures problems with a one-dimensional nonlinearity of arbitrary known complexity and with additional disturbance σ along the same direction as the nonlinearity. In the case that Q does not depend explicitly on t (this is the case considered in the numerical example in Section 4), the response is driven by the known signal σ . For n = 2, A = 0 1 −ω 2 0 −2ζ ω 0 and b = 0 1(2) (1) models a single-degree of freedom oscillator with natural frequency ω 0 and damping constant ζ that is acted upon by an additional nonlinearity and periodic excitation with angular frequency 2π/T . We may imagine the experimental determination of families of periodic responses under variations in T as one goal of the control-based continuation analysis. For mechanical systems with more than one degree of freedom, the form of (1) limits consideration to problems with only one source of nonlinearity and an excitation that is "parallel" to the nonlinearity. An example is given by the parametrically excited two-degree-of-freedom model [34] obtained with A =      0 1 0 0 − k 01,lin +k 12 m 1 − c 01 +c 12 m 1 k 12 m 1 c 12 m 1 0 0 0 1 k 12 m 2 c 12 m 2 − k 02 +k 12 m 2 − c 02 +c 12 m 2     (3) in terms of the linear stiffness coefficients k 02 , k 12 , and k 01,lin and damping coefficients c 02 , c 12 , and c 01 , b = 0 1 0 0 T and θ = − 1 m 1   k 01,nlin k PE,1,lin k PE,1,nlin  (4) in terms of the unknown stiffness coefficients k 01,nlin , k PE,1,lin , and k PE,1,nlin , Q(t, q) = q 3 1 q 1 cos Ω PE t q 3 1 cos Ω PE t T(5) in terms of the excitation frequency Ω PE , and σ ≡ 0. Here, m 1 and m 2 are two lumped masses along a clamped-free cantilever beam such that a harmonic current running through a coil imposes a time-varying, restoring force on the first mass that is nonlinear in displacement. Again, we may consider experimental continuation of periodic responses under variations in Ω PE . Control objectives Suppose that there exists a locally unique periodic (but a priori unknown) solution q * of period T to (1). Due to the nonlinearity, q * (t) generally contains frequencies other than ω = 2π/T . The stability of q * is determined by the eigenvalues (the Floquet multipliers of q * ) of the monodromy matrix Φ(T ), obtained froṁ Φ = A + bθ T Q q (t, q * (t))) Φ, Φ(0) = I,(6) where the subscript q denotes the Jacobian with respect to q. As long as these eigenvalues lie inside the unit circle in the complex plane, then q * is locally asymptotically stable. Asymptotic stability is not assumed, however, as we seek to use control-based continuation to locate and track such periodic solutions, even if unstable. Let r(t) be a reference periodic function of period T and define x = q − r. It follows thaṫ x = Ax + bθ T (Q(t, x + r) − Q(t, r)) + g,(7) where g = −ṙ + Ar + b θ T Q(t, r) + σ(8) is also periodic with period T and identically equal to 0 for r ≈ q * if and only if r = q * . By definition, the periodic function x * = q * − r satisfies (7) and is locally asymptotically stable if all eigenvalues of Φ(T ) are inside the unit circle in the complex plane. In the special case that r = q * , it follows that x * (t) is identically equal to 0. We consider the introduction of a matched scalar control input u as shown below, q = Aq + b u + θ T Q (t, q) + σ ,(9) with the aim of having u determined by q and r, such that u converges to a periodic steady-state signal that is uniquely determined by r and equal to 0 for r ≈ q * and q(0 ) ≈ q * (0) if and only if r = q * , in which case q(t) → q * (t) as t → ∞. We refer to such a control design as non-invasive along the sought periodic orbit. By definition of x, we obtaiṅ x = Ax + bθ T (Q(t, x + r) − Q(t, r)) + bu + g(10) and it follows that the construction of a non-invasive design along the sought periodic orbit needs to ensure that u(t) → 0 for r ≈ q * and x(0) ≈ 0 if and only if g(t) ≡ 0, and that x(t) → 0 in this case. Proportional feedback As an example, let u = −k T (Q(t, q) − Q(t, r))(11) for some to-be-determined constant vector k. Substitution yieldṡ x = Ax + b(θ − k) T (Q(t, x + r) − Q(t, r)) + g.(12) It follows that u(t) → 0 for r ≈ q * and x(0) ≈ 0 if g(t) ≡ 0 provided that all the eigenvalues of the monodromy matrix Φ(T ) lie inside the unit circle, where Φ(t) in this case is governed bẏ Φ = A + b(θ − k) T Q q (t, r) Φ, Φ(0) = I,(13) and that x(t) → 0 in this case. The control design in (11) is clearly non-invasive along the periodic orbit in the case when k = θ . Importantly, given the local character of this control law, there is no a priori degree of closeness that will guarantee the convergence of x and u, nor are bounds available on transient deviations from 0. It is clear that x(t) cannot converge to 0 if g = 0 under the control law (11). Under exceptional circumstances, it is still possible that u(t) → 0 if Q(t, x + r) − Q(t, r) converges to a signal in the orthogonal complement of k, in violation of our articulated objective. This possibility may be excluded on a case-by-case basis or, in the case of (11), eliminated entirely by requiring that x(t) → 0 if and only if g(t) ≡ 0. Model-reference adaptive control strategy In the absence of knowledge about θ , the selection of k in (11) such that all Floquet multipliers have magnitude less than 1 is, at best, trial-and-error. As an alternative, we consider a form of model-reference adaptive control [14] that relies on an adaptive estimate of θ to achieve the stated objective. We show that this is non-invasive along the sought periodic orbit under generic conditions. As a side benefit, we obtain guaranteed bounds on the transient and steadystate dynamics. Control design To this end, consider the control law u = −θ T (Q(t, q) − Q(t, r)) ,(14) whereθ (t) denotes a time-dependent estimate of θ , such thaṫ θ = −Γ e T PbQ(t, q),(15) defines the adaptive dynamics in terms of the adaptation gain Γ > 0. As usual, P is a positive definite matrix that satisfies the algebraic Lyapunov equation PA + A T P = −S for some positive definite matrix S. The prediction error e = x m − x is defined in terms of the reference state x m governed by the differential equatioṅ x m = Ax m + bθ T Q(t, r) + g,(16) whereθ =θ − θ is the estimation error. It follows thaṫ e = Ae + bθ T Q(t, q).(17) Notably, whileθ appears explicitly in (16), terms involving θ cancel out of the sum of the last two terms, ensuring that (15) and (16) are implementable. Lyapunov analysis Now let B denote a ball of Euclidean radius R such that θ ∈ B, and assume that the initial conditions are chosen so that x m (0) = x(0), i.e., e(0) = 0, andθ (0) ∈ B. Then, the Lyapunov function V = e T Pe + 1 Γθ Tθ (18) satisfieṡ V = −e T Se ≤ 0,(19) and, consequently, V (t) ≤ V (0) = 1 Γ θ (0) 2 2 ≤ 4R 2 Γ .(20) It follows that e(t) 2 ≤ 2R λ min (P)Γ ,(21) where λ min (P) is the smallest eigenvalue of P, and θ (t) 2 ≤ 2R.(22) Since Q(t, r) and g(t) are also uniformly bounded, it follows from (16) and the fact that A is Hurwitz that x m is bounded, and consequently, that x and q are bounded. Equation (14) then implies that u is bounded. By the above analysis, e andė are both bounded. This implies thatV is bounded and, by Barbalat's lemma [14], thatV (t) → 0, which in turn implies that e(t) → 0, i.e., that x m (t) → x(t) and, consequently,θ (t) → 0. Moreover, sincë e is bounded, it follows thatė(t) → 0 and, consequently, θ T (t)Q(t, q) → 0. Persistent excitation In order to conclude thatθ (t) → 0, suppose that the reference input r is chosen so that the signal Q(t, q) is persistently exciting [12,20], i.e., that the smallest eigenvalue of the (at least positive semi-definite) matrix t+T t Q(τ, q(τ))Q T (τ, q(τ)) dτ (23) is bounded from below by a positive number α for all t. Although we cannot confirm the persistence of excitation of the signal Q(t, q) a priori, we may consider the integral obtained by replacing q(t) with r(t) in (23), since r is assumed to be close to q * in practice. Since the integrand then becomes periodic, it suffices to compute its value for t = 0. It now follows from the mean-value theorem and the observation thatθ (t) → 0 that for every τ ∈ [t,t + T ], there ex- ists a σ ∈ [t,t + T ] such thatθ (τ) −θ (t) = τθ (σ ) ≤ T M(t) for some function M(t) → 0 as t → ∞. Consequently, t+T tθ T (τ)Q(τ, q(τ))Q T (τ, q(τ))θ (τ) dτ =θ T (t) t+T t Q(τ, q(τ))Q T (τ, q(τ)) dτ θ (t) + ε(t),(24) where ε(t) → 0 as t → ∞. Consequently, t+T tθ T (τ)Q(τ, q(τ))Q T (τ, q(τ))θ (τ) dτ − ε(t) ≥ α θ (t) 2 2 .(25) Since the left-hand side converges to 0 and the right-hand side is bounded from below by 0, it follows thatθ (t) → 0. Sinceθ (t) → 0 andθ is bounded, the classical result of Desoer [11] and Solo [31] applied to the governing equatioṅ x = Ax − bθ T (Q(t, x + r) − Q(t, r)) + g(26) implies that x(t) and, consequently, u(t) both converge to periodic steady-state responses that are uniquely determined by g, and that x(t) → 0 if and only if g ≡ 0. We conclude that the model-reference adaptive control design is non-invasive along the sought periodic orbit. Generically, we again expect that u(t) 0 when g = 0. Numerical simulations In this section, we explore the predictions from Section 3 regarding the boundedness of the prediction and estimation errors, as well as the convergence ofθ (t) to 0 under suitable conditions on the reference input r. As an example, consider the dynamical system (1) with A = 0 1 −1.5 −0.5 , b = 0 1 , θ =   0.5 0.4 −0.04   ,(27) Q(t, q) = q 1 q 2 q 3 1 T , and σ = sin ωt. This corresponds to a harmonically excited Duffing oscillator in which the damping, stiffness, and nonlinearity coefficients are assumed unknown to the control design. For ω = 1, there exists a periodic steady-state response q * given by q * 1 (t) ≈ −0.9928 cost + 2.9876 sint + 0.0336 cos 3t − 0.0255 sin 3t − 0.0005 cos 5t + 0.00002 sin 5t, (28) q * 2 (t) ≈ 2.9876 cost + 0.9928 sint − 0.0765 cos 3t − 0.1008 sin 3t + 0.0001 cos 5t + 0.0025 sin 5t. It is easy to check that Q(t, q * ) is persistently exciting since the integral in (23) is independent of t and positive definite with smallest eigenvalue approximately equal to 3.2. Without loss of generality, we restrict attention to functions r(t) chosen so thatṙ − Ar is parallel to b for all time, since this must be true of the desired reference input for which g ≡ 0. Consider the two choices of r(t) = q * (t) and r(t) = cost + sint cost − sint .(30) The steady-state solution toẋ = Ax + g is then given by x(t) = 0 and the periodic function shown in Fig. 1, respectively. The nonlinearity Q(t, x + r) is persistently exciting also in the latter case, since the smallest positive eigenvalue of the integral (23) is again independent of t and equal to 1.0. Steady-state response u Fig. 1 The steady-state responses of x(t) and u(t) under the proposed model-reference adaptive control strategy with r(t) given in (30). Suppose that q(0) = 0 andθ (0) = 0, and let P = 8/3 1/3 1/3 5/3 , Γ = 1.(31) The system response under the proposed model-reference adaptive control strategy with r(t) = q * (t) up to the fifth harmonic is shown in Fig. 2. It is seen that e(t) and θ (t) both go to 0 as t → ∞. Similarly, to within the resolution of the first five harmonics, x(t) and u(t) also converge to 0. The system response with r(t) given in (30) is shown in Fig. 3. It is seen that again, e(t) and θ (t) both converge to 0 as t → ∞. Since r deviates from a periodic steadystate response of the system, x(t) and u(t) converge to nonzero periodic responses, as predicted by the analysis in the previous section. For each of these two cases, Fig. 4 shows the time histories of the smallest eigenvalue of the integral in (23). Since these are bounded below by some positive numbers (and, indeed, converge to the predicted values obtained by substituting q = x + r with x being the unique solution toẋ = Ax + g), Q(t, q) is persistently exciting in both cases. This is also consistent with the general theory that ensures persistent excitation provided that the frequency content exceeds a multiple of the system dimension. Here, this is guaranteed by the nonlinearity. Control-based continuation The theoretical treatment in Section 3 shows that we may identify an a priori unknown periodic response q * by the System response kxk 2 kuk 2 Fig. 3 The system response under the proposed model-reference adaptive control strategy with r(t) given in (30). The predicted bounds on e(t) 2 and θ (t) 2 approximately equal 1.28 and 1.03, respectively. fact that u(t) → 0 as t → ∞ provided that the reference input r happens to equal q * . In this section, we use Newton's method to iteratively improve upon the reference input in order for the steady-state control input to fall within a threshold distance from 0, thereby obtaining an approximation of q * . As in the previous section, we restrict attention to r cho- Smallest eigenvalue r = q $ r 6 = q $ Fig. 4 The smallest eigenvalue of the integral in (23) as a function of t, with r(t) ≈ q * (t) given by the up-to-fifth-harmonic approximation in (28)-(29) and (30), respectively, under the model reference adaptive control. sen so thatṙ − Ar is parallel to b, i.e., such that r may be parameterized by a scalar periodic function of the same dimension as the control input. We proceed to use simulations of the closed-loop dynamics to estimate the coefficients of a truncated Fourier series of the steady-state control input and their derivatives with respect to the corresponding coefficients of r and modify the coefficients of r accordingly. We combine the application of Newton's method with a pseudo-arclength continuation algorithm [10] in order to trace q * under variations in a model parameter, say ω in the example in Section 4, also past geometric folds where the assumption of existence and local uniqueness of q * for fixed ω would fail. Here, an approximation to the tangent line of the graph of (ω, q * ) at a particular point on this graph is used to construct a predictor (ω, r) some distance h from (ω, q * ) along the tangent line. We proceed to require that all subsequent iterates of Newton's method lie on a line perpendicular to the tangent and intersecting the tangent at the predictor. We initialize the overall algorithm at some point (ω, q * ) obtained for example using forward simulation in the case that the corresponding q * is asymptotically stable. Importantly, convergence of the Newton iterations is independent of the open-loop stability of q * . Also, since r remains close to q * , g is close to 0 for all iterations and the adaptive control strategy ensures the desired convergence. Consistent with the implementation of the continuation algorithm in a physical experiment, we assume no direct control over the state q, for example its value at any moment in time. Other than the first simulation of the closed-loop dynamics, we initialize q at its terminal value in the preceding simulation. We similarly initializeθ for each simulation at its terminal value in the preceding simulation, thus ensuring close tracking by x of the reference state x m , since θ is assumed to vary smoothly with the model parameter. Since it is not possible to obtain an exact match of r and q * , given the presence of harmonics of all orders, we select a truncation order that (empirically) yields sufficient information about q * while also ensuring that Q(t, q) is persistently exciting. We avoid aliasing by applying the discrete Fourier transform to a fine sample of a period of the steady-state control input. Figure 5 shows the successful application of the controlbased continuation algorithm, implemented in the software package COCO [10], to the example in Section 4 under variations in ω and with control parameters P and Γ as given there. Throughout continuation, we approximate u by its truncated Fourier series up to the fifth harmonic and iteratively update the corresponding coefficients of r. In each iteration, and when approximating derivatives with respect to the Fourier coefficients of r, the Matlab integrator ode45 (with relative tolerance 10 −8 and absolute tolerance 10 −10 ) is used to simulate the closed-loop transient dynamics for 10 periods, followed by a sampling of the control input during one additional period of simulation. We consider r to have converged to q * when the norm of the Fourier coefficients of the steady-state control input is smaller than 10 −6 . The step size along the graph (ω, q * ) is adaptively determined by COCO using default settings. As seen in the figure, the algorithm is able to trace out the solution branch independent of the open-loop stability of the periodic solutions (which are unstable along the middle branch in the range of coexisting periodic solutions). kq $ 1 (!)k L 1 ; kr 1 (!)k L 1 Upward sweep Downward sweep MRAC Fig. 5 A comparison between branches of periodic solutions of the harmonically excited nonlinear Duffing oscillator obtained using upward and downward sweeps in the excitation frequency ω (represented by q * 1 (ω) L∞ ), and those obtained using the control-based continuation algorithm with the proposed model reference adaptive control strategy (represented by r 1 (ω) L∞ ). Robustness In practice, the presence of unmodeled disturbances may limit the utility of the proposed methodology, both in terms of the expected convergence of u(t) to 0 as t → ∞ when the reference signal r is chosen appropriately, and in terms of any guarantees on a bounded response. Consider, for example, the introduction in the closedloop dynamics of an additive, uniformly bounded, unknown disturbance h(t, q): q = Aq + b u + θ T Q (t, q) + σ + h(t, q),(32) where u is given in (14). With x = q − r, e = x m − x, and x m again governed bẏ x m = Ax m + bθ T Q(t, r) −ṙ + Ar + b θ T Q(t, r) + σ ,(33) it then follows thaṫ e = Ae + bθ T Q (t, q) − h(t, q).(34) Given the Lyapunov function in (18), the adaptation law (15) implies thaṫ V = −e T Se + 2e T Ph(t, q).(35) Because of the second term on the right-hand side, we can no longer claim thatV ≤ 0 or e → 0 as t → ∞. It is still the case that e is bounded, however. Indeed, if h b denotes an upper bound for h(t, q) , theṅ V (t) ≤ −λ min (S) e 2 + 2 e λ max (P)h b .(36) Consequently,V < 0 when e > 2 λ max (P) λ min (S) h b .(37) While this implies that e is eventually upper bounded by the right hand side of (37), boundedness ofθ does not follow. To achieve this, and by implication the boundedness of x, q, and u, we may modify the adaptation law using a projection operator [14] θ = Γ Proj B θ , −e T PbQ (t, q) ,θ (0) =θ 0 .(38) With this modification, the proposed model-reference adaptive control design guarantees a bounded response given a uniformly bounded additive disturbance h(t, q), a reassuring prediction for any actual implementation in a physical experiment. In the special case that h(t, q) is periodic in t with period T , the unique, locally attractive limit cycle dynamics of the closed-loop system obtained when h ≡ 0 and r ≈ q * persist for sufficiently small h ∞ . In this case, if r is chosen so thaṫ r = Ar + b θ T Q(t, r) + σ + h(t, r) (39) theṅ x = Ax − bθ T (Q(t, x + r) − Q(t, r)) + h(t, x + r) − h(t, r)(40) and we conclude that x(t) ≡ 0 along the perturbed limit cycle and q → r locally if and only if r is chosen in this way. In the case that h is not periodic in t, we cannot expect local persistence. In this case, the control-based continuation algorithm fails to trace out the solution branch, since it is no longer the case that u(t) converges to 0 as t → ∞ when r = q * if the disturbance persists. Given additional information about the disturbance, it may be possible to substitute the requirement that u(t) → 0 with a condition on a suitably filtered version of u(t). In either case, bounded performance is guaranteed. Systems with unmodeled nonlinearities In this section, we attempt to relax the expectation that the form of the nonlinearity be known to the control design and accessible to the feedback law. As a first step in this direction, consider the example systeṁ q = −q + sin q + sin ωt + u, q ∈ R.(41) When u = 0, there exists a periodic solution q * (t) for ω = 1 given by q * (t) ≈ −0.9849 cost + 0.1160 sint + 0.0053 cos 3t + 0.0115 sin 3t + 0.0002 cos 5t − 0.0003 sin 5t. Next, let x = q − r and u = −kx, and consider the closedloop dynamics of the systeṁ x = −(1 +k)x + g,k = Γ x 2(43) where g = sin q + sin ωt −ṙ − r for some periodic reference input r. We note that x ≡ 0 is a solution of this system provided that g| q=r ≡ 0, i.e., that r is a periodic solution of the open-loop dynamics. Indeed, when this is not the case,k is non-decreasing and must grow beyond all bounds as t → ∞. Given the Lyapunov function V = x 2 + 1 Γk 2 ,(45) it follows thaṫ V (t) = −2x 2 + 2xg.(46) Let g b denote an upper bound on the magnitude of g. Then, sincek is a non-decreasing function of time, |x(t)| ≤ max{g b , |x(0)|} + |k(0)|/ √ Γ .(47) Indeed, fork(0) > 0, |x(t)| cannot exceed max{g b , |x(0)|}, since |x(t * )| > g b at some instant t * implies thatV (t * ) < 0 and, consequently, that x(t * )ẋ(t * ) < 0. If, instead,k(0) < 0, then V cannot exceed max{g 2 b , x 2 (0)} +k 2 (0)/Γ and the bound follows from the inequality √ a 2 + b 2 ≤ |a|+|b|. Since r is bounded, this is also true of q. If g q=r (t) does not vanish identically,k must grow without bounds. In this case, x(t),ẋ(t) → 0 (due to the boundedness of g and its partial derivatives with respect to q and t) and, consequently, that q(t) → r(t) and u(t) → −g q=r (t) as t → ∞, independently of initial conditions. The same conclusions follow if g q=r (t) ≡ 0 although, in this case,k saturates at some finite positive value. These predictions are confirmed by the results of numerical simulations shown in Figs. 6 and 7 obtained with r(t) given by the right-hand side of (42) and r(t) = cos(t) + sin(t), respectively. In each case, Γ = 100 and q(0) =k(0) = 0. The observations for the example system (41) generalize to a scalar system of the forṁ q = aq + b (kq + f (t, q) + σ + u) ,(48) where a < 0 and b = 0 are known constants, k is an unknown constant, and f (t, q) represents a uniformly bounded, unmodeled nonlinearity with uniformly bounded first-order partial derivatives. This reduces to the example system in (41) when a = −1, b = 1, k = 0, and f (t, q) = sin q. System response With the introduction of a control input u = −kx in terms of the deviation x = q − r, it follows thaṫ x u u + gx = (a − bk)x + g,(49) wherek =k − k and g = (a + bk)r + b ( f (t, q) + σ ) −ṙ.(50) If we leṫ k = Γ bx 2(51) and define the Lyapunov function V = x 2 + 1 Γk 2 ,(52) it follows by the same argument as for the example that |x(t)| ≤ max{−g b /a, |x(0)|} + k (0) − k / √ Γ ,(53) where g b denotes the upper bound for |g| ∞ . By the uniform boundedness of f and its partial derivatives, we again conclude that x(t),ẋ(t) → 0 and, consequently, q(t) → r(t) and u(t) → −g q=r (t) as t → ∞. We proceed to consider the application of the controlbased continuation paradigm to the system (41) with Γ = 1. Here, the initial value of the adaptation parameterk(0) is set to 0 in each simulation. Throughout continuation, we approximate u by its truncated Fourier series up to the fifth harmonic and iteratively update the corresponding coefficients of r. In each iteration, and when approximating derivatives with respect to the Fourier coefficients of r, the Matlab integrator ode45 (with relative tolerance 10 −8 and absolute tolerance 10 −10 ) is used to simulate the closed-loop transient dynamics for 10 periods, followed by a sampling of the control input during one additional period of simulation. We consider r to have converged to q * when the norm of the Fourier coefficients of the steady-state control input is smaller than 10 −3 . The step size along the graph (ω, q * ) is adaptively determined by COCO using default settings. The result is shown in Fig. 8. Fig. 8 A comparison between the branch of periodic solutions of the scalar system defined in (43) obtained using an upward sweep in ω (represented by q * (ω) L∞ ), and that obtained using controlbased continuation under the proposed model-reference adaptive control strategy (represented by r(ω) L∞ ). Notably, the convergence criterion tolerance is here set orders of magnitude larger than the COCO default (which is 10 −6 ). Indeed, we observe that x(t) remains close to 0 throughout continuation, resulting in slow dynamics of the adaptation parameterk and, consequently, slow rates of convergence of u(t) to −g(t), as required by the control-based continuation paradigm. Although we might be able to improve upon this state of affairs by settingk(0) to a larger number, this would likely produce large transient dynamics, including in the control input, making this impractical in physical experiments. Conclusion As shown in the previous sections, although adaptive control designs may be proposed for control-based continuation of periodic orbits, significant effort may be required to prove their non-invasiveness and ensure bounded performance and robustness to disturbances, if at all possible. Nevertheless, the benefits over non-adaptive control designs may be equally significant, especially in examples where the linearized dynamics near the sought periodic orbits vary greatly over the parameter range of interest. It may be reasonably argued that a great number of challenges must be overcome in order to fully realize the potential of control-based continuation already without considering the complexity of self-tuning feedback control designs. Indeed, recent work by Renson, Barton, Sieber, and their collaborators has explored the merger of control-based continuation techniques with data-based approaches such as Gaussian process regression [21,26] for estimating the local manifold geometry and adaptive filters [2] that update the reference input on the fly. The development of the CON-TINEX toolbox [27] for COCO also highlighted the many challenges associated with continuation in the presence of measurement noise. It would be worthwhile to consider if algorithms for data-based adaptive accommodation of noise and uncertainty along families of periodic orbits could be co-designed with the adaptive feedback control used to locate individual orbits in order to improve overall performance. For those inclined to explore more general classes of nonlinear problems, for example multi-dimensional systems with unmodeled nonlinearities, we refer to preliminary work described in Chapter 5 of the first author's doctoral dissertation [16]. There, promising numerical results and guaranteed boundedness of the closed-loop system response (for some proposed choices of adaptive feedback designs) do not compensate for the lack of proofs of the existence of asymptotically stable limit cycle dynamics, nor the independence of the steady-state control input from initial conditions. Much work remains to be done. Fig. 2 2The system response under the proposed model-reference adaptive control strategy with r(t) ≈ q * (t) given by the up-to-fifth-harmonic approximation in (28)-(29). The predicted bounds on e(t) 2 and θ (t) 2 approximately equal 1.28 and 1.03, respectively. Fig. 6 6Time histories ofk(t), x(t), u(t), and u(t) + g(t) of the system defined in (41) at ω = 1 under the proposed model-reference adaptive control strategy, where r(t) ≈ q * (t) given by the up-to-fifth-harmonic approximation in (42). Fig. 7 7Time histories ofk(t), x(t), u(t), and u(t) + g(t) of the system defined in (41) at ω = 1 under the proposed model-reference adaptive control strategy, where r(t) = cos(t) + sin(t). Statements and DeclarationsFunding This work is supported by Agriculture and Food Research Initiative Competitive Grant no. 2014-67021-22109 from the USDA National Institute of Food and Agriculture. Part of the editing of this paper was performed while the second author served at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.Competing Interests The authors have no relevant financial or non-financial interests to disclose.Author ContributionsThe authors contributed equally to the conception and design of this research, and to the writing of the manuscript. Implementation of algorithms in code and generation of numerical results was performed by Yang Li. Both authors read and approved the final manuscript.Data Availability Datasets generated and analyzed during this study are available upon request from the authors. Matlab scripts sufficient to generate this data will be posted to an open-source archive. A consistency analysis of phase-locked-loop testing and controlbased continuation for a geometrically nonlinear frictional system. G Abeloos, F Müller, E Ferhatoglu, M Scheel, C Collette, G Kerschen, M Brake, P Tiso, L Renson, M Krack, DOI10.1016/j.ymssp.2022.108820Mechanical Systems and Signal Processing. 170Abeloos, G., Müller, F., Ferhatoglu, E., Scheel, M., Collette, C., Kerschen, G., Brake, M., Tiso, P., Renson, L., Krack, M.: A consistency analysis of phase-locked-loop testing and control- based continuation for a geometrically nonlinear frictional sys- tem. Mechanical Systems and Signal Processing 170 (2022). DOI 10.1016/j.ymssp.2022.108820 Stepped and swept control-based continuation using adaptive filtering. G Abeloos, L Renson, C Collette, G Kerschen, 10.1007/ s11071-021-06506-zNonlinear Dynamics. 1044Abeloos, G., Renson, L., Collette, C., Kerschen, G.: Stepped and swept control-based continuation using adaptive filtering. Nonlinear Dynamics 104(4), 3793-3808 (2021). DOI 10.1007/ s11071-021-06506-z Control-based continuation: Bifurcation and stability analysis for physical experiments. D Barton, 10.1016/j.ymssp.2015.12.039Mechanical Systems and Signal Processing. 84Barton, D.: Control-based continuation: Bifurcation and stability analysis for physical experiments. Mechanical Systems and Signal Processing 84, 54-64 (2017). DOI 10.1016/j.ymssp.2015.12.039 Numerical continuation in a physical experiment: Investigation of a nonlinear energy harvester. D Barton, S Burrow, 10.1115/1.4002380Journal of Computational and Nonlinear Dynamics. 61Barton, D., Burrow, S.: Numerical continuation in a physical ex- periment: Investigation of a nonlinear energy harvester. Journal of Computational and Nonlinear Dynamics 6(1) (2011). DOI 10.1115/1.4002380 Control-based continuation for investigating nonlinear experiments. D Barton, B Mann, S Burrow, DOI10.1177/1077546310384004Journal of Vibration and Control. 184Barton, D., Mann, B., Burrow, S.: Control-based continuation for investigating nonlinear experiments. Journal of Vibration and Control 18(4), 509-520 (2012). DOI 10.1177/1077546310384004 Systematic experimental exploration of bifurcations with noninvasive control. D Barton, J Sieber, DOI10.1103/PhysRevE.87.052916Nonlinear, and Soft Matter Physics. 875Physical Review E -StatisticalBarton, D., Sieber, J.: Systematic experimental exploration of bi- furcations with noninvasive control. Physical Review E -Sta- tistical, Nonlinear, and Soft Matter Physics 87(5) (2013). DOI 10.1103/PhysRevE.87.052916 Robustness of nonlinear parameter identification in the presence of process noise using control-based continuation. S Beregi, D Barton, D Rezgui, S Neild, 10.1007/s11071-021-06347-wNonlinear Dynamics. 1042Beregi, S., Barton, D., Rezgui, D., Neild, S.: Robustness of nonlin- ear parameter identification in the presence of process noise using control-based continuation. Nonlinear Dynamics 104(2), 885-900 (2021). DOI 10.1007/s11071-021-06347-w Experimental bifurcation analysis of an impact oscillator-determining stability. E Bureau, F Schilder, M Elmegård, I Santos, J Thomsen, J Starke, DOI10.1016/j.jsv.2014.05.032Journal of Sound and Vibration. 33321Bureau, E., Schilder, F., Elmegård, M., Santos, I., Thomsen, J., Starke, J.: Experimental bifurcation analysis of an impact oscillator-determining stability. Journal of Sound and Vibration 333(21), 5464-5474 (2014). DOI 10.1016/j.jsv.2014.05.032 Experimental bifurcation analysis of an impact oscillator -tuning a non-invasive control scheme. E Bureau, F Schilder, I Ferreira Santos, J Thomsen, J Starke, 10.1016/j.jsv.2013.05.033Journal of Sound and Vibration. 33222Bureau, E., Schilder, F., Ferreira Santos, I., Juel Thomsen, J., Starke, J.: Experimental bifurcation analysis of an impact oscilla- tor -tuning a non-invasive control scheme. Journal of Sound and Vibration 332(22), 5883-5897 (2013). DOI 10.1016/j.jsv.2013. 05.033 Recipes for Continuation. H Dankowicz, F Schilder, SIAM11Dankowicz, H., Schilder, F.: Recipes for Continuation, vol. 11. SIAM (2013) Slowly varying systemẋ = a(t)x. C Desoer, 10.1109/TAC.1969.1099336IEEE Transactions on Automatic Control. 146Desoer, C.: Slowly varying systemẋ = a(t)x. IEEE Transactions on Automatic Control 14(6), 780-781 (1969). DOI 10.1109/TAC. 1969.1099336 Convergence properties of adaptive systems and the definition of exponential stability. B Jenkins, A Annaswamy, E Lavretsky, T Gibson, 10.1137/15M1047805SIAM Journal on Control and Optimization. 564Jenkins, B., Annaswamy, A., Lavretsky, E., Gibson, T.: Conver- gence properties of adaptive systems and the definition of ex- ponential stability. SIAM Journal on Control and Optimization 56(4), 2463-2484 (2018). DOI 10.1137/15M1047805 Application of control-basedcontinuation for characterization of dynamic systems with stiffness and friction nonlinearities. G Kleyman, M Paehr, S Tatzko, DOI10.1016/j.mechrescom.2020.103520Mechanics Research Communications. 106Kleyman, G., Paehr, M., Tatzko, S.: Application of control-based- continuation for characterization of dynamic systems with stiff- ness and friction nonlinearities. Mechanics Research Communi- cations 106 (2020). DOI 10.1016/j.mechrescom.2020.103520 Robust and Adaptive Control: With Aerospace Applications. Advanced Textbooks in Control and Signal Processing. E Lavretsky, K Wise, SpringerLondonLavretsky, E., Wise, K.: Robust and Adaptive Control: With Aerospace Applications. Advanced Textbooks in Control and Sig- nal Processing. Springer London (2012) Model identification of a fluttering aerofoil using control-based continuation and normal form analysis. K Lee, D Barton, L Renson, Proceedings of the International Conference on Noise and Vibration Engineering, ISMA 2020, and the International Conference on Uncertainty in Structural Dynamics. the International Conference on Noise and Vibration Engineering, ISMA 2020, and the International Conference on Uncertainty in Structural DynamicsLeuven, Belgium2020Lee, K., Barton, D., Renson, L.: Model identification of a flut- tering aerofoil using control-based continuation and normal form analysis. In: Proceedings of the International Conference on Noise and Vibration Engineering, ISMA 2020, and the International Conference on Uncertainty in Structural Dynamics, USD 2020, Leuven, Belgium, pp. 261-268 (2020) Adaptive control for enhanced performance of devices and algorithms. Y Li, University of Illinois at Urbana-ChampaignPh.D. thesisLi, Y.: Adaptive control for enhanced performance of de- vices and algorithms. Ph.D. thesis, University of Illinois at Urbana-Champaign (2019). URL http://hdl.handle.net/ 2142/106152 Adaptive control designs for control-based continuation in a class of uncertain discrete-time dynamical systems. Y Li, H Dankowicz, 10.1177/1077546320913377Journal of Vibration and Control. 26Li, Y., Dankowicz, H.: Adaptive control designs for control-based continuation in a class of uncertain discrete-time dynamical sys- tems. Journal of Vibration and Control 26(21-22), 2092-2109 (2020). DOI 10.1177/1077546320913377 Adaptive control designs for control-based continuation of periodic orbits in a class of uncertain linear systems. Y Li, H Dankowicz, DOI10.1007/s11071-021-06216-6Nonlinear Dynamics. 1033Li, Y., Dankowicz, H.: Adaptive control designs for control-based continuation of periodic orbits in a class of uncertain linear sys- tems. Nonlinear Dynamics 103(3), 2563-2579 (2021). DOI 10.1007/s11071-021-06216-6 Event-driven feedback tracking and control of tapping-mode atomic force microscopy. S Misra, H Dankowicz, M Paul, 10.1098/ rspa.2007.0016Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 464Misra, S., Dankowicz, H., Paul, M.: Event-driven feedback track- ing and control of tapping-mode atomic force microscopy. Pro- ceedings of the Royal Society A: Mathematical, Physical and En- gineering Sciences 464(2096), 2113-2133 (2008). DOI 10.1098/ rspa.2007.0016 Persistent excitation in adaptive systems. K Narendra, A Annaswamy, DOI10.1080/00207178708933715International Journal of Control. 451Narendra, K., Annaswamy, A.: Persistent excitation in adaptive systems. International Journal of Control 45(1), 127-160 (1987). DOI 10.1080/00207178708933715 Identification of backbone curves and nonlinear frequency responses using control-based continuation and local gaussian process regression. L Renson, 10.1007/978-3-030-47626-713Conference Proceedings of the Society for Experimental Mechanics Series. Renson, L.: Identification of backbone curves and nonlinear fre- quency responses using control-based continuation and local gaus- sian process regression. In: Conference Proceedings of the Soci- ety for Experimental Mechanics Series, pp. 83-85 (2021). DOI 10.1007/978-3-030-47626-7 13 Experimental tracking of limitpoint bifurcations and backbone curves using control-based continuation. L Renson, D Barton, S Neild, DOI10.1142/S0218127417300026International Journal of Bifurcation and Chaos. 271Renson, L., Barton, D., Neild, S.: Experimental tracking of limit- point bifurcations and backbone curves using control-based con- tinuation. International Journal of Bifurcation and Chaos 27(1) (2017). DOI 10.1142/S0218127417300026 Connecting nonlinear normal modes to the forced response of a geometric nonlinear structure with closely spaced modes. L Renson, D Ehrhardt, D Barton, S Neild, J Cooper, Proceedings of the International Conference on Noise and Vibration Engineering, ISMA 2016, and the International Conference on Uncertainty in Structural Dynamics. the International Conference on Noise and Vibration Engineering, ISMA 2016, and the International Conference on Uncertainty in Structural DynamicsUSD; Leuven, BelgiumRenson, L., Ehrhardt, D., Barton, D., Neild, S., Cooper, J.: Con- necting nonlinear normal modes to the forced response of a geo- metric nonlinear structure with closely spaced modes. In: Proceed- ings of the International Conference on Noise and Vibration Engi- neering, ISMA 2016, and the International Conference on Uncer- tainty in Structural Dynamics, USD 2016, Leuven, Belgium, pp. 2775-2784 (2016) Robust identification of backbone curves using control-based continuation. L Renson, A Gonzalez-Buelga, D Barton, S Neild, 10.1016/j.jsv.2015.12.035Journal of Sound and Vibration. 367Renson, L., Gonzalez-Buelga, A., Barton, D., Neild, S.: Robust identification of backbone curves using control-based continua- tion. Journal of Sound and Vibration 367, 145-158 (2016). DOI 10.1016/j.jsv.2015.12.035 Application of control-based continuation to a nonlinear structure with harmonically coupled modes. L Renson, A Shaw, D Barton, S Neild, DOI10.1016/j.ymssp.2018.10.008Mechanical Systems and Signal Processing. 120Renson, L., Shaw, A., Barton, D., Neild, S.: Application of control-based continuation to a nonlinear structure with harmoni- cally coupled modes. Mechanical Systems and Signal Processing 120, 449-464 (2019). DOI 10.1016/j.ymssp.2018.10.008 Numerical continuation in nonlinear experiments using local gaussian process regression. L Renson, J Sieber, D Barton, A Shaw, S Neild, DOI10.1007/s11071-019-05118-yNonlinear Dynamics. 984Renson, L., Sieber, J., Barton, D., Shaw, A., Neild, S.: Numerical continuation in nonlinear experiments using local gaussian pro- cess regression. Nonlinear Dynamics 98(4), 2811-2826 (2019). DOI 10.1007/s11071-019-05118-y Experimental bifurcation analysis -continuation for noise-contaminated zero problems. F Schilder, E Bureau, I Santos, J Thomsen, J Starke, DOI10.1016/j.jsv.2015.08.008Journal of Sound and Vibration. 358Schilder, F., Bureau, E., Santos, I., Thomsen, J., Starke, J.: Exper- imental bifurcation analysis -continuation for noise-contaminated zero problems. Journal of Sound and Vibration 358, 251-266 (2015). DOI 10.1016/j.jsv.2015.08.008 Tracking controlled chaos: Theoretical foundations and applications. I Schwartz, T Carr, I Triandaf, DOI10.1063/1.166285Chaos. 74Schwartz, I., Carr, T., Triandaf, I.: Tracking controlled chaos: Theoretical foundations and applications. Chaos 7(4), 664-679 (1997). DOI 10.1063/1.166285 Experimental continuation of periodic orbits through a fold. J Sieber, A Gonzalez-Buelga, S Neild, D Wagg, B Krauskopf, 10.1103/ PhysRevLett.100.244101Physical Review Letters. 10024Sieber, J., Gonzalez-Buelga, A., Neild, S., Wagg, D., Krauskopf, B.: Experimental continuation of periodic orbits through a fold. Physical Review Letters 100(24) (2008). DOI 10.1103/ PhysRevLett.100.244101 Using feedback control and Newton iterations to track dynamically unstable phenomena in experiments. J Sieber, B Krauskopf, 10.3182/20090622-3-UK-3004.00041IFAC Proceedings Volumes. 42Sieber, J., Krauskopf, B.: Using feedback control and Newton iterations to track dynamically unstable phenomena in experi- ments. IFAC Proceedings Volumes 42(7), 211-216 (2009). DOI 10.3182/20090622-3-UK-3004.00041 On the stability of slowly time-varying linear systems. V Solo, DOI10.1007/BF01211523Mathematics of Control, Signals, and Systems. 74Solo, V.: On the stability of slowly time-varying linear systems. Mathematics of Control, Signals, and Systems 7(4), 331-350 (1994). DOI 10.1007/BF01211523 Bayesian model updating and class selection of a wing-engine structure with nonlinear connections using nonlinear normal modes. M Song, L Renson, B Moaveni, G Kerschen, 10.1016/j.ymssp.2021.108337Mechanical Systems and Signal Processing. 165Song, M., Renson, L., Moaveni, B., Kerschen, G.: Bayesian model updating and class selection of a wing-engine structure with non- linear connections using nonlinear normal modes. Mechanical Systems and Signal Processing 165 (2022). DOI 10.1016/j.ymssp. 2021.108337 Experimental bifurcation analysis of a wing profile. I Tartaruga, D Barton, D Rezgui, S Neild, Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, IFASD 2019. the International Forum on Aeroelasticity and Structural Dynamics, IFASD 2019Savannah, Georgia, USATartaruga, I., Barton, D., Rezgui, D., Neild, S.: Experimental bi- furcation analysis of a wing profile. In: Proceedings of the Interna- tional Forum on Aeroelasticity and Structural Dynamics, IFASD 2019, Savannah, Georgia, USA (2019) Parametrically excited nonlinear two-degree-of-freedom electromechanical systems. B Zaghari, T Kniffka, C Levett, E Rustighi, 10.1088/1742-6596/1264/1/012024Journal of Physics: Conference Series. 12641Zaghari, B., Kniffka, T., Levett, C., Rustighi, E.: Parametrically excited nonlinear two-degree-of-freedom electromechanical sys- tems. Journal of Physics: Conference Series 1264(1) (2019). DOI 10.1088/1742-6596/1264/1/012024
[]
[ "Exploiting Neighbor Effect: Conv-Agnostic GNN Framework for Graphs with Heterophily", "Exploiting Neighbor Effect: Conv-Agnostic GNN Framework for Graphs with Heterophily" ]
[ "Jie Chen ", "Shouzhen Chen ", "Junbin Gao ", "Zengfeng Huang ", "Member, IEEEJunping Zhang ", "Jian Pu " ]
[]
[ "IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS" ]
Due to the homophily assumption in graph convolution networks (GNNs), a common consensus in the graph node classification task is that GNNs perform well on homophilic graphs but may fail on heterophilic graphs with many interclass edges. However, the previous inter-class edges perspective and related homo-ratio metrics cannot well explain the GNNs performance under some heterophilic datasets, which implies that not all the inter-class edges are harmful to GNNs. In this work, we propose a new metric based on von Neumann entropy to re-examine the heterophily problem of GNNs and investigate the feature aggregation of inter-class edges from an entire neighbor identifiable perspective. Moreover, we propose a simple yet effective Conv-Agnostic GNN framework (CAGNNs) to enhance the performance of most GNNs on heterophily datasets by learning the neighbor effect for each node. Specifically, we first decouple the feature of each node into the discriminative feature for downstream tasks and the aggregation feature for graph convolution. Then, we propose a shared mixer module to adaptively evaluate the neighbor effect of each node to incorporate the neighbor information. The proposed framework can be regarded as a plug-in component and is compatible with most GNNs. The experimental results over nine wellknown benchmark datasets indicate that our framework can significantly improve performance, especially for the heterophily graphs. The average performance gain is 9.81%, 25.81%, and 20.61% compared with GIN, GAT, and GCN, respectively. Extensive ablation studies and robustness analysis further verify the effectiveness, robustness, and interpretability of our framework. Code is available at https://github.com/JC-202/CAGNN.
10.1109/tnnls.2023.3267902
[ "https://export.arxiv.org/pdf/2203.11200v3.pdf" ]
247,596,625
2203.11200
d3fb40fdca4162a5749132317c52d24359f92aba
Exploiting Neighbor Effect: Conv-Agnostic GNN Framework for Graphs with Heterophily Jie Chen Shouzhen Chen Junbin Gao Zengfeng Huang Member, IEEEJunping Zhang Jian Pu Exploiting Neighbor Effect: Conv-Agnostic GNN Framework for Graphs with Heterophily IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS XX1Index Terms-Graph neural networksNode classificationRepresentation learningHeterophilyHomophily Due to the homophily assumption in graph convolution networks (GNNs), a common consensus in the graph node classification task is that GNNs perform well on homophilic graphs but may fail on heterophilic graphs with many interclass edges. However, the previous inter-class edges perspective and related homo-ratio metrics cannot well explain the GNNs performance under some heterophilic datasets, which implies that not all the inter-class edges are harmful to GNNs. In this work, we propose a new metric based on von Neumann entropy to re-examine the heterophily problem of GNNs and investigate the feature aggregation of inter-class edges from an entire neighbor identifiable perspective. Moreover, we propose a simple yet effective Conv-Agnostic GNN framework (CAGNNs) to enhance the performance of most GNNs on heterophily datasets by learning the neighbor effect for each node. Specifically, we first decouple the feature of each node into the discriminative feature for downstream tasks and the aggregation feature for graph convolution. Then, we propose a shared mixer module to adaptively evaluate the neighbor effect of each node to incorporate the neighbor information. The proposed framework can be regarded as a plug-in component and is compatible with most GNNs. The experimental results over nine wellknown benchmark datasets indicate that our framework can significantly improve performance, especially for the heterophily graphs. The average performance gain is 9.81%, 25.81%, and 20.61% compared with GIN, GAT, and GCN, respectively. Extensive ablation studies and robustness analysis further verify the effectiveness, robustness, and interpretability of our framework. Code is available at https://github.com/JC-202/CAGNN. Fig. 1 : The identifiability of neighbors. Nodes with the same colors share the same class labels. The above neighbor distribution for the blue class is random, and the bottom is not random. Although many inter-class edges exist, the identifiable patterns from the non-random distribution of dissimilar neighbors may also help graph convolution classify the center node's label. be of the same class. The aggregation process over intraclass edges would smooth representations of nodes from the same class, thus benefiting the classification task and achieving promising results on homophilic graphs [4]- [6]. However, the opposite heterophily property is also observed in a wide range of real-world graphs, including dating [7], molecular [8]- [10], and transaction networks [11], [12]. Unfortunately, the applications of GNNs in heterophilic graphs whose connected nodes often have different labels are usually problematic [12]- [15]. Most literature agrees that massive inter-class edges in heterophilic graphs are harmful for aggregation since they may cause the graph convolution to over-smooth the representations of nodes belonging to different classes [12], [15], [16]. Furthermore, they measure the proportion of interclass edges in a graph as homo-ratio metrics to evaluate the "strength" of heterophily [7], [17], which can be used to guide the application of GNNs. However, such metrics from edge perspectives fail to explain the completely different node classification performance under some heterophilic graphs with a similar homo-ratio [7], [17], [18]. Therefore, it compels us to re-examine the heterophily problem of GNNs and answer the following two questions: (1) Is inter-class neighborhood aggregation truly harmful or unnecessary? (2) How to improve GNNs' performance on heterophilic graphs? As shown in Fig. 1, we first note that the inter-class edges may be beneficial to improve the node classification task if their neighbor distribution is identifiable instead of random. The critical point is to treat the neighbor as an entirety and measure the identifiability of each class neighbor distribution rather than simply considering the proportion of inter-class edges. A class with a random neighbor distribution is considered to have lower identifiability, e.g., all nodes are randomly connected to others without considering classes of nodes. arXiv:2203.11200v3 [cs.LG] 16 Apr 2023 Whereas a specific neighbor-connected pattern is considered to have higher identifiability. An extreme example is a bipartite graph. Although the graph is highly heterophilic (each edge connects two nodes of the opposite class), node features are still distinguishable after a simple mean aggregation operator since their neighbor's distribution is identifiable [19], [20]. In this work, we first present a novel metric to quantify the identifiability of the neighbor's distribution. Compared with the previous inter-class edges metrics, ours can better indicate the graph-level neighbor effect for classifications and understand the heterophily problem. Then, motivated by this metric, we propose a simple yet effective Conv-Agnostic GNN Framework (CAGNNs) to improve GNNs' performance on heterophilic graphs by learning the node-level neighbor effect. We summarize the main contribution as follows. (1) We present a new perspective of heterophily with neighbor identifiability and quantify it with the metric inspired by the von Neumann entropy. Specifically, we extract the label distribution matrix of neighbors at the class level and extend the von Neumann entropy [21] as a new metric to measure the identifiability of neighbors for the graph. Unlike the edge perspective, this metric demonstrates the importance of the neighbor effect for classifications in understanding heterophily problems. It can be used to explain the performance differences of GNNs among various datasets with the same homoratio. Furthermore, it motivates us to consider the node-level neighbor effect to improve classical GNNs. (2) We propose CAGNNs as a general framework to improve classical GNNs by learning the neighbor effect for each node. To consider the neighbor effect at the node level, we first decouple the features of a node into two parts: discriminative features used for downstream tasks and aggregated features from its neighbors via graph convolution. Then, we introduce a mixer module to fuse these two features into new discriminative features for classification via learning the node-level neighbor effect. Our framework can improve the performance of most GNNs under heterophily without modifying the graph convolution kernel due to the generic node-level neighbor effect learning mechanism. Thus, unlike other heterophilicoriented GNNs that need to modify the convolution kernel [7], [14], [17], our framework is conv-agnostic and more general. (3) We conduct extensive experiments on nine well-known benchmark datasets to verify the effectiveness, interpretability, and robustness of CAGNNs. We show that a simple sharedparameters mixer module can substantially enhance the performance of most GNNs on heterophilic graphs while maintaining the performance on homophilic graphs. The average performance gain is 9.81%, 25.81%, and 20.61% compared with GIN, GAT, and GCN, respectively. Moreover, our framework shows good interpretability for determining whether neighbor information is helpful for node classification tasks, and the framework is robust to over-smoothing and noisy edges. The paper is organized as follows. In Section II, we survey the related works. Section III briefly introduces the notations and background. In Section IV, we present the proposed novel metric for measuring neighbor identifiability for graphs with heterophily from a neighbor perspective. In Section V, we describe the proposed CAGNN framework and implementation in detail. Evaluation results on nine benchmark datasets and ablation studies are presented in Section VI to verify the effectiveness, interpretability, and robustness of the proposed framework. The final section concludes the paper. II. RELATED WORKS A. Graph Neural Networks GNN models can be roughly categorized into spectral and spatial methods. Early on, Bruna et al. [22] first propose a spectral graph-based extension of convolutional networks to graphs. In a follow-up work, ChebyNets [23] define graph convolutions using Chebyshev polynomials to remove the computationally expensive Laplacian eigendecomposition. GCNs [6] further simplify graph convolutions by stacking layers of first-order Chebyshev polynomial filters with a redefined propagation matrix. Also, the GCN bridges the spectral and spatial domain gap since it can also be regarded as a mean aggregator to aggregate neighbor information to each node. Furthermore, in the spatial domain, Graph Attention Network (GAT) applies the attention mechanism to learn edge weights to improve the aggregation step. Xu et al. [24] study the expressiveness of graph neural networks and introduces Graph Isomorphism Network (GIN), which is proven to be as powerful as the Weisfeiler-Lehman test. There are many other graph neural models [2], [25], [26]; we refer to [1], [3], [27] for a more comprehensive review. To investigate why and when graph neural networks work well for node classification, some researchers aim to understand the behavior of GNNs. Ni and Machara [5] indicate that graph neural networks only perform low-pass filtering on feature vectors and do not have the non-linear manifold learning property from a signal processing perspective. Li et al. [4] point out that the GCN model's graph convolution is actually a special form of Laplacian smoothing, which is consistent with the homophily assumption. However, it also brings potential concerns about making the features of connected nodes from different labels indistinguishable. On the other hand, Ma et al. [20] theoretically reveal that homophily is not a necessary assumption for the GCN model. Moreover, for attentionbased GNNs, Wang et al. [28] find that stacking multiple attention layers causes excessive smoothing of node features due to information exchange over inter-class edges. [29], [30] summarize the current popular message-passing scheme in GNNs and argue that the message between intra-class edges would help nodes receive information gain. In contrast, the inter-class edges may introduce negative disturbance. B. GNNs for Heterophily Recently, heterophilic graph learning has become an upward-trending research topic, and various specific structured GNNs have been proposed. Most argue that message passing during inter-class edges is harmful to the node classification task and try to avoid harmfulness. Current kinds of literature can be divided into three lines: (1) Some works deal with heterophily from the spectral domain. FAGCN [14] divides the message from each edge into low-frequency and high-frequency signals and shows that both the low-and high-frequency signals are necessary for heterophilic graph learning. In addition, some works aim at extracting high-order approximation with graph spectral filters. GPRGNN [19] modifies the convolution to the generalized page rank and learns an arbitrary K-order polynomial graph filter. GCNII [31] proposes the initial residual and identity mapping for vanilla GCN and theoretically proves that it can express a K-order polynomial filter with arbitrary coefficients. BernNet [32] learns arbitrary graph spectral filters via Bernstein approximation to oversimplified or ill-posed filters. ACM-GCN [18] modifies graph convolution by explicitly dividing it into low-pass, high-pass, and identity filters in each layer and adaptively fusing them for each node. However, these spectral-based methods need to specifically design and modify the graph convolution, which is not trivial to generalize to broader spatial GNNs like GAT and GIN. (2) Some works reorganize the graph structure to obtain a more homophilic signal. Geom-GCN [17] utilizes geometric aggregation to capture structural similarity in the latent space and long-range dependencies. NLGNN [33] leverages attention-guide sorting to generate a re-connected graph and conducts non-local aggregation. SLAPS [34] combines the self-supervised technique to infer a homophily latent structure. GDAMN [30] proposes the decoupling attention mechanism on both features and labels to increase the intra-class and reduce inter-class edge weights. (3) Some works aim to capture high-order neighbor information, which was proven to be homophily dominant [14]. MixHop [35] repeatedly mixes representations of multihop neighbors to achieve higher-order message passing. JK-Nets [36] jumps the intermediate representations to the last layer for better structure-aware representation. H2GCN [7] proposes three designs with separated ego and neighbors, hider-order neighbors, and a combination of intermediate representations to combine the message from neighbors. Unlike these specific GNN architectures to avoid the harmfulness of inter-class edges; instead, we consider the inter-class edges from an identifiable neighbor distribution perspective. Furthermore, we propose a simple and general Conv-Agnostic framework. This framework can be regarded as a plug-in component and is compatible with most GNNs to improve their performance on heterophilic graphs. III. PRELIMINARY A. Problem Setup Consider an undirected graph G = (V, E) with adjacency matrix A ∈ R N ×N and the diagonal degree matrix D of A, where V and E are the sets of nodes and edges, respectively. For each node v i ∈ V, we denote N (v i ) = {j : (i, j) ∈ E} as its neighbor set. Each node is given a m-dimensional feature representation x i and a c-dimensional one-hot class label y i . The feature inputs are then formed by X = [x 1 , · · · , x N ], and the labels are Y = [y 1 , · · · , y N ]. Given the labels Y L of a subset of nodes L ⊂ V, the task of semi-supervised node classification is to predict the labels Y U of the unlabeled nodes U = V \ L by exploiting the graph structure E and the features of all the nodes X. Models Aggregation for each layer l(1 ≤ l ≤ L) GCN h (l) v = σ v ∈Nv ∪{v} 1 (|Nv |+1)·(|N v |+1) · W (l−1) · h (l−1) v GIN h (l) v = MLP (l) 1 + (l) · h (l−1) v + v ∈N (v) h (l−1) v GAT h (l) v i = σ v j ∈Nv i ∪{v i } a (l−1) i,j · W (l−1) · h (l−1) v j B. Graph Neural Networks From a probabilistic view, most GNNs assume the local Markov property on node features, i.e., for each node v i , the label y i only depends on the node self-feature x i and its neighbor-features x j : j ∈ N (v i ). We use superscript l to indicate the layer index. For the l-th layer of a GNN, we use h l i to represent the embedding of node v i and h 0 i to represent x i or a projection of x i for dimension reduction. Then, the general l-th layer GraphConvolution(A, H l−1 ) for node i can be formulated as h l i = f h l−1 i , h l−1 j : j ∈ N (v i ) ,(1) where the graph convolution operator f can be implemented by a weighted sum of each node based on the adjacent matrix A as in GCN [6] and GIN [24] or the attention mechanism in GAT [37]. The formulations of three well-known graph convolution layers are summarized in Table I. The final output Z ∈ R N ×c of the label prediction is evaluated using a softmax function to embed the last layer H L . The objective function is the cross-entropy of the ground truth labels Y and the output of the network Z: O = − i∈L c j=1 Y ij ln Z ij . (2) C. Homophily/Heterophily Metrics on Graphs The homophily ratio h aims to measure the overall homophily level in a graph. The commonly used node-level [17] and edge-level [7] homophily metrics are usually defined by H node (G) = 1 |V| v∈V |{u | u ∈ N v , Y u = Y v }| d v ,(3)H edge (G) = |{e uv | e uv ∈ E, Y u = Y v }| |E| .(4) Such metrics measure the proportion of inter-class edges in a graph based on label consistency. There is another metric H agg (G) to consider the node features similarity from postaggregation perspective [18]. However, it is limited to the scenario where nodes have features. By definition, the ratio h ∈ [0, 1], graphs with h closer to 1 tend to have more intraclass edges indicating stronger homophily; on the other hand, graphs with h closer to 0 have more edges connecting different classes, which indicates stronger heterophily. However, as reported in previous literature [7], [17], [18], these metrics are not significantly relevant to the prediction performance of GCNs. For example, for the well-known heterophily datasets Chameleon and Actor, their homo-ratios metrics H edge are all 0.22. However, on the one hand, the reported accuracy of node classification for GCN varies, i.e., 60% and 30%, respectively [7]. On the other hand, the accuracy of corresponding Multi-Layer Perceptrons(MLPs) is 46% and 35%. The opposite performance gap between GCN and MLP in the low homophilic datasets shows that interclass edges can be either beneficial or harmful to classification, which motivates us to consider a new metric beyond the edge perspective to measure heterophily. IV. REVIEW HETEROPHILY FROM AN ENTIRE NEIGHBOR PERSPECTIVE A. Measure the Graph-level Neighbor Effect for Heterophily In this section, we present a new metric to better understand the heterophily problem and answer whether all inter-class edges are harmful. As noted in Fig. 1, when the label distribution of neighbors is random, i.e., every node is connected to other neighbors with random labels, there is no helpful information we can learn from the aggregation step. However, when the neighbor distribution of each class's nodes forms a certain identifiable distribution, regardless of whether the connected edges are intra-or inter-class, graph convolution can extract useful information from this non-random neighbor distribution for the downstream tasks. For instance, graph convolution can still achieve perfect performance on a bipartite graph [19], [20]. Therefore, instead of simply calculating the proportion of inter-class edges as a measurement for graphs with heterophily, we need to measure the randomness/identifiability of the entire neighbor distribution. We define the identifiability of neighbors as the information of the nodes' neighbor distribution, which can be seen as the graph-level neighbor effect for heterophilic datasets. As Fig. 2 shows, to measure the identifiability of neighbors of each class, we group the nodes by class and form a class-level neighbor's label distribution matrix A k N ∈ R n k ×C for each class k, where k = 1, ..., C for different classes and n k indicates the number of nodes with label k. Then, our task is to evaluate the information of the neighbors' label distribution matrix to Fig. 3: The relation between hom-metric and GCN's performance from [7] on different graph node classification datasets 1 . Our metric Hneighbor(G) is more monotonous with the models' performance. Moreover, compared with other metrics, it can distinguish the Actor dataset, which has a random neighbor distribution and the lowest accuracy. More details can be found in Section VI-C. quantify the identifiability of neighbors. Inspired by the von Neumann entropy in quantum statistical mechanics [21], which measures the pureness/information of a quantum-mechanical system by calculating the entropy of the eigenvalue distribution of a positive definite symmetric density matrix, we generalize this idea to our task of evaluating neighbors' identifiability. Specifically, since the neighbor distribution matrix A k N is not symmetric, we consider the entropy of the singular values distribution rather than the eigenvalues as an indicator of identifiability. This can be understood as a measurement of how many vectors (patterns of neighbors) are needed for an adequate explanation of the neighbors' label distribution matrix, indicating the richness/randomness of the neighbor distribution. Suppose σ k 1 , σ k 2 , ..., σ k C denote singular values of A k N , we then normalize them so that C i=1 σ k i = 1, where i = 1, . .., C for index of singular values. Then we calculate the entropy of class K by H k neighbor = − C i=1 σ k i log(σ k i ) log(C) .(5) The above metric ranges from [0, 1] and can be used to quantify the identifiability of neighbors for a specific class (the lower, the more identifiable). Considering the problem of class imbalance, we compute the weighted sum of class-level entropy to evaluate the neighbors' identifiability of a graph. H neighbor (G) = C k=1 n k N H k neighbor .(6) Our metric sheds new light on understanding the heterophily problem from an identifiable neighbor perspective. As shown in Fig. 3, compared with the node/edge-level homophily metrics, our measurement and the GCNs' performance for different datasets are more monotonous. Specifically, we can observe that the Actor, Chameleon, and Squirrel datasets have similar node/edge-level homophily metrics (proportion of inter-class edges) in the first and second parts of Fig. 3. However, the performance of these datasets is inconsistent, especially for the Actor dataset, which has the poorest performance (accuracy≈30%) and is totally different from the others. As we can see in the third part of Fig. 3, the H agg (G) still fail to distinguish the Actor from others. In contrast, as shown in the right of Fig. 3, our metric can distinguish the Actor dataset from others since it has a nearly random neighbor distribution (H neighbor =0.98). This indicates that the proposed metric can better explain the difference in model performance from a neighbor perspective. Furthermore, our metric reveals that the inter-class edges are not always harmful for the node classification during aggregation, and the entire local neighbor perspective can provide more information. B. Improve GNNs via Learning Node-level Neighbor Effect Unlike other specific heterophilic-oriented GNNs that need to modify graph convolution kernels, our neighbor perspective motivates us to learn the effectiveness of each node's local neighbors during aggregation to help traditional GNNs deal with heterophily in a general way. Most GNNs straightforwardly feed the current aggregation features to the following graph convolution layer and adopt the last layer representation for the downstream node classification task [6], [37]. However, the entanglement of aggregation and classification may lead to over-smoothing of node representations due to inter-class neighbors, resulting in a loss of discrimination in heterophilic graphs, despite the fact that neighbor information may be useful for downstream node classification. To adaptively combine neighbor information and enhance traditional GNNs, we propose decoupling discriminant representations of nodes from the aggregation. Then, it allows us to guide the aggregation and generate suitable representations of nodes for classification by learning the node-level neighbor effect. However, to guide the aggregation during the training process, we cannot directly utilize the entropy measurement and need to evaluate the node-level neighbor effect in another way. The reasons are twofold: (1) Similar to H edge and H node , the computation of the entropy H neighbor also requires the labels of all the nodes, which are unavailable in the training process. (2) The class-level neighbor distribution identifiability H c neighbor is not consistent with the entropy of the node-level label distribution. Namely, the entropy of a node does not represent the identifiability of the neighbor distribution of this class. In the next section, we will elaborate on how to adaptively learn the node-level neighbor effect from the downstream supervision signal and the features of a node with its neighbor. V. PROPOSED METHOD In this section, we propose the Conv-Agnostic GNN framework (CAGNNs) to improve traditional GNNs performance by adaptively learning the node-level neighbor effect. Then, we provide a spectral analysis to show the expressive power of the proposed framework on the node classification task. Finally, we describe the difference between our framework and the GNNs with decoupling design and skip connection. A. Conv-Agnostic GNN Framework The proposed CAGNNs aims to empower traditional GNNs to generate suitable representations for each node for both homophilic and heterophilic graphs. The core idea is to treat the node-local neighbor as an entirety and determine its efficacy during aggregation. To this end, we first decouple the representation of nodes into discrimination and aggregation. Then, we learn a mixer module that can adaptively evaluate each node's neighbor effect based on these two representations and determine whether to incorporate the information from neighbors. As shown in Fig. 4, our framework is composed of four major components: Encoder, Graph Convolution (GC), Mixer, and Decoder. Below, we elaborate on each component in order. Moreover, we append the normalization operation after the Encoder, GC, and Mixer to maintain the numerical stability, which will be discussed later. Encoder: We use a linear layer as the encoder to transform the node features X. Then, we feed it into two streams. One is the node's own feature S 0 ∈ R N ×d for downstreamtask discrimination, and the other is the aggregation feature H 0 ∈ R N ×d for the graph convolution. These two streams decouple the discriminant feature and the information from neighbors, which may prevent representations of nodes for classification from being over-smooth by their neighbors during aggregation. Moreover, this decoupling operation allows each node to evaluate the neighbor effect for downstream discrimination tasks via the mixer module in the following aggregation step. H 0 = S 0 = Norm(f encoder (X)).(7) Graph Convolution (GC): Since our framework is Conv-Agnostic, in this part, any standard graph convolution layers (e.g., GCN, GAT, and GIN) can be applied to aggregate each node's neighborhood information to update the aggregation representation H. H l+1 = Norm(GraphConvolution(A, H l )).(8) Moreover, this graph convolution layer can be stacked multiple times to enhance the receptive field of each node by considering the information of more neighbors. The embedding H l can also be regarded as the l-hop neighbors' information. However, most GNNs ignore the fact that the information from neighbors is not always beneficial for the classification task of each node. Hence, we propose the following mixer operator to determine the neighbor effect of each hop. Mixer: From the view of node v i at layer l, it needs to combine the discrimination feature S l i and l-hop neighbors feature H l i according to the neighbor effect to update the representative embedding for the downstream task. Therefore, the goal of the mixer function is to evaluate the neighbor effect of each node and then to selectively incorporate the neighbors' information. As discussed in Section IV, we do not need to inject the entropy function to evaluate the node-level neighbor effect. We have already encoded the neighbor information of each node into H l by the graph convolution. According to the universal approximation theorem [38], [39], we hypothesize that the MLP can adaptively learn the node-level neighbor effect based on S l−1 ∈ R N ×d and H l ∈ R N ×d and the downstream objective O. For simplicity, we implement the mixer by α l = σ(f mixer (S l−1 H l )),(9)S l = Norm((1 − α l ) * S l−1 + α l * H l ),(10) where is the concat operator and the function f mixer is a linear layer R N ×2d → R N ×1 . It maps the discrimination feature S l−1 and neighbor feature H l−1 at layer l to a vector α l ∈ R N ×1 . With the sigmoid function σ, each element of α l is normalized to an importance score ranging from 0 to 1 and can be regarded as the neighbor effect of the node. Based on the importance score α l , we use the convex combination of discriminant feature S l−1 and neighbor information H l−1 at each layer to adaptively update the discriminant feature S l . This convex combination strategy has the advantages of numerical stability and interpretability and has been widely used in modern deep learning, such as the Highway network [40] and Attention mechanism [41]. Note that the parameters of all the f mixer functions are shared across layers to learn the neighbor effect of each node. Such a parameter-efficient sharing mechanism can help reduce overfitting and improve generalization, similar to the gating function in Long Short-Term Memory [42]. Moreover, the mixer function maintains the expressive power of the chosen graph convolution, since it can easily degenerate to the normal graph convolution when α l = 1. As shown in the ablation study, our implementation of this mixer function is very effective with minimal parameter cost. Decoder: With the last layer discriminant S L at hand, the task of the decoder is to produce the final prediction Z for classification. For simplicity, we use the linear layer with softmax operator as our decoder f decoder . Z = softmax(f decoder (S L )).(11) Note that we also apply the Norm layer to maintain numerical stability at the end of the Encoder, GC and Mixer modules. Specifically, we use L2 normalization for each node. Compared with the widely used BatchNorm and LayerNorm, we experimentally show that the parameter-free L2 normalization can achieve better performance for our task. In summary, unlike other GNNs that explicitly modify graph filters or graph convolution kernels for heterophily [14], [18], [19], our node-level mixer is a convolution-agnostic technique. We divide the representation of the node into two parts: nodeself for discrimination and neighbor information from arbitrary graph convolution. Hence the original graph convolution operator may not directly interact with the node's discrimination feature and over-smooth its representation. Moreover, we use only one shared-parameterized layer to adaptively learn node-level neighbor effect and mix these two representations, which can be seen as adding a plug-in side layer to standard GNNs for generating more discriminative node representations. Therefore, our method is compatible with most graph convolution layers (GCN, GAT, GIN, etc.) and enhances their performance in a more general and parameter-efficient way. B. Spectral Analysis of CAGNN We then study the expressive power of CAGNNs from a spectral perspective. Recall the K-order polynomial graph filters with graph signal X and propagation matrix P as K l=0 θ l P l X, where P = D −1/2 AD −1/2 is the normalized adjacency matrix and the scalar θs are the polynomial coefficients. Note that using such a polynomial graph filter can derive either high-or low-pass filters [32], [43], maintaining the model's ability to deal with various label-connected patterns, which is essential to prevent over-smoothing [31] and to learn from heterophilic graphs [19]. However, most spectral GNNs learn the shared K-order polynomial scalar coefficients θ's for all nodes, which may limit the expressive power. We demonstrate that CAGNNs corresponds to a polynomial graph filter with different coefficients for each node on the graph spectral domain. Theorem 1: Considering the propagation matrix P used for the basic graph convolution layer and a graph signal X, a Klayer CAGNNs has the ability to express a K-order polynomial filter K l=0 θ l P l X with different coefficients θs for each node. The above theorem indicates the expressive power on node classification of our framework. Intuitively, the importance score αs evaluated by the Mixer allows CAGNNs to simulate different coefficient θs of the polynomial graph filter for each node. Note that with a proper choice of θs, the discriminant feature S K of each node can carry information from both the input feature and the high-order neighbor's information adaptively with the increment of the order K. Compared with other spectral GNNs that learn shared polynomial filters K l=0 θ l P l X for all nodes [19], [23], [31], our models can empower each node with distinct polynomial coefficients by vectorized θ l . This property can capture the nodes' distinct complex connected patterns, as in Section VI-B, our framework achieves higher performance than other spectral GNNs experimentally. The detailed proof is presented below. Proof. For simplicity, we neglect the L2 normalization at each layer because the simplified version also produces comparable performance. Moreover, we assume the input feature X to be non-negative and remove the nonlinear ReLU activation in the graph convolution layer [31], [44]. Then, for the simplified graph convolution layer, we have H l = PH l−1 W l (12) = P l H 0 l j=0 W j (13) = P l XW l , whereW l = l j=0 W j .(14) Furthermore, we can express the K-layer representation as S K = (1 − α K ) * S K−1 + α K * H K (15) = K l=0 α l K k=l+1 (1 − α k ) * H l (16) = K l=0 ϑ l * P l XW l , where ϑ l = α l K k=l+1 (1 − α k ).(17) However, ϑ l ∈ [0, 1] N ×1 which limits the expressive power of the polynomial filter. Thanks to the weight matrixW l , the coefficients of the polynomial can be extended to arbitrary values. Inspired by [31], we consider a weaker version of CAGNNs by fixing the weight matrixW l to be γ l , where γ l is a learnable parameter. We have S K = K l=0 ϑ l * P l Xγ l (18) = K l=0 θ l * P l X, where θ l = ϑ l γ l .(19) The polynomial coefficient θ l for each layer l can be set to desired values with the help of the scalable parameter γ l , which concludes the proof. C. Relation to other Decouple or Skip Connection GNNs In this section, we discuss the relation of our two techniques (decoupled operation and mixer module) in the framework with other models. Relation to GNNs with Decouple Design: We remark that the decoupling operation, which we study in this work, is a distinct concept from other decoupling GNNs. Well-known decoupled GNNs, such as APPNP [25], S 2 GC [45], and GPRGNN [19], aim to decouple the propagation and transformation of graph convolution. Without the loss of generalizability, they can be formulated as follows. H K = ( K l=0 θ l P l ) propagate transf orm H 0 W ,(20) where θ l is a scalar and P is the propagation matrix. This operation makes the node have the ability to receive highorder neighbor information in one layer. In addition, the learnable parameter θ l ensures that it can learn the arbitrary coefficient of polynomial graph filters beyond the low-pass filters and perform well in heterophily [19]. However, it needs to decouple the weight matrix W to reformulate the graph convolution operator, which restricts the adaptation to other graph convolutions. In contrast, our approach aims to decouple the discriminant feature and the aggregation feature, which can be easily compatible with most standard GNNs. Moreover, we theoretically prove that our framework can learn different arbitrary coefficients of K-order polynomial graph filters for each node and achieve higher performance experimentally. Relation to GNNs with Skip Connection: To better understand the mixer module, we compare it with respect to the DeepGNNs with skip connections. Note that we aim at the heterophily problem, and our approach is different from the DeepGNNs designed to reduce the over-smoothing problem [4]. To alleviate the over-smoothing problem when the model becomes deeper, most DeepGNNs equip the residual connection or initial connection to combine previous layers' features to prevent forgetting the original feature when models become deeper [31], [46]. The standard DeepGNN with skip connection can be formulated as follows. H l = σ(PH l W l + H l−1 /H 0 )(residual/init connection).(21) However, the DeepGNNs with skip connections are not designed for heterophilic graphs. Most skip connections combine previous features and cannot adaptively aggregate information from neighbors for each node. For instance, the recent wellknown SOTA of this type model is GCNII [31], where H l = σ 1 − α l PH l + α l H 0 1 − β l I + β l W l .(22) The scalar α l in GCNII is a manually chosen hyper-parameter that controls the strength of initial connection. However, since the α l is shared for all nodes and not learnable, it is not the best choice to guide how to aggregate neighbors' information on heterophilic datasets. In contrast, as shown in Equations (9), we use the mixer function to learn the importance score α to explicitly evaluate the node-level neighbor effect for feature fusion. Moreover, compared with the skip connection, we separate the discriminant features S of nodes at each layer and do not feed them into the next layer graph convolution component. VI. EXPERIMENTS In this section, we report and compare the results for node classification on both real-world heterophily and homophily datasets to investigate the effectiveness, robustness and interpretability of the proposed heterophily GNN framework CAGNNs. We also show the importance of the neighbor effect from the relationship between the model performance and metrics. A. Experimental Setup 1) Datasets: We evaluate the performance on nine wellknown real-world node classification datasets, including three homophily datasets (Citeseer, Pubmed, and Cora) and six heterophily datasets (Texas, Wisconsin, Actor, Squirrel, Chameleon, and Cornell). For all benchmarks, we use the same feature vectors, graph structure, class labels, and standard II: Performance comparison on various real-world heterophily and homophily datasets. Mean test accuracy and standard deviation are reported over 10 random data splits. The best performance is highlighted. "*" denotes the results obtained from [7]. 10 fixed random splits (48%/32%/20% of nodes per class for train/validation/test) provided in literature [7], [17]. • Homophily Datasets -Citeseer, Pubmed, Cora [6]: For the basic citation datasets [47], nodes correspond to papers; edges correspond to citation links; the sparse bag-of-words are the feature representation of each node. Finally, the label of each node represents the topic of the paper. • Heterophily Datasets -Texas, Wisconsin, Cornell [17]: Cornell, Texas, and Wisconsin are the web page networks captured from the computer science departments of these universities in the WebKB dataset. In these networks, nodes and edges represent the web pages and hyperlinks. Similar to the Citations networks, words in the web page represent the node features in the bag-of-word form. The web pages are labeled into five categories: student, project, course, staff, and faculty. -Squirrel, Chameleon [17]: Chameleon and Squirrel are web pages extracted from different topics in Wikipedia [48]. Similar to WebKB, nodes and edges denote the web pages and hyperlinks among them, respectively, and informative nouns in the web pages are employed to construct the node features in the bag-of-word form. Webpages are labeled in terms of the average monthly traffic level. -Actor [17]: The actor network contains the cooccurrences of actors in films, which are extracted from the heterogeneous information networks. It describes the complex relationships among films, direc-tors, actors and writers [49]. In this network, nodes and edges represent actors and their co-occurrences in films, respectively. The actor's Wikipedia page is exploited to extract features and node labels. 2) Baselines: We compare our method with the following baselines: (1) classical standard GNNs: GCN [6], GAT [37] and GIN [24]; (2) recent state-of-the-art GNNs of specific structure tackling heterophily: Geom-GCN [17], MixHop [35], H2GCN [7], GPRGNN [19], FAGCN [14], GCN-Cheby [23], JK-Net [36], GCNII [31]; and (3) standard 2-layer MLP. To show the effectiveness and generalizability of our framework, we choose three simple GCN, GAT, and GIN as the graph convolution component in CAGNNs. For ease of comparison, we use the reported results of baselines in the literature [7], [17]. Moreover, for the missing results under these splits, we rerun the released code 10 times and report the mean and standard deviation. 3) Hyper-parameters: We adopt the same set of default hyper-parameters (2 layers and 64 hidden dimensions) for GCN, GAT, and GIN and the corresponding CAGNNs, which is the most widely used hyper-parameters setting as [14], [31], [37]. For CAGNNs, we add only one linear layer with 128 hidden units. We employ the Adam optimizer and select the learning rate ∈ {0.001, 0.01, 0.05}, weight decay ∈ {0.00005, 0.0005} and dropout rate ∈ {0, 0.5} based on the validation sets. For other models, we utilize their best default parameters in the original papers. B. Performance Comparison with SOTA We report and compare the performance for the standard node classification task in Table II. For the classic GNNs (GCN, GAT and GIN), we first note that they outperform the MLP in the homophily datasets. It indicates that the homophily assumption and connected intra-class neighbors provide helpful node classification information. However, we also notice that they sometimes also perform better than the MLP under heterophily datasets. For instance, in the Actor dataset, the performance of MLP is approximately 36%, but the traditional GCN only obtains 30%. However, in the Chameleon dataset, the performance comparison between GCN and MLP is (60% vs. 46%). Hence, it implies that not all the inter-class edges in the heterophily datasets are harmful to the node classification. In addition, we note that the GNNs with specific designs for the heterophily datasets outperform the traditional GNNs with a large margin for six heterophily datasets. Most of them (e.g., MixHop, JK-Net, and H2GCN) explicitly aggregate high-order neighbors' information to avoid the harmfulness of inter-class edges. For instance, the strong baseline H2GCN proves that the neighbor's high-order information is expectedly homophilydominant and achieves 70.87% average performance. In com- parison, the average performance of the two-layer GCN is only 61.62%. In contrast, instead of considering that the inter-class edges are all harmful, we take each node's entire neighbor effect into account. As a result, compared with recently state-ofthe-art heterophily GNNs, our CAGNNs can help traditional standard GNNs achieve competitive results while maintaining the performance on three homophily datasets. Moreover, from the perspective of the spectral domain, CAGNN can help traditional GNNs adaptively learn different coefficients of Korder polynomial graph filters for each node. Therefore, our framework performs better than the spectral GNNs sharing the same polynomial filter coefficients for all nodes (e.g., GPRGNN and GCNII). Under our framework, the average performance on nine datasets of GIN, GAT, and GCN outperforms all the baselines, and the average performance gains are 9.81%, 25.81%, and 20.61%, respectively. Among them, the proposed CAGNNs with 2-layer GCN achieves the best average performance (74.32%) over all datasets. It verifies the effectiveness of decoupling design and consideration of the neighbor effect when performing graph convolution. C. Relation between the Metrics and Performance Table II also shows the different metrics for all datasets. All the metrics range from [0, 1] and a higher score of H node , H edge and H agg denotes higher homophily. However, a higher H neighbor means lower identifiability of neighbors' distribution and a more challenging dataset, i.e., the neighbor distribution provides less useful information for classification. Neighbor distribution identifiability offers an alternative perspective to the inter-class edges approach (H node and H edge ) for understanding the heterophily problem in GNNs. Note that only our H neighbor can distinguish the dataset Actor (0.98) from others, in which the neighbors' distribution is nearly random and the best test classification accuracy is very low, i.e., 35.86. Our metric shows that the GNN does not outperform MLP (35.73) in the Actor datasets with randomness neighbor distribution. Therefore, it may be used as a guide to choosing whether to employ GNNs rather than MLP. Moreover, for the graph-level metrics H neighbor , we also report the Kendall rank correlation coefficient between these metrics and the performance of CAGNN in Table III. The H neighbor is more correlated with the performance over different datasets. Therefore, the proposed metric can be considered a rough evaluator to measure the difficulty of graphs for the node classification task using GNNs, especially for medium-scale graphs with more than 500 nodes. We must have a sufficient number of nodes to have meaningful statistical information. For the class-level H c neighbor , we show the relation with our CAGNN's performance under various datasets in Fig. 5. Since the larger H c neighbor means the lower identifiability, for clarity, we apply the negative H c neighbor for each class to show a positive correlation between the model's class-wise performance and the class-wise neighbor distribution identifiability. At the class level, we can see that the performance of CAGNN GCN is highly consistent with the neighbor distribution identifiability for most datasets, which indicates that the neighbor perspective can help explain the model performance on various datasets. It also verifies that our framework can adaptively evaluate the neighbor effect and guide each node to absorb helpful information for the downstream node classification task. D. Ablation Study In this section, we compare CAGNN with its variants for Mixer and Normalization to validate the effectiveness of each component. When testing different Mixer variants, we fix the Normalization to L2. Also, we set the Mixer to linear to test different Normalization variants. We select the best hyper-parameters of each variant and run experiments for each dataset under 10 random splits to report the average performance on all datasets. Variants of Mixer. The mixer module is a critical part of our framework to evaluate the neighbor effect and feature fusion. We compared our results with six variants. • Add: The Add mixer is implemented by S l = Norm(S l−1 + H l ). • Concat: The Concat mixer is implemented by S l = Norm([S l−1 ||H l ]). Note that we only apply the Norm at the first and last layers for the Concat variant to maintain stability. • Global: The Global mixer is S l = Norm((1 − α l )S l−1 + α l H l ), where the learnable scalar α l is shared for all nodes. • Unshared: It replaces the shared-parameterized f mixer to unshared version for each layer. • MLP-2/3: It replaces the linear layer in f mixer with two or three layers of shared-parameterized MLP. From the results of these mixer variants in Table IV, we have the following observations: (1) All these variants of mixer can substantially improve the performance of standard GNNs, which verifies the effectiveness of the decoupled design. (2) The results of the global mixer are better than those of the Add and Concat mixer, indicating that the neighbor effect is diverse under different datasets. (3) Our 1-layer mixer and multilayer MLP variants achieve consistently better performance than the global model. These mixers can adaptively learn each node's neighbor effect, which verifies that the learnable node-level neighbor effect plays a more vital role than the global mixer. (4) Compared with the unshared-parameterized version and the multiple layer MLPs in the current f mixer module, the result shows that the current shared-parameterized one-layer mixer is simple yet effective. Variants of Normalization. We also compare the L2 norm for each node with three variants (None, BatchNorm [50] and LayerNorm [51]) to show its effectiveness. We first observe that the None normalization version achieves comparable performance, demonstrating that our framework is stable and competitive. However, the BatchNorm usually assumes the independent and identical distribution of each sample in deep learning, which may not be reasonable for the heterophilic graph and results in worse performance. Moreover, LayerNorm is also not beneficial since the layer norm introduces learning parameters that may be redundant for GNNs. Compared with these variants, our simple yet effective L2 Norm achieves the best average performance. E. Robustness Analysis To investigate whether the proposed CAGNN framework can help basic GNNs become more robust, we report the performance comparison between the basic GNNs and corresponding CAGNN in the over-smoothing and noisy edges scenario. Alleviating oversmooth. It is well known that the traditional graph convolution is sensitive to the number of convolution layers due to the over-smoothing problem [4]. Our CAGNNs are able to increase the robustness of traditional graph convolution to avoid over-smoothing. As shown in Fig. 6(a), when the number of layers increases, the performance of traditional GNNs drops rapidly due to over-smoothing. Moreover, due to the massive inter-class edges, the over-smoothing phenomenon occurs earlier in the heterophily datasets. In contrast, the methods under the proposed framework are more stable and more consistent on both the Homophily (Citeseer) and Heterophily (Chameleon) datasets. The reason is that our framework has the ability to avoid incorporating the over-smoothing features to maintain the discrimination power for each node. Alleviating noisy edges. Most GNNs are also sensitive to the noisy edges in graphs [52], [53]. To evaluate the robustness of the proposed framework on noisy graphs, we construct graphs with random edge addition according to the literature [54]. Specifically, we randomly add 25%∼500% edges in the original graphs. As shown in Fig. 6(b), our CAGNNs achieve significantly better prediction for noisy graphs compared with the basic GNNs. It also demonstrates that our decouple design and the mixer module are able to learn to discard the noisy features from neighbors. F. Visualization and Interpretability To verify whether CAGNNs can adaptively learn the different neighbor effect of each node, we visualize the neighbor importance score α distribution of each layers on both homophily and heterophily datasets, where the results are shown in Fig. 7. We have the following observation: (1) As we can see, for homophily datasets (Cora, Citeseer, and Pubmed), the coefficients are near 0.4 for most nodes. This indicates that the information from neighbors is helpful for the downstream classification, which is consistent with the homophily assumption. (2) However, we observe similar trends in heterophily datasets (Chameleon and Squirrel) that most nodes still absorb the neighbors' information. Moreover, the distribution of the second layer score shows that 1-hop neighbors are more important than 2-hop neighbors. This phenomenon is contrary to the previous studies that the inter-class edges are all harmful, which implies that graph convolution can still extract classification information from an inter-class neighbor with a non-random distribution. (3) The Actor dataset has a similar proportion of inter-class edges with Chameleon and Squirrel, but the neighbor distribution is nearly random (H neighbor is 0.98). The neighbor importance score of nodes tends to be 0, which verifies that the inter-class edges with random distribution are harmful and guides the model to discard neighbors' information when aggregation. G. Efficiency Finally, we investigate the complexity of the proposed framework. The complexity of computing the neighbor importance score and mixing for all nodes is O(N d) where N is the total number of nodes, and d is the dimension of the hidden nodes. The computational complexity is on par with the neighborhood aggregation operation in GNNs, which is also O(N d). Hence, the complexity of our method and classic GNNs are on the same level. We also report the average training time of each epoch over all datasets of the standard graph convolution and our method in Fig. 8. It indicates that our model scales similarly to the basic GNNs with a small computation cost for the mixer module. VII. CONCLUSION In this paper, we have investigated the neighbor effect on heterophilic datasets. Unlike previous works that argue that inter-class edges are harmful for node classification, we find that inter-class edges can be helpful when the neighbor distribution is identifiable. Furthermore, instead of computing node/edge-level metrics for a graph with heterophily, we propose a new measurement from the entire neighbor-level perspective via von Neumann entropy. The proposed new metric sheds new light on the heterophily problem. It enables us to explain the performance variation of GNNs for different datasets and can be used to guide the application of GNNs. We also proposed a simple yet effective heterophily GNN framework, CAGNN, which adds just one mixer layer to enhance the performance of conventional GNNs. The node features are first decoupled into discrimination and aggregation parts, and then adaptively fused according to the neighbor effect of each node. Our experiments on nine well-known benchmark datasets not only demonstrate the effectiveness of the proposed framework as a plug-in to consistently improve existing GNNs on heterophilic graphs, but also indicate good interpretability for determining whether neighbor information is helpful for the downstream node classification task. Similar to other node-and edge-level metrics, our metrics are constrained by the requirement for node label information. Therefore, it is non-trivial to explicitly combine these metrics to guide the aggregation process during training without label information. One of our future works is to incorporate the pseudo-label with these metrics to improve the aggregation of GNNs. Besides, beyond node classification, exploiting the connection between neighbor identifiability with the edge or graph-level task is also an interesting topic for future work. Fig. 2 : 2Class-level von Neumann entropy, which measures the information of neighbors' label distribution matrix. This metric ranges from [0, 1] and can quantify the identifiability of neighbors for a specific class (a lower number indicates higher identifiability of neighbors). Fig. 4 : 4The GC operator indicates any Graph Convolutions. The shared-parameter Mixer function can help each node determine whether to absorb neighbors' information by considering the neighbor effect based on the node's feature. • Classical GNNs -GCN[6]: GCN can be seen as a Laplacian smoother since it uses the mean aggregator to smooth each node and its neighbor's features. -GIN[24]: GIN utilizes the MLP to model the injective function when aggregation and generalizes the WL test. -GAT[37]: GAT is a graph neural network that applies the attention mechanism on node features to learn edge weights for aggregation.• Heterophily GNNs -H2GCN [14]: H2GCN proposed three designs with separate ego and neighbors, hider-order neighbors and a combination of intermediate representations to combine the message from neighbors. -FAGCN [14]: FAGCN divides the message from each edge into low-frequency and high-frequency signals during aggregation. -GCN-Cheby [23]: GCN-Cheby combines higherorder neighbor information with Chebyshev polynomials from the spectral domain. -GEOM-GCN [17]: GEOM-GCN utilizes structural similarity to capture the smooth structure in the latent space and long-range dependencies. -MixHop [35]: MixHop repeatedly mixes feature representations of neighbors at various distances to achieve higher-order message passing. -GPRGNN [19]: GPRGNN modifies the convolution to the generalized page rank and learned an arbitrary polynomial graph filter to incorporate multi-scale information. -JK-Net [36]: JK-Net combines intermediate node representations from each layer by concatenating them in the final layer. -GCNII [31]: The state-of-the-art deep model combines initial connections and identity mapping to train a very deep GCN. Fig. 5 : 5Class-level comparison between our CAGNNGCN performance (red) and the negative class-level H c neighbor (blue) for Homophily (Cora, Citeseer, and Pubmed) and Heterophily (Chameleon, Squirrel, and Actor). The class-level performance of CAGNNGCN is highly consistent with the neighbor distribution identifiability for most datasets. Fig. 6 : 6Performance comparison with traditional GNNs and the variants under the proposed CAGNNs framework in terms of (a) number of layers, and (b) ratio of adding noisy edges on Homophily (Citeseer) and Heterophily (Chameleon) datasets. Fig. 7 : 7Visualization of nodes' neighbor importance scores of the first and second layers on different datasets. Fig. 8 : 8Average running time per epoch (ms) TABLE I : IThe different neighborhood aggregation schemes. Here σ denotes the ReLu activation, a denotes attention weights, W denotes the weight matrix, and MLP denotes multiple layer perceptron. TABLE TABLE III : IIIKendall correlation between different metrics and the performance of CAGNNGCN . The higher coefficient and the lower p-value are better and more significant.Datasets Kendall H node H edge Hagg H neighbor >500 nodes coefficient 0.733 0.828 0.467 0.867 p-value 0.056 0.022 0.27 0.017 All datasets coefficient 0.11 0.25 0.44 0.59 p-value 0.7 0.34 0.12 0.02 TABLE IV : IVThe average performance (test accuracy) over all datasets for the ablation study of different types of Mixers and Normalization.Variants CAGNNGIN CAGNNGAT CAGNNGCNMixer Add 71.91 70.63 71.41 Concat 69.50 71.27 72.23 Global 72.17 72.52 73.43 Unshared 72.82 73.06 73.88 MLP-2 72.84 73.15 74.38 MLP-3 72.57 73.47 74.20 Norm None 72.29 73.03 73.01 BatchNorm 65.11 61.61 64.92 LayerNorm 72.41 72.19 72.45 Ours 73.28 73.67 74.32 We ignore the datasets that have fewer than 500 nodes. Their performances are highly sensitive to the data splits. A comprehensive survey on graph neural networks. Z Wu, S Pan, F Chen, G Long, C Zhang, P S Yu, IEEE Transactions on Neural Networks and Learning Systems. 321Z. Wu, S. Pan, F. Chen, G. Long, C. Zhang, and P. S. Yu, "A comprehensive survey on graph neural networks," IEEE Transactions on Neural Networks and Learning Systems, vol. 32, no. 1, pp. 4-24, 2021. Inductive representation learning on large graphs. W L Hamilton, R Ying, J Leskovec, Advances in Neural Information Processing Systems. W. L. Hamilton, R. Ying, and J. Leskovec, "Inductive representation learning on large graphs," in Advances in Neural Information Processing Systems, 2017, pp. 1025-1035. A gentle introduction to deep learning for graphs. D Bacciu, F Errica, A Micheli, M Podda, Neural Networks. 129D. Bacciu, F. Errica, A. Micheli, and M. Podda, "A gentle introduction to deep learning for graphs," Neural Networks, vol. 129, pp. 203-221, 2020. Deeper insights into graph convolutional networks for semi-supervised learning. Q Li, Z Han, X.-M Wu, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence33Q. Li, Z. Han, and X.-M. Wu, "Deeper insights into graph convolutional networks for semi-supervised learning," in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 33, no. 1, 2018, pp. 3538- 3545. Revisiting graph neural networks: All we have is low-pass filters. H Nt, T Maehara, arXiv:1905.09550preprintH. Nt and T. Maehara, "Revisiting graph neural networks: All we have is low-pass filters," preprint arXiv:1905.09550, 2019. Semi-supervised classification with graph convolutional networks. T N Kipf, M Welling, International Conference on Learning Representations. T. N. Kipf and M. Welling, "Semi-supervised classification with graph convolutional networks," in International Conference on Learning Rep- resentations, 2017. Beyond homophily in graph neural networks: Current limitations and effective designs. J Zhu, Y Yan, L Zhao, M Heimann, L Akoglu, D Koutra, Advances in Neural Information Processing Systems. 33J. Zhu, Y. Yan, L. Zhao, M. Heimann, L. Akoglu, and D. Koutra, "Beyond homophily in graph neural networks: Current limitations and effective designs," in Advances in Neural Information Processing Sys- tems, vol. 33, 2020. Deep graph translation. X Guo, L Wu, L Zhao, IEEE Transactions on Neural Networks and Learning Systems. X. Guo, L. Wu, and L. Zhao, "Deep graph translation," IEEE Transac- tions on Neural Networks and Learning Systems, 2022. Graph polish: a novel graph generation paradigm for molecular optimization. C Ji, Y Zheng, R Wang, Y Cai, H Wu, IEEE Transactions on Neural Networks and Learning Systems. C. Ji, Y. Zheng, R. Wang, Y. Cai, and H. Wu, "Graph polish: a novel graph generation paradigm for molecular optimization," IEEE Transactions on Neural Networks and Learning Systems, 2021. Explaining deep graph networks via input perturbation. D Bacciu, D Numeroso, IEEE Transactions on Neural Networks and Learning Systems. D. Bacciu and D. Numeroso, "Explaining deep graph networks via input perturbation," IEEE Transactions on Neural Networks and Learning Systems, 2022. Item relationship graph neural networks for e-commerce. W Liu, Y Zhang, J Wang, Y He, J Caverlee, P P Chan, D S Yeung, P.-A Heng, IEEE Transactions on Neural Networks and Learning Systems. W. Liu, Y. Zhang, J. Wang, Y. He, J. Caverlee, P. P. Chan, D. S. Yeung, and P.-A. Heng, "Item relationship graph neural networks for e-commerce," IEEE Transactions on Neural Networks and Learning Systems, 2021. Graph neural networks for graphs with heterophily: A survey. X Zheng, Y Liu, S Pan, M Zhang, D Jin, P S Yu, arXiv:2202.07082preprintX. Zheng, Y. Liu, S. Pan, M. Zhang, D. Jin, and P. S. Yu, "Graph neural networks for graphs with heterophily: A survey," preprint arXiv:2202.07082, 2022. Two sides of the same coin: Heterophily and oversmoothing in graph convolutional neural networks. Y Yan, M Hashemi, K Swersky, Y Yang, D Koutra, IEEE International Conference on Data Mining. IEEEY. Yan, M. Hashemi, K. Swersky, Y. Yang, and D. Koutra, "Two sides of the same coin: Heterophily and oversmoothing in graph convolutional neural networks," in IEEE International Conference on Data Mining. IEEE, 2022, pp. 1287-1292. Beyond low-frequency information in graph convolutional networks. D Bo, X Wang, C Shi, H Shen, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence35D. Bo, X. Wang, C. Shi, and H. Shen, "Beyond low-frequency infor- mation in graph convolutional networks," in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, no. 5, 2021, pp. 3950- 3957. Graph neural networks with heterophily. J Zhu, R A Rossi, A Rao, T Mai, N Lipka, N K Ahmed, D Koutra, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence35J. Zhu, R. A. Rossi, A. Rao, T. Mai, N. Lipka, N. K. Ahmed, and D. Koutra, "Graph neural networks with heterophily," in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, no. 12, 2021, pp. 11 168-11 176. Memory-based message passing: Decoupling the message for propagation from discrimination. J Chen, W Liu, J Pu, IEEE International Conference on Acoustics, Speech and Signal Processing. IEEEJ. Chen, W. Liu, and J. Pu, "Memory-based message passing: De- coupling the message for propagation from discrimination," in IEEE International Conference on Acoustics, Speech and Signal Processing. IEEE, 2022, pp. 4033-4037. Geom-gcn: Geometric graph convolutional networks. H Pei, B Wei, K C Chang, Y Lei, B Yang, International Conference on Learning Representations. H. Pei, B. Wei, K. C.-C. Chang, Y. Lei, and B. Yang, "Geom-gcn: Geometric graph convolutional networks," in International Conference on Learning Representations, 2020. Revisiting heterophily for graph neural networks. S Luan, C Hua, Q Lu, J Zhu, M Zhao, S Zhang, X.-W Chang, D Precup, Advances in Neural Information Processing Systems. S. Luan, C. Hua, Q. Lu, J. Zhu, M. Zhao, S. Zhang, X.-W. Chang, and D. Precup, "Revisiting heterophily for graph neural networks," in Advances in Neural Information Processing Systems, 2022. Adaptive universal generalized pagerank graph neural network. E Chien, J Peng, P Li, O Milenkovic, International Conference on Learning Representations. E. Chien, J. Peng, P. Li, and O. Milenkovic, "Adaptive universal generalized pagerank graph neural network," in International Conference on Learning Representations, 2021. Is homophily a necessity for graph neural networks?. Y Ma, X Liu, N Shah, J Tang, in International Conference on Learning Representations. Y. Ma, X. Liu, N. Shah, and J. Tang, "Is homophily a necessity for graph neural networks?" in International Conference on Learning Representations, 2022. Geometry of quantum states: an introduction to quantum entanglement. I Bengtsson, K Życzkowski, Cambridge university pressI. Bengtsson and K.Życzkowski, Geometry of quantum states: an introduction to quantum entanglement. Cambridge university press, 2017. Spectral networks and locally connected networks on graphs. J Bruna, W Zaremba, A Szlam, Y Lecun, arXiv:1312.6203J. Bruna, W. Zaremba, A. Szlam, and Y. LeCun, "Spectral networks and locally connected networks on graphs," preprint arXiv:1312.6203, 2013. Convolutional neural networks on graphs with fast localized spectral filtering. M Defferrard, X Bresson, P Vandergheynst, Advances in Neural Information Processing Systems. 29M. Defferrard, X. Bresson, and P. Vandergheynst, "Convolutional neural networks on graphs with fast localized spectral filtering," Advances in Neural Information Processing Systems, vol. 29, 2016. How powerful are graph neural networks?. K Xu, W Hu, J Leskovec, S Jegelka, in International Conference on Learning Representations. K. Xu, W. Hu, J. Leskovec, and S. Jegelka, "How powerful are graph neural networks?" in International Conference on Learning Represen- tations, 2018. Predict then propagate: Graph neural networks meet personalized pagerank. J Klicpera, A Bojchevski, S Günnemann, International Conference on Learning Representations. J. Klicpera, A. Bojchevski, and S. Günnemann, "Predict then propagate: Graph neural networks meet personalized pagerank," in International Conference on Learning Representations, 2019. A fair comparison of graph neural networks for graph classification. F Errica, M Podda, D Bacciu, A Micheli, International Conference on Learning Representations. F. Errica, M. Podda, D. Bacciu, and A. Micheli, "A fair comparison of graph neural networks for graph classification," in International Conference on Learning Representations, 2020. Relational inductive biases, deep learning, and graph networks. P W Battaglia, J B Hamrick, V Bapst, A Sanchez-Gonzalez, V Zambaldi, M Malinowski, A Tacchetti, D Raposo, A Santoro, R Faulkner, arXiv:1806.01261preprintP. W. Battaglia, J. B. Hamrick, V. Bapst, A. Sanchez-Gonzalez, V. Zam- baldi, M. Malinowski, A. Tacchetti, D. Raposo, A. Santoro, R. Faulkner et al., "Relational inductive biases, deep learning, and graph networks," preprint arXiv:1806.01261, 2018. Improving graph attention networks with large margin-based constraints. G Wang, R Ying, J Huang, J Leskovec, arXiv:1910.11945preprintG. Wang, R. Ying, J. Huang, and J. Leskovec, "Improving graph attention networks with large margin-based constraints," preprint arXiv:1910.11945, 2019. Measuring and improving the use of graph information in graph neural networks. Y Hou, J Zhang, J Cheng, K Ma, R T B Ma, H Chen, M.-C Yang, International Conference on Learning Representations. Y. Hou, J. Zhang, J. Cheng, K. Ma, R. T. B. Ma, H. Chen, and M.-C. Yang, "Measuring and improving the use of graph information in graph neural networks," in International Conference on Learning Representations, 2020. Graph decoupling attention markov networks for semisupervised graph node classification. J Chen, S Chen, M Bai, J Pu, J Zhang, J Gao, IEEE Transactions on Neural Networks and Learning Systems. J. Chen, S. Chen, M. Bai, J. Pu, J. Zhang, and J. Gao, "Graph decoupling attention markov networks for semisupervised graph node classification," IEEE Transactions on Neural Networks and Learning Systems, pp. 1-15, 2022. Simple and deep graph convolutional networks. M Chen, Z Wei, Z Huang, B Ding, Y Li, International Conference on Machine Learning. PMLR, 2020. M. Chen, Z. Wei, Z. Huang, B. Ding, and Y. Li, "Simple and deep graph convolutional networks," in International Conference on Machine Learning. PMLR, 2020, pp. 1725-1735. Bernnet: Learning arbitrary graph spectral filters via bernstein approximation. M He, Z Wei, H Xu, Advances in Neural Information Processing Systems. 34M. He, Z. Wei, H. Xu et al., "Bernnet: Learning arbitrary graph spectral filters via bernstein approximation," Advances in Neural Information Processing Systems, vol. 34, 2021. Non-local graph neural networks. M Liu, Z Wang, S Ji, IEEE Transactions on Pattern Analysis and Machine Intelligence. M. Liu, Z. Wang, and S. Ji, "Non-local graph neural networks," IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021. Slaps: Self-supervision improves structure learning for graph neural networks. B Fatemi, L El Asri, S M Kazemi, Advances in Neural Information Processing Systems. 34B. Fatemi, L. El Asri, and S. M. Kazemi, "Slaps: Self-supervision improves structure learning for graph neural networks," Advances in Neural Information Processing Systems, vol. 34, 2021. Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing. S Abu-El-Haija, B Perozzi, A Kapoor, N Alipourfard, K Lerman, H Harutyunyan, G Ver, A Steeg, Galstyan, International Conference on Machine Learning. PMLRS. Abu-El-Haija, B. Perozzi, A. Kapoor, N. Alipourfard, K. Lerman, H. Harutyunyan, G. Ver Steeg, and A. Galstyan, "Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing," in International Conference on Machine Learning. PMLR, 2019, pp. 21-29. Representation learning on graphs with jumping knowledge networks. K Xu, C Li, Y Tian, T Sonobe, K Kawarabayashi, S Jegelka, International Conference on Machine Learning. PMLRK. Xu, C. Li, Y. Tian, T. Sonobe, K.-i. Kawarabayashi, and S. Jegelka, "Representation learning on graphs with jumping knowledge networks," in International Conference on Machine Learning. PMLR, 2018, pp. 5453-5462. Graph attention networks. P Veličković, G Cucurull, A Casanova, A Romero, P Liò, Y Bengio, International Conference on Learning Representations. P. Veličković, G. Cucurull, A. Casanova, A. Romero, P. Liò, and Y. Bengio, "Graph attention networks," in International Conference on Learning Representations, 2018. Multilayer feedforward networks are universal approximators. K Hornik, M Stinchcombe, H White, Neural networks. 25K. Hornik, M. Stinchcombe, and H. White, "Multilayer feedforward networks are universal approximators," Neural networks, vol. 2, no. 5, pp. 359-366, 1989. Approximation capabilities of multilayer feedforward networks. K Hornik, Neural networks. 42K. Hornik, "Approximation capabilities of multilayer feedforward net- works," Neural networks, vol. 4, no. 2, pp. 251-257, 1991. Highway networks. R K Srivastava, K Greff, J Schmidhuber, arXiv:1505.00387preprintR. K. Srivastava, K. Greff, and J. Schmidhuber, "Highway networks," preprint arXiv:1505.00387, 2015. Attention is all you need. A Vaswani, N Shazeer, N Parmar, J Uszkoreit, L Jones, A N Gomez, Ł Kaiser, I Polosukhin, Advances in Neural Information Processing Systems. A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, "Attention is all you need," in Advances in Neural Information Processing Systems, 2017, pp. 5998-6008. Long short-term memory. S Hochreiter, J Schmidhuber, Neural Computation. 98S. Hochreiter and J. Schmidhuber, "Long short-term memory," Neural Computation, vol. 9, no. 8, pp. 1735-1780, 1997. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. D I Shuman, S K Narang, P Frossard, A Ortega, P Vandergheynst, IEEE Signal Processing Magazine. 303D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Van- dergheynst, "The emerging field of signal processing on graphs: Ex- tending high-dimensional data analysis to networks and other irregular domains," IEEE Signal Processing Magazine, vol. 30, no. 3, pp. 83-98, 2013. Simplifying graph convolutional networks. F Wu, A Souza, T Zhang, C Fifty, T Yu, K Weinberger, International Conference on Machine Learning. PMLRF. Wu, A. Souza, T. Zhang, C. Fifty, T. Yu, and K. Weinberger, "Sim- plifying graph convolutional networks," in International Conference on Machine Learning. PMLR, 2019, pp. 6861-6871. Simple spectral graph convolution. H Zhu, P Koniusz, International Conference on Learning Representations. H. Zhu and P. Koniusz, "Simple spectral graph convolution," in Inter- national Conference on Learning Representations, 2021. Deepgcns: Can gcns go as deep as cnns. G Li, M Muller, A Thabet, B Ghanem, Proceedings of the IEEE/CVF international conference on computer vision. the IEEE/CVF international conference on computer visionG. Li, M. Muller, A. Thabet, and B. Ghanem, "Deepgcns: Can gcns go as deep as cnns?" in Proceedings of the IEEE/CVF international conference on computer vision, 2019, pp. 9267-9276. Collective classification in network data. P Sen, G Namata, M Bilgic, L Getoor, B Galligher, T Eliassi-Rad, AI magazine. 293P. Sen, G. Namata, M. Bilgic, L. Getoor, B. Galligher, and T. Eliassi- Rad, "Collective classification in network data," AI magazine, vol. 29, no. 3, pp. 93-93, 2008. Multi-scale attributed node embedding. B Rozemberczki, C Allen, R Sarkar, Journal of Complex Networks. 92B. Rozemberczki, C. Allen, and R. Sarkar, "Multi-scale attributed node embedding," Journal of Complex Networks, vol. 9, no. 2, 2021. Social influence analysis in large-scale networks. J Tang, J Sun, C Wang, Z Yang, Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining. the 15th ACM SIGKDD international conference on Knowledge discovery and data miningJ. Tang, J. Sun, C. Wang, and Z. Yang, "Social influence analysis in large-scale networks," in Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, 2009, pp. 807-816. Batch normalization: Accelerating deep network training by reducing internal covariate shift. S Ioffe, C Szegedy, arXiv:1502.03167preprintS. Ioffe and C. Szegedy, "Batch normalization: Accelerating deep network training by reducing internal covariate shift," preprint arXiv:1502.03167, 2015. Layer normalization. J Ba, J R Kiros, G E Hinton, arXiv:1607.06450preprintJ. Ba, J. R. Kiros, and G. E. Hinton, "Layer normalization," preprint arXiv:1607.06450, 2016. Learning discrete structures for graph neural networks. L Franceschi, M Niepert, M Pontil, X He, International Conference on Machine Learning. PMLRL. Franceschi, M. Niepert, M. Pontil, and X. He, "Learning discrete structures for graph neural networks," in International Conference on Machine Learning. PMLR, 2019, pp. 1972-1982. Topology attack and defense for graph neural networks: An optimization perspective. K Xu, H Chen, S Liu, P.-Y Chen, T.-W Weng, M Hong, X Lin, International Joint Conference on Artificial Intelligence. K. Xu, H. Chen, S. Liu, P.-Y. Chen, T.-W. Weng, M. Hong, and X. Lin, "Topology attack and defense for graph neural networks: An optimization perspective," in International Joint Conference on Artificial Intelligence, 2019. Iterative deep graph learning for graph neural networks: Better and robust node embeddings. Y Chen, L Wu, M Zaki, Advances in Neural Information Processing Systems. 33Y. Chen, L. Wu, and M. Zaki, "Iterative deep graph learning for graph neural networks: Better and robust node embeddings," Advances in Neural Information Processing Systems, vol. 33, 2020.
[ "https://github.com/JC-202/CAGNN." ]
[ "A Variational Perspective on Generative Flow Networks", "A Variational Perspective on Generative Flow Networks" ]
[ "Heiko Zimmermann [email protected] \nDivision of Statistics and Machine Learning\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nUniversity of Amsterdam\nLinköping University\nUniversity of Amsterdam\nUniversity of Amsterdam\n\n", "Fredrik Lindsten [email protected] \nDivision of Statistics and Machine Learning\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nUniversity of Amsterdam\nLinköping University\nUniversity of Amsterdam\nUniversity of Amsterdam\n\n", "Jan-Willem Van De Meent [email protected] \nDivision of Statistics and Machine Learning\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nUniversity of Amsterdam\nLinköping University\nUniversity of Amsterdam\nUniversity of Amsterdam\n\n", "Christian A Naesseth [email protected] \nDivision of Statistics and Machine Learning\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nUniversity of Amsterdam\nLinköping University\nUniversity of Amsterdam\nUniversity of Amsterdam\n\n" ]
[ "Division of Statistics and Machine Learning\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nUniversity of Amsterdam\nLinköping University\nUniversity of Amsterdam\nUniversity of Amsterdam\n", "Division of Statistics and Machine Learning\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nUniversity of Amsterdam\nLinköping University\nUniversity of Amsterdam\nUniversity of Amsterdam\n", "Division of Statistics and Machine Learning\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nUniversity of Amsterdam\nLinköping University\nUniversity of Amsterdam\nUniversity of Amsterdam\n", "Division of Statistics and Machine Learning\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nAmsterdam Machine Learning Lab\nUniversity of Amsterdam\nLinköping University\nUniversity of Amsterdam\nUniversity of Amsterdam\n" ]
[]
Generative flow networks (GFNs) are a class of models for sequential sampling of composite objects, which approximate a target distribution that is defined in terms of an energy function or a reward. GFNs are typically trained using a flow matching or trajectory balance objective, which matches forward and backward transition models over trajectories. In this work, we define variational objectives for GFNs in terms of the Kullback-Leibler (KL) divergences between the forward and backward distribution. We show that variational inference in GFNs is equivalent to minimizing the trajectory balance objective when sampling trajectories from the forward model. We generalize this approach by optimizing a convex combination of the reverse-and forward KL divergence. This insight suggests variational inference methods can serve as a means to define a more general family of objectives for training generative flow networks, for example by incorporating control variates, which are commonly used in variational inference, to reduce the variance of the gradients of the trajectory balance objective. We evaluate our findings and the performance of the proposed variational objective numerically by comparing it to the trajectory balance objective on two synthetic tasks.
10.48550/arxiv.2210.07992
[ "https://export.arxiv.org/pdf/2210.07992v1.pdf" ]
252,907,672
2210.07992
bc63bccfa8d1246fddb20f935ea9bb6ea56be4fb
A Variational Perspective on Generative Flow Networks Heiko Zimmermann [email protected] Division of Statistics and Machine Learning Amsterdam Machine Learning Lab Amsterdam Machine Learning Lab Amsterdam Machine Learning Lab University of Amsterdam Linköping University University of Amsterdam University of Amsterdam Fredrik Lindsten [email protected] Division of Statistics and Machine Learning Amsterdam Machine Learning Lab Amsterdam Machine Learning Lab Amsterdam Machine Learning Lab University of Amsterdam Linköping University University of Amsterdam University of Amsterdam Jan-Willem Van De Meent [email protected] Division of Statistics and Machine Learning Amsterdam Machine Learning Lab Amsterdam Machine Learning Lab Amsterdam Machine Learning Lab University of Amsterdam Linköping University University of Amsterdam University of Amsterdam Christian A Naesseth [email protected] Division of Statistics and Machine Learning Amsterdam Machine Learning Lab Amsterdam Machine Learning Lab Amsterdam Machine Learning Lab University of Amsterdam Linköping University University of Amsterdam University of Amsterdam A Variational Perspective on Generative Flow Networks Generative flow networks (GFNs) are a class of models for sequential sampling of composite objects, which approximate a target distribution that is defined in terms of an energy function or a reward. GFNs are typically trained using a flow matching or trajectory balance objective, which matches forward and backward transition models over trajectories. In this work, we define variational objectives for GFNs in terms of the Kullback-Leibler (KL) divergences between the forward and backward distribution. We show that variational inference in GFNs is equivalent to minimizing the trajectory balance objective when sampling trajectories from the forward model. We generalize this approach by optimizing a convex combination of the reverse-and forward KL divergence. This insight suggests variational inference methods can serve as a means to define a more general family of objectives for training generative flow networks, for example by incorporating control variates, which are commonly used in variational inference, to reduce the variance of the gradients of the trajectory balance objective. We evaluate our findings and the performance of the proposed variational objective numerically by comparing it to the trajectory balance objective on two synthetic tasks. Introduction Generative flow networks (GFNs) (Bengio et al., 2021a;b) have recently been proposed as a computationally efficient method for sampling composite objects such as molecule strings (Bengio et al., 2021a), DNA sequences or graphs (Deleu et al., 2022). To generate such objects, GFNs sample a trajectory along a directed acyclic graph (DAG) in which edges correspond to actions that modify the object. A trajectory sequentially constructs an object by transitioning from a root node (initial object, or null state) to a terminating node (final composite object) which is scored according to a reward signal. While sampling sequences of actions has been well studied in the reinforcement learning literature (Sutton and Barto, 2018), the objective is typically to find a policy which maximizes the expected reward of the trajectory. By contrast, GFNs are trained to learn a policy that solves a planning-as-inference problem (Toussaint et al., 2006) by learning a distribution over trajectories ending in a terminating state with probability proportional to the reward assigned to it. This is done by optimizing objectives which aim to satisfy a flow matching or detailed balance condition (Bengio et al., 2021a). has since found that these objectives are prone to ineffective credit propagation across trajectories and proposes an alternative objective based on a trajectory balance (TB) condition to alleviate these problems. Most recently, Madan et al. (2022) proposed an objective that can be optimized on partial trajectories and (Do et al., 2022) proposed an optimal-transport-based objective to further improve generalization and exploration. A positive reward function can be interpreted as an unnormalized distribution, which one wishes to generate samples from. In this view, we are interested in sequentially sampling form a factorized joint distibution, such that the marginal distribution of the final state is approximately equal to the corresponding normalized distribution. Generating approximate samples from an unnormalized target distribution is a common task in probabilistic inference, for which many methods have been developed. Examples include methods based on MCMC (Hoffman and Gelman, 2014;Salimans et al., 2015;Li et al., 2017;Hoffman, 2017;Naesseth et al., 2021;Zhang et al., 2022c), importance sampling Neal (2001); Del Moral et al. (2006); Naesseth et al. (2019) and variational inference (Blei et al., 2017;Naesseth et al., 2018;Maddison et al., 2017;Le et al., 2018;Zimmermann et al., 2021). Recent work on GFNs by takes a similar view by treating the reward function as an energy-based model, which can be trained to maximize the data likelihood following a contrastive divergence-based approach (Hinton, 2002), while the forward-and backward transition models are trained by optimizing the TB objective. In this work we show that, in certain settings, optimizing the TB objective is indeed equivalent to optimizing a forward-or reverse Kullback-Leibler divergence. To this end we compare the TB objective when optimized with samples generated form the forward transition model, backward transition model, or a mixture of both, to training with a corresponding variational objective, which takes the form of a convex combination of a forward-and reverse Kullback-Leibler divergence. We identify cases in which the TB objective is equivalent to the corresponding variational objective and leverage this insight to employ variance reduction techniques from variations inference. Finally, we run experiments, to evaluate our theoretical findings and the empirical performance of the trajectory balance and corresponding variational objective. Zhang et al. (2022a) identifies equivalences between GFNs and certain classes of generative models. The authors observe that hierarchical variational auto-encoders are equivalent to a special class of GFNs, and that training hierarchical latent variable models with the forward KL divergence between the full backward-and forward transition model of the GFN is equivalent to training a hierarchical VAE by maximizing its ELBO. Related Work Recent work by In concurrent and independent work, derive the same equivalences between optimizing the TB objective and forward-and reverse KL divergence that we establish in this work. The difference with our work is that we propose a novel composite objective based on a convex combination of the reverse and forward Kullback-Leibler divergences. Furthermore, we discuss and study this objective in context of learning energy-based models. Finally, we also study the differences between variational inference and trajectory balance optimization when the forward and backward trajectory distributions share parameters. Generative Flow Networks Generative flow networks (Bengio et al., 2021a;b) generate trajectories τ = (s 0 , s 1 , . . . , s T , s f ) along the edges of a directed acyclic graph G = (S, E). Each trajectory starts in the root, s 0 , and terminates in a terminating state, s T , before transitioning to a special final state, s f , which is the single leaf node of G. A non-negative reward signal R(s T ) is assigned to each terminating state s T . The task is to learn a sampling procedure, or flow, for simulating trajectories, such that the marginal distribution of reaching the terminating state s T is proportional to R(s T ). We adopt the convention that s f = s T +1 . The structure of the DAG imposes a partial order, <, on states s, s ∈ S such that s < s if s is an ancestor of s . Hence, any trajectory satisfies s j < s k for 0 ≤ j < k ≤ T + 1 and consequently does not contain loops. Trajectory Flows A trajectory flow is a non-negative function F G : T → R + on complete trajectories T , i.e. trajectories starting in a initial state s 0 and ending in the final state s f associated with a DAG G. Below, we drop the graph subscript for notational convenience. A trajectory flow defines a probability measure P over complete trajectories, such that for any event A ⊆ T P (A) = F (A) Z , F (A) = τ ∈A F (τ ), Z = τ ∈T F (τ ), where Z can be interpreted as the total amount of flow. The flow F (s) through a state and the flow F (s → s ) along an edge (s, s ) are denoted by F (s) := F ({τ ∈ T : s ∈ τ }), F (s → s ) := F ({τ ∈ T | ∃t ∈ N : s = s t , s = s t+1 ∈ τ }). The probability of a trajectory containing the state s, and the forwardand backward transition probabilities are denoted by P (s) := F (s) Z , P F (s | s) := P (s → s | s) := F (s → s ) F (s) , P B (s | s ) := P (s → s | s) = F (s → s ) F (s ) . A flow is referred to as a Markovian flow if its corresponding probability measure satisfies P (s → s | τ ) = P (s → s | s) for any consecutive states s, s and partial trajectory τ = (s 0 , . . . , s) ending in s. For a Markovian flow and complete trajectory τ ∈ T we have, P (τ ) = T t=0 P F (s t+1 | s t ) = T t=0 P B (s t | s t+1 ). For a comprehensive study of flows and generative flow networks we refer to Bengio et al. (2021b). Training Generative Flow Networks We are considering GFNs, which parameterize a Markovian flow on a DAG by modeling forward transition probabilities P F (s |s; φ), together with a normalizing constant Z ψ which can be interpreted as an approximation to the total amount of flow. The trajectory flow is F (τ ; φ, ψ) = Z ψ T t=0 P F (s t+1 | s t ; φ) = T t=0 F (s t → s t+1 ; φ, ψ) T −1 t=0 F (s t+1 ; φ, ψ) = Z ψ T t=0 P B (s t | s t+1 ; φ), . For a reward function R, the goal is to find transition probabilities such that P B (s T | s f ; φ) = R(s T )/Z.In some scenarios we want to fix the backward transition model, e.g. a uniform distribution model can be advantageous for exploration, or parameterize it with a distinct set of parameters θ. In this case, the forward and backward transition probabilities do not correspond to the same flow and, under slight overload of notation, we refer to P B (s | s ; θ) as the backward transition probabilities. Bengio et al. (2021a) originally proposed objectives to train GFNs based on the flow matching conditions and a detailed balance condition. observe that optimizing these may lead to inefficient credit propagation to early transitions, especially for long trajectories. To alleviate this, propose an alternative TB objective for complete trajectories L TB (τ, λ) = log Z ψ T t=0 P F (s t+1 |s t ; φ) R(s T ) T −1 t=0 P B (s t |s t+1 ; θ) 2 = log Z ψ Q(τ ; φ) ZP (τ ; θ) 2 ,(1) where λ = (φ, θ, ψ) and we define P (τ ; θ) := R(s T ) Z T −1 t=0 P B (s t | s t+1 ; θ), Q(τ ; φ) := T t=0 P F (s t+1 | s t ; φ). Trajectories τ are sampled from a proposal distribution q with full support over the space of trajectories T . The TB objective is optimized using stochastic gradient descent. The gradient w.r.t. all parameters λ = (φ, θ, ψ) is computed as the average over a batch of S i.i.d. samples. Solutions correspond to fixed points of the (negative) expected gradient E τ ∼q d dλ L TB (τ, λ) = 0. We can compute an unbiased estimate of this gradient using samples from the proposal distribution, g TB (λ) := 1 S S s=1 d dλ L TB (τ s , λ), τ s ∼ q. In section 4, we show how optimizing GFNs using the TB objective corresponds to variational inference on complete trajectories. Going forward, we refer to the probability mass functions Q(τ ; φ) and P (τ ; θ) over complete trajectories as forward and backward model, respectively. Variational Inference The problem of finding corresponding forward and backward transition probabilities can alternatively be phrased as a variational inference problem. The goal is to find parameters φ and θ such that the difference between the forward and backward transition probabilities, measured by a suitable divergence, is minimized. Two commonly used divergence measures are the forward Kullback-Leibler divergence (FKL) and reverse Kullback-Leibler divergence (RKL), L RKL (φ, θ) := KL(Q(· ; φ) | P (· ; θ)) = E τ ∼Q log Q(τ ; φ) P (τ ; θ) = E τ ∼Q [− log w] ,(2)L FKL (φ, θ) := KL(P (· ; θ) | Q(· ; φ)) = E τ ∼P log P (τ ; θ) Q(τ ; φ) = E τ ∼P [log w] ,(3) with the importance weights w := P (τ ; θ)/Q(τ ; φ). The divergences can be optimized using stochastic gradient descent with gradients estimated from samples from the forward model Q and backward model P , respectively. In most setting, samples from P are not readily available and one has to resort other techniques to generate approximate samples, e.g. using importance sampling or MCMC. Computing the derivative of L RKL w.r.t. parameters θ of the backward transition model is straightforward, the dependence only appears in the log-weights. We can approximate the resulting expected gradient using S samples from the forward model, d dθ L RKL (φ, θ) = E τ ∼Q − d dθ log P (τ ; θ) ≈ g θ RKL (φ, θ) := 1 S S s=1 − d dθ log P (τ s ; θ), τ s ∼ Q(·; φ). Similarly, the derivative of L FKL w.r.t. parameters φ of the forward transition model and corresponding gradient estimator g φ RKL are d dφ L FKL (φ, θ) = E τ ∼P − d dφ log Q(τ ; φ) ≈ g φ FKL (φ, θ) := 1 S S s=1 − d dφ log Q(τ s ; φ), τ s ∼ P (·; θ). Computing derivative of L RKL w.r.t. φ and derivative of L FKL w.r.t. θ on the other hand involves computing a so-called score-function gradient, d dφ L RKL (φ, θ) = τ ∈T log Q(τ ; φ) P (τ ; θ) d dφ Q(τ ; φ) + d dφ log Q(τ ; φ) P (τ ; θ) Q(τ ; φ) = τ ∈T log Q(τ ; φ) P (τ ; θ) Q(τ ; φ) d dφ log Q(τ ; φ) + Q(τ ; φ) d dφ log Q(τ ; φ) = E τ ∼Q (− log w + 1) d dφ log Q(τ ; φ) = E τ ∼Q − log w d dφ log Q(τ ; φ) Importantly, we can cancel-out the additional score-function term (last equality of above equation) as E τ ∼Q [a d dφ log Q(τ ; φ)] = 0 for any constant a. The corresponding score-function gradient estimator is thus g φ RKL (φ, θ) := 1 S S s=1 − log w s d dφ log Q(τ s ; φ), w s := P (τ s ; θ) Q(τ s ; φ) , τ s ∼ Q(·; φ). Analogously, we can compute a score function gradient of L FKL w.r.t. θ and corresponding estimator E τ ∼P log w d dθ log P (τ ; θ) ≈ g θ FKL (φ, θ) := 1 S S s=1 log w s d dθ log P (τ s ; θ), τ s ∼ P (·; θ). Score-function gradient estimators can exhibit high variance (Ranganath et al., 2013), which can be problematic for learning variational approximations via stochastic gradient descent, and hence it is often essential to employ additional variance reduction techniques. Variance reduction techniques for score-function estimators A commonly used technique to reduce the variance of score-function estimators is to use a control variate h (Ross, 1997) to replace the gradient estimator g with the modified estimator g = g + c(h − E h ), where c is a scaling parameter. Control variates leave the expected value of the gradient estimator g unchanged, E[g] = E[g ] , but has the potential to reduce the variance. Indeed, for a given control variate h we can minimize the variance of g Var[g ] = Var[g] + c 2 Var[h] − 2cCov[g, h](4) with respect to the scaling c: c * = arg min c Var[g ] = Cov[g , h] Var[h] . The score function d dφ log Q(τ ; φ) (Ranganath et al., 2013) is a useful and easy to compute control variate when optimizing the reverse KL divergence, which we will use as our running example. Using the score function as a control variate simplifies the expression of the resulting gradient estimator such that the scaling c can simply be added to the (negative) log-importance weight, g = 1 S S i=s − log w s d dφ log Q(τ s ; φ) g +c d dφ log Q(τ s ; φ) − E d dφ log Q(τ ; φ) =0 = 1 S S i=s − log w s + c d dφ log Q(τ s ; φ). Monte Carlo Estimation. We can estimate the optimal scaling with the same S i.i.d. samples τ s ∼ Q(τ s ; θ) used to estimate g. However, in order for the gradient estimator to remain unbiased, we have to employ a leave-one-out (LOO) estimatorĉ s (Mnih and Rezende, 2016), which only makes use of samples {τ s | s = s}, such that E 1 S S i=s − log w s +ĉ s d dφ log Q(τ s ; φ) =E − log w s d dφ log Q(τ s ; φ) . The leave-on-out estimate of the optimal scaling for the d-th dimension of c * iŝ c * d,s = Cov s [g d , h d ] Var s [h d ] , where Cov s [·, ·], and Var s [·] are empirical LOO covariance and variance estimates, respectively. Note that estimating the optimal scaling requires access to per-sample gradients and hence requires S forward-backward passes on the computations graph in many reverse-mode automatic differentiation frameworks. Two popular non-optimal scaling choices that are easily computed and do not require access to gradient information are c log w = E[log w] and c log Z = log E[w] with corresponding LOO estimatorŝ c log w s := 1 S − 1 S s =1,s =s log w s ĉ log Z s := log 1 S − 1 S s =1,s =s w s . Interestingly, forĉ log w one can show that it is sufficient to only compute the fixed scalingĉ log w and instead correct by a factor S−1 S to obtain an unbiased estimate of g , 1 S S i=1 − log w s +ĉ log w s d dφ log Q(τ s ; φ) = 1 S − 1 S i=1 − log w s + 1 S S j=1 log w j ĉ log w d dφ log Q(τ s ; φ). In Section 4 we show how we can leverage these variance reduction techniques for training GFNs by identifying scenarios in which training GFNs with the TB objective is equivalent to performing variational inference with a score-function gradient estimator. Variational Inference for Generative Flow Networks The trajectory balance objective and variational objectives, introduced in 3.1, all try to find a forward model Q and backward model such that P (τ ; θ) = π T (s T ) T −1 t=0 P B (s t | s t+1 ; θ) ≈ T t=0 P F (s t+1 | s t ; φ) = Q(τ ; φ), and hence terminating states s T which are approximately distributed according to π T , which is proportional to the reward R. While the TB objective can be optimized with samples from any proposal distribution that has full support on T , it is commonly optimized with samples from either the forward model τ F ∼ Q or the backward model τ B ∼ P . Similarly, variational inference commonly optimizes the RKL divergence or FKL divergence, which can be estimated by sampling from the forward model and reverse model, respectively. propose a special case of the trajectory balance objective using a proposal that first samples a Bernoulli random variable u ∼ B(α). This variable then determines whether the trajectory samples are drawn from the forward model or the backward model. The corresponding expected gradient is E u∼B(α) [u = 0]E τ ∼P (·;θ) d dλ L TB (τ, λ) + [u = 1]E τ ∼Q(·;φ) d dλ L TB (τ, λ) (5) =αE τ B ∼P (·;θ) d dλ L TB (τ, λ) + (1 − α)E τ ∼Q(·;φ) d dλ L TB (τ, λ) .(6) We can approximate the expected gradient by approximating the expectation w.r.t. the forward and backward model for any backward ratio α ∈ [0, 1], which is equivalent to optimizing a weighted sum of TB objectives, L αTB (τ F , τ B , λ) := αL TB (τ B , λ) + (1 − α)L TB (τ F , λ), where τ F ∼ Q(·; φ) and τ B ∼ P (·; θ). We can similarly define a convex combination of the two KL divergences, which penalizes the RKL objective and FKL objective with (1 − α) and α, respectively, L αKL (φ, θ, α) =(1 − α)L RKL (φ, θ) + αL FKL (φ, θ). Like RKL and FKL, this is a divergence which is non-negative and zero if and only if P = Q. We are now equipped to compare the various objectives for different setting of α and different parameterizations of the forward and backward model. Specifically, we will differentiate between two settings: (1) the setting where P F and P B (and hence Q and P ) have distinct parameters φ and θ respectively, and (2) the setting where P F and P B share parameters η = φ = θ. The expected gradient of L αTB can be computed as the convex combination of the expected gradient of the TB objective w.r.t. samples from the forward model and the expected gradient w.r.t. samples from the backward model (see Equation 5). Similarly, L αKL can be computed as convex combination of L RKL and L F KL . Thus, in the following we study the cases α = 0 and α = 1 separately and results for 0 < α < 1 follow accordingly. Forward model and backward model with shared parameters If the forward and reverse model share parameters η = (φ, θ), e.g. when they are parameterized by the same GFN, the expected gradient of the TB objective (Equation 1) takes the form E τ ∼q(·;η) d dλ L TB (τ, λ) = −2E τ ∼q(·;η) log w + log Z Z ψ d dψ log Z ψ + d dη log Q(τ ; η) − d dη log P (τ ; η) , where the proposal q is either the forward model Q(τ ; φ) (α = 0) or backward model P (τ ; θ) (α = 1). The corresponding gradients of the RKL and FKL divergences are d dη L RKL (η) = −E τ ∼Q(·;η) log w + c d dη log Q(τ ; η) + d dη log P (τ ; η) , d dη L FKL (η) = E τ ∼P (·;η) log w + c d dη log P (τ ; η) − d dη log Q(τ ; η) , where c is a scaling parameter as discussed in Section 3.1. Forward model and backward model with distinct parameters Sampling from the forward model (α = 0). In the case where we are using samples from the forward model τ ∼ Q(·; φ) only, the expected TB gradients reduce to E τ ∼Q(·;φ) d dφ L TB (τ, λ) = −2E τ ∼Q(·;φ) log w + log Z Z ψ d dφ log Q(τ ; φ) , E τ ∼Q(·;φ) d dθ L TB (τ, λ) = 2E τ ∼Q(·;φ) log w + log Z Z ψ d dθ log P (τ ; θ) , E τ ∼Q(·;φ) d dψ L TB (τ, λ) = 2E τ ∼Q(·;φ) log w + log Z Z ψ d dψ log Z ψ . Interestingly, the expected gradient w.r.t. φ does not depend on log Z ψ and is proportional to the gradient of the standard score-function gradient for the reverse KL-divergence d dφ L RKL (φ, θ) = −E τ ∼Q(·;φ) log w + c d dφ log Q(τ ; φ) = 1 2 E τ ∼Q(·;φ) d dφ L TB (τ, λ) . Hence, solutions of the corresponding optimization problem correspond to fixed points of the (negative) expected gradient. Moreover, the term log Z/Z ψ can be interpreted as a learned scaling parameter c ψ for variance reduction similar to the control variates discussed in section 3.1. Optimizing the TB objective w.r.t. parameters of the forward model is equivalent to optimizing a RKL divergence using a score-function estimator with a learned scaling parameter c ψ , updated according to the gradient described above. This insight also suggests that the control variate described in Section 3.1 can be used as an alternative to the learned baseline to reduce the variance of the expected gradient estimates of the trajectory balance objective. The expression of the gradient of the RKL w.r.t. parameters of the backward model θ differs from the expected gradient of the corresponding TB objective d dθ L RKL (φ, θ) = −E τ ∼Q(·;φ) d dθ log P (τ ; θ) . The integrand differs by a multiplicative factor log w + c ψ . Intuitively, if the likelihood of a sample is higher under the backward transition model P than under the forward transition model Q by more than predicted by −c ψ = log(Z ψ /Z), then log w + c ψ < 0 and the TB objective tries to increase the likelihood of the sample under P and vice versa. In contrast, the gradient of the RKL objective tries to always maximize the likelihood of samples under the backward transition model, which achieves its global maximum for P = Q. Due to the fact that τ P (τ ; θ) = 1, increasing the probability of P (τ ; θ) for some τ decreases the probability of other trajectories indirectly. Hence, while both objectives have the same global minima for flexible enough Q and P , their optimization dynamics may differ. Sampling from the backward model (α = 1). When samples are taken from the backward model τ ∼ P (·; θ) the expected TB gradients reduce to E τ ∼P (·;θ) d dφ L TB (τ, λ) = −2E τ ∼P (·;θ) log w + log Z Z ψ d dφ log Q(τ ; φ) , E τ ∼P (·;θ) d dθ L TB (τ, λ) = 2E τ ∼P (·;θ) log w + log Z Z ψ d dθ log P (τ ; θ) , E τ ∼P (·;θ) d dψ L TB (τ, λ) = 2E τ ∼P (·;θ) log w + log Z Z ψ d dψ log Z ψ . Here, a similar observation holds. The expected gradient, w.r.t. θ, of the TB objective is proportional to the corresponding gradient of the forward KL-divergence w.r.t. parameters θ d dθ L FKL (φ, θ) = E τ ∼P (·;θ) log w d dθ log P (τ ; θ) = 1 2 E τ ∼P (·;θ) d dθ L TB (φ, θ, τ ) . Again, solutions of the corresponding optimization problem correspond to fixed points of the (negative) expected gradient. Moreover, analogously to the previous case, optimizing the TB objective w.r.t. θ is equivalent to optimizing a FKL divergence w.r.t. θ using a score-function estimator with a learned scaling parameter c ψ . The expression of the gradient of the FKL w.r.t. parameters of the forward model φ analogously differs from the expected gradient of the corresponding TB objective by a factor log w + c ψ in the integrand, d dφ L FKL (φ, θ) = −E τ ∼P (·;θ) d dφ log Q(τ ; φ) . Observing the expected gradients of the TB objective and corresponding gradients of the RKL and FKL shows that in certain cases optimizing the TB objective is equivalent to variational inference using reverse or forward KL divergences. This observation also suggests that we can leverage the various variance reduction techniques for score-function estimators developed in the variational inference literature. Experiments We have shown that for certain settings, optimizing the αTB objective is equivalent to optimizing the αKL objective, in the sense that the fixed points are the same and the expected gradient of the αTB objective is proportional to the gradient of the αKL objective. In these settings we can use the variance reduction techniques for score-function gradient estimators to reduce the variance of the expected gradients of the TB objective. In settings where optimizing the αTB objective and αKL objective is not equivalent, it is not immediately clear if optimizing the αKL objective is advantageous over optimizing the αTB objective, or vice versa. In the following we compare the performance of the αTB and αKL objective with a learned baselinê c φ := log Z φ or LOO baselineĉ log Z s for different values of α. Then, we can estimate the marginal likelihood of the data under the the forward model using importance sampling, 1 N N i=1 P F (x|s i T −1 ; φ)P F (s i 1:T −1 |s i 0 ; φ) P B (s i 0:T −1 | x; θ) , s i 0:T −1 ∼ P B (s 0:T −1 | x; θ).(7) If no data is available we will report the expected log-weight E τ ∼Q(·;φ) [log w] ≤ log Z. where |s| denotes the number of set bits in s. With these definitions on place we define a DAG G(S, E) that specifies the structure of the state space. For mathematical convenience, we map the states s to numeric representationss in which ∅, 0 and 1 are replaced by 0, −1 and 1 respectively. This allows us to compute the number of set bits |s| = d |s d |, and the location and type of the bit added by a transition s → s as the signed one-hot vectors −s. We can also compute state ¬s (s, s ) =s − (s −s) that results from flipping the newly added bit in s . These operations are useful for defining the transition model. Transition model. We consider a fixed backward transition model P B (s t | s t+1 ) which uniformly at random select a set bit and replaces it with ∅. The forward transition model P F (s t+1 | s t ; φ) uniformly at random selects ∅-bit and and replaces it with a bit value sampled from a Bernoulli distribution whose (logit) parameters are the output of a function f φ : S × S → R + . The corresponding probability mass functions of the forward-and backward transition model are P B (s t | s t+1 ) = 1 |s t+1 | , P F (s t+1 |s t ; φ) = 1 D − |s t | f φ (s t ) f φ (s t ) + f φ (¬s (s t , s t+1 )) . In practice f φ : R D → R D×2 is a vector valued function parameterized by an Multilayer Perceptron (MLP) with weights φ. Given a state s t , it produces D pairs of logits associated with positions in the state vector. The state s t+1 is required only to compute the position d of the added bit, which is used to select the corresponding logits f φ (s) d ∈ R 2 . Synthetic densities To model a discrete target distribution π T over terminating states we follow Dai et al. (2020); and discretize a continuous distribution π cont GT : R 2 → R + into 2 16 equally sized grid cells along each dimension. The cells are remapped to Gray code such that neighbouring grid cells differ in exactly one bit and the resulting pair of 16-bit vectors is concatenated to obtain a single 32-bit vector. We are interested in two settings: (1) Learning a forward model Q(τ ; φ) such that its marginal distribution Q T (s T ; φ) approximates a fixed distribution π T (s T ) over terminating states, and (2) learning a forward model jointly with an energy function ξ : {0, 1} 32 → R such that the discretized ground truth density π GT ≈ π T (s T ; θ) ∝ exp(−ξ(s T , θ)). We optimize the energy function by maximizing the negative loglikelihood via stochastic gradient descent, interleaving gradient updates to the forward model and energy function. We approximate the gradient of the log-marginal likelihood using a contrastive divergence-based approach (Hinton, 2002), which replaces the expectation w.r.t. π T with an expectation w.r.t. the marginal distribution of a K-step Metropolis-Hastings (MH) chain m(x | x) initialized at data x, − d dθ log π T (s T ; θ) = d dθ (ξ(s T ; θ) + log Z θ ) = d dθ ξ(s T ; θ) − E s T ∼π T (·;θ) [ξ(s T ; θ)]E x∼U (X ) d dθ ξ(x; θ) − E x ∼m(x |x) [ξ(x ; θ)] . The MH updates uses the GFN to construct proposals . For K → ∞ this gradient update recovers the expected gradient of the log-marginal likelihood. We evaluate the αTB objective and αKL objective for different values of α and two different control variates, a learned (LRN) control variate c ψ and estimated control variate c log Z using a leave-one-out estimator (LOO). For each α we consider two settings: 1) jointly learning the energy function and parameters of the GFN, and 2) using a previously learned fixed energy function and learning parameters of the GFN only. We find that, unsurprisingly, for α = 0, in which case optimizing the αKL objective is equivalent to optimizing the αKL objective with a learned control variate, both objectives perform comparably (numbers within one standard deviation) in terms of negative log-likelihood (see Table 1). For 0 < α < 1, both objective perform similarly, with αTB having a slight edge over αKL in terms of negative log-likelihood. Interestingly, for α = 1, i.e. when sampling from backward model only, the performance of αTB drops significantly while the performance of the αKL objective remains stable. Ising model We are modeling a discrete distribution over terminating states s T ∈ {−1, 1} D corresponding to the grid cells of an Ising model, where A is the adjacency matrix of a N × N (D = N 2 ) grid with periodic boundary conditions, and β is interaction strength. In this setting, to obtain a suitable numeric representationss of the states s we only need to map ∅-bit to 0. π T (s T ) ∝ exp(−βH(s T )), H(s T ) = − 1 2 s T A N s T ,(8) As we do not have access to ground truth samples from the Ising model, we are training the GFN with α = 0. In this setting, optimizing the αTB objective and αKL objective is equivalent and hence we focus on the effect of replacing the learned baseline, log Z φ used in the original αTB objective, with a LOO control variate typically used to reduce the variance in score-function estimators. We report the expected log-weights (see Table 2) for different values of β (averaged over 10 trained GFNs), and show samples from a GFN and samples generated by running a MH chain for qualitative comparison in Figure 5.2. We find no significant difference in performance between the learned baseline and LOO control variate across different values of β. Conclusion In this paper, we draw connections between the recent literature on generative flow networks and the literature on variational inference methods. We observe that GFNs can be trained using variational objectives that minimize a divergence between a forward and a backward distribution over trajectories. When minimizing the reverse Kullback-Leibler divergence, the objective is analogous to that used in standard variational inference methods that maximize a lower bound on the log-marginal likelihood (Blei et al., 2017). When minimizing the forward Kullback-Leibler divergence, we obtain a variant of the objective that is commonly used in wake-sleep methods and related approaches (Hinton et al., 1995;Bornschein and Bengio, 2015;Naesseth et al., 2021). It is also possible to optimize a convex combination of the two. These objectives are closely related to the trajectory-balance objective that is typically used when training GFNs. Specifically, the gradient of the RKL is proportional to computing the expected gradient of the TB objective with respect to trajectories that are sampled from the forward distribution. Evaluations on synthetic densities and an Ising model demonstrate that variational objectives for GFNs achieve a comparable performance in terms of the expected log weight relative to variants of the trajectory balance objective. This observation opens up opportunities to explore new variational objectives for GFNs that incorporate credit assignment methods Schulman et al. (2015) as well as importance sampling methods for GFNs based on e.g. variational sequential Monte Carlo (Naesseth et al., 2018) or nested variational inference (Zimmermann et al., 2021). Evaluation metrics .PP metricsIf samples from the target distribution π T are available we can sample trajectories from the backward model conditioned on x. Let P B (s 0:T −1 | s T ; θ) B (s t | s t+1 ; θ) and P F (s 1:T −1 |s 0 ; φ) F (s t+1 |s t ; φ). Structure and representation of the state space Following Zhang et al. (2022b) we target a discrete distribution over terminating states on S T = {0, 1} D by consecutively sampling values in {0, 1} for each step. To this end we define the state space S = {∅, 0, 1} D ∪ {s f }, where ∅ indicates that no bit value has been sampled for the corresponding position yet. We further define edges E = {(s, s ) : s ∈ S \ {s f } ∧ s ∈ S (s)} ∪ {(s, s f ) : s ∈ S T }, S (s) = {s ∈ S \ {s f } : |s| = |s | − 1}, Figure 1 : 1Negative log-likelihood during training for a fixed energy function (pre-trained on 2spirals) and different values of α. For α = 1 the αTB objective performs significantly worse than the αKL divergence. Figure 2 : 2Approximate samples from Ising model running MH chains and forward model of a trained GFN Table 1 : 1Negative log-likelihood of test data under GFN for learned baseline and different backward ratio α.Method 2spirals 8gaussians 2spirals (fixed ξ) Table 2 : 2Expected log-weights of αTB with different control variates for ten Ising models with different interaction strengths β.β=-1. β=-0.8 β=-0.6 β=-0.4 β=-0.2 αTB (LRN, α=0.0) 183.997±22.010 153.512±13.550 112.511±3.967 42.454±1.905 -60.925±0.388 αTB (LOO, α=0.0) 174.101±41.934 144.964±20.893 102.232±21.148 42.742±1.984 -62.970±0.276 β=0.2 β=0.4 β=0.6 β=0.8 β=1 αTB (LRN, α=0.0) -60.900±0.389 40.707±3.733 112.189±4.139 144.608±21.020 174.262±23.712 αTB (LOO, α=0.0) -62.844±0.405 40.932±1.941 97.109±24.809 153.425±13.999 190.531±19.436 Flow Network based Generative Models for Non-Iterative Diverse Candidate Generation. Emmanuel References, Moksh Bengio, Maksym Jain, Doina Korablyov, Yoshua Precup, Bengio, Advances in Neural Information Processing Systems. Curran Associates, Inc34References Emmanuel Bengio, Moksh Jain, Maksym Korablyov, Doina Precup, and Yoshua Bengio. Flow Network based Generative Models for Non-Iterative Diverse Candidate Generation. In Advances in Neural Information Processing Systems, volume 34, pages 27381-27394. Curran Associates, Inc., 2021a. URL https:// proceedings.neurips.cc/paper/2021/hash/e614f646836aaed9f89ce58e837e2310-Abstract.html. . Yoshua Bengio, Tristan Deleu, Edward J Hu, Salem Lahlou, Mo Tiwari, Emmanuel Bengio, arXiv:2111.09266arXiv: 2111.09266GFlowNet Foundations. cs, statYoshua Bengio, Tristan Deleu, Edward J. Hu, Salem Lahlou, Mo Tiwari, and Emmanuel Bengio. GFlowNet Foundations. arXiv:2111.09266 [cs, stat], November 2021b. URL http://arxiv.org/abs/2111.09266. arXiv: 2111.09266. Variational inference: A review for statisticians. M David, Alp Blei, Jon D Kucukelbir, Mcauliffe, Journal of the American statistical Association. 112518David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians. Journal of the American statistical Association, 112(518):859-877, 2017. Reweighted wake-sleep. Jörg Bornschein, Yoshua Bengio, International Conference on Learning Representations. Jörg Bornschein and Yoshua Bengio. Reweighted wake-sleep. In International Conference on Learning Representations, 2015. Learning Discrete Energybased Models via Auxiliary-variable Local Exploration. Hanjun Dai, Rishabh Singh, Bo Dai, Charles Sutton, Dale Schuurmans, Advances in Neural Information Processing Systems. H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. LinCurran Associates, Inc33Hanjun Dai, Rishabh Singh, Bo Dai, Charles Sutton, and Dale Schuurmans. Learning Discrete Energy- based Models via Auxiliary-variable Local Exploration. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 10443-10455. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper/2020/file/ 7612936dcc85282c6fa4dd9d4ffe57f1-Paper.pdf. Sequential Monte Carlo samplers. Pierre Del Moral, Arnaud Doucet, Ajay Jasra, 10.1111/j.1467-9868.2006.00553.x._eprintJournal of the Royal Statistical Society: Series B (Statistical Methodology). 683Pierre Del Moral, Arnaud Doucet, and Ajay Jasra. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(3):411-436, 2006. ISSN 1467-9868. doi: 10.1111/j.1467-9868.2006.00553.x. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467- 9868.2006.00553.x. Bayesian Structure Learning with Generative Flow Networks. Tristan Deleu, António Góis, Chris Chinenye Emezue, Mansi Rankawat, Simon Lacoste-Julien, Stefan Bauer, Yoshua Bengio, The 38th Conference on Uncertainty in Artificial Intelligence. Tristan Deleu, António Góis, Chris Chinenye Emezue, Mansi Rankawat, Simon Lacoste-Julien, Stefan Bauer, and Yoshua Bengio. Bayesian Structure Learning with Generative Flow Networks. In The 38th Conference on Uncertainty in Artificial Intelligence, June 2022. URL https://openreview.net/forum?id=HElfed8j9g9. Improving Generative Flow Networks with Path Regularization. Anh Do, Duy Dinh, Tan Nguyen, Khuong Nguyen, Stanley Osher, Nhat Ho, arXiv:2209.15092cs, statAnh Do, Duy Dinh, Tan Nguyen, Khuong Nguyen, Stanley Osher, and Nhat Ho. Improving Generative Flow Networks with Path Regularization, September 2022. URL http://arxiv.org/abs/2209.15092. arXiv:2209.15092 [cs, stat]. Training products of experts by minimizing contrastive divergence. Geoffrey E Hinton, 10.1162/089976602760128018Neural Computation. 148Geoffrey E. Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation, 14(8):1771-1800, August 2002. ISSN 0899-7667. doi: 10.1162/089976602760128018. URL https://doi. org/10.1162/089976602760128018. The" wake-sleep" algorithm for unsupervised neural networks. Geoffrey E Hinton, Peter Dayan, J Brendan, Radford M Frey, Neal, Science. 2685214Geoffrey E Hinton, Peter Dayan, Brendan J Frey, and Radford M Neal. The" wake-sleep" algorithm for unsupervised neural networks. Science, 268(5214):1158-1161, 1995. Learning Deep Latent Gaussian Models with Markov Chain Monte Carlo. Matthew D Hoffman, PMLRProceedings of the 34th International Conference on Machine Learning. the 34th International Conference on Machine LearningMatthew D. Hoffman. Learning Deep Latent Gaussian Models with Markov Chain Monte Carlo. In Proceedings of the 34th International Conference on Machine Learning, pages 1510-1519. PMLR, July 2017. URL https://proceedings.mlr.press/v70/hoffman17a.html. ISSN: 2640-3498. The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo. D Matthew, Andrew Hoffman, Gelman, Journal of Machine Learning Research. 15Matthew D Hoffman and Andrew Gelman. The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15, 2014. Biological Sequence Design with GFlowNets. Moksh Jain, Emmanuel Bengio, Alex-Hernandez Garcia, Jarrid Rector-Brooks, F P Bonaventure, Chanakya Dossou, Jie Ekbote, Tianyu Fu, Micheal Zhang, Dinghuai Kilgour, Lena Zhang, Payel Simine, Yoshua Das, Bengio, arXiv:2203.04115cs, q-bioMoksh Jain, Emmanuel Bengio, Alex-Hernandez Garcia, Jarrid Rector-Brooks, Bonaventure F. P. Dossou, Chanakya Ekbote, Jie Fu, Tianyu Zhang, Micheal Kilgour, Dinghuai Zhang, Lena Simine, Payel Das, and Yoshua Bengio. Biological Sequence Design with GFlowNets, March 2022. URL http://arxiv.org/abs/ 2203.04115. arXiv:2203.04115 [cs, q-bio]. Auto-encoding sequential monte carlo. Maximilian Tuan Anh Le, Tom Igl, Tom Rainforth, Frank Jin, Wood, International Conference on Learning Representations. Tuan Anh Le, Maximilian Igl, Tom Rainforth, Tom Jin, and Frank Wood. Auto-encoding sequential monte carlo. In International Conference on Learning Representations, 2018. Approximate Inference with Amortised MCMC. Yingzhen Li, Richard E Turner, Qiang Liu, arXiv:1702.08343cs, statYingzhen Li, Richard E. Turner, and Qiang Liu. Approximate Inference with Amortised MCMC, May 2017. URL http://arxiv.org/abs/1702.08343. arXiv:1702.08343 [cs, stat]. Learning GFlowNets from partial episodes for improved convergence and stability. Kanika Madan, Jarrid Rector-Brooks, Maksym Korablyov, Emmanuel Bengio, Moksh Jain, Andrei Nica, Tom Bosc, Yoshua Bengio, Nikolay Malkin, arXiv:2209.12782cs, statKanika Madan, Jarrid Rector-Brooks, Maksym Korablyov, Emmanuel Bengio, Moksh Jain, Andrei Nica, Tom Bosc, Yoshua Bengio, and Nikolay Malkin. Learning GFlowNets from partial episodes for improved convergence and stability, September 2022. URL http://arxiv.org/abs/2209.12782. arXiv:2209.12782 [cs, stat]. Filtering variational objectives. J Chris, John Maddison, George Lawson, Nicolas Tucker, Mohammad Heess, Andriy Norouzi, Arnaud Mnih, Yee Doucet, Teh, Advances in Neural Information Processing Systems. Chris J Maddison, John Lawson, George Tucker, Nicolas Heess, Mohammad Norouzi, Andriy Mnih, Arnaud Doucet, and Yee Teh. Filtering variational objectives. In Advances in Neural Information Processing Systems, pages 6573-6583, 2017. Nikolay Malkin, Moksh Jain, Emmanuel Bengio, Chen Sun, Yoshua Bengio, arXiv:2201.13259arXiv: 2201.13259Trajectory Balance: Improved Credit Assignment in GFlowNets. cs, statNikolay Malkin, Moksh Jain, Emmanuel Bengio, Chen Sun, and Yoshua Bengio. Trajectory Balance: Improved Credit Assignment in GFlowNets. arXiv:2201.13259 [cs, stat], January 2022a. URL http: //arxiv.org/abs/2201.13259. arXiv: 2201.13259. GFlowNets and variational inference. Nikolay Malkin, Salem Lahlou, Tristan Deleu, Xu Ji, Edward Hu, Katie Everett, Dinghuai Zhang, Yoshua Bengio, arXiv:2210.00580cs, stat] version: 1Nikolay Malkin, Salem Lahlou, Tristan Deleu, Xu Ji, Edward Hu, Katie Everett, Dinghuai Zhang, and Yoshua Bengio. GFlowNets and variational inference, October 2022b. URL http://arxiv.org/abs/2210.00580. arXiv:2210.00580 [cs, stat] version: 1. Variational inference for monte carlo objectives. Andriy Mnih, Danilo Rezende, International Conference on Machine Learning. PMLRAndriy Mnih and Danilo Rezende. Variational inference for monte carlo objectives. In International Conference on Machine Learning, pages 2188-2196. PMLR, 2016. Variational sequential Monte Carlo. C A Naesseth, S W Linderman, R Ranganath, D M Blei, Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS). the 21st International Conference on Artificial Intelligence and Statistics (AISTATS)Lanzarote, SpainC. A. Naesseth, S. W. Linderman, R. Ranganath, and D. M. Blei. Variational sequential Monte Carlo. In Proceedings of the 21st International Conference on Artificial Intelligence and Statistics (AISTATS), Lanzarote, Spain, Apr 2018. Elements of sequential Monte Carlo. Foundations and Trends® in Machine Learning. C A Naesseth, F Lindsten, T B Schön, Now Publishers, Inc12C. A. Naesseth, F. Lindsten, and T. B. Schön. Elements of sequential Monte Carlo. Foundations and Trends® in Machine Learning, 12(3):307-392, November 2019. Now Publishers, Inc. Markovian Score Climbing: Variational Inference with KL(p||q). Christian A Naesseth, Fredrik Lindsten, David Blei, arXiv:2003.10374arXiv: 2003.10374cs, statChristian A. Naesseth, Fredrik Lindsten, and David Blei. Markovian Score Climbing: Variational Inference with KL(p||q). arXiv:2003.10374 [cs, stat], February 2021. URL http://arxiv.org/abs/2003.10374. arXiv: 2003.10374. Annealed importance sampling. M Radford, Neal, 10.1023/A:1008923215028Statistics and Computing. 112Radford M. Neal. Annealed importance sampling. Statistics and Computing, 11(2):125-139, April 2001. ISSN 1573-1375. doi: 10.1023/A:1008923215028. URL https://doi.org/10.1023/A:1008923215028. . Rajesh Ranganath, Sean Gerrish, David M Blei, arXiv:1401.0118arXiv: 1401.0118Black Box Variational Inferencecs, statRajesh Ranganath, Sean Gerrish, and David M. Blei. Black Box Variational Inference. arXiv:1401.0118 [cs, stat], December 2013. URL http://arxiv.org/abs/1401.0118. arXiv: 1401.0118. . M Sheldon, Ross, Simulation. academic pressSheldon M Ross. Simulation. academic press, 1997. Markov Chain Monte Carlo and Variational Inference: Bridging the Gap. Tim Salimans, Diederik Kingma, Max Welling, PMLRProceedings of the 32nd International Conference on Machine Learning. the 32nd International Conference on Machine LearningTim Salimans, Diederik Kingma, and Max Welling. Markov Chain Monte Carlo and Variational Inference: Bridging the Gap. In Proceedings of the 32nd International Conference on Machine Learning, pages 1218-1226. PMLR, June 2015. URL https://proceedings.mlr.press/v37/salimans15.html. ISSN: 1938-7228. Gradient estimation using stochastic computation graphs. John Schulman, Nicolas Heess, Theophane Weber, Pieter Abbeel, Advances in Neural Information Processing Systems. 28John Schulman, Nicolas Heess, Theophane Weber, and Pieter Abbeel. Gradient estimation using stochastic computation graphs. Advances in Neural Information Processing Systems, 28, 2015. Reinforcement Learning, second edition: An Introduction. Richard S Sutton, Andrew G Barto, 978-0-262-35270-3MIT PressRichard S. Sutton and Andrew G. Barto. Reinforcement Learning, second edition: An Introduction. MIT Press, November 2018. ISBN 978-0-262-35270-3. Probabilistic inference for solving (PO)MDPs. Marc Toussaint, Stefan Harmeling, Amos Storkey, Neural Computation. 31Marc Toussaint, Stefan Harmeling, and Amos Storkey. Probabilistic inference for solving (PO)MDPs. Neural Computation, 31(December):357-373, 2006. Unifying Generative Models with GFlowNets. Dinghuai Zhang, Ricky T Q Chen, Nikolay Malkin, Yoshua Bengio, arXiv:2209.02606cs, statDinghuai Zhang, Ricky T. Q. Chen, Nikolay Malkin, and Yoshua Bengio. Unifying Generative Models with GFlowNets, September 2022a. URL http://arxiv.org/abs/2209.02606. arXiv:2209.02606 [cs, stat]. Generative Flow Networks for Discrete Probabilistic Modeling. Dinghuai Zhang, Nikolay Malkin, Zhen Liu, Alexandra Volokhova, Aaron Courville, Yoshua Bengio, PMLRProceedings of the 39th International Conference on Machine Learning. the 39th International Conference on Machine LearningDinghuai Zhang, Nikolay Malkin, Zhen Liu, Alexandra Volokhova, Aaron Courville, and Yoshua Bengio. Generative Flow Networks for Discrete Probabilistic Modeling. In Proceedings of the 39th International Conference on Machine Learning, pages 26412-26428. PMLR, June 2022b. URL https://proceedings. mlr.press/v162/zhang22v.html. ISSN: 2640-3498. Transport score climbing: Variational inference using forward KL and adaptive neural transport. Liyi Zhang, David M Blei, Christian A Naesseth, arXiv:2202.01841Liyi Zhang, David M. Blei, and Christian A. Naesseth. Transport score climbing: Variational inference using forward KL and adaptive neural transport. arXiv:2202.01841, 2022c. Nested Variational Inference. Heiko Zimmermann, Hao Wu, Babak Esmaeili, Jan-Willem Van De Meent, Advances in Neural Information Processing Systems. 3420423Heiko Zimmermann, Hao Wu, Babak Esmaeili, and Jan-Willem van de Meent. Nested Varia- tional Inference. In Advances in Neural Information Processing Systems, volume 34, pages 20423-
[]
[ "Dozer: Migrating Shell Commands to Ansible Modules via Execution Profiling and Synthesis", "Dozer: Migrating Shell Commands to Ansible Modules via Execution Profiling and Synthesis" ]
[ "Eric Horton [email protected] ", "North Carolina ", "Chris Parnin [email protected] ", "North Carolina ", "\nState University Raleigh\nNCUSA\n", "\nState University Raleigh\nPittsburghNC, PAUSA, USA\n" ]
[ "State University Raleigh\nNCUSA", "State University Raleigh\nPittsburghNC, PAUSA, USA" ]
[]
Software developers frequently use the system shell to perform configuration management tasks. Unfortunately, the shell does not scale well to large systems, and configuration management tools like Ansible are more difficult to learn. We address this problem with Dozer, a technique to help developers push their shell commands into Ansible task definitions. It operates by tracing and comparing system calls to find Ansible modules with similar behaviors to shell commands, then generating and validating migrations to find the task which produces the most similar changes to the system. Dozer is syntax agnostic, which should allow it to generalize to other configuration management platforms. We evaluate Dozer using datasets from open source configuration scripts.
10.1109/icse-seip55303.2022.9793935
[ "https://arxiv.org/pdf/2203.12065v1.pdf" ]
247,619,047
2203.12065
8029519204bbcfcd1d2e8d4374996a3911291ea5
Dozer: Migrating Shell Commands to Ansible Modules via Execution Profiling and Synthesis Eric Horton [email protected] North Carolina Chris Parnin [email protected] North Carolina State University Raleigh NCUSA State University Raleigh PittsburghNC, PAUSA, USA Dozer: Migrating Shell Commands to Ansible Modules via Execution Profiling and Synthesis 10.1145/3510457.3513060ACM Reference Format: Eric Horton and Chris Parnin. 2022. Dozer: Migrating Shell Commands to Ansible Modules via Execution Profiling and Synthesis. In 44nd International Conference on Software Engineering: Software Engineering in Practice (ICSE-SEIP '22), May 21-29, 2022, Pittsburgh, PA, USA. ACM, New York, NY, USA, 2 pages. https://MigrationConfiguration ManagementShellAnsibleSystem CallStraceLinux Software developers frequently use the system shell to perform configuration management tasks. Unfortunately, the shell does not scale well to large systems, and configuration management tools like Ansible are more difficult to learn. We address this problem with Dozer, a technique to help developers push their shell commands into Ansible task definitions. It operates by tracing and comparing system calls to find Ansible modules with similar behaviors to shell commands, then generating and validating migrations to find the task which produces the most similar changes to the system. Dozer is syntax agnostic, which should allow it to generalize to other configuration management platforms. We evaluate Dozer using datasets from open source configuration scripts. DOZER Using Bash scripts to manage infrastructure is, according to Netflix engineer Lorin Hochstein, "like the dark side of the force: quicker, easier, and more seductive, but not the right way to go" [7]. Despite this, developers frequently begin their configuration management journey with the shell because it is a familiar environment that provides them with a "quick and dirty" solution to their configuration management needs [4,12]. Many of these developers will eventually discover that the shell is not without its growing pains and seek to integrate a full configuration management system like Ansible [2,3,8]. This happened to NASA when they migrated 65 legacy applications to the cloud and realized that their shell-based process made it difficult to do seemingly simple tasks like managing user accounts [5]. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. Dozer 1 is designed to address the often recommended approach of converting an existing shell script into an Ansible playbook: manually, one command at a time [1,6,9,10]. It does so by accepting a shell command and returning an Ansible task that makes similar configuration changes to the system. Dozer is based on the insight that shell commands and Ansible modules can only change the system state by communicating with the kernel via the system call (syscall) interface. This shared interface provides an opportunity to observe and compare the behavior of different executables. Before the Dozer migration pipeline begins, we collect system call traces (straces) for many Ansible module executions. These execution definitions and straces form the core of the Dozer knowledge base. When a developer wishes to migrate a shell command to Ansible, Dozer will compare the strace of the shell command to those in its knowledge base to find a collection of Ansible modules with similar behavior. We use a comparison scheme that attempts to map shell command parameters to Ansible module parameters for the best overall match, since recorded executions in the knowledge base likely used different parameters than the command to migrate, and weights syscall matches based on their information content to emphasize more infrequent syscalls [11]. Figure 1 gives a high-level view of how two straces are compared. Finally, Dozer uses the similar modules and parameter mappings discovered during the comparison step to generate different migrations of the shell command into an Ansible module. The shell command and each migration are executed against the same Docker image, and the migration with the most similar resulting system changes is selected. EVALUATION We evaluated Dozer on its ability to migrate 62 common shell commands found in open source Dockerfiles to Ansible modules. The Dockerfiles were sourced from projects on GitHub. Ansible modules were traced using the DebOps suite. 2 Migrations were assessed on Dozer's ability to select the correct Ansible module, select the correct module parameters, and to correctly map source to target parameters if applicable. Overall, Dozer successfully chose the correct target module and parameters for 38 of the 62 commands being migrated. Figure 2 shows an example of one such migration of an echo shell command into an Ansible module (Figure 2a). Dozer first finds the definition of an Ansible module with similar behavior by comparing the strace of the shell command to the recorded straces of Ansible modules in its knowledge base. The similar module and inferred parameter mapping are used to generate the final migration to Ansible's Figure 1: A high-level depiction of the strace comparison between the shell command rm and the Ansible module file. Dozer first searches for instances of parameters within syscalls, then determines the mapping that will result in the best score. Finally, it matches equivalent syscalls between the straces and assigns an overall comparison score based on the weighted scores of the matched syscalls. lineinfile module (Figure 2c). Note that lineinfile will append to the end of the file if its regexp parameter is not matched, so the final migration has the same effect on the system in a clean starting environment. Some unsuccessful migrations were a result of Ansible not having a module that directly supported the same behavior as the shell command or were a result of the correct module not appearing in Dozer's knowledge base because it was not used in the DebOps suite we traced. In other cases, unsuccessful migrations resulted from Dozer being unable to detect similar behavior. For example, the shell command mkdir -p <path> splits the path argument into its component parts while the Ansible file module uses absolute paths. This mismatch adversely affects Dozer's comparison procedure. DISCUSSION Dozer presents a novel approach to migrating individual configuration tasks. We believe that this is a critical first step towards being able to migrate entire configuration scripts, since modern configuration management languages like Ansible, Puppet, Chef, etc. are composed of individual building blocks (Ansible modules, Puppet/Chef resources, . . . ) that operate at approximately the same level of scope. However, there are additional challenges that need to be addressed in order to scale up to full configuration scripts. Dozer works by profiling a program's behavior based on its interaction with the syscall interface, allowing it to operate without an explicit domain knowledge of the underlying configuration script. This approach collects very little information about the system itself or the changes being made (outside of the final validation for similarity). We expect that migrations could be improved by incorporating additional information about changes to the system state into the search process. Additional work is needed to solve the problem of composing configuration tasks, which is necessary when a task in one configuration language is equivalent to a sequence of two or more tasks in another. Notable challenges with composition include the correct propagation of information as outputs and inputs, selecting tasks that are "compatible" such that they don't conflict with each other or overwrite desired changes, and preserving control flow and error handling. ICSE-SEIP '22, May 21-29, 2022, Pittsburgh, PA, USA © 2022 Copyright held by the owner/author(s). Publication rights licensed to ACM. ACM ISBN 978-1-4503-9226-6/22/05. . . $15.00 https://doi.org/10.1145/3510457.3513060 echo 'daemon off;' >> /etc/nginx/nginx.conf(a) An echo shell command that writes a configuration value to a line in a file. lineinfile: dest: '/root/.profile' regexp: '^.*mesg n.*$' line: 'tty -s && mesg n || true' state: 'present' (b) An Ansible module with detected similar behavior to the shell command in Figure 2a. lineinfile: dest: '/etc/nginx/nginx.conf' regexp: '^.*mesg n.*$' line: 'daemon off;' state: 'present' (c) Dozer's final migration of the shell command in Figure 2a into an Ansible module. Figure 2 2Figure 2 https://github.com/config-migration/dozer 2 https://docs.debops.org/en/stable-1.2/ ACKNOWLEDGMENTSThis work is funded in part by the NSF SHF grant #1814798. Bash scripts to Ansible. 2018. Bash scripts to Ansible. https://www.reddit.com/r/ansible/comments/ a1qpr0/bash_scripts_to_ansible/. Ansible versus BASH script. 2020. Ansible versus BASH script. https://www.reddit.com/r/linuxadmin/ comments/emcuqm/ansible_versus_bash_script/. NASA: INCREASING CLOUD EFFICIENCY WITH ANSIBLE AND ANSIBLE TOWER. Ansible, Ansible. 2016. NASA: INCREASING CLOUD EFFICIENCY WITH ANSIBLE AND ANSIBLE TOWER. https://www.ansible.com/hs-fs/hub/330046/file-1649288715- pdf/Whitepapers__Case_Studies/nasa_ansible_case_study.pdf. Shell Scripts to Ansible. Allen Eastwood, Allen Eastwood. 2018. Shell Scripts to Ansible. https://www.ansible.com/blog/ shell-scripts-to-ansible. . Lorin Hochstein, Lorin Hochstein. 2015. https://twitter.com/norootcause/status/ 679731676193230849. Configuration Management is an Antipattern. Jonah Horowitz, Jonah Horowitz. 2017. Configuration Management is an Antipattern. https:// hackernoon.com/configuration-management-is-an-antipattern-e677e34be64c. Matt Jaynes, Shell Scripts vs Ansible: Fight!. Matt Jaynes. 2013. Shell Scripts vs Ansible: Fight! https://hvops.com/articles/ ansible-vs-shell-scripts/. How to get started using Ansible. Luke Rawlins, Luke Rawlins. 2018. How to get started using Ansible. https://sudoedit.com/how- to-get-started-using-ansible/. A mathematical theory of communication. C E Shannon, The Bell System Technical Journal. 27C. E. Shannon. 1948. A mathematical theory of communication. The Bell System Technical Journal 27, 3 (1948), 379-423. Tortoise: Interactive System Configuration Repair. Aaron Weiss, Arjun Guha, Yuriy Brun, Proceedings of the 32Nd IEEE/ACM International Conference on Automated Software Engineering. the 32Nd IEEE/ACM International Conference on Automated Software EngineeringUrbana-Champaign, IL, USA; Piscataway, NJ, USAIEEE PressAaron Weiss, Arjun Guha, and Yuriy Brun. 2017. Tortoise: Interactive System Configuration Repair. In Proceedings of the 32Nd IEEE/ACM International Con- ference on Automated Software Engineering (Urbana-Champaign, IL, USA) (ASE 2017). IEEE Press, Piscataway, NJ, USA, 625-636. http://dl.acm.org/citation.cfm? id=3155562.3155641
[ "https://github.com/config-migration/dozer" ]
[ "Karina Reshetova Quantum Algorithm for Researching the Nearest (QARN)", "Karina Reshetova Quantum Algorithm for Researching the Nearest (QARN)" ]
[]
[]
[]
Processing large amounts of data to this day causes difficulties due to the lack of power resources. Classical algorithms implement a chain of actions, requiring a certain time to execute, as well as space in the form of RAM. Parallelization, if it can be used, allows to gain time, but also needs buffering of all parallel actions. Quantum computing acts as an attractive alternative to parallel computing with qubits, qudits and their distinctive properties. The quantum algorithm proposed in this paper allows to search for the best (closest to a given) element in a random data array by storing all its initial elements in a superposition. This allows to perform the search operations on all elements at the same time and due to the same to save the amount of RAM.
null
[ "https://export.arxiv.org/pdf/2304.10976v1.pdf" ]
258,291,418
2304.10976
8fc7d89766ae5c04a174cb0e48c6aef2a169d244
Karina Reshetova Quantum Algorithm for Researching the Nearest (QARN) Karina Reshetova Quantum Algorithm for Researching the Nearest (QARN) Processing large amounts of data to this day causes difficulties due to the lack of power resources. Classical algorithms implement a chain of actions, requiring a certain time to execute, as well as space in the form of RAM. Parallelization, if it can be used, allows to gain time, but also needs buffering of all parallel actions. Quantum computing acts as an attractive alternative to parallel computing with qubits, qudits and their distinctive properties. The quantum algorithm proposed in this paper allows to search for the best (closest to a given) element in a random data array by storing all its initial elements in a superposition. This allows to perform the search operations on all elements at the same time and due to the same to save the amount of RAM. Introduction Quantum computing has been fascinated by the prospect of using quantum effects for decades [1]. In particular, Shor's algorithm, based on quantum Fourier transform, demonstrates the use of qubit phase as an additional source for storing information [2,3]. The same technique is implemented in the phase estimation algorithm [4] and the algorithm for solving systems of linear equations, known as HHL [5]. These and other quantum algorithms have their own field of application and related limitations, but the tools implemented in their schemes can be combined and used autonomously to solve different problems: the use of phase, superposition, qubit entanglement, transition to multilevel qudits [1,[6][7][8]. All these are powerful tools of quantum technologies, allowing to obtain advantages in comparison with classical computations. As for the data search problem, the most well-known approach to its optimization contains Grover's algorithm [9]. However, even here we have to face a number of difficulties in implementation, such as determination of the number of calls to this algorithm, which affects its accuracy; the number of possible solutions, which must also be taken into account, and also the presence of a hypothetical oracle function, or black box (also used in Deutsch-Jozsa algorithm and the so-called Simon problem [10,11]), whose action is described only in the abstract. This scheme is an excellent illustrative example of quantum superiority and is often used as demonstration and training material. As mentioned above, Grover's algorithm searches for the desired element in a data array K and works with a certain structure of it containing the whole range of values from k0 = 0 to kn-1 = 2 n -1 (K = [ki, ki+2, …, kn-3], where n is the number of array element bits). This method imply that at least one element that exactly matches the sought one must exist. In practice, the array can contain only a part of the full spectrum (K' = [ki, ki+2, …, kn-3]), and if the very existence of the value exactly coinciding with the desired solution is not guaranteed, it is necessary to find the nearest, most suitable solution in the database with a minimum error. For this purpose, the following method is proposed and containing the following actions: a) receiving as inputs an array A of m elements of size n and a reference value B of size n; b) creating, with the help of quantum logic gates, a superposition of all m elements of the array A in a special ancilla of n qubits and l d-level qudits. The qubits serve as a buffer for copying and storing the array A and occupy a memory size equal to one element of the array being copied. The qudits act as an auxiliary tool to create superposition: each state of the qudits is entangled with a single copy element of the array A; c) the bitwise implementation of finding the element nearest to B occurs simultaneously over all m elements, and also allows us to ignore the influence of the sign of the calculated difference. The result of comparing each element is recorded as a change in the probability of getting a qudit state entangled with the given element: the smaller the element matches the search conditions, the smaller the probability of getting a qudit state entangled with this element when measured; d) the measurement of the qudits of ancilla. The resulting state will indicate the number of the element being searched for. The essence of the problem Let there exist an array A = [A0, A1, …, Am-1] and some value B, for which it is required to find in the array A either an exact match, or the nearest. Obviously, you should subtract B from each element in turn, fix the value of difference at each step, at the end choose the smallest difference and by its index refer to the element of the array A, taken by this way the nearest to the reference value B. In the classical version of calculations it is necessary to create a local copy (A') for array A of identical size. When searching for the smallest difference it would be necessary to perform successive calculations on each element of the copy and in addition to that to take into account the sign of the difference, which requires additional resources in the form of time. Execution of QARN Figure 1 shows a schematic diagram, implemented on a quantum computer, for finding the nearest value (to a given B) in an array A, where each of m elements consists of n qubits. The reference value B also consists of n qubits. Figure 1 To create a local copy of the array A = ∑ , we will use ancillas C and D, where ancilla C occupies n qubits and ancilla D contains l d-level qudits (D0, D1, … Dl-1). The number of possible states of the ancilla D is proportional to the array A, i.e., d l = m. All qubits and qudits of the ancilla are initiated by zeros, the memory blocks A0…Am and B receive at input the values which are required to be processed by the task condition. The scheme works as follows. The first step creates a superposition of all possible states in ancilla D by means of Hadamard gates, each of which is described by the formula [12]: H = 1 √ , ( )( ) | ⟩⟨ | Thus a superposition of all states is formed in the ancilla D: 1 (|0 … 0 0 ⟩ + |0 … 0 1 ⟩ + ⋯ + |( − 1) … ( − 1) ( − 1) ⟩) For convenience, each of the ancillary states D will be denoted by d'j, j ∈ {0 … d l −1}. The first step is necessary to entangle one element of a copy of the array A (placed in the ancilla C) with each of the obtained states of the auxiliary ancilla D. Which is why d l must be equal to m. The entanglement of the qubits is carried out in step I ( Figure 1) by means of controlling gates, where the controlling states are the values |1⟩ of the qubits of element Ak and the states d'k of the ancilla D such that Ck (the k th superposition state of ancilla C) becomes equal to Ak: | ⟩| ⟩| ⟩ … | ⟩|0 0 … 0 ⟩|0 0 … 0 ⟩ → ⊗ → ⊗ 1 2 | ⟩| ⟩| ⟩ … | ⟩|0 0 … 0 ⟩(|0 0 … 0 ⟩ + |0 0 … 1 ⟩ + | … ⟩) → → 1 2 | ⟩| ⟩| ⟩ … | ⟩(| ⟩|0 0 … 0 ⟩ + | ⟩|0 0 … 1 ⟩ + ⋯ + | ⟩ | … ⟩) In step II, a sequence of control gates making turns in the ancilla D is invoked. Here the main task is to find the closest value by calculating the difference between B and each element of the array A stored in the ancilla C. In doing so, the probability amplitude for each state d'j entangled with Cj decreases in proportion to the difference B -Cj obtained. An abstraction of the described action is shown in Figure 2. Figure 2 If we represent all N = 2 n possible states of some system as basic orthogonal vectors of the hyperspace of states, the superposition vector will be the result vector equidistant from the basic ones, respectively, when measuring this system the probability of getting each of its possible states is the same and equal to : | ⟩ = 1 2 (|0 0 … 0 ⟩ + |0 0 … 1 ⟩ + ⋯ + |1 1 … 1 ⟩) = | ⟩ The weakening of the probability of the k th state | ⟩ means the shift of the result vector | ⟩ towards the hyperplane α of all other basis vectors (∑ | ⟩ − | ⟩) and away from the vector | ⟩. The difference B -Ak = B -Ck is written as a rotation around the X-axis in ancilla D using rotation gates such that the probability of a state | ⟩ entangled with | ⟩ decreases in proportion to the value of B -Ak. The comparison is bitwise, so the i th bit of the n-bit value of B is subtracted from the i th bit of the n-bit ancilla C. The value of the rotation is discrete, depends on the significance of the bits being compared and is equal to : the difference of the higher bits is written as a rotation by π/2, the next bits by π/4, etc. The difference of the low bits will rotate the state of the ancilla D by . As mentioned above, due to the superposition of states concentrated in ancilla C, the difference B -Aj is counted simultaneously over all elements. At the end, ancilla D is measured, and the resulting state entangled with a particular element of the array will point to that sought element. A general view of the space rotation matrix for the attenuated probability amplitude Cj is shown below: Example for 2-level qudit Figure 3 details a scheme for the case of n=3, m=2 and a two-level qudite D (d = 2) corresponding to a classical qubit. Figure 3 The superposition creation function is a sequence of Toffoli and Pauli X gates that rotate the state of the qubit around the X axis by π: ( ) = cos 2 − sin 2 − sin 2 cos 2 The transformations after step I are as follows: | ⟩| ⟩| ⟩|000⟩|0⟩ → √ | ⟩| ⟩| ⟩|000⟩(|0⟩ + |1⟩) → √ | ⟩| ⟩| ⟩(| ⟩|0⟩ + | ⟩|1⟩), where the ancilla CD will have state √ (| ⟩|0⟩ + | ⟩|1⟩). Then it is necessary to determine nearest to | ⟩ and strengthen the probability to get the state | ⟩ ⟩ when measuring ancilla D by writing the difference for each B -Aj as a rotation around the X axis with controlled gates Rx. The gate provides a rotation only if the values of the qubits with the same significance are not equal to each other. That is, if you compare states 101 and 011, the qubit rotation will be at the expense of the high and middle bits, since the values of the low bits coincide and are equal to 1. The direction of rotation is initialized in any convenient way, only the sign of the difference determines the change of direction: if at a negative result the rotation is chosen clockwise, then at a positive one must be chosen counterclockwise, and vice versa. This is necessary for the correct calculation of the difference, so 101 -011 = 010, which in this example is equivalent to + π/2 -π/4 = π/4. If all turns are made in the same direction, the angle will be equal to 3π/4, which is equivalent to 110, and this is incorrect. Thus, if the state Aj is entangled with the state of a special D qubit was |0⟩ with each turn the probability of the value |1⟩ will increase in proportion to the difference B -Aj. It does not matter whether the total (by all digits) turn occurs clockwise or counterclockwise (more is subtracted from less or vice versa), because eventually in both cases the probability will "move away" from |0⟩ and get closer to |1⟩. Consequently, the need to consider the sign is discarded. The following values will be taken as an example: | ⟩ = |101⟩, | ⟩ = |010⟩, | ⟩ = |110⟩; − = 101 − 010 = 011. The result 011 should be written in qubit D as a rotation around the X-axis, the abstraction of the amplitude amplification | ⟩|1⟩ is shown in Figure 4. Figure 4 In calculating − = 101 − 110 = −001 the difference is negative, which does not affect the absolute value of the probability amplitude of state | ⟩|0⟩ corresponding to a rotation angle of π/4 ( Figure 4). As you can see, there is no need to consider the sign of the difference, because in this case it only affects the direction of motion: clockwise or counterclockwise (+π/2 or -π/2). 101 -010 will be equal to 011 -110: the probability of getting |0⟩ when measuring the ancilla D will decrease towards |1⟩ in both cases equally. The matrix of the rotation gates for this example will have the form shown below: Conclusion Taking into account all the arithmetic calculations after measuring the ancilla D the probability of getting |0⟩ is ~37% and probability of getting |1⟩ is ~63%. So, with a probability of ~63% depending on the proportions of differences − and − , the nearest to B value will be determined by the element A1, which means that the goal by using the proposed scheme is achieved. Gratitude I want to express my gratitude to the free application «Quantum Computing» by hex@dec: https://play.google.com/store/apps/details?id=hu .hexadecimal.quantum&hl=en_AU&gl=US This application allowed me to perform verification calculations to make sure that my algorithm works. I also want to express my gratitude to Yuri Ozhigov for his basic knowledge in this field. given | ⟩ = |101⟩, | ⟩ = |010⟩, | ⟩ = |110⟩ the operation of the circuit shown inFigure 4is described by equation: Three principles of quantum computing. Yuri Ozhigov, Quantum Inf. Comput. 22Yuri Ozhigov, Three principles of quantum computing, Quantum Inf. Comput. Vol.22 P Shor, Algorithms for Quantum Computation: Discrete Logarithms and Factoring, Foundations of Computer Science, 35th Annual Symposium on -IEEE.  [2] P. Shor, Algorithms for Quantum Computation: Discrete Logarithms and Factoring, Foundations of Computer Science, 35th Annual Symposium on -IEEE (1994), pp.124- 134. An approximate Fourier transform useful in quantum factoring. D Coppersmith, IBM Research ReportD. Coppersmith, An approximate Fourier transform useful in quantum factoring, IBM Research Report (1994). A Kitaev, arXiv:quant-ph/9511026Quantum measurements and the Abelian Stabilizer Problem. A. Kitaev, Quantum measurements and the Abelian Stabilizer Problem, arXiv:quant- ph/9511026 (1995). Quantum algorithm for solving linear systems of equations. Aram W Harrow, Avinatan Hassidim, Seth Lloyd, Phys. Rev. Lett. 10315Aram W. Harrow, Avinatan Hassidim, Seth Lloyd, Quantum algorithm for solving linear systems of equations, Phys. Rev. Lett., 103 (15) (2008). Asymptotically improved circuit for a d-ary Grover's algorithm with advanced decomposition of the nqudit Toffoli gate. Amit Saha, Ritajit Majumdar, Debasri Saha, Amlan Chakrabarti, Susmita Sur-Kolay, Phys. Rev. A. 10562453Amit Saha, Ritajit Majumdar, Debasri Saha, Amlan Chakrabarti, and Susmita Sur-Kolay, Asymptotically improved circuit for a d-ary Grover's algorithm with advanced decomposition of the n- qudit Toffoli gate, Phys. Rev. A 105, 062453 (2022). Qudits and High-Dimensional Quantum Computing. Yuchen Wang, Zixuan Hu, Barry C Sanders, Sabre Kais, Front. Phys. Yuchen Wang, Zixuan Hu, Barry C. Sanders, Sabre Kais, Qudits and High-Dimensional Quantum Computing, Front. Phys (2020). The Deutsch-Josza algorithm for n-qudits. G Mogos, 7th WSEAS International Conference on APPLIED COMPUTER SCIENCE. Venice, ItalyG. Mogos, The Deutsch-Josza algorithm for n-qudits, 7th WSEAS International Conference on APPLIED COMPUTER SCIENCE, Venice, Italy (2007). A fast quantum mechanical algorithm for database search. K Lov, Richard Grover ; David Deutsch, Jozsa, Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing. STOC '96. the Twenty-Eighth Annual ACM Symposium on Theory of Computing. STOC '96Philadelphia, PennsylvaniaRapid solutions of problems by quantum computationLov K. Grover, A fast quantum mechanical algorithm for database search, Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing. STOC '96. Philadelphia, Pennsylvania (1996), pp.212-219.  [10] David Deutsch, Richard Jozsa, Rapid solutions of problems by quantum computation, Proceedings of the Royal Society of London A. (1992), pp.553-558. On the Power of Quantum Computation. Daniel R Simon, SIAM J. Comput. 265Daniel R. Simon, On the Power of Quantum Computation, SIAM J. Comput. 26 (5) (1997). . M Chizzini, L Crippa, A Chiesa, F Tacchino, F Petiziol, I Tavernelli, P Santini, S , M. Chizzini, L. Crippa, A. Chiesa, F. Tacchino, F. Petiziol, I. Tavernelli, P. Santini, and S. Molecular nanomagnets with competing interactions as optimal units for qudit-based quantum computation. Carretta, Phis. Rev. Research. 443135Carretta, Molecular nanomagnets with competing interactions as optimal units for qudit-based quantum computation, Phis. Rev. Research 4, 043135 (2022).
[]
[ "A topological perspective on singular canards for critical sets with transverse intersections", "A topological perspective on singular canards for critical sets with transverse intersections" ]
[ "Riccardo Bonetto ", "Hildeberto Jardón-Kojakhmetov " ]
[]
[]
This paper gives a new perspective on singular canards, which is topological in flavor. One key feature is that our construction does not rely on coordinates; consequently, the conditions for the existence of singular canards that we provide are purely geometric. The singularities we study originate at the self-intersection of curves of equilibria of the unperturbed system. Our contribution even allows us to consider degenerate cases of multiple pairwise transverse intersecting branches of the critical set. We employ stratification theory and algebraic geometric properties to provide sufficient conditions leading to the presence of singular canards. By means of two examples, we corroborate our findings using the well-known blow-up technique.2020 Mathematics Subject Classification. ...
null
[ "https://export.arxiv.org/pdf/2304.10822v1.pdf" ]
258,291,601
2304.10822
1b0001bb2686afcdc673130b6a167b182534160c
A topological perspective on singular canards for critical sets with transverse intersections Riccardo Bonetto Hildeberto Jardón-Kojakhmetov A topological perspective on singular canards for critical sets with transverse intersections This paper gives a new perspective on singular canards, which is topological in flavor. One key feature is that our construction does not rely on coordinates; consequently, the conditions for the existence of singular canards that we provide are purely geometric. The singularities we study originate at the self-intersection of curves of equilibria of the unperturbed system. Our contribution even allows us to consider degenerate cases of multiple pairwise transverse intersecting branches of the critical set. We employ stratification theory and algebraic geometric properties to provide sufficient conditions leading to the presence of singular canards. By means of two examples, we corroborate our findings using the well-known blow-up technique.2020 Mathematics Subject Classification. ... Introduction The set of zeros, or equilibria, of a vector field is one of the most fundamental objects to study in order to understand the behaviour of integral curves, or solutions. For this reason, one is also interested in the effect of small perturbations of the former vector field on the set of equilibria. Looking back in history, Fenichel contributed to the development of the theory for normally hyperbolic manifolds and their persistence [10,11,19]. A remarkable result from Fenichel's theory is that hyperbolic equilibria perturb to invariant objects with the same stability properties. We can therefore say that the behaviour near hyperbolic equilibria is well-understood. However, zeros are not always hyperbolic. So, a further aim of geometric singular perturbation theory (GSPT) has been to understand the properties of non-hyperbolic equilibria. It is worth mentioning that, if a zero is semihyperbolic by a centre manifold reduction [3,27] one can reduce the study to a fully non-hyperbolic point. A particular class of non-hyperbolic equilibria on which we will focus is the nilpotent one 1 . The presence of a nilpotent singularity can induce the exceptional appearance of canard solutions, i.e., solutions that unexpectedly follow a repelling set for a non-negligible amount of time (more detailed explanations will be given later in the text.) Canards have a long history as well [2,8,7,25], that reach modern times with the application of the blow-up (see appendix B). Our aim is to give a coordinate-free description, in line with coordinate-free GSPT [28], of canard solutions crossing singularities arising at the intersection of two, or more, curves of equilibria. We adapt tools and results from algebra and topology to the problem under consideration. A coordinate-free and topological perspective could set the path to further generalisation of the theory to settings that appear to be challenging under current techniques including e.g., high-dimensions, and degenerate problems. Singular Canards We consider a two-dimensional real polynomial vector field (1) X = X 0 + X 1 , where is a small parameter, 0 < 1. We omit higher order terms of the form n X n , n > 1, as they do not contribute to the following analysis. The critical variety is the set defined by C := {p ∈ R 2 | X 0 (p) = 0}. The vector field X 0 has two orthogonal components, let us call them A and B. That is, in Cartesian coordinates one would have X 0 = A ∂ ∂x + B ∂ ∂y . Let V (A), V (B) be the algebraic curves defined by setting to zero the polynomials A, B; notice that C = V (A) ∩ V (B). Let us consider the factorisation into irreducible (real) polynomials A = k A k , B = l B l such that V (A) = k V (A k ), V (B) = l V (B l ). Since we are interested in critical varieties arising from the above factorisation, we have the following definition. Definition 2.1. A and B have a common component if ∃ k, l such that A k = B l modulo constants. In particular, we will deal with singular perturbations, which in the current setting appear when the critical variety has dimension 1. Proposition 2.3. The perturbation problem defined by X = X 0 + X 1 is singular only if either of the following hold: (1) A and B are nontrivial and have at least one common component, say A k , B l such that dim (V (A k )) = dim (V (B l )) = 1, (2) A is trivial and dim (V (B)) = 1 (or vice versa). Proof. If A and B are nontrivial and have no common components then V (A) ∩ V (B) is a finite set of points [13]. Then A and B must have a common component. Moreover, to satisfy dim(C) = 1 such a component must be one-dimensional, leading to the first condition. If A is trivial and dim(V (B)) = 1 then the system is in the standard form [24]. Remark 2.4. A relevant case where A and B can have a common component is when there is a Z 2 symmetry of the vector field X 0 . Such a case, together with the higher-dimensional counterparts, arises naturally in the study of coupled cell networks [16], i.e., networked dynamical systems inheriting the symmetry properties of the graph structure. Let us consider the case of singular perturbations. We assume that C has a point, p s , with intersection number equal to one [13,14]. In two dimensions, this last condition prescribes the typical geometry of (nondegenerate) transcritical and pitchfork singularities (later in section 2.1 we consider degenerate cases where at p s several curves intersect). For the rest of the paper, we restrict ourselves to a neighbourhood of p s where there are no other intersection points. Given the assumptions above, the polynomials A and B have two common components F 1 , F 2 such that V (F 1 )∩V (F 2 ) = p s and V (F 1 )∪V (F 2 ) = C. Roughly speaking, V (F 1 ) and V (F 2 ) are two algebraic curves intersecting transversely at the point p s . We define the polynomial F := F 1 F 2 such that V (F ) = C. The set of singular points of V (F ) is denoted by ΣV (F ), and it is given by {p s }. Let us now consider the perturbation term X 1 , for which we assume that X 1 (p s ) = O(1), in order to avoid the emergence of new equilibria in a neighbourhood of p s for 0 < 1. Moreover, we assume that X 1 is such that there are no equilibria of the reduced flow in a sufficiently large neighbourhood of p s , see appendix A. Depending on the structure of the perturbation, and on the stability properties of the critical variety, the system can behave differently around p s . Generically, integral curves that at first are close to an attracting branch of C, when crossing (a neighbourhood of) p s can either follow another attracting branch or be repelled away. However, there is the possibility that a solution follows a repelling branch of C for a non-negligible amount of time; such solutions are called canards. A similar behaviour can be observed for solution transitioning from a repelling branch to an attracting one; those are usually called faux-canards. We are not going to distinguish between the two possible situations. In the following, we state the standard definition of singular canard. Definition 2.5 ( [28,25]). A singular canard is a solution of the reduced problem passing through the singular point with (nonzero) finite speed. However, there is a problem: in our setting, at p s the tangent space of the critical variety is not defined. In order to obtain a well-defined problem, we introduce some concepts from stratification theory, which we adapt to the context of the paper. Definition 2.6 ([4]). A stratification of C is a collection X of smooth manifolds such that (1) X is a local partition of C. (2) ∀S ∈ X, the closure of S in C is the union of S and {S ∈ X | dim(S) = 0}. Definition 2.7 ( [15]). A Whitney stratification is a stratificationX that satisfies the Whitney regularity condition. The Whitney regularity condition ensures that along a stratumS ∈X the topological type ofS does not change; in terms of the map X 0 , the rank of DX 0 |S is maximal along each stratum, where DX 0 denotes the Jacobian. We say that a stratification is minimal if it is refined by any other, unless otherwise stated we always refer to minimal stratifications. For such a reason, the word minimal will be omitted. We recall that any algebraic set in R n admits a Whitney stratification [18]. We define a map, ρ, acting on vector fields on the Whitney stratification,X, such that for the one-dimensional strata it coincides with the projection π (see appendix A) of a vector field along fast fibres, while for the zero-dimensional stratum, it returns the restriction of the vector field to the point. Let us notice that ρ • X 1 gives a stratified vector field on the one-dimensional (Whitney) strata. Next, let us relax the Whitney condition on the stratification of C, and let us consider stratifications X = {S 1 , S 2 , S 3 }, where one of the strata, say S 3 , is now given by the union of two one-dimensional Whitney strata with the zero-dimensional stratum, e.g., S 3 =S 2 ∪S 4 ∪S 5 , see figure 1. Let us remark that, by the definition of stratification, we consider only the combinations that give rise to smooth strata (and are minimal). Figure 1. Stratifications of the critical variety. On the left, the critical variety. In the centre, the Whitney stratification,X, of C. On the right, a possible stratification, X, of C when the Whitney condition is relaxed. Proposition 2.9. A singular canard exists if the vector field ρ•X 1 is a stratified vector field on the relaxed stratification X. C C →XS 2 S 3S 1 S 4 S 5X → X S 1 S 3 S 2 Proof. Let us start with the Whitney stratificationX, and without loss of generality suppose we want to connect the two strataS 2 ,S 4 . We recall that ρ • X 1 is already a stratified vector field onS 2 ,S 4 . At the singular point, p s , the map ρ gives the vector X 1 (p s ). In order to connect with "nonzero velocity" the two stratã S 2 ,S 4 it is necessary that the vector X 1 (p s ) is both tangent at the intersection of the closure of the two strata and compatible with the vector fields on the strata. In terms of stratifications, one considers the stratum S 3 :=S 2 ∪S 4 ∪S 5 belonging to a relaxed stratification X and ask that ρ • X 1 induces a stratified vector field, in particular on S 3 . In this case, S 3 is invariant under the (reduced) flow of ρ • X 1 making S 3 a singular canard according to definition 2.5. Remark 2.10. To better understand the construction of proposition 2.9, we show, in figure 2, a couple of cases where the singular canard connection is not present or not possible. The notation used in the proof of proposition 2.9 has been adapted to correspond to the one used in figures 1 and 2. Essentially, proposition 2.9 states that a singular canard exists when the vector field X 1 at p s connects smoothly the stratified vector fields on the two strata we are connecting. Of course, such stratum corresponds to the solution of the reduced problem crossing the singular point with finite speed. By means of the stratification, the ambiguity arising at the intersection point is removed. Let us recall that the critical variety C can be seen as the union of two smooth Figure 2. Figure 2a shows a case where the vector X 1 (p s ) fails to connect the two strata because it is not tangent to the curve. Indeed the vector field obtained via ρ • X 1 is not a vector field on the smooth curve considered. In figure 2b we can see a case where there exist no vector at p s inducing a smooth vector field, under the condition X 1 (p s ) = O(1). algebraic curves, C = V (F 1 ) ∪ V (F 1 ), Lemma 2.11. If the vector field X 1 satisfies the condition X 1 (p s ) ∧ T ps V (F 1 ) = 0, where ∧ is the exterior product, then (1) admits singular canard solutions. Similarly for V (F 2 ). Remark 2.12. In the standard form, conditions for singular canards in the case of transcritical and pitchfork singularities have been exploited in [23]. A non-standard approach has been used in [5] to prove existence of canards for the unfolding of the vector field considered in [23]. Our condition is in agreement with the results in the aforementioned manuscripts. The analysis conducted in [23,5] not only provides conditions for singular canards but also proves their persistence under (appropriate) small perturbations. The persistence of canard solutions is a relevant result within the topic while relying on a detailed description of the system and rather quantitative techniques. In fact, proving the existence of canards for the unfolded problem usually relies on implicit function theorem arguments, while more quantitative estimates require, for example, an adaptation of Melnikov's method [17]. Our approach is not quantitative in nature and we consider the problem from a wider perspective. For such reasons, there is no straightforward generalisation to the conditions of persistence (which we postpone to future works). Nevertheless, our method easily extends to degenerate singularities with pairwise transversal intersections, which we discuss in the following. 2.1. Degeneracy. In this section, we address some degenerate cases. Let us start by pointing out a desingularisation procedure. Let F be the polynomial associated with the critical variety. In the previous section, we assumed a factorisation into irreducible components of the form F = F i F j . Since we assumed that the system has a singularity with an intersection number equal to one, then the corresponding critical variety is hyperbolic away from the singularity. However, if a term of the factorisation is of the form (F i ) n , n > 1, although the curve described is geometrically the same as the one given by F i , the linearisation is not of full rank along the curve. Lemma 2.13. Let (F i ) n be a term of the factorisation of F , with n = 2k + 1, k ∈ N. The rescaling F → F/(F i ) 2k desingularises the system. Proof. We want to show that near a section of the critical variety induced by (F i ) n , with n = 2k + 1, away from the singularity, the system is topologically equivalent to the regularised system. Considering a small enough section, we can study the system given by (2)ẋ = x 2k+1 + ẏ = α , where α is a real parameter. Equation (2) is obtained by considering the fact that the perturbation in a small enough neighbourhood of the critical variety can be considered constant. The section of the critical variety has been stretched to a straight line, and by a rotation the system is orthogonal to the fast-foliation. Let us point out that for completeness one should also study the cases where the equation forẋ is given by x 2k+1 − and x 2k+1 respectively. However, there is no difference in the analysis we are going to show, for such reason these cases are omitted in the proof. We perform a blow-up of system (2) along the x and directions. The blow-up technique is briefly introduced in appendix B. So, ∀y the singular point (x, ) = (0, 0) become a circle. After desingularisation the motion on the circle is given by (3)ψ = (2k + 1) sin ψ cos 2k+1 (ψ) + sin ψ k cos(2ψ) − k − 1 , where ψ ∈ [0, π], i.e., ψ is restricted to the values corresponding to positive . The equilibria corresponding to the fast-foliation are given by ψ = 0 and ψ = π, these equilibria are obtained by setting to zero the factor sin ψ. We are left to study the equation cos 2k (ψ) + tan ψ = 0. Let us notice that the function cos 2k (ψ) is positive semi-definite, and tan ψ is a monotone function in ] − π/2, π/2[, with image R. Therefore, there is a unique solution in the interval ] − π/2, π/2[. By noticing that the equation cos 2k (ψ) + tan ψ = 0 is periodic, with period π, we can conclude that there is a unique equilibrium for each interval with length π. So, we have that there is a unique equilibrium, ψ * , in [0, π], ∀k. By computing the derivative of (3) we can study the stability properties. The equilibria 0, π are both stable and hyperbolic ∀k. As there is a unique equilibrium on the surface of the blown-up cylinder, we can further deduce that the equilibrium ψ * is a hyperbolic source. So, we can see that system (2) is topologically equivalent to the desingularised version, see figure 3. The case where the equilibria are sinks is analogous. Briefly speaking, terms of the form (F i ) n with odd powers can be easily desingularised and then our results apply. In contrast, even powers lead to a more problematic situation, that is not desingularisable as in the odd case. Heuristically, we could think of the equation x 2 + a, where a is a real constant. Depending on the value of a we can have 0, 1, or 2 solutions, where the unique solution is obtained only for a = 0. So, we deduce that, in general, sections of the critical variety corresponding to terms of the form (F i ) 2k do not persist, but either they disappear, or double. So, even powers require a dedicated analysis that we do not perform here. Now, let us consider the case where the polynomial F admits the factorisation F 1 . . . F N , with N ≥ 2, such that V (F 1 )∪· · ·∪V (F N ) = C, V (F 1 )∩· · ·∩V (F N ) = p s . Definition 2.14. Given N algebraic curves intersecting at p s , we say that the intersection is pairwise transversal if each pair of curves, V (F i ), V (F j ) has intersection number one at p s , ∀i, j ∈ {1, . . . , N }. Pairwise transversality ensures that there are no curves of equilibria having the same tangent at p s . So, under the setting of definition 2.14, we can repeat the procedure done for the case of two curves. The potential singular canards now are N , namely V (F 1 ), . . . , V (F N ). Condition 2.11 will read as follows. Proposition 2.15. If the vector field X 1 satisfies the condition X 1 (p s )∧T ps V (F i ) = 0, then the system admits a singular canard along V (F i ), where i = 1, . . . , N . Remark 2.16. In this example, the existence of the singular canard could be also proven by noticing that the curve y = x/2 is invariant under the flow of X. Considering higher order terms, e.g., 2 X 2 + 3 X 3 + . . . the line y = x/2 would not be invariant anymore. However, our claims and the blow-up picture still apply. For the second example, we consider a degenerate pitchfork singularity given by the unperturbed vector field (6) X 0 = (x + y/2)(x − y/2)(y − x 2 ), 0 . The critical variety is given by a parabola and two straight lines intersecting at the origin. This case is interesting because, unlike the nondegenerate pitchfork, the parabola changes stability. So, we look for the canard following the parabola transitioning from the attracting to the repelling section. Since the tangent to the parabola at the origin is parallel to the x axis, a simple choice for the perturbation satisfying proposition 2.15 is given by (7) X 1 = − (1, x) , Discussion and perspectives In this paper, we studied the presence of singular canards for polynomial vector fields of the from X = X 0 + X 1 , when a singularity due to the intersection of several branches of the critical variety is present. Let us remark that our results could also be applied to smooth (not necessarily polynomial) vector fields by taking the zeroth and first jets in and restricting the domain to a neighbourhood of the singularity. Also, one can extend the results to (smooth) vector fields on Riemannian manifolds, if there exists a local chart covering a neighbourhood of the singularity. Let us recall that canard solutions act as a separatrix between generic behaviours. Therefore, knowing the conditions for canards allows us also to predict the behaviour of the system under a given perturbation. Singularities arising from tangency with the fast-foliation, such as in the case of the fold, have been omitted. In principle, in the case of fold singularities the concept of stratification is not necessary, since there is only one critical curve. By comparison with the results in the literature [22], one can check that the tangency condition we found, see lemma 2.11, works also in the case of the fold. Indeed, one would ask the perturbation to be tangent to the parabola at the fold point. However, some new aspects arise. Notice that, generically, in an -neighbourhood of the fold, the vector field X 0 is of order , since only one curve of equilibria passes through the singular point. Such a property allows coordinate transformations to absorb part of the perturbation terms, leading to potentially more general conditions on the perturbation. For such a reason, we omitted the treatment here and postpone Figure 4a is associated with the vector field X 0 + X 1 given by (4), (5). While figure 4b is associated with the vector field X 0 + X 1 given by (6), (7). The pictures below (4c, 4d) represent the blow-up of the singularities. In particular, we are showing the hemispheres corresponding to positive values of . Along the (invariant) equatorial line the equilibria emerging from the desingularisation process are displayed together with the polar coordinate. The equilibria at 0 and π correspond with the fast-foliation, while the other equilibria can be traced back to the different branches of the critical variety. In figure 4c, we can see the blow-up of the degenerate transcritical singularity for the system displayed in fig.4a. The canard is given by the highlighted solution connecting the attracting equilibrium at π + arctan 1/2 to the repelling equilibrium at arctan 1/2. In figure 4d, we can see the blow-up of the degenerate pitchfork singularity for the system displayed in in fig.4b. Let us notice that in this case one spherical blow-up it is not enough to fully desingularise the system, indeed for a detailed local behaviour near the points ±π/2 further blow-ups are necessary. At any rate, the blow-up we performed is enough to establish the connection (highlighted curve) between the attracting and repelling branches of the parabola, respectively at arctan 1/ϕ and π − arctan 1/ϕ, where ϕ is the golden ratio. the problem to future works. As a further development, we can consider those degenerate cases where the intersection number between two curves is greater than one. For example, when the intersection of two curves is not transversal and so the two curves share the tangent space at the intersection. Unfolding the degenerate problems we have discussed can also be interesting. We conclude by briefly digressing on slow-fast discrete-time planar maps. 3.1. Discrete planar maps. The theory of slow-fast maps is considerably limited compared to the continuous-time counterpart. Nevertheless, in this section, we argue that the theory developed in the main part of this paper holds as well for singularly perturbed discrete-time maps. Particularly relevant for the coming digression are [1,9], which deal with pitchfork and transcritical singularities, respectively. For a recent development towards Discrete GSPT see [21]. Moreover, [26,Appendix A] is particularly useful within the context of normal hyperbolicity of 2-dimensional slow-fast maps. Given the planar vector field (1), we can consider the discrete map obtained via Euler discretisation, namely a map P : q →q defined by (8)q = q + (X 0 + X 1 ) δ, where q ∈ R 2 , δ is the discretisation step, and abusing notation we re-use the notation X i also for the discrete maps. In this setting, the critical set is C = p ∈ R 2 | X 0 (p) = 0 , which geometrically coincides with its continuous-time counter part. In what follows, just as in the main text, we assume that p s ∈ C is an isolated singular point where the branches of the critical set have pairwise intersection number one. The stratification arguments hold the same with the appropriate adaptation for the Whitney regularity condition, that is whenever the linear map DP | C, =0 = Id + δDX 0 has no multipliers on the unit circle. Naturally, for δ = 0 this coincides with the continuous-time regularity condition. In other words, the Whitney stratification of C is the same for the continuous and for the discrete-time problems. We, therefore, use the same notatioñ X and X to denote the Whitney and the relaxed stratifications that we introduced in section 2. Regarding geometric properties, one can verify that in the setting of [21], the one-dimensional strata of C are, as in the continuous-time case, normally hyperbolic. As shown in [21], many of the concepts of the continuous-time setting have discretetime analogues. In particular, one keeps important objects such as the fast-foliation, the layer map, and the reduced map. What is most relevant for our discussion, is that the reduced map in the discrete-time setting has the exact same geometric interpretation as for the continuous-time one. Regarding singular canards, and completely analogous to definition 2.5, we shall say that, for a slow-fast discrete-time map, a singular canard is an orbit, of iterations of the reduced map, passing through the singular point with nonzero finite speed. Moreover, in turn, analogous to definition 2.8, we shall say that a stratified map is a map that assigns to each stratum a smooth map. If we use the exact same notation ρ for the projection along the fast-foliation, we see that the reduced map defined by ρ • X 1 is a stratified map. So, we can now argue that as in proposition 2.9, for discrete-time slow-fast maps a singular canard exists if ρ • X 1 is a stratified map on the (relaxed) stratification X, see also lemma 2.11. Appendix A. Projections and the reduced problem Let us recall here some key concept from GSPT [28]. The vector field X 0 defines a foliation of the phase-space that generically is transverse to the critical variety C, this is the so-called fast-foliation. The reduced problem, or reduced flow, of a singularly perturbed system is defined via the unique projection map, π : T C R 2 → T C, which takes vectors in R 2 with base point in C and projects them onto the tangent space of the critical variety along the fast-foliation generated by X 0 . So, the reduced problem is defined by the vector field π • X 1 on (regular sections of) C. As immediate consequence of the definition of the projection, we have that such map is not well-defined when the fast-foliation is tangent to the critical variety. Such a property can be used to define, for example, fold points and folded singularities. Let us notice that, for the class of problems under our consideration, the singularities arise from the geometrical properties of the critical variety, i.e., there is a point of (self-)intersection. So, in such cases, and in contrast to for example fold points, the projection is not defined because there is no well-defined tangent space of the critical variety at the intersection point. Moreover, from the fact that Whitney strata have maximal rank, we can exclude the possibility of a whole branch of the critical manifold being aligned with the fast-foliation. Appendix B. Blow-Up Given a nilpotent equilibrium point of a vector field, the blow-up is a desingularisation process that transforms the singular point in a higher-dimensional manifold with the aim to retrieve 'more' hyperbolicity. Such a technique has been widely studied and applied with success to many problems concerning singular perturbations [20,24,12,23,5,25,22,6]. Let us briefly describe the spherical blow-up for the setting of this paper. First, we consider not as a parameter, but as a coordinate. So, our two-dimensional vector field, X, now becomes a three-dimensional vector fieldX. The singular point is given by the coordinate of p s together with = 0, we call this nilpotent point p s . The spherical blow-up is given by the transformation φ : R 2+1 → S 2 ps × R + , where S 2 ps is the two-sphere centred atp s . The restriction of the blown-up system to S 2 ps × {0} corresponds to the singular point. However, the transformation φ by itself is not enough to obtain more hyperbolicity on the sphere S 2 ps × {0}. In order to desingularise the system it is necessary to perform a conformal transformation such that the system on S 2 ps × {0} is non-trivial, and in the best scenario fully hyperbolic. +8 cos φ(sin θ + 2 cos θ) . Setting φ = π/2 in (10), one can check that the equatorial line is invariant, and the motion on the equator is given by (11)θ = − 1 16 sin θ(−6 cos(2θ) + 5 cos(4θ) + 5). The equilibria of (11) are θ = 0, π, ±π/4, ±3/4π, ± arctan 1/2, ±(π − arctan 1/2). By setting θ = −π+arctan 1/2 in (10), we obtain the connection between the points ±(π − arctan 1/2) corresponding to the canard. The second example is given by (12)ẋ = (x + y/2)(x − y/2)(y − x 2 ) + ẏ = ẋ = 0 . In this case, we choose a spherical blow-up where the powers of r are calibrated to compensate the parabola's equation, i.e., x → r cos θ sin φ, y → r 2 sin θ sin φ, → r 4 cos φ. The desingularisation is given once again dividing by r 3 . So, the vector field on the sphere at r = 0 reads (13)θ = − 1 2 (cos(2θ) − 8 csc 2 (φ) + 5) 2 sin θ cos 4 (θ)(5 sin φ + sin(3φ)) − 4 cos 2 (θ) sin 2 (θ) cos(2φ) + cos φ + 4 cos φ cot 2 (φ) + 4 sin θ(cos(2φ) + 3) cot φ csc 2 (φ) − 3 sin 2 (2θ) φ = − 16 cos θ cos φ −2 cos(2θ) sin 2 (φ) + 5 cos(2φ) + 11 cos 4 (θ) sin 4 (φ) − sin θ cos 2 (θ) sin 3 (φ) + sin θ sin φ cos φ + cos φ Of course, the equator is invariant and the vector field on it is (14)θ = − −3 sin 2 (2θ) + 8 sin θ cos 4 (θ) + 4 sin 2 (θ) cos 2 (θ) 2(cos(2θ) − 3) . The equilibria of (14) are given by θ = 0, π, ±π/2, arctan 1/ϕ, π − arctan 1/ϕ, where ϕ is the golden ratio. For this example, it is not possible to obtain an analytical expression for the canard. However, one can check that (13) has a symmetry that enforces the presence of the canard. Let us consider the reflection with respect to the meridian passing trough θ = π/2. In terms of coordinates, we first translate the system so that the axis θ = π/2 is at θ = 0 and then we apply the reflection R, such that Rθ → −θ. Let (θ,φ) be the translated system, then the action of the reflection on the vector field gives R(θ,φ) = (θ, −φ). Such a symmetry implies that integral curves are symmetric with respect to reflection along the meridian passing through θ = π/2. Consequently, the canard connection is established. Definition 2.2 ([28]). The perturbation problem (1) is singular if dim(C) = 1. Considering the filtration V (F ) ⊇ ΣV (F ), the Whitney stratification of the critical variety C is given byX = {S 1 ,S 2 ,S 3 ,S 4 ,S 5 }, where, without loss of generality, we can assume that dim(S i ) = 1, for i = 1, 2, 3, 4 and dim(S 5 ) = 0, see figure 1.Clearly, the stratumS 5 is the point of intersection, {p s } = ΣV (F ). Definition 2.8 ([4]). A stratified vector field on a stratification is a map that assigns to each stratum a smooth vector field. intersecting transversality at the singularity. Given the Whitney stratificationX = {S 1 ,S 2 ,S 3 ,S 4 ,S 5 }, there are two possible (relaxed) stratifications X: {V (F 1 ),S 2 ,S 4 }, {V (F 1 ),S 1 ,S 3 } (other combinations would not be smooth). So, from proposition 2.9 we deduce the following condition. Figure 3 . 3A section of V (F 2k+1 i ) after the blow-up. We show the fast-foliation together with the motion on the cylindrical blown-up space. 2. 2 . 2Examples. As we already mentioned in the previous sections, the generic cases of transcritical and pitchfork singularities have been widely studied. As a consequence, a check of agreement between our results and the known conditions for singular canards in the non-degenerate setting is straightforward. Hereby we present two examples of degenerate singularities: a degenerate transcritical and a degenerate pitchfork, both satisfying the conditions of definition 2.14. We are going to verify that, given the conditions of proposition 2.15, we can actually connect an attracting branch of the critical variety with a repelling one. To verify our claims, we employ a known technique: the blow-up, see appendix B. Let us consider the unperturbed vector field (4) X 0 = ((y − x)(y + x)(y − x/2)(y + x/2), 0) The point p s = (0, 0) is at the intersection of four lines of equilibria. Suppose we are interested in the canard following the branch y = x/2, which is attracting for x negative and repelling for x positive. Then, following the prescription of proposition 2.15 we consider the perturbation term (5) X 1 = (1, 1/2) .So, the vector field we study is given by X = X 0 + X 1 , see figure 4a. By using a spherical blow-up on the singular point (0, 0), after desingularisation it is possible to identify the desired connection between the attracting and the repelling branch of the critical variety. Such a connection appears as a geodesic solution on the blow-up sphere, see figure 4c. where the minus sign gives the correct direction, and the y component provides a compatible flow on the normally hyperbolic sections without introducing equilibria of the reduced flow. Again, by studying the vector field X = X 0 + X 1 , see figure 4b, via a spherical blow-up we can confirm the presence of a canard connection between the attracting and repelling branches of the critical parabola, see figure 4d. For more details on the blow-up computations of both examples we refer to section B.1 of the appendix. Degenerate pitchfork -Blow-up Figure 4 . 4The upper pictures (4a, 4b) are representative phase portraits where the fast-foliation (double-arrow black lines) and the reduced flow (single-arrow red lines) are shown. A neighbourhood of the singularity has been highlighted by a light blue disk. B. 1 .. 1Blow-ups of examples of section 2.2. The first example leads toThe spherical blow-up is given by the following transformation: x → r cos θ sin φ, y → r sin θ sin φ, → r 4 cos φ, where the powers of r have been chosen calibrating the powers in theẋ equation. The appropriate desingularisation is obtained dividing the blown-up vector field by r 3 . At this point, by setting r to zero we obtain the desingularised vector field on the blown-up sphere, θ(6 cos(2θ) − 5(cos(4θ) + 1)) sin 3 (φ) + 8 cot φ(cos θ − 2 sin θ) φ = cos φ 6 cos(2φ) + 10 cos θ(−6 cos(2θ) + 5 cos(4θ) + 5) sin 4 (φ) An equilibrium point of a vector field is called nilpotent if the linearisation of the vector field at such a point has only real zero eigenvalues. Discretized fast-slow systems near pitchfork singularities. L Arcidiacono, M Engel, C Kuehn, Journal of Difference Equations and Applications. 25L. Arcidiacono, M. Engel, and C. Kuehn, Discretized fast-slow systems near pitchfork singularities, Journal of Difference Equations and Applications, 25 (2019), pp. 1024-1051. . E Benoît, J L Callot, F Diener, M Diener, Chasse Au Canard, Collectanea Mathematica, E. Benoît, J. L. Callot, F. Diener, and M. Diener, Chasse au canard, Collectanea Mathematica, (1981). J Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences. New YorkSpringerJ. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences, Springer New York, 1981. R Cushman, L Bates, Global Aspects of Classical Integrable Systems. BaselSpringerR. Cushman and L. Bates, Global Aspects of Classical Integrable Systems, Springer Basel, 2015. Planar canards with transcritical intersections. P De Maesschalck, Acta Applicandae Mathematicae. 137P. De Maesschalck, Planar canards with transcritical intersections, Acta Applicandae Mathematicae, 137 (2015), pp. 159-184. F Dumortier, R Roussarie, Canard cycles and center manifolds. American Mathematical Soc577F. Dumortier and R. Roussarie, Canard cycles and center manifolds, vol. 577, American Mathematical Soc., 1996. F A G Dumortier, R Roussarie, Canard cycles and center manifolds. F. A. G. Dumortier and R. Roussarie, Canard cycles and center manifolds, 1996. Relaxation oscillations including a standard chase on french ducks. W Eckhaus, Asymptotic Analysis II. Berlin, Heidelberg; Berlin HeidelbergSpringerW. Eckhaus, Relaxation oscillations including a standard chase on french ducks, in Asymp- totic Analysis II -, Berlin, Heidelberg, 1983, Springer Berlin Heidelberg, pp. 449-497. Discretized fast-slow systems near transcritical singularities. M Engel, C Kuehn, Nonlinearity. 322365M. Engel and C. Kuehn, Discretized fast-slow systems near transcritical singularities, Non- linearity, 32 (2019), p. 2365. Persistence and smoothness of invariant manifolds for flows. N Fenichel, Indiana Univ. Math. J. 21N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1972), pp. 193-226. Geometric singular perturbation theory for ordinary differential equations. Journal of Differential Equations. 31, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), pp. 53-98. A survey on the blow up technique. A Ferragut, X Jarque, M Alvarez, International Journal of Bifurcation and Chaos. 21A. Ferragut, X. Jarque, and M. Alvarez, A survey on the blow up technique, International Journal of Bifurcation and Chaos, 21 (2011). Algebraic Curves: An Introduction to Algebraic Geometry. W Fulton, W. Fulton, Algebraic Curves: An Introduction to Algebraic Geometry, 2008. Ergebnisse der Mathematik und ihrer Grenzgebiete. Intersection Theory. New YorkSpringer, Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer New York, 2012. C Gibson, K Wirthmüller, A Du Plessis, E Looijenga, Topological Stability of Smooth Mappings. SpringerC. Gibson, K. Wirthmüller, A. Du Plessis, and E. Looijenga, Topological Stability of Smooth Mappings, Lecture Notes in Mathematics, Springer, 1976. Nonlinear dynamics of networks: The groupoid formalism. M Golubitsky, I Stewart, Bulletin of the American Mathematical Society. 43M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bulletin of the American Mathematical Society, 43 (2006). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. J Guckenheimer, P Holmes, Applied Mathematical Sciences. SpringerJ. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifur- cations of Vector Fields, Applied Mathematical Sciences, Springer New York, 1983. Subanalytic sets. H Hironaka, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki. Kinokuniya, TokyoH. Hironaka, Subanalytic sets, in Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453-493. M Hirsch, C Pugh, M Shub, Invariant Manifolds. Berlin HeidelbergSpringerM. Hirsch, C. Pugh, and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Springer Berlin Heidelberg, 2006. A survey on the blow-up method for fast-slow systems. H Jardón-Kojakhmetov, C Kuehn, IV Meeting Reunión de Matemáticos Mexicanos en el Mundo ; Conference date. AMS Press775H. Jardón-Kojakhmetov and C. Kuehn, A survey on the blow-up method for fast-slow systems, vol. 775, AMS Press, 2021, pp. 115-160. IV Meeting Reunión de Matemáticos Mex- icanos en el Mundo ; Conference date: 10-06-2018 Through 15-06-2018. Discrete geometric singular perturbation theory, Discrete and Continuous Dynamical Systems. S Jelbart, C Kuehn, 43S. Jelbart and C. Kuehn, Discrete geometric singular perturbation theory, Discrete and Continuous Dynamical Systems, 43 (2023), pp. 57-120. Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions. M Krupa, P Szmolyan, Society for Industrial and Applied Mathematics. 33M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyper- bolic points-fold and canard points in two dimensions, Society for Industrial and Applied Mathematics, 33 (2001), pp. 286-314. Extending slow manifolds near transcritical and pitchfork singularities. M Krupa, P Szmolyan, Nonlinearity. 141473M. Krupa and P. Szmolyan, Extending slow manifolds near transcritical and pitchfork singularities, Nonlinearity, 14 (2001), p. 1473. C Kuehn, Multiple Time Scale Dynamics. Springer International PublishingC. Kuehn, Multiple Time Scale Dynamics, Applied Mathematical Sciences, Springer Inter- national Publishing, 2016. Canards in R 3. P Szmolyan, M Wechselberger, Journal of Differential Equations. 177P. Szmolyan and M. Wechselberger, Canards in R 3 , Journal of Differential Equations, 177 (2001), pp. 419-453. Relaxation oscillations in r3. P Szmolyan, M Wechselberger, Journal of Differential Equations. 200P. Szmolyan and M. Wechselberger, Relaxation oscillations in r3, Journal of Differential Equations, 200 (2004), pp. 69-104. Centre Manifolds, Normal Forms and Elementary Bifurcations. A Vanderbauwhede, Vieweg+Teubner VerlagWiesbadenA. Vanderbauwhede, Centre Manifolds, Normal Forms and Elementary Bifurcations, Vieweg+Teubner Verlag, Wiesbaden, 1989, pp. 89-169. M Wechselberger, address: [email protected] University of Groningen -Bernoulli Institute for Mathematics. 9747AG, Groningen, The Netherlands EmailSpringer International Publishing9University of Groningen -Bernoulli Institute for MathematicsComputer Science and Artificial Intelligence. Nijenborgh 9, 9747AG, Groningen, The Netherlands Email address: [email protected]. Wechselberger, Geometric Singular Perturbation Theory Beyond the Standard Form, Frontiers in Applied Dynamical Systems: Reviews and Tutorials, Springer International Pub- lishing, 2020. University of Groningen -Bernoulli Institute for Mathematics, Computer Sci- ence and Artificial Intelligence; Nijenborgh 9, 9747AG, Groningen, The Netherlands Email address: [email protected] University of Groningen -Bernoulli Institute for Mathematics, Computer Sci- ence and Artificial Intelligence; Nijenborgh 9, 9747AG, Groningen, The Netherlands Email address: [email protected]
[]
[ "First Light And Reionisation Epoch Simulations (FLARES) VI: The colour evolution of galaxies z = 5 − 15", "First Light And Reionisation Epoch Simulations (FLARES) VI: The colour evolution of galaxies z = 5 − 15" ]
[ "Stephen M Wilkins \nAstronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK\n\nInstitute of Space Sciences and Astronomy\nUniversity of Malta\nMsida MSD 2080Malta\n", "Aswin P Vijayan \nAstronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK\n\nCosmic Dawn Center (DAWN)\n\n\nDTU-Space\nTechnical University of Denmark\nElektrovej 327DK-2800Kgs. LyngbyDenmark\n", "Christopher C Lovell \nAstronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK\n\nCentre for Astrophysics Research\nSchool of Physics, Engineering & Computer Science\nUniversity of Hertfordshire\nAL10 9ABHatfieldUK\n", "William J Roper \nAstronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK\n", "Dimitrios Irodotou \nAstronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK\n\nDepartment of Physics\nUniversity of Helsinki\nGustaf Hällströmin katu 2FI-00014HelsinkiFinland\n", "Joseph Caruana \nInstitute of Space Sciences and Astronomy\nUniversity of Malta\nMsida MSD 2080Malta\n\nDepartment of Physics\nUniversity of Malta\nMsida MSD 2080Malta\n", "Louise T C Seeyave \nAstronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK\n", "Jussi K Kuusisto \nAstronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK\n", "Peter A Thomas \nAstronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK\n" ]
[ "Astronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK", "Institute of Space Sciences and Astronomy\nUniversity of Malta\nMsida MSD 2080Malta", "Astronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK", "Cosmic Dawn Center (DAWN)\n", "DTU-Space\nTechnical University of Denmark\nElektrovej 327DK-2800Kgs. LyngbyDenmark", "Astronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK", "Centre for Astrophysics Research\nSchool of Physics, Engineering & Computer Science\nUniversity of Hertfordshire\nAL10 9ABHatfieldUK", "Astronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK", "Astronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK", "Department of Physics\nUniversity of Helsinki\nGustaf Hällströmin katu 2FI-00014HelsinkiFinland", "Institute of Space Sciences and Astronomy\nUniversity of Malta\nMsida MSD 2080Malta", "Department of Physics\nUniversity of Malta\nMsida MSD 2080Malta", "Astronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK", "Astronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK", "Astronomy Centre\nUniversity of Sussex\nBN1 9QHFalmer, BrightonUK" ]
[ "MNRAS" ]
With its exquisite sensitivity, wavelength coverage, and spatial and spectral resolution, the James Webb Space Telescope (JWST) is poised to revolutionise our view of the distant, high-redshift (z > 5) Universe. While Webb's spectroscopic observations will be transformative for the field, photometric observations play a key role in identifying distant objects and providing more comprehensive samples than accessible to spectroscopy alone. In addition to identifying objects, photometric observations can also be used to infer physical properties and thus be used to constrain galaxy formation models. However, inferred physical properties from broadband photometric observations, particularly in the absence of spectroscopic redshifts, often have large uncertainties. With the development of new tools for forward modelling simulations it is now routinely possible to predict observational quantities, enabling a direct comparison with observations. With this in mind, in this work, we make predictions for the colour evolution of galaxies at z = 5 − 15 using the First Light And Reionisation Epoch Simulations (FLARES) cosmological hydrodynamical simulation suite. We predict a complex evolution with time, driven predominantly by strong nebular line emission passing through individual bands. These predictions are in good agreement with existing constraints from Hubble and Spitzer as well as some of the first results from Webb. We also contrast our predictions with other models in the literature: while the general trends are similar we find key differences, particularly in the strength of features associated with strong nebular line emission. This suggests photometric observations alone should provide useful discriminating power between different models and physical states of galaxies.
10.1093/mnras/stac2548
[ "https://export.arxiv.org/pdf/2207.10920v2.pdf" ]
251,018,500
2207.10920
3858a0a90b485103859d4f8e1f4b583669870761
First Light And Reionisation Epoch Simulations (FLARES) VI: The colour evolution of galaxies z = 5 − 15 2022 Stephen M Wilkins Astronomy Centre University of Sussex BN1 9QHFalmer, BrightonUK Institute of Space Sciences and Astronomy University of Malta Msida MSD 2080Malta Aswin P Vijayan Astronomy Centre University of Sussex BN1 9QHFalmer, BrightonUK Cosmic Dawn Center (DAWN) DTU-Space Technical University of Denmark Elektrovej 327DK-2800Kgs. LyngbyDenmark Christopher C Lovell Astronomy Centre University of Sussex BN1 9QHFalmer, BrightonUK Centre for Astrophysics Research School of Physics, Engineering & Computer Science University of Hertfordshire AL10 9ABHatfieldUK William J Roper Astronomy Centre University of Sussex BN1 9QHFalmer, BrightonUK Dimitrios Irodotou Astronomy Centre University of Sussex BN1 9QHFalmer, BrightonUK Department of Physics University of Helsinki Gustaf Hällströmin katu 2FI-00014HelsinkiFinland Joseph Caruana Institute of Space Sciences and Astronomy University of Malta Msida MSD 2080Malta Department of Physics University of Malta Msida MSD 2080Malta Louise T C Seeyave Astronomy Centre University of Sussex BN1 9QHFalmer, BrightonUK Jussi K Kuusisto Astronomy Centre University of Sussex BN1 9QHFalmer, BrightonUK Peter A Thomas Astronomy Centre University of Sussex BN1 9QHFalmer, BrightonUK First Light And Reionisation Epoch Simulations (FLARES) VI: The colour evolution of galaxies z = 5 − 15 MNRAS 0002022Accepted XXX. Received YYY; in original form ZZZPreprint 7 September 2022 Compiled using MNRAS L A T E X style file v3.0galaxies: general -galaxies: evolution -galaxies: formation -galaxies: high-redshift -galaxies: photometry With its exquisite sensitivity, wavelength coverage, and spatial and spectral resolution, the James Webb Space Telescope (JWST) is poised to revolutionise our view of the distant, high-redshift (z > 5) Universe. While Webb's spectroscopic observations will be transformative for the field, photometric observations play a key role in identifying distant objects and providing more comprehensive samples than accessible to spectroscopy alone. In addition to identifying objects, photometric observations can also be used to infer physical properties and thus be used to constrain galaxy formation models. However, inferred physical properties from broadband photometric observations, particularly in the absence of spectroscopic redshifts, often have large uncertainties. With the development of new tools for forward modelling simulations it is now routinely possible to predict observational quantities, enabling a direct comparison with observations. With this in mind, in this work, we make predictions for the colour evolution of galaxies at z = 5 − 15 using the First Light And Reionisation Epoch Simulations (FLARES) cosmological hydrodynamical simulation suite. We predict a complex evolution with time, driven predominantly by strong nebular line emission passing through individual bands. These predictions are in good agreement with existing constraints from Hubble and Spitzer as well as some of the first results from Webb. We also contrast our predictions with other models in the literature: while the general trends are similar we find key differences, particularly in the strength of features associated with strong nebular line emission. This suggests photometric observations alone should provide useful discriminating power between different models and physical states of galaxies. INTRODUCTION The study of the distant, high-redshift (z > 5) Universe stands on the cusp of a revolution thanks to the James Webb Space Telescope. Webb's combination of infrared coverage, sensitivity, and spectroscopic capabilities should ultimately enable the accurate identification of statistical samples of star forming galaxies to z > 10 (Robertson 2021), and the measurement of many key properties including star formation rates, stellar masses, metallicities, and rest-frame optical morphologies. A pillar of Webb's exploration of the distant Universe will be its broadband photometric observations obtained by NIRCam, MIRI, and NIRISS. In cycle 1 alone Webb will acquire > 1 deg 2 of NIR-Cam imaging, with the deepest observations approaching 31 mag. E-mail: [email protected] These observations will enable precise measurement of the restframe UV luminosity function, particularly at the faint end, allowing the determination of a faint-end turnover. Photometric observations will also enable the measurement of key physical properties such as stellar masses, star formation rates, ages, and dust attenuation. However, the physical properties inferred from photometric observations alone yield large uncertainties (see, e.g., Whitler et al. 2022), reducing their usefulness in terms of constraining models. One critical cause of this uncertainty is the impact of nebular line emission (see e.g Zackrisson et al. 2008;Schaerer & de Barros 2009;Stark et al. 2013;Wilkins et al. 2013b) which can shift predicted colours by up to 1 mag Wilkins et al. (e.g 2020). However, with the development of sophisticated forward modelling pipelines -which are used to create synthetic observations -(e.g. Camps & Baes 2015;Narayanan et al. 2021) it is increasingly possible to directly compare observed and simulated individual galaxies and populations. While there remain uncertain elements of this modelling -in particular the modelling of dust and nebular emission -we are now at the point where this forward modelling process is comparable, or perhaps even simpler, than the reverse. Efforts to produce synthetic observations are now ubiquitous in the modelling community (e.g Wilkins et al. 2013b;Trayford et al. 2015;Wilkins et al. 2016b;Vogelsberger et al. 2019;Vijayan et al. 2021) with the ability to make observational predictions across the electromagnetic spectrum incorporating a range of physical processes, including nebular emission from HII regions (e.g. Wilkins et al. 2013b;Orsi et al. 2014;Wilkins et al. 2020;Vijayan et al. 2021) and the ISM (e.g. Lagache et al. 2018;Katz et al. 2019;Popping et al. 2019;Leung et al. 2020;Kannan et al. 2022), and dust attenuation (e.g. Wilkins et al. 2018;Vogelsberger et al. 2019;Vijayan et al. 2021Vijayan et al. , 2022 and emission (e.g. Wilkins et al. 2018;Ma et al. 2019;Lovell et al. 2021b;Vijayan et al. 2022). Beyond integrated photometry/spectroscopy it is now also possible to produce synthetic imaging (e.g. Trayford et al. 2015;Snyder et al. 2015;Ma et al. 2018;Roper et al. 2022;Marshall et al. 2022), enabling self-consistent comparisons of morphological metrics. Forward modelling has been applied to models covering a wide range of scales, from high-resolution simulations of individual halos (e.g. Ma et al. 2019) to the construction of synthetic lightcones encompassing hundreds of square arcminutes (e.g. Williams et al. 2018;Laigle et al. 2019;Davidzon et al. 2019;Somerville et al. 2021;Yung et al. 2022;Drakos et al. 2022), enabling a direct comparison with observed galaxy populations. In this work we make predictions for the redshift and luminosity evolution of Hubble, Spitzer, and JWST colours at z 5 using the First Light And Reionisation Epoch Simulations (FLARES;Lovell et al. 2021a;Vijayan et al. 2021) cosmological hydrodynamical simulation suite. The current FLARES simulations combine the z = 0 validated EAGLE (Schaye et al. 2015;Crain et al. 2015) physics model with an innovative simulation strategy, resulting in a large effective volume, and thus dynamic range, of stellar masses and luminosities. This article is organised as follows: we begin, in Section 2, by exploring predictions for the evolution of NIRCam and MIRI colours using a simple toy model. In Section 3 we then briefly describe the FLARES project before, in Section 4, presenting predictions including a comparison with existing observations ( §4.1) and other models ( §4.2). Finally, in Section 5 we present our conclusions. THEORETICAL BACKGROUND To obtain an understanding for the physical effects that drive the colour evolution of galaxies in the distant Universe, in this section we explore predictions from a simple toy model utilising simple star formation and metal enrichment histories and dust modelling. In this toy model composite spectral energy distributions (SEDs) are created by combining age/metallicity SED grids with a parametric star formation and metal enrichment history. Specifically, we employ the same stellar population synthesis model (SPS): version 2.2.1 of BPASS: Binary Population And Spectral Synthesis (BPASS; Stanway & Eldridge 2018), and initial mass function (IMF): Chabrier (2003), as used by FLARES. To account for nebular continuum and line emission we process the pure stellar SED grids using the cloudy photoionisation model (Ferland et al. 2017). This follows the same approach as Wilkins et al. (2020) and that utilised by FLARES. In short, each pure stellar SED is associated with a HII region with the same metallicity with scale solar composition, a cov-ering fraction of 1, and assuming a reference 1 ionisation parameter of log 10 U = −2. A key difference between this modelling and FLARES, however, is the treatment of dust. In the absence of spatially resolved stellar populations and dust distributions in the toy model, we assume a simple screen model, while FLARES employs a line-of-sight model, in principle assigning a unique attenuation to every star particle. We begin by using this model to generate spectra of an unobscured (i.e. τ V = 0) composite stellar population, with mass M = 10 8 M , a 100 Myr continuous star formation history, and metallicity Z= 0.001 at z ∈ {5, 7, 10, 15}, shown in Figure 1. For this fiducial model we assume f esc,LyC = 0, i.e. maximising the contribution of nebular emission. On this figure we also add predicted broadband fluxes for each of the NIRCam wide filters (F070W, F090W, F115W, F150W, F200W, F277W, F356W, F444W) in addition to the MIRI F560W and F770W bands. Immediately evident in this figure is the impact of the Lyman-limit/α break, the largely smooth UV continuum, and the impact of strong nebular line emission, particularly from [OII]λ λ 3726, 3729Å, [OIII]λ 5007Å, and Hβ . To further aid in the understanding of these predictions, in Fig. 2 we show the restframe wavelength probed by the same NIRCam and MIRI filters as a function of redshift, highlighting key emission lines and spectral features. This further reinforces that strong line emission will play an important role in driving NIRCam and MIRI colours at high redshift. In the following sections we discuss the implications of different model components on the color evolution shown in Figure 3. Impact of nebular emission Firstly, as denoted by the grey line, we show our fiducial model (100 Myr constant star formation, Z= 0.001) but with no reprocessing by dust (i.e. τ V = 0) or gas (i.e. f esc,LyC = 1) -that is, pure stellar emission. The resulting colours evolve smoothly remaining relatively blue (A − B ≈ 0), except when encompassing the Lyman or Balmer breaks (e.g. F277W-F356W at z ≈ 7.5) with the latter shifting the colour by up to ≈ 0.5 mag. We next show our fiducial model (as previous, but with f esc = 0) as the solid black line. The result is a complex colour evolution with rapid changes, coinciding with strong line emission falling within one of the bands. In this model colours shift by up to 0.5 mag relative to pure stellar colours. This can also lead to colours changing by up-to 0.7 mag across small redshift intervals (e.g. F444W-F560W at z ≈ 9). As we will see in Section 4 these shifts are even more pronounced when medium and wide filters are combined (e.g. F430M-F444W) with shifts up to 1 mag predicted for FLARES galaxies. Where the colour is probing the UV continuum the result is also a shift to redder colours, caused by nebular continuum emission (Wilkins et al. 2013a). Star formation and metal enrichment history We next consider models with different star formation histories: a short 10 Myr episode of continuous star formation and a maximalaged burst 2 . For the short burst the result is to further enhance the impact of nebular line emission, due to the increased ratio of ionising to optical photons for younger stellar populations. In this model colours now change by up to 1 mag over short redshift intervals. For the maximal-aged burst the resulting colours are consistently redder, due to (increasing) lack of massive hot short lived stars. As the ionising photon luminosity drops rapidly in the first few million years (Wilkins et al. 2020) the contribution of nebular emission to these models is extremely limited resulting in little rapid evolution of the colour. In the rest-frame UV the shift from our fiducial model is ≈ 0.5 mag, growing to ≈ 1 mag when encompassing the age sensitive Balmer break feature. As galaxies in FLARES span a range of metallicities Wilkins et al. (2022b), we also explore the impact of increasing the metallicity of our fiducial model to Z = 0.01. Increasing the metallicity both makes the pure stellar SED redder but also reduces the ionising photon luminosity, reducing the overall contribution of nebular emission. Changing the metallicity will also impact individual line ratios leading to a complex impact on colours, especially when one or more strong line is present. However, for the most part the effect of metallicity is fairly subtle, shifting colours by < 0.1 mag. Reprocessing by dust Finally, we consider the impact of dust, applying a screen model assuming a simple λ −1 attenuation law parameterised using the optical depth in the V-band: τ V . Because this simple model leaves nebular equivalent widths unchanged, where the colours are dominated by nebular emission (e.g. F444W-F560W at z = 5 − 10) the result is only a weak shift to redder colours. In the rest-frame UV however, the shift from the fiducial model can be more dramatic, with the colours increasing by ≈ 1 mag for a model with τ V = 1. Together, the results in this section highlight the significant impact of many modelling assumptions on the colour evolution of galaxies in the epoch of reionisation, and demonstrates how colours can be used as a key constraint on these assumptions. THE FIRST LIGHT AND REIONISATION EPOCH SIMULATIONS In this rest of this study, we make use of the core suite of simulations from the first phase of FLARES: the First Light And Reionisation Epoch Simulations. The core suite and its initial processing are described in Lovell et al. (2021a) and Vijayan et al. (2021), while predictions at the redshift frontier (z > 10) are presented in Wilkins et al. (2022a), and we refer the reader to those articles for a detailed introduction. In short, the core FLARES suite is a set of 40 spherical re-simulations, 14 h −1 cMpc in radius, of regions selected from a large (3.2 cGpc) 3 dark matter only simulation. The regions selected to re-simulate span a range of environments: (at (2021a) this strategy allows us to efficiently sim-ulate a much larger dynamic range in mass (or luminosity) than a traditional periodic box for the same computational resources. We adopt the AGNdT9 variant of the EAGLE simulation project (Schaye et al. 2015;Crain et al. 2015) with identical resolution and cosmology to the fiducial EAGLE simulation. This allows us to resolve galaxies with stellar masses M > 10 8 M corresponding to intrinsic rest-frame far-UV absolute magnitudes of M UV −18.4 (L FUV 10 28 erg s −1 Hz −1 ) and m 29.1 at z = 10 (). Spectral Energy Distribution Modelling To produce galaxy observables we process the outputs with a custom pipeline with the approach described in Vijayan et al. (2021), broadly following the approach developed by Wilkins et al. (2013bWilkins et al. ( , 2016bWilkins et al. ( , 2018Wilkins et al. ( , 2020, with modifications to the dust treatment. In short, we begin by associating each star particle with a pure stellar spectral energy distribution (SED) using v2. in paper, which parameterises the dust-to-metal ratio as a function of the mass-weighted age of the stellar population and the gas-phase metallicity). For the attenuation due to the birth cloud component, we scale it with the star particle metallicity, thus assuming a constant dust-to-metal ratio. For more details see Section 2.4 in Vijayan et al. (2021). As demonstrated in Fig. 3, broadband colours can evolve rapidly due to the presence of strong emission lines and continuum breaks. However, FLARES, like many hydrodynamical simulations, only produces outputs at discrete snapshots, in FLARES' case being integer redshifts from z = 15 → 5. To provide continuous redshift coverage we re-compute observed frame colours from the rest-frame SEDs using perturbed redshifts extending ±0.5 around the snapshot redshift. For example, galaxies in the z = 10 snapshot are used to produce predicted colours over the redshift range z = [9.5, 10.5). In practice, we re-sample every galaxy 10 times across the redshift interval, though ensure that no galaxy appears more than once in each δ z = 0.1 interval. The downside of this approach is that it assumes no evolution in the physical properties of galaxies between snapshots. However, if there was strong evolution between snapshots this would lead to discontinuities at the boundaries between snapshot redshift ranges. As we will see in Figure 4, while discontinuities exist they are small. PREDICTIONS We begin by presenting the predictions for the average colour, in various NIRCam and MIRI bands, as a function of redshift and restframe far-UV absolute magnitude in Figure 4. Similar predictions for two Hubble/Spitzer colours are presented in Figures 6 and 7 in the context of the comparison with current observational constraints described in §4.1. As described in the Data Availability section, we make these results publicly available for comparison with observations or other models. Most notable in Figure 4 is the complex redshift evolution of most colours, with this variability driven by strong nebular line emission and break features (e.g. F150W−F200W at z > 10). To see the impact of nebular line emission more clearly in Figure 5 we also show the evolution of pure stellar and unattenuated colours. This reveals that the impact of nebular emission is to shift colours by up 0.7 mag, with the largest shifts for F430M−F444W. As strong lines cross to adjacent filters this can also cause sharp redshift evolution of colours, with e.g. F430M-F444W shifting by > 1 mag from z = 7.5 → 8.5. This analysis also reveals that nebular emission is often important even in the absence of strong line emission. For example, in colours probing the rest-frame UV continuum (e.g. F356W−F444W at z > 12) the impact of nebular emission reddens colours by up to 0.3 mag. Figure 4 also reveals that the most luminous galaxies are typically redder, albeit the shift is small relative to the variability introduced by nebular emission. As this trend largely disappears for intrinsic colours, we attribute this to the impact of dust attenuation. This is consistent with wider predictions from FLARES which broadly predicts increasing attenuation with observed UV absolute magnitude (see Vijayan et al. 2021) 4 In Figure 4, in addition to the median colour, we also show the central 68% range of colours predicted over the full absolute magnitude range. This range is often very narrow, though does increase to up to 0.7 mag where nebular emission line emission is important. This arises due to the strong sensitivity of the ionising photon luminosity, and thus the contribution of line emission, to the recent star formation histories of galaxies. Comparison with observations At the time of writing, the first constraints on galaxy colours at high-redshift have just become available (Naidu et al. 2022) based on observations from the GLASS and CEERS Early Release Surveys. Naidu et al. (2022) report the discovery of two promising highredshift galaxy candidates at z ≈ 11 (GL-z11) and z ≈ 13 (GL-z11). The reported colours of these sources are included in Figure 4. The F277W−F356W and F356W−F444W colours of both candidates are consistent with the FLARES predictions. For F200W−F277W GL-z11 falls slightly above our predictions and GL-z13 falls slightly below. The handful of observations available, however, do not provide statistically useful constraints. Instead we compare with comprehensive observed samples Hubble and Spitzer. In Figure 6 Stefanon et al. (2021) consistently reduced Spitzer/IRAC imaging across the Great Observatories Origins Deep Survey (GOODS)-N and GOODS-S fields. This data-set was then used to measure the IRAC fluxes of almost 10,000 galaxies at 3.5 < z < 10 based on the catalogue of Bouwens et al. (2015). The Bouwens et al. (2015) sample was selected and assigned photometric redshifts using Hubble imaging. The resulting colour evolution of this sample, matched to have the same rest-frame UV luminosity limit (M UV < −18.4) is shown in Figure 6. This reveals a median observed colour providing a close match to the FLARES predictions. The impact of [OIII] and Hβ exiting the IRAC/[3.6µm] band and entering the [4.5µm] band at z = 6.5 − 7.5 can be clearly discerned. While the median colour is well matched, the scatter in the observations is up to 5× larger. While photometric scatter and redshift uncertainties will drive some of this difference it is possible this may reflect a real difference between the observations and FLARES. While we have applied a consistent luminosity limit to both samples the Bouwens et al. Figure 7 we calculate the average colour in bins of luminosity and redshift and contrast FLARES and the observations. This is inevitably more noisy but doesn't appear to reveal any systematic bias. In Figure 6 we also compare against the sample of Endsley et al. (2021). Endsley et al. (2021) select a sample of galaxies at z ∼ 6.5 − 7 from ground based imaging of the COSMOS and XMM1 fields. They use a colour selection employing Subaru/Hyper Suprime-Cam NB921 narrow-band imaging to yield precise photometric redshifts. This in turn results in clean constraints on the [OIII]+Hβ equivalent widths of the sources from the Spitzer/IRAC photometry. The resulting [3.6µm] -[4.5µm] colours at z ∼ 7 closely match the FLARES predictions, in particular the very blue colours predicted at z ≈ 6.8 and strong subsequent evolution to z > 7. Comparison with other models We now compare our predictions to mock catalogues from the phenomenological models of Williams et al. (2018) (also known as the JAGUAR: JAdes extraGalactic Ultradeep Artificial Realizations package) and DREaM: the Deep Realistic Extragalactic Model (Drakos et al. 2022), in addition to the Santa Cruz semi-analytical model (Somerville et al. 2021;Yung et al. 2022). The colour evolution of these models are contrasted with FLARES in Figure 9 and Figure 8 for JWST and Hubble + Spitzer colours respectively. While the evolution of colours in all three sets of predictions is qualitatively similar there are some important differences, particularly around the impact of nebular emission. Specifically, FLARES consistently predicts a stronger contribution from nebular emission, resulting in more extreme variation of colours. While, at present, it is not possible to definitively claim one model provides better agreement with the observations, FLARES does appear to better reproduce the magnitude of the observed dip in the average [3.6µm] -[4.5µm] colour at z ≈ 6.8. Due to the multitude of differences between the three models, the exact cause of this discrepancy is difficult to ascertain. Possibilities include the presence of more stochastic, young, or rapidly increasing star formation in FLARES or simply the choice of stellar population synthesis model, initial mass function, and/or photoionisation modelling assumptions (see e.g. Wilkins et al. 2013bWilkins et al. , 2016aWilkins et al. , 2020. Critical to ascertaining the cause of this discrepancy is applying a consistent approach to the choice of SPS model, IMF, and photoionisation assumptions should ultimately help diagnose the cause of this discrepancy. At the same time, results from Webb are now beginning to provide the observational constraints to differentiate between different sets of predictions. CONCLUSIONS In this work we have presented theoretical predictions for the colour evolution of galaxies at z = 5−15 from the FLARES: First Light And Reionisation Epoch Simulations. These predictions enable direct comparison with observational constraints, allowing both FLARES and the underlying model to be tested in a new regime. Our major findings are: • The predicted galaxy colours show complex evolution with rapid changes (up to ≈ 1 mag) in the average colours over short redshift intervals. This is due to the presence of strong nebular line emission moving through individual bands. In addition, our predicted colours show a trend, albeit modest, with luminosity, attributed to the increasing impact of dust in the most luminous galaxies. • FLARES predictions currently closely match, with the possible exception of the scatter, recent observational constraints at highredshift using Hubble and Spitzer from Stefanon et al. (2021) (based on the galaxy sample identified by Bouwens et al. 2015) and Endsley et al. (2021), in addition to early results from Webb (Naidu et al. 2022). • FLARES predictions qualitatively match other available predictions including the phenomenological models of Williams et al. (2018) and Drakos et al. (2022) and the Somerville et al. (2021); Yung et al. (2022) semi-analytical model. However, FLARES predicts larger variations due to stronger nebular line emission. The exact cause of this difference may lie in the fundamental physical properties of galaxies in the models but may also reflect different modelling assumptions -e.g. choice of stellar population synthesis model and initial mass function -for the forward modelling. With the imminent explosion of constraints from Webb we will soon be in a position to differentiate between these and other models, providing insights into the physics driving the physical and ob- Lovell C. C., Vijayan A. P., Thomas P. A., Wilkins S. M., Barnes D. J., Irodotou D., Roper W., 2021a, MNRAS, 500, 2127 Lovell C. C., Geach J. E., Davé R., Narayanan D., Li Q., 2021b, MNRAS, 502, 772 Ma X., et al., 2018, MNRAS, 477, 219 Ma X., et al., 2019, MNRAS, 487, 1844Marshall M. A., et al., 2022, arXiv e-prints, p. arXiv:2206.08941 Naidu R. P., et al., 2022, arXiv e-prints, p. arXiv:2207.09434 Narayanan D., et al., 2021 Orsi Á., Padilla N., Groves B., Cora S., Tecce T., Gargiulo I., ( + ) + , + + Figure 9. Predictions for the Hubble colour evolution of galaxies with M FUV < −18 from FLARES (thick grey line) and JAGUAR (Williams et al. 2018). DREaM and the Santa Cruz SAM are omitted as they do not provide both Hubble and Spitzer photometry. Figure 1 . 1The observed spectral energy distribution of a star forming galaxy at z = 5 − 15 alongside key JWST/NIRCam, and JWST/MIRI filter transmission functions. Coloured points denote the predicted fluxes in each of the NIRCam and MIRI bands, highlighting the impact of nebular emission in the rest-frame optical. Figure 2 . 2The rest-frame wavelength probed by selected NIRCam and MIRI filters at z = 5 − 15. The two dashed horizontal lines denote the location of the Balmer (3646Å) and Lyman-α (1216Å) break while solid lines denote strong nebular emission lines with line thickness and opacity indicating the equivalent width for a simple star forming model. Figure 3 . 3Predicted colours for a range of simple models. Our default model, denoted by the solid black line assumes 100 Myr constant star formation, nebular emissions assuming f esc = 0 and no dust. Coloured solid lines show the same model but with increasing amount of dust attenuation. The solid grey line instead assumes no nebular emission ( f esc = 1). The dashed line assumes 10 Myr constant star formation. The dotted line assumes Z = 0.01. The dot dashed line denotes a maximal-aged burst of star formation; effectively the reddest intrinsic colour possible. z ≈ 4.7) log 10 (1 + δ 14 ) = [−0.3, 0.3] 3 with over-representation of the extremes of the density contrast distribution. As demonstrated in Lovell et al. we compare FLARES predictions for the Hubble/WFC3/F160W -Spitzer/IRAC/[3.6µm] and Spitzer/IRAC/[3.6µm] -Spitzer/IRAC/[4.5µm] colours with recent measurements from Stefanon et al. (2021) (based on the sample identified by Bouwens et al. (2015)), Endsley et al. (2021), Hashimoto et al. (2018), and Laporte et al. (2021). (2015)/Stefanon et al. (2021) sample do not cover the same range of luminosities. To provide a fairer comparison in Figure 4 . 4FLARES predictions for the evolution of NIRCam and MIRI colours of galaxies at z = 5 − 15. The left-hand side plot shows the average colour in bins of redshift and rest-frame far-UV luminosity. The right-hand side plot shows both the median colour for galaxies in four absolute magnitude bins and and central 68% range for the entire sample. Also included in the right-hand side figure are recent observational constraints from Naidu et al.(2022). Figure 5 .Figure 6 . 56FLARES predictions for the pure stellar (dotted line), stellar + nebular (dashed line), and observed: stellar + nebular + dust (solid thin line) colour evolution of galaxies with M FUV < −18, and in the latter scenario also the brightest (M FUV < −21) galaxies (solid thick line).FLARES predictions, and observations, for the Hubble/WFC3/F160W -Spitzer/IRAC/[3.6] and Spitzer/IRAC/[3.6] -Spitzer/IRAC/[4.5] colour evolution of galaxies with M FUV < −18. The shaded band shows the central 68% range of all galaxies M FUV < −18. The lines denote the median in the various M FUV bins. Figure 7 .Figure 8 . 78FLARES predictions for the colour evolution of galaxies with M FUV < −18. The shaded band shows the central 68% range of all galaxies M FUV < −18. The lines denote the median in the various M FUV bins. servational properties of galaxies in the early Universe. Mock observations like those presented here, and in e.g. Williams et al. (2018), Somerville et al. (2021), Drakos et al. (2022) and Yung et al. (2022), provide a critical resource to test and refine photometric redshift selection techniques and/or the inference of physical properties from broadband photometry. Predictions for the JWST colour evolution of galaxies with M FUV < −18 from FLARES (thick grey line), the Santa Cruz SAM (Somerville et al. 2021; Yung et al. 2022), JAGUAR (Williams et al. 2018), and DREaM Drakos et al. (2022). and the wider interstellar medium (ISM). For the latter, we employ a simple line-of-sight attenuation model, similar to that described inWilkins et al. (2018), but using the fitting function for the dust-to-metal ratio presented inVijayan et al. (2019) (Equation 15 2.1 of the Binary Pop- ulation and Spectral Synthesis (BPASS; Stanway & Eldridge 2018) stellar population synthesis model assuming a Chabrier (2003) IMF according to its age and metallicity. We then associate each star par- ticle with a HII region giving rise to nebular continuum and line emission following the approach detailed in Wilkins et al. (2020). We then account for the effect of dust, both in the birth clouds of young stellar populations (with age less than 10 Myr, follow- ing Charlot & Fall 2000, that birth clouds disperse along these timescales) Stephen M.Wilkins et al. This defines the ionisation parameter at the reference age (1 Myr) and metallicity (Z = 0.01). The ionisation parameter at other ages and metallicities are scaled according to the ionising photon luminosity. 2 That is, a model in which the stellar age is equivalent to the age of the Universe at that redshift.MNRAS 000, 1-10(2022) MNRAS 000, 1-10 (2022) Where δ 14 is the density contrast measured within the re-simulation volume size. While there is a broad trend of increasing attenuation with observed UV luminosity, the most heavily attenuated galaxies have more modest UV luminosities due to the effects of attenuation dominating. This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000, 1-10(2022) ACKNOWLEDGEMENTSWe dedicate this article to healthcare and other essential workers, the teams involved in developing the vaccines, and to all the parents who found themselves having to home-school children while holding down full-time jobs. We thank Rychard Bouwens, Ryan Endsley, and Mauro Stefanon for providing machine readable catalogues of their observations for comparison. We thank the EAGLE team for their efforts in developing the EAGLE simulation code. 140. We also wish to acknowledge the following open source software packages used in the analysis: NUMPY(Harris et al. 2020), SCIPY(Virtanen et al. 2020), and MATPLOTLIB(Hunter 2007). This research made use of ASTROPY http://www.astropy.org a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013. Parts of the results in this work make use of the colormaps in the CMASHER package (van der Velden 2020).DATA AVAILABILITY STATEMENTThe 2.2, 15.8, 50, 84.2, and 97.8 percentiles of the colour distribution in redshift and rest-frame far-UV luminosity bins are available in the astropy Enhanced Character Separated Values (.ecsv) table format at https://github.com/stephenmwilkins/flares_ colours_data and as part of the wider First Light and Assembly of Galaxies model predictions repository available at https://github.com/stephenmwilkins/flags_data. Data from the wider FLARES project is available at https:// flaresimulations.github.io/data.html. If you use data from this paper please also cite Lovell et al. (2021a) andVijayan et al. (2021). . 10.1051/0004-6361/201322068A&A. 55833Astropy Collaboration et al., 2013, A&A, 558, A33 . 10.3847/1538-3881/aabc4fAJ. 156123Astropy Collaboration et al., 2018, AJ, 156, 123 . R J Bouwens, 10.1088/0004-637x/803/1/34ApJ. 80334Bouwens R. J., et al., 2015, ApJ, 803, 34 . P Camps, M Baes, 10.1016/j.ascom.2014.10.004Astron. Comput. 920Camps P., Baes M., 2015, Astron. Comput., 9, 20 . G Chabrier, 10.1086/376392PASP. 115763Chabrier G., 2003, PASP, 115, 763 . Charlot S Fall, S M , 10.1086/309250ApJ. 539718Charlot S., Fall S. M., 2000, ApJ, 539, 718 . R A Crain, 10.1093/mnras/stv725MNRAS. 4501937Crain R. A., et al., 2015, MNRAS, 450, 1937 . I Davidzon, 10.1093/mnras/stz2486MNRAS. 4894817Davidzon I., et al., 2019, MNRAS, 489, 4817 . N E Drakos, 10.3847/1538-4357/ac46fbApJ. 926194Drakos N. E., et al., 2022, ApJ, 926, 194 . R Endsley, D P Stark, J Chevallard, S Charlot, 10.1093/mnras/staa3370MNRAS. 5005229Endsley R., Stark D. P., Chevallard J., Charlot S., 2021, MNRAS, 500, 5229 . G J Ferland, Rev. Mex. Astron. Astrofis. 53385Ferland G. J., et al., 2017, Rev. Mex. Astron. Astrofis., 53, 385 . C R Harris, 10.1038/s41586-020-2649-2Nature. 585357Harris C. R., et al., 2020, Nature, 585, 357 . T Hashimoto, 10.1038/s41586-018-0117-zNature. 557392Hashimoto T., et al., 2018, Nature, 557, 392 . J D Hunter, 10.1109/MCSE.2007.55Computing in Science & Engineering. 990Hunter J. D., 2007, Computing in Science & Engineering, 9, 90 . R Kannan, A Smith, E Garaldi, X Shen, M Vogelsberger, R Pakmor, V Springel, L Hernquist, 10.1093/mnras/stac1557MNRAS. 5143857Kannan R., Smith A., Garaldi E., Shen X., Vogelsberger M., Pakmor R., Springel V., Hernquist L., 2022, MNRAS, 514, 3857 . H Katz, 10.1093/mnras/stz1672MNRAS. 4875902Katz H., et al., 2019, MNRAS, 487, 5902 . G Lagache, M Cousin, M Chatzikos, 10.1051/0004-6361/201732019A&A. 609130Lagache G., Cousin M., Chatzikos M., 2018, A&A, 609, A130 . C Laigle, 10.1093/mnras/stz1054MNRAS. 4865104Laigle C., et al., 2019, MNRAS, 486, 5104 . N Laporte, R A Meyer, R S Ellis, B E Robertson, J Chisholm, G W Roberts-Borsani, 10.1093/mnras/stab1239MNRAS. 5053336Laporte N., Meyer R. A., Ellis R. S., Robertson B. E., Chisholm J., Roberts- Borsani G. W., 2021, MNRAS, 505, 3336 . T K D Leung, K P Olsen, R S Somerville, R Davé, T R Greve, C C Hayward, D Narayanan, G Popping, 10.3847/1538-4357/abc25eApJ. 905102Leung T. K. D., Olsen K. P., Somerville R. S., Davé R., Greve T. R., Hayward C. C., Narayanan D., Popping G., 2020, ApJ, 905, 102 . G Popping, D Narayanan, R S Somerville, A L Faisst, M R Krumholz, 10.1093/mnras/sty2969MNRAS. 4824906Popping G., Narayanan D., Somerville R. S., Faisst A. L., Krumholz M. R., 2019, MNRAS, 482, 4906 . B E Robertson, arXiv:2110.13160astro-phRobertson B. E., 2021, arXiv:2110.13160 [astro-ph] . W J Roper, C C Lovell, A P Vijayan, M A Marshall, D Irodotou, J K Kuusisto, P A Thomas, S M Wilkins, 10.1093/mnras/stac1368MNRAS. 5141921Roper W. J., Lovell C. C., Vijayan A. P., Marshall M. A., Irodotou D., Kuu- sisto J. K., Thomas P. A., Wilkins S. M., 2022, MNRAS, 514, 1921 . D Schaerer, S De Barros, 10.1051/0004-6361/200911781A&A. 502423Schaerer D., de Barros S., 2009, A&A, 502, 423 . J Schaye, 10.1093/mnras/stu2058MNRAS. 446521Schaye J., et al., 2015, MNRAS, 446, 521 . G F Snyder, 10.1093/mnras/stv2078MNRAS. 4541886Snyder G. F., et al., 2015, MNRAS, 454, 1886 . R S Somerville, 10.1093/mnras/stab231MNRAS. 5024858Somerville R. S., et al., 2021, MNRAS, 502, 4858 . E R Stanway, J J Eldridge, 10.1093/mnras/sty1353MNRAS. 47975Stanway E. R., Eldridge J. J., 2018, MNRAS, 479, 75 . D P Stark, M A Schenker, R Ellis, B Robertson, R Mclure, J Dunlop, 10.1088/0004-637X/763/2/129ApJ. 763129Stark D. P., Schenker M. A., Ellis R., Robertson B., McLure R., Dunlop J., 2013, ApJ, 763, 129 . M Stefanon, 10.3847/1538-4365/ac2498ApJS. 25768Stefanon M., et al., 2021, ApJS, 257, 68 . J W Trayford, 10.1093/mnras/stv1461MNRAS. 4522879Trayford J. W., et al., 2015, MNRAS, 452, 2879 . A P Vijayan, S J Clay, P A Thomas, R M Yates, S M Wilkins, B M Henriques, 10.1093/mnras/stz1948MNRAS. 4894072Vijayan A. P., Clay S. J., Thomas P. A., Yates R. M., Wilkins S. M., Hen- riques B. M., 2019, MNRAS, 489, 4072 . A P Vijayan, C C Lovell, S M Wilkins, P A Thomas, D J Barnes, D Irodotou, J Kuusisto, W J Roper, 10.1093/mnras/staa3715MNRAS. 5013289Vijayan A. P., Lovell C. C., Wilkins S. M., Thomas P. A., Barnes D. J., Irodotou D., Kuusisto J., Roper W. J., 2021, MNRAS, 501, 3289 . A P Vijayan, 10.1093/mnras/stac338MNRAS. 5114999Vijayan A. P., et al., 2022, MNRAS, 511, 4999 . P Virtanen, 10.1038/s41592-019-0686-2Nature Methods. 17261Virtanen P., et al., 2020, Nature Methods, 17, 261 . M Vogelsberger, arXiv:1904.07238arXiv e-printsVogelsberger M., et al., 2019, arXiv e-prints, 1904, arXiv:1904.07238 . L Whitler, D P Stark, R Endsley, J Leja, S Charlot, J Chevallard, arXiv:2206.05315Whitler L., Stark D. P., Endsley R., Leja J., Charlot S., Chevallard J., 2022, arXiv e-prints, p. arXiv:2206.05315 . S M Wilkins, A Bunker, W Coulton, R Croft, T Di Matteo, N Khandai, Y Feng, 10.1093/mnras/stt096MNRAS. 4302885Wilkins S. M., Bunker A., Coulton W., Croft R., di Matteo T., Khandai N., Feng Y., 2013a, MNRAS, 430, 2885 . S M Wilkins, 10.1093/mnras/stt1471MNRAS. 4352885Wilkins S. M., et al., 2013b, MNRAS, 435, 2885 . S M Wilkins, Y Feng, T Di-Matteo, R Croft, E R Stanway, R J Bouwens, P Thomas, 10.1093/mnrasl/slw007MNRAS. 4586Wilkins S. M., Feng Y., Di-Matteo T., Croft R., Stanway E. R., Bouwens R. J., Thomas P., 2016a, MNRAS, 458, L6 . S M Wilkins, Y Feng, T Di-Matteo, R Croft, E R Stanway, A Bunker, D Waters, C Lovell, 10.1093/mnras/stw1154MNRAS. 4603170Wilkins S. M., Feng Y., Di-Matteo T., Croft R., Stanway E. R., Bunker A., Waters D., Lovell C., 2016b, MNRAS, 460, 3170 . S M Wilkins, Y Feng, Di Matteo, T Croft, R Lovell, C C Thomas, P , 10.1093/mnras/stx2588MNRAS. 4735363Wilkins S. M., Feng Y., Di Matteo T., Croft R., Lovell C. C., Thomas P., 2018, MNRAS, 473, 5363 . S M Wilkins, 10.1093/mnras/staa649MNRAS. 4936079Wilkins S. M., et al., 2020, MNRAS, 493, 6079 . S M Wilkins, arXiv:2204.09431arXiv e-printsWilkins S. M., et al., 2022a, arXiv e-prints, p. arXiv:2204.09431 . S M Wilkins, arXiv:2208.00976arXiv e-printsWilkins S. M., et al., 2022b, arXiv e-prints, p. arXiv:2208.00976 . C C Williams, 10.3847/1538-4365/aabcbbApJS. 23633Williams C. C., et al., 2018, ApJS, 236, 33 . L Y A Yung, arXiv:2206.13521arXiv e-printsYung L. Y. A., et al., 2022, arXiv e-prints, p. arXiv:2206.13521 . E Zackrisson, N Bergvall, E Leitet, 10.21105/joss.02004The Journal of Open Source Software. 676ApJZackrisson E., Bergvall N., Leitet E., 2008, ApJ, 676, L9 van der Velden E., 2020, The Journal of Open Source Software, 5, 2004
[ "https://github.com/stephenmwilkins/flares_", "https://github.com/stephenmwilkins/flags_data." ]
[ "HAUSDORFF DIMENSION IN STOCHASTIC DISPERSION", "HAUSDORFF DIMENSION IN STOCHASTIC DISPERSION" ]
[ "D Dolgopyat ", "V Kaloshin ", "L Koralov " ]
[]
[]
We consider the evolution of a connected set in Euclidean space carried by a periodic incompressible stochastic flow. While for almost every realization of the random flow at time t most of the particles are at a distance of order √ t away from the origin [DKK1], there is an uncountable set of measure zero of points, which escape to infinity at the linear rate[CSS1]. In this paper we prove that this set of linear escape points has full Hausdorff dimension.
10.1023/a:1019731212422
[ "https://export.arxiv.org/pdf/math/0205032v1.pdf" ]
1,507,322
math/0205032
d9cee731496a02a50978110741f1ce7a2fa74cfa
HAUSDORFF DIMENSION IN STOCHASTIC DISPERSION 3 May 2002 D Dolgopyat V Kaloshin L Koralov HAUSDORFF DIMENSION IN STOCHASTIC DISPERSION 3 May 2002 We consider the evolution of a connected set in Euclidean space carried by a periodic incompressible stochastic flow. While for almost every realization of the random flow at time t most of the particles are at a distance of order √ t away from the origin [DKK1], there is an uncountable set of measure zero of points, which escape to infinity at the linear rate[CSS1]. In this paper we prove that this set of linear escape points has full Hausdorff dimension. Dedicated to our teacher Yakov Sinai on occasion of his 65th birthday. Introduction. One of the greatest achievements in mathematics of the second half of the last century was creation of the theory of hyperbolic dynamical systems in works of Anosov, Bowen, Ruelle, Sinai, Smale and many others. The importance of this theory is not so much in that it allows one to get new information about a large class of ordinary differential equations but in that it provides a paradigm for understanding irregular behavior in a large class of natural phenomena. From the mathematical point of view it means that the theory should be useful in many branches of mathematics beyond the study of finite-dimensional dynamical systems. The aim of this note is to illustrate this on a simple example. Namely, we show how the theory of nonuniformly hyperbolic systems, i.e. systems with non-zero Lyapunov exponents, can explain ballistic behavior in a problem of passive transport in random media. This paper concerns the long time behavior of a passive substance (say an oil spill) carried by a stochastic flow. Various aspects of such behavior have been a subject of a number of recent papers (see [Cm,CC,CGXM,CS,CSS1,CSS2,DKK1,DKK2,LS,SS,ZC1,ZC2] etc.) Consider an oil spill at the initial time concentrated in a domain Ω. Let Ω evolve in time along trajectories of the stochastic flow and Ω t be its image at time t. The papers mentioned study the rate of stretching of the boundary ∂Ω t , growth of the diameter and the "shape" of Ω t , distribution of mass of Ω t , and many other related questions. In this D.D. was partially supported by NSF and Sloan Foundation, V. K. was partially supported by American Institute of Mathematics Fellowship and Courant Institute, and L. K. was partially supported by NSF postdoctoral fellowship. paper we model the stochastic flow by a stochastic differential equation driven by a finite-dimensional Brownian motion {θ(t) = (θ 1 (t), . . . , θ d (t)) ∈ R d } t≥0 (1) dx t = X 0 (x t )dt + d k=1 X k (x t ) • dθ k (t) x ∈ R N where {X k } d k=0 are C ∞ -smooth space periodic divergence free vector fields on R N . Alternatively one can regard this system as a flow on T N = R N /Z N . Below we impose certain nondegeneracy assumptions on vector fields {X k } d k=1 from [DKK1]. These assumptions hold on an dense open set of C ∞ -smooth divergence free vector fields on T N or satisfied generically. An interesting feature of the flow (1) is the dichotomy between growth of the mass and shape of the spill Ω t . On one hand, most of the points of the tracer Ω t move at distance of order √ t at time t. More precisely, let ρ be a smooth metric on T N , naturally lifted to R N and ν be a measure of a finite energy, i.e. for some positive p we have dν(x)dν(y) ρ p (x, y) < ∞. In particular, ν can be the Lebesgue probability measure supported on an open set Ω, which also supports the initial oil spill. Let ν t be its image under the flow (1) andν t be rescaling of ν t , defined by as follows: for a Borel set Ω ⊂ R N putν t (Ω) = ν t ( √ tΩ). Theorem 1. ( [DKK1]) For almost every realization of the Brownian motion {θ(t)} t≥0 the measureν t weakly converges to a Gaussian measure on R N as t → ∞. Remark 1. Notice that this is the Central Limit Theorem with respect to randomness in initial conditions, not with respect to randomness of the Brownian motion {θ(t)} t≥0 . On the other hand, there are many points with linear growth. Fix a realization of the Brownian motion {θ(t)} t≥0 . Let L θ denote the random set of points with a linear escape L θ = x ∈ R N : lim inf t→+∞ |x t | t > 0 . The following result is a special case of [SS] (see also [CSS1]). Theorem 2. Let S be a connected set containing at least two points. Then for almost every realization of the Brownian motion {θ(t)} t≥0 the set L θ S is uncountable. In fact in dimension 2 there is a limiting shape of the rescaled contaminated area. Namely, let Ω be a bounded open set, Ω t be its image under the flow (1) and C t = 0≤s≤t Ω s . In other words, we call a point x contaminated by the time t if there is a trajectory from our curve which has passed through x before time t. The Shape Theorem. ( [DKK2]) If N = 2, then there exists a convex compact set B ⊂ R 2 such that for almost every realization of the Brownian motion {θ(t)} t≥0 and any δ > 0 there exists T = T (δ) such that for all t > T (1 − δ) B ⊂ C t t ⊂ (1 + δ) B. Remark 2. The "shape" B ⊂ R 2 is independent of the initial spill Ω. Moreover, an open set Ω can be replaced by a smooth curve γ for the Shape Theorem to hold true. In view of Theorems 1, 2, and the Shape Theorem it is interesting to see how large is the set of points with linear growth. In this paper we first prove the following Theorem 3. Let γ be a smooth curve on R 2 . Then for almost every realization of the Brownian motion {θ(t)} t≥0 we have HD(L θ γ) = 1. Then in Section 8 using this Theorem we derive the following main result of the paper Theorem 4. (Main Result) For almost every realization of the Brownian motion {θ(t)} t≥0 we have that points of the flow (1) with linear escape to infinity L θ form a dense set of full Hausdorff dimension HD (L θ ) = N. By Theorem 1 for most points x 0 = x in R N its trajectory x t is of order √ t away from the origin at time t. Also, the Law of Iterated Logarithm for functionals of diffusion processes and Fubini Theorem imply that the set of points L θ with linear escape has measure zero. This Corollary says that L θ is the "richest" possible set of measure zero in R N , namely, is of full Hausdorff dimension N. Nondegeneracy assumptions. In this section we formulate a set of assumptions on the vector fields, which in particular imply the Central Limit Theorem for measures, the estimates on the behavior of the characteristic function of a measure carried by the flow (see [DKK1]), and large deviations estimates (see [BS]). Such estimates are essential for the proof of our results. Recall that X 0 , X 1 , . . . , X d are assumed to be C ∞ -smooth, periodic and divergence free. (A) (hypoellipticity for x t ) For all x ∈ R N we have Lie(X 1 , . . . , X d )(x) = R N . Denote the diagonal in T N × T N by ∆ = {(x 1 , x 2 ) ∈ R N × R N : x 1 = x 2 (mod 1)}. (B) (hypoellipticity for the two-point motion) The generator of the two-point motion {(x 1 t , x 2 t ) : t > 0} is nondegenerate away from the diagonal ∆, meaning that the Lie brackets made out of (X 1 (x 1 ), X 1 (x 2 )), . . . , (X d (x 1 ), X d (x 2 )) generate R N × R N . To formulate the next assumption we need additional notations. Let Dx t : T x 0 R N → T xt R N be the linearization of x t at t. We need the hypoellipticity of the process {(x t , Dx t ) : t > 0}. Denote by T X k the derivative of the vector field X k thought as the map on T R 2 and by SR N = {v ∈ T R N : |v| = 1} the unit tangent bundle on R N . If we denote byX k (v) the projection of T X k (v) onto T v SR N , then the stochastic flow (1) on R N induces a stochastic flow on the unit tangent bundle SR N , defined by the following equation: dx t = d k=1X k (x t ) • dθ k (t) +X 0 (x t )dt. With these notations we have condition (C) (hypoellipticity for (x t , Dx t )) For all v ∈ SR N we have Lie(X 1 , . . . ,X d )(v) = T v SR N . For measure-preserving stochastic flows with conditions (C) Lyapunov exponents λ 1 , . . . , λ N exist by multiplicative ergodic theorem for stochastic flows of diffeomorphisms (see [Cv], Thm. 2.1). Moreover, the sum of Lyapunov exponents N j=1 λ j should be zero (see e.g. [BS]). Under conditions (A)-(C) the leading Lyapunov exponent is positive λ 1 = lim t→∞ log |dϕ t (x)(v)| t > 0 ,(2) where dϕ t (x) is the linearization matrix of the flow (1) integrated from 0 to t at the point x. Indeed, Theorem 6.8 of [Bx] states that under condition (A) the maximal Lyapunov exponent λ 1 can be zero only if for almost every realization of the flow (1) one of the following two conditions is satisfied (a) there is a Riemannian metric ρ ′ on T N , invariant with respect to the flow (1) or (b) there is a direction field v(x) on T N invariant with respect to the flow (1). However (a) contradicts condition (B). Indeed, (a) implies that all the Lie brackets of {(X k (x 1 ), X k (x 2 ))} d k=1 are tangent to the leaves of the foliation {(x 1 , x 2 ) ∈ T N × T N : ρ ′ (x 1 , x 2 ) = Const } and don't form the whole tangent space. On the other hand (b) contradicts condition (C), since (b) implies that all the Lie brackets are tangent to the graph of v. This positivity of λ 1 is crucial for our approach. Remark 3. Let us mention an important difference between deterministic and stochastic dynamics. Most of the results dealing with statistical properties of deterministic systems assume that all Lyapunov exponents are non-zero. By contrast we need only one positive exponent. This is because in the random situation hypoellipticity condition (C) implies that growth rate of any deterministic vector is given by the largest exponent (see equation (19). This allows us to get our results without assuming that all the exponents are non-zero. We further require that the flow has no deterministic drift, which is expressed by the following condition (E) (zero drift) T 2 d k=1 L X k X k + X 0 (x)dx = 0 , where L X k X k (x) is the derivative of X k along X k at the point x. Notice that d k=1 L X k X k + X 0 is the deterministic components of the stochastic flow (1) rewritten in Ito's form. The Central Limit Theorem for measures was formulated in [DKK1] under an additional assumption T 2 X k (x)dx = 0 , k = 1, ..., d .(3) This assumption is not needed for the proof of Theorem 3 and as the result for the proof of the Main Theorem. However, in order to simplify the proof, i.e. use the results of [DKK1] without technical modifications, we shall assume (3) to hold. 3. Idea of the proof. 3.1. A Model Example. Below we define a random dynamical system on R which models the motion of the projection of the spill Ω t onto a fixed line l ⊂ R N . Introduce notations: I(b; a) = [b − a/2, b + a/2] -the segment on R centered at b of length a; s ∈ {0, 1} Z + a semiinfinite sequence of 0's and 1's, s k ∈ {0, 1} k a set of k numbers 0 or 1, {{θ s k (t)} s k ∈{0,1} k } k∈Z + countable number of standard i.i.d. Brownian motions on R indexed by binary sequences. Let τ be positive. The random dynamical system is defined as follows. Let I ∅ = I(0; 1). Then σ θ 0 : I ∅ → R stretches I ∅ uniformly by 2 around its center and shifts it randomly by θ ∅ (τ ). Divide σ θ 0 (I ∅ ) in two equal parts I 0 and I 1 σ θ 0 (I ∅ ) = I 0 ∪ I 1 = I(θ ∅ (τ ) − 1/2; 1) ∪ I(θ ∅ (τ ) + 1/2; 1). (4) Now σ θ 1 acts on each {I i } i=0,1 independently by stretching each I i 's uniformly by 2 around its center and shifting by θ 0 (τ ) and θ 1 (τ ) respectively. σ θ 1 • σ θ 0 (I ∅ ) = (I 00 ∩ I 01 ) ∪ (I 10 ∪ I 11 ) = I([θ ∅ (τ ) − 1/2] + [θ 0 (τ ) − 1/2]; 1) ∪ I([θ ∅ (τ ) − 1/2] + [θ 1 (τ ) + 1/2]; 1) ∪ I([θ ∅ (τ ) + 1/2] + [θ 0 (τ ) − 1/2]; 1) ∪ I([θ ∅ (τ ) + 1/2] + [θ 1 (τ ) + 1/2]; 1),(5) and so on. Let n ∈ Z + . Then at the n-th stage "after time nτ " the image of the initial unit interval I ∅ = [−1/2, 1/2] consists of 2 n unit intervals. The preimage of each of those unit intervals is an interval of length 2 −n uniformly contracted. Let's give a different definition of the random dynamical system (4)-(5). Consider an isomorphism of the dynamical system on the unit interval I = I ∅ + 1/2 = [0, 1] given by φ : x → 2x (mod 1) and the one sided Bernoulli shift on two symbols, say 0 and 1. Such an isomorphism is given by s : x → s(x) = {s k (x)} ∞ k=0 ∈ {0, 1} Z + , where for each k ∈ Z + s k (x) = 0 if φ n (x) < 1/2 s k (x) = 1 otherwise. (6) Let η n (x) = #{k ≤ n : s k (x) = 1}. Notice now that σ θ n • σ θ n−1 • · · · • σ θ 0 (x) = n k=0 θ s k (τ ) + (η n+1 (x) − (n + 1)/2)/2,(7) where θ s k (τ )'s are i.i.d. Brownian motions. Define η − (x) = lim inf n→∞ η n (x)/n. Then for almost all points x ∈ I we have η − (x) = lim n→∞ η n (x)/n = 1/2. Let us show however that there is full Hausdorff dimension set of points in the interval I such that frequency of 0's is less than frequency of 1's, i.e. HD{x ∈ I : η(x) > 1/2} = 1. Since n k=0 θ s k (τ )/n → 0 almost surely this would imply that the set of points in I ∅ with a nonzero drift for the random dynamical system, defined by (4)-(5), has full Hausdorff dimension almost surely, but is of measure zero. We shall justify the fact that HD{x ∈ I : η(x) > 1/2} = 1. Points with a nonzero drift. Fix an arbitrary small positive ε. The goal is to find a fractal set of points I ∞ ⊂ I ∅ and a probability measure µ ∞ supported on I ∞ such that µ ∞ -a.e. point x ∈ I ∞ has a nonzero drift to the right, i.e. lim inf n→∞ σ θ n • · · · • σ θ 0 (x)/n > 0. Moreover, HD(µ ∞ ) tends to 1 as ε tends to 0. Construction of the set I ∞ and of the measure µ ∞ is inductive. I ∞ is defined as a countable intersection of a nested sequence of compact sets and µ ∞ is given as a weak limit of Lebesgue measures supported on those sets. We describe the base of the induction and the inductive steps. • For n = 1 we have σ θ 0 (I ∅ ) is a segment of length 2 or union of two segments I 0 and I 1 of length 1 each. Cut off the bottom ε-segment from each segment. This corresponds to cutting off ε-segments [−1/2, −1/2 + ε] and [0, ε] from I ∅ . Denote the surgery result by I 0 ⊂ I ∅ and by µ 0 the Lebesgue probability measure supported on whole I 0 . Notice that µ 0 {x ∈ I 0 : σ θ 1 (x) = 1} > µ 0 {x ∈ I 0 : σ θ 1 (x) = 0} (8) creates a nonzero drift up, since frequency of 1's exceeds frequency of 0's. • Suppose I n−1 and µ n−1 are constructed. To construct I n and µ n consider the image σ θ n (I n−1 ). It consists of 2 n segments of equal length close to 1. Cut off the bottom ε-segment from each. This corresponds to cutting off 2 n segments of length 2 −n ε from I n−1 ⊂ I ∅ . The result of the surgery is denoted by I n and by µ n we denote the Lebesgue probability measure supported on the whole I n . Again the surgery increases probability of s n (x) being 1 over s n (x) being 0. Thus, this creates a positive drift. The intersection I ∞ = ∩ n I n is a fractal set and the weak limit measure µ ∞ = lim µ n has Hausdorff dimension approaching 1 as ε tends to zero. It follows from the construction that for µ ∞ -almost every point η − (x) = µ 0 {x ∈ I 0 : σ θ 1 (x) = 1} > 1/2. 3.3. Difficulties in extending of the Model Example to the case of the flow. Let γ ⊂ R N be a smooth curve, l ∈ R N be a line, and π l : R N → l be an orthogonal projection onto l. Suppose at the initial moment of time π l (γ) = I ∅ is the unit interval. If not, then rescale it and shift it to the origin. The most subtle element in extending the Model Example is defining the stopping (stretching) time τ or deciding when to stop γ t and how to cut off some parts of γ t in order to create a nonzero drift as in (8). Such a stopping time needs to have several important features 1 . 1. Stretching property of the stopping time: It is not difficult to show that if |π l (γ 0 )| = 1, then the stopping time τ γ = inf{t ≥ 0 : |π l (γ t )| = 2} (9) has finite expectation and exponential moments uniformly bounded over all compact curves with projection of length 1 (see e.g. [CSS1]). The analogy between this τ γ and the model τ is clear. However, the geometry of γ τγ in R N might become quite complicated (γ τγ might spin, bend, fold, and so on see computer simulations in [CC]) so it is not reasonable to stop all parts of the curve γ simultaneously and perform the surgery (cut off of "bottom" parts as in the passage preceding (8)). For this reason at the first stage of a partition/cut off process we split γ τγ not in two parts as in the Model Example, but in a number (may be countable) of random parts γ = ∪ j∈J γ j and each part γ j will have its own stopping time τ j . 2. A countable Partition of γ: We shall partition γ into at most countable number of segments γ = ∪ j∈J γ j (see Section 5). Each γ j has its own stopping time τ j so that the image ϕ τ j γ j under the flow (1) is not too folded (see condition (b) of Theorem 5). Moreover, such a stopping time τ j still has finite expectation and exponential moments (see condition (e) of Theorem 5). Now if we have that the image ϕ τ j γ j is "regular" it does not reflect dynamics on γ j . In order to imitate the Model Example's cut off construction we need to stop ϕ t γ j at the moment when ϕ t | γ j is more or less uniformly expanding on γ j forward in time or (ϕ t ) −1 | ϕtγ is uniformly contracting on ϕ t γ backward in time (see conditions (a), (c), and (d) of Theorem 5). For example, if there is no backward contraction by dynamics of ϕ −1 τ j on ϕ τ j γ j , then if we cut off an ε-part of ϕ τ j γ j its preimage in γ j might not be small compare to length of γ j . As we explain in more details below we need this smallness to estimate Hausdorff dimension of remaining points in γ ⊃ γ j after the surgery. The property of uniformness of distortion of a dynamical system is usually called: 3. A bounded distortion property: In the Model Example, Section 3.1 we have uniform backward contraction of intervals: at stage n (after time τ n) by a factor 2 −n . So, when we cut off an ε-part of an interval at stage n, it corresponds to 2 −n ε-part of the initial segment I ∅ . This remark makes an estimate of Hausdorff dimension of the set I ∞ or of the measure µ ∞ supported on I ∞ trivial, because the sets {I n } n∈Z + have selfsimilar structure. Certainly, this is no longer true for the evolution of γ under the flow (1). Some parts of γ expanded by ϕ t | γ expanded more than others. Condition (c) of Theorem 5 makes sure that there is a backward contraction in time and condition (d) of the same theorem says that rate of backward contraction Holder regularly depends on a point on a short interval γ j . Thus, backward contraction is sufficiently uniform on γ j 's. In the theory of deterministic dynamical systems with non-zero Lyapunov exponents the set of points satisfying uniform estimates for forward and backward expansion (as well as uniform estimates for angles between stable and unstable manifolds) are called Pesin sets and times when an orbit visits given Pesin set are called hyperbolic times. The existence of Pesin sets follows from abstract ergodic theory (see [P1]). Understanding the geometry of these sets in concrete examples is an important but often difficult task. In this paper we describe some properties of Pesin sets for stochastic flows. This description plays a key role in the proof of Theorem 3 and we also think it can be useful in many other questions about stochastic flows. In particular let us mention that the estimates similar to ones given in Section 4 play important role in many other questions in the theory of deterministic systems such as periodic orbit estimates [K] and constructions of maximal measures [N], etc. Our arguments in this paper are quite similar to [D1], [KM] even though the control of the geometry of images of curves is much more complicated in our case. Some interesting formulas for dimensions of nontypical points can be found in [BaSc]. We also refer the readers to the survey [Sz] and the book [P3] for more results about dimensions of dynamically defined sets. The rest of the paper is organized as follows. In Section 4 in Theorem 5 we define a stopping time τ and prove that it has finite expectation and exponential moments. In Section 4.1 we investigate expansion properties of the flow (1) at the stopping time τ and complete the proof of Theorem 5. Recall that section 3.2 above was devoted to the construction of points with a nonzero drift. Namely, we need to construct a Cantor set I ∞ and a measure µ ∞ supported on I ∞ so that µ ∞ -a.e. point has a nonzero drift. First, in Section 5 we present an algorithm of construction of a random Cantor set I inside the initial curve γ. Then, in Section 6 we define a probability measure µ supported on I with almost sure nonzero drift. Hausdorff dimension of such a measure is estimated in Section 7. Main Result (Theorem 4) is derived from Theorem 3 in Section 8. Auxiliary lemmas are in the Appendix at the end of the paper. Hyperbolic moments. Control of the smoothness. Introduce notations. Denote by ϕ t 1 ,t 2 a diffeomorphism of T N , obtained by solving (1) on the time interval [t 1 , t 2 ], and by ϕ t the diffeomorphism ϕ 0,t . The flow (1) can also be thought as the product of independent diffeomorphisms {ϕ n,n+1 : T N → T N } n∈Z + . Given positive numbers K and α we say that a curve γ is (K, α)-smooth if in the arclength parameterization the following inequality holds dγ ds (s 1 ) − dγ ds (s 2 ) ≤ Kρ α (s 2 , s 1 ) for each pair of points s 1 , s 2 ∈ γ. In all the inequalities which appear below the distance ρ between the points on γ or its images ϕ t γ's is measured in the arclength metric induced on γ or ϕ t γ from the ambient space. In order to do not overload notations we omit dependence on γ or ϕ t γ when it is clear from the context which curve we use. The goal of this section is to show that for a sufficiently small α and a sufficiently large K, starting with an arbitrary point x on a (K, α)-smooth curve γ, the part of image of this curve in a small neighborhood of the image of x is often smooth. More precisely, we prove the following statement. Let λ 1 be the largest Lyapunov exponent of the flow (1) which is positive see (2). Theorem 5. For any 0 < λ ′ 1 < λ 1 there exist sufficiently small r > 0, α ∈ (0, 1), and sufficiently large K > 0 and n 0 ∈ Z + with the following properties: For any (K, α)-smooth γ of length between r 100 and 100r and each point x ∈ γ there is a stopping time τ = τ (x), divisible by n 0 , such that (a) dϕ τ |T γ(x) > 100 and length of the corresponding curve l(ϕ τ γ) ≥ r; Denote byγ r a curve inside ϕ τ γ of radius r with respect to induced in ϕ τ γ length centered at ϕ τ (x). Then (b)γ r is (K, α)-smooth and for each pair of points y 1 , y 2 ∈γ r the following holds (c) for each integer 0 ≤ k ≤ τ n 0 we have ρ(ϕ τ,τ −kn 0 y 1 , ϕ τ,τ −kn 0 y 2 ) ≤ e −λ ′ 1 kn 0 ρ(y 1 , y 2 ); (d) |ln dϕ −1 τ |Tγ r (y 1 ) − ln dϕ −1 τ |Tγ r (y 2 )| ≤ Const ρ α (y 1 , y 2 ); Moreover, for such a stopping time τ (x) we have (e) E τ (x) ≤ C 0 ; P{τ (x) > T } ≤ C 1 e −C 2 T for any T > 0, All the above constants depend only on vector fields {X k } d k=0 and λ 1 , but independent of the curve γ. Remark 4. Choosing integer n 0 is only for our convenience. Requirement that τ is divisible by n 0 will be used for construction of partition of γ in Section 5. This is also indication of flexibility in choice of both constants. The choice of constants 100, 1000, etc. in this paper is more or less arbitrary. Any constant greater than 1 would suffice. Proof. The leading idea of the proof is that with probability close to 1 for a sufficiently large n 0 the diffeomorphism ϕ t,t+n 0 : T N → T N gets close to its asymptotic behavior. In particular, the norm of the linearization dϕ t,t+n 0 (x) as the matrix is ∼ exp(λ 1 n 0 +o(n 0 )) as the top Lyapunov exponent predicts. Moreover, the linearization dominates higher order terms of ϕ t,t+n 0 (x) and, therefore, determines local dynamics in a neighborhood of x. Thus, to some extend for large periods of time the flow (1) behaves similarly to uniformly hyperbolic system, for which properties of the Theorem are easy to verify. Now we start the proof. First we construct a stopping time τ as a first moments satisfying a certain number of regularity inequalities (see (11)-(15)). This inequalities would include K, r, n 0 and some other parameters. Then we show that for any ε > 0 these parameters can be adjusted so that probability that the number of times each inequality is violated up to time T at least εT times decay exponentially in T. This would guarantee condition (e). Finally, we show that these inequalities imply conditions (a)-(d) and as the result prove the Theorem. In the Appendix we obtain large deviation estimates necessary for the proof below. Our first goal is to control distortion of the unit tangent vector to images ϕ t γ of γ as time t evolves. Consider a collection of subsets of γ indexed by j B T,jn 0 (x) = {y ∈ γ : ρ(ϕ n 0 k y, ϕ n 0 k x) ≤ re −λ ′ 1 (T −n 0 k) for 0 ≤ k ≤ j}, where j varies from 1 to T /n 0 . We would like to find an integer moment of time τ , divisible by n 0 , such that (10) ϕ jn 0 B τ,τ (x) is (Ke ǫ(τ −jn 0 ) , α) − smooth for all j = 0, . . . , τ n 0 . The rest of this section is devoted to showing that the set of those T , divisible by n 0 for which (10) holds with T = τ has density close to 1 if K is sufficiently large. Given T denote by K j the α-Holder norm of ϕ jn 0 B T,jn 0 (x). We would like to derive an inductive in j formula relating K j and K j+1 so that in T /n 0 steps we get a required statement. Let z 1 , z 2 be two points on ϕ jn 0 B T,jn 0 (x) and r is sufficiently small, then ρ(ϕ jn 0 ,(j+1)n 0 z 1 , ϕ jn 0 ,(j+1)n 0 z 2 ) ≥ 1 2 inf ϕ jn 0 B T,jn 0 (x) dϕ jn 0 ,(j+1)n 0 |T γ ρ(z 1 , z 2 ). Let d 2 ϕ jn 0 ,(j+1)n 0 be the Hessian matrix consisting of second derivatives of the diffeomorphism ϕ jn 0 ,(j+1)n 0 . Assuming now that for each integer j < T n 0 and some R > 0 we have (11) d 2 ϕ jn 0 ,(j+1)n 0 ≤ Re ǫ(T −jn 0 ) and that condition (10) holds true up for each j ≤ j * . Then we get ρ(ϕ jn 0 ,(j+1)n 0 z 1 , ϕ jn 0 ,(j+1)n 0 z 2 ) ≥ 1 2 dϕ jn 0 ,(j+1)n 0 |T γ (ϕ jn 0 x) − rRKe −(λ ′ 1 n 0 −2ǫ)(T −j * ) ρ(z 1 , z 2 ). We would like to prove that (10) holds true for j = j * + 1. Assume also that for each j < T /n 0 we have (13) rRKe −(λ ′ 1 n 0 −2ǫ)(T −jn 0 ) ≤ dϕ jn 0 ,(j+1)n 0 |T γ (ϕ jn 0 x) 4 then we get ρ(ϕ j * n 0 ,(j * +1)n 0 z 1 , ϕ j * n 0 ,(j * +1)n 0 z 2 ) ≥ dϕ j * n 0 ,(j * +1)n 0 |T γ (ϕ j * n 0 x) 4 ρ(z 1 , z 2 ). Let v 1 and v 2 be directions of the tangent vectors to ϕ j * n 0 γ at z 1 and z 2 respectively, then ρ(dϕ j * n 0 ,(j * +1)n 0 (z 1 )v 1 , dϕ j * n 0 ,(j * +1)n 0 (z 2 )v 2 ) ≤ ρ(dϕ j * n 0 ,(j * +1)n 0 (z 1 )v 1 , dϕ j * n 0 ,(j * +1)n 0 (z 1 )v 2 ) +ρ(dϕ j * n 0 ,(j * +1)n 0 (z 1 )v 2 , dϕ j * n 0 ,(j * +1)n 0 (z 2 )v 2 ). Denote the first and the second terms by I and II respectively. If (11) holds, then II ≤ Re ǫ(T −j * n 0 ) ρ(z 1 , z 2 ). Now since v 1 and v 2 are close T (ϕ j * n 0 γ)(x) and z 1 is close to ϕ j * n 0 x we have dϕ j * n 0 ,(j * +1)n 0 (z 1 )v 1 − dϕ j * n 0 ,(j * +1)n 0 (z 1 )v 2 ≈ d v dϕ j * n 0 ,(j * +1)n 0 (x)(v 1 − v 2 ). Let us give more precise estimates. Notice that if A is a linear map, then its action on the projective space satisfies dA(v)δv = Π (Av) ⊥ Aδv Av ≤ A Av , where Π (Av) ⊥ is the orthogonal projection onto the direction Av and δv is an element of T v T x M. Therefore, we can assume that for a positive integer T /n 0 and each j * < T /n 0 the following inequality is satisfied (14) ρ(Av 1 , Av 2 ) ρ(v 1 , v 2 ) ≤ 2 dϕ j * n 0 ,(j * +1)n 0 dϕ j * n 0 ,(j * +1)n 0 |T ϕ j * n 0 γ , for any linear map A such that A − dϕ j * n 0 ,(j * +1)n 0 ≤ rRe (λ ′ 1 n 0 −ǫ)(T −j * n 0 ) and for any pair (v 1 , v 2 ) of tangent vectors such that ρ (v 1 , v 2 ) ≤ Ke −(λ ′ 1 n 0 α−ǫ)(T −j * n 0 ) . Thus I ≤ 2K j * ρ α (z 1 , z 2 ) dϕ j * n 0 ,(j * +1)n 0 dϕ j * n 0 ,(j * +1)n 0 |T ϕ j * n 0 γ . Hence (11)-(14) imply that K j * +1 ≤ 8K j * dϕ j * n 0 ,(j * +1)n 0 dϕ j * n 0 ,(j * +1)n 0 |T ϕ j * n 0 γ 1+α + 2Re −(λ ′ 1 n 0 (1−α)−ǫ)(T −j * n 0 ) dϕ j * n 0 ,(j * +1)n 0 |T ϕ j * n 0 γ α If T is chosen so that (15) dϕ j * n 0 ,(j * +1)n 0 |T ϕ j * n 0 γ ≥ Re ǫ(T −j * n 0 ) −1 , then the last inequality becomes (16) K j * +1 ≤ 8K j * dϕ j * n 0 ,(j * +1)n 0 dϕ j * n 0 ,(j * +1)n 0 |T ϕ j * n 0 γ 1+α + 2R 2 e −(λ ′ 1 n 0 (1−α)−2ǫ)(T −j * n 0 ) Let us summarize what we have learned so far. Lemma 1. For n 0 as above suppose that T is such that for every j such that jn 0 ≤ T estimates (11)-(15) hold true and also the solution of (17)K j+1 = 4K j dϕ jn 0 ,(j+1)n 0 dϕ jn 0 ,(j+1)n 0 |T ϕ jn 0 γ 1+α + 2R 2 ,K 0 =K satisfies (18)K j ≤Ke (T −jn 0 )ǫ then inequality (10) holds. Now we want to show that the set of points where either (11)-(15) or (18) fail has density less than εT except on a set of exponentially small probability. The result for (18) follows from Proposition 8 applied to lnK j . To see that the conditions of this proposition are satisfied if α is sufficiently small it is enough to verify that lnK j has uniform drift to the left. By Carverhill's extension of Oseledets' Theorem [Cv] for every point x on M and every unit vector v in T x M (19) 1 n 0 E ln dϕ n 0 (x)v → λ 1 uniformly as n 0 → ∞ and [BS] provides exponential estimate for probabilities of large deviations. Since dϕ n 0 (x) ≤ N j=1 dϕ n 0 (x)v j where {v j } N j=1 is any orthonormal frame, the above mentioned results of [BS] imply that 1 n 0 E ln | dϕ n 0 (x) → λ 1 as n 0 → ∞ with exponential bound for large deviations. Thus Proposition 8 from the Appendix applies to lnK j for a large enough n 0 . The fact that (11)-(15) fail rarely if n 0 is sufficiently large and r is sufficiently small follows from Lemma 10. 4.1. Hyperbolic moments. Control of expansion. We now define the stopping time τ as the first moment when (10), (11), and (15) are satisfied as well as (20) dϕ τ | T γ (x) ≥ 1000 and for each positive integer j ≤ τ /n 0 and some constant 0 <λ 1 < λ 1 we have (21) dϕ τ,τ −jn 0 | T ϕ τ γ ≤ e −λ 1 jn 0 . Then the large deviations estimates of [BS] guarantee that property (20) has density close to 1. Lemma 2. For any ε > 0 and any 0 <λ 1 < λ 1 there exists a positive integer n 0 such that with probability exponentially approaching to 1 the fraction of integers τ , divisible by n 0 , with the linearization dϕ τ,τ −jn 0 |T ϕ τ γ contracting exponentially backward in time for all integer j between 0 and τ /n 0 tends to 1. More precisely, P #{S ≤ L : ∀0 ≤ j ≤ S, τ = Sn 0 dϕ τ,τ −jn 0 |T ϕ τ γ ≤ e −λ 1 jn 0 } L ≤ 1 − ε decays exponentially in L. Proof. We first show how to prove a weaker statement with "∃ε" instead of "∀ε" (which is enough to prove Theorem 5) and then explain briefly the changes needed to prove the sharp result. Let τ 1 be the first moment such that for each integer j ≤ τ n 0 (22) dϕ τ 1 ,τ 1 −jn 0 |T ϕ τ γ ≤ e −λ 1 jn 0 . We claim that τ 1 has exponential tail. Indeed, let Y j = Y j (θ) = dϕ jn 0 |T γ (x) e −(λ 1 +ε)jn 0 θ , Y 0 = 1, and Z j = dϕ jn 0 |T γ (x) e −λ 1 jn 0 , Z 0 = 1. Then [BS] shows that if n 0 is sufficiently large and ε, θ are sufficiently small, then Y j is a submartingale. Thus the first momentĵ such that Zĵ > 10 has exponential tail. But there is at least one maximumj of Z j between 0 andĵ. Thenj satisfies (22). Now define τ k inductively so that τ k+1 > τ k is the first moment such that for every j ≤ τ k+1 −τ k n 0 dϕ τ k+1 ,τ k+1 −jn 0 |T ϕ τ k+1 γ ≤ e −λ 1 n 0 j . Then τ k+1 − τ k have exponential tails, so by Lemma 9 there exists c such that P{ τ k k ≥ C} decays exponentially in k. However all τ k satisfy (22). This proves the result with ε = 1 − 1 C . To get the optimal result one should note that P{τ 1 = n 0 } → 1 as n 0 → ∞ and apply the arguments of Lemma 9. We leave the details to the reader. Now we want to verify conditions (b), (c), and (d) of Theorem 5 withγ r replaced byγ = ϕ τ B τ,τ (x). Once we prove this we get from (c) that the main restriction on B τ,τ (x) is for k = τ so thatγ r =γ and then (a) will also be true. Now (b) is true by Lemma 1. We will establish (c) and (d) by induction. Namely we suppose that (c) is true for k ≥ k 0 . Then for every y ∈γ ln dϕ τ,τ −(k 0 +1)n 0 |Tγ (x) − ln dϕ τ,τ −(k 0 +1)n 0 |Tγ (y) ≤ k 0 m=0 ln dϕ τ −mn 0 ,τ −(m+1)n 0 |Tγ (x) − ln dϕ τ −mn 0 ,τ −(m+1)n 0 |Tγ (y) ≤ k 0 m=0 ln dϕ τ −mn 0 ,τ −(m+1)n 0 |Tγ (x) − ln dϕ τ −mn 0 ,τ −(m+1)n 0 |Tγ (y) + k 0 m=0 ln dϕ τ −mn 0 ,τ −(m+1)n 0 |Tγ (y) − ln dϕ τ −mn 0 ,τ −(m+1)n 0 |Tγ (y) . Denote the left term by I and the right term by II respectively. Now by (10) and (11) I ≤ k 0 m=0 Re ǫm Ke ǫm ρ α (ϕ τ,τ −mn 0 x, ϕ τ,τ −mn 0 y) ≤ k 0 m=0 KRe −(λ ′ 1 αn 0 −2ǫ)m ρ α (x, y) ≤ Const r α .(23) On the other hand II ≤ k 0 m=0 d 2 ϕ τ −mn 0 ,τ −(m+1)n 0 dϕ τ −mn 0 ,τ −(m+1)n 0 ρ(ϕ τ,τ −mn 0 x, ϕ τ,τ −mn 0 y) ≤ k 0 m=0 R 2 e −(λ ′ 1 n 0 −2ǫ)m ρ(x, y) ≤ Const r.(24) Hence (11) and (15) (25) ln dϕ τ,τ −(k 0 +1)n 0 |Tγ (x) − ln dϕ τ,τ −(k 0 +1)n 0 |Tγ (y) ≤ C(R)ρ(y 1 , y 2 ). Thus for all y (26) implies that (c) is valid for k 0 − 1. Thus, we obtain (c) for all k. Now repeating the proof of (25) with x and y replaced by y 1 and y 2 (and using (26) instead of (21)) we obtain (d). This completes the proof of Theorem 5. (26) dϕ τ −(k 0 +1)n 0 ,τ |T ϕ τ −(k 0 +1)n 0 γ (y) ≥ exp λ 1 kn 0 − C(R)r ≥ exp (λ ′ 1 kn 0 ) ifλ 1 − λ ′ 1 ≥ C(R)r. Remark 5. The term hyperbolic time was introduced in [A] but the notion itself was used before, e.g. in [P1, P2, J, Y]. Considerations of this section are similar to [ABV,D2] but the additional difficulty is that in those papers the analogue of (10) was true by the general theory of partially hyperbolic systems [HPS] whereas here additional arguments in spirit of [P1, P2] were needed to establish it. One interesting question is how large can α be so that Theorem 5 still holds. We note that α appears in (16) twice. So we want α to be as large as possible to control the first part and we want α to be small to control the second term. In general, the optimal choice of α should depend on the ratio of leading exponents. We refer to [CL, L, PSW, JPL] for the discussion of this question. Construction of the partition. We are now ready to describe a partition γ = j∈Z + γ(j). It will be defined inductively. Each of γ(j)'s is a finite union of intervals. As j tends to infinity size of intervals tends to zero and they fill up γ. To simplify the notation we assume that Theorem 5 is true with n 0 = 1. This can be achieved by rescaling the time. Fix an orientation from left to right on γ. Suppose γ(1), γ(2), . . . , γ(m) are already defined in an F m -measurable way. Let K m+1 = {x ∈ γ : τ (x) = m + 1}.(27) By definition K m+1 is a finite union of intervals. Let U m+1 = ϕ m+1 K m+1 . We call an obstacle any point on the boundary of either K m+1 , m j=1 γ(j) or γ. Fix r satisfying Theorem 5. Let C be a connected component of U m+1 and a and b be its left and right endpoints with respect to left-right orientation induced by ϕ m+1 . If distance from b to the closest image of an obstacle to the right on ϕ m+1 (γ) is less than r 2 and b ′ is this image, then putb = b ′ . Otherwise letb be a point at distance r 100 from b. Defineã similarly. Consider the set W m+1 = Cãb . Divide W m+1 into the segments of lengths between r 100 and r 50 and denote this partition by V m+1 . Now we define partition of a subset of γ \ m j=1 γ(j) by pulling back along ϕ −1 m+1 the partition V m+1 γ(m + 1) = ϕ −1 m+1 V m+1 . To justify that this algorithm produces a partition which covers all of K m+1 we need to check that length of each component is at least r 100 . To do this we argue by contradiction. Otherwise, there would be two obstacles x ′ , x ′′ neither of which is from K m+1 such that ρ(ϕ m+1 x ′ , ϕ m+1 x ′′ ) ≤ r 100 and a point from U m+1 between them. At least one of the obstacles would have to come from m j=1 γ(j). Let x ′ be such an obstacle. Since both points are close to U m for each n ≤ m + 1 we have ρ(ϕ n x ′ , ϕ n x ′′ ) ≤ r 100 e −λ ′ 1 (m−n) . But in this case the interval [x ′ , x ′′ ] in γ with endpoints x ′ and x ′′ would be added to our partition at a previous step of the algorithm. Denote by γ = j∈Z + γ j the partition which is made out of the partition γ = j∈Z + γ(j) by renumerating intervals of this partition in length decreasing order. Let us summarize the outcome. Proposition 6. We can partition γ = j∈Z + γ j in such a way that (a) there exists a positive integer n j such that dϕ n j |T γ ≥ 100 and length l(ϕ n j γ j ) ≥ r 100 ; (b) for each positive integer m ≤ n j and lengths of the corresponding curves we have l(ϕ m γ j ) ≤ l(ϕ n j γ j )e −λ ′ 1 (n j −m) ; (c) ln dϕ n j |T γ (x ′ ) − ln dϕ n j |T γ (x ′′ ) ≤ Const ρ α (ϕ n j x ′ , ϕ n j x ′′ ) for every pair x ′ , x ′′ ∈ γ j ; (d) for some α > 0 and each pair x ′ , x ′′ ∈ γ j we have |v(x ′ , n j ) − v(x ′′ , n j )| ≤ Const ρ α (ϕ n j x ′ , ϕ n j x ′′ ) , where v(x, n) denote the unit tangent vector to ϕ n γ at ϕ n (x); (e) Let j(x) be such that x ∈ γ j(x) . Then E n j(x) ≤ Const and P{n j(x) > T } ≤ C 1 e −C 2 T for some positive C 1 , C 2 and any T > 0; This Proposition is designed to allow application of Theorem 5 so that we can use regularity and geometric properties of γ j 's at stopping times τ j 's. 6. Construction of the measure with almost sure nonzero drift. Now we construct a random Cantor set I ⊂ γ and a probability measure µ supported on I such that µ-almost all points have a nonzero drift. This construction goes along the same line with the construction in Section 3.2 of the Cantor set I ∞ in the unit interval and a probability measure µ ∞ on I such that µ ∞ -almost all points have nonzero drift. Choose a direction e ∈ R N . Let θ be a small parameter which we let to zero in the next section. We say that a curve is e-monotone if its projection to e is monotone. Now we describe construction of a Cantor set I ⊂ γ and a probability measure µ on I by induction. This Cantor set I at k-th step of induction consists of countable number of segments numerated by k-tuples of positive integers. Denote k-tuples (j 1 , · · · , j k ) ∈ Z k + and (n 1 , · · · , n k ) ∈ Z k + by J k and N k respectively. Let |N k | = k j=1 n j . The first step of induction goes as follows. Let γ j , n j be the sequence of pairs: a curve and an integer, described in Proposition 6. Let θ be a small positive number. If ϕ n j γ j is e-monotone put σ(j) equal ϕ n j γ j without the segment of length θr, which we cut off from the e-bottom point of ϕ n j γ j . Otherwise σ(j) = ϕ n j γ j with no cut off. Let γ(J 1 ) = ϕ −1 n j σ(j) and N 1 (J 1 ) = n j for J 1 = j Suppose a collection of disjoint segments {γ(J k )} J k ∈Z k + ⊂ γ is defined as above and multiindices N k (resp. J k ) are defined as the corresponding set of hyperbolic times multiindexed by J k segments. Then I k = ∪ J k ∈Z k + γ(J k ) ⊂ I k−1 ⊂ · · · ⊂ I 1 ⊂ γ (29) is the k-th order of construction of the random Cantor set I (cf. with an open set I k from Section 3.2). The (k + 1)-st step goes as follows. Pick a segment γ(J k ) of partition (29). Consider the partition of the curve ϕ |N J k | γ(J k ) = j k+1 ∈Z +γ (J k , j k+1 )(30) defined in Section 5 and let n J k ,j k+1 be the corresponding hyperbolic times forγ(J k , j k+1 ) from Proposition 6. For brevity denote |N k (J k )| by n (k) and |N k (J k )| + n (J k ,j k+1 ) by n (k+1) . If the curve ϕ n (k) ,n (k+1)γ (J k ,j k+1 ) is e-monotone we let σ(J k , j k+1 ) be ϕ n (k) ,n (k+1)γ (J k ,j k+1 ) with cut off of the segment of length θ starting from the e-bottom. Otherwise, σ(J k , j k+1 ) equal ϕ n (k) ,n (k+1)γ (J k , j k+1 ) with no cut off. Then a segment γ(J k , j k+1 ) = ϕ −1 n (k+1) σ(J k , j k+1 )(31) with j k+1 ∈ Z + this defines the (k + 1)-st order partition {γ (J k+1 )} J k+1 ∈Z k+1 + ⊂ γ and the k-order set I k+1 = ∪ J k+1 ∈Z k+1 + γ(J k+1 ) ⊂ γ. We now describe a sequence of measures µ k 's on I k ⊂ γ with k ∈ Z + respectively. Let µ 0 be the arclength on γ. Suppose µ k is already defined on I k . Consider {γ(J k+1 )} J k+1 ∈Z k+1 + . If ϕ n (k+1) γ(J k+1 ) is not e-monotone we let µ k+1 | γ(J k+1 ) = µ k | γ(J k+1 ) . Otherwise, µ k+1 | γ(J k+1 ) = ρ jk µ k | γ(J k+1 ) , where ρ jk is a normalizing constant. Lemma 3. Let k be an integer. If r is sufficiently small and γ ⊂ R N is (K, α)smooth as in Theorem 5, then if we consider partition of γ up to order k + 1, then for each multiindex J k ∈ Z k + the corresponding k-th order curve γ(J k ) ⊂ γ satisfy the property: for any positive integer j k+1 the (k + 1)-st order curve γ(J k+1 ) ⊂ γ(J k ) has e-monotone with positive probability, i.e. P{ϕ n (k+1) γ (J k+1 ) is e − monotone | F n (k) ,n (k+1) } > c for some positive c and c is uniform for all (K, α)-smooth curves. Proof. Pick a point x ∈ γ(J k+1 ). By assumption (D) of hypoellipticity on the unit tangent bundle SM for the flow (1) probability that the angle between e and T ϕ n k+1 γ(x) makes less than 1 • is positive. By definition ϕ n (k+1) γ(J k+1 ) is (K, α)-smooth. Thus if r is small enough, then the tangent vectors to ϕ n k+1 γ are close to T ϕ n k+1 γ(x) with large probability, where x is a point on γ (J k+1 ). This completes the proof. Recall that θ > 0 is a fraction of ϕ n j γ j we cut off from ϕ n j γ j on the j-th step, provided ϕ n j γ j is e-monotone. Let µ = µ(θ) denote the weak limit of µ k 's µ = lim k→∞ µ k . Proof. The first step is to show that for any s for almost all realizations of the Brownian motion {θ(t)} t≥0 (32) lim inf t→∞ x (k) n (k) , e n (k) > 0 Applying Proposition 6 (e) and Lemma 9 we get that there exists a constant C > 0 such that lim sup k→∞ n (k) k < C almost surely. Therefore, to prove (32) it suffices to show that (33) lim inf k→∞ x (k) n (k) , e k > 0 However by Lemma 3 there exists c such that E x (k+1) n (k+1) − x (k) n (k) , e > c uniformly in k, s. (This is because E x (k) n (k+1) − x (k) n (k) , e = 0 and x (k) n (k+1) , e − x (k) n (k) , e ≥ 0 with strict inequality having positive probability by Lemma 3.) Hence (33) follows by Lemma 9. Therefore (32) is established. Now we apply the following estimate. Lemma 5. ( [CSS2], Theorem 1) Let Φ s,t = sup s≤τ ≤t |x τ − x s |,Φ s,t = Φ s,t max(1, t − s) then there exists a constant C such that for all s and t E exp Φ 2 s,t max(1, ln 3Φ s,t ) < C. Combining this lemma with Proposition 6 (e) we obtain that there are positive constants α and D such that E exp α sup n (k) <τ <n (k+1) |x τ (s) − x n (k) | < D. Using Borel-Cantelli's lemma we derive from this that almost surely lim sup k→∞ sup n (k) <τ <n (k+1) |x τ (s) − x n (k) | ln k < +∞. In this section we complete the proof of Theorem 3 by establishing the following fact. Recall that θ > 0 is a fraction of ϕ n j γ j we cut off from ϕ n j γ j on the j-th step, provided ϕ n j γ j is e-monotone. Consider the measure µ we constructed in the previous Section. Proposition 7. With notations above we have that as θ → 0 Hausdorff dimension of the measure µ = µ(θ) tends to 1: HD(µ(θ)) → 1. Let us recall the following standard principle. Lemma 6 (Mass distribution principle). Let S be a compact subset of a Euclidean (or metric) space such that there exists a probability measure ν such that ν(S) = 1 and for each x we have ν(B(x, r)) ≤ Cr s for some positive C and s. Then HD(S) ≥ s. Proposition 7 is a direct consequence of the following statements. Lemma 7. Let γ be a smooth curve in R N . Suppose there exist a nested sequence of partitions γ ⊃ J 1 ∈Z 1 + γ(J 1 ) ⊃ · · · ⊃ J k ∈Z k + γ(J k ) ⊃ . . .(34) and probability measures µ 0 , µ 1 , . . . , µ k , . . . supported on γ, J 1 ∈Z 1 + γ(J 1 ), . . . , J k ∈Z k + γ(J k ), . . . respectively such that µ 0 is the normalized arclength on γ and so on µ k is the normalized arclength on J k ∈Z k + γ(J k ). Then if we have (b) for all J k ∈ Z k + length of the corresponding interval γ(J k ) is bounded by l(γ(J k )) ≤ 100 −k ; (b) for each l > k we have µ l (γ(J k )) = µ k (γ(J k )); (c) dµ k+1 dµ k (x) ≤ ρ k (x) for every point x ∈ J k ∈Z k + γ(J k ), where ρ k < 1 + δ. Let µ = lim k→∞ µ k in the sense of weak limit. Then HD(µ) ≥ d(δ), where d(δ) → 1 as δ → 0. Lemma 8. For each δ > 0 there exists θ > 0 such that the densities of dµ k+1 dµ k (x) used to define measures µ k+1 knowing µ k satisfy condition (c) of Lemma 7. Proof of Lemma 7. We prove that for any segment I we have µ(I) ≤ Const |I| 1−β , where β → 0 as δ → 0. Let k(I) = | ln |I ln 100 and a and b be the endpoints of I. Letã be the left endpoint of the k-th partition containing a and b be the right endpoint of the k-th partition containing b. Then (35) µ(I) ≤ µ([ã,b]) = µ k ([ã,b]) ≤ (1 + δ) k µ 0 ([ã,b]) ≤ 3(1 + δ) k |I| ≤ 3|I| 1−β where β = ln(1+δ) ln 100 . Thus, β → 0 as δ → 0. Application of the mass distribution principle implies that HD(µ) ≥ 1 − β. Proof of Lemma 8. Recall the notation of Section 6. We need to show that (36) sup J k+1 |ϕ −1 n (k+1) σ(J k+1 )| |ϕ −1 n (k) γ(J k+1 )| → 1, θ → 0 By construction |ϕ n (k+1) ,n (k) σ(J k+1 )| |γ(J k+1 )| ≥ 1 − θ r/100 . Hence to prove (36) it is enough to show that there is a constant C independent of j, k, l such that for any interval I ⊂ γ(J k+1 ) |ϕ −1 n (k) j I| |ϕ −1 n (k) j γ(J k+1 )| ≤ C |I| |γ(J k+1 )| . To do so it is enough to show that there is a constantC such that for every pair y 1 , y 2 ∈ γ(J k+1 ) dϕ −1 n (k) j |T γ(J k+1 ) (y 1 ) dϕ −1 n (k) j |T γ(J k+1 ) (y 2 ) ≤C. But by Proposition 6 there are constants C 1 , C 2 , and C 3 such that ln dϕ −1 n (k) j |T γ(J k+1 ) (y 1 )− ln dϕ −1 n (k) j |T γ(J k+1 ) (y 2 ) ≤ k m=1 ln dϕ n (m) ,n (m−1) | T ϕ n (m) (ϕ n (k) ,n (m) y 1 ) − ln dϕ n (m) ,n (m−1) |T ϕ n (m) (ϕ n (k) ,n (m) (y 2 )) ≤ C 1 m ρ α (ϕ n (k) ,n (m) y 1 , ϕ n (k) ,n (m) (y 2 )) ≤ C 2 m 100 (m−k)α ρ α (y 1 , y 2 ) ≤ C 3 r α . This completes the proof. Proof of Theorem 4. Let G denote the foliation of T N by curves {x 1 = c 1 , x 2 = c 2 . . . x N −1 = c N −1 }. By (35) for each β > 0 and each leaf γ c of G almost surely there exists a measure µ c on γ c such that µ c (I) ≤ 3|I| 1−β and µ c (L θ ) = 1. Let µ = µ c dc. Then by Fubini Theorem almost surely for any cube C of side r we have µ(C) ≤ 3r N −β and µ(L θ ) = 1. The application of the mass distribution principle completes the proof. Appendix A. Large deviations. Here we collect some estimates used throughout the proof of Theorem 3. Lemma 9. Let F j be a filtration of σ-algebras and {ξ j , } be a sequence of F jmeasurable random variables such that (a) there exist C 1 , λ such that for every |s| ≤ λ we have E(e sξ j+1 |F j ) ≤ C 1 ; (b) there exists C 2 such that E(ξ j+1 |F j ) ≤ C 2 . Then for each ǫ > 0 the probability P N −1 j=0 ξ j ≥ (C 2 + ǫ)N decays exponentially in N. Proof. Consider Φ n (s) = exp n−1 j=0 ξ j − C 2 + ǫ 2 n s . Then (a) and (b) imply that Φ n (s) is a supermartingale if s is sufficiently small. Hence EΦ n (s) ≤ EΦ 0 (s) = 1, and so E exp n−1 j=0 ξ j − (C 2 + ǫ)n s ≤ exp − nǫs 2 , which proves the lemma. Lemma 10. Let F j be a filtration of σ-algebras and {ξ j , } be a sequence of F j -measurable random variables such that there exists constant C 1 such that (37) E(ξ j+1 |F j ) ≤ C 1 then for every ǫ, ε > 0 there is R > 0 such that P #{n ≤ N : ξ j ≤ Re ǫ(n−j) for all 0 ≤ j < n} N ≤ 1 − ε tends to zero exponentially fast in N. Proof. We say that a pair (j, n) is R-bad if ξ j > Re ǫ(n−j) . By (37) (38) P{(j, n) is R-bad} ≤ C 1 R e −ǫ(n−j) . Now given k let B R (k) be the number of n > k such that (k, n) is R-bad. By Proposition 8. Let x j be (in general, non-homogeneous) random walk on Z. Suppose that there exist constants C 1 , C 2 , C 3 such that (a) there exist m such that for every x j > m we have E(x j+1 − x j |x j ) ≤ −C 1 ; (b) for every x j and every ζ < C 2 we have E(e ζ(x j+1 −x j ) |x j ) ≤ C 3 . Fix δ > 0. Let F (M) denote the set of j such that for all k < j x k ≤ max(x j , m) + M + δ(j − k). Proof. Let τ 1 < τ 2 · · · < τ k < . . . be the consecutive returns of x j to {x ≤ m}. Let t j = τ j+1 − τ j , X j = max τ j−1 <l<τ j x l . Lemma 11. t j and X j have exponential tails. Proof. It suffices to prove it for t 1 and X 1 and the assumption that x 0 ≤ m. Clearly it suffices to condition on x 1 > m since otherwise t 1 = 1, X 1 ≤ m. Then (b) implies that for small ε 1 , ε 2 y j = e ε 1 x j +ε 2 j 1 {j≤τ 1 } is a supermartingale. Thus (39) E y j ≤ E y 1 ≤ C 4 . On the other hand E y j ≥ P{τ 1 > j} e ε 1 m+ε 2 j . Hence P{τ 1 > j} ≤ C 5 e −ε 2 j where C 5 = C 4 e −ε 1 m . Now E e ε 1 X 1 ≤ E τ 1 j=1 e ε 1 x j ≤ ∞ j=1 Ee ε 1 x j ∞ k=j P{τ 1 > k} ≤ C 4 j C 5 e −ε 2 j 1 − e −ε 2 j < ∞. This completes the proof. The rest of the proof of Proposition 8 is similar to the proof of Lemma 10. We say that the pair (k, j) is bad if x k > max(x j , m) + M + δ(j − k). If (k, j) is bad then j − k < x k − m + M δ . Let B l (M) be the number of bad pairs (k, j) such that τ l−1 < k < τ l . By the previous lemma E B l (M) < ∞ and so by dominated convergence theorem E B l (M) → 0 as M → ∞. Hence by Lemma 9 the number of bad pairs such that k < τ N is less than εN except on a set of exponentially small probability. Since τ N ≥ N the proposition follows. Lemma 4 . 4For almost every realization of the Brownian motion {θ(t)} t≥0 and µ-almost every x ( 38 ) 38E(B R (k + 1)|F k ) ≤ C 2 e −ǫ R(1 − e −ǫ )→ 0 as R → ∞. Thus by Lemma 9 there exists R such thatP N k=1 B R (k) ≥ εNdecays exponentially in N. This completes the proof of the lemma. Therefore for any s and for almost all realizations of the Brownian motion {θ(t)} t≥0 we have lim inf By Fubini Theorem we have that for almost every realization of the Brownian motion {θ(t)} t≥0 the setτ →∞ x τ (s), e τ > 0. s : lim inf τ →∞ x τ (s), e τ > 0 has full measure. 7. Hausdorff dimension of µ. In the Model Example τ is a constant SRB measures for non-hyperbolic systems with multidimensional expansion. J Alves, Ann. Sci. Ecole Norm. Sup. 33J. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. Ecole Norm. Sup., 33, (2000), 1-32; SRB measures for partially hyperbolic systems whose central direction is mostly expanding. J Alves, C Bonatti, &amp; M Viana, Invent. Math. 140J. Alves, C. Bonatti & M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140, (2000), 351-398; Sets of "non-typical" points have full topological entropy and full Hausdorff dimension. L Barreira, &amp; J Schmeling, Israel J. Math. 116L. Barreira & J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116, (2000), 29-70; Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms. P Baxendale, Probab. Th. & Rel. Fields. 81P. Baxendale Lyapunov exponents and relative entropy for a stochastic flow of dif- feomorphisms, Probab. Th. & Rel. Fields 81 (1989) 521-554. Large deviations and stochastic flows of diffeomorphisms. P Baxendale, &amp; D Stroock, Prob. Th. & Rel. Fields. 80P. Baxendale & D. Stroock, Large deviations and stochastic flows of diffeomorphisms, Prob. Th. & Rel. Fields, 80, (1988), 169-215; R Carmona, Transport properties of Gaussian velocity fields. Boca Raton, FLCRCR. Carmona, Transport properties of Gaussian velocity fields, in Real and stochastic analysis (Ed. M. M. Rao), 9-63, Probab. Stochastics Ser., CRC, Boca Raton, FL, 1997; Flows of stochastic dynamical systems: ergodic theory. A , Stochastics. 14A. Carverhill, Flows of stochastic dynamical systems: ergodic theory, Stochastics, 14, 273-317, (1985); R A Carmona, &amp; F Cerou, Transport by incompressible random velocity fields: simulations & mathematical conjectures, in Stochastic partial differential equations: six perspectives. Ed. R. Carmona & B. RozovskiiProvidence, RIAmer. Math. Soc64R. A. Carmona & F. Cerou, Transport by incompressible random velocity fields: simulations & mathematical conjectures, in Stochastic partial differential equations: six perspectives (Ed. R. Carmona & B. Rozovskii) 153-181, Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI, 1999; Surface stretching for Ornstein Uhlenbeck velocity fields. R Carmona, S Grishin, L Xu, &amp; S Molchanov, El. Comm. Prob. 2R. Carmona, S. Grishin, L. Xu & S. Molchanov, Surface stretching for Ornstein Uhlenbeck velocity fields, El. Comm. Prob., 2, (1997), 1-11; Asymptotic curvature for stochastic dynamical systems. M Cranston &amp; Y. Le Jan, Stochastic dynamics. Bremen, 1997, Ed. H. Crauel & M. GundlachNew YorkSpringerM. Cranston & Y. Le Jan, Asymptotic curvature for stochastic dynamical systems, in Stochastic dynamics (Bremen, 1997, Ed. H. Crauel & M. Gundlach), 327-338, Springer, New York, 1999; M Cranston, &amp; M Schuetzow, Dispersion rates for Kolmogorov flows. preprintM. Cranston & M. Schuetzow, Dispersion rates for Kolmogorov flows, preprint; Linear expansion of isotropic Brownian flows. M Cranston, M Schuetzow, &amp; D Steinsaltz, El. Comm. Prob. 4M. Cranston, M. Schuetzow & D. Steinsaltz, Linear expansion of isotropic Brownian flows, El. Comm. Prob., 4, (1999); 91-101; Linear bounds for stochastic dispersion. M Cranston, M Schuetzow, &amp; D Steinsaltz, Ann. Prob. 28M. Cranston, M. Schuetzow & D. Steinsaltz. Linear bounds for stochastic dispersion, Ann. Prob., 28, (2000), 1852-1869; Bounded orbits of Anosov flows. D Dolgopyat, Duke Math. J. 87D. Dolgopyat, Bounded orbits of Anosov flows, Duke Math. J., 87, (1998), 87-114; On Dynamics of mostly contracting systems. D Dolgopyat, Comm. Math. Phys. 2131D. Dolgopyat, On Dynamics of mostly contracting systems, Comm. Math. Phys., 213, (2000), no. 1, 181-201; Sample path properties of stochastic flows. D Dolgopyat, V Kaloshin, &amp; L Koralov, preprintD. Dolgopyat, V. Kaloshin & L. Koralov Sample path properties of stochastic flows, preprint. A limit shape theorem for periodic stochastic dispersion. D Dolgopyat, V Kaloshin, &amp; L Koralov, preprintD. Dolgopyat, V. Kaloshin & L. Koralov A limit shape theorem for periodic stochastic dispersion, preprint. Invariant manifolds. M Hirsch, C Pugh, &amp; M Shub, Lect. Notes in Math. 583SpringerM. Hirsch, C. Pugh & M. Shub, Invariant manifolds, Lect. Notes in Math., 583, Springer Berlin-New York, 1977; Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. M Jakobson, Comm. Math. Phys. 81M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys., 81, (1981), 39-88; On the integrability of intermediate distributions for Anosov diffeomorphisms. M Jiang, Ya Pesin &amp; R. De La Llave, Erg. Th. & Dyn. Sys. 15M. Jiang, Ya. Pesin & R. de la Llave, On the integrability of intermediate distribu- tions for Anosov diffeomorphisms, Erg. Th. & Dyn. Sys., 15, (1995), 317-331; Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. A Katok, Publ. IHES. 51A. Katok Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. IHES 51 (1980), 137-173; Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaȋ's Moscow Seminar on Dynamical Systems. D Kleinbock, &amp; G Margulis, Amer. Math. Soc. Transl. Ser. L. A. Bunimovich, B. M. Gurevich & Ya. B. Pesin2Amer. Math. SocD. Kleinbock & G. Margulis, Bounded orbits of nonquasiunipotent flows on ho- mogeneous spaces, in Sinaȋ's Moscow Seminar on Dynamical Systems (Ed. L. A. Bunimovich, B. M. Gurevich & Ya. B. Pesin) 141-172, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996; Invariant jets of a smooth dynamical system. S Lemaire, Bull. Soc. Math. France. 129S. Lemaire Invariant jets of a smooth dynamical system, Bull. Soc. Math. France 129, (2001), 379-448; Linear bounds and Gaussian tails in a stochastic dispersion model. H Lisei, &amp; M Scheutzow, Stoch. Dyn. 13H. Lisei & M. Scheutzow, Linear bounds and Gaussian tails in a stochastic dispersion model, Stoch. Dyn. 1 (2001), no. 3, 389-403; Continuity properties of entropy. S Newhouse, Ann. of Math. 129S. Newhouse, Continuity properties of entropy, Ann. of Math. 129, (1989), 215-235; Characteristic Liapunov exponents, and smooth ergodic theory. Ya, Pesin, Russ. Math. Surveys. 32Ya. Pesin, Characteristic Liapunov exponents, and smooth ergodic theory, Russ. Math. Surveys, 32, (1977), 55-114; Families of invariant manifolds that correspond to nonzero characteristic exponents. Ya, Pesin, Math. USSR-Izvestiya. 40Ya. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, Math. USSR-Izvestiya, 40, (1976), 1261-1305; Dimension theory in dynamical systems. Ya, Pesin, Chicago Lect. Math. U. Chicago PressYa. Pesin, Dimension theory in dynamical systems, Chicago Lect. Math., U. Chicago Press, Chicago-London, 1997; Holder foliations. C Pugh, M Shub, &amp; A Wilkinson, Duke Math. J. 86C. Pugh, M. Shub & A. Wilkinson, Holder foliations, Duke Math. J., 86, (1997), 517-546. Correction: 105, (2000), 105-106; M Schuetzow, &amp; D Steinsaltz, Chasing balls through martingale fields. preprintM. Schuetzow & D. Steinsaltz, Chasing balls through martingale fields, preprint; Ball-avoiding theorems. D Szasz, Erg. Th. & Dyn. Sys. 20D. Szasz, Ball-avoiding theorems, Erg. Th. & Dyn. Sys., 20, (2000), 1821-1849; Statistical properties of dynamical systems with some hyperbolicity. L S Young, Ann. of Math. 147L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math., 147, (1998), 585-650; Mass transport by Brownian flows. C Zirbel, &amp; E Cinlar, Stochastic models in geosystems. S. A. Molchanov & W. A. WoyczynskiMinneapolis, MN; New YorkSpringer85IMAC. Zirbel & E. Cinlar, Mass transport by Brownian flows, in Stochastic models in geosystems (Minneapolis, MN, 1994, Ed. S. A. Molchanov & W. A. Woyczynski), 459-492, IMA Vol. Math. Appl., 85, Springer, New York, 1997; Dispersion of particle systems in Brownian flows. C Zirbel, &amp; E Cinlar, Adv. in Appl. Prob. 28C. Zirbel & E. Cinlar, Dispersion of particle systems in Brownian flows, Adv. in Appl. Prob., 28, (1996), 53-74.
[]
[ "Bioinspired Composite Learning Control Under Discontinuous Friction for Industrial Robots ⋆", "Bioinspired Composite Learning Control Under Discontinuous Friction for Industrial Robots ⋆" ]
[ "Yongping Pan ", "Kai Guo [email protected] ", "Tairen Sun ", "Mohamed Darouach [email protected] ", "\nSchool of Computer Science and Engineering\nSchool of Mechanical Engineering\nSun Yat-sen University\n510006GuangzhouChina\n", "\nSchool of Medical Instrument and Food Engineering\nShandong University\n250061JinanChina\n", "\nUniversity of Shanghai for Science and Technology\n200093ShanghaiChina\n", "\nCRAN\nUMR 7035\nCNRS\nUniversity of Lorraine\nCosnes et Romain54400France\n" ]
[ "School of Computer Science and Engineering\nSchool of Mechanical Engineering\nSun Yat-sen University\n510006GuangzhouChina", "School of Medical Instrument and Food Engineering\nShandong University\n250061JinanChina", "University of Shanghai for Science and Technology\n200093ShanghaiChina", "CRAN\nUMR 7035\nCNRS\nUniversity of Lorraine\nCosnes et Romain54400France" ]
[]
Adaptive control can be applied to robotic systems with parameter uncertainties, but improving its performance is usually difficult, especially under discontinuous friction. Inspired by the human motor learning control mechanism, an adaptive learning control approach is proposed for a broad class of robotic systems with discontinuous friction, where a composite error learning technique that exploits data memory is employed to enhance parameter estimation. Compared with the classical feedback error learning control, the proposed approach can achieve superior transient and steady-state tracking without high-gain feedback and persistent excitation at the cost of extra computational burden and memory usage. The performance improvement of the proposed approach has been verified by experiments based on a DENSO industrial robot.
10.1016/j.ifacol.2022.07.292
[ "https://arxiv.org/pdf/2206.12195v1.pdf" ]
250,048,455
2206.12195
1cae7c61b25989e49623482979bfadb9c9479275
Bioinspired Composite Learning Control Under Discontinuous Friction for Industrial Robots ⋆ 24 Jun 2022 Yongping Pan Kai Guo [email protected] Tairen Sun Mohamed Darouach [email protected] School of Computer Science and Engineering School of Mechanical Engineering Sun Yat-sen University 510006GuangzhouChina School of Medical Instrument and Food Engineering Shandong University 250061JinanChina University of Shanghai for Science and Technology 200093ShanghaiChina CRAN UMR 7035 CNRS University of Lorraine Cosnes et Romain54400France Bioinspired Composite Learning Control Under Discontinuous Friction for Industrial Robots ⋆ 24 Jun 2022Robot controlAdaptive controldiscontinuous frictiondata memoryfeedback error learningparameter learning Adaptive control can be applied to robotic systems with parameter uncertainties, but improving its performance is usually difficult, especially under discontinuous friction. Inspired by the human motor learning control mechanism, an adaptive learning control approach is proposed for a broad class of robotic systems with discontinuous friction, where a composite error learning technique that exploits data memory is employed to enhance parameter estimation. Compared with the classical feedback error learning control, the proposed approach can achieve superior transient and steady-state tracking without high-gain feedback and persistent excitation at the cost of extra computational burden and memory usage. The performance improvement of the proposed approach has been verified by experiments based on a DENSO industrial robot. INTRODUCTION The feedback error learning (FEL) framework is a computational model of human motor learning control in the cerebellum, where the limitations on feedback delays and small feedback gains in biological systems can be overcome by internal forward and inverse models, respectively (Kawato et al., 1988). The name "FEL" emphasizes the usage of a feedback control signal for the heterosynaptic learning of internal neural models. There are two key features for the FEL: Internal dynamics modeling and hybrid feedbackfeedforward (HFF) control, which are well supported by much neuroscientific evidence (Thoroughman et al., 2000;Wolpert et al., 1998;Morasso et al., 2005). A simplified FEL architecture without the internal forward model has been extensively studied for robot control (Kawato et al., 1988;Tolu et al., 2012;Gomi et al., 1993;Hamavand et al., 1995;Talebi et al., 1998;Topalov et al., 1998;Teshnehlab et al., 1996;Kalanovic et al., 2000;Kurosawa et al., 2005;Neto et al., 2010;Jo et al., 2011). However, stability guarantee relies on a precondition that the controlled plant can be stabilized by linear feedback without feedforward control in Tolu et al. (2012); Kawato et al. (1988); Teshnehlab et al. (1996); Kalanovic et al. (2000); Kurosawa et al. (2005); Neto et al. (2010); Jo et al. (2011), which may not be satisfied for many control problems such as robot tracking control; in Gomi et al. (1993); Hamavand et al. (1995); Talebi et al. (1998); Topalov et al. (1998), internal inverse models are implemented in the feedback loops, which violates the original motivation of proposing FEL. stability analysis of FEL control for a class of nonlinear systems was investigated in Nakanishi et al. (2004), where the feedback gain is required to be sufficiently large to compensate for plant uncertainty so as to guarantee closed-loop stability. The approach of Nakanishi et al. (2004) was applied to the rehabilitation of Parkinson's disease in Rouhollahi et al. (2017). However, the accurate capture of the plant dynamics is not fully investigated and discontinuous friction is largely neglected in existing FEL robot control methods. This paper proposes a bioinspired adaptive learning control approach for a broad class of robotic systems with discontinuous friction, where a composite error learning (CEL) technique exploiting data memory is applied to enhance parameter estimation. The word "composite" refers to the composite exploitation of instantaneous data and data memory, and the composite exploitation of the tracking error and a generalized predictive error for parameter learning. Compared with the classical FEL control, the proposed approach can achieve superior transient and steady-state tracking without high-gain feedback and per-sistent excitation (PE). Exponential convergence of both the tracking error and the parameter estimation error is guaranteed under an interval-excitation (IE) condition that is much weaker than PE. In the HFF control, the use of the desired output as the regressor input leads to two attractive merits (Sadegh et al., 1990): 1) The regressor output can be calculated and stored offline to significantly reduce the amount of online calculations; 2) the noise correlation between the parameter estimation error in the parameter update law and the adaptation signal (i.e. an infinite gain phenomenon) can be removed to enhance estimation robustness. Additional advantages of HFF control based on neural networks can be referred to Pan et al. (2016aPan et al. ( , 2017a. This study is based on our previous studies in composite learning control (Pan et al., 2016b(Pan et al., ,c, 2017b(Pan et al., , 2018Guo et al., 2019Guo et al., , 2020Guo et al., , 2022a, in which the methods of Pan et al. (2016bPan et al. ( ,c, 2018Pan et al. ( , 2019 do not consider discontinuous friction, the methods of Pan et al. (2016bPan et al. ( ,c, 2017b consider only the case of M (q) in (1) being a known constant, and all the above methods do not resort to the HFF scheme. In the rest of this article, the control problem is formulated in Sec. 3; the CEL control is presented in Sec. 4; experimental results are given in Sec. 5; conclusions are drawn in Sec. 6. Throughout this paper, R, R + , R n and R n×m denote the spaces of real numbers, positive real numbers, real nvectors and real n × m-matrices, respectively, x denotes the Euclidian norm of x, L ∞ denotes the space of bounded signals, tr(A) denotes the trace of A, λ min (A) denotes the minimal eigenvalue of A, diag(·) denotes a diagonal matrix, min(·), max(·) and sup(·) denote the operators of minimum, maximum and supremum, respectively, B c := {x| x ≤ c} is the ball of radius c, and C k represents the space of functions whose k-order derivatives all exist and are continuous, where c ∈ R + , x ∈ R n , A ∈ R n×n , and n, m and k are natural numbers. In the subsequent sections, for the sake of brevity, the argument(s) of a function may be omitted while the context is sufficiently explicit. BIOLOGICAL BACKGROUND The FEL control process for human voluntary movements is described the following procedure (Wolpert et al., 1998): (1) The association cortex sends a desired output q d in the body coordinates to the motor cortex; (2) A motor command u is computed in the motor cortex and is transmitted to muscles via spinal motoneurons to generate a control torque τ ; (3) The musculoskeletal system realizes an actual movement q by interacting with its environment; (4) The q is measured by proprioceptors and is fed back to the motor cortex via the transcortical loop; (5) The spinocerebellum-magnocellular red nucleus system requires an internal neural model of the musculoskeletal system (i.e. internal forward model) while monitoring u and q to predictq, where the predictive errorq is transmitted to the motor cortex via the ascending pathway and to muscles through the rubrospinal tract as the modification of u; (6) The cerebrocerebellum-parvocellular red nucleus system requires an internal neural model for the inverse modeling of the musculoskeletal system (i.e. internal Fig. 1. A cerebellar neural circuit of simplified FEL control which is redrawn according to Wolpert et al. (1998), where the internal forward model is omitted. inverse model) while receiving the desired output q d and a feedback command u FB ; (7) As the motor learning proceeds, a feedforward command u FF generated by the internal inverse model gradually takes place of u FB as the main command; (8) Once the internal inverse model is learnt, it generates the motor command u directly using q d to perform various tasks precisely without external feedback. The cerebellar neural circuit of a simplified FEL framework without the internal forward model is demonstrated in Fig. 1, in which the simple spikes of Purkinje cells represent u FF , the parallel fiber inputs receive q d , the climbing fiber inputs receive u FB , and the complex spikes of Purkinje cells activated by the climbing fiber inputs represent sensory error signals in motor command coordinates. PROBLEM FORMULATION Consider a general class of robotic systems described by an Euler-Lagrange formulation (Pan et al., 2016a) 1 : M (q)q + C(q,q)q + G(q) + F (q) = τ (1) where q(t) = [q 1 (t) , q 2 (t), · · · , q n (t)] T ∈ R n is a joint angle vector, M (q) ∈ R n×n is an inertia matrix, C(q,q) ∈ R n×n is a centripetal-Coriolis matrix, G(q) ∈ R n , F (q) ∈ R n and τ (t) ∈ R n denote gravitational, friction, and control torque vectors, respectively, and n is the number of links. It is assumed that F (q) can be expressed as follows: F (q) = F vq + F c sgn(q) (2) with F v := diag(k v1 , k v2 , · · · , k vn ), F c := diag(k c1 , k c2 , · · · ,k cn ), and sgn(q) := [sgn(q 1 ), sgn(q 2 ), · · · , sgn(q n )] T , where k vi ∈ R + are coefficients of viscous friction, k ci ∈ R + are coefficients of Coulomb friction, and i = 1 to n. For facilitating presentation, define a lumped uncertainty H(q,q, v,v) := M (q)v + C(q,q)v + G(q) (3) with v ∈ R n an auxiliary variable. The following properties from Pan et al. (2018) and definitions from Kingravi et al. (2012) are introduced to facilitate control design. Property 1: M (q) is a symmetric and positive-definite matrix which satisfies m 0 ζ 2 ≤ ζ T M (q)ζ ≤m ζ 2 , ∀ ζ ∈ R n , in which m 0 ,m ∈ R + are some constants. Property 2:Ṁ (q)−2C(q,q) is skew-symmetric such that ζ T (Ṁ (q) −2C(q,q))ζ = 0, ∀ζ ∈ R n , implying the internal forces of the robot do no work. Property 3: F (q) can be parameterized by F (q) = Φ T f (q)W f = Φ T v (q)W v + Φ T c (q)W c (4) F vq F c sgn(q) where Φ f := [Φ T v , Φ T c ] T ∈ R 2n×n is a known regressor, W f := [W T v , W T c ] T ∈ B c f ⊂ R 2n is an unknown parameter vector, c f ∈ R + is a certain constant, and W v := [k v1 , k v2 , · · · , k vn ] T , W c := [k c1 , k c2 , · · · , k cn ] T , Φ v := diag(q 1 ,q 2 , · · · ,q n ), Φ c := diag(sgn(q 1 ), sgn(q 2 ), · · · , sgn(q n )). Property 4: H(q,q, v,v) can be parameterized by H(q,q, v,v) = Φ T h (q,q, v,v)W h (5) where Φ h : R 4n → R N ×n is a known C 1 regressor, W h ∈ B c h ⊂ R N is an unknown parameter vector, c h ∈ R + is a certain constant, and N is the dimension of W h . Definition 1: A bounded signal Φ(t) ∈ R N ×n is of IE if ∃ T e , τ d , σ e ∈ R + such that Te Te−τ d Φ(τ )Φ T (τ )dτ ≥ σ e I. Definition 2: A bounded signal Φ(t) ∈ R N ×n is of PE if ∃ σ e , τ d ∈ R + such that t t−τ d Φ(τ )Φ T (τ )dτ ≥ σ e I, ∀t ≥ 0. Let c w := (c 2 h + c 2 f ) 1/2 . It follows from (3)-(5) that the left side (1) can be written as a parameterized form: H(q,q,q,q) + F (q) = Φ T (q,q,q)W (6) in which Φ(q,q,q) : = [Φ T h (q,q,q,q), Φ T v (q), Φ T c (q)] T ∈ R (N +2n)×n and W := [W T h , W T v , W T c ] T ∈ B cw ⊂ R N +2n . LetŴ h (t) ∈ R N ,Ŵ f (t) ∈ R 2n ,Ŵ v (t) ∈ R n andŴ c (t) ∈ R n denote estimates of W h , W f , W v and W c , respectively. Define a parameter estimation errorW := W −Ŵ = [W T h , W T v ,W T c ] T , whereŴ := [Ŵ T h ,Ŵ T v ,Ŵ T c ] T ∈ R N +2n ,W h := W h −Ŵ h ,W v := W v −Ŵ v , andW c := W c −Ŵ c . Let x d (t) := [q T d (t),q T d (t),q T d (t)] T ∈ R 3n be of L ∞ with q d := [q d1 , q d2 , · · · , q dn ] T ∈ R n a desired output. Define a position tracking error e 1 := q d − q and a "reference velocity" tracking error e 2 :=ė 1 + Λ 1 e 1 =q r −q, where Λ 1 ∈ R n×n denotes a positive-definite diagonal matrix, andq r :=q d + Λ 1 e 1 denotes a "reference velocity". Let e = [e T 1 , e T 2 ] T ∈ R 2n . The objective of this study is to develop an adaptive control strategy for the system (1), such that the closed-loop system is stable with guaranteed convergence of both e andW . ROBOT CONTROL DESIGN Hybrid Feedback-Feedforward Structure In this subsection, we have v =q r in (5). Taking the time derivative of e 2 and multiplying M (q), one obtains M (q)ė 2 = M (q)(q d + Λė 1 ) − M (q)q. Substituting the expression of M (q)q by (1) into the foregoing equality, one gets the tracking error dynamics M (q)ė 2 = M (q)q r + C(q,q)q r + G(q) + F (q) − τ . Applying (3) with v =q r to the above result leads to M (q)ė 2 = H(q,q,q r ,q r ) + F (q) − C(q,q)e 2 − τ . (7) It follows from the definitions of e 1 , e 2 andq r that H(q, q,q r ,q r ) and its corresponding Φ h (q,q,q r ,q r ) can be denoted as H(x d , e) and Φ h (x d , e), respectively. Subtracting and adding H(x d , 0) at the right side of (7) yields M (q)ė 2 =H(x d , e) + H(x d , 0) + F (q) − C(q,q)e 2 − τ(8) whereH : R 3n × R 2n → R n is given bỹ H(x d , e) := H(x d , e) − H(x d , 0). As H is of C 1 as in Property 4, there is a globally invertible and strictly increasing function ρ : R + → R + so that the following bound condition holds (Xian et al., 2004): H (x d , e) ≤ ρ( e ) e . (9) Applying (5) to H(x d , 0) and using (4), (8) becomes M (q)ė 2 =H(x d , e) + Φ T h (x d )W h + Φ T f (q)W f − C(q,q)e 2 − τ (10) where Φ T h (x d )W h = H(x d , 0) with Φ h : R 3n → R N ×n . Inspired by the human motor learning control mechanism, the control torque τ is designed as follows: τ = K c e + Φ T h (x d )Ŵ h + Φ T f (q r )Ŵ f (11) τ FB τ FF τ FC with K c := [I, Λ 2 ], in which Λ 2 ∈ R n×n denotes a positivedefinite diagonal matrix of control gains, τ FB and τ FF are PD feedback and adaptive feedforward parts, respectively, and τ FC is applied to compensate for the friction F (q). Replacingq byq r in τ FC can make it less noise-sensitive (asq d and e 1 (q r =q d + Λ 1 e 1 ) are usually less noisy thaṅ q) but still maintains the stability requirement (Slotine & Li, 1991). Substituting (11) into (10), one obtains the closed-loop tracking error dynamics M (q)ė 2 =H(x d , e) + Φ T h (x d )W h + Φ T f (q)W f − Φ T f (q r )Ŵ f − C(q,q)e 2 − K c e.(12) Composite Error Learning Technique In this subsection, we have v =q in (5). Applying (6) to (1), one gets a parameterized robot model τ (t) = Φ T (q(t) ,q(t),q(t))W (13) To eliminate the necessity ofq in parameter estimation, a linear filter α s+α is applied to each side of (13) resulting in τ F (t) = Φ T F (q(t),q(t))W (14) where s denotes a complex variable, α ∈ R + is a filtering parameter, Φ F := αe −αt * Φ and τ F := αe −αt * τ are filtered counterparts of Φ and τ , respectively, and " * " is the convolution operator. A predictive model is given bŷ τ F (t) = Φ T F (q(t),q(t))Ŵ (t)(15) in whichτ F ∈ R n is a predicted counterpart of τ F . To facilitate presentation, define an excitation matrix (17) which is a filtered, regressor-extended and integrated form of (13). Define a generalized predictive error Θ(t) := t t−τ d Φ F q(τ ),q(τ ) Φ T F q(τ ),q(τ ) dτ.(Θ(t)W = t t−τ d Φ F q(τ ),q(τ ) τ F (τ )dτξ(t) := Θ(t)W − Θ(t)Ŵ (t), t < T e Θ(T e )W − Θ(T e )Ŵ (t), otherwise(18) where ΘW is obtainable by (17). A CEL law with switching σ-modification is designed as follows: W = γ Φ(x d ,q r )e 2 + κξ − σ sŴ (19) with Φ(x d ,q r ) = [Φ T h (x d ), Φ T f (q r )] T ∈ R (N +2n)×n , where γ ∈ R + is a learning rate, κ ∈ R + is a weight factor, and σ sŴ is a C 0 switching "leaky" term with σ s (t) :=    0, if Ŵ < c w σ 0 , if Ŵ > 2c w σ 0 Ŵ /c w − 1 , otherwise and σ 0 ∈ R + a constant design parameter. The switching σ-modification is used to guarantee closed-loop stability with bounded parameter estimation under perturbations if no excitation exists during control (Ioannou& Sun, 1996). The overall closed-loop system that combines the tracking error dynamics with the parameter estimation error dynamics is presented according to (12) and (19) as follows:       ė 1 = e 2 − Λ 1 e 1 e 2 = M −1 (q) H (x d , e) − C(q,q)e 2 − K c e +Φ T h (x d )W h + Φ T f (q)W f − Φ T f (q r )Ŵ f Ẇ = −γ Φ(x d ,q r )e 2 + κξ − σ sŴ .(20) A block diagram of the CEL robot control scheme is given in Fig. (2). If there exists T e , σ e , τ d ∈ R + such that the IE condition Θ(T e ) ≥ σ e I holds, the control parameters Λ 1 , Λ 2 , γ and σ 0 can be properly selected so that the closedloop system has semiglobal stability in the sense that all signals are of L ∞ and e(t) asymptotically converges to 0 on t ∈ [0, ∞), and both e(t) andW (t) exponentially converge to 0 on t ∈ [T e , ∞). The above results can be proven based on the Filippov's theory of differential inclusions and the LaSalle-Yoshizawa corollaries for nonsmooth systems (Fischer et al., 2013), where the details are omitted here due to the page limitation. INDUSTRIAL ROBOT APPLICATION The proposed CEL controller is implemented on a DENSO robot arm with a Quanser real-time control module [see Fig. 3]. Each joint of the robot arm is driven by an AC servo motor with a speed reducer, where the gear ratios of the three joints used in the experiments are 160, 120, and 100, respectively. A 17-bit absolute rotary encoder is used to measure the angle of each motor. Therefore, the resolutions of the three joints are 3.0 × 10 −7 rad, 4.0 × 10 −7 rad and 4.8×10 −7 rad, respectively. The sampling time of the real-time control module is 1 ms. The robot regression model in Xin et al. (2007) is introduced for implementation. The proposed CEL control law comprised of (11) and (19) is rewritten as follows: τ = K c e + Φ T (x d ,q r )Ŵ W = γ Φ(x d ,q r )e 2 + κξ − σ sŴ where the values of the control parameters are selected as Λ 1 = diag(4, 4, 8) and Λ 2 = diag(6, 6, 1.5) in (11), α = 5 in (14), τ d = 4 in (16), and γ = 0.15, κ = 0.5, σ 0 = 0.1, c w = 5 andŴ (0) = 0 in (19). A baseline controller is chosen as the classical FEL control law as follows: τ = K c e + Φ T (x d ,q r )Ŵ W = γ Φ(x d ,q r )e 2 − σ sŴ where the control parameters are selected to be the same values as the proposed control law for fair comparison. To verify the learning ability of the proposed controller, the desired joint position q d is expected to be simple. Consider a regulation problem with q d being generated by Table I provides control results of the two controllers for both the first (before learning) and the last tasks (after learning). For the classical FEL control, there exists a large tracking deviation between the desired position q d and the actual position q for the first task. After the learning for Table I. q dï q di = 0 1 −36 −12 q di q di + 0 36 q ci with i = 1 to 3 andq d (0) = 0, where q ci (t) It is also demonstrated in Table I that the maximal control torques τ of the proposed CEL control are much smaller than those of the classical FEL control for all joints and tasks, which implies that the proposed CEL control is able to achieve much better tracking accuracy even using much smaller control gains and much less energy cost. This is because the improved feedforward control resulting from accurate parameter estimation is beneficial for reducing feedback control torques. The control torque τ of joint 3 shows slight oscillations compared with those of joints 1 and 2 due to its more significant joint elasticity caused by the unique synchronous belt drive mechanism. This is also the reason why the performance improvement of joint 3 by the proposed CEL control shown in Table I is not as significant as those of joints 1 and 2. Fig. 4 provides performance comparisons of the two controllers. For the classical FEL control, as the PE condition does not hold for the entire control process, no parameter convergence is shown [see Fig. 4(b)]. In sharp contrast, the proposed CEL control achieves fast convergence of Ŵ to a certain constant [see Fig. 4(b)]. This is consistent with the theoretical analysis: The proposed CEL control only requires the much weaker IE condition for parameter convergence, which can be satisfied during the transient process of the first task. Also, the CEL control achieves a smaller e 1 than the FEL control even from the first control task owing to the predictive error feedback in the CEL law, and maintains the superior tracking performance during the entire control process [see Fig. 4(a)]. CONCLUSIONS In this paper, a novel CEL framework has been developed for robot control under discontinuous friction. Compared with the classical FEL control, the distinctive features of the proposed approach include: 1) Semiglobal stability of the closed-loop system is ensured without high feedback gains; 2) exact robot modeling is ensured by the weaken IE condition. The proposed approach has been applied to a DENSO industrial robot, and experimental results have shown that it is superior with respect to tracking accuracy and control energy compared with the classical FEL control. Future work would focus on the optimization of the experimental setup to speed up parameter convergence and the applications of the proposed approach to more real-world robotic systems (Liu et al., 2021a,b). ⋆ This work was supported in part by the Guangdong Pearl River Talent Program of China under Grant No. 2019QN01X154 (Corresponding author: Yongping Pan). Fig. 3 . 3Experimental setup: A 6-axis articulated robot. (a) A Denso robot arm (Type: VP6242G). (b) A Quanser open architecture real-time control module. is a step trajectory that repeats every 50 s. The experiments last for 250s, and thus, there are five control tasks. The units of joint position and torque are rad and N.m, respectively. Let e 1 = [e 11 , e 12 , e 13 ] T , where e 1i ∈ R denotes the position tracking error for Joint i with i = 1 to 3. Fig. 4 . 4A comparison of control trajectories for the two controllers. (a) The norm of the tracking error e. (b) The norm of the parameter estimateŴ . Table 1 . 1A comparison of performance indices for the two controllers 200 s, the tracking performance is slightly improved for the last task. As an example, the range of the tracking error e 11 after learning for Joint 1 is reduced from [−1.266, 2.016] (before learning) to [−0.679, 1.044] (≈52% of that before learning) under the classical FEL control.For the proposed CEL control, a large tracking error e 1 still exists before learning. During the first task, the IE condition is met. After the learning for 200 s, the proposed CEL control improves significantly in tracking accuracy compared with that before learning. For example, the range of e 11 after learning for Joint 1 is reduced from [−0.921, 0.912] (before learning) to[−0.284, 0.152] (only ≈24% of that before learning) under the proposed FEL control. The strong learning capacity of the proposed CEL control is also clearly shown by comparing the ranges of e 1 under the two controllers inRanges of e 1 and τ The classical FEL Control The proposed CEL control Joint 1 Joint 2 Joint 3 Joint 1 Joint 2 Joint 3 e 1 before learning ( • ) [−1.266, 2.016] [−1.775, 1.976] [−3.876, 2.297] [−0.921, 0.912] [−1.590, 1.911] [−2.973, 0.908] e 1 after learning ( • ) [−0.679, 1.044] [−1.519, 0.969] [−3.136, 1.509] [−0.284, 0.152] [−0.415, 0.098] [−1.708, 1.406] τ before learning (N.m)[−5.478, 11.08] [−11.21, 6.738] [−10.51, 8.289] [−1.307, 7.059] [−6.246, 3.358] [−7.832, 5.458] τ after learning (N.m) [−5.165, 7.909] [−9.304, 8.865] [−10.90, 6.932] [−1.837, 6.604] [−7.927, 3.721] [−7.861, 5.774] It can be regarded as a simplified model of the musculoskeletal system where the muscle dynamics is ignored resulting in τ = u. . N Fischer, R Kamalapurkar, W E Dixon, Fischer, N., Kamalapurkar, R., and Dixon, W. E. (2013). LaSalle-Yoshizawa corollaries for nonsmooth systems. IEEE Transactions on Automatic Control. 589LaSalle-Yoshizawa corollaries for nonsmooth systems. IEEE Transactions on Automatic Control, 58(9), 2333- 2338. Neural network control for a closed-loop system using feedback-error-learning. H Gomi, M Kawato, Neural Networks. 67Gomi, H., and Kawato, M. (1993). Neural network control for a closed-loop system using feedback-error-learning. Neural Networks, 6(7), 933-946. Composite learning robot control with friction compensation: A neural network-based approach. K Guo, Y Pan, Yu , H , IEEE Transactions on Industrial Electronics. 6610Guo, K., Pan, Y., and Yu, H. (2019). Composite learn- ing robot control with friction compensation: A neural network-based approach. IEEE Transactions on Indus- trial Electronics, 66(10), 7841-7851. Composite learning control of robotic systems: A least squares modulated approach. K Guo, Y Pan, D Zheng, Yu , H , ID: 108612Automatica. 111Guo, K., Pan, Y., Zheng, D., and Yu, H. (2020). Composite learning control of robotic systems: A least squares mod- ulated approach. Automatica, 111, Article ID: 108612. Adaptive tracking control of hydraulic systems with improved parameter convergence. K Guo, M Li, W Shi, Y Pan, IEEE Transactions on Industrial Electronics. 697Guo, K., Li, M., Shi, W., and Pan, Y. (2022). Adaptive tracking control of hydraulic systems with improved pa- rameter convergence. IEEE Transactions on Industrial Electronics, 69 (7), 7140-7150. Locally weighted learning robot control with improved parameter convergence. K Guo, Y Liu, B Xu, Y Xu, Y Pan, 10.1109/TIE.2022.3140503IEEE Transactions on Industrial Electronics. Guo, K., Liu, Y., Xu, B., Xu, Y., and Pan, Y. (2022). Locally weighted learning robot control with improved parameter convergence. IEEE Transactions on Indus- trial Electronics, DOI: 10.1109/TIE.2022.3140503. Trajectory control of robotic manipulators by using a feedbackerror-learning neural network. Z Hamavand, H M Schwartz, Robotica. 13Hamavand, Z. and Schwartz, H. M. (1995). Trajectory control of robotic manipulators by using a feedback- error-learning neural network. Robotica, 13, 449-459. Robust Adaptive Control. P A Ioannou, J Sun, Prentice HallEnglewood Cliffs, NJ, USAIoannou, P. A., and Sun, J. (1996). Robust Adaptive Control. Prentice Hall, Englewood Cliffs, NJ, USA. A computational neuromusculoskeletal model of human arm movements. S Jo, International Journal of Control Automation and Systems. 95Jo, S. (2011). A computational neuromusculoskeletal model of human arm movements. International Journal of Control Automation and Systems, 9(5), 913-923. Feedback error learning neural network for transfemoral prosthesis. V D Kalanovic, D Popovic, N T Skaug, IEEE Transactions on Neural Systems and Rehabilitation Engineering. 81Kalanovic, V. D., Popovic, D., and Skaug, N. T. (2000). Feedback error learning neural network for trans- femoral prosthesis. IEEE Transactions on Neural Sys- tems and Rehabilitation Engineering, 8(1), 71-80. Hierarchical neural network model for voluntary movement with application to robotics. M Kawato, Y Uno, M Isobe, R Suzuki, IEEE Control Systems Magazine. 82Kawato, M., Uno, Y., Isobe, M., and Suzuki, R. (1988). Hi- erarchical neural network model for voluntary movement with application to robotics. IEEE Control Systems Magazine, 8(2), 8-15. Reproducing kernel Hilbert space approach for the online update of radial bases in neuroadaptive control. H A Kingravi, G Chowdhary, P A Vela, Johnson , E N , IEEE Transactions on Neural Networks and Learning Systems. 237Kingravi, H. A., Chowdhary, G., Vela, P. A., and John- son, E. N. (2012). Reproducing kernel Hilbert space approach for the online update of radial bases in neuro- adaptive control. IEEE Transactions on Neural Net- works and Learning Systems, 23(7), 1130-1141. Joint angle control by FES using a feedback error learning controller. K Kurosawa, R Futami, T Watanabe, N Hoshimiya, IEEE Transactions on Neural Systems and Rehabilitation Engineering. 133Kurosawa, K., Futami, R., Watanabe, T., and Hoshimiya, N. (2005). Joint angle control by FES using a feedback error learning controller. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 13(3), 359-371. Preliminary evaluation of composite learning tracking control on 7-DoF collaborative robots. X Liu, Z Li, Y Pan, IFAC-PapersOnLine. 5414Liu, X., Li, Z., and Pan, Y. (2021). Preliminary evaluation of composite learning tracking control on 7-DoF collab- orative robots. IFAC-PapersOnLine, 54(14), 470-475. X Liu, Z Li, Y Pan, Experiments of composite learning admittance control on 7-DoF collaborative robots. International Conference on Intelligent Robotics and Applications. Yantai, ChinaLiu, X., Li, Z., and Pan, Y. (2021). Experiments of compos- ite learning admittance control on 7-DoF collaborative robots. International Conference on Intelligent Robotics and Applications, Yantai, China, 532-541. Preflexes and internal models in biomimetic robot systems. P Morasso, A Bottaro, M Casadio, V Sanguineti, Cognitive Processing. 61Morasso, P., Bottaro, A., Casadio, M., and Sanguineti ,V. (2005). Preflexes and internal models in biomimetic robot systems. Cognitive Processing, 6(1), 25-36. Feedback error learning and nonlinear adaptive control. J Nakanishi, S Schaal, Neural Networks. 1710Nakanishi, J., and Schaal, S. (2004). Feedback error learn- ing and nonlinear adaptive control. Neural Networks, 17(10), 1453-1465. Accumulative learning using multiple ANN for flexible link control. A D Neto, L C S Goes, C L Nascimento, IEEE Transactions on Aerospace and Electronic Systems. 462Neto, A. D., Goes, L. C. S., and Nascimento, C. L. (2010). Accumulative learning using multiple ANN for flexible link control. IEEE Transactions on Aerospace and Electronic Systems, 46(2), 508-524. Hybrid feedback feedforward: An efficient design of adaptive neural network control. Y Pan, Y Liu, B Xu, Yu , H , Neural Networks. 76Pan, Y., Liu, Y., Xu, B., and Yu, H. (2016a). Hybrid feedback feedforward: An efficient design of adaptive neural network control. Neural Networks, 76, 122-134. Composite learning from adaptive dynamic surface control. Y Pan, Yu , H , IEEE Transaction on Automatic Control. 619Pan, Y., and Yu, H. (2016b). Composite learning from adaptive dynamic surface control. IEEE Transaction on Automatic Control, 61(9), 2603-2609. Model reference composite learning control without persistency of excitation. Y Pan, J Zhang, Yu , H , IET Control Theory and Applications. 1016Pan, Y., Zhang, J., and Yu, H. (2016c). Model reference composite learning control without persistency of exci- tation. IET Control Theory and Applications, 10(16), 1963-1971. Biomimetic hybrid feedback feedforward neural network learning control. Y Pan, Yu , H , IEEE Transactions on Neural Networks and Learning Systems. 286Pan, Y., and Yu, H. (2017a). Biomimetic hybrid feedback feedforward neural network learning control. IEEE Transactions on Neural Networks and Learning Systems, 28(6), 1481-1487. Composite learning from adaptive backstepping neural network control. Y Pan, T Sun, Y Liu, H Yu, Neural Networks. 95Pan, Y., T. Sun, Y. Liu and H. Yu (2017b). Composite learning from adaptive backstepping neural network control. Neural Networks, 95: 134-142. Composite learning robot control with guaranteed parameter convergence. Automatica. Y Pan, Yu , H , 89Pan, Y., and Yu, H. (2018). Composite learning robot control with guaranteed parameter convergence. Auto- matica, 89, 398-406. Efficient learning from adaptive control under sufficient excitation. Y Pan, S Aranovskiy, A Bobtsov, Yu , H , International Journal of Robust and Nonlinear Control. 2910Pan, Y., Aranovskiy, S., Bobtsov, A., and Yu, H. (2019). Efficient learning from adaptive control under sufficient excitation. International Journal of Robust and Nonlin- ear Control, 29(10), 3111-3124. Design of robust adaptive controller and feedback error learning for rehabilitation in Parkinson's disease: A simulation study. K Rouhollahi, M E Andani, S M Karbassi, I Izadi, IET Systems Biology. 111Rouhollahi, K., Andani, M. E., Karbassi, S. M., and Izadi, I. (2017). Design of robust adaptive controller and feedback error learning for rehabilitation in Parkinson's disease: A simulation study. IET Systems Biology, 11(1), 19-29. Stability and robustness analysis of a class of adaptive controllers for robotic manipulators. N Sadegh, R Horowitz, International Journal of Robotics Research. 93Sadegh, N., and Horowitz, R. (1990). Stability and robustness analysis of a class of adaptive controllers for robotic manipulators. International Journal of Robotics Research, 9(3), 74-92. Applied Nonlinear Control. J.-J E Slotine, Li , W , Prentice HallEnglewood Cliffs, NJ, USASlotine, J.-J. E., and Li, W. (1991). Applied Nonlinear Control. Prentice Hall, Englewood Cliffs, NJ, USA. Neural network based control schemes for flexible-link manipulators: Simulations and experiments. H A Talebi, K Khorasani, R V Patel, Neural Networks. 117-8Talebi, H. A., Khorasani, K., and Patel, R. V. (1998). Neural network based control schemes for flexible-link manipulators: Simulations and experiments. Neural Networks, 11(7-8), 1357-1377. Neural network controller with flexible structure based on feedbackerror-learning approach. M Teshnehlab, K Watanabe, Journal of Intelligent and Robotic Systems. 154Teshnehlab, M. and Watanabe, K. (1998). Neural network controller with flexible structure based on feedback- error-learning approach. Journal of Intelligent and Robotic Systems, 15(4), 367-387. Learning of action through adaptive combination of motor primitives. K A Thoroughman, R Shadmehr, Nature. 6805Thoroughman, K. A. and Shadmehr, R. (2000). Learning of action through adaptive combination of motor primi- tives. Nature, 407(6805), 742-747. Bio-inspired adaptive feedback error learning architecture for motor control. S Tolu, M Vanegas, N R Luque, J A Garrido, Ros , E , Biological Cybernetics. 1068-9Tolu, S., Vanegas, M., Luque, N. R., Garrido, J. A., and Ros, E. (2012). Bio-inspired adaptive feedback error learning architecture for motor control. Biological Cybernetics, 106(8-9), 507-522. Fuzzy-net control of non-holonomic mobile robot using evolutionary feedback-error-learning. A V Topalov, J H Kim, T P Proychev, 23Robotics and Autonomous SystemsTopalov, A. V., Kim, J. H., and Proychev, T. P. (1998). Fuzzy-net control of non-holonomic mobile robot using evolutionary feedback-error-learning. Robotics and Au- tonomous Systems, 23(3), 187-200. Internal models in the cerebellum. D M Wolpert, R C Miall, M Kawato, Trends in Cognitive Science. 29Wolpert, D. M., Miall, R. C., and Kawato, M. (1998). Internal models in the cerebellum. Trends in Cognitive Science, 2(9), 338-347. A continuous asymptotic tracking control strategy for uncertain nonlinear systems. B Xian, D M Dawson, M S De Queiroz, Chen , J , IEEE Transactions on Automatic Control. 497Xian, B., Dawson, D. M., de Queiroz, M. S., and Chen, J. (2004). A continuous asymptotic tracking control strat- egy for uncertain nonlinear systems. IEEE Transactions on Automatic Control , 49(7), 1206-1211. Swing-up control for a 3-DOF gymnastic robot with passive first joint: design and analysis. X Xin, M Kaneda, IEEE Transactions on Robotics. 236Xin, X. and Kaneda, M. (2007). Swing-up control for a 3- DOF gymnastic robot with passive first joint: design and analysis. IEEE Transactions on Robotics, 23(6), 1277- 1285.
[]
[ "Quantum AI simulator using a hybrid CPU-FPGA approach", "Quantum AI simulator using a hybrid CPU-FPGA approach" ]
[ "Teppei Suzuki [email protected] \nResearch and Development Center\nSCSK Corporation\nToyosu Front\n3-2-20 Toyosu, Koto-ku135-8110Tokyo\n", "Tsubasa Miyazaki \nResearch and Development Center\nSCSK Corporation\nToyosu Front\n3-2-20 Toyosu, Koto-ku135-8110Tokyo\n", "Toshiki Inaritai \nResearch and Development Center\nSCSK Corporation\nToyosu Front\n3-2-20 Toyosu, Koto-ku135-8110Tokyo\n", "Takahiro Otsuka \nResearch and Development Center\nSCSK Corporation\nToyosu Front\n3-2-20 Toyosu, Koto-ku135-8110Tokyo\n", "Japan \nResearch and Development Center\nSCSK Corporation\nToyosu Front\n3-2-20 Toyosu, Koto-ku135-8110Tokyo\n" ]
[ "Research and Development Center\nSCSK Corporation\nToyosu Front\n3-2-20 Toyosu, Koto-ku135-8110Tokyo", "Research and Development Center\nSCSK Corporation\nToyosu Front\n3-2-20 Toyosu, Koto-ku135-8110Tokyo", "Research and Development Center\nSCSK Corporation\nToyosu Front\n3-2-20 Toyosu, Koto-ku135-8110Tokyo", "Research and Development Center\nSCSK Corporation\nToyosu Front\n3-2-20 Toyosu, Koto-ku135-8110Tokyo", "Research and Development Center\nSCSK Corporation\nToyosu Front\n3-2-20 Toyosu, Koto-ku135-8110Tokyo" ]
[]
The quantum kernel method has attracted considerable attention in the field of quantum machine learning. However, exploring the applicability of quantum kernels in more realistic settings has been hindered by the number of physical qubits current noisy quantum computers have, thereby limiting the number of features encoded for quantum kernels. Hence, there is a need for an efficient, application-specific simulator for quantum computing by using classical technology. Here we focus on quantum kernels empirically designed for image classification and demonstrate a field programmable gate arrays (FPGA) implementation. We show that the quantum kernel estimation by our heterogeneous CPU-FPGA computing is 470 times faster than that by a conventional CPU implementation. The co-design of our application-specific quantum kernel and its efficient FPGA implementation enabled us to perform one of the largest numerical simulations of a gate-based quantum kernel in terms of features, up to 780-dimensional features. We apply our quantum kernel to classification tasks using the Fashion-MNIST dataset and show that our quantum kernel is comparable to Gaussian kernels with the optimized hyperparameter.OPEN
10.1038/s41598-023-34600-2
[ "https://export.arxiv.org/pdf/2206.09593v2.pdf" ]
249,889,646
2206.09593
8a83c7118d72d909473d3dd036ae0930b488579b
Quantum AI simulator using a hybrid CPU-FPGA approach 0123456789 Teppei Suzuki [email protected] Research and Development Center SCSK Corporation Toyosu Front 3-2-20 Toyosu, Koto-ku135-8110Tokyo Tsubasa Miyazaki Research and Development Center SCSK Corporation Toyosu Front 3-2-20 Toyosu, Koto-ku135-8110Tokyo Toshiki Inaritai Research and Development Center SCSK Corporation Toyosu Front 3-2-20 Toyosu, Koto-ku135-8110Tokyo Takahiro Otsuka Research and Development Center SCSK Corporation Toyosu Front 3-2-20 Toyosu, Koto-ku135-8110Tokyo Japan Research and Development Center SCSK Corporation Toyosu Front 3-2-20 Toyosu, Koto-ku135-8110Tokyo Quantum AI simulator using a hybrid CPU-FPGA approach 012345678910.1038/s41598-023-34600-21 Scientific Reports | (2023) 13:7735 | https:// The quantum kernel method has attracted considerable attention in the field of quantum machine learning. However, exploring the applicability of quantum kernels in more realistic settings has been hindered by the number of physical qubits current noisy quantum computers have, thereby limiting the number of features encoded for quantum kernels. Hence, there is a need for an efficient, application-specific simulator for quantum computing by using classical technology. Here we focus on quantum kernels empirically designed for image classification and demonstrate a field programmable gate arrays (FPGA) implementation. We show that the quantum kernel estimation by our heterogeneous CPU-FPGA computing is 470 times faster than that by a conventional CPU implementation. The co-design of our application-specific quantum kernel and its efficient FPGA implementation enabled us to perform one of the largest numerical simulations of a gate-based quantum kernel in terms of features, up to 780-dimensional features. We apply our quantum kernel to classification tasks using the Fashion-MNIST dataset and show that our quantum kernel is comparable to Gaussian kernels with the optimized hyperparameter.OPEN Quantum computing 1 is an emerging technology that could transform many areas of industries and scientific research, including finance 2 , chemistry 3 , and machine learning (ML) 4,5 . In particular, quantum machine learning (QML) [4][5][6][7][8][9][10][11][12][13][14][15][16][17] has received considerable attention at a rapid rate, indicating that QML is a plausible candidate for the practical application of near-term quantum devices. While early fault-tolerant quantum computing has been demonstrated recently 18 , noisy intermediate-scale quantum (NISQ) processors are currently available through various hardware platforms with ∼ 10-100 physical qubits. However, the number of physical qubits today's NISQ computers have is generally insufficient to explore practical applications of QML. Therefore, there is a need for application-specific quantum computing simulators to explore and validate the practicality of QML in real-world settings. The quantum kernel method is a NISQ algorithm in the framework of the hybrid quantum-classical approach 19,20 and can also be feasible on current NISQ computers with shallow quantum circuits 9,12,13,16,17 . In the quantum kernel method, a quantum feature map can be described explicitly by a quantum circuit and the quantum kernel entry can be estimated by measuring the inner product of the quantum feature map 8,9 . The calculation of quantum kernels when using real devices or general-purpose simulators based on quantum assembly language (QASM) requires a number of measurements to obtain the quantum kernel entries (note that measurements are a key part of the QASM simulator, which handles measurements by collapsing the state of the qubit according to the probabilities predicted by quantum mechanics). Commonly used quantum kernels inspired by instantaneous quantum polynomials (IQP) 9 can be computationally prohibitive on classical computers as the number of qubits increases; for instance, the number of entangled qubits in the simulation of quantum kernels using state-of-the-art classical platforms is 30 11 . On the other hand, it becomes challenging to reliably estimate such quantum kernels using near-term quantum devices with increasing size in circuits, owing to expensive gate costs, low gate fidelities, and different qubit connectivities. The above points can be a drawback in exploring practical applications of quantum kernels since machine learning models typically improve performance by increasing training data samples or expanding the number of input features. There is still a gap between theoretical developments and practical applications in the quantum kernel method. To bridge the gap between theory and practice in the quantum kernel method, in this paper, we focus on an application-specific quantum kernel that can be applied to image data with a large number of features. To this end, we demonstrate an implementation of an efficient quantum AI simulator by using a heterogeneous classical computing platform. Our approach is highly customized for our specific tasks at the hardware level and the main objective of our simulator differs from a general-purpose quantum simulator, which is designed to be versatile and to perform a range of quantum algorithms. Until now, there have been considerable efforts to develop quantum computing simulators [21][22][23][24][25][26][27] www.nature.com/scientificreports/ arrays (FPGA) are one of the desirable platforms, because FPGA has the properties of efficient parallelism, low latency, and customization. FPGAs comprise programmable logic blocks that can be interconnected to perform parallel processing, allowing each logic block to perform a specific task simultaneously. FPGAs can also be customized to perform specific tasks using hardware description languages such as Verilog. Herein, we co-design application-specific quantum kernels and our FPGA architecture, which allows efficient numerical simulations. FPGA has been successfully applied to fault-tolerant quantum algorithms such as Grover's algorithm [28][29][30] , quantum Fourier transform [28][29][30][31] , and Deutsch's algorithm 32 . However, an FPGA implementation of quantum kernels has been unexplored and the present study is the first demonstration of a gate-based quantum kernel simulator using an FPGA platform. The rest of the paper is organized as follows. We provide a brief introduction to support vector machine (SVM) and describe our quantum feature map that is useful for image classification. Then we explain the overview of our quantum AI simulator using a heterogeneous CPU-FPGA computing. From an algorithmic point of view, the quantum kernel method can be divided into the quantum kernel estimation and the rest of the tasks. The simulation of the quantum kernel can be computationally demanding; hence, the workload can be accelerated by FPGA hardware. On the other hand, the rest of the tasks, such as dimensionality reduction and the optimization of machine learning parameters, can be efficiently performed on the CPU using existing classical libraries. The FPGA implementation of the quantum kernel is checked in terms of both numerical precision and hardware acceleration. We apply our quantum kernel simulator to binary and multiclass classification for a range of input features using the Fashion-MNIST dataset. Then we summarize our conclusions. Results Quantum support vector machine. The quantum kernel method is one of the most important algorithms in QML techniques and many studies have been reported 4,5,8,9,[11][12][13][14][15][16][17] . In the classical kernel method 33,34 the inner product of the feature map is represented by kernel functions, which implicitly use the Hilbert space; on the other hand, the quantum kernel explicitly defines a quantum feature map by means of a quantum state |φ(x)� for d-dimensional input vectors x ∈ R d . The quantum kernel matrix K x, x ′ can be estimated by calculating the inner product of the quantum feature map 8,9 : For binary classification in the framework of SVM, one can obtain a support vector classifier that estimates the label for a new datum x: where y i ∈ {+1, −1} and parameters {α * i } and b * are the optimal parameters obtained in the training phase 34 . In the hybrid quantum-classical algorithm, the training phase can be performed on classical computers, whereas the quantum kernel entries can be computed by NISQ computers or quantum computing simulators; such methodology is called the quantum SVM (QSVM). The NISQ computation of the quantum kernel requires many quantum measurements to obtain a quantum kernel entry with statistically reliable accuracy. For example, a value for each computational-basis measurement is zero or one. For each quantum kernel entry, O N 2 shots are required with respect to the number of data samples N , resulting in the computational complexity of O N 4 /ε 2 operations with the maximum error ε , in order to obtain all the quantum kernel entries 9 . Such computational complexity prohibits us from developing and validating quantum kernels as the number of data samples grows. Also, the number of entangling qubits with different connectivities in the previously proposed quantum kernels is increased with qubit count 9 , which requires additional computational resources. To address these issues, here we introduce a shallow, fixed-depth quantum circuit that can be applied to a quantum kernel for a larger number of input features. In the previously proposed quantum kernels based on IQP circuits 9 , the number of dimensional features is typically set to the number of entangled qubits 9,11,14,15 . IQP circuits are a subclass of quantum circuits that cannot be classically efficiently simulated unless the polynomialtime hierarchy collapses to the third level 35 . Here, an IQP circuit is a circuit where a Hadamard gate is applied to each qubit at the beginning and end of the computation, but the rest of the gates are diagonal. In the context of the quantum kernel method, researchers have typically used a more specific type of IQP, called the ZZ feature map 9 . In the ZZ feature map, the connectivity of qubits is achieved in a pair-wise manner, resulting in n(n − 1)/2 interactions, where n is the number of qubits. This leads to a rapid expansion of expressibility and results in a deterioration of generalization performance as qubit count increases 11,14,15 . Our approach aims to simplify the quantum feature map, limit the extent to which qubits are entangled, and control the capacity of our QML model, while increasing the number of input features. This framework can handle several hundreds of input features in QSVM. For the mn-dimensional input vector x = [s 1 , s 2 , · · · , s m ] T ∈ R mn , where s b is the n-dimensional vector s b = s b,1 , s b,2 , · · · , s b,n T , we consider a block product state (BPS) wavefunction 36 : where (1) K x, x ′ = φ(x)|φ(x ′ ) 2 . (2) y = sgn www.nature.com/scientificreports/ and In the BPS wavefunction, a modest number of qubits can be entangled within each block (in our numerical simulations, n was varied from 2, 3, and 6); and for the wavefunction |ψ b (s b )� , each component s b,q is encoded three times as the input angle for the ration operator gates (i.e., s b,q is encoded in the R z gate, in the R y gate, and again in the R z gate in Eq. (4)). Such kind of redundant encoding leads to the better performance of QML models based on angle encoding 37 . The state |ψ b (s b )� is related to matrix product states, a class of tensor networks that have been used for the study of ground states of quantum systems and recently for machine learning. The connectivity of qubits in Eq. (4) is limited to their nearest neighbors, resulting in (n − 1) interactions. The idea of BPS has been originally used for ML models based on tensor networks 36 ; yet, to our knowledge, this kind of BPS-based quantum feature map has not been applied to quantum kernels. In this work, we will show that such a feature map can be used for QSVM. The kernel associated with the quantum feature map defined by Eq. (3) can be given by I y i α * i K x (i) , x + b * ,(3)� BPS (x)� = |ψ 1 (s 1 )� ⊗ |ψ 2 (s 2 )� ⊗ · · · ⊗ |ψ m (s m )� , (4) |ψ b (s b )� = ⊗ n q=1 R z s b,q U ent 2 n ⊗ n q=1 R y s b,q R z (s b,q )H 0 ⊗n � , The number of blocks m can be varied in order to allow a larger number of input features depending on different datasets. Another interesting aspect is that the quantum kernel is not translation invariant, which means that the quantum kernel does not depend solely on the distance of input vectors, in contrast with Gaussian kernels. A computational benefit of our approach is that the calculation of the quantum kernel can be divided into m computational tasks, allowing an efficient computation on classical computers. In particular, �ψ i b |ψ j b � 2 in Eq. (6) can be computed separately; hence, each task can now be efficiently simulated through FPGA acceleration and the multiplication can then be performed on CPU. Quantum AI simulator using a hybrid CPU-FPGA approach. By co-designing FPGA architecture and a quantum kernel given by a shallow quantum circuit, we implemented a fast and efficient quantum AI simulator using a heterogeneous computing approach (Fig. 1a). Details of computational resources in the cloud system (FPGA and CPU) are given in Methods. To begin with, using the principal component analysis (PCA) method 38 we conducted the dimensionality reduction of the 28 × 28 image data from the Fashion-MNIST dataset 39 ; then the number of input features can be varied from d = 4 to d = 780 . After obtaining PCA-reduced input vectors x (i) ∈ R d , the input data are sent from CPU to the internal memory of an FPGA hardware via PCI express. Then, for each block wavefunction |ψ b (s b ) �(b = 1, · · · , m) of the quantum feature map, we calculate the square of the norm of the inner products �ψ i b |ψ j b � 2 (which is depicted in Fig. 1b) on our FPGA architecture in the following procedure: First, the sine and cosine of the input angles for quantum gates are computed using the COordinate Rotational DIgital Computer (CORDIC) algorithm 40 . Second, the square of the norm of the inner product can be calculated using the unitary matrices in Eq. (4), together with an efficient implementation of nqubit entanglement. (The procedure is described in great detail in Methods and Supplementary Notes 1 and 2.) This process can be repeated for all the pairs of data samples, namely, for N 2 /2 cycles. The processed, real-valued data are sent back to the CPU. The kernel matrix element will thus be calculated by the multiplication of m blocks. After all the kernel entries are obtained, the training phase of the SVM can be performed on the CPU platform. In the test process ( Fig. 1c), prediction can be done using the same FPGA acceleration with O(ND) operations, where D is the number of test data. FPGA implementation: numerical precision and acceleration. Herein we validate our FPGA implementation in terms of numerical precision and acceleration. We begin by comparing the quantum kernel values obtained by the FPGA platform and those obtained by the CPU platform (Fig. 2a-c). The norms of inner prod- ucts �ψ i b |ψ j b � 2 have values between 0 and 1. Such property along with a shallow circuit depth is amenable to the use of 16-bit fixed-point arithmetic in our FPGA architecture, which in turn makes the calculation faster with efficient hardware utilization. We also employed 64-bit floating-point arithmetic in the CPU platform to validate our FPGA implementation. The parity plot suggests the success of our FPGA implementation of the quantum kernel (Fig. 2c). The numerical deviation between the two hardware platforms was ± ∼ 0.095%, indicating that there was a negligible loss of numerical accuracy. Next, we compare the execution time for computing a kernel matrix (in the case of 6 entangled qubits) using the FPGA platform with the one obtained by our CPU implementation, as well as the one obtained by Qiskit Aer 21 , a QASM quantum computing simulator (Fig. 2d). Measurement is a vital aspect of the simulation process in the QASM simulator, which handles measurements by collapsing the state of the qubit according to the probabilities determined by the state of the qubit. Therefore, in the QASM simulator, a number of shots are required to obtain the expectation value. In our CPU implementation, the kernel matrix entry is obtained directly by calculating the inner product of the state vectors. In particular, we used NumPy 41 , which is a popular library for scientific computing and data analysis (note that the core of NumPy is implemented in C Language). For our particular tasks (in the case of 6 entangled qubits), the execution time by our CPU implementation is likely to be somewhat faster than that by the state-vector simulator; this is because the state-vector simulator tracks the www.nature.com/scientificreports/ quantum state of the system as it evolves through the circuit, resulting in a slowing down of the execution time. Thus, the plot for the state-vector simulation would be the upper side of the plot denoted by orange in Fig. 2d. In our FPGA architecture, once the data are sent to the FPGA architecture, we used only the internal memory of the FPGA hardware without accessing the external (off-chip) memory, which circumvents the associated communication overhead (for more details of our FPGA architecture, see Supplementary Note 2). In addition, two more factors are responsible for the FPGA acceleration. First, an FPGA allows each programable logic block to perform a specific task simultaneously in an efficient manner. Second, an FPGA can be customized to perform specific tasks using the hardware description language, resulting in faster performance in comparison with CPUs. (5) U ent 2 n := n−1 q=1 CNOT q,q+1 . (6) K x (i) , x (j) = � BPS x (i) |� BPS x (j) 2 = m b=1 ψ b s (i) b |ψ b s (j) b 2 . In our FPGA implementation, all the kernel entries were computed in 4.1 ms at N = 1000 ; and the execution time including CPU-FPGA communication overhead was 15.4 ms at N = 1000 . In other words, our FPGA implementation achieved a 1784 × improvement in comparison with the CPU counterpart. Also, the execution time including the communication overhead was 472 times faster (Fig. 2d); moreover, in comparison with the execution by a QASM simulator (assuming that the computation cost grows as O N 4 /ε 2 operations), a 10 million times speedup was accomplished at N = 400 (Fig. 2d). The results show that our FPGA implementation is highly efficient in terms of the number of data samples, with a modest number of entangling qubits (up to 6 qubits) being used in our quantum feature map. Owing to the limitation of the internal memory and digital single processors within an FPGA, however, our implementation technique will be prohibitive for n large than 8. Nonetheless, for our machine-learning tasks, this can be overcome by dividing input features into a number of blocks, and each block's quantum kernel can be efficiently computed in FPGA. Thus, the FPGA-based simulator accelerates the numerical simulations of QSVM using our quantum kernel and allows us to validate its applicability to much larger features in quantum kernel methods. We calculate the square of the norm of the inner products for each block wavefunction |ψ b � of the quantum feature map. This process is repeated for all the pairs of the data points (i.e., N 2 /2 times). The data are then sent back to the CPU. A kernel matrix value can be obtained by multiplying m blocks. After all the quantum kernel entries are computed, the SVM algorithm is performed on the CPU. (b) 2-qubit example of a quantum circuit that performs the estimation of the quantum kernel element. For the entangling gate, the CNOT gate is used. The quantum circuit is simulated on FPGA using the procedure described in the text (see Methods). (c) Test process: the decision function can be computed using the hybrid CPU-FPGA scheme. Details of computational resources in the cloud system (FPGA and CPU) are given in Methods. . 4), and coat versus shirt (4 vs. 6) classification tasks were somewhat difficult to distinguish (e.g., the images of pullovers are more similar to those of coats than to those of trousers). Hence, we focused on the three binary classification tasks and investigate the performance in detail (Fig. 3). The performance of our quantum kernel without introducing any hyperparameter was comparable to that of the Gaussian kernel exp −γ �x (i) − x (j) � 2 with the optimized bandwidth γ , for dimensions smaller than ∼ 300 (Fig. 3a). Here, a key hyperparameter in the Gaussian kernel is the kernel bandwidth γ , which is known to affect the performance of kernel-based methods such as SVMs and is routinely optimized when SVMs are used in practice. The hyperparameter γ controls the smoothness of the decision boundary in the SVM. Analogously, we introduced a scaling hyperparameter (i.e., x (i) ← x (i) in the quantum circuit) to improve the performance of QSVM. The role of appears to be similar to the classical counterpart. The hyperparameter can calibrate the angles of the rotation gates and directly affect the quantum feature map in the Hilbert space. From a physical point of view, changing the hyperparameter in the quantum kernel is related to changing the total evolution time in the Hamiltonian evolution 15 . The best test accuracy for the quantum kernel was 0.87 at d = 180, 190, 200 ; www.nature.com/scientificreports/ whereas that for the classical kernel with the optimal bandwidth was 0.88 at d = 190 . We found that introducing the scaling parameter improved the performance of our quantum kernel for larger dimensions ( d >∼ 300 ), maintaining its comparable performance to the classical kernel, which is indicated by the blue dotted line in Fig. 3a (for the grid search over the hyperparameters of the classical and quantum kernels, see Supplementary Note 3). The test accuracy obtained by our quantum kernel was improved by increasing the number of data samples N (Fig. 3b). In particular, as the number N was increased, the test accuracies for higher dimensional vectors tended to improve gradually (Fig. 3b). But the relatively sharp drop for dimensions higher than ∼ 300 was difficult to overcome just by increasing N ; nonetheless, the dimension d that gave the best test accuracy was typically in the range between 100 and 200 for this particular application. We note that the drop in the test accuracy for higher dimensions can be overcome by optimizing the aforementioned scaling parameter (which will be discussed in multiclass classification). using 1000 data samples. The coat versus shirt (4 vs. 6) classification task was used. The performance of the Gaussian kernel with the optimized bandwidth for each dimension (train: red triangle; test: yellow triangle) is compared with that of the quantum kernel (train: green circle; test: blue circle). Note that the performance of our quantum kernel without introducing any hyperparameter is comparable to that of the classical kernel with the optimized bandwidth for the dimension d smaller than ∼ 300. Introducing the scaling parameter improved the performance of our quantum kernel for the dimension d larger than ∼ 300, which is indicated by the blue dotted line. (b) Test accuracies obtained by the quantum kernel for a range of features with varying the number of data samples ( N = 20, 100, 500, 1000, 4000 ). The results were averaged over the three tasks: pullover versus shirt (2 vs. 6), pullover versus coat (2 vs. 4), and coat versus shirt (4 vs. 6) classification tasks. (c) Best test accuracies with respect to the number of data samples N , up to N = 4000 for the best classical (red triangle) and the quantum (blue circle) kernels. Each plot represents the best test accuracy for a given N . The results were averaged over the same three tasks as in (b). (d) The effect of quantum entanglement within a block (the block size n = 2, 3, 6 ). The coat versus shirt (4 vs. 6) classification task was used. The shaded regions indicate the standard deviation over 6 independent runs. www.nature.com/scientificreports/ The performance of our hyperparameter-free quantum kernel was competitive with the Gaussian kernel with the optimized bandwidth at N > 1500 (Fig. 3c), which might be beneficial for practical applications. The best test accuracies at N = 2000 and N = 3000 were 0.89 and 0.90, respectively, for both of the two kernels. For smaller numbers of data samples ( N < 1000 ), the performance of our quantum kernel was slightly lower than the best classical counterpart. To understand the role of quantum entanglement, we investigated the effects of enlarging the number of entangled qubits. Increasing the number of entangled qubits (from 2 to 6 qubits per block) did not significantly change the performance for PCA-reduced input vectors (Fig. 3d); this kind of insensitiveness to quantum entanglement has been previously reported in an ML model based on tensor networks using BPS 36 . Our results probably indicate that the capacity of our quantum feature map is already sufficiently high even in the case of n = 2 . However, this may not necessarily mean that quantum entanglement is unimportant; the CNOT entangling gate can make the quantum feature map more complex in comparison with no quantum entanglement. Overall, the behavior of our quantum kernel is quite different from the previously used quantum kernels 9,[11][12][13][14][15] . The results suggest that our quantum kernel is comparable to the best classical kernel with good generalization performance for a range of features. Multiclass classification on Fashion-MNIST dataset. We also show the numerical results for 10-class classification on Fashion-MNIST. We trained our multiclass QSVMs using a one-versus-rest strategy. As was found in binary classification tasks, our quantum kernel was comparable to the best classical kernel (Fig. 4a). www.nature.com/scientificreports/ For multiclass classification using the quantum kernel, we found that it was important to introduce the scaling parameter. Hence, we performed a grid search for the scaling parameter for a range of features ( 4 < d < 340 ) (Fig. 4b). The optimal value for was 0.6 at d = 330 . On the other hand, the optimal value for γ of the Gaussian kernel was 2.5 at d = 330 . The confusion matrices for QSVM and SVM were similar to each other (Fig. 4c,d). The performance metrics for the quantum (classical) kernel were the following: accuracy, 0.855 (0.855); precision, 0.850 (0.853); recall, 0.855 (0.855); F-measure, 0.848 (0.851). We note that, among 45 pairs generated by 10 categories of Fashion-MNIST, about half the pairs of classification tasks were relatively easy to distinguish; hence, the difference in the test accuracy between the classical and the quantum kernels tended to be decreased. The results suggest that our quantum kernel performed competitively with the best classical kernel in the multiclass classification task. Discussion In this study, we have implemented an application-specific quantum AI simulator using a heterogeneous CPU-FPGA computing, which was achieved by co-designing the FPGA architecture and our quantum kernel. To this end, we have introduced a BPS structure as a quantum feature map for QSVM, where a small number of qubits are entangled in each block. This is the first demonstration of the FPGA implementation of a gated-based quantum kernel. The co-design of the quantum kernel and its efficient FPGA implementation have enabled us to perform one of the largest numerical simulations of QSVM in terms of input features, up to 780-dimensional data. In the literature, one of the largest simulations of quantum kernels in terms of qubit count was performed by Huang et al. 11 . The number of qubits in their study is 30. For our particular study, increasing the number of entangled qubits is not a practical direction. Instead, our strategy is to divide input features into a number of blocks, and each block's quantum kernel can be efficiently computed in FPGA. By doing this, hundreds of features can be handled. Our approach is highly customized for our specific tasks at the hardware level; the focus of our simulator differs from that of a general-purpose quantum simulator, which is designed to be flexible and to perform various quantum algorithms. An application of our quantum kernel to dimensional features larger than ∼ 1000 would be more challenging because off-diagonal kernel values could become much smaller. This limitation is related to our formalism of the quantum kernel, owing to the multiplication of many values that are less than one in Eq. (6). Nevertheless, the FPGA-based quantum kernel simulator has significantly accelerated our numerical simulations and allowed us to validate the applicability to QSVM with hundreds of input features. The quantum circuit presented in this work might have implications for co-designing quantum software and hardware and for developing application-specific quantum computers 42,43 . We have demonstrated that the FPGA-based quantum kernel simulator was 470 times faster than that obtained by the CPU implementation, without loss of accuracy. The numerical simulations show that our FPGA implementation is highly efficient in terms of the number of data samples (up to 4000), with a modest number of entangling qubits being used in the quantum feature map. We have applied our quantum kernel to image classification using Fashion-MNIST for a wide range of PCA-reduced features. The results suggest that our quantum kernel is comparable to the best classical kernel, with similar generalization performance for binary and multiclass classification tasks. In binary classification, our hyperparameter-free quantum kernel was comparable to the Gaussian kernels; whereas, in multiclass classification, the scaling parameter played a significant role in improving the performance of our quantum kernel, in line with recent studies 15,44 . Whether quantum kernels could perform better than classical kernels or have a practical advantage in realworld settings is still an open question. Our quantum kernel may be helpful for understanding the applicability of quantum kernels as well as their limitations. While our quantum kernel was applied to classification, the quantum kernel could be used for other kernel-based ML tasks, such as regression, spectral clustering, Gaussian process 17 , and causal discovery 45 . With hundreds of input features being handled in our quantum kernel, other possible applications might include financial data, cheminformatics, and medical data. There is room for improvement in our quantum feature map. For instance, a recent approach based on the automatic design of quantum feature maps 46 may possibly improve our quantum feature map or reduce the number of quantum gates required. Nonetheless, our results might have implications for developing quantum-inspired algorithms and designing practical quantum kernels in the NISQ era. Methods FPGA implementation of the quantum kernel. We describe an approach for efficient simulation of our quantum kernel, which is particularly designed for our FPGA architecture. The quantum kernel is given by the inner product of the quantum feature map, which in principle requires O 2 3n operations, owing to the multiplication of 2 n × 2 n matrices to generate the quantum feature map. Such computational complexity becomes prohibitive for efficient FPGA implementation of quantum kernels, because FPGA architecture is memorybound and the number of complex multipliers is limited. For that reason, efficient resource utilization of FPGA was crucial for calculating our quantum kernel. In this work, we employed a shallow quantum circuit so that we were able to calculate the quantum kernel with O(2 n ) operations, as we will see below. This enabled efficient parallelization and the use of internal memory in FPGA. We consider the following quantum state: where U 1 , U 2 , · · · , U n and V 1 , V 2 , · · · , V n are single-qubit gates and U ent 2 n := n−1 q=1 CNOT q,q+1 represents n-qubit entanglement operation. For the sake of our discussion, it is convenient to rewrite |ψ� as f = VU ent 2 n Uf 0 with f 0 being a vector [1, 0, · · · , 0] T , where U := U 1 ⊗ · · · ⊗ U n and V := V 1 ⊗ · · · ⊗ V n . First, we note that, in the (7) |ψ� = (V 1 ⊗ V 2 ⊗ · · · ⊗ V n )U ent 2 n (U 1 ⊗ U 2 ⊗ · · · ⊗ U n ) 0 ⊗n �. T and the first column vector of U as u = [u 1 , u 2 , · · · , u 2 n ] T ∈ C 2 n , then we have This calculation can be performed by 4 · (2 n−1 − 1) operations using complex multipliers in FPGA (more details are given in Supplementary Fig. 5). The feature map can thus be rewritten as f = VU ent 2 n u . Next, we note that V is a diagonal matrix in our quantum circuit and that U ent 2 n is a sparse matrix, in which each row vector contains only one non-zero entry. By denoting the diagonal elements {V kk } as v = [v 1 , v 2 , · · · , v 2 n ] T ∈ C 2 n , we can calculate f as Here ξ k is the index of the non-zero element in the i th row of U ent 2 n (e.g., for n = 2 , then ξ 1 = 1 , ξ 2 = 2 , ξ 3 = 4 , and ξ 4 = 3 ). In general, U ent 2 n can be calculated recursively by where U ent 2 and Y 2 denote the 2 × 2 identity matrix and the Pauli X matrix, respectively, and O 2 n denotes the 2 n × 2 n zero matrix. The proof of the recurrence relation is given in Supplementary Note 1. The indices {ξ k } in Eq. (9) can be determined once U ent 2 n is obtained. Finally, the inner product �ψ i |ψ j � can be calculated by k f * k s (i) f k s (j) . Details of computational resources. Our Machine learning. Here we provide the details of our ML models. Preprocessing was applied to the original data to make them suitable for quantum angle encoding: PCA was used to reduce the dimension of the 28 × 28 original image data to d-dimensional input vectors x (i) ∈ R d (where d was varied from 4 to 780), which were then transformed such that x (i) ∈ [−1, 1] . In the training of support vector classifiers, hinge loss was used for the loss function. Throughout the paper, the regularization parameter C for soft margin SVM 47 was set to 1.0 for both classical and quantum ML models. For the multiclass classification task shown in Fig. 4, a one-versus-rest strategy was employed. To compare the performance of our quantum kernel with the classical counterpart, we used the Gaussian kernel, which is given by exp −γ �x (i) − x (j) � 2 , with γ being a hyperparameter. To obtain the optimal test accuracy, we performed a grid search over the bandwidth. It is also possible to introduce a hyperparameter in our quantum feature map � BPS (x) . In this work, we consider that the input vector x can be scaled by (i.e., x (i) ← x (i) ), which is similar to an approach by recent work 15 . Thus, we performed a grid search over the scaling parameter. The effect of the scaling parameter was somewhat different from that of γ . In particular, we found that, for binary classification, the case of = 1 typically gave the near-optimal performance (see also Supplementary Note 3), implying that our quantum kernel gave a reasonable performance without introducing any hyperparameter. Nonetheless, to further optimize the value for , we narrowed the range for and performed another grid search over the scaling parameter (8) Uf 0 = u =                  χ (1) 1 · · · · · χ (n−2) 1 · χ (n−1) 1 · χ (n) 1 χ (1) 1 · · · · · χ (n−2) 1 · χ (n−1) 1 · χ (n) 2 χ (1) 1 · · · · · χ (n−2) 1 · χ (n−1) 2 · χ (n) 1 χ (1) 1 · · · · · χ (n−2) 1 · χ (n−1) 2 · χ (n) 2 . . . χ (1) 2 · · · · · χ (n−2) 2 · χ (n−1) 1 · χ (n) 1 χ (1) 2 · · · · · χ (n−2) 2 · χ (n−1) 1 · χ (n) 2 χ (1) 2 · · · · · χ (n−2) 2 · χ (n−1) www.nature.com/scientificreports/ We found that the test accuracy was slightly improved from 0.870 to 0.875 in binary classification (see also Supplementary Note 4) and that the use of the scaling parameter played an important role in multiclass classification. 2 · χ (n) 1 χ (1) 2 · · · · · χ (n−2) 2 · χ (n−1) 2 · χ (n) 2                  . (9) f k = v k u ξ k . (10) U ent 2 n+1 = U ent 2 n O 2 n O 2 n Y 2 n ; Y 2 n+1 = O 2 n U ent 2 n Y 2 n O 2 n (n ≥ 1) .(11) Data availability All the datasets used in the present study are publicly available at https:// github. com/ zalan dores earch/ fashi onmnist. We cited the reference of the data in the manuscript. Figure 1 . 1Schematic representation of our quantum AI simulator using a hybrid CPU-FPGA approach. (a) PCA is used to reduce the dimension of the original data from Fashion-MNIST; a range of features from d = 4 to d = 780 can be used for machine learning. Then PCA-reduced features are sent from the CPU to the FPGA. Figure 2 . 2FPGA implementation of the quantum kernel and its execution time. The numerical simulations were performed on a 6-qubit quantum circuit that estimates the quantum kernel element. (a) Quantum kernel matrix obtained by an FPGA platform (16-bit fixed-point arithmetic). (b) Quantum kernel matrix obtained by a CPU platform (64-bit floating-point arithmetic). (c) Parity plot for the quantum kernel values obtained by CPU and FPGA platforms. Inset shows small differences between the two: the error between the two hardware platforms was ± ∼ 0.095%. (d) Execution time with respect to the number of data N for different platforms: FPGA, blue; FPGA (including CPU-FPGA communication overhead; denoted by the asterisk), green; CPU, orange; QASM quantum simulator (Qiskit Aer), red. Note that the FPGA execution including communication overhead was 472 times faster than that for the CPU counterpart at N = 1000. Scientific Reports | (2023) 13:7735 | https://doi.org/10.1038/s41598-023-34600-2 Figure 3 . 3Train and test accuracies of the quantum kernel on the Fashion-MNIST dataset. (a) Training and test accuracies with a range of features from d = 4 to d = 780 Figure 4 . 4Multiclass classification on Fashion MNIST dataset. (a) Test accuracies for a range of PCA-reduced features from d = 4 to d = 390 (the block size n = 2 ) for the best classical (red triangle) and the quantum (blue circle) kernels. The number of data samples was 1000. For the classical kernel, we used Gaussian kernels with the optimized bandwidth for each dimension. For the quantum counterpart, the optimal scaling parameter ( x (i) ← x (i) ) was used for each dimension. The quantum kernel with the optimal scaling parameter is competitive with the classical counterpart.(b) Grid search over the scaling parameter for a range of features. The scaling parameter that gave the optimal test accuracy is indicated by the red open square: * = 0.6 at d = 330 . (c) Confusion matrix obtained from the best classical kernel (dimension d = 330 ). The performance metrics were as follows: accuracy, 0.855; precision, 0.853; recall, 0.855; F-measure, 0.851. (d) Confusion matrix obtained by the quantum kernel with the optimized scaling parameter (dimension d = 330 ). The performance metrics were as follows: accuracy, 0.855; precision, 0.850, recall, 0.855; F-measure, 0.848. Scientific Reports | (2023) 13:7735 | https://doi.org/10.1038/s41598-023-34600-2 .5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1, 1.05, Scientific Reports | (2023) 13:7735 | https://doi.org/10.1038/s41598-023-34600-2 . Among hardware implementations, field programmable gateScientific Reports | (2023) 13:7735 | https://doi.org/10.1038/s41598-023-34600-2 www.nature.com/scientificreports/ calculation of Uf 0 , only the first column of U is needed; hence, Uf 0 can be obtained without the need for fully conducting tensor operations. By denoting the first column vector of each 2 × 2 unitary matrix U q as χScientific Reports | (2023) 13:7735 | https://doi.org/10.1038/s41598-023-34600-2 (q) 1 , χ (q) 2 quantum AI simulator based on a hybrid CPU-FPGA system is implemented on the Amazon Web Services (AWS) Elastic Computing Cloud (EC2) platform, in which AWS EC2 F1 instances of AMD Xilinx FPGA hardware are accessible. In particular, we used the f1.2xlarge instance size, which has 1 FPGA, 8 vCPUs, and 122 GB of memory. More specifically, we used AMD Xilinx Virtex™ UltraScale + ™ VU19P FPGA and Intel Xeon™ E5-2686 v4 with a base clock speed of 2.3 GHz as vCPU. The details of our FPGA architecture and block diagrams are provided in Supplementary Note 2. Received: 21 September 2022; Accepted: 4 May 2023 © The Author(s) 2023 AcknowledgementsWe thank Hideki Asoh (National Institute of Advanced Industrial Science and Technology) for useful discussions.Author contributionsCompeting interestsThe authors declare no competing interests.Additional informationSupplementary Information The online version contains supplementary material available at https:// doi. org/ 10. 1038/ s41598-023-34600-2.Correspondence and requests for materials should be addressed to T.S.Reprints and permissions information is available at www.nature.com/reprints.Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. M A Nielsen, I L Chuang, Quantum Computing and Quantum Information, 10th Anniversary. Cambridge University PressNielsen, M. A. & Chuang, I. L. Quantum Computing and Quantum Information, 10th Anniversary (Cambridge University Press, 2010). Quantum risk analysis. S Woerner, D J Egger, 10.1038/s41534-019-0130-6s41534-019-0130-6NPJ Quantum Inf. 515Woerner, S. & Egger, D. J. Quantum risk analysis. NPJ Quantum Inf. 5, 15. https:// doi. org/ 10. 1038/ s41534-019-0130-6 (2019). Quantum chemistry in the age of quantum computing. Y Cao, Chem. Rev. 119Cao, Y. et al. Quantum chemistry in the age of quantum computing. Chem. Rev. 119, 10856-10915 (2019). Quantum support vector machine for big data classification. P Rebentrost, M Mohseni, S Lloyd, Phys. Rev. Lett. 113130503Rebentrost, P., Mohseni, M. & Lloyd, S. Quantum support vector machine for big data classification. Phys. Rev. Lett. 113, 130503 (2014). A rigorous and robust quantum speed-up in supervised machine learning. Y Liu, S Arunachalam, K Temme, Nat. Phys. 17Liu, Y., Arunachalam, S. & Temme, K. A rigorous and robust quantum speed-up in supervised machine learning. Nat. Phys. 17, 1013-1017 (2021). Quantum machine learning. J Biamonte, Nature. 549Biamonte, J. et al. Quantum machine learning. Nature 549, 195-202 (2017). Quantum circuit learning. K Mitarai, M Negoro, M Kitagawa, K Fujii, Phys. Rev. A. 9832309Mitarai, K., Negoro, M., Kitagawa, M. & Fujii, K. Quantum circuit learning. Phys. Rev. A 98, 032309 (2018). Quantum machine learning in feature Hilbert spaces. M Schuld, N Killoran, Phys. Rev. Lett. 12240504Schuld, M. & Killoran, N. Quantum machine learning in feature Hilbert spaces. Phys. Rev. Lett. 122, 040504 (2019). Supervised learning with quantum-enhanced feature spaces. V Havlíček, Nature. 567Havlíček, V. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209-212 (2019). Parameterized quantum circuits as machine learning models. M Benedetti, E Lloyd, S Sack, M Fiorentini, Quantum Sci. Technol. 443001Benedetti, M., Lloyd, E., Sack, S. & Fiorentini, M. Parameterized quantum circuits as machine learning models. Quantum Sci. Technol. 4, 043001 (2019). Power of data in quantum machine learning. H.-Y Huang, 10.1038/s41467-021-22539-9Nat. Commun. 12Huang, H.-Y. et al. Power of data in quantum machine learning. Nat. Commun. 12, 2631. https:// doi. org/ 10. 1038/ s41467-021- 22539-9 (2021). Machine learning of high dimensional data on a noisy quantum processor. E Peters, 10.1038/s41534-021-00498-9s41534-021-00498-9NPJ Quantum Inf. 7Peters, E. et al. Machine learning of high dimensional data on a noisy quantum processor. NPJ Quantum Inf. 7, 161. https:// doi. org/ 10. 1038/ s41534-021-00498-9 (2021). Training quantum embedding kernels on near-term quantum computers. T Hubregtsen, Preprint atHubregtsen, T. et al. Training quantum embedding kernels on near-term quantum computers. Preprint at https:// arxiv. org/ abs/ 2105. 02276 (2021). Quantum machine learning beyond kernel methods. S Jerbi, Preprint atJerbi, S. et al. Quantum machine learning beyond kernel methods. Preprint at https:// arxiv. org/ abs/ 2110. 13162 (2021). Importance of kernel bandwidth in quantum machine learning. R Shaydulin, S M Wild, Preprint atShaydulin, R. & Wild, S. M. Importance of kernel bandwidth in quantum machine learning. Preprint at https:// arxiv. org/ abs/ 2111. 05451 (2021). Experimental quantum kernel trick with nuclear spins in a solid. T Kusumoto, K Mitarai, K Fujii, M Kitagawa, M Negoro, 10.1038/s41534-021-00423-0s41534-021-00423-0NPJ Quantum Inf. 7Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M. & Negoro, M. Experimental quantum kernel trick with nuclear spins in a solid. NPJ Quantum Inf. 7, 94. https:// doi. org/ 10. 1038/ s41534-021-00423-0 (2021). Error mitigation for quantum kernel based machine learning methods on IonQ and IBM quantum computers. S Moradi, Preprint atMoradi, S. et al. Error mitigation for quantum kernel based machine learning methods on IonQ and IBM quantum computers. Preprint at https:// arxiv. org/ abs/ 2206. 01573 (2022). Demonstration of fault-tolerant universal quantum gate operations. L Postler, Nature. 605Postler, L. et al. Demonstration of fault-tolerant universal quantum gate operations. Nature 605, 675-680 (2022). Quantum computing in the NISQ era and beyond. J Preskill, 10.22331/q-2018-08-06-792018-08-06-79279Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79. https:// doi. org/ 10. 22331/q-2018-08-06-79 (2018). Noisy intermediate-scale quantum algorithms. K Bharti, Rev. Mod. Phys. 9415004Bharti, K. et al. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys. 94, 015004 (2022). Qiskit: An open-source framework for quantum computing. G Aleksandrowicz, 10.5281/ZENODO.2562111Aleksandrowicz, G. et al. Qiskit: An open-source framework for quantum computing. Qiskit. https:// doi. org/ 10. 5281/ ZENODO. 25621 11 (2019). General-purpose quantum circuit simulator with projected entangled-pair states and the quantum supremacy frontier. C Guo, Phys. Rev. Lett. 123190501Guo, C. et al. General-purpose quantum circuit simulator with projected entangled-pair states and the quantum supremacy frontier. Phys. Rev. Lett. 123, 190501 (2019). A quantum circuit simulator and its applications on. Z Wang, 10.1038/s41598-020-79777-ySunway TaihuLight supercomputer. Sci. Rep. 11Wang, Z. et al. A quantum circuit simulator and its applications on Sunway TaihuLight supercomputer. Sci. Rep. 11, 355. https:// doi. org/ 10. 1038/ s41598-020-79777-y (2021). Qulacs: A fast and versatile quantum circuit simulator for research purpose. Y Suzuki, 10.22331/q-2021-10-06-5595559Suzuki, Y. et al. Qulacs: A fast and versatile quantum circuit simulator for research purpose. Quantum 5, 559. https:// doi. org/ 10. 22331/q-2021-10-06-559 (2021). Qibo: A framework for quantum simulation with hardware acceleration. S Efthymiou, Quantum Sci. Technol. 715018Efthymiou, S. et al. Qibo: A framework for quantum simulation with hardware acceleration. Quantum Sci. Technol. 7, 015018 (2022). Jet: Fast quantum circuit simulations with parallel task-based tensor-network contraction. T Vincent, 10.22331/q-2022-05-09-7092022-05-09-7096Vincent, T. et al. Jet: Fast quantum circuit simulations with parallel task-based tensor-network contraction. Quantum 6, 709. https:// doi. org/ 10. 22331/q-2022-05-09-709 (2022). Tensor network quantum virtual machine for simulating quantum circuits at exascale. T Nguyen, Preprint atNguyen, T. et al. Tensor network quantum virtual machine for simulating quantum circuits at exascale. Preprint at https:// arxiv. org/ abs/ 2104. 10523 (2021). FPGA emulation of quantum circuits. A U Khalid, Z Zilic, K Radecka, IEEE International Conference on Computer Design: VLSI in Computers and Processors (ICCD). Khalid, A. U., Zilic, Z. & Radecka, K. FPGA emulation of quantum circuits. In IEEE International Conference on Computer Design: VLSI in Computers and Processors (ICCD) (2004). An FPGA-based quantum computing emulation framework based on serial-parallel architecture. Y H Lee, M Khalil-Hani, M N Marsono, Int. J. Reconfigurable Comput. 5718124Lee, Y. H., Khalil-Hani, M. & Marsono, M. N. An FPGA-based quantum computing emulation framework based on serial-parallel architecture. Int. J. Reconfigurable Comput. 2016, 5718124 (2016). A scalable high-precision and high-throughput architecture for emulation of quantum algorithms. N Mahmud, E El-Araby, 31st IEEE International System-on-Chip Conference (SOCC). Mahmud, N. & El-Araby, E. A scalable high-precision and high-throughput architecture for emulation of quantum algorithms. In 2018 31st IEEE International System-on-Chip Conference (SOCC) (2018). Scaling reconfigurable emulation of quantum algorithms at high precision and high throughput. N Mahmud, E El-Araby, D Caliga, 10.1002/que2.19Quantum Eng. 1, e19.Mahmud, N., El-Araby, E. & Caliga, D. Scaling reconfigurable emulation of quantum algorithms at high precision and high throughput. Quantum Eng. 1, e19. https:// doi. org/ 10. 1002/ que2. 19 (2019). An FPGA-based real quantum computer emulator. J Pilch, J Długopolski, J. Comput. Electron. 18Pilch, J. & Długopolski, J. An FPGA-based real quantum computer emulator. J. Comput. Electron. 18, 329-342 (2019). Support-vector networks. C Cortes, V Vapnik, Mach. Learn. 20Cortes, C. & Vapnik, V. Support-vector networks. Mach. Learn. 20, 273-297 (1995). Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. B Schölkopf, A J Smola, MIT PressSchölkopf, B. & Smola, A. J. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond (MIT Press, 2002). Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. M J Bremner, R Jozsa, D J Shepherd, Proc. R. Soc. A. 467459Bremner, M. J., Jozsa, R. & Shepherd, D. J. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proc. R. Soc. A 467, 459 (2011). Entanglement and tensor networks for supervised image classification. J Martyn, G Vidal, C Roberts, S Leichenauer, Preprint atMartyn, J., Vidal, G., Roberts, C. & Leichenauer, S. Entanglement and tensor networks for supervised image classification. Preprint at https:// arxiv. org/ abs/ 2007. 06082 (2020). Predicting toxicity by quantum machine learning. T Suzuki, M Katouda, 10.1088/2399-6528/abd3d8J. Phys. Commun. 4125012Suzuki, T. & Katouda, M. Predicting toxicity by quantum machine learning. J. Phys. Commun. 4, 125012. https:// doi. org/ 10. 1088/ 2399-6528/ abd3d8 (2020). EEG signal classification using PCA, ICA, LDA and support vector machines. A Subasi, M I Gursoy, Expert Syst. Appl. 37Subasi, A. & Gursoy, M. I. EEG signal classification using PCA, ICA, LDA and support vector machines. Expert Syst. Appl. 37, 8659-8666 (2010). . 10.1038/s41598-023-34600-2137735| (2023) 13:7735 | https://doi.org/10.1038/s41598-023-34600-2 Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. H Xiao, K Rasul, R Vollgraf, Preprint atXiao, H., Rasul, K. & Vollgraf, R. Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. Preprint at https:// arxiv. org/ abs/ 1708. 07747 (2017). The CORDIC trigonometric computing technique. J E Volder, IRE Trans. Electron. Comput. 3Volder, J. E. The CORDIC trigonometric computing technique. IRE Trans. Electron. Comput. 3, 330-334 (1959). Array programming with NumPy. C R Harris, Nature. 585Harris, C. R. et al. Array programming with NumPy. Nature 585, 357-362 (2020). On the co-design of quantum software and hardware. G Li, Proceedings of the Eight Annual ACM International Conference on Nanoscale Computing and Communication. the Eight Annual ACM International Conference on Nanoscale Computing and CommunicationLi, G. et al. On the co-design of quantum software and hardware. In Proceedings of the Eight Annual ACM International Conference on Nanoscale Computing and Communication (2021). . T Tomesh, M Martonosi, Quantum Codesign, IEEE Micro. 41Tomesh, T. & Martonosi, M. Quantum codesign. IEEE Micro 41, 33-40 (2021). Bandwidth enables generalization in quantum kernel models. A Canatar, E Peters, C Pehlevan, S M Wild, R Shaydulin, Preprint atCanatar, A., Peters, E., Pehlevan, C., Wild, S. M. & Shaydulin, R. Bandwidth enables generalization in quantum kernel models. Preprint at https:// arxiv. org/ abs/ 2206. 06686 (2022). Application of quantum computing to a linear non-Gaussian acyclic model for novel medical knowledge discovery. H Kawaguchi, Preprint atKawaguchi, H. Application of quantum computing to a linear non-Gaussian acyclic model for novel medical knowledge discovery. Preprint at https:// arxiv. org/ abs/ 2110. 04485 (2021). Automatic design of quantum feature maps. S Altares-López, A Ribeiro, J J García-Ripoll, Quantum Sci. Technol. 645015Altares-López, S., Ribeiro, A. & García-Ripoll, J. J. Automatic design of quantum feature maps. Quantum Sci. Technol. 6, 045015 (2021). A library for support vector machines. C C Chang, C J Lin, Libsvm, ACM Trans. Intell. Syst. Technol. 2Chang, C. C. & Lin, C. J. LIBSVM: A library for support vector machines. ACM Trans. Intell. Syst. Technol. 2, 1-27 (2011).
[]
[ "Accurate screened exchange LDA band structures for transition metal monoxides MnO, FeO, CoO and NiO", "Accurate screened exchange LDA band structures for transition metal monoxides MnO, FeO, CoO and NiO" ]
[ "Roland Gillen \nDepartment of Engineering\nUniversity of Cambridge\nCB3 0FACambridgeUnited Kingdom\n", "John Robertson \nDepartment of Engineering\nUniversity of Cambridge\nCB3 0FACambridgeUnited Kingdom\n" ]
[ "Department of Engineering\nUniversity of Cambridge\nCB3 0FACambridgeUnited Kingdom", "Department of Engineering\nUniversity of Cambridge\nCB3 0FACambridgeUnited Kingdom" ]
[]
We report calculations of the band structures and density of states of the four transition metal monoxides MnO, FeO, CoO and NiO using the hybrid density functional sX-LDA. Late transition metal oxides are prototypical examples of strongly correlated materials, which pose challenges for electronic structure methods. We compare our results with available experimental data and show that our calculations generally yield accurate predictions for the fundamental band gaps and valence bands, in favourable agreement with previously reported theoretical studies. For MnO, the band gaps are still underestimated, suggesting additional many-body effects that are not captured by our screened hybrid functional approach. arXiv:1208.0786v2 [cond-mat.mtrl-sci]
10.1088/0953-8984/25/16/165502
[ "https://arxiv.org/pdf/1208.0786v2.pdf" ]
7,730,395
1208.0786
ce2363c5385f75ffe742c1ff386b446c0513b3dd
Accurate screened exchange LDA band structures for transition metal monoxides MnO, FeO, CoO and NiO (Dated: May 2, 2014) Roland Gillen Department of Engineering University of Cambridge CB3 0FACambridgeUnited Kingdom John Robertson Department of Engineering University of Cambridge CB3 0FACambridgeUnited Kingdom Accurate screened exchange LDA band structures for transition metal monoxides MnO, FeO, CoO and NiO (Dated: May 2, 2014) We report calculations of the band structures and density of states of the four transition metal monoxides MnO, FeO, CoO and NiO using the hybrid density functional sX-LDA. Late transition metal oxides are prototypical examples of strongly correlated materials, which pose challenges for electronic structure methods. We compare our results with available experimental data and show that our calculations generally yield accurate predictions for the fundamental band gaps and valence bands, in favourable agreement with previously reported theoretical studies. For MnO, the band gaps are still underestimated, suggesting additional many-body effects that are not captured by our screened hybrid functional approach. arXiv:1208.0786v2 [cond-mat.mtrl-sci] I. INTRODUCTION The oxides of late transition metals are classic examples of strongly correlated systems. While appearing to be simple compounds, the partially filled d shells of the metal ions make them challenging materials for electronic structure theory. The exact origin, nature and size of the fundamental gaps of the four oxides has been subject of debate for many years, among both experimentalists [1][2][3][4][5][6][7][8][9][10] and theorists [11][12][13][14][15][16][17][18][19][20] . Despite its successes, density functional theory (DFT) in the framework of the local density approximation (LDA) and the generalized gradient approximation (GGA) is inapplicable to strongly correlated systems. In the case of the MnO and NiO, the predicted band gaps are too small compared to experiments, while CoO and FeO are predicted to be antiferromagnetic but metallic. The reason lies in the incorrect treatment of the exchange interaction in these approximations, which do not sufficiently cancel the electron self-interaction 21 . This leads to an underestimation of exchange splitting and the energies of unoccupied states. Clearly, this calls for alternative methods offering a better treatment of exchange effects. The semi-empirical LDA+U 22 method modifies LDA by adding an on-site Coulomb interaction (the Hubbard U) to correct the energies of localized semi-core electrons. This method is related to addition of dynamical local correlation effects within dynamic mean field theory (LDA+DMFT), which has been applied to selected transition metals oxides in the past, e.g. in Ref. 23. However, the choice of the correct U, whether self-consistently or empirically, and the treatment of double-counting is not straight-forward and electrons are not treated on equal footing. An alternative, non-local, treatment of electronelectron interaction are quasiparticle calculations at the GW level of approximation 24 . These are usually performed by taking the wavefunctions and eigenvalues of a previous DFT calculation as an input and either calculating a one-shot self-energy correction (G 0 W 0 ), iteratively updating the Green function G (GW 0 ) or updating both Green function G and screened Coulomb interaction W (scGW) until self-consistency is obtained. While these methods have been used on NiO and MnO, the predictions depend on basis set and underlying approximations [12][13][14][15][16]19,20,25 , which make it difficult to draw a conclusive picture. Despite their wide success, quasiparticle methods are difficult for a variety of reasons: The energy dependence of the electron-electron interaction makes GW calculations computationally costly and finding a proper way to achieve self-consistency is non-trivial. On the other hand, the perturbative nature of G 0 W 0 , while being computationally more favourable, does only permit access to properties related to the electronic band structure, and its results ultimatively depend on the quality of the input wavefunctions and energies. It is thus imperative to find a method that can reasonably describe the electronic structure of strongly correlated materials and that is capable of computationally economic self-consistent calculation of groundstate properties. Hybrid functionals are an interesting choice, as they incorporate non-local exchange-correlation effects, while maintaining the possibility of variational total energy calculations at moderate computational cost and thus allowing for self-consistent optimization of geometries. They are firmly rooted within DFT via the framework of generalized Kohn-Sham schemes 26 , which allow for explicitly orbital-dependent non-local exchange-correlation potentials. A fraction of Hartree-Fock exchange is mixed into LDA or GGA, which remedies most of the short-comings of the underlying local functional due to the improved (non-local) treatment of the electron exchange interaction. This can be likened to the self-energy in the wellknown COHSEX approximation 24 , where the exchange interaction is statically screened by a Coulomb hole. Hybrid functionals have been shown to be versatile methods for the study of correlated materials, often on par with quasiparticle methods [27][28][29] , while generally performing great for sp semiconductors and materials of practical interest, such as GaN or ZnO 27 . A number of hybrid functional studies on late transition metal oxides has been reported 17,20,[30][31][32][33][34][35][36][37][38] , with band gaps varying with the fraction of included Hartree-Fock exchange. Most notably, Rödl et al. 20 showed that the screened hy-brid functional HSE03, which they used as input for their G 0 W 0 calculations, yields good band gaps on its own. We recently reported that the screened hybrid functional 'screened-exchange LDA' (sX-LDA) successfully reproduces the electronic properties of CeO 2 , as well as those of lanthanide (X=La,Ce,...,Lu) 29 and transition metal (X=Ti, Cr, Fe) 38 sesquioxides X 2 O 3 . Interestingly, we observed that variations in treatment of the screened exchange interaction in the hybrid functional approach can lead to qualitative differences of the predicted properties. Motivated by this success, we here report the results from our sX-LDA calculations on the transition metal monooxides MnO, FeO, CoO and NiO. We confirm that the inclusion of non-local exchange interaction in the functional is sufficient to restore the insulating nature of all four oxides, while the screening leads to overall excellent agreement with experiment, particularly for the valence band density of states and valence band widths, and compares favourably with other theoretical methods. II. METHOD The presented results have been calculated by the use of hybrid functionals in the generalized Kohn-Sham (GKS) formalism 26 of density functional theory. Here, the self-energy of an electron in the crystal is described by a linear combination of an orbital-dependent Hartree-Fock (HF) exchange-type term and a density-dependent local term. In practice, the long-range contribution of the HF exchange can be approximated very well by the long-range contribution of a local functional 39 , giving 'screened' hybrid functionals. This form is favorable for the use in solids due to the slow convergence of the longrange HF contribution if periodic boundary conditions are used and thus allows for more efficient calculations. In this work, we used the hybrid functional 'screened exchange-LDA' (sX-LDA) 26,40 as implemented in the planewave code CASTEP 27,41 . It is given by E sX-LDA xc [φ] = E HF,SR x [φ] − E LDA,SR x [n] + E LDA xc [n] where the Hartree-Fock exchange is screened by a Thomas-Fermi dielectric function, i.e ρ ij (r) = φ * i (r)φ j (r) E HF,SR x [φ] ∝ occ i,j ρ ij (r) e −ks|r−r | ρ * ij (r ) |r − r | d 3 r d 3 r E LDA,SR x [ρ] = F [γ]E LDA x F [γ] = 1 − 4 3 γ arctan 2 γ − γ 2 6 1 − γ 2 4 + 3 ln 1 + 4 γ 2 , with γ = k s /k F , where k s is the inverse screening length of the exact exchange contribution and k F is the Fermi direction. To confirm that the AFII phase indeed is the global ground state, we also tested the ferromagnetic and the AFI phases and found the AFII spin-ordering to correspond to the lowest total energy. We chose to neglect rhombohedral distortions in this work for computational reasons and fitted the total energies to a BirchŰMurnaghan equation of state while keeping the cell angles fixed. Table I shows the obtained theoretical lattice constants and bulk moduli. The atomic cores of Mn, Co and Ni were described by standard normconserving pseudopotentials from the CASTEP database, while we generated a pseudopotential for Fe using the OPIUM 44 code. We treat the (3d,4s,4p) states of the four considered transition metals as valence electrons by plane waves with a cutoff energy of 750 eV. On GGA level, inclusion of semi-core 3s and 3p electrons in the calculations did not noticeably affect on the electronic properties. A Monkhorst-Pack grid of 4x4x4 points in the Brillouin zone is sufficient to yield total energies converged to 0.005 eV. We used a grid of 12x12x12 k-points for the calculation of the density of states plots. Figure 1 shows the band structures from our calculations using the hybrid functional sX-LDA, together with the calculated density of states and reported xray photoemission spectroscopy-bremsstrahlung isochromat spectroscopy (XPS-BIS) spectra 8,10 . In accordance with the previous reports, all of our investigated materials are predicted to be insulating in the antiferromagnetic phase. In all four cases, the valence band top consists of weakly dispersive metal d states with some O 2p mixed in. The octahedral crystal field leads to a splitting of the five-fold degenerate d orbitals into two degenerate The valence band maximum for MnO and NiO is located at the Z point, whereas we find it between the Z and the L point and at the F point in case of FeO and CoO, respectively. The conduction band minimum for all oxides is a parabolic band of metal 4s states, which is centered at Γ. Correspondingly, the minimum direct band gap is at the Γ point. Table II shows the band gaps compared to those in previous calculations and from various experimental methods. For experimental lattice constants, our sX-LDA results are quite similar to those given by Rödl et al. for HSE03, even though the difference between indirect and direct band gaps is lower in our sX-LDA calculations. We will discuss our results for the different oxides in comparison with other theoretical methods and experiments in the following. III. RESULTS AND DISCUSSION FeO Unfortunately, the experimental preparation of pure FeO samples is difficult due to Fe segregation 4 , which hampers comparison of theoretical calculations with experimental spectra. This most likely causes the bad agreement of the experimental XPS-BIS spectrum with theoretical density-of-states plots, see Fig. 1, which is compatible with the reports from other groups 20,31,52 . The bad experimental situation also raises the question of the accuracy of the one reported experimental estimate for the fundamental band gap of FeO 4 , 2.4 eV from optical absorption measurements. From the theoretical point of view, the fundamental band gap is between a single band of t 2g character, which is detached from a block of hybridized O 2p and Fe 3d bands at lower energies, and the parabolic Fe 4s band. Our calculations predict the energy difference between the single d band and the top of the 'bulk' valence band to be approximately 1.1 eV. The (indirect) fundamental band gap of 2.45 eV is in slightly better agreement with the experimental value than those from HSE03 and G 0 W 0 @HSE03 calculations 20 . We believe that this good agreement together with the reported weak quasiparticle correction to HSE03 suggests that the experimental band gap value is indeed of the found magnitude. CoO For CoO, most experimental reports point towards a band gap size of 2.5-2.8 eV, but several other studies 47,51 found significantly higher values. Kang et al. 51 reported a band gap of 5.43 eV, which they obtained by a 'standard critical point' (SCP) fitting procedure to their ellipsometry spectra. However, they also report a signifi-cant optical structure at 2.72 eV, which they attribute to intra-atomic d-d transitions. A Tauc plot of the absorption spectrum we calculated from their measured dielectric function yields an indirect band gap of 2.8 eV and a direct band gap of ∼5 eV. Our calculated cross-section weighted density of states in Fig. 1 shows very good agreement with the reported XPS and BIS spectra for CoO 8,10 . The band structure calculations suggest a fundamental band gap of 2.4 eV, between a valence band top of hybridized Co 3d and O 2p states and the minimum of the parabolic band at Γ. This value is comparable to those reported for recent EXX-OEP and G 0 W 0 @LDA+U calculations and is close to the range of experimental band gap values. The HSE03 and G 0 W 0 @HSE03 study by Roedl et al. 20 yields considerably higher band gaps of magnitude 3.2 eV and 3.4 eV, respectively. This is surprising considering the overall close agreement of the results from sX-LDA and HSE03 for the other three oxides, in case experimental lattice constants are used. We thus performed our own HSE03 calculations to test whether the observed divergence is caused by differences in the Co pseudopotentials. Our calculated indirect band gap of 3.15 eV indeed is in good agreement with the value from Ref. 20, we thus rule out effects from the pseudopotentials and attribute the differences to details in the structure of the two hybrid functionals. A detailed investigation might be subject of later work. Finally, we note that Engel et al. 18 found a significant energy gap of approximately 2 eV between the d bands at the valence band top and the rest of the valence band, similar to the case of FeO. However, we cannot find traces of such a gap neither in the reported photoemission spectra 6,10 , nor in other theoretical work on CoO 17,20,32 . MnO and NiO Compared to FeO and CoO, MnO and NiO have been investigated more extensively. LDA and GGA predict both materials to be insulators, where the band gap of MnO arises from exchange splitting of the d bands and the band gap of NiO arises from a combination of exchange splitting and additional crystal-field splitting between e g and t 2g states. The fundamental band gap of NiO was experimentally established to be 3.7-4.3 eV 1,3,6,48,51 and attributed to transitions from O 2p or Ni 3d states at the valence band top to either unoccupied Ni 3d or Ni 4s orbitals 3 . In our case, the final state of the fundamental transition of 3.85 eV is clearly in the parabolic Ni 4s band, while the onset of the unoccupied d bands is pushed up to 5.9 eV above the valence band maximum (VBM), see the band structure in Fig. 1. The parabolic band is of pure Ni 4s character at the Γ-point and of mixed Ni 4s and 4p character with a minor contribution from O 2p away from Γ. The valence band top arises from a strong hybridization of oxygen 2p and nickel 3d states, see Fig. 2, and is in good agreement with the widely accepted cluster model 53 for the electronic structure of NiO and previous studies on GW and hybrid functional level. We find a spectral weight of 16% oxygen states and 84% nickel 3d states for the bands within VBM-2 eV. These values are comparable to the results from GW@GGA calculations by Li et al. 15 , who reported a 20% contribution of oxygen states at the valence band top at Γ. Our results would thus support both proposed models for the fundamental transition of NiO. For the conduction band, a balanced description of the itinerant "4s" bands and the unoccupied 3d bands is necessary. Independent of the method, the inclusion of non-local exchange interaction generally leads to an improved splitting of the occupied and unoccupied d bands in NiO compared to LDA/GGA. On the other hand, the "correction" of the itinerant "4s" bands seems to be strongly influenced by the underlying wavefunctions. Li et al 15 reported GW@GGA calculations, where the unoccupied d bands are shifted to an energy of ∼4.2 eV above the valence band maximum, while the parabolic band was only weakly shifted compared to GGA. The corresponding DOS of the unoccupied levels is in excellent agreement with BIS 10 spectra and can account for both the dominant peak at 4.3 eV and the onset of strong optical absorption and at ∼3 eV. In contrast, the scGW calculations by Faleev et al. 14 , which feature fully selfconsistent wavefunctions, predicted a much smaller energy difference between the minima of the itinerant band (VBM+4.8 eV) and the 3d states (VBM+5 eV), with both being at considerably higher energies than in GW@GGA. Faleev et al. 14 attributed their results to underestimated screening from neglecting higher order correlation effects in their calculations, the favourable results from Li et al. 15 might thus benefit from error cancellation between the RPA level correlation and the GGA wavefunctions. Using HSE03 wavefunctions and eigenvalues for a G 0 W 0 one-shot calculation seems to have a similar effect as the fully self-consistent scheme of Faleev et al 14 and leads to an overestimated indirect band gap of 4.7 eV. In contrast, G( 0 )W 0 on top of LDA+U wavefunctions and pure HSE03 calculations predict considerably smaller gaps of 3.75 eV and 4.1 eV, respectively. Our obtained value of 3.85 eV falls right in the middle and is in excellent agreement with the experimental values. We note that the use of the slightly smaller experimental lattice constants leads to a larger band gap of 4.04 eV, which is very close to the HSE03 prediction for the same cell volume. On the other hand, the splitting of unoccupied and occupied d states in our sX-LDA is even more pronounced than those reported from GW calculations and clearly too strong if one compares the calculated DOS with experimental XPS and BIS spectra for NiO (Fig. 1) and the HSE03 calculations. We note that the strong shift of unfilled semi-core states compared to HSE03 and HSE06 is a known behaviour of sX-LDA 29,54 and is likely caused by the different inclusion of non-local exchange. While both functional types are conceptually similar in the sense that they employ a range separation scheme to screen the non-local exchange and generally yield similar band gaps, sX-LDA uses 100% of Thomas-Fermi screened Hartree-Fock exchange, whereas HSE only incorporates only 25% Hartree-Fock exchange, but with Error function screening and a weaker screening length compared to sX-LDA. The very strong contribution of Hartree-Fock exchange in short-range within sX-LDA particularly affects more localized orbitals, such as semi-core d electrons, and often leads to strong renormalizations of the corresponding energy levels. On the other hand, the typically stronger screening of the non-local exchange contribution in sX-LDA compared to HSE often leads to a weaker shift of the itinerant states in the conduction band, e.g. the "4s" states in case of the transition metal monooxides shown in this work, the Cu p states in copper-based transparent conducting oxides 54 , or the antibonding sp 3 states in silicon. In contrast, HSE03 predicts the d states at significantly lower energies. As a result, the minima of both the "4s" band and the "3d" band are at about 4.1-4.2 eV, so that 3d → 4s, 3d → 3d and 2p → 3d transitions are candidates for the fundamental transition. In this sense, the predictions of sX-LDA are a quantitative mixture of the 4s band from HSE03 and the strong shift of the d states from quasiparticle methods. We find a similar 'intermediate' prediction between HSE03 and G 0 W 0 for the case of MnO. HSE03 and sX-LDA both yield fundamental band gaps, which are considerably smaller than the experimental values of 3.6-4 eV, see Table II. We obtain a fundamental band gap of 2.8 eV, which is somewhat larger than the reported HSE03 value of 2.6 eV. However, we note that the prediction from sX-LDA is very close to the HSE03 gap if the experimental lattice constants are used. The quasiparticle corrections Rödl et al. found for the Mn 4s band in MnO are quite high (0.8 eV). This might indicate that the contribution of middle-range Hartree-Fock exchange in sX-LDA and HSE, which mainly leads to a rigid shift of the unoccupied states, is not strong enough in the case of MnO. An indicator is the fact that the splitting of occupied and unoccupied d bands in the hybrid functional calculations is considerably weaker than in GW, even though the relative energies of the minima of the 4s band and the d bands is 2-2.5 eV in all methods, see Table III, compared to the indirect band gaps in Table II. Also, the band gaps of B3LYP, a non-screened hybrid functional, and EXX-OEP, which essentially is a localized version of the pure Hartree-Fock exchange potential, are considerably larger and well within the range of experimental values. A more suitable range separation scheme of the Hartree-Fock exchange might lead to better band gaps for MnO. However, it is not clear to what extent the transition from the weakly dispersive band at the valence band top to the parabolic band contributes to the low energy optical absorption. In a free ion, a transition from d to s orbitals is symmetry-forbidden, as both intial and end states are even under inversion. This constraint is lifted in the monoxides, where the parabolic 4s band has a considerable contribution from Mn 4p states away from the Γ point. These p states are odd under inversion and thus allow for transitions to the conduction band minimum. At the same time, the valence band top has contributions from O 2p, which allows for charge transfer from oxygen states to Ni 4s states. However, it is possible that these transitions are rather weak due to the mixed nature of the valence band top and the 4s band. As for NiO, the predicted energies of the unoccupied d states in sX-LDA are higher than those from HSE03 and similar to those from G 0 W 0 @HSE03. Our cross-weighted DOS nicely reproduces the three dominating peaks found in the experimental XPS spectrum 10 . The peak at the valence band top arises from d states of e g character, which are hybridized with O 2p states and split from the second, t 2g dominated, peak by an energy of 1.8 eV. This splitting is considerably higher than the splitting in LDA (1.0 eV) and in excellent agreement with the reported value from scGW of 1.7 eV and with G 0 W 0 @HSE03. Van Elp et al. 9 reported a peak splitting of 1.9 eV in their photoemission experiments. The splitting in EXX-OEP 18 is approximately 1.4 eV. The main peak in the conduction band arises from a convolution of weakly dispersive states of mainly Mn 3d character at VBM+7 eV (t 2g ) and VBM+8 eV (e g ) and is predicted to lie slightly higher in energy than the BIS peak by about 0.5 eV. Exp (UPS) 10 7.5-8.5 8-9 8.5-9 8.5-9.5 Comparison with XES-XAS spectra Lastly, we compare the partial density of states of the oxygen 2p electrons with recently reported oxygen x-ray emission spectroscopy and x-ray absorption spectroscopy (XES-XAS) measurements 49 on MnO, CoO and NiO, in Fig. 3. The shape of the experimental spectra is quite similar for all three materials. The main feature is a sharp peak at 4.0-4.5 eV below the valence band maximum (VBM), which corresponds to a flat band in the valence band from a mixture of oxygen 2p and a metal 3d state of e g symmetry. Three smaller features exist at -1 eV, -2.3 eV and -6.5 eV. Our calculations exhibit all four peaks, albeit at different energies. While the hump from the flat hybrid 2p/3d band at the valence band maximum is well reproduced by a peak in all our theoretical spectra, the other three features are predicted at lower energies compared to experiments. For MnO, the agreement is quite good, with the dominant peak being shifted to about -5.5 eV. For CoO and NiO, the peak is being shifted even stronger, to -6.5 eV and -6 eV, respectively. The energies of the oxygen 2p states are strongly influenced by their hybridization with metal d electrons. The behaviour of these occupied states mirrors our observations for the unfilled states with strong 3d contribution. The non-local exchange causes a noticeable down-shift of the occupied d states compared to GGA. Table IV compares the valence band widths obtained from different theoretical methods with the XES results and ultraviolet photoemission spectroscopy (UPS). Our results from sX-LDA are generally at the upper end of the theoretical values and fit very well to the experimental spectra. The reported HSE03 valence band widths for MnO and NiO agree with our own HSE03 calculations. G 0 W 0 @HSE03 inherits the compressed valence bands from HSE03 and introduces only a tiny renormalization. IV. CONCLUSION Hybrid functional density functional theory was used to calculate the electronic band structures and density of states of the four transition metal oxides MnO, FeO, CoO and NiO. We conclude that the screened hybrid functional sX-LDA can successfully predict the electronic properties of all four materials with overall similar or greater accuracy than the established hybrid functional HSE03. This confirms that a correct description of the exchange interaction is of foremost priority for these materials. On the other hand, the effect of many-body effects can be reasonably approximated by a suitable screening of the electron exchange. A more sophisticated treatment of correlation effects might lead to even better results. Combined with the capability of selfconsistent total energy calculations and relatively low computational cost, this makes screened hybrid functionals interesting alternatives to quasiparticle methods for the simulation of defect properties. FIG. 1 : 1(Color online) Calculated electronic band structures and corresponding density of states (coloured lines) and cross-section weighted total density of states (dashed grey line) for MnO, FeO, CoO and NiO. The experimental XPS and BIS spectra (grey circles) are taken from Refs. 10 and 8. The cross-sectional DOS was calculated by summing the angular momentum channels weighted by their cross-sections 45 for the Al Kα line and using a Gaussian broadening of the peaks of 0.5 eV (0.1 eV for the non-weighted DOS) for better comparability with the XPS and BIS spectra. The energy zero of the experimental spectra was aligned to the calculated Fermi energy. e g orbitals and three-fold degenerate t 2g orbitals. For MnO, all five d spin-up orbitals are filled and the top d band is of e g character. Along the series, d orbitals of t 2g are successively occupied with minority spin electrons and are shifted down to the valence band top, as seen in the orbital-resolved LDOS in Fig. 2. For NiO, only two d bands of e g remain in the conduction band. F e p x 2 F e s x 2 FN i s x 2 FIG. 2 : 2222Engel et al.18 recently reported the band structure of FeO from optimized effective potential (EXX-OEP) calculations. While their band structures are qualitatively similar to ours, there are three noticeable differences: In their calculations, the energy difference between the top of the 'bulk' valence band and the conduction band min-(Color online) Angular momentum channel-and spin-resolved local density of states for MnO, FeO, CoO and NiO from sX-LDA calculations (solid lines). The d states were decomposed into contributions from e g (light grey area) and t 2g (dark grey area)orbitals. The peaks were broadened by a Gaussian of width 0.1 eV. imum, 4.6 eV, is about 0.8 eV larger than in our results. At the same time, the energy of the single d band is predicted to be about 0.75 eV closer to the conduction bands than in our calculations, closing the fundamental band gap to 1.7 eV. The third difference is the energy of the unoccupied d bands. Our calculations predict a d band very close to the conduction band minimum, while EXX-OEP shifts them to about 4 eV above the valence band. A low energy of the unoccupied d states is also suggested by the BIS spectrum in Ref.10. FIG. 3 : 3(Color online) Comparison of the calculated energy density of oxygen 2p states of MnO (blue solid line), CoO (purple solid line) and NiO (red solid line) with XES-XAS measurements (grey circles and triangles) taken from Ref. 49. The experimental XES and XAS contributions were scaled by two factors to match the peak heights of our calculations. As before, we used a Gaussian broadening of 0.5 eV. TABLE I : ILattice constants and bulk moduli of the cubic antiferromagnetic unit cell from sX-LDA calculations. MnO FeO CoO NiO a sX-LDA 8.67 8.53 8.65 8.47 (in Å) Exp. 17 8.89 8.668 8.534 8.342 ∆ -2.5% -1.6% +1.4% +1.5% B0 sX-LDA 181.02 207.66 214.92 230.72 (in GPa) Exp. 147 42 ,153 43 174 42 181 42 190-220 42 wave vector. We use a fixed value of k s =0.76 bohr −1 , which works well for sp semiconductors, in all our cal- culations The four transition metal oxides MnO, NiO, FeO and CoO were modelled by a rhombohedral unit cell containing two metal and two oxygen atom, respectively. All calculations converged to the AFII phase, i.e. with antiferromagnetic spin ordering along the crystal [111] TABLE II : IIMinimum direct and indirect band gaps (in eV) of four transition metal oxides as obtained from different theoretical and experimental methods.Method\Compound MnO FeO CoO NiO indir dir indir dir indir dir indir dir PBE 0.8 1.1 - - - - 0.7 0.9 sX-LDA 2.8 3.3 2.45 2.67 2.4 3.4 3.85 4.1 sX-LDA (exp. lat. const.) 2.5 3.0 2.3 2.4 2.7 3.7 4.04 4.3 HSE03 20 2.6 3.2 2.1 2.2 3.2 4.0 4.1 4.5 B3LYP 3.92 32 3.70 31 3.73 31 3.5 30 , 3.63 32 4.2 30,33 EXX-OEP 18 3.85 4.25 1.66 1.7 2.66 3.5 4.1 4.5 G0W0@LDA+U 2.34 25 ,3.05 19 3.51 19 0.95 25 2.47 25 3.75 25 , 3.46 19 3.97 19 G0W0@HSE03 20 3.4 4.0 2.2 2.3 3.4 4.5 4.7 5.2 Exp (conductivity) 3.8-4.2 46 3.6 47 3.7 48 Exp (XAS-XES) 4.1 49 2.6 49 4.0 49 Exp (PES-BIS) 3.9 9 2.5 8 4.3 6 Exp (absorption) 2.0 2 ,3.6-3.8 50 2.4 4 2.8 1 4.0 1 Exp (reflectance) 2.7 3 , 5.4 51 3.7 3 , 3.9 51 TABLE III : IIIMinimum splitting (in eV) of the occupied and unoccupied d-levels for different theoretical methods. We derived the values for GW 0 @LDA+U from the DOS plots for U=5.4 eVMnO FeO CoO NiO sX-LDA 5 2.7 4.1 5.5 sX-LDA (exp. lat. const.) 4.75 2.6 4.4 5.9 HSE03 20 4.5 3 4 4.1 B3LYP 4.7 32 3.7 31 3.4 32 4.2 33 EXX-OEP 18 6 4 4 6.5 GW@GGA 15 4.2 scGW 14 6 5 GW0@LDA+U 25 5.75 3.5 3.75 4.25 G0W0@HSE03 20 5.5-6 4.5 3.8 5-5.5 TABLE IV : IVValence band widths (in eV) from different theoretical methods compared to experiment.MnO FeO CoO NiO sX-LDA 8.25 8.7 8.5 7.5 sX-LDA (exp. lat. const.) 7.5 8.3 8.6 8 HSE03 20 7.1 8.3 8 7.4 EXX-OEP 18 ∼6 6-8.5 ∼8.5 ∼8 B3LYP 6.8 32 7.5 31 8.2 32 7.5 33 scGW 14 7.25 ∼7.8 GW0@LDA+U 25 7.5 9.5 8.5 8 G0W0@HSE03 20 7 8.8 8.1 7.4 Exp (XES) 49 7.5-8 8-9 8-8.5 . J G W Pratt, R Coelho, Phys. Rev. 116281J. G. W. Pratt and R. Coelho, Phys. Rev. 116, 281 (1959). . D R Huffman, R L Wild, M Shinmei, J. Chem. Phys. 504092D. R. Huffman, R. L. Wild, and M. Shinmei, J. Chem. Phys. 50, 4092 (1969). . R J Powell, W E Spicer, Phys. Rev. B. 22182R. J. Powell and W. E. Spicer, Phys. Rev. B 2, 2182 (1970). . H K Bowen, D Adler, B H Auker, J. Solid State Chem. 12355H. K. Bowen, D. Adler, and B. H. Auker, J. Solid State Chem. 12, 355 (1975). . G Van Der Laan, C Westra, C Haas, G A Sawatzky, Phys. Rev. B. 234369G. van der Laan, C. Westra, C. Haas, and G. A. Sawatzky, Phys. Rev. B 23, 4369 (1981). . G A Sawatzky, J W Allen, Phys. Rev. Lett. 532339G. A. Sawatzky and J. W. Allen, Phys. Rev. Lett. 53, 2339 (1984). . S Hüfner, F Hulliger, J Osterwalder, T Riesterer, Sol. Stat. Comm. 5083S. Hüfner, F. Hulliger, J. Osterwalder, and T. Riesterer, Sol. Stat. Comm. 50, 83 (1984). . J Van Elp, J L Wieland, H Eskes, P Kuiper, G A Sawatzky, F M F , T S Turner, Phys. Rev. B. 446090J. van Elp, J. L. Wieland, H. Eskes, P. Kuiper, G. A. Sawatzky, F. M. F. deGroot, and T. S. Turner, Phys. Rev. B 44, 6090 (1991). . J Van Elp, R H Potze, H Eskes, R Berger, G A Sawatzky, Phys. Rev. B. 441530J. van Elp, R. H. Potze, H. Eskes, R. Berger, and G. A. Sawatzky, Phys. Rev. B 44, 1530 (1991). . P Zimmermann, P Steinert, R Claessen, F Reinert, S Hüfner, P Blaha, P Dufek, J. Phys.: Condens. Matter. 111657P. Zimmermann, P. Steinert, R. Claessen, F. Reinert, S. Hüfner, P. Blaha, and P. Dufek, J. Phys.: Condens. Matter 11, 1657 (1999). . K Terakura, A R Williams, T Oguchi, J Kübler, Phys. Rev. Lett. 521830K. Terakura, A. R. Williams, T. Oguchi, and J. Kübler, Phys. Rev. Lett. 52, 1830 (1984). . S Massidda, A Continenza, M Posternak, A Baldereschi, Phys. Rev. Lett. 742323S. Massidda, A. Continenza, M. Posternak, and A. Baldereschi, Phys. Rev. Lett. 74, 2323 (1995). . F Aryasetiawan, O Gunnarsson, Phys. Rev. Lett. 743221F. Aryasetiawan and O. Gunnarsson, Phys. Rev. Lett. 74, 3221 (1995). . S V Faleev, M Van Schilfgaarde, T Kotani, Phys. Rev. Lett. 93126406S. V. Faleev, M. van Schilfgaarde, and T. Kotani, Phys. Rev. Lett 93, 126406 (2004). . J.-L Li, G.-M Rignanese, S G Louie, Phys. Rev. B. 71193102J.-L. Li, G.-M. Rignanese, and S. G. Louie, Phys. Rev. B 71, 193102 (2005). . C H Patterson, Int. J. Quant. Chem. 1063383C. H. Patterson, Int. J. Quant. Chem. 106, 3383 (2006). . F Tran, P Blaha, K Schwarz, P Novak, Phys. Rev. B. 74155108F. Tran, P. Blaha, K. Schwarz, and P. Novak, Phys. Rev. B 74, 155108 (2006). . E Engel, R N Schmid, Phys. Rev. Lett. 10336404E. Engel and R. N. Schmid, Phys. Rev. Lett. 103, 036404 (2009). . S Kobayashi, Y Nohara, S Yamamoto, T Fujiwara, Phys. Rev. B. 78155112S. Kobayashi, Y. Nohara, S. Yamamoto, and T. Fujiwara, Phys. Rev. B 78, 155112 (2008). . C Rödl, F Fuchs, J Furthmüller, F Bechstedt, Phys. Rev. B. 79235114C. Rödl, F. Fuchs, J. Furthmüller, and F. Bechstedt, Phys. Rev. B 79, 235114 (2009). . J Perdew, A Zunger, Phys. Rev. B. 235048J. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). . V I Anisimov, J Zaanen, O K Andersen, Phys. Rev. B. 44943V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44, 943 (1991). Consistent lda'+dmft approach to electronic structure of transition metal oxides: charge transfer insulators and correlated metals. I A Nekrasov, N S Pavlov, M V Sadovskii, I. A. Nekrasov, N. S. Pavlov, and M. V. Sadovskii, "Consis- tent lda'+dmft approach to electronic structure of transi- tion metal oxides: charge transfer insulators and correlated metals," (2012), http://arxiv.org/abs/1208.4732. . L Hedin, Phys. Rev. 139796L. Hedin, Phys. Rev. 139, A796 (1965). . H Jiang, R I Gomez-Abal, P Rinke, M Scheffler, Phys. Rev. B. 8245108H. Jiang, R. I. Gomez-Abal, P. Rinke, and M. Scheffler, Phys. Rev. B 82, 045108 (2010). . A Seidl, A Görling, P Vogl, J A Majewski, M Levy, Phys. Rev. B. 533764A. Seidl, A. Görling, P. Vogl, J. A. Majewski, and M. Levy, Phys. Rev. B 53, 3764 (1996). . S J Clark, J Robertson, Phys. Rev. B. 8285208S. J. Clark and J. Robertson, Phys. Rev. B 82, 085208 (2010). . F Trani, J Vidal, S Botti, M A L Marques, Phys. Rev. B. 8285115F. Trani, J. Vidal, S. Botti, and M. A. L. Marques, Phys. Rev. B 82, 085115 (2010). The nature of the electronic band gap in lanthanide oxides. R Gillen, S J Clark, J Robertson, Submitted to PRBR. Gillen, S. J. Clark, and J. Robertson, "The nature of the electronic band gap in lanthanide oxides," Submitted to PRB (2012), http://arxiv.org/abs/1208.0503. . T Bredow, A R Gerson, Phys. Rev. B. 615194T. Bredow and A. R. Gerson, Phys. Rev. B 61, 5194 (2000). . M Alfredsson, G D Price, C R A Catlow, S C Parker, R Orlando, J P Brodholt, Phys. Rev. B. 70165111M. Alfredsson, G. D. Price, C. R. A. Catlow, S. C. Parker, R. Orlando, and J. P. Brodholt, Phys. Rev. B 70, 165111 (2004). . X B Feng, Phys. Rev. B. 69155107X. B. Feng, Phys. Rev. B 69, 155107 (2004). . X.-B Feng, N M Harrison, Phys. Rev. B. 6935114X.-B. Feng and N. M. Harrison, Phys. Rev. B 69, 035114 (2004). . C Franchini, V Bayer, R Podloucky, J Paier, G Kresse, Phys. Rev. B. 7245132C. Franchini, V. Bayer, R. Podloucky, J. Paier, and G. Kresse, Phys. Rev. B 72, 045132 (2005). . I C Gerber, J G Angyan, M Marsman, G Kresse, J. Chem. Phys. 12754101I. C. Gerber, J. G. Angyan, M. Marsman, and G. Kresse, J. Chem. Phys. 127, 054101 (2007). . M Marsman, J Paier, A Stroppa, G Kresse, J. Phys.: Condens. Matter. 2064201M. Marsman, J. Paier, A. Stroppa, and G. Kresse, J. Phys.: Condens. Matter 20, 064201 (2008). . F Tran, Phys. Lett. A. 376879F. Tran, Phys. Lett. A 376, 879 (2012). . Y Guo, S J Clark, J Robertson, J. Phys.: Condens. Mat. 24325504Y. Guo, S. J. Clark, and J. Robertson, J. Phys.: Condens. Mat. 24, 325504 (2012). . J Heyd, G E Scuseria, M Ernzerhof, J. Chem. Phys. 1188207J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003). . D M Bylander, L Kleinman, Phys. Rev. B. 417868D. M. Bylander and L. Kleinman, Phys. Rev. B 41, 7868 (1990). . S J Clark, M D Segall, C J Pickard, P J Hasnip, M J Probert, K Refson, M C Payne, Zeitschrift für Kristallographie. 220567S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, and M. C. Payne, Zeitschrift für Kristallographie 220, 567 (2005). . S Ohnishi, H Mizutani, J. Geophys. Res. 831852S. Ohnishi and H. Mizutani, J. Geophys. Res. 83, 1852 (1978). . Y Makino, S Miyake, J. All. Comp. 313235Y. Makino and S. Miyake, J. All. Comp. 313, 235 (2000). . N J Ramer, A M Rappe, Phys. Rev. B. 5912471N. J. Ramer and A. M. Rappe, Phys. Rev. B 59, 12471 (1999). Atomic Data and Nuclear Data Tables. J J Yeh, I Lindau, 321J. J. Yeh and I. Lindau, Atomic Data and Nuclear Data Tables 32, 1 (1985). . I A Drabkin, L T Emel&apos;yanova, R N Iskenderov, Y M Ksendzov, Fiz. Tverd. Tela (Leningrad). 103082I. A. Drabkin, L. T. Emel'yanova, R. N. Iskenderov, and Y. M. Ksendzov, Fiz. Tverd. Tela (Leningrad) 10, 3082 (1968). . M Gvishi, D S Tannhauser, J. Phys. Chem. Solids. 33893M. Gvishi and D. S. Tannhauser, J. Phys. Chem. Solids 33 33, 893 (1972). . Y M Ksendzov, I A Drabkin, Fiz. Tverd. Tela (Leningrad). 71884Y. M. Ksendzov and I. A. Drabkin, Fiz. Tverd. Tela (Leningrad) 7, 1884 (1965). . E Z Kurmaev, R G Wilks, A Moewes, L D Finkelstein, S N Shamin, J Kunes, Phys. Rev. B. 77165127E. Z. Kurmaev, R. G. Wilks, A. Moewes, L. D. Finkelstein, S. N. Shamin, and J. Kunes, Phys. Rev. B 77, 165127 (2008). . R N Iskenderov, I A Drabkin, L T , Y M Ksendzov, Fiz. Tverd. Tela (Leningrad). 102573R. N. Iskenderov, I. A. Drabkin, L. T. Emel'yanova, and Y. M. Ksendzov, Fiz. Tverd. Tela (Leningrad) 10, 2573 (1968). . T D Kang, H S Lee, H Lee, J. Korean Phys. Soc. 50632T. D. Kang, H. S. Lee, and H. Lee, J. Korean Phys. Soc. 50, 632 (2007). Spectrum of extended systems from reduced density matrix functional theory. S Sharma, S Shallcross, J K Dewhurst, E K U Gross, S. Sharma, S. Shallcross, J. K. Dewhurst, and E. K. U. Gross, "Spectrum of extended systems from reduced density matrix functional theory," (2012), http://arxiv.org/abs/0912.1118. . S Asada, S Sugano, J. Phys. Soc. Japan. 411921S. Asada and S. Sugano, J. Phys. Soc. Japan 41, 1921 (1976); . S Larsson, Chem. Phys. Lett. 40362S. Larsson, Chem. Phys. Lett. 40, 362 (1976). . R Gillen, J Robertson, Phys. Rev. B. 8435125R. Gillen and J. Robertson, Phys. Rev. B 84, 035125 (2011).
[]
[ "Morphological variations to ptychographic algorithm", "Morphological variations to ptychographic algorithm" ]
[ "F Salinas \nDepartamento de Ciencias Físicas\nFacultad de Ingeniería y Ciencias\nUniversidad de La Frontera\" Avenida Francisco Salazar 01145\nTemucoChile\n", "M A Solís-Prosser \nDepartamento de Ciencias Físicas\nFacultad de Ingeniería y Ciencias\nUniversidad de La Frontera\" Avenida Francisco Salazar 01145\nTemucoChile\n" ]
[ "Departamento de Ciencias Físicas\nFacultad de Ingeniería y Ciencias\nUniversidad de La Frontera\" Avenida Francisco Salazar 01145\nTemucoChile", "Departamento de Ciencias Físicas\nFacultad de Ingeniería y Ciencias\nUniversidad de La Frontera\" Avenida Francisco Salazar 01145\nTemucoChile" ]
[]
Ptychography is a technique widely used in microscopy for achieving high-resolution imaging. This method relies on computational processing of images gathered from diffraction patterns produced by several partial illuminations of a sample. In this work, we numerically studied the effect of using different shapes for illuminating the aforementioned sample: convex shapes, such as circles and regular polygons, and unconnected shapes that resemble a QR code. Our results suggest that the use of unconnected shapes seems to outperform convex shapes in terms of convergence and, in some cases, accuracy.©2022 Optica Publishing Group. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibited.http://dx.
10.1364/ao.462173
[ "https://export.arxiv.org/pdf/2209.02148v1.pdf" ]
250,472,974
2209.02148
9100a7bc116748d6c8f75eb4552925271b21fcb3
Morphological variations to ptychographic algorithm F Salinas Departamento de Ciencias Físicas Facultad de Ingeniería y Ciencias Universidad de La Frontera" Avenida Francisco Salazar 01145 TemucoChile M A Solís-Prosser Departamento de Ciencias Físicas Facultad de Ingeniería y Ciencias Universidad de La Frontera" Avenida Francisco Salazar 01145 TemucoChile Morphological variations to ptychographic algorithm Compiled September 7, 2022Research Article Applied Optics 1 See published version at https://doi.org/10.1364/AO.462173 Ptychography is a technique widely used in microscopy for achieving high-resolution imaging. This method relies on computational processing of images gathered from diffraction patterns produced by several partial illuminations of a sample. In this work, we numerically studied the effect of using different shapes for illuminating the aforementioned sample: convex shapes, such as circles and regular polygons, and unconnected shapes that resemble a QR code. Our results suggest that the use of unconnected shapes seems to outperform convex shapes in terms of convergence and, in some cases, accuracy.©2022 Optica Publishing Group. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibited.http://dx. INTRODUCTION Every time one tries to recover optical information about an object, one may think in taking a high-resolution picture, which will be able to inform us about some features of the aforementioned object: color and transparency. However, the phase is a nontrivial piece of information missing from a single picture, which is very important in some fields such as optical microscopy [1,2], electron microscopy [3][4][5][6][7][8] and X-ray imaging [9][10][11][12][13][14][15]. In this context, the field of phase retrieval aims to obtain the phase of a complex-valued function that describes either a wave field or the transmission function of an object. One of the first approaches, if not the first, was introduced by Gerchberg and Saxton in 1972 [16], which requires a picture of the object and a picture of its Fourier transform. This algorithm, also known as the GS algorithm, is guaranteed to converge, although very slowly, and it is not free of inaccuracies. Based on the GS algorithm, Fienup proposed an algorithm that only needs information of the diffraction pattern of the object [17,18]. These algorithms have been mathematically analysed in terms of convex optimization [19] and, very recently, Zhao and Chi [20] introduced modifications to the GS algorithm that improved its convergence and accuracy, and studied their feasibility for optical cryptography. Other widely used technique is ptychography, which is a phase retrieval method that allows us to retrieve both amplitude and phase of a sample object function using data from several diffraction patterns, each obtained by illuminating a subregion of the object being reconstructed [21][22][23] and applying an iterative numerical algorithm on the obtained images, which is known as Ptychographical Iterative Engine (PIE) [24]. For this method to work properly, it is crucial that the different regions illuminated in the sample have a significant overlap between each other [25,26]. As the GS algorithm can reconstruct imagesalthough with some drawbacks-from only two images, one may see that a dataset used for ptychography exhibits redundancy whenever more than two diffraction patterns are recorded. This redundancy, far from being undesired, allows one to achieve superresolved imaging [27,28]. Ptychography has benefited from several improvements and modifications, including-but not limited to-enhanced algorithms such as extended PIE (ePIE) [29,30], combination with a Hybrid Input-Output approach [31], a reciprocal approach in which the illumination beam is tilted instead of displaced on a sample, also known as Fourier ptychography [32][33][34][35][36][37], among others [38][39][40][41]. This technique and its variants have already found application in the context of optical imaging [42][43][44][45][46][47][48], X-ray microscopy [27,41,[49][50][51][52][53][54][55][56][57][58][59], electron microscopy [27,[60][61][62][63][64][65], optical encryption [66][67][68], and recent demonstrations show promising applications in Quantum Information Science [69][70][71]. Noteworthily, the partial illumination of the sample is circularly shaped in most works of the literature. One of the few works that studied other possibilities is the one of Ref. [72], which included hexagonal and square shapes. The work of Ref. [73] explored the impact of overlap uniformness in the quality of the reconstruction. Besides from these studies, and up to our knowledge, irregularly-shaped partial illuminations have not yet been explored. Moreover, as most iterative algorithms, convergence and accuracy might depend on the choice of an initial guess. In this work, we explored the effects of considering different shapes for the partial illuminations of the sample on both accuracy and convergence when the PIE algorithm is used. Through simulations, we considered squares, regular hexagons, circles, and irregular unconnected regions resembling a QR code. As a strategy to avoid reaching to misleading conclusions, we performed every reconstruction with 50 different initial guesses in order to have statistically significant results. Consequently, our results showed that continuous regions (polygons and circles) exhibit significant differences in performances only in a handful of cases. Unexpectedly, the use of unconnected regions, in general, outperformed the use of continuous regions. This article is organized as follows. Section 2 explains the method in detail. Particularly, Subsection A introduced the images being used as the optical object to be reconstructed and the different shapes of the illumination functions; Subsection B introduces useful notation for this article and explains how the initial guessed function are dealt with; Subsection C shows an overview of the algorithm used and the figures of merit used to assess performance. Section 3 shows the results of our study. Section 4 concludes the paper. METHOD A. Illumination functions and optical objects Let O( r) be a 2D-transmission function of an arbitrary optical object (sample). For this work, we will also assume this object will be illuminated by a coherent monochromatic plane-wave electromagnetic field. This light field can be modulated through diffractive devices. In this context, let us define a set of illumination functions A j ( r) N j=1 which will describe the incident light field being shaped in order to illuminate different parts of the sample. In this context, we have tested two classes of illumination functions: (a) regular convex figures, and (b) unconnected sets. On one hand, in (a) we used N illumination functions shaped as a regular figure, distributed among N positions on the sample (see Figure 1, left, for an example with N = 4). For this purpose, we compared circular, hexagonal and square shapes. Let R be the radius of the circles. Two values of R were used in this study: 40 and 80 pixels. It is important to note that a regular polygon with radius R will always have a smaller area than that of a circle of the same radius (considering the radius of a regular polygon as the distance between its center and any of its vertices). As the purpose of this work is to compare the same method using different types of illuminated regions, it becomes necessary to build figures with the same illuminated area over the sample in order to avoid a bias towards circles. For this reason, radii for polygons (R pol ) were computed in such a way their areas are the closest possible to the area of a circle of radius R. Thus, by imposing the area of the polygon to be equal to πR 2 , we obtain that R pol = 2π/K sin(2π/K) R,(1) where K is the number of sides the polygon has (4 and 6 for squares and hexagons, respectively). In general, K is lowerbounded by 3 (triangles) and has no upper bounds since a polygon may have any number of sides . Moreover, according K are being used as illumination functions. The radius of each hexagon is such that its area is the closest possible to one of a circle of R = 80 pixels as radius. Lower half: unconnected shapes (4 different ones in this example) are being used as illumination functions. For this particular example, = 8 pixels. The number of squares is such that the illuminated area in each A j ( r) is the closest possible to one of a circle of R = 80 pixels as radius. moves toward infinity, R pol becomes closer to R. Figure 2 shows and example of a circle of radius equal to 100 px together with a square and a hexagon whose radii R pol were computed using Eq. (1), ensuring that each shape encloses the same area. In (b), instead, we used unconnected regions. There were modelled as a plate with the same size of the sample, containing a number of small transparent squares whose sides have length equal to pixels. These squares are randomly distributed on the object, resembling a QR code (see Figure 1, right, for an example). This resembles the array of N × N pinholes in Ref. [74], but in this study there is more than one plate and the positions of orifices is random. In order to compare the results with the ones attainable from the aforementioned convex figures of radius R, the number of squares is also adjusted in such a way the total transparent area in each A j ( r) is the closest possible to the one of a circle of radius R. Consequently, every illumination function A j ( r) has πR 2 / 2 squares. Figure 3 shows a superposition of all illumination functions for every shape used in this work. For this work, we considered N = 9, 16, and 25 illumination functions S({A j }). Each value of R was adapted in order to be the smallest possible subject to have the complete image illuminated at least once. Consequently, we tested 9 functions with R = 93 px, 16 functions with R = 74 px, and 25 functions with R = 62 px. These minimal radii were chosen in order to make the overlaps between different values of N more uniform. In this study, optical objects are described by transmission functions O( r), which are also known as target functions since the reconstruction algorithm must aim to reconstruct a function like those. The target function is built from two images of 256 × 256 pixels each, one of them will be used for the amplitude and the other one for the phase, so the object will be described by a transmission function given by O( r) = |O( r)| e 2πiϕ( r) .(2) To prevent dependency on the use of the same images, we have selected three different target functions, constructed from different images in grayscale, as Figure 4 shows. These grayscale values are used to encode values between 0 (black) and 1 (white). The images being used were chosen because they have diverse features that are useful for testing the algorithms: thick and thin stripes; coarse and fine details; well-focused and blurred backgrounds; high and low contrast. B. Reconstructed functions and initial guessed function In ptychography, one may set a fixed number of iterations for the algorithm to run on, or to define stopping criteria. As For each case, the radius was adjusted to the minimum one that allows the complete image to be illuminated at least once. our goal is to compare performance between several choices of parameters, we decided to use the same number of iterations for every of the possible shapes being used as illumination functions regardless of the target function. Thus, 200 iterations were used. Let n be the number of iterations the algorithm has reached, with 1 ≤ n ≤ 200. After n iterations of the PIE algorithm, a reconstructed function O g,n ( r) will be obtained. For this purpose, PIE starts with an initial guessed function O g,0 ( r) which can be defined, for instance, as a random function or as a constant function. Naturally, it may happen that O g,n ( r) exhibits an implicit dependence on the choice of the initial guessed function O g,0 ( r) and, consequently, the quality of the reconstruction may be strongly conditioned by such a choice. For this reason, we ran the PIE 50 times for every target function and for every shape of illumination functions, each time using a different initial guessed function. C. Overview of the algorithm and figures of merit Let us recall Figure 1. For every illumination function A j ( r), the transmitted electromagnetic field will be described by O( r)A j ( r). In an experimental situation, a detection system will be able to retrieve intensity distributions I j ( u) from the Fourier plane, where u is the transverse position vector in the Fourier plane. These distributions are the experimental inputs the algorithm needs. In our case, these I j ( u) are computed via FFT. Once the initial guessed function is defined, the algorithm may start. Let us also recall that O g,n ( r) is the reconstructed function after n iterations. Our implementation of PIE is mostly based on Refs. [23,24,27] and summarized in Algorithm 1. The inputs the algorithm needs are the intensities I j ( u) from the Fourier plane and the list of illumination functions. Before elaborating details about the figures of merit used, it is necessary to define a matrix norm. Particularly, the following definition will be used, B = ∑ p,q∈S |B p,q | 2 ,(3) where the sum is performed on the pixels comprising the image (S) to be reconstructed. Two parameters were used to assess performance: convergence and accuracy. Convergence (∆) is studied in terms of the difference between the last two estimated functions for each iteration. This parameter should decrease with increasing iterations, as after each iteration these functions should become similar. This parameter is given by ∆ = κO g,n ( r) − O g,n−1 ( r) κO g,n ( r) O g,n−1 ( r) ,(4) where the denominator has been included as normalization factor in order to avoid image size dependence and to address ∆ as a relative-difference coefficient. As the matrix difference in the numerator might be artificially increased by a global phase or a global scaling factor, a proportionality constant κ has been included in order to minimize this effect. After an optimization, it is possible to show that κ = ∑ p,q∈S O g,n * p,q O g,n−1 p,q ∑ p,q∈S O g,n * p,q O g,n p,q ,(5) is the value that assures a minimum of the numerator of Eq. (4) with respect to global scaling factors. On the other hand, accuracy (d) is studied in terms of the difference between the last estimated function and the target function. This parameter indicates the quality of the retrieval, as it indicates how much the nth estimated function resembles the target function: d = O ( r) − µO g,n ( r) O ( r) µO g,n ( r) , µ = ∑ p,q∈S O g,n * p,q (O) p,q ∑ p,q∈S O g,n * p,q O g,n p,q ,(6) where a normalization factor has also been included here to avoid image size dependence and to address d also as a relativedifference coefficient. A proportionality constant µ was included to remove effects of global phases or global scaling factors as well. This scaling factor µ was computed in an analogous way as with κ in Equation (5). For both d and ∆, the closer to zero they are, the better the performance is. If a little abuse of terminology is tolerated, we may name ∆ directly as convergence, and d as accuracy throughout this document. The use of multiplicative constants, such as κ and µ, to avoid the effect of global phases was already proposed in [75]. Both ∆ and d are used in this work to assess the performance of the method and the use of every geometry. In an experimental situation, convergence can be used also as a stopping criterion. Accuracy, on the other hand, is not usable in most experimental situations, but rather a figure of merit that can be used mostly for assessing algorithmic performance. D. Finite-sized pixels and noise Finally, we set up physical parameters in order to include effects from a realistic experimental scenario. Firstly, we now consider the finite size of the detector that can be used in an experiment. That is, the fact that a CCD/CMOS pixel is not exactly a pointlike detector, but rather a small bucket detector capturing light over the complete area each pixel covers. For this reason, although the objects we aim to reconstruct are 256 × 256, we increased the number of points each FFT/IFFT uses in order to integrate over each pixel. That is, for each illumination function A j ( r), we have an expected field intensity I j ( r) and a expected retrieved distribution I j ( r). We used 16 points to model each CCD/CMOS pixel. The expected pictures were computed by integrating the expected field intensity over each pixel. For this experimental-case simulation, we considered camera pixels 3.45 µm-wide and the object to be composed by 8.00 µm-wide pixels, so the object is, approximately, 2.05 mm-wide. The Fourier transform is performed by a lens Algorithm 1. Summary of PIE algorithm, as shown in Refs. [23,24,27] and in the way it was used on this work. N is the number of PIE iterations, which is equal to 200 along this report. 1: procedure PIE({I j ( u)} N j=1 , {A j ( r)} N j=1 ) 2: Define O g,0 ( r) This is the initial guessed function 3: for n = 1, . . . , N do Loop along PIE iterations 4: O g,n ( r) ← O g,n−1 ( r) 5: for j = 1, . . . , N do Loop along illumination functions 6: ψ g ( r) ← A j ( r) O g,n ( r) 7: Ψ g ( u) ← F ψ g ( r) ( u) 8: Ψ c ( u) ← I j ( u) exp i arg Ψ g ( u) Correct amplitude by using data 9: ψ c ( r) ← F −1 [Ψ c ( u)] ( r) 10: U j ( r) ← |A j ( r)| max r |A j ( r)| A * j ( r) |A j ( r)| 2 + δ δ ∼ 10 −7 11: O g,n ( r) ← O g,n ( r) + U j ( r) ψ c ( r) − ψ g ( r) Update reconstruction 12: Compute figures of merit regarding O g,n ( r) 13: if (any stopping criterion is met) then 14: return with focal length f = 0.1 m in our simulation and we considered illumination from a coherent monochromatic light source of 565.25 nm as wavelength. Thus, the object (modeled as a 256 × 256-sized matrix) was padded with zeros in order to obtain a 8192 × 8192 matrix. Thus, each expected field intensity (I j ) was a 8192 × 8192 matrix and each expected retrieved distribution (I j ) was a 2048 × 2048 matrix. Secondly, we incorporated noise to each expected retrieved distribution. For this purpose, for each shape under consideration, we normalized each dataset {I j } N j=1 such that their maximum is equal to 1. Afterwards, speckle noise was added to each I j . We tested speckle noise variances equal to 0 (noiseless case) and 0.20. Figure 5 shows some samples of the object (target functions of Figure 4) under partial illumination, the expected ideal intensity distribution (noiseless) and one obtained after having applied speckle noise with variance equal to 0.20. Since the diffraction patterns are normalized to have a maximum value of 1, a value of 0.20 as noise variance seems to be relevant. RESULTS For a better comparison of the results both parameters have been plotted over the number of iterations performed in the algorithm, stopping the algorithm after N = 200 iterations. Additionally, as aforementioned, each case was studied with 50 choices of initial guessed functions. Thus, our results show bands comprising the central 95% of the results surrounding the mean values of the 50 first guesses. This selection has been made in order to avoid the effect of outliers in our conclusions. As the computational demand increased largely due to the size of the matrices under consideration, we resorted to Single-precision floating-point arithmetic for the computations. As such, we would expect the convergence to end, at best, around 10 −7 since the machine epsilon for single-precision floating-point format is approximately 1.1921 × 10 −7 for Matlab/Octave. For starters, Figure 6 shows the results achieved when the algorithm reconstructed target function 1, using N = 9 illumination functions and R = 93 px. It can be seen that the convergence attained by convex figures is dwarfed by the one attained by unconnected regions, which converge much faster. The accuracy reaches to final values much faster when discontinuous shapes are used instead of convex ones: less than 20 PIE iterations using discontinuous shapes lead to the same accuracy that continuous shapes achieve after more than 70 PIE iterations. Moreover, for both figures of merit, convex shapes exhibit great dependence in terms of the first guess. Instead, the choice of the first guess seems to be completely irrelevant when discontinuous shapes are used. Figure 7 shows our results for reconstruction of target function 2 using N = 16 illumination functions and R = 74 px. The use of more illumination regions, although smaller ones, leads to faster results (in terms of PIE iterations needed) when compared with the previous case of a smaller number of larger illumination regions. In terms of noise, all shapes seem to be very noise-resistant, but the results from convex regions still depend very strongly on the choice of the first guess-although in a lesser degree than the one observed for N = 9. Circles now exhibit a performance comparable to the one attained with discontinuously-shaped illumination. These results and the previous ones indicate that, among continuous shapes, circles exhibit the best results. For discontinuous shapes, = 4 px and = 8 px perform almost identically. Finally, Figure 8 shows convergence and accuracy, respectively, when target function 3 is reconstructed using N = 25 and R = 62 px. Although the illumination is more uniform in this case (see Figure 3), convergence is now slower for circles: they needed around 60 PIE iterations to reach a final result (convergence) when N = 16, but need almost 80 PIE iterations when N = 25. Discontinuous shapes with = 16 px also decreased their performance when noise is present: from less than 60 PIE iterations in N = 16 to almost 70 in N = 25. On the other hand, = 4 px and = 8 px perform almost identically in every configuration, needing around 30 PIE iterations regardless the value of N. Accuracy, unlike convergence, seems to benefit from increasing the number of illumination functions regardless of the shrinking radii. On one hand, this would be expected since ptychography may be used for achieving superresolution as consequence of information redundancy [27,28]. It is natural to think more redundancy would lead to better performance and these results seem to agree with that. On the other hand, convergence might need a trade-off between the number of illumination functions used and their width. This is not completely unexpected since Ref. [26] already showed that performance is non-monotonically linked to the overlap between illumination functions. Although it is easy to quantify an overlap between two functions, a study on the overlap between N functions for different radius and shapes lies beyond the scope of the current work and can be addressed in a future study. All results indicate that unconnected regions constructed from smaller squares perform better than the ones built from larger squares. A possible explanation lies in the fact that, for a fixed area, the smaller squares gather information from a more diverse set of regions on the image than convex shapes. In order to appreciate the results of ptychographic reconstruction, Figure 9 shows some reconstructed images compared with their respective target functions. Figure 9a shows the result of a reconstruction using large squares (N = 9) on target function 1. One may see a kind of artefact on the reconstructed images which is not seen on the other shapes. Remarkably, Figure 7 showed squares struggled to converge. Figure 9b shows reconstruction after using circles (N = 16) on target function 2, leading to good results. Finally, Figure 9c shows reconstruction of target function 3 after using discontinuous illumination functions (N = 25, = 8 px). leading to seemingly high-quality results. CONCLUSION We used three different complex target functions that exhibit several diverse features (coarse and fine details, high and low contrast, etc) in order to test the scope of our conclusions. Additionally, as it could be expected, the use of more illumination functions leads to more experimental information, which leads to better results in terms of the number of iterations needed to achieve a final accuracy level. However, and perhaps unexpectedly, the use of unconnected illumination functions resembling QR codes seems to outperform the use of regular shapes by a substantial margin in some cases. Moreover, the effect of speckle noise for these shapes seemed not so relevant. In this regard, ptychography might be very noise-resistant against speckle noise regardless of the shape of the illumination functions. The results suggest that gathering information from more diverse regions on the image lead to better results. For this reason, unconnected regions built from smaller squares performed better than the ones produced from larger squares, which, in turn, outperformed convex shapes. The use of unconnected regions is something it could be implemented through the current technology of spatial light modulators if visible light is used. However, we acknowledge smaller squares might be experimentally more challenging to implement than larger ones. As the advantages were more relevant for smaller number of illuminations used, these shapes could be used when data acquisition must be done very quickly, like in a biologically active sample. We anticipate these results can be useful for any topic in which ptychography has been used, mostly microscopy, as well as in any field in which high-resolution imaging is not only necessary, but difficult, such as observational astronomy. authors upon reasonable request. Fig. 1 . 1Simplified depiction of a ptychographic scheme. Upper half: convex regular shapes (4 hexagons in this example) Fig. 2 . 2Example of three figures constructed from a given radius (R = 100 px in this example). Upper panels: each figure shown separately. For the polygonal shapes, their radii R pol are computed using Eq. (1). Lower panel: comparison of the three figures by superimposition. Up to rounding errors and pixelation, every shape encloses an area of 10000π px 2 . Fig. 3 . 3Superposition of illumination functions for the three cases under study: N = 9, 16, and 25 illumination functions. Fig. 4 . 4Amplitude and phase of the functions O( r) used as target functions. The size of each image, in pixels, is 256 × 256. Fig. 5 .Fig. 6 . 56Example of illumination functions acting on different target functions, their noiseless diffraction pattern and the noisy diffraction pattern. Upper row: N = 9 and R = 93 px using square illumination functions. Middle row: N = 16 and R = 74 px using circular illumination functions. Lower row: N = 25 and R = 62 px using discontinuous illumination functions with = 8 px. Convergence (upper panels) and accuracy (lower panels) obtained for reconstruction of target function 1 using N = 9 illumination functions and R = 93 px. Thick lines represent average over the 50 times the algorithm was applied on different initial guesses. The bands represent the central 95% of the results. The method, regardless of the shape employed, seems to be noiseresistant. However, the use of discontinuous shapes seems to outperform the use of continuous shapes in terms of convergence and necessary PIE iterations. Fig. 7 . 7Convergence (upper panels) and accuracy (lower panels) obtained for reconstruction of target function 2 using N = 16 illumination functions and R = 74 px. For this configuration, circularly-shaped illumination vastly outperforms polygonallyshaped illumination. Discontinuous illuminations still show better results than convex illuminations, although the advantages are not so evident now. To ease observations, only the first 150 PIE iterations are shown. Fig. 8 . 8Convergence (upper panels) and accuracy (lower panels) obtained for reconstruction of target function 3 using N = 25 illumination functions and R = 62 px. To ease observations, only the first 120 PIE iterations are shown. Regarding convergence and the number of iterations needed, polygonally-shaped illuminations still underperform circularly-shaped illumination which, in turn, underperforms discontinuous illumination. In terms of convergence, however, only squares seem to underperform the other shapes by a relevant margin. Funding. DIUFRO grant DI20-0154 Acknowledgments. F.S. acknowledges partial financial support from the Master of Science in Physics program at Universidad de La Frontera. M.A.S.-P. acknowledges funding from Universidad de La Frontera through DIUFRO grant DI20-0154. The authors also would like to thank Fabián Torres Ruiz and Leonardo Teixeira Neves for fruitful conversations. Disclosures. The authors declare no conflicts of interest. Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained Target function 3, using = 8 px (N = 25, R = 62 px). Fig. 9 . 9Some of the reconstructed figures using different illumination shapes. For every pair of images, left one represents the target function. Right one is the result of ptychographic reconstruction. They correspond to the same configurations exemplified inFigure 5. Phase contrast, a new method for the microscopic observation of transparent objects. F Zernike, Physica. 9F. Zernike, "Phase contrast, a new method for the microscopic obser- vation of transparent objects," Physica 9, 686-698 (1942). N T Shaked, Z Zalevsky, L L Satterwhite, Biomedical Optical Phase Microscopy and Nanoscopy. OxfordAcademic Press1st edN. T. Shaked, Z. Zalevsky, and L. L. Satterwhite, eds., Biomedical Optical Phase Microscopy and Nanoscopy (Academic Press, Oxford, 2013), 1st ed. The phase problem, microdiffraction and wavelengthlimited resolution -a discussion. J Rodenburg, Ultramicroscopy. 27J. Rodenburg, "The phase problem, microdiffraction and wavelength- limited resolution -a discussion," Ultramicroscopy 27, 413-422 (1989). Emission microscopy and related techniques: Resolution in photoelectron microscopy, low energy electron microscopy and mirror electron microscopy. G F Rempfer, O. Hayes Griffith, Ultramicroscopy. 47G. F. Rempfer and O. Hayes Griffith, "Emission microscopy and related techniques: Resolution in photoelectron microscopy, low energy elec- tron microscopy and mirror electron microscopy," Ultramicroscopy 47, 35-54 (1992). Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy. W Coene, G Janssen, M Op De Beeck, D Van Dyck, Phys. Rev. Lett. 69W. Coene, G. Janssen, M. Op de Beeck, and D. Van Dyck, "Phase retrieval through focus variation for ultra-resolution in field-emission transmission electron microscopy," Phys. Rev. Lett. 69, 3743-3746 (1992). Wave function reconstruction in HRTEM: The parabola method. M Op De Beeck, D Van Dyck, W Coene, Ultramicroscopy. 64M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function re- construction in HRTEM: The parabola method," Ultramicroscopy 64, 167-183 (1996). Inversion of dynamical electron diffraction data including absorption. L J Allen, H M L Faulkner, H Leeb, Acta Crystallogr. Sect. A: Foundations Crystallogr. 56L. J. Allen, H. M. L. Faulkner, and H. Leeb, "Inversion of dynamical electron diffraction data including absorption," Acta Crystallogr. Sect. A: Foundations Crystallogr. 56, 119-126 (2000). Phase retrieval and aberration correction in the presence of vortices in highresolution transmission electron microscopy. L J Allen, H M L Faulkner, M P Oxley, D Paganin, Ultramicroscopy. 88L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, "Phase retrieval and aberration correction in the presence of vortices in high- resolution transmission electron microscopy," Ultramicroscopy 88, 85- 97 (2001). Phase retrieval in crystallography and optics. R P Millane, JOSA A. 7R. P. Millane, "Phase retrieval in crystallography and optics," JOSA A 7, 394-411 (1990). Phase-Sensitive X-Ray Imaging. R Fitzgerald, Phys. Today. 53R. Fitzgerald, "Phase-Sensitive X-Ray Imaging," Phys. Today 53, 23-26 (2000). The phase problem. G Taylor, Acta Crystallogr. Sect. D: Biol. Crystallogr. 59G. Taylor, "The phase problem," Acta Crystallogr. Sect. D: Biol. Crystal- logr. 59, 1881-1890 (2003). Medical phase contrast x-ray imaging: Current status and future prospects. R A Lewis, Phys. Medicine Biol. 49R. A. Lewis, "Medical phase contrast x-ray imaging: Current status and future prospects," Phys. Medicine Biol. 49, 3573-3583 (2004). X-ray phase-attenuation duality and phase retrieval. X Wu, H Liu, A Yan, Opt. Lett. 30X. Wu, H. Liu, and A. Yan, "X-ray phase-attenuation duality and phase retrieval," Opt. Lett. 30, 379-381 (2005). Phase retrieval in X-ray phase-contrast imaging suitable for tomography. A Burvall, U Lundström, P A C Takman, D H Larsson, H M Hertz, Opt. Express. 19A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, "Phase retrieval in X-ray phase-contrast imaging suitable for tomography," Opt. Express 19, 10359-10376 (2011). Enhanced phase retrieval via deep concatenation networks for in-line X-ray phase contrast imaging. Y Wu, L Zhang, S Guo, L Zhang, F Gao, M Jia, Z Zhou, Phys. Medica. 95Y. Wu, L. Zhang, S. Guo, L. Zhang, F. Gao, M. Jia, and Z. Zhou, "Enhanced phase retrieval via deep concatenation networks for in-line X-ray phase contrast imaging," Phys. Medica 95, 41-49 (2022). A practical algorithm for the determination of phase from image and diffraction plane pictures. R W Gerchberg, W O Saxton, Optik. 35R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972). Reconstruction of an object from the modulus of its Fourier transform. J R Fienup, Opt. Lett. 327J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," Opt. Lett. 3, 27 (1978). Phase retrieval algorithms: A comparison. J R Fienup, Appl. Opt. 212758J. R. Fienup, "Phase retrieval algorithms: A comparison," Appl. Opt. 21, 2758 (1982). Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization. H H Bauschke, P L Combettes, D R Luke, JOSA A. 19H. H. Bauschke, P. L. Combettes, and D. R. Luke, "Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization," JOSA A 19, 1334-1345 (2002). Modified Gerchberg-Saxton (G-S) Algorithm and Its Application. T Zhao, Y Chi, Entropy. 221354T. Zhao and Y. Chi, "Modified Gerchberg-Saxton (G-S) Algorithm and Its Application," Entropy 22, 1354 (2020). Trace structure analysis, ptychography, phase tomography. W Hoppe, Ultramicroscopy. 10W. Hoppe, "Trace structure analysis, ptychography, phase tomography," Ultramicroscopy 10, 187-198 (1982). Movable Aperture Lensless Transmission Microscopy: A Novel Phase Retrieval Algorithm. H M L Faulkner, J M Rodenburg, Phys. Rev. Lett. 9323903H. M. L. Faulkner and J. M. Rodenburg, "Movable Aperture Lensless Transmission Microscopy: A Novel Phase Retrieval Algorithm," Phys. Rev. Lett. 93, 023903 (2004). A phase retrieval algorithm for shifting illumination. J M Rodenburg, H M L Faulkner, Appl. Phys. Lett. 85J. M. Rodenburg and H. M. L. Faulkner, "A phase retrieval algorithm for shifting illumination," Appl. Phys. Lett. 85, 4795-4797 (2004). Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy. H Faulkner, J Rodenburg, Ultramicroscopy. 103H. Faulkner and J. Rodenburg, "Error tolerance of an iterative phase re- trieval algorithm for moveable illumination microscopy," Ultramicroscopy 103, 153-164 (2005). Ptychography and Related Diffractive Imaging Methods. J Rodenburg, Advances in Imaging and Electron Physics. Elsevier150J. Rodenburg, "Ptychography and Related Diffractive Imaging Methods," in Advances in Imaging and Electron Physics, vol. 150 (Elsevier, 2008), pp. 87-184. Influence of the overlap parameter on the convergence of the ptychographical iterative engine. O Bunk, M Dierolf, S Kynde, I Johnson, O Marti, F Pfeiffer, Ultramicroscopy. 108O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, "Influence of the overlap parameter on the convergence of the ptycho- graphical iterative engine," Ultramicroscopy 108, 481-487 (2008). Transmission microscopy without lenses for objects of unlimited size. J Rodenburg, A Hurst, A Cullis, Ultramicroscopy. 107J. Rodenburg, A. Hurst, and A. Cullis, "Transmission microscopy with- out lenses for objects of unlimited size," Ultramicroscopy 107, 227-231 (2007). Superresolution imaging via ptychography. A M Maiden, M J Humphry, F Zhang, J M Rodenburg, J. Opt. Soc. Am. A. 28604A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, "Su- perresolution imaging via ptychography," J. Opt. Soc. Am. A 28, 604 (2011). An improved ptychographical phase retrieval algorithm for diffractive imaging. A M Maiden, J M Rodenburg, Ultramicroscopy. 109A. M. Maiden and J. M. Rodenburg, "An improved ptychographical phase retrieval algorithm for diffractive imaging," Ultramicroscopy 109, 1256-1262 (2009). Ptychographic transmission microscopy in three dimensions using a multi-slice approach. A M Maiden, M J Humphry, J M Rodenburg, J. Opt. Soc. Am. A. 291606A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, "Ptychographic transmission microscopy in three dimensions using a multi-slice ap- proach," J. Opt. Soc. Am. A 29, 1606 (2012). Combining ptychographical algorithms with the Hybrid Input-Output (HIO) algorithm. A Konijnenberg, W Coene, S Pereira, H Urbach, Ultramicroscopy. 171A. Konijnenberg, W. Coene, S. Pereira, and H. Urbach, "Combining pty- chographical algorithms with the Hybrid Input-Output (HIO) algorithm," Ultramicroscopy 171, 43-54 (2016). Wide-field, high-resolution Fourier ptychographic microscopy. G Zheng, R Horstmeyer, C Yang, Nat. Photonics. 7G. Zheng, R. Horstmeyer, and C. Yang, "Wide-field, high-resolution Fourier ptychographic microscopy," Nat. Photonics 7, 739-745 (2013). Experimental robustness of Fourier ptychography phase retrieval algorithms. L.-H Yeh, J Dong, J Zhong, L Tian, M Chen, G Tang, M Soltanolkotabi, L Waller, Opt. Express. 23L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, "Experimental robustness of Fourier ptychography phase retrieval algorithms," Opt. Express 23, 33214- 33240 (2015). Symmetric illumination in Fourier ptychography. Zhang Lei-Lei, Tang Li-Jin, Zhang Mu-Yang, Liang Yan-Mei, Acta Phys. Sinica. 66224201Zhang Lei-Lei, Tang Li-Jin, Zhang Mu-Yang, and Liang Yan-Mei, "Sym- metric illumination in Fourier ptychography," Acta Phys. Sinica 66, 224201 (2017). Fourier ptychography: Current applications and future promises. P C Konda, L Loetgering, K C Zhou, S Xu, A R Harvey, R Horstmeyer, Opt. Express. 289603P. C. Konda, L. Loetgering, K. C. Zhou, S. Xu, A. R. Harvey, and R. Horstmeyer, "Fourier ptychography: Current applications and future promises," Opt. Express 28, 9603 (2020). Miscalibration-Tolerant Fourier Ptychography. V Bianco, B Mandracchia, J Bhal, D Barone, P Memmolo, P Ferraro, IEEE J. Sel. Top. Quantum Electron. 27V. Bianco, B. Mandracchia, J. Bhal, D. Barone, P. Memmolo, and P. Ferraro, "Miscalibration-Tolerant Fourier Ptychography," IEEE J. Sel. Top. Quantum Electron. 27, 1-17 (2021). Rapid misalignment correction method in reflective fourier ptychographic microscopy for full field of view reconstruction. H Lee, B Chon, H Ahn, Opt. Lasers Eng. 138106418H. Lee, B. Chon, and H. Ahn, "Rapid misalignment correction method in reflective fourier ptychographic microscopy for full field of view re- construction," Opt. Lasers Eng. 138, 106418 (2021). Parallel ptychographic reconstruction. Y S G Nashed, D J Vine, T Peterka, J Deng, R Ross, C Jacobsen, Opt. Express. 2232082Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacob- sen, "Parallel ptychographic reconstruction," Opt. Express 22, 32082 (2014). Ptychographic coherent diffractive imaging with orthogonal probe relaxation. M Odstrcil, P Baksh, S A Boden, R Card, J E Chad, J G Frey, W S Brocklesby, Opt. Express. 248360M. Odstrcil, P. Baksh, S. A. Boden, R. Card, J. E. Chad, J. G. Frey, and W. S. Brocklesby, "Ptychographic coherent diffractive imaging with orthogonal probe relaxation," Opt. Express 24, 8360 (2016). Further improvements to the ptychographical iterative engine. A Maiden, D Johnson, P Li, Optica. 4736A. Maiden, D. Johnson, and P. Li, "Further improvements to the ptycho- graphical iterative engine," Optica 4, 736 (2017). Broadband X-ray ptychography using multi-wavelength algorithm. Y Yao, Y Jiang, J Klug, Y Nashed, C Roehrig, C Preissner, F Marin, M Wojcik, O Cossairt, Z Cai, S Vogt, B Lai, J Deng, J. Synchrotron Radiat. 28Y. Yao, Y. Jiang, J. Klug, Y. Nashed, C. Roehrig, C. Preissner, F. Marin, M. Wojcik, O. Cossairt, Z. Cai, S. Vogt, B. Lai, and J. Deng, "Broadband X-ray ptychography using multi-wavelength algorithm," J. Synchrotron Radiat. 28, 309-317 (2021). Probe retrieval in ptychographic coherent diffractive imaging. P Thibault, M Dierolf, O Bunk, A Menzel, F Pfeiffer, Ultramicroscopy. 109P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, "Probe retrieval in ptychographic coherent diffractive imaging," Ultramicroscopy 109, 338-343 (2009). Optical ptychography: A practical implementation with useful resolution. A M Maiden, J M Rodenburg, M J Humphry, Opt. Lett. 352585A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, "Optical ptychog- raphy: A practical implementation with useful resolution," Opt. Lett. 35, 2585 (2010). Dual wavelength optical metrology using ptychography. D Claus, D J Robinson, D G Chetwynd, Y Shuo, W T Pike, J J De J Toriz, J M Garcia, Rodenburg, J. Opt. 1535702D. Claus, D. J. Robinson, D. G. Chetwynd, Y. Shuo, W. T. Pike, J. J. De J Toriz Garcia, and J. M. Rodenburg, "Dual wavelength optical metrology using ptychography," J. Opt. 15, 035702 (2013). Translation position determination in ptychographic coherent diffraction imaging. F Zhang, I Peterson, J Vila-Comamala, A Diaz, F Berenguer, R Bean, B Chen, A Menzel, I K Robinson, J M Rodenburg, Opt. Express. 2113592F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, "Translation position determination in ptychographic coherent diffraction imaging," Opt. Express 21, 13592 (2013). Ptychographic microscope for three-dimensional imaging. T M Godden, R Suman, M J Humphry, J M Rodenburg, A M Maiden, Opt. Express. 2212513T. M. Godden, R. Suman, M. J. Humphry, J. M. Rodenburg, and A. M. Maiden, "Ptychographic microscope for three-dimensional imaging," Opt. Express 22, 12513 (2014). Imaging Correlography Using Ptychography. Wen Li, Song, Jiang, Liu Zhang, Wei , Appl. Sci. 94377Li, Wen, Song, Jiang, Zhang, Liu, and Wei, "Imaging Correlography Using Ptychography," Appl. Sci. 9, 4377 (2019). Single-shot ptychography with highly tilted illuminations. C Chang, X Pan, H Tao, C Liu, S P Veetil, J Zhu, Opt. Express. 2828441C. Chang, X. Pan, H. Tao, C. Liu, S. P. Veetil, and J. Zhu, "Single-shot ptychography with highly tilted illuminations," Opt. Express 28, 28441 (2020). Hard-X-Ray Lensless Imaging of Extended Objects. J M Rodenburg, A C Hurst, A G Cullis, B R Dobson, F Pfeiffer, O Bunk, C David, K Jefimovs, I Johnson, Phys. Rev. Lett. 9834801J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, "Hard-X-Ray Lensless Imaging of Extended Objects," Phys. Rev. Lett. 98, 034801 (2007). High-Resolution Scanning X-ray Diffraction Microscopy. P Thibault, M Dierolf, A Menzel, O Bunk, C David, F Pfeiffer, Science. 321P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, "High-Resolution Scanning X-ray Diffraction Microscopy," Science 321, 379-382 (2008). Ptychographic X-ray computed tomography at the nanoscale. M Dierolf, A Menzel, P Thibault, P Schneider, C M Kewish, R Wepf, O Bunk, F Pfeiffer, Nature. 467M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, "Ptychographic X-ray computed tomography at the nanoscale," Nature 467, 436-439 (2010). Ptychographic coherent diffractive imaging of weakly scattering specimens. M Dierolf, P Thibault, A Menzel, C M Kewish, K Jefimovs, I Schlichting, K König, O Bunk, F Pfeiffer, New J. Phys. 1235017M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlicht- ing, K. von König, O. Bunk, and F. Pfeiffer, "Ptychographic coherent diffractive imaging of weakly scattering specimens," New J. Phys. 12, 035017 (2010). Sampling in x-ray ptychography. T B Edo, D J Batey, A M Maiden, C Rau, U Wagner, Z D Pešić, T A Waigh, J M Rodenburg, Phys. Rev. A. 8753850T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, "Sampling in x-ray ptychography," Phys. Rev. A 87, 053850 (2013). Soft X-ray spectromicroscopy using ptychography with randomly phased illumination. A Maiden, G Morrison, B Kaulich, A Gianoncelli, J Rodenburg, Nat. Commun. 41669A. Maiden, G. Morrison, B. Kaulich, A. Gianoncelli, and J. Roden- burg, "Soft X-ray spectromicroscopy using ptychography with randomly phased illumination," Nat. Commun. 4, 1669 (2013). X-Ray Near-Field Ptychography for Optically Thick Specimens. M Stockmar, I Zanette, M Dierolf, B Enders, R Clare, F Pfeiffer, P Cloetens, A Bonnin, P Thibault, Phys. Rev. Appl. 314005M. Stockmar, I. Zanette, M. Dierolf, B. Enders, R. Clare, F. Pfeiffer, P. Cloetens, A. Bonnin, and P. Thibault, "X-Ray Near-Field Ptychogra- phy for Optically Thick Specimens," Phys. Rev. Appl. 3, 014005 (2015). X-ray ptychography using randomized zone plates. G R Morrison, F Zhang, A Gianoncelli, I K Robinson, Opt. Express. 2614915G. R. Morrison, F. Zhang, A. Gianoncelli, and I. K. Robinson, "X-ray ptychography using randomized zone plates," Opt. Express 26, 14915 (2018). X-ray ptychography. F Pfeiffer, Nat. Photonics. 12F. Pfeiffer, "X-ray ptychography," Nat. Photonics 12, 9-17 (2018). Three-dimensional imaging of integrated circuits with macro-to nanoscale zoom. M Holler, M Odstrcil, M Guizar-Sicairos, M Lebugle, E Müller, S Finizio, G Tinti, C David, J Zusman, W Unglaub, O Bunk, J Raabe, A F J Levi, G Aeppli, Nat. Electron. 2M. Holler, M. Odstrcil, M. Guizar-Sicairos, M. Lebugle, E. Müller, S. Finizio, G. Tinti, C. David, J. Zusman, W. Unglaub, O. Bunk, J. Raabe, A. F. J. Levi, and G. Aeppli, "Three-dimensional imaging of integrated circuits with macro-to nanoscale zoom," Nat. Electron. 2, 464-470 (2019). Multi-slice ptychography enables high-resolution measurements in extended chemical reactors. M Kahnt, L Grote, D Brückner, M Seyrich, F Wittwer, D Koziej, C G Schroer, Sci. Reports. 111500M. Kahnt, L. Grote, D. Brückner, M. Seyrich, F. Wittwer, D. Koziej, and C. G. Schroer, "Multi-slice ptychography enables high-resolution measurements in extended chemical reactors," Sci. Reports 11, 1500 (2021). Electron Ptychography. I. Experimental Demonstration Beyond the Conventional Resolution Limits. P D Nellist, J M Rodenburg, Acta Crystallogr. Sect. A Foundations Crystallogr. 54P. D. Nellist and J. M. Rodenburg, "Electron Ptychography. I. Exper- imental Demonstration Beyond the Conventional Resolution Limits," Acta Crystallogr. Sect. A Foundations Crystallogr. 54, 49-60 (1998). Atomic Structure Imaging Beyond Conventional Resolution Limits in the Transmission Electron Microscope. S J Haigh, H Sawada, A I Kirkland, Phys. Rev. Lett. 103126101S. J. Haigh, H. Sawada, and A. I. Kirkland, "Atomic Structure Imaging Beyond Conventional Resolution Limits in the Transmission Electron Microscope," Phys. Rev. Lett. 103, 126101 (2009). Wave-front phase retrieval in transmission electron microscopy via ptychography. F Hüe, J M Rodenburg, A M Maiden, F Sweeney, P A Midgley, Phys. Rev. B. 82121415F. Hüe, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, "Wave-front phase retrieval in transmission electron microscopy via ptychography," Phys. Rev. B 82, 121415 (2010). Probe position recovery for ptychographical imaging. A C Hurst, T B Edo, T Walther, F Sweeney, J M Rodenburg, J. Physics: Conf. Ser. 24112004A. C. Hurst, T. B. Edo, T. Walther, F. Sweeney, and J. M. Rodenburg, "Probe position recovery for ptychographical imaging," J. Physics: Conf. Ser. 241, 012004 (2010). Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging. M Humphry, B Kraus, A Hurst, A Maiden, J Rodenburg, Nat. Commun. 3730M. Humphry, B. Kraus, A. Hurst, A. Maiden, and J. Rodenburg, "Pty- chographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging," Nat. Commun. 3, 730 (2012). Contrast transfer and noise considerations in focusedprobe electron ptychography. C M O&apos;leary, G T Martinez, E Liberti, M J Humphry, A I Kirkland, P D Nellist, Ultramicroscopy. 221113189C. M. O'Leary, G. T. Martinez, E. Liberti, M. J. Humphry, A. I. Kirkland, and P. D. Nellist, "Contrast transfer and noise considerations in focused- probe electron ptychography," Ultramicroscopy 221, 113189 (2021). Optical image encryption via ptychography. Y Shi, T Li, Y Wang, Q Gao, S Zhang, H Li, Opt. Lett. 381425Y. Shi, T. Li, Y. Wang, Q. Gao, S. Zhang, and H. Li, "Optical image encryption via ptychography," Opt. Lett. 38, 1425 (2013). Optical image encryption via photon-counting imaging and compressive sensing based ptychography. N Rawat, I.-C Hwang, Y Shi, B.-G Lee, J. Opt. 1765704N. Rawat, I.-C. Hwang, Y. Shi, and B.-G. Lee, "Optical image encryption via photon-counting imaging and compressive sensing based ptychog- raphy," J. Opt. 17, 065704 (2015). High-capacity encryption system based on single-shot-ptychography encoding and QR code. Y Zhu, W Xu, Y Shi, Opt. Commun. 435Y. Zhu, W. Xu, and Y. Shi, "High-capacity encryption system based on single-shot-ptychography encoding and QR code," Opt. Commun. 435, 426-432 (2019). Phase and amplitude imaging with quantum correlations through Fourier Ptychography. T Aidukas, P C Konda, A R Harvey, M J Padgett, P.-A Moreau, Sci. Reports. 910445T. Aidukas, P. C. Konda, A. R. Harvey, M. J. Padgett, and P.-A. Moreau, "Phase and amplitude imaging with quantum correlations through Fourier Ptychography," Sci. Reports 9, 10445 (2019). Ptychography of pure quantum states. M F Fernandes, L Neves, Sci. Reports. 916066M. F. Fernandes and L. Neves, "Ptychography of pure quantum states," Sci. Reports 9, 16066 (2019). Ptychographic reconstruction of pure quantum states. M F Fernandes, M A Solís-Prosser, L Neves, Opt. Lett. 456002M. F. Fernandes, M. A. Solís-Prosser, and L. Neves, "Ptychographic reconstruction of pure quantum states," Opt. Lett. 45, 6002 (2020). Research on the key parameters of illuminating beam for imaging via ptychography in visible light band. Wang Ya-Li, Shi Yi-Shi, Li Tuo, Gao Qian-Kun, Xiao Jun, Zhang San-Guo, Acta Phys. Sinica. 6264206Wang Ya-Li, Shi Yi-Shi, Li Tuo, Gao Qian-Kun, Xiao Jun, and Zhang San-Guo, "Research on the key parameters of illuminating beam for imaging via ptychography in visible light band," Acta Phys. Sinica 62, 064206 (2013). Optimization of overlap uniformness for ptychography. X Huang, H Yan, R Harder, Y Hwu, I K Robinson, Y S Chu, Opt. Express. 2212634X. Huang, H. Yan, R. Harder, Y. Hwu, I. K. Robinson, and Y. S. Chu, "Optimization of overlap uniformness for ptychography," Opt. Express 22, 12634 (2014). Single-shot ptychography. P Sidorenko, O Cohen, Optica. 39P. Sidorenko and O. Cohen, "Single-shot ptychography," Optica 3, 9 (2016). Invariant error metrics for image reconstruction. J R Fienup, Appl. Opt. 368352J. R. Fienup, "Invariant error metrics for image reconstruction," Appl. Opt. 36, 8352 (1997).
[]
[ "ELECTRICITY INTRADAY PRICE MODELING WITH MARKED HAWKES PROCESSES", "ELECTRICITY INTRADAY PRICE MODELING WITH MARKED HAWKES PROCESSES" ]
[ "Thomas Deschatre ", "Pierre Gruet " ]
[]
[]
We consider a 2-dimensional marked Hawkes process with increasing baseline intensity in order to model prices on electricity intraday markets. This model allows to represent different empirical facts such as increasing market activity, random jump sizes but above all microstructure noise through the signature plot. This last feature is of particular importance for practitioners and has not yet been modeled on those particular markets. We provide analytic formulas for first and second moments and for the signature plot, extending the classic results of Bacry et al.[2]in the context of Hawkes processes with random jump sizes and time dependent baseline intensity. The tractable model we propose is estimated on German data and seems to fit the data well. We also provide a result about the convergence of the price process to a Brownian motion with increasing volatility at macroscopic scales, highlighting the Samuelson effect.Mathematics Subject Classification (2020): 60G55, 62M10, 91G30
10.1080/1350486x.2023.2180399
[ "https://arxiv.org/pdf/2103.07407v2.pdf" ]
232,222,868
2103.07407
88d98de774afbd6969aa3be7efb9f66af1cc2a15
ELECTRICITY INTRADAY PRICE MODELING WITH MARKED HAWKES PROCESSES Thomas Deschatre Pierre Gruet ELECTRICITY INTRADAY PRICE MODELING WITH MARKED HAWKES PROCESSES Electricity intraday pricesMicrostructure noiseHawkes processesHigh-frequency statistics We consider a 2-dimensional marked Hawkes process with increasing baseline intensity in order to model prices on electricity intraday markets. This model allows to represent different empirical facts such as increasing market activity, random jump sizes but above all microstructure noise through the signature plot. This last feature is of particular importance for practitioners and has not yet been modeled on those particular markets. We provide analytic formulas for first and second moments and for the signature plot, extending the classic results of Bacry et al.[2]in the context of Hawkes processes with random jump sizes and time dependent baseline intensity. The tractable model we propose is estimated on German data and seems to fit the data well. We also provide a result about the convergence of the price process to a Brownian motion with increasing volatility at macroscopic scales, highlighting the Samuelson effect.Mathematics Subject Classification (2020): 60G55, 62M10, 91G30 Introduction In Europe, The electricity intraday markets are quickly developing as they meet increasing trading needs from the actors in the field: according to EPEX SPOT, which runs those markets in Western Europe, the German yearly exchanged volume has grown from 2 TWh in 2008 to 54 TWh in 2019. To a large extent, this development is caused by the increasing share of renewable power plants in the energy system: the output of such plants is hard to forecast (one may refer to Giebel et al. [11] for a thorough review on this topic), so that their owners face imbalances between their production and the consumption in their perimeter. For each delivery hour (24 in a day), an auction called "day-ahead market" is held at noon the day before: it reflects the supply-demand equilibrium that holds, based on the anticipations of market agents. Yet, those agents can also suffer from random events happening between the day-ahead auction and the electricity delivery, so that they need to rebalance their positions before the delivery. Electricity intraday markets provide them with a way to trade deliveries of electricity for the upcoming hours in order to reduce their imbalances. As regulation varies across market places, the intraday markets may change a bit from one country to another. They however share common features: they offer the possibility to exchange power for the next, say, 9 to 32 hours (in Germany, for instance). Three hours after the day-ahead market is cleared, the intraday market for the next day opens. Then trades occur as in the regular order book markets that are well known to finance practitioners. It is possible to trade up to 5 minutes before the physical delivery of electricity actually occurs. The traded products are much standardized: for instance in Germany, one may exchange hourly, half-hourly and quarter-hourly contracts for, respectively, the 24 hours, the 48 half-hours and the 96 quarters of an hour that exist each day. Because the electricity markets expanded quite lately compared to more conventional financial markets, practitioners in the former already benefit from the modeling tools developed by academic researchers and practitioners in the latter. Yet the nature of the underlying, which is not storable at large scale, has motivated the development of some modeling artifacts that are specific to the field. Kiesel and Paraschiv [17] look at the fundamental drivers of intraday prices: they can understand how the intraday prices react to, for instance, errors in the renewable production forecast. In order to gain some more insight about the endogeneous mechanisms driving the electricity intraday markets, Narajewski and Ziel [18] examine the transaction arrival times in this market to understand the behavior of market participants and to describe the increase in the transaction rhythm as maturity gets closer, which is a largely observed stylized fact in the intraday market. Aïd et al. [1] determine the optimal strategy of a trader aiming at reducing her imbalance thanks to the intraday market, additionally featuring market impact leading to additional transaction costs. Recently, Hawkes processes have proved to be relevant modeling tools in electricity intraday markets. For an extensive review on those processes, the interested reader may refer to Bacry et al. [4]. They are well known to finance practitioners as they allow to represent self-exciting behaviors and clustering of events; for instance Gao et al. [10] use them to model the trades arrival in dark pool trading. Back to the context of electricity intraday markets, Favetto [9] fits an univariate Hawkes process to describe the market activity of a product in the intraday market. Graf von Luckner and Kiesel [12] work with a 2-dimensional Hawkes process in the intraday market. Our model shares a lot of features with the model of Graf von Luckner and Kiesel [12]: we also use a 2-dimensional Hawkes process with exponential baseline function and exponential excitation functions to define the positive and the negative jumps of prices in the intraday market. Within this model, the occurrence of positive jumps has an influence on the intensity of negative jumps. The same holds for the occurrence of negative jumps, which have an influence on the intensity of positive jumps. Whereas Graf von Luckner and Kiesel [12] focus on the statistical selection of models to describe the occurrence of jumps at best, we choose to go one step further on the modeling path by also selecting the distribution of the price after a thorough statistical analysis, and then we define the price on the intraday market as being the sum of positive and negative jumps triggered by the 2-dimensional marked Hawkes process. Being able to model the price and analyse its theoretical features is a great improvement for the practitioner. A very interesting empirical fact captured by our model is the presence of microstructure noise. We generalize the signature plot formulas of Bacry et al. [2] to the case where the baseline intensity is not constant and the Hawkes processes are marked. We also extend the results of Bacry et al. [3] concerning limit theorems at macroscopic scale to our framework. To our knowledge, we are proposing the first price model for electricity intraday prices featuring microstructure effects, with a focus on sticking to the actual signature plot, which represents the realized volatility of prices against the sampling frequency. We care for the statistical properties of the price process: the computation of the first and second moments allows to compare the theoretical signature plots to the ones we compute on the market. By doing so, we provide some evidence of patterns that are well known in classic financial markets, and we reproduce them within the model. This is especially interesting as the intraday market of electricity is rather uncommon in the landscape of financial markets. We also examine the behavior of prices in a low-frequency asymptotics. The outline of the paper is organized as follows. In Section 2, we perform a statistical analysis to show some evidence of the increase of the intensity of jumps over the trading sessions. We also study the distribution of jumps sizes, and we look at the signature plot computed on market data. In Section 3, we explain how the features we examine in Section 2 translate into modeling choices. We also state the theoretical moments of the price and the signature plot formula within our model. We then estimate the model on German data and we study the adequacy of the model to the data. Section 4 examines the behavior of prices at the macroscopic scale and the representation of the Samuelson effect, according to which the prices tend to be more volatile as time to maturity decreases. Section 5 contains the conclusion, and proofs of our propositions are deferred to Sections 6 and 7. Empirical stylized facts In this section, we first describe the dataset and some empirical stylized facts which we want to reproduce within our model. These facts are mainly related to the jumps one may observe in the intraday prices: we will estimate their arrival intensity and comment on the distributions that may fit the sizes of the positive and negative price jumps. Then we will identify the presence of microstructure noise through a signature plot of electricity intraday prices. These features are of particular interest from the perspective of a practitioner, as they provide guidance for modeling in order to trade on the intraday market. 2.1. Description of the dataset. We consider German electricity intraday mid-prices (arithmetic average between bid and ask prices) between July and September 2017 for products with a delivery period of one hour. The mid-prices are built using order book data from EPEX Spot, which is the organizer of the short-term market in Western Europe. There is one time series per trading session and delivery product. Let us recall that one trading session starts at 3 pm the day before maturity and ends 5 minutes before maturity for hourly products, that are the only ones we consider here. Therefore, the duration of observation changes from a maturity to another. Consider one trading session: each time an order is inserted, we update the mid-price. Prices are then given with a precision up to the millisecond. To simplify the different numerical computations, we build the prices data set with a time step of one second, which is enough for our study. If there are one or several price changes over one second, the average volume weighted price is taken for that second. One hour before delivery, cross-border trading is not possible anymore. Thirty minutes before delivery, transactions are only possible into each of the four control areas in Germany and not across them, meaning the bidding zone then becomes smaller than before entering this 30 minutes delay. Those changes in the market can have an impact on the liquidity and then in the price behavior. For this reason, we only consider data up to one hour before maturity. Furthermore, as the price is not defined at the beginning of the session (at least one buy and one sell order are needed), the price we register at 3 pm is the first existing price (which is thus defined in the future). This does not affect the results as most of them concern the returns and times of price changes. Figure 1 represents prices for deliveries beginning at 18h, 19h and 20h on July 11 th , 2017 and on August 30 th , 2017. In the rest of the paper, we focus on those three maturities but only for a better visualization and understanding: results are robust to the choice of the maturity (with a delivery period of one hour). 2.2. Arrival times of price changes distribution. In this section, we study statistical properties of the price changing times, that is when there is an upward or downward move in the price. Intensity is first studied, followed by the (non) adequacy of the arrival times to an inhomogeneous Poisson process. Increasing intensity. Looking at Figure 1, the first empirical finding is that market activity increases when trading time approaches maturity. Let N c t be the number of price changes between 0 and t. Figure 2 represents the path of (N c t ) t for one trading session but also the average for all trading sessions. N c t is an estimator of the cumulative intensity up to time t. If the intensity were constant, (N c t ) t would be a linear function of time. Let (λ c t ) t be the intensity associated to the jump process (N c t ) t . We pointly estimate (λ c t ) t using the kernel estimator λ c t = T 0 K h (t − s)dN c s T 0 K h (t − s)ds with T the last date of observation (that is one hour before maturity), K h (u) = 1 h K( u h ) for u ∈ R, h > 0 and K a kernel function. The denominator allows to avoid boundary effects when the number of data diminishes. Figure 3 shows the intensity process for a single trading session and the average of all the intensity processes using the Epanechnikov kernel K(u) = 3 4 (1−u 2 )1 |u|≤1 and the bandwidth h = 300 seconds. Typically, market activity is almost null during the first hours of the trading session, and at some point, intensity begins increasing exponentially as trading time comes closer to maturity. The quality of renewable production and consumption forecasts improves only a few hours before the delivery, so that market actors do not really have enough information to reduce their imbalances during the first hours of the trading session. Those empirical findings are consistent with the ones of [9,12] showing an increasing market activity in the order book with trading time. Non Poissonian arrival times. Let us assume that (N c t ) t is an inhomogeneous Poisson process with deterministic intensity (λ c t ) t . Thus, the variables Λ c (τ c i ) − Λ c (τ c i−1 ), where Λ c (t) = t 0 λ c u du and (τ c i ) i≥0 are the jump times of N c , are identically and independently distributed (i.i.d.) and follow an exponential distribution with parameter 1. It is then possible to compute the quantile-quantile plot of those variables against an exponential distribution, see Figure 4 for the quantile-quantile plot corresponding to one trading session and maturity 18h. Results are similar for maturities 19h and 20h. The function Λ c at time t is estimated considering the empirical average of N c t across the different days. The process is clearly not an inhomogeneous Poisson process, which argues in favor of a more sophisticated modeling. While very liquid markets present tick by tick data, that is prices moving by one tick up or down, the electricity intraday market features jumps with different sizes. The purpose of this section is to study the distribution of the jump heights. We distinguish positive and negative jumps. When we study the negative jumps, we look at their absolute value. The distributions of positive and negative jumps are represented in Figure 5 (left column) for maturities 18h, 19h and 20h. First, we can observe jump sizes up to 100 e/MWh. Those jumps are probably triggered by the lack of liquidity at the beginning of the trading session. As noticed by Balardy [5], the bid-ask spread is large when time is far from maturity. One order can then move the price strongly. If we consider only price moves less than 9 hours before maturity, the jump sizes are lower as seen in the histogram in Figure 5 (right column). Figure 6 represents the mean and standard deviation of jump sizes when we stick to the jumps happening less than x hours before the delivery starts, x being Confidence intervals for first and second order moments of negative and positive price jumps at level 95% for the different maturities and considering jumps happening less than 9 hours before maturity the value on the x-axis. The mean and the standard deviation decrease when time approaches maturity, which is consistent with the increasing liquidity and decreasing bid-ask spread. There is a stabilization of those two moments between 8 and 10 hours before maturity. From now on, and in the rest of the paper, particular attention is paid to the price distribution starting 9 hours before maturity. Second, we can observe that positive and negative jumps seem to have a similar distribution in Figure 5. The Kolmogorov-Smirnov test to compare the two distributions, with null hypothesis being that the two sequences of samples are drawn for the same distribution [14], is not rejected at level 95% for maturity 18h, but it is for maturities 19h and 20h, considering the jump sizes from 9 hour before maturity. Performing this test on each day, it is not rejected for maturity 18h in 64.1 % of the days in the dataset, 67.4 % of them for maturity 19h and 77.2 % of them for maturity 20h, at level 95%. Results are similar considering the whole trading session. Table 1 gives confidence intervals for first and second orders moments at level 95% for positive and negative jumps happening less than 9 hours before maturity. Intervals linked to positive and negative jumps intersect for both moments. The same table for the whole trading session could be displayed but it gives very wide intervals for the second order moments caused by the very large jumps. While we cannot perfectly conclude on the equality in distribution of positive and negative jumps, it seems to be a reasonable assumption if modeling the trading session starting 9 hours before maturity, especially as properties linked to second order moments will be of particular interest in our model. 2.4. Signature plot. One of the main interesting facts concerning high-frequency data studied those last years is the presence of microstructure effects and in particular the existence of some patterns in the signature plot: the estimated realized volatilitŷ C(T, δ) = 1 T T δ i=1 (f iδ − f (i−1)δ ) 2 , δ > 0, decreases with the estimation frequency, where (f t ) t is the price process observed between 0 and T , and δ represents the estimation time step (inverse of the frequency). While it is natural to consider the largest frequency to estimate quadratic variation with a high precision, the estimator can increase a lot and be unstable for large frequencies. The signature plot is the function C(T, δ) = E Ĉ (T, δ) , δ > 0. This phenomenon is called microstructure noise and it is caused by the fact that at high frequencies, prices have a mean reverting behavior ; a positive jump is often followed by a negative one and this round trip is not observed at lower frequencies. The objective is then to have a consistent model (a) All trading sessions, maturity 18h (b) 9 hours before maturity, maturity 18h (c) All trading sessions, maturity 19h (d) 9 hours before maturity, maturity 19h (e) All trading sessions, maturity 20h (f) 9 hours before maturity, maturity 20h Figure 5. Positive and negative jump size distributions with a log scale on the y-axis, for maturities 18h, 19h and 20h for all the trading sessions (left), keeping only jumps that happen less than 9 hours before maturity (right) that can explain this volatility change: this is not the case if we consider a Brownian motion which is invariant by scale change and has a flat signature plot. The signature plot for the electricity intraday prices is represented in Figure 7 considering the price process less than 9 hours before maturity for one trading session (left) and on average (right). This shape is common in the highfrequency finance literature, with a fast decrease when frequency goes high and a stabilization as it becomes low. Yet, to our knowledge, we are the first to exhibit such behavior for electricity Price model Let us consider the sequence of arrival times 0 < τ 1 < τ 2 < . . . defined on a rich enough probability space (Ω, F, P) endowed with a right continuous and complete filtration (F t ) t≥0 . Let (J i ) i≥1 be a sequence of positive i.i.d. random variables such that J i is F τi measurable for i ≥ 1. We assume that they have the same law as a random variable J defined on (Ω, F, P) with E(J 2 ) < ∞. Marking the arrival times (τ i ) i , one can construct two sequences (τ + i ) i and (τ − i ) i , associated with two sequences of jump sizes (J + i ) i and (J − i ) i , and define the bivariate point process N = (N + , N − ) with N + t = ∞ n=1 1 τ + n ≤t , N − t = ∞ n=1 1 τ − n ≤t . From now on, we work on the finite time horizon [0, T ], T > 0. We model the point process N as a bivariate Hawkes process whith intensity depending on the marks (J i ) i : (1) λ + t λ − t = µ t T 1 1 + t 0 ϕ(t − s) J s dN + s J s dN − s with µ : [0, 1] → R + a non decreasing bounded function and ϕ : R + → R 2,2 a locally bounded function with positive components such that ρ(K) < 1, where K = E(J) ∞ 0 |ϕ(u)|du and ρ(K) is the spectral radius of K. Recall that the standard notation t 0 h(s)J s dN + s (resp. t 0 h(s)J s dN − s ) for a measurable function h stands for N + t i=1 J + i h(τ + i ) (resp. N + t i=1 J − i h(τ − i ) ). Let us also consider the marked version of N , (f + , f − ) , that models the upward and downward jumps in the price by (2) f + t f − t = t 0 J s dN + s J s dN − s . The price is then given by (3) f t = f 0 + f + t − f − t with f 0 ∈ R the initial value of the price. Remark 1. The marked Hawkes process could also be defined using random measure notations, see Jacod and Shiryaev [15,Chapter 2]. Let N (dt, dx) = (N + (dt, dx), N − (dt, dx)) be the 2- dimensional random Poisson measure with compensator ν(dt, dx) = λ t dt ⊗ µ J (dx) on [0, T ] × R + where µ J (dx) is a probability measure representing the law of J and λ t = µ t T 1 1 + t 0 R+ xϕ(t − s)N (ds, dx). The components of the price process are then defined by f + t f − t = t 0 R+ xN (ds, dx) and f t = f 0 + t 0 R+ x N + (ds, dx) − N − (ds, dx) . One can then naturally define the integral, for a measurable function h, t 0 R+ h(s, x)N (ds, dx) = N + t i=1 h(τ + i , J + i ) N − t i=1 h(τ − i , J − i ) . While this notation can seem more convenient than the notation J s dN s , we prefer keeping the latter as it is more used in financial price modeling or insurance risk modeling. A priori, our model allows to represent the different empirical facts identified on data in Section 2: (i) the baseline intensity depends on time, in order to model the increasing market activity over a trading session identified in Section 2.2; (ii) the Hawkes modeling encompasses the inhomogeneous Poisson framework which does not fit the data well, see Section 2.2; (iii) positive and negative jump heights in price are random with the same distribution, see Section 2.3; (iv) the mutual excitation between positive and negative jumps is expected to represent the signature plot described in Section 2.4 well, as in Bacry et al. [2]. Remark 2. The time in the baseline function is normalized by the time horizon T . The normalization allows to have a bounded intensity even when the time horizon becomes large. It is then possible to study the limit behavior of the model when T → ∞ (macroscopic scale), which is done in Sections 3.1 and 4. This normalization is used by Duval and Hoffmann [7] to study the limit behavior of an inhomogeneous compound Poisson process when T → ∞. We choose to set the model as in Assumption 3 in order to have analytical and tractable formulas while still allowing to represent the different empirical facts. Our model presents two main differences with the classic Hawkes model used by Bacry et al. [2], which are the presence of random jumps, both in prices and intensities, and the time dependent baseline intensity. This parameterization is the same as the one in Graf von Luckner and Kiesel [12] to model the order book activity. The mid-price considered here being strongly related to the order book activity, the empirical study in [12] is in favor of the choice of the parameterization 3. The main difference made by our model is the presence of the jump size in the intensity and the null diagonal in ϕ : this last assumption allows us to have a tractable model with closed formulas for the moments and the signature plot, as shown in Section 3.1. Assumption 3. The baseline intensity and the excitation function are given by (i) µ(t) = µ 0 e κt with µ 0 , κ > 0; (ii) ϕ = 0 ϕ exp ϕ exp 0 with ϕ exp : t → αe −βt , α, β > 0 ; the condition ρ(K) < 1 becomes αE(J) < β. 3.1. Theoretical properties. In this section, one confirms theoretically that our model is able to feature an increasing market activity, by computing E(λ + t + λ − t ) using Proposition 4, and that it can provide a good signature plot representation using Proposition 6, under Assumption 3. First and second order moment properties for the price are also given in Proposition 4 and Proposition 5. Proposition 4 gives the expectation of the positive and negative price changes together with the expectation of the intensities. Proof is given in Section 6.2. The market activity intensity is equal to E(λ + t + λ − t ), which is increasing exponentially with time at a rate κ T : our model can reproduce the increase in market activity intensity. E(f + t ) = E(f − t ) = µ 0 E(J) β + κ T κ T β − αE(J) + κ T e κ t T + αE(J) (β − αE(J)) β − αE(J) + κ T e −(β−αE(J))t − β κ T (β − αE(J) ) and E(λ + t ) = E(λ − t ) = µ 0 β + κ T β − αE(J) + κ T e κ t T − αE(J) β − αE(J) + κ T e −(β−αE(J))t . Proposition 5 gives the second order moment of the price. Proof is given in Section 6.3. E(f 2 t ) = f 2 0 + 2µ 0 E(J 2 ) (C 1 + C 2 + C 3 + C 4 ) e κ t T − C 1 e −(β−αE(J))t −C 2 e −2(β+αE(J))t − C 3 e −(β+αE(J))t − C 4 with C 1 = −α 2 E(J) 2 (β − αE(J)) (β + 3αE(J)) β − αE(J) + κ T , C 2 = α 2 E(J) 2 (β + 2αE(J)) (β + αE(J)) 2 (β + 3αE(J)) 2β + 2αE(J) + κ T , C 3 = αβE(J) (β + αE(J)) 2 β + αE(J) + κ T , C 4 = β 3 κ T (β + αE(J)) 2 (β − αE(J)) . In Proposition 6, we give a tractable formula that allows to compute the signature plot. Proof is given in Section 6.4. We have E (f t − f s ) 2 = E(f 2 t ) − E(f 2 s ) − (1 − e −(β+αE(J))(t−s) ) β + αE(J) dE(f 2 s ) ds (s) − 2E(J 2 )E(λ + s ) with E(λ + s ) given in Proposition 4 and E(f 2 s ) given in Proposition 5. The signature plot for t ∈ [0, T ], δ > 0, given by C(t, δ) = 1 t E   t δ i=1 f iδ − f (i−1)δ 2   , can be computed directly from the result of Proposition 6: (4) C(t, δ) = 1 t E(f 2 t δ δ ) − f 2 0 − 1 t (1 − e −(β+αE(J))δ ) (β + αE(J)) t δ −1 i=0 dE(f 2 s ) ds (iδ) − 2E(J 2 )E(λ + iδ ) . C(t, δ) can then be computed explicitly directly from Equation (4). In particular, the following two regimes are specifically interesting to understand the evolution of the signature plot: • In the microscopic regime, that is δ → 0, the signature plot converges to (5) C micro (t) = 2E(J 2 ) E t 0 λ + s ds t . • In the macroscopic regime, that is δ → ∞ and δ t → 0 (while t ≤ T ), we have (6) C macro (t) ∼ 2E(J 2 ) 1 + αE(J) β 2 1 − αE(J) β t 0 µ( s T )ds t . The microscopic and macroscopic signature plots evolve at the same speed as the intensity with respect to time, that is exponentially with a rate κ T . This volatility increase, present at both microscopic and macroscopic scales, is induced by the time-dependent baseline and allows to represent the so-called Samuelson effect, which is well known for electricity forward prices, see for instance Jaeck and Lautier [16] : volatility increases when time to maturity decreases. One can also give intuition about the formula considering the asymptotics t → ∞ (while t ≤ T ) : C(t, δ) ∼ 2E(J 2 ) t 0 µ( s T )ds t 1 − αE(J) β    1 1 + αE(J) β 2 +   1 − 1 1 + αE(J) β 2    1 − e −(β+αE(J))δ (β + αE(J)) δ    . In this asymptotics, the signature plot is a decreasing function of δ with a term having an exponential decay and a constant term. This theoretical shape can fit the empirical signature plot described in Section 2.4 for t = T as we have C(T, δ) ∼ 2E(J 2 ) 1 0 µ(s)ds 1 − αE(J) β    1 1 + αE(J) β 2 +   1 − 1 1 + αE(J) β 2    1 − e −(β+αE(J))δ (β + αE(J)) δ    . In the case where µ is constant and equal to µ 0 , and the regime is stationary, we find a signature plot equal to (7) C(T, δ) = 2µ 0 E(J 2 ) 1 − αE(J) β    1 1 + αE(J) β 2 +   1 − 1 1 + αE(J) β 2    1 − e −(β+αE(J))δ (β + αE(J)) δ    , and moreover, in this stationary case, C micro = 2µ 0 E(J 2 ) 1 − αE(J) β , C macro = 2µ 0 E(J 2 ) 1 + αE(J) β 2 1 − αE(J) β . The structure of the signature plot is the same as the one computed by Bacry et al. [2], the main difference being the multiplicative term E(J 2 ) accounting for the random jump size and α which is multiplied by E(J) accounting for the presence of the jump size in the intensity. L = T 0 log(λ t )dN t + T 0 (1 − λ t )dt. The log-likelihood for one observation (that is, for one trading session) is then equal to the sum of L + = N + T i=1 log µ 0 e κ τ + i T + N − τ + i j=1 αJ − j e −β(τ + i −τ − j ) + T − µ 0 T κ (e κ − 1) − N − T i=1 α β J − i 1 − e −β(T −τ − i ) Maturity µ 0 (h −1 ) κ α (h −1 ) β (h −1 ) E(J) (L − = N − T i=1 log µ 0 e κ τ − i T + N + τ − i j=1 αJ + j e −β(τ − i −τ + j ) + T − µ 0 T κ (e κ − 1) − N + T i=1 α β J + i 1 − e −β(T −τ + i ) . If we dispose of continuous independent observations of the price process f (and equivalently of observations of N + , N − and J at jump times) on [0, T ], the log-likelihood is equal to the sum of the likelihood of each observation. We perform the estimation on the whole dataset (prices between July and September) for maturities 18h, 19h and 20h and considering data from 9 hours to 1 hour before maturity (we then assume that T = 8 hours). We do not fit a model for the jump sizes but only estimate their first two moments by using empirical averages. To initialize the parameters of the minimization, except κ, we use estimates from the minimization of the distance between the averaged empirical signature plot and the signature plot associated to the stationary model when κ = 0, given by Equation (7) for δ = 1, . . . , 300. The initial value for κ is chosen equal to 0.1. Parameters are given in Table 2. Values for κ are very close to each other, which could be interesting for a multidimensional model in term of parameter numbers. 3.3. Analysis. First, simulated prices are given in Figure 8 for maturities 18h, 19h and 20h using parameters of Table 2 and jumps simulated from the empirical distributions. We also plot a price series from the dataset in each sub-figure. Prices are simulated using the well known thinning algorithm for Poisson processes given in [19]. It is difficult to distinguish simulations from the real prices and, at first sight, the model seems to reproduce the different stylized facts of the price, in particular the increase of market activity over the trading session. This intuition is confirmed by Figure 9 where empirical moments are compared to theoretical ones for the different considered maturities. The model succeeds in reproducing the shape of the different moments of the model with a low number of parameters. The expectation that reproduces the market activity can sometimes be slightly underestimated or overestimated. Of course one could improve these curves by estimating the baseline non parametrically but it would increase the complexity in the model a lot. Our main concern is the reproduction of the signature plot C(t, δ), t ∈ [0, T ], δ > 0. Empirical signature plot (average of the different signature plot for the different trading days) and theoretical one are given in Figure 10 for different times. For every considered time, one observes the usual shape of the signature plot that exists in classic financial markets, with a very strong value at high frequency, a strong decrease then a stabilization. The model reproduces this shape very well. At last date T , the theoretical values are very close to the empirical ones. An interesting fact is the translation of the signature plot when time increases and gets closer to maturity : the whole curve goes upwards. For maturities 19h and 20h, theoretical curves values are above the empirical ones when time is equal to T minus one or two hours. As at very high frequency, the signature plot is proportional to the expectation of the integrated intensity, see Equation (5), equal to E(f + t ) E(J) which is overestimated for those two maturities at the end, see Figure 9. To conclude, our model represents the different empirical facts described in Section 2 well while being tractable and providing formulas for different quantities of interest for a practitioner. The model could be improved for instance by considering a full matrix for ϕ, or a more complex baseline µ however losing the analytical results. Price at macroscopic scale and Samuelson effect In this section, we aim at defining the behavior of prices at large scale. Precisely, as done by Bacry et al. [3], we provide a law of large numbers and a functional central limit theorem in our setting, as T is sent to infinity. In the context of intraday trading sessions, of which duration is finite, this should be understood as having trading sessions of arbitrary length. The analysis of the prices at macroscopic scale will allow us to observe the Samuelson effect on intraday markets. In order to provide an adequate time normalization, the time index t will be denoted as vT , where v ∈ [0, 1]. First we establish a law of large numbers for the sums of jump sizes to the power 0, 1 and 2 respectively. Its proof is in Section 7.1. Proposition 7. Let us consider the model (1), let N i t = t 0 (J s ) i dN s = N + t j=1 (J + j ) i , N − t j=1 (J − j ) i for i ∈ {0, 1, 2}. For i ∈ {0, 1, 2}, if E(J 2i ) < ∞, sup v∈[0,1] T −1 N i vT − (I 2 − K) −1 1 1 E(J i ) v 0 µ(s)ds → 0 as T → ∞ almost-surely and in L 2 (P). This proposition gives the uniform convergence of N i ·T to a limit function of v, featuring the integral of the baseline intensity. In this respect, it extends Theorem 1 of Bacry et al. [3] in which the baseline intensity is constant and the jump sizes are equal to 1. Now we provide a functional central limit theorem for the process N 1 met in Proposition 7, which is the couple (f + , f − ) . Under an additional integrability condition, it gives the convergence in law of the quantity in Proposition 7 after a suitable renormalization. The convergence holds for the Skorokhod topology, see Skorokhod [20] for the exact definition. The proof is in Section 7.2. (1) 1] in law for the Skorokhod topology when T → ∞ where W is a 2-dimensional Brownian motion. Adding the assumption 1] in law for the Skorokhod topology when T → ∞. Proposition 8. Let us consider the model -(2), let N 1 t = (f + t , f − t ) and assume E(J 4 ) < ∞. Let Σ denote the diagonal matrix with Σ j,j = (I 2 − K) −1 (1, 1) j , j = 1, 2. We have 1 √ T N 1 vT − E N 1 vT v∈[0,1] → (I 2 − K) −1 Σ 1/2 E(J 2 ) v 0 µ(s)dW s v∈[0,∞ 0 ϕ(t)t 1 2 dt < ∞ componentwise, we have √ T N 1 vT T − (I 2 − K) −1 1 1 v 0 µ(s)ds v∈[0,1] → (I 2 − K) −1 Σ 1/2 E(J 2 ) v 0 µ(s)dW s v∈[0, Finally, under the same assumptions as in the above Proposition 8, we can state the limit law of the price process in Corollary 9. Table 2, together with one sample Table 2 Corollary 9. Let us consider the model (1)-(2)-(3) under Assumption 3 and assume E(J 4 ) < ∞. We have The price process converges to a Brownian motion with a time dependent volatility at large scale : this volatility increases at the same speed as the baseline intensity when time gets closer to maturity. The instantaneous squared volatility of the Brownian limit is equal to 1 √ T (f vT − f 0 ) v∈[0,1] →    2E(J 2 ) 1 + αE(J) β 2 1 − αE(J) β v 0 µ(s)dW s    v∈[0,1] (a) 18h (b) 19h (c) 20h(σ macro ) 2 (t) = 2E(J 2 )µ 0 e κt 1 + αE(J) β 2 1 − αE(J) β which is consistent with the macroscopic realized volatility given in Equation (6). Conclusion and perspectives In this article, we first state some empirical facts about electricity intraday markets. In particular, we highlight an increasing price changing activity and the presence of microstructure noise through the empirical signature plot. Numerical illustrations on the German market are provided. A simple model based on marked Hawkes processes is introduced. Generalizing the results of [2], we provide theoretical properties within our model, that allow us to reproduce the stylized facts quite well. In particular, such modeling allows us to fit the empirical signature plot. Similarly to electricity forward markets on which volatility increases as time gets closer to maturity, one also observes a Samuelson effect than can be reproduced within our model, in which the whole signature plot curve increases when approaching maturity. Finally, we examine the behavior of prices at macroscopic scale that confirms the existence of this Samuelson effect : the price process converges to a Brownian motion with increasing instantaneous volatility. Future research on this subject might focus on a multidimensional modeling with a specific focus on the Epps effect, according to which the correlation between two prices increases when the sampling frequency of estimation increases, as illustrated in Figure 11. This is a well known effect in classic finance and would deserve being investigated in electricity intraday markets. We also think it would be meaningful to lead a large-scale study on the sizes of jumps in order to be able to use an even more representative distribution. 6. Proofs for Section 3 6.1. Preliminary results. The following two results are required for the computation of the different moments. The first result, in Proposition 10, gives the characteristic function of the multivariate marked Hawkes process (2). The moments will be derived from this function. Proposition 10. Let (f + , f − ) defined the bivariate marked Hawkes process defined by Equation (2). Its characteristic function L(a + , a − , t) = E e ia+f + t +ia−f − t for t ∈ [0, T ], a + ∈ R, a − ∈ R, is given by L(a + , a − , t) = exp t 0 µ s T 1 1 C(a + , a − , t − s) − 1 1 ds with C solution of the integro-differential equation C(a + , a − , t) = E exp iJ a + a − + J t 0 ϕ(s) C(a + , a − , t − s) − 1 1 ds . Proof. The proof is similar to the one of [10, Proposition 5] in a multidimensional setting. It also extends the one of [8, Theorem 3.1] including marks. Let us write ϕ = ϕ +,+ ϕ +,− ϕ −,+ ϕ −,− . In the following, we consider the cluster representation of Hawkes processes introduced in [13]. The first generation of migrants with type + and the first generation of migrants with type − both appears as an inhomogeneous Poisson process with intensity µ( · T ). Each new migrant is associated to a random jump mark with law J, and the marks are i.i.d. Migrants of type + (resp. −) of the first generation give birth to migrants of the second generation of type + as a inhomogeneous Poisson process with intensity xϕ +,+ (· − t) (resp. xϕ +,− (· − t) and of type − with intensity xϕ −,+ (· − t) (resp. xϕ −,− (· − t)) if migrant arrived at time t and is associated to a mark x > 0. Migrants of the second generation give birth to migrants of the third generation in the same way, and so on. N + t and N − t correspond respectively to the total number of migrants of type + and of type − up to time t and f + t and f − t are the sum of their marks up to time t. For m ∈ {+, −} and a given mark x > 0, one considers the point process Ñ m,+ x,t ,Ñ m,− x,t t defined with a first generation that arrives as an inhomogeneous Poisson process with rate xϕ +,m for positive migrants and xϕ −,m for negative migrants, and migrants are associated with an i.i.d. sequence with same law as J that represents their mark. Migrants of type j ∈ {+, −} with mark y > 0 born at time t give birth to children of type l ∈ {+, −} as an inhomogeneous Poisson process with intensity yϕ l,j (· − t). Each born migrant is associated with a mark having same law as J, and the marks are independent. The next generations are created in the same way as the second one. with mark x, the sum of the marks of positive (resp. negative) migrants induced by its progeny has the same law asf m,+ x,t−s (resp.f m,− x,t−s ). Therefore, we have the equality in law (8) f + t f − t = f +,0 t f −,0 t + N −,0 t i=1 f −,+,i J −,0 i ,t−τ −,0 ĩ f −,−,i J −,0 i ,t−τ −,0 i + N +,0 t i=1 f +,+,i J +,0 i ,t−τ +,0 ĩ f +,−,i J +,0 i ,t−τ +E e a+f + t +a−f − t |N 0 = m∈{+,−} N m,0 t i=1 C m (a + ,E e a+f + t +a−f − t |N 0 t = m∈{+,−} t 0 C m (a + , a − , t − s) µ s T t 0 µ s T ds ds N 0,m t . Finally, as N 0,+ and N 0,− are independent inhomogeneous Poisson processes, we get E e a+f + t +a−f − t = exp   m∈{+,−} t 0 (C m (a + , a − , t − s) − 1) µ s T ds   . Conditionally on a given mark J, f l,+ J ,f l,− J is also a bivariate marked Hawkes process defined in the same way as (f + , f − ) but with baseline equal to (Jϕ +,l , Jϕ −,l ) for l ∈ {+, −}. Therefore we can find, for l ∈ {+, −}, E e a+f l,+ J +a−f l,− J |J = exp   m∈{+,−} t 0 J(C m (a + , a − , t − s) − 1)ϕ m,l (s)ds   and C l (a + , a − , t) is equal to E e a l J+a+f l,+ J +a−f l,− J = E   exp   a l J + m∈{+,−} J t 0 (C m (a + , a − , t − s) − 1)ϕ m,l (s)ds     . Second result is Lemma 11. The equation in this lemma is a Volterra equation of the second kind that often appears in moments computation, and the proof follows directly from [3, Lemma 3]. Lemma 11. Let a, β ∈ R, |a| < |β|, ϕ : x → ae −βx and f : R + → R some measurable and locally bounded function. The unique solution of Ψ(t) = f (t) + t 0 Ψ(t − s)ϕ(s)ds with unknown Ψ is given by Ψ(t) = f (t) +L (a + , a − , t) = exp t 0 (C + (a + , a − , s) − 1) µ t − s T ds + t 0 (C − (a + , a − , s) − 1) µ t − s T ds with C + and C − solutions of (10)    C + (a + , a − , t) = E exp iJa + + t 0 JC − (a + , a − , t − s) ϕ exp (s) ds C − (a + , a − , t) = E exp iJa − + t 0 JC + (a + , a − , t − s) ϕ exp (s) ds . By differentiating (9) and (10) with respect to a + and a − , we find (11) E(f + t ) = t 0 Ψ + (s)µ t − s T ds,(12)E(f − t ) = t 0 Ψ − (s)µ t − s T ds with Ψ + (t) = −i ∂(C + +C − ) ∂a+ (0, 0, t) and Ψ − (t) = −i ∂(C + +C − ) ∂a− (0, 0, t) and we have the system of equations (13)                −i ∂C + ∂a+ (0, 0, t) = E(J) + t 0 E(J) −i ∂C − ∂a+ (0, 0, t − s) ϕ exp (s)ds −i ∂C − ∂a− (0, 0, t) = E(J) + t 0 E(J) −i ∂C + ∂a− (0, 0, t − s) ϕ exp (s)ds −i ∂C + ∂a− (0, 0, t) = t 0 E(J) −i ∂C − ∂a− (0, 0, t − s) ϕ exp (s)ds −i ∂C − ∂a+ (0, 0, t) = t 0 E(J) −i ∂C + ∂a+ (0, 0, t − s) ϕ exp (s)ds with solutions found using Lemma 11 : (14)            −i ∂C + ∂a+ (0, 0, t) = βE(J) 2(β−αE(J)) + βE(J) 2(β+αE(J)) − αE(J) 2 2(β−αE(J)) e −(β−αE(J))t + αE(J) 2 2(β+αE(J)) e −(β+αE(J))t −i ∂C − ∂a− (0, 0, t) = −i ∂C + ∂a+ (0, 0, t) −i ∂C + ∂a− (0, 0, t) = βE(J) 2(β−αE(J)) − βE(J) 2(β+αE(J)) − αE(J) 2 2(β−αE(J)) e −(β−αE(J))t − αE(J) 2 2(β+αE(J)) e −(β+αE(J))t −i ∂C − ∂a+ (0, 0, t) = −i ∂C + ∂a− (0, 0, t) . We then have (15) Ψ + (t) = Ψ − (t) = −αE(J) 2 β − αE(J) e −(β−αE(J))t + βE(J) β − αE(J) and E(f + t ), E(f − t ) can be derived from (11) and (12). The computation of E(λ + t ) and E(λ − t ) follows from E(f + t ) = E(J)E( t 0 λ + s ds) and E(f − t ) = E(J)E( t 0 λ − s ds).E((f + t ) 2 ) = t 0 Ψ + (s)µ t − s T ds 2 + t 0 Ψ 2,+ (s)µ t − s T ds with Ψ + defined in (15) and Ψ 2 (t) = − ∂ 2 (C + +C − ) ∂a 2 + (0, 0, t) solution of Ψ 2,+ (t) =E(J 2 ) 1 + t 0 −i ∂C − ∂a + (0, 0, t − s)ϕ exp (s)ds 2 + E(J 2 ) t 0 −i ∂C + ∂a + (0, 0, t − s)ϕ exp (s)ds 2 + t 0 Ψ 2,+ (t − s)ϕ exp (s)ds, which can be rewritten using (13) : Ψ 2,+ (t) = E(J 2 ) E(J) 2 −i ∂C + ∂a + (0, 0, t) 2 + E(J 2 ) E(J) 2 −i ∂C − ∂a + (0, 0, t) 2 + t 0 Ψ 2,+ (s)ϕ exp (t − s)ds. We easily show that E((f − t ) 2 ) = E((f + t ) 2 ). The cross moment can be computed in the same way and it is equal to E(f + t f − t ) = t 0 Ψ + (s)µ t − s T ds 2 + t 0 Ψ 2,+,− (s)µ t − s T ds with Ψ 2,+,− (t) = − ∂C + +C − ∂a + ∂a − (0, 0, t) solution of Ψ 2,+,− (t) = 2 E(J 2 ) E(J) 2 −i ∂C + ∂a + (0, 0, t) −i ∂C − ∂a + (0, 0, t) + E(J) t 0 Ψ 2,+,− (s)ϕ exp (t − s)ds. Therefore, E(f 2 t ) = f 2 0 + 2 E f + t 2 − E f + t f − t = f 2 0 + 2 t 0 Ψ 2 (s)µ t − s T ds (16) with Ψ 2 = Ψ 2,+ − Ψ 2,+,− solution of Ψ 2 (t) = E(J 2 ) E(J) 2 −i ∂C + ∂a + (0, 0, t) − −i ∂C − ∂a + (0, 0, t) 2 + E(J) t 0 Ψ 2 (s)ϕ exp (t − s)ds =Ψ(t) + E(J) t 0 Ψ 2 (s)ϕ exp (t − s)ds withΨ (t) = E(J 2 ) β (β + αE(J)) + αE(J) (β + αE(J)) e −(β+αE(J))t 2 . Lemma 11 implies that Ψ 2 (t) =Ψ(t) + t 0Ψ (s)E(J)ϕ exp (t − s)e αE(J)(t−s) ds =E(J 2 ) α 2 E(J) 2 (β + αE(J)) 2 e −2(β+αE(J))t + β 2 (β + αE(J)) 2 + 2 αβE(J) (β + αE(J)) 2 e −(β+αE(J))t + α 3 E(J) 3 (β + αE(J)) 2 (β + 3αE(J)) e −(β−αE(J))t − e −2(β+αE(J))t + β 2 αE(J) (β + αE(J)) 2 (β − αE(J)) 1 − e −(β−αE(J))t + βαE(J) (β + αE(J)) 2 e −(β−αE(J))t − e −(β+αE(J))t =E(J 2 ) C 1 e −(β−αE(J))t +C 2 e −2(β+αE(J))t +C 3 e −(β+αE(J))t +C 4 withC 1 = −α 2 E(J) 2 (β − αE(J))(β + 3αE(J)) , C 2 = α 2 E(J) 2 (β + 2αE(J)) (β + αE(J)) 2 (β + 3αE(J)) , C 3 = αβE(J) (β + αE(J)) 2 , C 4 = β 3 (β + αE(J)) 2 (β − αE(J)) . Finally, from (16), we obtain E(f 2 t ) = f 2 0 +2µ 0 E(J 2 ) (C 1 + C 2 + C 3 + C 4 )e κ t T − C 1 e −(β−αE(J))t − C 2 e −2(β+αE(J))t − C 3 e −(β+αE(J))t − C 4 with C 1 =C 1 β − αE(J) + κ T , C 2 =C 2 2β + 2αE(J) + κ T , C 3 =C 3 β + αE(J) + κ T , C 4 =C 4 κ T . 6.4. Proof of Proposition 6. Let 0 ≤ s ≤ t ≤ T . To compute E((f t − f s ) 2 ), one needs to know E(f t f s ) which is equal to the sum of f 2 0 and E((f + t − f − t )(f + s − f − s )) = E(f + t f + s ) + E(f − t f − s ) − E(f + t f − s ) − E(f − t f + s ). Conditioning with respect to F s gives E f + t f + s = E f + s 2 + E (J) E t s E λ + u |F s duf + s , E f + t f − s = E f + s f − s + E (J) E t s E λ + u |F s duf − s , E f − t f − s = E f − s 2 + E (J) E t s E λ − u |F s duf − s , E f − t f + s = E f − s f + s + E (J) E t s E λ − u |F s duf + s . Hence E (f t f s ) = E f 2 s + E (J) E t s E λ + u − λ − u |F s du f + s − f − s . The conditional intensities are such that E λ + u |F s = µ u T + s 0 ϕ exp (u − v) J v dN − v + E (J) u s ϕ exp (u − v) E λ − v |F s dv = µ u T + e −β(u−s) λ + s − µ s T + E(J) u s ϕ exp (u − v) E λ − v |F s dv and E λ − u |F s = µ u T + e −β(u−s) λ − s − µ s T + E(J) u s ϕ exp (u − v) E λ + v |F s dv. Applying Lemma 11, we find (17) E λ + u − λ − u |F s = λ + s − λ − s e −(β+αE(J))(u−s) . As E f + s 2 = E J 2 E s 0 λ + v dv + 2E (J) E s 0 f + v λ + v dv , E f − s 2 = E J 2 E s 0 λ − v dv + 2E (J) E s 0 f − v λ − v dv , and E f + s f − s = 2E (J) E s 0 f − v λ + v dv = 2E (J) E s 0 f + v λ − v dv , we have, using also Equation (17), E(f t f s ) = E(f 2 s ) + 1 2 dE((f + s ) 2 ) + E((f − s ) 2 ) − 2E(f + s f − s ) ds − E(J 2 )E(λ + s ) (1 − e −(β+αE(J))(t−s) ) β + αE(J) which is equal to E(f 2 s ) + 1 2 dE(f 2 s ) ds − E J 2 E λ + s 1 − e −(β+αE(J))(t−s) β + αE(J) . We obtain E (f t − f s ) 2 = E(f 2 t ) − E(f 2 s ) − 1 − e −(β+αE(J))(t−s) β + αE(J) dE(f 2 s ) ds − 2E J 2 E λ + s , which achieves the proof. Computation of the expectation. First step consists in computing the expectation of N 0 t = N t . We easily find that, for t ∈ [0, T ], E(N t ) = t 0 µ s T ds + t 0 Ψ(t − s) s 0 µ u T du ds where Ψ is defined by n≥1 ϕ n and ϕ n is defined recursively by ϕ 1 = E(J)ϕ ϕ n+1 (t) = t 0 E(J)ϕ(t − s)ϕ n (s)ds, n ≥ 1. The sum is well defined: as ∞ 0 ϕ n (s)ds = K n and ρ(K) < 1, n≥1 ϕ n converges in L 1 (dt). Furthermore, ∞ 0 Ψ(s)ds = (I 2 − K) −1 − I 2 . By doing a simple change of variable in the integral, we have for v ∈ [0, 1], E(N T v ) T = v 0 µ(s)ds + 1 T T v 0 Ψ(T v − s) s 0 µ u T du ds = v 0 µ(s)ds + 1 T T v 0 Ψ(s) T v−s 0 µ u T du ds = I 2 + T v 0 Ψ(s)ds v 0 µ(s)ds − 1 T T v 0 Ψ(s) s 0 µ v − u T du ds.(18) Approximation of E N 1 t − E(N 1 t ) = M 1 t + E(J) t 0 s 0 ϕ(s − u) dN 1 u − E(dN 1 u ) ds. Then, applying Fubini theorem, we have N 1 t − E(N 1 t ) = M 1 t + E(J) t 0 t u ϕ(s − u)ds dN 1 u − E(dN 1 u ) so that using integration by parts, similarly to the proof of [3, Lemma 2], one finds (21) N 1 t − E(N 1 t ) = M 1 t + E(J) t 0 ϕ(t − s) N 1 s − E(N 1 s ) ds. Applying [3, Lemma 3], we obtain Thus, integrating by parts again, N i t − E(N i t ) = M i t + E(J i ) t 0 ϕ(t − s) N 1 s − E(N 1 s ) ds and we have, using (21) and (22), N i t − E(N i t ) = M i t + E(J i ) E(J) t 0 Ψ(t − s)M 1 s ds. Remaining of the proof is the same as the one of [3, Theorem 1] : • if E(J 2i ) < ∞, we can use Doob's inequality to bound E((sup t∈[0,T ] M i t ) 2 ) by a constant times T (the quadratic variation of M i being N 2i on the diagonal and 0 otherwise, and we can easily show that E(N 2i T ) is bounded by T as µ is bounded) and the convergence of sup v∈[0,1] T −1 N i T v − E(N T v ) toward 0 in L 2 (P) follows ; • the arguments for the almost-sure convergence remain true. . • sup v∈[0,1] T −1 N i T v − E(N i T v ) The quadratic variation matrix is equal on the diagonal to 1 T N 2 T v and 0 otherwise, and then converges to C v = E(J 2 )Σ v 0 µ(s)ds, with Σ i,i = (I 2 − K) −1 (1, 1) i , i = 1, 2, in L 2 (P) according to Proposition 7. The jump measure compensator of M 1,(T ) is ν (T ) (dt, dx) = T λ tT dt ⊗ µ J (dx √ T ) where µ J is the probability measure associated to J. Let t ∈ [0, 1], > 0. The integral with W a 2-dimensional Brownian motion. Remaining of the proof is similar to the one of [3,Theorem 2] Prices on July 11 th , 2017 (b) Prices on August 30 th , 2017 Figure 1 . 1Intraday mid-prices for deliveries at 18h, 19h and 20h up to 1 hour before maturity for two trading sessions Figure 2 . 2Number of price changes for deliveries at 18h, 19h and 20h up to 1 hour before maturity for one trading session and on average over all the dataset Intensity on August, 30 th , 2017 (b) Average Figure 3 . 3Intensity of price changes for deliveries at 18h, 19h and 20h up to 1 hour before maturity for one trading session and on average over all the dataset Figure 4 . 4Quantile-quantile plot between the time-changed jump time intervals and an exponential distribution for the trading session of August, 30 th , 2017 and for maturity 18h in order to test if the process of price changes is an inhomogenous Poisson process 2.3. Jump sizes distribution. Figure 6 . 6Mean and standard deviation of jump sizes (positive and negative considered indifferently) against time to maturity: x-axis corresponds to the number of hours before maturity at which the estimation starts(a) August 30 th , 2017 (b) Average Figure 7 . 7Signature plot for different maturities estimated from 9 hours before maturity for one trading session, and on average over all trading sessions intraday prices. The signature plot still has this shape when considering all the trading sessions but it is less smooth because of the big jumps happening at the beginning of the trading session. Proposition 4 . 4Let us consider the model (1)-(2)-(3) under Assumption 3. We have for t ∈ [0, T ] Proposition 5 . 5Let us consider the model (1)-(2)-(3) under Assumption 3. We have for t ∈ [0, T ] Proposition 6 . 6Let us consider the model (1)-(2)-(3) under Assumption 3 and 0 ≤ s < t ≤ T . 3. 2 . 2Estimation. Let us estimate the parameters of the model on the data, using likelihood maximization. Observing continuously the price process f corresponds to a continuous observation of the process (N + , N − ) and of the jump sizes (J + , J − ) at jump times. If N is a Poisson process with intensity λ, the log-likelihood is equal to (see Daley and Vere-Jones [6, Proposition 7.2III]) Figure 8 . 8Price simulations, starting from the same initial value, for maturities 18h, 19h and 20h with estimated parameters in Second order moment, 18h (c) Expectation 19h (d) Second order moment, 19h (e) Expectation, 20h (f) Second order moment, 20h Figure 9 . 9Empirical and theoretical expectation of f + and f − and second order moment for price and for maturities 18h, 19h and 20h with estimated parameters in Figure 10 . 10Empirical and theoretical signature plot C(t, δ) at different times for maturities 18h, 19h and 20h with estimated parameters ofTable 2(a) August 30 th , 2017 (b) Average Figure 11 . 11Epps effect for different maturities estimated from 9 hours before the nearest maturity for one trading session and on average over all trading sessions in law for the Skorokhod topology when T → ∞, where W is a 1-dimensional Brownian motion. Letf m,+ andf m,− be respectively the sum of the mark of all positive and negative migrants. Let C m (a + , a − , t) = E exp ia m J + ia +f m,+ J,t + ia −f m,− J,t for a + , a − ∈ R and t ∈ [0, T ] be the characteristic function of J1 m=+ +f m,+ J,t , J1 m=− +f m,− J,t for m ∈ {+, −} and let us consider the conditional version C c m (a + , a − , t, x) = E exp ia m J + ia +f number of migrants of type + and of type − of the first generation at time t, that have arrived at times (τ the marks of migrants of type + (resp. −) is then equal to the sum of f +the sum of the marks of first generation, and of the sum of the marks of positive migrants from the second generation. For a first generation migrant of type m ∈ {+, −} born at time s ≤ t and , 0 i 0where, for m = +, −, for i = 1, ..., N a − , t − τ mT the order statistics of i.i.d. random variables with density s → )ds . We then have t 0 f 0(s)ϕ(t − s)e a(t−s) ds. 6.2. Proof of Proposition 4. Let t ∈ [0, T ].The expectations of f + t and f − t are respectively given by −i ∂L ∂a+ (0, 0, t) and −i ∂L ∂a− (0, 0, t) with L the characteristic function of (f + t , f + t 6. 3 . 3Proof of Proposition 5. Let t ∈ [0, T ]. The expectation of (f + t ) 2 , (f − t ) 2 and f + t f − t are respectively given by − ∂ t) and − ∂ 2 L ∂a+∂a− (0, 0, t) with L defined in (9)-(10). Thus, . Proof of Proposition 7. The proof is very similar to the one of [3, Theorem 1]. Let us recall the main steps and identify the differences. In the following, one denotes by µ the function t → µ(t) (1, 1) . s)sds µ * with µ * an upper bound for µ componentwise on [0, 1]. The right hand side of (19) is equal tov (I 2 − K) −1 µ * − E(Ñ T v ) TwhereÑ is a Hawkes process with constant baseline intensity µ * and excitation kernel E(J)ϕ. By [3,Lemma 5], it converges to 0 uniformly on [0, 1] as T → ∞, as T → ∞. As E(N i t ) = E(J i )E(N t ), we obtain directly thatE(N i T v ) Tconverges uniformly on [0, 1] as T → ∞ to E(J i )(I 2 − K) −1 v 0 µ(s)ds. Let us now consider the case i ∈ {0, 2}. We have ∈ {0, 1, 2} almost-surely and in L 2 (P) as T → ∞, ending the proof. 7.2. Proof of Proposition 8. The proof relies on the convergence of the martingale M 1,(T ) = T N T t ) T which is bounded componentwise by E(J 4 )T −1 −2 E(N T t )T as can be seen by applying Cauchy-Schwarz inequality to E(J 2 1 J> √ T ) and then Markov inequalityP(J > √ T ) ≤ E(J 4 ) T 2 4 .The bound converges to 0 when T → ∞ using(20) from the proof of Proposition 7. Then t 0 R+ x 2 1 x> ν (T ) (ds, dx) converges in probability to 0 when T → ∞ (as it is positive) for every t ∈ [0, 1], > 0. From [15, Section VIII, Theorem 3.22], the quadratic variation convergence implies the convergence in law of M 1,(T ) to a martingale with quadratic variation C, that is to E(J 2 )Σ Acknowledgements. This research is supported by the FiME (Finance for Energy Markets) Research Initiative. An optimal trading problem in intraday electricity markets. René Aïd, Pierre Gruet, Huyên Pham, Mathematics and Financial Economics. 101René Aïd, Pierre Gruet, and Huyên Pham. An optimal trading problem in intraday electricity markets. Math- ematics and Financial Economics, 10(1):49-85, 2016. Modelling microstructure noise with mutually exciting point processes. Emmanuel Bacry, Sylvain Delattre, Marc Hoffmann, Jean-François Muzy, Quantitative finance. 131Emmanuel Bacry, Sylvain Delattre, Marc Hoffmann, and Jean-François Muzy. Modelling microstructure noise with mutually exciting point processes. Quantitative finance, 13(1):65-77, 2013. Some limit theorems for Hawkes processes and application to financial statistics. Emmanuel Bacry, Sylvain Delattre, Marc Hoffmann, Jean-François Muzy, Stochastic Processes and their Applications. 123Emmanuel Bacry, Sylvain Delattre, Marc Hoffmann, and Jean-François Muzy. Some limit theorems for Hawkes processes and application to financial statistics. Stochastic Processes and their Applications, 123(7):2475-2499, 2013. Hawkes processes in finance. Emmanuel Bacry, Iacopo Mastromatteo, Jean-François Muzy, Market Microstructure and Liquidity. 1011550005Emmanuel Bacry, Iacopo Mastromatteo, and Jean-François Muzy. Hawkes processes in finance. Market Mi- crostructure and Liquidity, 1(01):1550005, 2015. German continuous intraday market: Orders book's behavior over the trading session. Clara Balardy, Meeting the Energy Demands of Emerging Economies, 40th IAEE International Conference. International Association for Energy EconomicsClara Balardy. German continuous intraday market: Orders book's behavior over the trading session. In Meet- ing the Energy Demands of Emerging Economies, 40th IAEE International Conference, June 18-21, 2017. International Association for Energy Economics, 2017. An Introduction to the Theory of Point Processes. Volume I: Elementary Theory and Methods, Second Edition. Daryl J Daley, David Vere, -Jones , SpringerDaryl J. Daley and David Vere-Jones. An Introduction to the Theory of Point Processes. Volume I: Elementary Theory and Methods, Second Edition. Springer, 2003. Statistical inference across time scales. Céline Duval, Marc Hoffmann, Electronic journal of statistics. 5Céline Duval and Marc Hoffmann. Statistical inference across time scales. Electronic journal of statistics, 5:2004-2030, 2011. The characteristic function of rough Heston models. Omar El Euch, Mathieu Rosenbaum, Mathematical Finance. 291Omar El Euch and Mathieu Rosenbaum. The characteristic function of rough Heston models. Mathematical Finance, 29(1):3-38, 2019. The European intraday electricity market: a modeling based on the Hawkes process. Available on hal.archives-ouvertes.fr. Benjamin Favetto, Benjamin Favetto. The European intraday electricity market: a modeling based on the Hawkes process. Avail- able on hal.archives-ouvertes.fr, 2019. Transform analysis for Hawkes processes with applications in dark pool trading. Xuefeng Gao, Xiang Zhou, Lingjiong Zhu, Quantitative Finance. 182Xuefeng Gao, Xiang Zhou, and Lingjiong Zhu. Transform analysis for Hawkes processes with applications in dark pool trading. Quantitative Finance, 18(2):265-282, 2018. The stateof-the-art in short-term prediction of wind power. a literature overview. Gregor Giebel, Caroline Draxl, Richard Brownsword, Georges Kariniotakis, Michael Denhard, Advanced Tools for the Management of Electricity Grids with Large-Scale Wind Generation. Gregor Giebel, Caroline Draxl, Richard Brownsword, Georges Kariniotakis, and Michael Denhard. The state- of-the-art in short-term prediction of wind power. a literature overview. In Advanced Tools for the Management of Electricity Grids with Large-Scale Wind Generation, 2011. Modeling market order arrivals on the intraday market for electricity deliveries in Germany with the Hawkes process. Nikolaus Graf Von Luckner, Rüdiger Kiesel, Available at SSRNNikolaus Graf von Luckner and Rüdiger Kiesel. Modeling market order arrivals on the intraday market for electricity deliveries in Germany with the Hawkes process. Available at SSRN, 2020. A cluster process representation of a self-exciting process. Alan Geoffrey Hawkes, David Oakes, Journal of Applied Probability. Alan Geoffrey Hawkes and David Oakes. A cluster process representation of a self-exciting process. Journal of Applied Probability, pages 493-503, 1974. The significance probability of the Smirnov two-sample test. John L Hodges, Arkiv för Matematik. 35John L. Hodges. The significance probability of the Smirnov two-sample test. Arkiv för Matematik, 3(5):469- 486, 1958. Limit theorems for stochastic processes. Jean Jacod, Albert Shiryaev, Springer Science & Business Media288Jean Jacod and Albert Shiryaev. Limit theorems for stochastic processes, volume 288. Springer Science & Business Media, 2013. Volatility in electricity derivative markets: The Samuelson effect revisited. Édouard Jaeck, Delphine Lautier, Energy Economics. 59Édouard Jaeck and Delphine Lautier. Volatility in electricity derivative markets: The Samuelson effect revisited. Energy Economics, 59:300-313, 2016. Econometric analysis of 15-minute intraday electricity prices. Rüdiger Kiesel, Florentina Paraschiv, Energy Economics. 64Rüdiger Kiesel and Florentina Paraschiv. Econometric analysis of 15-minute intraday electricity prices. Energy Economics, 64:77-90, 2017. Estimation and simulation of the transaction arrival process in intraday electricity markets. Florian Micha L Narajewski, Ziel, Energies. 12234518Micha l Narajewski and Florian Ziel. Estimation and simulation of the transaction arrival process in intraday electricity markets. Energies, 12(23):4518, 2019. On Lewis' simulation method for point processes. Yosihiko Ogata, IEEE transactions on information theory. 27Yosihiko Ogata. On Lewis' simulation method for point processes. IEEE transactions on information theory, 27(1):23-31, 1981. Limit theorems for stochastic processes. Anatoliy Volodymyrovych, Skorokhod , Theory of Probability & Its Applications. 13Anatoliy Volodymyrovych Skorokhod. Limit theorems for stochastic processes. Theory of Probability & Its Applications, 1(3):261-290, 1956. . Thomas Deschatre, 91120EDF Lab Paris-Saclay and FiMe, Laboratoire de Finance des Marchés de l'EnergieThomas Deschatre, EDF Lab Paris-Saclay and FiMe, Laboratoire de Finance des Marchés de l'Energie, 91120 . France Palaiseau, Email, [email protected], France Email address: [email protected] . Pierre Gruet, EDF Lab Paris-Saclay and FiMe, Laboratoire de Finance des Marchés de l'EnergiePierre Gruet, EDF Lab Paris-Saclay and FiMe, Laboratoire de Finance des Marchés de l'Energie, . France Palaiseau, Email, address: [email protected], France Email address: [email protected]
[]
[ "On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential", "On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential" ]
[ "Steffen Betsch \nInstitute of Stochastics\nKarlsruhe Institute of Technology\n76131KarlsruheGermany\n", "Günter Last \nInstitute of Stochastics\nKarlsruhe Institute of Technology\n76131KarlsruheGermany\n" ]
[ "Institute of Stochastics\nKarlsruhe Institute of Technology\n76131KarlsruheGermany", "Institute of Stochastics\nKarlsruhe Institute of Technology\n76131KarlsruheGermany" ]
[]
We prove that the distribution of a Gibbs process with non-negative pair potential is uniquely determined as soon as an associated Poisson-driven random connection model (RCM) does not percolate. Our proof combines disagreement coupling in continuum (established in[16,23]) with a coupling of a Gibbs process and a RCM. The improvement over previous uniqueness results is illustrated both in theory and simulations.
10.1214/22-aihp1265
[ "https://arxiv.org/pdf/2108.06303v1.pdf" ]
237,048,436
2108.06303
468d97cd57c802982146e166b3ab46946dd85a56
On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential August 16, 2021 Steffen Betsch Institute of Stochastics Karlsruhe Institute of Technology 76131KarlsruheGermany Günter Last Institute of Stochastics Karlsruhe Institute of Technology 76131KarlsruheGermany On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential August 16, 2021arXiv:2108.06303v1 [math.PR] 13 Aug 2021Gibbs processuniqueness of Gibbs distributionspair potentialsdisagreement couplingPoisson embeddingrandom connection modelpercolation AMS MSC 2010: 60K3560G5560D05 We prove that the distribution of a Gibbs process with non-negative pair potential is uniquely determined as soon as an associated Poisson-driven random connection model (RCM) does not percolate. Our proof combines disagreement coupling in continuum (established in[16,23]) with a coupling of a Gibbs process and a RCM. The improvement over previous uniqueness results is illustrated both in theory and simulations. Introduction Let (X, d) be a complete separable metric space, denote its Borel-σ-field by X , and let λ be a locally finite measure on X. Denote by v : X × X → [0, ∞] a (non-negative) pair potential, that is, a measurable and symmetric function. A Gibbs process with pair potential v and reference measure λ is a point process η on X satisfying, loosely speaking, P η(dx) > 0 | η X\{x} = exp − X v(x, y) dη(y) dλ(x), for all x ∈ X. Here we interpret a point process η as a random measure on X, which is integer-valued on bounded sets, and we write η B for the restriction of η to a set B ∈ X . For more details on the point process approach to Gibbs processes we refer to [30,27,5] and to Subection 2.1. We let G(v, λ) denote the space of distributions of Gibbs processes with pair potential v and reference measure λ. Our assumption below will imply that X 1 − e −v(x,y) dλ(y) < ∞, λ-a.e. x ∈ X. (1.1) It is then known that G(v, λ) = ∅, referring to [35] for the case X = R d and to [19] for the present generality. In this paper we establish an explicit criterion for the uniqueness of a Gibbs distribution. To do so we consider the random connection model (RCM) (see e.g. [31,29]) with connection function ϕ := 1−exp(−v) based on a Poisson process Φ with intensity measure λ. In a RCM every pair x, y of distinct points from Φ (taking into account multiplicities) is connected with probability ϕ(x, y), independently for different pairs. This gives a random graph Γ whose vertices are the points of Φ. For each x ∈ X we consider a RCM Γ x with connection function ϕ based on the Poisson process Φ augmented by the point x, as detailed in Subsection 2.2. Let C x denote the cluster (component) of this graph containing x, interpreted as a point process on X. We say that the RCM is subcritical if these clusters have only a finite number of points, that is, P C x (X) < ∞ = 1, λ-a.e. x ∈ X. (1.2) In this case we also say that the pair (v, λ) is subcritical. Note that the degree of x in the RCM Γ x has a Poisson distribution with parameter X 1 − e −v(x,y) dλ(y). Therefore (1.2) implies the integrability condition (1.1). The following theorem is the main result of our paper. It gives an affirmative answer to a question asked in [19]. By |A| we denote the cardinality of a set A. If X = R d , λ is the Lebesgue measure, and v is translation invariant, then it is known that 0 < γ c < ∞, see [29]. But even in this case Theorem 1.1 raises the question for sufficient conditions implying that (v, λ) is subcritical. By comparison with a branching process (similar to [31]) we prove in Appendix A that ess sup x∈X X ϕ(x, y) dλ(y) < 1 is sufficient for subcriticality (where the essential supremum is with respect to λ). Hence we obtain the following corollary, still on the general state space. Then |G(v, γλ)| = 1. Corollary 1.3 is the main result in the recent preprint [18], proved there under two additional assumptions by applying the classical Dobrushin method [9] to a suitable discretization. Our Theorem 1.1 generalizes this in several way. First of all we work on a general space X and not just on R d . Second we do not need any hard core assumption on v. Neither do we need another technical assumption made in [18]. Finally, and perhaps most importantly, the simulations in Section 5 show that we extend the bounds on the uniqueness region much beyond the branching bound (1.4). Theorem 1.1 also considerably generalizes the percolation criteria derived in [16] for translation invariant pair potentials in R d with a finite range R > 0, that is, v(x) = 0 for |x| > R. Indeed, the uniqueness result in [16,Subsection 3.2.1] requires the RCM with connection function ϕ R (x) := 1{|x| ≤ R} to be subcritical. Since ϕ ≤ ϕ R , the critical intensity associated with ϕ is larger than the one associated with ϕ R . As our bound takes into account the full information contained in the pair potential and not just its range, the difference can be very significant as illustrated in Section 5. The RCM associated with ϕ R is referred to as the Gilbert graph, see e.g. [24]. The corresponding pair potential is v R (x) = ∞ · 1{|x| ≤ R}, and describes hard spheres of radius R/2 (in equilibrium), where ∞·0 = 0·∞ := 0. A non-Poisson version of disagreement percolation was applied in [7] to prove uniqueness of the Widom-Rowlinson process for a certain range of parameters. This Gibbs process is not governed by a pair potential but enjoys nice and rather specific monotonicity properties. We think of Corollary 1.2 as of an explicit lower bound on the range of uniqueness. Even though γ c is not explicitly known, it admits a clear probabilistic meaning and the Poisson-based RCM can be easily simulated. To get an impression of the critical intensities, we have simulated three examples, including the well-studied Gilbert graph as a benchmark model as well as two interaction functions from the physics literature. The results are presented in Section 5. The Dobrushin criterion (mentioned above) was also used in other papers to establish uniqueness of Gibbs measures, for instance in [11] (without giving the details) and in [3]. The results of these papers allow the pair potential to take negative values but do not provide explicit information on the domain of uniqueness. A further drawback is the assumption of a finite range, a severe restriction of generality. Another method for proving uniqueness of Gibbs distributions are fixed point methods based on Kirkwood-Salsburg integral equations and the fact that the correlation functions determine the distribution of a point process under suitable assumptions. This method can be traced back to [34]. For some recent contributions we refer to [19], which might also serve as a good survey, and [42]. The uniqueness intensity region identified by this method is characterized by the contractivity of certain integral operators and does not seem to have an explicit probabilistic interpretation. In view of the method and the special cases discussed in [19], we expect this region to be comparable with the branching bound (1.4) or its generalization in Appendix A. Still another method for proving uniqueness is to identify a Gibbs distribution as a stationary and reversible measure with respect to a suitable Markovian dynamics, cf. [15,37]. Uniqueness follows if the so-called ancestor clans, coming from an embedding into a space-time Poisson process, are finite. The resulting bounds on the domain of uniqueness are not explicit but might be comparable with the branching bounds (see also the discussion in [1]). Our main tool for proving Theorem 1.1 is the disagreement coupling from [16,23], which was inspired by the results of [41] in the discrete setting, combined with an (approximative) simultaneous coupling with the associated RCM. The connection function ϕ = 1 − e −v was used in [14,11] for a Fortuin-Kasteleyn type coupling of a Potts model (allowing for general pairwise interactions between particles of different types) and the continuum cluster model. While this coupling proceeds by conditioning, we use a marked Poisson process for embedding two Gibbs processes with different boundary conditions and, at the same time, for constructing a RCM. This way we can directly refer to the percolation properties of a Poissondriven RCM. We need to assume that v ≥ 0. Apart from that, and the subcriticality, we do not make any further assumptions such as finite range or strict repulsion. We believe that our approach will be useful for analyzing further properties of repulsive Gibbs processes, even though its details are a bit technical. Preliminaries In this section we collect a few basic definitions and facts on Gibbs point processes and the random connection model. A reader familiar with these topics might skip this section at first reading. Gibbs processes Let (X, X ) denote a Borel space and λ a σ-finite measure on X. Let B 1 ⊂ B 2 ⊂ · · · be measurable sets of finite λ-measure such that ∞ ℓ=1 B ℓ = X. Let X b denote the system of all bounded Borel sets, meaning the collection of all sets which are contained in one of the B ℓ . A measure ν on (X, X ) which is finite on X b is called locally finite. Let N(X) be the space of all locally finite measures on X which are N 0 -valued on X b , and let N (X) denote the smallest σ-field such that µ → µ(B) is measurable for all B ∈ X . A point process on X is a random element η of N(X), defined over some fixed probability space (Ω, F, P). The intensity measure of η is the measure E[η] defined by E[η](B) := E[η(B)], B ∈ X . By our assumption on X, every point process η is proper, that is, η = η(X) n=1 δ Xn , where {X n : n ≥ 1} is a collection of random elements with values in X, see [24, Section 6.1] for more details. Given a measure ν on X and B ∈ X , we write ν B to indicate the trace measure D → ν(D ∩ B). For µ ∈ N(X) we write x ∈ µ if µ({x}) > 0. Let κ : X × N(X) → [0, ∞) be measurable and satisfy the cocycle relation, κ(x, µ) · κ(y, µ + δ x ) = κ(y, µ) · κ(x, µ + δ y ), x, y ∈ X, µ ∈ N(X). A point process η on X is called a Gibbs process with Papangelou intensity (PI) κ (and reference measure λ) if E X f (x, η) dη(x) = E X f (x, η + δ x ) κ(x, η) dλ(x) (2.1) for each measurable f : X × N(X) → [0, ∞). The latter are the GNZ equations named after Georgii, Nguyen and Zessin [10,30]. If κ ≡ 1 this is the Mecke equation, characterizing a Poisson process with intensity measure λ, cf. [24,Theorem 4.1]. In that case the distribution of η is denoted by Π λ . Equation (2.1) can be generalized. For m ∈ N we define a measurable map κ m : X m × N(X) → [0, ∞) by κ m (x 1 , . . . , x m , µ) := κ(x 1 , µ) · κ(x 2 , µ + δ x 1 ) · · · κ(x m , µ + δ x 1 + · · · + δ x m−1 ). Note that κ 1 = κ. If η is a Gibbs process with PI κ then we have for each m ∈ N and for each measurable f : X m × N(X) → [0, ∞) that E X m f (x 1 , . . . , x m , η) dη (m) (x 1 , . . . , x m ) = E X m f (x 1 , . . . , x m , η + δ x 1 + · · · + δ xm ) · κ m (x 1 , . . . , x m , η) dλ m (x 1 , . . . , x m ) ,(2.2) where µ (m) is the m-th factorial measure of µ, consulting [24] for the general definition. This follows by induction, using that κ m+1 (x 1 , . . . , x m+1 , µ) = κ m (x 1 , . . . , x m , µ) · κ(x m+1 , µ + δ x 1 + · · · + δ xm ). By the cocycle assumption on κ, κ m is symmetric in the first m arguments. The Hamiltonian H : N(X) × N(X) → (−∞, ∞] (based on κ) is defined by H(µ, ν) :=      0, if µ(X) = 0, − log κ m (x 1 , . . . , x m , ν), if µ = δ x 1 + · · · + δ xm , ∞, if µ(X) = ∞. For B ∈ X b the partition function Z B : N(X) → [0, ∞] is defined by Z B (ν) := e λ(B) N(X) e −H(µ,ν) dΠ λ B (µ) = 1 + ∞ m=1 1 m! B m κ m (x 1 , . . . , x m , ν) dλ m (x 1 , . . . , x m ), (2.3) where [24, Exercise 3.7] establishes the equality. We clearly have that Z B (ν) ≥ 1. For ν ∈ N(X) and B ∈ X b , the Gibbs measure P B,ν on N(X) is defined by P B,ν := Z B (ν) −1 e λ(B) N(X) 1{µ ∈ ·} e −H(µ,ν) dΠ λ B (µ), provided that Z B (ν) < ∞, and an expansion similar to (2.3) is possible. If Z B (ν) = ∞ we set P B,ν ≡ 0. The measure P B,ν is concentrated on N B (X), the set of all measures µ ∈ N(X) with µ(B c ) = 0, where B c := X \ B, and P B,ν is the distribution of a Gibbs process with PI κ (B,ν) (x, µ) := κ(x, ν + µ) 1 B (x) (and reference measure λ). It was proved in [27,30] that if η is a Gibbs process with PI κ then P Z B (η B c ) < ∞ = 1, B ∈ X b , and, for each measurable f : N(X) → [0, ∞), E f (η B ) | η B c = N(X) f (µ) dP B,η B c (µ), B ∈ X b ,(2.4) where relations involving conditional expectations are assumed to hold almost surely. These are the DLR-equations, cf. [35,21,26]. The Gibbs distributions we deal with are based on a non-negative pair potential v : X × X → [0, ∞], a symmetric and measurable function. In this case we define the corresponding PI κ by κ(x, µ) := exp − X v(x, y) dµ(y) , x ∈ X, µ ∈ N(X). If (1.1) holds, then a Gibbs process with this PI is known to exist, referring to [35] for the case X = R d , and to [19] for the more general case considered here. If v is allowed to take negative values the existence proofs become more complicated, see e.g. [35,26,6,8]. The random connection model In the setting of Section 2.1, suppose that ϕ : X × X → [0, 1] is a measurable and symmetric function. Let Φ = Φ(X) n=1 δ Xn be a point process. Let U i,j , i, j ∈ N, be independent random variables, uniformly distributed on the unit interval [0, 1], and such that the double sequence (U i,j ) i,j∈N is independent of Φ. Let ≺ be an order on X with {(x, y) ∈ X 2 : x ≺ y} ∈ X ⊗2 . Let X [2] denote the space of all sets e ⊂ X containing exactly two elements, which is a measurable space in its own right. We define a point process Γ on X [2] by Γ := Φ(X) i,j=1 1 X i ≺ X j , U i,j ≤ ϕ(X i , X j ) · δ {X i ,X j } . (2.5) This is the random connection model (RCM) (based on Φ) with connection function ϕ. We interpret Γ as a random graph with vertex set Φ. As a rule, there are isolated points from Φ with no emanating edges. While the definition of Γ depends on the ordering of the points of Φ, its distribution does not. We say that x, y ∈ Φ are connected via Γ if either x = y or there exist n ∈ N and e 1 , . . . , e n ∈ Γ such that x ∈ e 1 , y ∈ e n and e i ∩ e i+1 = ∅ for i ∈ {1, . . . , n − 1}. In this case we write x Γ ← → y. Let Γ z be a random connection model based on Φ z := Φ + δ z . The cluster of z in Γ z is the point process C z on X given by C z := X 1{x ∈ ·} 1{z Γ z ← → x} dΦ z (x). It charges z and all points from Φ which are connected to z via Γ z . Let us now assume that Φ is a Poisson process with intensity measure λ. Then we say that the RCM is subcritical if (1.2) holds. In this case we also say that the pair (ϕ, λ) is subcritical. If ϕ = 1 − e −v for a pair potential v, then we say that the pair (v, λ) is subcritical. Remark 2.1. Assume that Φ is a Poisson process with a non-diffuse intensity measure λ. Then some of the points of Φ come with multiplicities, and there are other, more refined, ways to define a RCM. Indeed when connecting points (taking into account multiplicities) we may introduce multiple edges between their positions. And we may also allow for (multiple) loops. The result would be a random multigraph. To avoid such technicalities we stick to the more simple and intuitive definition (2.5) and the notion of connectedness introduced thereafter. Proof of Theorem 1.1 The existence part of Theorem 1.1 is settled by Theorem B.1 of [19]. Therefore, it suffices to show that |G(v, λ)| ≤ 1. We first argue that we can assume, without loss of generality, that λ is diffuse. To do so, we use randomization, a well-known technique in point process theory, which was already tailored to Gibbs processes in [38]. We consider the spaceX := X × [0, 1] and equip it with the product metric and the productλ of λ and the Lebesgue measure on [0, 1]. Defineṽ : X ×X → [0, ∞] byṽ (x, r), (y, s) := v(x, y). LetC (z,r) denote the cluster of (z, r) ∈X in a RCM based onΦ (z,r) , whereΦ := Φ(X) n=1 δ (Xn,Un) is a uniform randomization of the Poisson process Φ = Φ(X) n=1 δ Xn on X with intensity measure λ. Here the randomization is defined with the help of independent random variables U n , n ∈ N, that are uniformly distributed on [0, 1], with the whole sequence (U n ) n∈N being independent of Φ. The processΦ is a Poisson process onX with intensity measureλ, and we havẽ C (z,r) (X) ≤ C z (X) P-a.s., with C z defined in Section 2.2. Hence, if (v, λ) is subcritical then so is (ṽ,λ) . Moreover, if η is a Gibbs process in X with PI κ and reference measure λ, it is easy to show via the GNZ-equations (2.1) that these equations hold with (η, λ, κ) replaced by (η,λ,κ), whereη is a uniform randomization of η and κ (x, r), ψ = κ x, ψ(· × [0, 1]) . In particular, each uniform randomization of η is a Gibbs process with PIκ and reference measureλ. Thus, if η is a point process with P η ∈ G(v, λ), where P X denotes the distribution of a random element X, then Pη ∈ G(ṽ,λ) for any uniform randomizationη of η. Now, if (v, λ) is subcritical and Theorem 1.1 holds for diffuse reference measures, then |G(ṽ,λ)| = 1 as (ṽ,λ) is subcritical. Consequently, uniform randomizationsη,η ′ of two point processes η, η ′ with P η , P η ′ ∈ G(v, λ) satisfy Pη = Pη ′ , and we obtain, for each A ∈ N (X), P(η ∈ A) = P η(· × [0, 1]) ∈ A = P η ′ (· × [0, 1]) ∈ A = P(η ′ ∈ A). We conclude that P η = P η ′ , that is, |G(v, λ)| = 1. In the remainder of the section we let the space (X, d) and the pair (v, λ) be as in the introduction and assume, in addition, that λ is diffuse. The definition of N(X) (see Section 2.1) is based on the collection X b of d-bounded Borel subsets of X. For the moment we dispense with the subcriticality assumption on (v, λ) and suppose that only the weaker assumption (1.1) holds. It is an easy exercise to construct sets B 1 ⊂ B 2 ⊂ · · · in X b such that ∞ ℓ=1 B ℓ = X and B ℓ X 1 − e −v(x,y) dλ(y) dλ(x) < ∞, ℓ ∈ N. (3.1) We denote by X * b the collection of all sets from X which are contained in one of the B ℓ . The collection X * b is a ring over X with σ(X * b ) = X . Moreover, the algebra Z := C∈X * b N C (X) generates N (X), where we denote by N C (X) the sub-σ-field of N (X) generated by all maps µ → µ(D) for D ∈ X with D ⊂ C. This precise construction of X * b is motivated by [19,Equation (B.6)] and it proves useful later on in order to avoid additional integrability assumptions. Put B 0 := ∅. For each ℓ ∈ N we consider a Borel isomorphism ι ℓ from the Borel space B ℓ \ B ℓ−1 , X ∩ (B ℓ \ B ℓ−1 ) onto a Borel subset of (ℓ − 1, ℓ]. Define the injective and measurable map ι : X → (0, ∞) as ι(x) := ∞ k=1 ι k (x) 1 B k \B k−1 (x) and define an order on X via x ≺ y iff ι(x) < ι(y). Observe that for x ∈ B ℓ and y ∈ B c ℓ we always have x ≺ y. Put ϕ := 1 − e −v as a function on X × X. In order to apply the disagreement coupling from [23], we want to approximate the RCM with connection function ϕ by suitably interpreting a Poisson process on a rich product space. To this end, we consider as a mark space M := [0, 1] N×N , the space of doubly indexed sequences in [0, 1], endowed with the product σ-field, and denote by Q the probability measure on M given as an infinite product of uniform distributions on [0, 1]. Moreover, we need to be able to separate the points in our space with the help of countable partitions. Thus, for each δ > 0, we let D δ 1 , D δ 2 , . . . ∈ X be a partition of X such that for any two points x, y ∈ X there exists some δ 0 > 0 so that x and y are separated by the δ-partition for every δ < δ 0 , where the points being separated means that they lie in different sets of the partition. Such directed partitions can always be constructed in separable metric spaces. Define the measurable map R δ : (X × M) 2 → [0, 1] as R δ (x, r, y, s) := ∞ i,j=1 1 D δ i (x)1 D δ j (y) 1{x ≺ y} · r i,j + 1{y ≺ x} · s j,i as well as a relation ∼ δ on X × M via (x, r) ∼ δ (y, s) ⇐⇒ R δ (x, r, y, s) ≤ ϕ(x, y). Similar to N(X), denote by N(X × M) the set of all measures ψ on X × M such that ψ(B × M) ∈ N 0 for each B ∈ X b , endowed with the apparent σ-field. Whenever ψ is a measure on X × M we writē ψ := ψ(· × M) for its projection onto X. Conversely, if µ is a counting measure on X, we construct a measureμ on X × M by endowing each point of µ with a fixed (but arbitrary) mark s ∈ M. Given some set B ∈ X and a measure ψ on X × M, we write ψ B := ψ · ∩(B × M) for the restriction of ψ onto B × M, and we denote in a generic way by N f s the set of finite simple counting measures. Here a counting measure is understood to be simple if it assigns to one-point-sets measure either 0 or 1. For a point (x, r) ∈ X × M and a set S ⊂ X × M we write (x, r) ∼ δ S if (x, r) ∼ δ (y, s) for some (y, s) ∈ S, and likewise (x, r) ∼ δ S if (x, r) is not connected to any point in S via ∼ δ . We also use this notation for ψ ∈ N(X × M), formally meaning that S is chosen as S = supp ψ := (y, s) ∈ X × M : ψ {(y, s)} > 0 . We say that (x, r) and (y, s) are connected via ψ (and ∼ δ ), written as (x, r) ψ ∼ δ (y, s), if there exist k ∈ N 0 and (y 1 , s 1 ), . . . , (y k , s k ) ∈ ψ such that (y j , s j ) ∼ δ (y j+1 , s j+1 ) for j = 0, . . . , k, with (y 0 , s 0 ) := (x, r) and (y k+1 , s k+1 ) := (y, s). For δ > 0, let κ δ : X × M × N(X × M) → [0, ∞) be given through κ δ (x, r, ψ) := 1 (x, r) ∼ δ ψ = exp − X×M − log 1 (x, r) ∼ δ (y, s) dψ(y, s) . The map κ δ is obviously measurable and corresponds to the PI of a pair interaction Gibbs process with hard core type pair potential (x, r), (y, s) → ∞·1 (x, r) ∼ δ (y, s) . For (x, r) ∈ X×M and ψ ∈ N(X×M) we call C δ (x, r, ψ) := X×M 1 (y, s) ∈ · 1 (x, r) ψ ∼ δ (y, s) dψ(y, s) the ψ-cluster of (x, r) with respect to ∼ δ . It is easy to see that (x, r, ψ) → C δ (x, r, ψ) ∈ N(X × M) is a measurable mapping. Note that (x, r) ∼ δ ψ iff C δ (x, r, ψ) = 0, where 0 denotes the null measure on X × M. Therefore κ δ (x, r, ψ) = κ δ x, r, C δ (x, r, ψ) . Define κ : X × N(X) → [0, ∞), κ(x, µ) := exp − X v(x, y) dµ(y) , the PI of a Gibbs process on X with pair potential v. As in Section 2.1, we denote by P B,ν the distribution of a (finite) Gibbs process with PI κ (B,ν) (and reference measure λ), where B ∈ X b and ν ∈ N(X). We now prove, in two steps, a projection property of Gibbs processes. More specifically, we show that (in the limit δ ↓ 0) the projection of a Gibbs process on X × M with PI κ δ onto X gives a Gibbs process with PI κ. The projection property that is established in Proposition 2.1 of [11], though dealing specifically with the Potts model in R d , is conceptually related. Lemma 3.1. Let ℓ ∈ N and ψ ∈ N f s (X × M) with ψ(B ℓ × M) = 0. For each δ > 0, let ξ δ be a Gibbs process on X × M with PI κ (B×M,ψ) δ and reference measure λ ⊗ Q. Then, for every set E ∈ N (X), lim δ↓0 P ξ δ ∈ E = P B ℓ ,ψ (E). Proof. For notational convenience we abbreviate B := B ℓ . For δ > 0 and E ∈ N (X), the probability P ξ δ ∈ E is given by 1 Z δ,B×M (ψ) 1 E (0) + ∞ m=1 1 m! B m 1 E m j=1 δ x j · M m κ δ,m (x 1 , r 1 , . . . , x m , r m , ψ) dQ m (r 1 , . . . , r m ) dλ m (x 1 , . . . , x m ) , where Z δ,B×M is the partition function corresponding to the PI κ δ and the measure λ ⊗ Q. Denote by y 1 , . . . , y k ∈ B c the points ofψ. For x 1 , . . . , x m ∈ B with x 1 ≺ . . . ≺ x m we have x m ≺ y j for each j ∈ {1, . . . , k}, and we can find δ 0 > 0 such that the points x 1 , . . . , x m , y 1 , . . . , y k lie in different sets of the δ-partition for each δ < δ 0 . By definition of κ δ , for each such choice we get M m κ δ,m (x 1 , r 1 , . . . , x m , r m , ψ) dQ m (r 1 , . . . , r m ) = 1≤i<j≤m 1 − ϕ(x i , x j ) m i=1 k j=1 1 − ϕ(x i , y j ) = κ m (x 1 , . . . , x m ,ψ). With the symmetry properties of κ δ,m and the fact that λ is diffuse, dominated convergence (using κ δ,m ≤ 1) implies for each F ∈ N (X) that lim δ↓0 ∞ m=1 1 m! B m 1 F m j=1 δ x j · M m κ δ,m (x 1 , r 1 , . . . , x m , r m , ψ) dQ m (r 1 , . . . , r m ) dλ m (x 1 , . . . , x m ) = ∞ m=1 1 m! B m 1 F m j=1 δ x j κ m (x 1 , . . . , x m ,ψ) dλ m (x 1 , . . . , x m ). Applied to F = N(X) this yields lim δ↓0 Z δ,B×M (ψ) = Z B (ψ). A further application of the limit relation (to F = E), and the observation from the beginning of this proof, imply the claim. Let P δ B ℓ ×M,ψ denote the distribution of a (finite) Gibbs process with PI κ (B ℓ ×M,ψ) δ and reference measure λ ⊗ Q. Then the previous lemma reads as lim δ↓0 P δ B ℓ ×M,ψ ν ∈ N(X × M) :ν ∈ E = P B ℓ ,ψ (E), E ∈ N (X). However, this relation being true only for finite boundary conditions ψ is not enough to consider infinite range interactions. Fortunately the integrability assumption on the pair potential v allows us to extract more information. Recall that we associate with each µ ∈ N(X) a measureμ ∈ N(X × M) by endowing each point of µ with a fixed mark s ∈ M. Also, keep in mind that any Gibbs process with a diffuse reference measure is simple, which follows immediately from the DLR equations (2.4) and the fact that a Poisson process with diffuse intensity measure is simple (cf. Proposition 6.9 of [24]). Lemma 3.2. Let ℓ ∈ N. Let η denote a Gibbs process on X with PI κ and reference measure λ. Then we have for each E ∈ N (X) that By monotone convergence we have lim n→∞ κ(x, µ Bn ) = κ(x, µ) for all x ∈ X and µ ∈ N(X). Thus, by definition of κ m and dominated convergence (using that κ m ≤ 1), lim δ↓0 E P δ B ℓ ×M,η B c ℓ ν ∈ N(X × M) :ν ∈ E − P B ℓ ,η B c ℓ (E) = 0.lim n→∞ E κ m (x 1 , . . . , x m , η B c ) − κ m (x 1 , . . . , x m , η Bn\B ) = 0. For each fixed n > ℓ, the proof of Lemma 3.1 and dominated convergence yield r 1 , . . . , x m , r m ,η Bn\B ) dQ m (r 1 , . . . , r m ) = 0. lim δ↓0 E κ m (x 1 , . . . , x m , η Bn\B ) − M m κ δ,m (x 1 , As for the last term in (3.2), recall that κ δ,m is nothing but an indicator function. More precisely, κ δ,m (x 1 , r 1 , . . . , x m , r m ,η B c ) is equal to 1 if there is no ∼ δ -connection between any of the points (x 1 , r 1 ), . . . , (x m , r m ) and none of these points is connected toη B c , and it is equal to 0 otherwise. If κ δ,m (x 1 , r 1 , . . . , x m , r m ,η B c ) = 1 then clearly also κ δ,m (x 1 , r 1 , . . . , x m , r m ,η Bn\B ) = 1. Hence the only situation in which the difference appearing in the last term of (3.2) can give a value different from 0 is if one of the points (x 1 , r 1 ), . . . , (x m , r m ) is connected toη B c n . Hence, we obtain for each δ > 0 and n ∈ N that I δ,n := M m E κ δ,m (x 1 , r 1 , . . . , x m , r m ,η Bn\B ) − κ δ,m (x 1 , r 1 , . . . , x m , r m ,η B c ) dQ m (r 1 , . . . , r m ) ≤ M m E 1 (x j , r j ) ∼ δηB c n for at least one j ∈ {1, . . . , m} dQ m (r 1 , . . . , r m ) ≤ m j=1 E B c n ×M M m 1 (x j , r j ) ∼ δ (x, r) dQ m (r 1 , . . . , r m ) dη(x, r) . By construction, the marks ofη are not used in any decision about connections in the above term as x 1 , . . . , x m are points in B and thus always smaller than points in η B c n with respect to the order ≺ on X, so only the marks r 1 , . . . , r m matter. Thus, we arrive at I δ,n ≤ m j=1 E B c n M m 1 (x j , r j ) ∼ δ (x, s) dQ m (r 1 , . . . , r m ) dη(x) = m j=1 E B c n ϕ(x j , x) dη(x) ≤ m j=1 B c n ϕ(x j , x) dλ(x) using that η is a Gibbs process with PI κ (and κ ≤ 1). This last term, however, goes to 0 as n → ∞ by (1.1), and so does I δ,n (uniformly in δ). Therefore, the left-hand side of (3.2) converges to 0 as δ ↓ 0. Dominated convergence implies for each E ∈ N (X) that L 1 (P) −→ ∞ m=1 1 m! B m 1 E m j=1 δ x j κ m (x 1 , . . . , x m , η B c ) dλ m (x 1 , . . . , x m ) as δ ↓ 0. It follows immediately that Z δ,B×M (η B c ) L 1 (P) −→ Z B (η B c ) as well as, for each E ∈ N (X), Z δ,B×M (η B c ) · P δ B×M,η B c ν ∈ N(X × M) :ν ∈ E L 1 (P) −→ Z B (η B c ) · P B,η B c (E) both as δ ↓ 0. Using that the occurring partition functions are always ≥ 1 and bounded by e λ(B) , the assertion follows from dominated convergence. Apart from this projection property of the hard core type Gibbs processes in the extended state space, the construction via the relation ∼ δ has another useful feature. It allows an approximation of the RCM by considering a Poisson process on X × M and constructing the connections via ∼ δ . In the following result we show that in the limit δ ↓ 0 the RCM is indeed recovered. Ψ B ℓ + µ + δ x . Then, lim δ↓0 M P (x, r) Ψ B ℓ ∼ δμ dQ(r) = P x Γ x,µ B ℓ ← − → µ . Proof. Set B := B ℓ . Recalling that a Poisson process with diffuse intensity measure is simple, we have that, for PΨ B -a.e. ν ∈ N f s (X) ∩ N B (X) (write ν = k j=1 δ x j ) the conditional distribution of Ψ B given Ψ B = ν is M k 1 k j=1 δ (x j ,r j ) ∈ · dQ k (r 1 , . . . , r k ). Choose δ > 0 so small that the (finitely many) points in µ + ν are separated by the δ-partition. By definition of ∼ δ (rendering the marks ofμ irrelevant) and of the RCM we have M P (x, r) Ψ B ∼ δμ |Ψ B = ν dQ(r) = P x Γ x,µ B ← − → µ |Ψ B = ν . Dominated convergence gives lim δ↓0 M P (x, r) Ψ B ∼ δμ dQ(r) = lim δ↓0 N(X) M P (x, r) Ψ B ∼ δμ |Ψ B = ν dQ(r) dPΨ B (ν) = N(X) P x Γ x,µ B ← − → µ |Ψ B = ν dPΨ B (ν) = P x Γ x,µ B ← − → µ , as asserted. Just like for the projection property, a suitable approximation allows to consider in Lemma 3.3 infinite boundary conditions (coming from a Gibbs process). The proof of the following result shows why we have introduced the collection X * b . Lemma 3.4. Let Ψ be a Poisson process on X × M with intensity measure λ ⊗ Q. Let ℓ ∈ N and C ∈ X with C ⊂ B ℓ . Suppose that η is a Gibbs process on X with PI κ (i.e., with pair potential v) and reference measure λ, such that η is independent of Ψ and independent of the double sequence that is used to construct the RCMs. For each x ∈ X we let Γ x,η ℓ be a RCM with connection function ϕ = 1 − e −v based onΨ B ℓ + η B c ℓ + δ x . Then, lim δ↓0 C×M P (x, r) Ψ B ℓ ∼ δηB c ℓ d(λ ⊗ Q)(x, r) = C P x Γ x,η ℓ ← − → η B c ℓ dλ(x). Proof. For n ∈ N with n > ℓ we write Γ x,η ℓ,n for the restriction of Γ x,η ℓ onto B n , meaning that only vertices inside B n and their connections among each other remain. For such n, we have C×M P (x, r) Ψ B ℓ ∼ δηB c ℓ d(λ ⊗ Q)(x, r) − C P x Γ x,η ℓ ← − → η B c ℓ dλ(x) ≤ C×M P (x, r) Ψ B ℓ ∼ δηB c ℓ − P (x, r) Ψ B ℓ ∼ δηB n\Bℓ d(λ ⊗ Q)(x, r) + C×M P (x, r) Ψ B ℓ ∼ δηB n\Bℓ d(λ ⊗ Q)(x, r) − C P x Γ x,η ℓ,n ← − → η Bn\B ℓ dλ(x) + C P Γ x,η ℓ,n ← − → η Bn\B ℓ − P x Γ x,η ℓ ← − → η B c ℓ dλ(x). (3.3) We consider the three terms that appear in (3.3) separately. Let δ > 0. As for the first term, note that C×M P (x, r) Ψ B ℓ ∼ δηB c ℓ − P (x, r) Ψ B ℓ ∼ δηB n\Bℓ d(λ ⊗ Q)(x, r) ≤ C×M E 1 (x, r) Ψ B ℓ ∼ δηB c ℓ − 1 (x, r) Ψ B ℓ ∼ δηB n\Bℓ d(λ ⊗ Q)(x, r) ≤ C×M E 1 (x, r) Ψ B ℓ ∼ δηB c n d(λ ⊗ Q)(x, r). If (x, r) ∈ C × M is ∼ δ -connected via Ψ B ℓ toη B c n , then either (x, r) ∼ δ (y, s) for some point y ∈ η B c n or one of the Poisson points is connected toη B c n . Thus, the previous term is bounded by C×M E B c n 1 (x, r) ∼ δ (y, s) dη(y) d(λ ⊗ Q)(x, r) + λ(C) · E B ℓ ×M B c n 1 (z, t) ∼ δ (y, s) dη(y) dΨ(z, t) . Using that η is independent of Ψ and that the PI κ is bounded by 1, the above is bounded by C×M B c n 1 (x, r) ∼ δ (y, s) dλ(y) d(λ ⊗ Q)(x, r) + λ(C) B ℓ ×M B c n 1 (z, t) ∼ δ (y, s) dλ(y) d(λ ⊗ Q)(z, t). As B ℓ ⊂ B n , the points in B ℓ are always smaller (w.r.t. ≺) than the points in B c n , so by construction of ∼ δ (and Q), the latter sum equals C B c n ϕ(x, y) dλ(y) dλ(x) + λ(C) B ℓ B c n ϕ(z, y) dλ(y) dλ(z), which converges to 0 as n → ∞ by dominated convergence, using (3.1). Hence, the first term on the right hand side of (3.3) converges to 0 as n → ∞ uniformly in δ. The second term in (3.3) converges to 0 as δ ↓ 0 (for each fixed n > ℓ) by the independence of η and Ψ, dominated convergence, and Lemma 3.3. As for the third term on the right hand side of (3.3), note that C P x Γ x,η ℓ,n ← − → η Bn\B ℓ − P x Γ x,η ℓ ← − → η B c ℓ dλ(x) ≤ C E 1 x Γ x,η ℓ,n ← − → η Bn\B ℓ − 1 x Γ x,η ℓ ← − → η B c ℓ dλ(x). The difference of the indicator functions appearing in the expectation can only be distinct from 0 if there exists a connection from x ∈ C to η B c n (via Γ x,η ℓ ) which uses no point in η Bn\B ℓ . Thus, either x or one of the Poisson points in B ℓ has to be connected (directly) to one of the points in η B c n . Together with the given independence properties, the quantity is therefore further bounded by C E B c n ϕ(x, y) dη(y) dλ(x) + C E B ℓ B c n ϕ(z, y) dη(y) dΨ(z) dλ(x) ≤ C B c n ϕ(x, y) dλ(y) dλ(x) + λ(C) B ℓ B c n ϕ(z, y) dλ(y) dλ(z). By choice of C and B ℓ , referring to (3.1), dominated convergence implies that the above term converges to 0 as n → ∞. Summarizing, we see that the left hand side of (3.3) tends to zero as δ ↓ 0. Before we prove our main result, we need to investigate the behavior of the RCM in the subcritical regime. More specifically, we need to establish that the probability of a point x being connected to a (pair potential-) Gibbs process point in B c ℓ via the RCM based on a Poisson process on B ℓ goes to 0 as ℓ → ∞. Lemma 3.5. Assume that (v, λ) is subcritical. Let Φ be a Poisson process on X with intensity measure λ. Suppose that η is a Gibbs process with PI κ and reference measure λ, such that η is independent of Φ and independent of the double sequence that is used to construct the RCMs. For each x ∈ X let Γ x,η ℓ be a RCM with connection function ϕ = 1 − e −v and vertex set Φ B ℓ + η B c ℓ + δ x . Then, lim ℓ→∞ P x Γ x,η ℓ ← − → η B c ℓ = 0, λ-a.e. x ∈ X. Proof. First of all, observe that since κ ≤ 1, we can assume without loss that η ≤ Φ ′ almost surely, where Φ ′ is a Poisson process with intensity measure λ independent of Φ and independent of the double sequence used for the RCM. This follows from an extension of Example 2.1 in [12] to unbounded λ, explicitly using that X is a complete separable metric space (comparable to Lemma 5.3 in [23]). Thus, for each x ∈ X and each ℓ ∈ N, we have that P x Γ x,η ℓ ← − → η B c ℓ ≤ P x Γ x,Φ ′ ℓ ← −− → Φ ′ B c ℓ , where Γ x,Φ ′ ℓ denotes the RCM with connection function ϕ based on Φ B ℓ + Φ ′ B c ℓ + δ x . Since the two Poisson processes are independent, Φ B ℓ + Φ ′ B c ℓ is (for every ℓ ∈ N) a Poisson process on X with intensity measure λ B ℓ + λ B c ℓ = λ. Therefore (again using the independence), we can replace Φ ′ B c ℓ by Φ B c ℓ and Γ x,Φ ′ ℓ by Γ x (the RCM based on Φ + δ x ) in the above probability, which yields P x Γ x,η ℓ ← − → η B c ℓ ≤ P x Γ x ← → Φ B c ℓ . However, if x is connected via Γ x to Φ B c ℓ , then the cluster of x in Γ x has at least one point in B c ℓ . Thus, the probability in question is bounded by P C x (B c ℓ ) > 0 . Since (v, λ) is assumed to be subcritical the latter probability tends to 0 as ℓ → ∞ for λ-a.e. x ∈ X. We proceed to prove our main result. Though the technical details harnessed by the previous lemmata as well as our formal description differ from the proofs of existing uniqueness results, the last steps taken in the following proof retain a conceptual similarity to equation (4.4) of [41], where the disagreement coupling was originally introduced for the discrete setting. Proof of Theorem 1.1. Fix ℓ ∈ N and let Ψ be a Poisson process on X × M with intensity measure λ ⊗ Q. For each ψ, ψ ′ ∈ N B c ℓ ×M (X × M) and every δ > 0, Theorem 6.3 of [23] provides us with a Gibbs process ξ δ on X × M with PI κ (B ℓ ×M,ψ) δ and a Gibbs process ξ ′ δ in X × M with PI κ (B ℓ ×M,ψ ′ ) δ such that ξ δ ≤ Ψ and ξ ′ δ ≤ Ψ almost surely, and such that each point in |ξ δ − ξ ′ δ | is ∼ δ -connected via ξ δ + ξ ′ δ to some point in ψ + ψ ′ . Hereby |ν| denotes the total variation measure of a signed measure ν on X × M, so, in the case of two counting measures ν, ν ′ ∈ N f s (X × M), the measure |ν − ν ′ | ∈ N f s (X × M) comprises those points in which ν and ν ′ differ. For any E ∈ N C (X) with C ∈ X and C ⊂ B ℓ , we obtain P δ B ℓ ×M,ψ ν ∈ N(X × M) :ν ∈ E − P δ B ℓ ×M,ψ ′ ν ∈ N(X × M) :ν ∈ E = P ξ δ ∈ E − P ξ ′ δ ∈ E ≤ max P ξ δ ∈ E,ξ ′ δ / ∈ E , P ξ δ / ∈ E,ξ ′ δ ∈ E . Since E ∈ N C (X), the events in the probability measures on the right hand side can only occur if the restrictions ofξ δ andξ ′ δ onto C differ, so the term is bounded by P (ξ δ ) C = (ξ ′ δ ) C ≤ P |ξ δ − ξ ′ δ |(C × M) > 0 . Since each point in |ξ δ − ξ ′ δ | ≤ Ψ is (∼ δ -)connected via Ψ B ℓ to some point in ψ + ψ ′ , a further bound is given through E C×M 1 (x, r) Ψ B ℓ ∼ δ (ψ + ψ ′ ) dΨ(x, r) = E C×M 1 C δ (y, t, Ψ B ℓ ) {(x, r)} > 0 for some (y, t) ∈ (ψ + ψ ′ ) dΨ(x, r) . By Mecke's equation, [24,Theorem 4.1], this last term equals C×M P C δ y, t, Ψ B ℓ + δ (x,r) {(x, r)} > 0 for some (y, t) ∈ (ψ + ψ ′ ) d(λ ⊗ Q)(x, r) = C×M P (x, r) Ψ B ℓ ∼ δ (ψ + ψ ′ ) d(λ ⊗ Q)(x, r). Now, let η and η ′ be two Gibbs processes on X with PI κ and reference measure λ. Assume, without loss of generality, that (η, η ′ ) is independent of Ψ and independent of the double sequence used to define the RCMs. Let C ∈ X * b be arbitrary and choose ℓ large enough so that C ⊂ B ℓ . Take E ∈ N C (X). By the DLR-equation (2.4), Lemma 3.2, and the above bound, we obtain P η (E) − P η ′ (E) = E P B ℓ ,η B c ℓ (E) − E P B ℓ ,η ′ B c ℓ (E) ≤ E P B ℓ ,η B c ℓ (E) − P B ℓ ,η ′ B c ℓ (E) = lim δ↓0 E P δ B ℓ ×M,η B c ℓ ν ∈ N(X × M) :ν ∈ E − P δ B ℓ ×M,η ′ B c ℓ ν ∈ N(X × M) :ν ∈ E ≤ lim sup δ↓0 C×M P (x, r) Ψ B ℓ ∼ δ (η B c ℓ +η ′ B c ℓ ) d(λ ⊗ Q)(x, r) ≤ lim sup δ↓0 C×M P (x, r) Ψ B ℓ ∼ δηB c ℓ d(λ ⊗ Q)(x, r) + lim sup δ↓0 C×M P (x, r) Ψ B ℓ ∼ δη ′ B c ℓ d(λ ⊗ Q)(x, r). Applying Lemma 3.4 and dominated convergence to each of the two terms, we arrive at P η (E) − P η ′ (E) ≤ C P x Γ x,η ℓ ← − → η B c ℓ dλ(x) + C P x Γ x,η ′ ℓ ← −− → η ′ B c ℓ dλ(x). (3.4) Using Lemma 3.5 and dominated convergences (for ℓ → ∞) twice, the right hand side of (3.4) is seen to converge to 0. As C ∈ X * b and E ∈ N C were arbitrary, the measures P η and P η ′ agree on the algebra Z which generates N . Hence, P η ≡ P η ′ and the proof is complete. Comments and examples In this section we first work in the general setting of the introduction, that is we fix a complete separable metric space (X, d) equipped with a locally finite measure λ, and let v be a non-negative pair potential. The next result is an immediate consequence of Corollary 1.3. Then |G(v, γλ)| = 1 for all sufficiently small γ ≥ 0. v(x, y) dλ(y) < ∞. Then |G(βv, λ)| = 1 for all sufficiently small β ≥ 0. Proof. Since 1 − e −βv ≤ βv the result follows from Corollary 1.3 and dominated convergence. The constant β in Corollary 4.2 can be interpreted as inverse temperature. Example 4.3. Suppose that X equals the space C d of all compact subsets of R d , equipped with the Hausdorff metric and a translation invariant locally finite measure λ, cf. [36]. Let V : C d → [0, ∞] be measurable with V (∅) = 0. For instance V could be the volume or, if λ is concentrated on the convex bodies, a linear combination of the intrinsic volumes. Assume that the pair potential is given by v(K, L) = V (K ∩ L), K, L ∈ C d . As a percolation model, the associated RCM with connection function ϕ = 1 − e v is considerably more general than the (Poisson driven) Boolean model, studied (for spherical bodies), for instance, in [29]. The latter arises in the special case V (K) = ∞ · 1{K = ∅}. Then the connection function is given by ϕ ∞ (K, L) := 1{K ∩ L = ∅}, so that the connections do not involve any additional randomness. The corresponding Gibbs model are hard particles in equilibrium, while the case of a general V could be addressed as soft particles in equilibrium, at least if V is translation invariant. Theorem 1.1 requires (ϕ, λ) to be subcritical, while the previous results from [16,1] (when specialized to non-negative pair potentials) require the Boolean model (ϕ ∞ , λ) to be subcritical. Since ϕ(K, L) ≤ ϕ ∞ (K, L) our result gives better bounds on the uniqueness region. In particular, ϕ(K, L) < ϕ ∞ (K, L) whenever V (K ∩ L) > 0. If, for instance, V is continuous at ∅, ϕ(K, L) can be arbitrarily small, and still ϕ ∞ (K, L) = 1. Remark 4.4. Since we can allow for a non-diffuse intensity measure, our results cover the case of a discrete graph G = (V, E). We may then take X = V and λ = γλ 0 , where λ 0 is the counting measure on V and γ > 0. A possible choice of a connection function is ϕ(x, y) := p if {x, y} ∈ E and ϕ(x, y) = 0, otherwise, where p is a given probability. The resulting RCM is then a Poisson version of a mixed percolation model, see [2]. However, ϕ could also be long-ranged as in [4]. In fact, it easy to come up with a version of the model in [4] driven by a Poisson process on Z d . In principle it might be possible to apply our uniqueness results to discrete models of statistical physics. We leave this for future research. Remark 4.5. It is believed (see e.g. [5] and the references given there) that in many Gibbs models there exist γ * > 0 such that |G(v, γλ)| ≥ 2 provided that γ > γ * . We expect γ c , as defined by (1.3), to be (much) smaller than γ * . A careful analysis of our proofs suggests that a possible (but rather implicit) approximation of γ * is a critical intensity, which is defined in terms of RCMs based on suitable finite volume versions of a Gibbs process with pair potential v, see e.g. [11,Proposition 3.1]. This is also supported by the discussion in [13]. Simulation results for the critical thresholds Upon comparing the RCM with a branching process, our general Theorem 1.1 implies Corollary 1.3 which corresponds with the uniqueness result from [18]. However, it is known from simulations that for the Gilbert graph, which corresponds to the hard sphere model, the branching bounds are widely off the actual critical intensity in lower dimensions. For an overview, we refer to [43]. Thus, we expect Theorem 1.1 to yield substantial improvements of Corollary 1.3 in low dimensions. To illustrate this point, we provide simulation results that give a rough overview on this difference in various models. To approximate the true critical intensity of the RCM, we proceed as follows. For a given connection function in R d with finite range R > 0, for instance one coming from a (finite range) pair potential, in a given dimension d and for a given intensity γ > 0, we fix a large system size S > R. In the ball of radius S around the origin we now construct the cluster of the origin in the RCM based on a stationary Poisson process with intensity γ augmented by the origin. We start by simulating Poisson points (according to the given intensity) in the ball of radius R around the origin, corresponding to all points which could possibly be connected to the origin and we check each of those points for such a connection (only to the origin). In the following, we keep track of three types of points, namely saturated points which are part of the cluster and whose perspective has already been taken (which after the first step includes only the origin), those points which are part of the cluster but around which we might still have to simulate new Poisson points, and those Poisson points which are not yet connected to the cluster. Then we proceed algorithmically as follows. Of those cluster points from whose perspective we have not yet simulated we choose that point x which is furthest away from the origin to take its perspective, meaning that we check if that particular point connects to any of the Poisson points which already exist but are not yet part of the cluster, and then proceed to simulate new Poisson points (according to the given intensity) in that part of B(x, R) which was not yet covered in previous steps and we check if any of the new points connects to x. Note that we do not look for connections between two cluster points as we already know that both are part of the cluster and an additional connection between them does not change the size of the cluster. Also notice that when we first generate new points, we do not check for connections among them immediately (and only for connections to the center of the given step), but as soon as we take the perspective of any of the cluster points we check for connections to the points in its neighborhood and thus miss no relevant connection. The algorithm terminates as soon as all cluster points are saturated (and hence the construction of the cluster has died out within B(0, S)) or if the cluster connects to the complement of B(0, S), meaning that some point in the cluster has a norm larger than S. To make a decision whether the RCM percolates for a given intensity, we construct the cluster 5, 000 times. If the cluster connects to the complement of B(0, S) a single time, we count that as percolation, even though, of course, a larger initial choice of the system size might have revealed that the cluster is actually finite. If the cluster lies within B(0, S) in each of the 5, 000 runs, we count this as no percolation. To find a rough approximation of the critical intensity, we start at the branching lower bound, which is easily calculated for a given model, and increase the intensity by 10% as long as our algorithm decides that no percolation occurs. As soon as we first encounter percolation, we accept that intensity as an upper bound and the last intensity at which no percolation occurred as a lower bound. To refine the approximation further, we then slice the resulting interval by half two times, investigating the middle between the two bounds for percolation and adjusting the upper and lower bound accordingly. Note that the choices for S and the number of runs where made according to our computational resources, larger values for both quantities will surely lead to better estimates, but one has to observe that there is no theoretical guarantee that the simulation will provide lower bounds for the critical intensities (see the discussion below). Note that a highly related algorithmic way for exploring a cluster is explained in Chapter 5.2 of [20] for the discrete setting with finitely many vertices. To establish that the simulations yield plausible and useful results, we consider as a benchmark the hard sphere model with pair potential v R (x, y) = ∞ · 1{|x − y| ≤ R}, x, y ∈ R d , which corresponds to the Gilbert graph with interaction range R, whose connection function is given through ϕ R (x, y) = 1 − exp − v(x, y) = 1 |x − y| ≤ R . The percolation properties of the Gilbert graph, in turn, are exactly those of the Boolean model with grains being balls with fixed radius R/2. The branching lower bound on the critical intensity is sup x∈R d R d ϕ R (x, y) dy −1 = 1 V d B(0, R) . In the simulations, we fix R = 2 and compare the approximation of our method of simulation with the branching bounds and some of the best approximations for the critical intensity from the literature, namely the results from [39]. Note that it is common in physics to investigate the percolation behavior in terms of the reduced number density, so we had to convert the values from [39] into corresponding critical intensities by dividing with the volume of the unit ball in the given dimension. Throughout we round all values to five significant digits. The comparison between our approximation and the lower bounds from [39] (with the correction reported in [40]), which are known to be very precise, show that our simulations, even with the very manageable choice of the simulation parameters, are reasonably well calibrated in that they provide conservative lower bounds for the critical intensity which are not wide off the mark and thus provide a solid reference for the order of magnitude of the critical intensity. The critical intensities of the Gilbert graph, the table immediately shows, are substantially larger than the branching lower bounds, which implies that our bounds on the region of uniqueness in the hard sphere model is substantially larger than the bounds by [18]. For the hard spheres model this is not a new observation since our results agree (in this specific model) with the earlier disagreement percolation results by [16]. This is due to the fact that, as discussed in Section 1, the range of the potential is the only information taken into account by [16] and for the hard spheres model it happens to be the only relevant parameter. In the upcoming examples a further improvement can be observed. Next we consider a modification of the hard sphere model, where an arbitrary overlap of spheres is possible, namely the penetrable spheres model considered by [25]. Let 0 < c < ∞ and consider v(x, y) = c · 1{|x − y| ≤ R}, x, y ∈ R d . The parameter c (which in the hard sphere model is ∞) gives a measure on how valiantly spheres resist an overlap, but as c is a fixed constant, the manner of the overlap plays no role in the spheres resistance of it. The RCM corresponding to this interaction function has connection function ϕ(x, y) = (1 − e −c ) · 1{|x − y| ≤ R}. As we simulate from the RCM perspective, we parameterize p = 1 − e −c ∈ (0, 1) (in our case p = 0.5 and p = 0.75) which is then the probability that any two points with distance less than R connect. Hence, the model can be interpreted as a modified Gilbert graph with an adjusted connection probability. The branching lower bounds for this RCM are simply those from the Gilbert graph divided by p. In order to be able to compare the simulation results for the different models, we again fix R = 2. Table 2: Probability-adjusted Gilbert graph with p = 0.5 The observations in the penetrable spheres model are similar to those in the case of hard spheres. Even our conservative approximations of the critical intensity, and hence the region of uniqueness, improve the branching bounds (or Dobrushin method, [18]) by factors larger than 3 in two dimensions. In five dimensions the improvement is still by factors of more than 1.5. Also the approximated values for γ c are substantially larger than in the Gilbert graph, which is an improvement over the classical disagreement Table 3: Probability-adjusted Gilbert graph with p = 0.75 percolation approach in [16], where only the range of v is taken into account. As a last model, we consider a pair interaction considered in the physics literature, namely the softsphere or inverse-power potential, which can be traced back at least to [33]. Its interaction function is given through v(x, y) = β · R n |x − y| n · 1{|x − y| ≤ R}, x, y ∈ R d , where β > 0 is the characteristic energy and n ∈ N the hardness parameters. We fix β = 1 and consider n ∈ {6, 12}. The connection function of the corresponding RCM is ϕ(x, y) = 1 − exp − R n |x − y| n · 1 |x − y| ≤ R , and the branching lower bounds for the critical intensity thus calculate as 2π d/2 Γ d 2 R 0 1 − exp − R n r n · r d−1 dr −1 . To ensure comparability, we consider R = 2. Table 5: Soft-sphere model with n = 12 The simulations show that in the soft-sphere model, with the specific parameter specifications, Theorem 1.1 yields improvements on the region of uniqueness qualitatively similar to the penetrable spheres model. The results of our simulations lead to the following observations. In the dimensions we consider, the true critical intensities are larger than the branching bounds by factors between 1.4 and 4.5, depending on the model and (mostly) on the dimension. Thus, Theorem 1.1 improves the regions of uniqueness of the corresponding Gibbs process by those very same factors as compared to [18]. Moreover, for the penetrable and soft spheres, the critical values are (in some cases significantly) higher than those of the hard spheres which stands for an according improvement of the results by [16]. While it is known from equation (6) of [32] that (for the Gilbert graph) the branching bound improves as the dimension grows, it also seems to improve (if much less so) if the overall connection probability in the RCM decreases and the critical intensity thus rises. Note that our approximation approach is really just an approximation and not founded on a solid theoretical basis. The benchmark model (hard spheres) and the corresponding existing simulations indicate that a further improvement of our approximations (by somewhere around 2-5%) is possible, but to our knowledge our simulations are the first for general RCMs. A different approach which would lead to approximations that come with confidence intervals is to prove a mean-field lower bound for the RCM in lines with equation (5.1) of [43] which holds for the Gilbert graph. This is an open problem that is certainly beyond the scope of this paper, but we strongly believe that a bound like this holds, at least for sufficiently regular connection functions. Still, our simulations for the Gilbert graph, where reference values from other simulations are available, suggest that our hands-on approach provides fairly solid and conservative approximations of the critical intensity of the RCM. Indeed, as we stay on the conservative side in every choice of simulation parameters, the results should slightly underestimate the corresponding true critical intensities but give a very good idea of their overall magnitude, in particular compared to the branching bounds. A Branching bounds on the random connection model In this section we provide a rigorous result as to when the RCM is subcritical. The proof is established via a bound on a suitable branching construction. We essentially work in the setting of Section 2.2, that is, we consider a Borel space (X, X ) with a σ-finite measure λ on X and localizing structure B 1 ⊂ B 2 ⊂ · · · of sets with finite λ-measure. Let ϕ : X × X → [0, 1] be a measurable and symmetric function and denote by Φ a Poisson process on X with intensity measure λ. As before, we write Γ (or Γ x , or Γ x,y ) for the RCM with connection function ϕ and vertex set Φ (or Φ + δ x , or Φ + δ x + δ y ). We define the pair connectedness function τ : X × X → [0, 1], τ (x, y) := P x Γ x,y ← − → y . We can immediately state the main result of this section, without introducing any further notation. Theorem A.1. Assume there exists a measurable function g : X → [0, ∞) such that X ϕ(x, y) dλ(y) + X ϕ(x, y) g(y) dλ(y) ≤ g(x), λ-a.e. x ∈ X. (A.1) Then, (ϕ, λ) is subcritical. Before we prove this result, we give some remarks. Remark A.2. It is easy to rewrite the theorem at hand in terms of v if ϕ is given through such a pair potential as in Section 3. A resemblance to (1.1) then becomes obvious. In fact, we can strengthen assumption (1.1) such that it implies the condition in the theorem. More precisely, assume that q := ess sup x∈X X ϕ(x, y) dλ(y) < 1. Then, choosing g ≡ q 1−q , we obtain X ϕ(x, y) dλ(y) + X ϕ(x, y) g(y) dλ(y) ≤ q + q 1 − q · q = q 1 − q = g(x), λ-a.e. x ∈ X. This is essentially the assumption in [18]. The theorem in discussion is also weaker than assumption (KPU t ) in [19]. Indeed, if there exists a measurable function g : X → [0, ∞) and some t ≥ 0 such that e t X ϕ(x, y) e g(y) dλ(y) ≤ g(x), λ-a.e. x ∈ X, then (A.1) follows from e g(y) ≥ 1 + g(y) and e t ≥ 1. In particular, Theorem A.1 improves Corollary C.1 in [19]. Proof of Theorem A.1. First of all, notice that by an extension of Mecke's equation for the RCM, as stated in (4.1) of [22], we have E C x (X) − 1 = E X 1 x Γ x ← → y dΦ(y) = X P x Γ x,y ← − → y dλ(y) = X τ (x, y) dλ(y). Observe that if two points x, y ∈ X are connected via Γ x,y , then either x and y are directly connected, or there lies at least one Poisson point in between them. Therefore, τ (x, y) ≤ ϕ(x, y) + E X ϕ(x, z) 1 z Γ y ← → y dΦ(z) = ϕ(x, y) + X ϕ(x, z) τ (z, y) dλ(z). Defining a convolution type operator in the obvious way, this inequality reads as τ ≤ ϕ + ϕ * τ. Iteration of this inequality yields that τ ≤ ϕ + ϕ * 2 + . . . + ϕ * n + (ϕ * n * τ ) for every n ∈ N. By a similar iteration of (A.1), we see that g(x) ≥ n k=1 X ϕ * k (x, y) dλ(y) + X ϕ * n (x, y) g(y) dλ(y), λ-a.e. x ∈ X, for any n ∈ N. In particular, we have g(x) ≥ ∞ k=1 X ϕ * k (x, y) dλ(y), λ-a.e. x ∈ X. For each such x ∈ X we thus have lim k→∞ X ϕ * k (x, y) dλ(y) = 0. Combining our previous observations, we find that, for λ-a.e. x ∈ X and all n, ℓ ∈ N, 1 + B ℓ τ (x, y) dλ(y) ≤ 1 + n k=1 B ℓ ϕ * k (x, y) dλ(y) + B ℓ X ϕ * n (x, z) τ (z, y) dλ(z) dλ(y) ≤ 1 + n k=1 B ℓ ϕ * k (x, y) dλ(y) + λ(B ℓ ) X ϕ * n (x, z) dλ(z). Letting n → ∞, we obtain 1 + B ℓ τ (x, y) dλ(y) ≤ 1 + ∞ k=1 B ℓ ϕ * k (x, y) dλ(y) for λ-a.e. x ∈ X and each ℓ ∈ N. With monotone convergence (let ℓ → ∞) we arrive at E C x (X) = 1 + X τ (x, y) dλ(y) ≤ 1 + ∞ k=1 X ϕ * k (x, y) dλ(y) ≤ 1 + g(x) < ∞ for λ-a.e. x ∈ X. In particular, the cluster of each such x (in Γ x ) is finite almost surely, that is to say, (ϕ, λ) is subcritical. Theorem 1 . 1 . 11Assume that (v, λ) is subcritical. Then |G(v, λ)| = 1.From a percolation perspective it might be helpful to consider for each γ ≥ 0 a RCM with connection function ϕ based on a Poisson process with intensity measure γλ. Then γ can be interpreted as an intensity (or activity). Define a critical intensity byγ c := sup{γ ≥ 0 : (v, γλ) is subcritical}. (1.3)Theorem 1.1 immediately implies the following result.Corollary 1.2. Assume that γ < γ c . Then |G(v, γλ)| = 1. Proof. As in the previous proof we abbreviate B := B ℓ . First of all, note that, for any n > ℓ and (λ m -almost) all x 1 , . . . , x m ∈ B,E κ m (x 1 , . . . , x m , η B c ) − M m κ δ,m (x 1 , r 1 , . . . , x m , r m ,η B c ) dQ m (r 1 , . . . , r m ) ≤ E κ m (x 1 , . . . , x m , η B c ) − κ m (x 1 , . . . , x m , η Bn\B ) + E κ m (x 1 , . . . , x m , η Bn\B ) −M m κ δ,m (x 1 , r 1 , . . . , x m , r m ,η Bn\B ) dQ m (r 1 , . . . , r m ) + M m E κ δ,m (x 1 , r 1 , . . . , x m , r m ,η Bn\B ) − κ δ,m (x 1 , r 1 , . . . , x m , r m ,η B c ) dQ m (r 1 , . . . , r m ). (3.2) ,m (x 1 , r 1 , . . . , x m , r m ,η B c ) dQ m (r 1 , . . . , r m ) dλ m (x 1 , . . . , x m ) Lemma 3. 3 . 3Let Ψ be a Poisson process on X × M with intensity measure λ ⊗ Q. Let ℓ ∈ N, x ∈ B ℓ and µ ∈ N f s (X) ∩ N B c ℓ (X). Denote by Γ x,µ B ℓ the RCM with connection function ϕ = 1 − e −v based on Corollary 4 . 1 . 41Assume that ess sup x∈X X 1 − e −v(x,y) dλ(y) < ∞. Table 4 : 4Soft-sphere model with n = 6d S Runs Approximation of γ c Branching bound 2 1000 5000 0.35272 0.082379 3 500 5000 0.083445 0.031387 4 300 5000 0.027011 0.013523 5 200 5000 0.010873 0.00643 AcknowledgmentsWe thank Sabine Jansen for several fruitful discussions of the topics surrounding our research. Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics. V Beneš, C Hofer-Temmel, G Last, J Večeřa, J. Appl. Probab. 573Beneš, V., Hofer-Temmel, C., Last, G. and Večeřa, J. (2020). Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics. J. Appl. Probab. 57 (3), 928-955. Mixed percolation as a bridge between site and bond percolation. L Chayes, R H Schonmann, Ann. Appl. Probab. 104Chayes, L. and Schonmann, R.H. (2000). Mixed percolation as a bridge between site and bond percolation. Ann. Appl. Probab. 10 (4), 1182-1196. Gibbs states of continuum particle systems with unbounded spins: existence and uniqueness. D Conache, A Daletskii, Y Kondratiev, T Pasurek, J. Math. Phys. 5913507Conache, D., Daletskii, A., Kondratiev, Y. and Pasurek. T. (2018). Gibbs states of continuum particle systems with unbounded spins: existence and uniqueness. J. Math. Phys. 59, 013507. Scale-free percolation. M Deijfen, R Van Der Hofstad, G Hooghiemstra, Ann. Inst. H. Poincaré Probab. Statist. 493Deijfen, M., van der Hofstad, R. and Hooghiemstra, G. (2013). Scale-free percolation. Ann. Inst. H. Poincaré Probab. Statist. 49 (3), 817-838. Introduction to the Theory of Gibbs Point Processes. D Dereudre, Stochastic geometry : Modern research frontiers. Coupier, D.ChamSpringerDereudre, D. (2019). Introduction to the Theory of Gibbs Point Processes. In Coupier, D. (Ed.) Stochastic geometry : Modern research frontiers (pp. 181-226). Lecture Notes in Mathematics 2237. Springer, Cham. Existence of Gibbsian point processes with geometry-dependent interactions. D Dereudre, R Drouilhet, H.-O Georgii, Probab. Theory Relat. Fields. 153Dereudre, D., Drouilhet, R. and Georgii, H.-O. (2012). Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Relat. Fields 153, 643-670. Sharp phase transition for the continuum Widom-Rowlinson model. D Dereudre, P Houdebert, Ann. Inst. Henri Poincaré Probab. Stat. 571Dereudre, D. and Houdebert, P. (2021). Sharp phase transition for the continuum Widom-Rowlinson model. Ann. Inst. Henri Poincaré Probab. Stat. 57 (1), 387-407. Existence of Gibbs point processes with stable infinite range interaction. D Dereudre, T Vasseur, J. Appl. Probab. 573Dereudre, D. and Vasseur, T. (2020). Existence of Gibbs point processes with stable infinite range interaction. J. Appl. Probab. 57 (3), 775-791. The description of a random field by means of conditional probabilities and conditions of its regularity. R L Dobrushin, Theory Probab. Appl. 132Dobrushin, R.L. (1968). The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab. Appl. 13 (2), 197-224. Canonical and grand canonical Gibbs states for continuum systems. H.-O Georgii, Commun. Math. Phys. 481Georgii, H.-O. (1976). Canonical and grand canonical Gibbs states for continuum systems. Commun. Math. Phys. 48 (1), 31-51. Phase transition in continuum Potts models. H.-O Georgii, O Häggström, Commun. Math. Phys. 181Georgii, H.-O. and Häggström, O. (1996). Phase transition in continuum Potts models. Commun. Math. Phys. 181, 507-528. Stochastic comparison of point random fields. H.-O Georgii, T Küneth, J. Appl. Probab. 344Georgii, H.-O. and Küneth, T. (1997). Stochastic comparison of point random fields. J. Appl. Probab. 34 (4), 868-881. The continuum Potts model at the disorder-order transition-a study by cluster dynamics. H.-O Georgii, J Lőrinczi, J Lukkarinen, J. Stat. Mech. 6011Georgii, H.-O., Lőrinczi, J. and Lukkarinen, J. (2005). The continuum Potts model at the disor- der-order transition-a study by cluster dynamics. J. Stat. Mech. P06011. The Kirkwood-Salsburg equations for continuum percolation. J A Given, G Stell, J. Stat. Phys. 593-4Given, J.A. and Stell, G. (1990). The Kirkwood-Salsburg equations for continuum percolation. J. Stat. Phys. 59 (3-4), 981-1018. Perfect simulation for interacting point processes, loss networks and Ising models. P A Ferrari, R Fernández, N L Garcia, Stoch. Process. Their Appl. 1021Ferrari, P.A., Fernández, R. and Garcia, N.L. (2002). Perfect simulation for interacting point pro- cesses, loss networks and Ising models. Stoch. Process. Their Appl. 102 (1), 63-88. Disagreement percolation for Gibbs ball models. C Hofer-Temmel, P Houdebert, Stoch. Process. Their Appl. 12910Hofer-Temmel, C. and Houdebert, P. (2019). Disagreement percolation for Gibbs ball models. Stoch. Process. Their Appl. 129 (10), 3922-3940. Disagreement percolation for the hard-sphere model. C Hofer-Temmel, Electron. J. Probab. 24Hofer-Temmel, C. (2019). Disagreement percolation for the hard-sphere model. Electron. J. Probab. 24, 1-22. An explicit Dobrushin uniqueness region for Gibbspoint processes with repulsive interactions. P Houdebert, A Zass, arXiv:2009.06352Houdebert, P. and Zass, A. (2021). An explicit Dobrushin uniqueness region for Gibbspoint processes with repulsive interactions. arXiv:2009.06352. Cluster expansions for Gibbs point processes. S Jansen, Adv. Appl. Probab. 514Jansen, S. (2019). Cluster expansions for Gibbs point processes. Adv. Appl. Probab. 51 (4), 1129- 1178. S Janson, T Luczak, A Ruciński, Random Graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. New YorkJohn Wiley & Sons, IncJanson, S., Luczak, T. and Ruciński, A. (2000). Random Graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., New York. O Kallenberg, Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling 77. ChamSpringerKallenberg, O. (2017). Random Measures, Theory and Applications. Probability Theory and Stochas- tic Modelling 77. Springer, Cham. The random connection model and functions of edgemarked Poisson processes: Second order properties and normal approximation. G Last, F Nestmann, M Schulte, Ann. Appl. Probab. 311Last, G., Nestmann, F. and Schulte, M. (2021). The random connection model and functions of edge- marked Poisson processes: Second order properties and normal approximation. Ann. Appl. Probab. 31 (1), 128-168. Disagreement coupling of Gibbs processes with an application to Poisson approximation. G Last, M Otto, arXiv:2104.00737Last, G. and Otto, M. (2021). Disagreement coupling of Gibbs processes with an application to Poisson approximation. arXiv:2104.00737 G Last, M Penrose, Lectures on the Poisson Process. CambridgeCambridge University PressLast, G. and Penrose, M. (2017). Lectures on the Poisson Process. Cambridge University Press, Cambridge. Freezing and clustering transitions for penetrable spheres. C N Likos, M Watzlawek, H Löwen, Phys. Rev. E. 583Likos, C.N., Watzlawek, M. and Löwen, H. (1998). Freezing and clustering transitions for penetrable spheres. Phys. Rev. E 58 (3), 3135-3144. Marked Gibbs processes and asymptotic normality of maximum pseudo-likelihood estimators. S Mase, Math. Nachr. 209Mase, S. (2000). Marked Gibbs processes and asymptotic normality of maximum pseudo-likelihood estimators. Math. Nachr. 209, 151-169. Bemerkungen zu einer Arbeit von Nguyen Xuan Xanh und Hans Zessin. K Matthes, W Warmuth, J Mecke, Math. Nachr. 88Matthes, K., Warmuth, W. and Mecke, J. (1979). Bemerkungen zu einer Arbeit von Nguyen Xuan Xanh und Hans Zessin. Math. Nachr. 88, 117-127. Equality of critical densities in continuum percolation. R Meester, J. Appl. Probab. 321Meester, R. (1995). Equality of critical densities in continuum percolation. J. Appl. Probab. 32 (1), 90-104. R Meester, R Roy, Continuum Percolation. CambridgeCambridge University PressMeester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press, Cambridge. Integral and differential characterizations of the Gibbs process. X X Nguyen, H Zessin, Math. Nachr. 88Nguyen, X.X. and Zessin, H. (1979). Integral and differential characterizations of the Gibbs process. Math. Nachr. 88, 105-115. On a continuum percolation model. M D Penrose, Adv. Appl. Probab. 233Penrose, M.D. (1991). On a continuum percolation model. Adv. Appl. Probab. 23 (3), 536-556. Continuum percolation and Euclidean minimal spanning trees in high dimensions. M D Penrose, Ann. Appl. Probab. 62Penrose, M.D. (1996). Continuum percolation and Euclidean minimal spanning trees in high dimen- sions. Ann. Appl. Probab. 6 (2), 528-544. The statistical mechanics of systems with steep intermolecular potentials. J S Rowlinson, Mol. Phys. 82Rowlinson, J.S. (1964). The statistical mechanics of systems with steep intermolecular potentials Mol. Phys. 8 (2), 107-115. Statistical Mechanics: Rigorous Results. D Ruelle, Mathematical Physics Monograph Series. Ruelle, D. (1969). Statistical Mechanics: Rigorous Results. Mathematical Physics Monograph Series. Benjamin, New York. Superstable interactions in classical statistical mechanics. D Ruelle, Commun. Math. Phys. 182Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18 (2), 127-159. Stochastic and Integral Geometry. Probability and Its Applications. R Schneider, W Weil, SpringerBerlinSchneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Probability and Its Applica- tions. Springer, Berlin. Limit theorems for geometric functionals of Gibbs point processes. T Schreiber, J E Yukich, Ann. Inst. H. Poincaré Probab. Statist. 494Schreiber T., Yukich J.E. (2013). Limit theorems for geometric functionals of Gibbs point processes. Ann. Inst. H. Poincaré Probab. Statist. 49 (4), 1158-1182. Gibbs point process approximation: Total variation bounds using Stein's method. D Schuhmacher, K Stucki, Ann. Probab. 425Schuhmacher, D. and Stucki, K. (2014). Gibbs point process approximation: Total variation bounds using Stein's method. Ann. Probab. 42 (5), 1911-1951. Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses. S Torquato, Y Jiao, J. Chem. Phys. 137774106Torquato, S. and Jiao, Y. (2012). Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses. J. Chem. Phys. 137 (7), 074106. Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses. S Torquato, Y Jiao, J. Chem. Phys. 13715159901J. Chem. Phys.Torquato, S. and Jiao, Y. (2014). Erratum: "Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses" [J. Chem. Phys. 137, 074106 (2012)] J. Chem. Phys. 141 (15), 159901. Disagreement percolation in the study of Markov fields. J Van Den Berg, C Maes, Ann. Probab. 222Van den Berg, J. and Maes, C. (1994). Disagreement percolation in the study of Markov fields. Ann. Probab. 22 (2), 749-763. Gibbs point processes on path space: existence, cluster expansion and uniqueness. A Zass, arXiv:2106.14000Zass, A. (2021). Gibbs point processes on path space: existence, cluster expansion and uniqueness. arXiv:2106.14000. Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on R d. S Ziesche, Ann. Inst. H. Poincaré Probab. Statist. 542Ziesche, S. (2018). Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on R d . Ann. Inst. H. Poincaré Probab. Statist. 54 (2), 866-878.
[]
[ "UPPER BOUND THEOREM FOR ODD-DIMENSIONAL FLAG MANIFOLDS", "UPPER BOUND THEOREM FOR ODD-DIMENSIONAL FLAG MANIFOLDS" ]
[ "Michał Adamaszek ", "Jan Hladký " ]
[]
[]
We prove that among all flag triangulations of manifolds of odd dimension 2r − 1 with sufficiently many vertices the unique maximizer of the entries of the f -, h-, g-and γ-vector is the balanced join of r cycles. Our proof uses methods from extremal graph theory.UPPER BOUND THEOREM FOR ODD-DIMENSIONAL FLAG MANIFOLDS
10.1112/s0025579316000115
[ "https://arxiv.org/pdf/1503.05961v4.pdf" ]
119,318,432
1503.05961
ea89886f8f9c9cb658e21f04f542016d99489d2f
UPPER BOUND THEOREM FOR ODD-DIMENSIONAL FLAG MANIFOLDS 19 Mar 2015 Michał Adamaszek Jan Hladký UPPER BOUND THEOREM FOR ODD-DIMENSIONAL FLAG MANIFOLDS 19 Mar 2015arXiv:1503.05961v1 [math.CO] We prove that among all flag triangulations of manifolds of odd dimension 2r − 1 with sufficiently many vertices the unique maximizer of the entries of the f -, h-, g-and γ-vector is the balanced join of r cycles. Our proof uses methods from extremal graph theory.UPPER BOUND THEOREM FOR ODD-DIMENSIONAL FLAG MANIFOLDS INTRODUCTION The classification of face numbers ( f -vectors) of various classes of simplicial complexes, especially triangulations of spheres, balls and manifolds, is a classical topic in enumerative combinatorics. The Charney-Davis conjecture [5] and its generalization by Gal [10] sparkled the interest in similar questions for the class of flag triangulations. In this paper we prove a general upper bound theorem for flag triangulations of odd-dimensional manifolds. A simplicial complex K is flag if every set of vertices pairwise adjacent in the 1-skeleton of K spans a face of K or, equivalently, if K is the clique complex of its 1-skeleton. Flag complexes appear prominently in Gromov's approach to nonpositive curvature (see [13] and [4] for an exposition). In this context Charney and Davis proposed their famous conjecture [5] that a certain linear combination of the face numbers of any odd-dimensional flag homology sphere is non-negative. Subsequently, Gal [10] introduced a modification of the f -vector called the γ-vector, which seems well-suited to the study of flag homology spheres, and is conjecturally non-negative. Since then a number of conjectures have been made about the structure of γ-vectors of flag spheres, with many of them verified in special cases [2,3,10,14,16,18,19]. Note that a flag complex is completely determined by its 1-skeleton, and thus its face vector is the clique vector of the underlying graph. Paradoxically, this only adds to the complexity of the problem. For example, face vectors of arbitrary simplicial complexes are characterized by the Kruskal-Katona theorem, while the clique vectors of general graphs are not so well understood [9]. Our contribution is an upper bound theorem for odd-dimensional flag homology manifolds, a class which includes flag simplicial manifolds and flag homology spheres. We exhibit a unique maximizer of any reasonable linear combination of face numbers. For any r ≥ 1 and n ≥ 4r let J r (n) be the n-vertex flag complex obtained as a join of r copies of the circle S 1 , each one a cycle with ⌊ n r ⌋ or ⌈ n r ⌉ vertices. This complex is a flag simplicial (2r − 1)-sphere. To phrase our main theorem we say that a real-valued function F defined on simplicial complexes is a face MA In this context the standard statistics based on face numbers are the f -vector ( f −1 , f 0 , . . . , f d−1 ), the h-vector (h 0, h 1, . . . , h d ), the g-vector (g 0 , g 1 , . . . , g d 2 ) and the γ-vector (γ 0 , γ 1 , . . . , γ d 2f i (M) ≤ f i (J d 2 (n)) for 1 ≤ i ≤ d − 1, h i (M) ≤ h i (J d 2 (n)) for 2 ≤ i ≤ d − 2, g i (M) ≤ g i (J d 2 (n)) for 2 ≤ i ≤ d 2 , γ i (M) ≤ γ i (J d 2 (n)) for 2 ≤ i ≤ d 2 . Moreover, equality in any of these inequalities implies that M is isomorphic to J d 2 (n). For all other values of the index i, as well as for face functions in dimension 0 or −1 in Theorem 1, the corresponding inequalities are trivially satisfied with equality for all M. The only previously known case of Corollary 2 was d = 4 (for any n) due to Gal [10], with the uniqueness part (for large n) following from [2]. In this case all inequalities follow from f 1 (M) ≤ f 1 (J 2 (n)). Our upper bound for the γ-vector confirms for large n a conjecture of Lutz and Nevo [17,Conjecture 6.3] and provides supporting evidence for a question of Nevo and Petersen [18,Problem 6.4] (see Section 5 for details). We also previously conjectured the upper bound on f 1 for arbitrary even d in [2]. For arbitrary (not necessarily flag) odd-dimensional homology manifolds tight upper bounds for f i were obtained by Novik [20,Theorem 1.4]. In this case the maximum is attained by the boundary of the d-dimensional cyclic polytope with n vertices (the maximizer is not unique). For the subclass of simplicial spheres this had been known before by the celebrated upper bound theorem of Stanley [23]. In the flag case our result is new even for flag simplicial spheres. PRELIMINARIES We recommend the reader [24] and [20] as references for face numbers of triangulations of manifolds and spheres. An abstract simplicial complex K with vertex set V is a collection K ⊆ 2 V such that σ ∈ K and τ ⊆ σ imply τ ∈ K. The elements of K are called faces. The dimension of σ is |σ| − 1 and the dimension of K is the maximal dimension of any of its faces. The link of a face σ is the subcomplex lk K (σ) = {τ ∈ K : τ ∩ σ = ∅, τ ∪ σ ∈ K}. A simplicial complex K is a simplicial manifold (resp. simplicial sphere) of dimension q if the geometric realization |K| is homeomorphic to a connected, compact topological q-manifold without boundary (resp. to the sphere S q ). Most known results involving face numbers of simplicial manifolds hold for more general objects, which we now introduce. A simplicial complex K is a homology manifold if for any point p ∈ |K| and any i = dim K, H i (|K|, |K| − p; Z) = 0 and H dim K (|K|, |K| − p; Z) = Z. This is equivalent to saying that for every nonempty face σ ∈ K the link lk K (σ) has the homology of the sphere S q−|σ| (equivalence follows from the excision axiom, see [15,Lemma 3.3]). A homology sphere is a homology manifold K such that K itself has the homology of a sphere. It is easy to see that if K is a homology q-manifold then for every nonempty face σ ∈ K the link lk K (σ) is a homology (q − |σ|)-sphere. Clearly every simplicial manifold (resp. simplicial sphere) is a homology manifold (resp. homology sphere). A complex K of dimension q is called Eulerian if for every face σ ∈ K (including the empty one) the link lk K (σ) has the same Euler characteristic as the sphere S q−|σ| . Every homology manifold satisfies Poincaré duality; as a consequence the Euler characteristic of an odd-dimensional homology manifold M equals 0 and so M is Eulerian. For a (d − 1)-dimensional complex K with n vertices let f i (K) be the number of i-dimensional faces. The vector ( f −1 , f 0 , . . . , f d−1 ) is called the f -vector of K (note that f −1 (K) = 1 and f 0 (K) = n). The h-vector (h 0 , h 1 , . . . , h d ) of K is a convenient modification of the f -vector defined by the identity (2.1) d ∑ i=0 f i−1 x i (1 − x) d−i = d ∑ i=0 h i x i . Note h 0 (K) = 1 and h 1 (K) = n − d. An Eulerian simplicial complex satisfies the Dehn-Sommerville equations h i (K) = h d−i (K) for 0 ≤ i ≤ d. In that case one can define the γ-vector (γ 0 , . . . , γ ⌊ d 2 ⌋ ) of K by the identity (2.2) d ∑ i=0 h i x i = ⌊d/2⌋ ∑ i=0 γ i x i (x + 1) d−2i . Here γ 0 (K) = 1 and γ 1 (K) = n − 2d. The γ-vector was first introduced by Gal [10] for flag homology spheres, for which it is conjectured to be non-negative. This conjecture generalizes the Charney-Davis conjecture, which in this language asserts that γ d 2 (K) is non-negative for a (d − 1)-dimensional flag homology sphere K with d even. Another classical invariant, studied mostly for simplicial spheres and balls, is the g-vector (g 0 , g 1 , . . . , g ⌊ d 2 ⌋ ) given by g 0 = 1 and Let us now move towards flag complexes. If G = (V, E) is a finite, simple, undirected graph then the clique number ω = ω(G) of G is the cardinality of the largest clique (complete subgraph) in G and the clique vector of G is the sequence (e 0 (G), e 1 (G), . . . , e ω (G)), where e i (G) is the number of cliques of cardinality i (in particular e 0 (G) = 1, e 1 (G) = |V| and e 2 (G) = |E|). The clique complex of G, denoted X(G), is the simplicial complex with vertex set V whose faces are all cliques in G. We have dim g i = h i − h i−1 for 1 ≤ i ≤ ⌊ d 2 ⌋.≤ i ≤ d we have h i = ∑ i j=0 (−1) j−i ( d−j i−j ) f j−1 , so h i (M)f i = ∑ i+1 j=0 ( d−j i+1−j )h j . If d 2 ≤ i ≤ d − 1X(G) = ω(G) − 1 and f i (X(G)) = e i+1 (G). Note that the 1-skeleton of X(G) is G. A simplicial complex is flag if it is the clique complex of a graph. A flag homology manifold (resp. flag homology sphere) is a flag complex which is a homology manifold (resp. a homology sphere). By abuse of language we will say that G triangulates a homology manifold (resp. sphere) if X(G) is a flag homology manifold (resp. sphere). Fix n, r ∈ N. We write T r (n) for the r-partite Turán graph of order n, that is a graph with n vertices partitioned into sets V 1 , V 2 , . . . , V r , each of size either ⌊ n r ⌋ or ⌈ n r ⌉, with no edge inside any V i and with a complete bipartite graph between every two V i and V j , i = j. Further, for n ≥ 4r we define J r (n) to be the graph obtained from T r (n) by declaring that each of the parts V i induces a cycle of length |V i |. The condition n ≥ 4r guarantees that each part is a cycle of length at least 4, hence a flag triangulation of S 1 . Of course we have J r (n) = X(J r (n)) and this complex is a flag simplicial (2r − 1)-sphere. We say that a real-valued function F defined on graphs is a clique function of order k, if F can be written as F(G) = c k e k (G) + c k−1 e k−1 (G) + · · · + c 1 e 1 (G) + c 0 where c i ∈ R and c k > 0. Theorem 1 can be equivalently rephrased as follows. Theorem 3 (Main Theorem, Graph formulation). For every r ≥ 2 and every clique function F of order k, where 2 ≤ k ≤ r, there exists a constant n 0 for which the following holds. If G is a graph with n ≥ n 0 vertices which triangulates a (2r − 1)-dimensional homology manifold then F(G) ≤ F(J r (n)) and equality holds if and only if G is isomorphic to J r (n). Let us first fix some additional notation. The neighborhood of a vertex v in a graph G is the set N G (v) = {w : vw ∈ E(G)} and for a clique σ in G we define the link of σ in G as the induced subgraph lk G (σ) = G[ v∈σ N G (v)]. This notation is designed so that X(lk G (σ)) = lk X(G) (σ). For a vertex v ∈ V(G) and a subset W ⊂ V(G) we write deg G (v) = |N G (v)| and deg G (v, W) = |N G (v) ∩ W|. The subscript G will be omitted if there is no risk of confusion. 2.1. Properties of flag homology manifolds. Below, we record two basic properties of flag homology manifolds that we need for our proof of the Main Theorem. Lemma 4. For every r ≥ 1 there is a constant C r such that every n-vertex graph G triangulating a (2r − 1)-dimensional homology manifold satisfies e r+1 (G) ≤ C r n r . Proof. The Dehn-Sommerville relation h r+1 (X(G)) = h r−1 (X(G)) expressed in terms of face numbers implies that f r (X(G)) is a linear combination of entries of the vector ( f r−1 (X(G)), . . . , f −1 (X(G))) = (e r (G), . . . , e 0 (G)) with coefficients depending only on r. Since e i (G) ≤ ( n i ) ≤ n r for 0 ≤ i ≤ r, we get e r+1 (G) = f r (X(G)) ≤ C r n r for a suitable C r . Let K r 3 := T r (3r) denote the complete r-partite graph with all parts of size 3. A graph G is H-free if it does not contain H as a subgraph. The crucial geometric ingredient of our arguments is provided by the next lemma. Lemma 5. Fix r ≥ 1. If G triangulates a homology sphere of dimension 2(r − 1) then G is K r 3 -free. Proof. By a result of Galewski and Stern [11, Corollary 1.9], if X(G) is a homology 2(r − 1)-sphere then the double suspension Σ 2 X(G) is homeomorphic to S 2r . Now if G contained K r 3 then Σ 2 X(G) would contain an embedded X(K r+1 3 ) , formed by the original K r 3 and any three of the four suspending vertices. That contradicts the theorem of van Kampen and Flores [25,8] (see also [26,Section 2.4 ]) that X(K r+1 3 ) is not embeddable in S 2r . In our arguments we are going to apply Lemma 5 to links of faces in a homology manifold. For example, we get that if G triangulates a homology (2r − 1)-manifold then for every vertex v the link lk G (v) is K r 3 -free. 2.2. Extremal graph theory. The remaining tools for our proof come entirely from extremal graph theory. An approach to face enumeration via extremal graph theory was pioneered in [2] where we classified all flag homology 3-manifolds M with a sufficiently large number n of vertices which are almost extremal for f 1 or γ 2 . Thus, the main technical contribution of our current work is in connecting further tools from extremal graph theory (namely Zykov's inequalities (Theorem 8) and the Removal lemma (Theorem 9)) to the area of face enumeration. The following definition introduces a distance -sometimes called the edit distance -on the set of n-vertex graphs. Definition 6. We say that two graphs with the same number of n vertices are ǫclose if there exists an identification of their vertex sets, so that then one graph can be obtained from the other by editing (i.e., adding or deleting) less than ǫn 2 edges. The celebrated Stability Theorem of Erdős and Simonovits [6,22] below says that a K r+1 -free graph whose number of edges is close to the Turán bound must actually be close to the Turán graph in the edit distance. Theorem 7. Suppose that r ≥ 2 and ǫ > 0 are given. Then there exists δ > 0 such that whenever H is an n-vertex, K r+1 -free graph with e 2 (H) > (1 − δ)e 2 (T r (n)) then H is ǫ-close to T r (n). We will also make use of the following result. Theorem 8. Let r ≥ 1 and suppose than H is an n-vertex, K r+1 -free graph. Then we have 1 = e 1 (H) e 1 (T r (n)) ≥ e 2 (H) e 2 (T r (n)) ≥ . . . ≥ e r (H) e r (T r (n)) . Theorem 8 generalizes the result of Zykov [27] that e k (H) ≤ e k (T r (n)), which in turn generalizes Turán's Theorem stating e 2 (H) ≤ e 2 (T r (n)). A nice proof of Theorem 8 using symmetrization can be found in [12, Theorem 3.1]. Let us now motivate the Removal lemma. A graph of order n can contain at most ( n r+1 ) = Θ(n r+1 ) copies of K r+1 . If the graph is not complete then of course it contains less copies. However, we think of the graph H as "essentially K r+1 -free" if e r+1 (H) = o(n r+1 ). It is then tempting to say that by removing a few edges we can delete all the copies of K r+1 . This is true, yet far from trivial, and a subject of the famous Removal lemma, a form of which first appeared in [21], and which was later formulated in its full strength in [7]. Theorem 9. Suppose that r ≥ 1 and α > 0 are given. Then there exists β > 0 such that whenever H is an n-vertex graph with e r+1 (H) ≤ βn r+1 then by deleting a suitable set of less than αn 2 edges H can be made K r+1 -free. 2.3. Outline of the proof of Theorem 3. Suppose G triangulates a homology (2r − 1)-manifold and the number of vertices n is large. First, note that if k ≤ r then e k (T r (n)) ≈ ( r k ) n r k and e k (J r (n)) = e k (T r (n)) (T r (n)), for a fixed (but arbitrary) α > 0 then the inequality F(G) ≤ F(J r (n)) follows just by comparing the terms of order n k in F. That leaves us only with the case where e k (G) ≈ e k (T r (n)), in which case we can also deduce e 2 (G) ≈ e 2 (T r (n)) by Theorem 8. By Lemma 4, G is "sparse in (r + 1)-cliques", i.e., it has only O(n r ) = o(n r+1 ) many K r+1 's, yet at the same time very dense (close to the maximal number of edges allowed for a K r+1 -free graph by Turán's Theorem). At this point Theorem 7 shows that G must be similar to T r (n). Additional geometric properties of X(G) allow us to conclude from there that F(G) is maximized by F(J r (n)). + O(n k−1 ) = ( r k ) n r k + O(n k−1 ). Now if G is such that e k (G) ≤ (1 − α)e k 2.4. Organisation of the paper. As said earlier, the difficult cases Theorem 3 are those when G is close to T r (n). We will analyze their structure more closely in the next section. In Section 4 we give a proof of Theorem 3. We then conclude with open problems stemming from this work in Section 5. ANALYSIS OF ALMOST EXTREMAL GRAPHS ⊂ V(H), X ∩ Y = ∅. In this section we deal with almost extremal cases, that is, with triangulations of homology (2r − 1)-manifolds that are close to T r (n). These graphs fall into the class of (η, r)-extremal graphs introduced below. Definition 10. Let 0 ≤ η < 1 and r ≥ 1 be given. We say that an n-vertex graph H is (η, r)-extremal if the vertices of H can be partitioned into sets V 0 , V 1 , . . . , V r such that (a) |V 0 | ≤ 1 30r r ηn and ⌊(1 − 1 30r η) n r ⌋ ≤ |V i | ≤ ⌈(1 + 1 30r η) n r ⌉ for i ∈ [r], (b) H[V i ] is triangle-free, for i ∈ [r],((v, V g ) ≤ 2 and deg H (v, V h ) ≤ (1 − 1 2 η)|V h |, and it is of Type 2 if there exist two distinct indices g, h ∈ [r] such that deg H (v, V g ) ≤ 3rη|V g | and deg H (v, V h ) ≤ 3rη|V h |. For small η, graphs with the above structure resemble J r (n) up to some error. That is, we allow that a small fraction of edges missing in each H[V i , V j ], that the parts are slightly unbalanced and we admit a small set of exceptional vertices V 0 . In the next definition we introduce a class of graphs that resemble J r (n) even better. Definition 11. We say that a graph is r-radical if it is (0, r)-extremal, and for each i ∈ [r] every vertex of H[V i ] has degree 2. If H is r-radical then V 0 = ∅, each V i is of size ⌊ n r ⌋ or ⌈ n r ⌉ for i ∈ [r] and each graph H[V i , V j ] is complete bipartite for i, j ∈ [r], i = j. An r-radical graph is (η, r)extremal for any 0 ≤ η < 1. Note that if H is any n-vertex r-radical graph then F(H) = F(J r (n)) for every clique function F. Lemma 12. If H is an r-radical graph with n vertices which triangulates a homology (2r − 1)-manifold then H is isomorphic to J r (n). Proof. For all i = 1, . . . , r − 1 pick any edge in H[V i ]. The endpoints of these r − 1 edges form a clique of order 2r − 2 whose link is H[V r ]. However, in a homology (2r − 1)-manifold the link of a face of size 2r − 2 is a homology 1-sphere, that is a cycle. It means that H[V r ] is a cycle. The same argument shows that all H[V i ] are cycles and therefore H is isomorphic to J r (n). The next lemma is used to find copies of K r 3 in (η, r)-extremal graphs. Lemma 13. Fix r ≥ 1 and η > 0. Suppose H is a graph with n ≥ 2rη −1 vertices and a partition V(H) = V 0 ⊔ V 1 ⊔ . . . ⊔ V r which satisfies conditions (a) and (d) of Definition 10. Let w 1 , w 2 , w 3 ∈ V 1 be any three fixed vertices. For i ∈ {2, . . . , r} let A i ⊆ V i be sets with |A i | ≥ 3rη|V i |. Then the subgraph of H induced by {w 1 , w 2 , w 3 } ∪ r i=2 A i contains a K r 3 with 3 vertices in each part V i , i ∈ [r]. Proof. We will construct by induction 3-element subsets {w i 1 , w i 2 , w i 3 } ⊆ A i such that for each l ∈ [r] the subgraph of H induced by w i j with j = 1, 2, 3 and i = 1, . . . , l contains K l 3 . For l = r this proves the lemma. When i = 1 the vertices w 1 j = w j are already given. Suppose we have constructed the vertices {w i 1 , w i 2 , w i 3 } l i=1 for some l ≤ r − 1. By condition (d) the common neighborhood N l+1 of these 3l vertices satisfies |N l+1 ∩ V l+1 | ≥ (1 − 3lη)|V l+1 |. It follows that |A l+1 ∩ N l+1 | ≥ |A l+1 | − |V l+1 \ N l+1 | ≥ 3rη|V l+1 | − 3lη|V l+1 | ≥ 3η|V l+1 | ≥ 3η n 2r ≥ 3, where the last line uses condition (a) of Definition 10 and the bound n ≥ 2rη −1 . It means that we can pick three distinct vertices w l+1 1 , w l+1 2 , w l+1 3 ∈ A l+1 ∩ N l+1 and the induction step is complete. We can now prove that graphs triangulating homology (2r − 1)-manifolds are (η, r)-extremal as soon as they are sufficiently close to T r (n). Lemma 14. For every r ≥ 2 and 0 < η < 1 7r set ǫ = η 2 120r r+3 . If a graph H with n ≥ 2rη −1 vertices triangulates a homology (2r − 1)-manifold and H is ǫ-close to T r (n) then H is (η, r)-extremal. Proof. If H is ǫ-close to T r (n) then the vertices of H can be partitioned into r sets X 1 , . . . , X r , each of size ⌊ n r ⌋ or ⌈ n r ⌉, such that ∑ i<j e 2 (H[X i , X j ]) + ∑ i e 2 (H[X i ]) ≤ ǫn 2 . Here e 2 (H[X i , X j ]) is the number of edges missing between X i and X j , that is e 2 (H[X i , X j ]) = |X i | · |X j | − e 2 (H[X i , X j ]). For every i = j let X i,j = {v ∈ X i : deg(v, X j ) ≤ (1 − 2 3 η)|X j |}. Every vertex in X i,j contributes to the number of missing edgesē(H[X i , X j ]) as follows ǫn 2 ≥ e 2 (H[X i , X j ]) ≥ ∑ v∈X i,j (|X j | − deg(v, X j )) ≥ 2 3 η|X j | · |X i,j | ≥ 1 2 η n r · |X i,j |, hence |X i,j | ≤ 2ǫrη −1 n. Consider a new partition V(H) = Y 0 ⊔ Y 1 ⊔ . . . ⊔ Y r , Y 0 = i =j X i,j , Y i = X i \ Y 0 for i ∈ [r]. We have |Y 0 | ≤ r 2 · 2ǫrη −1 n = 1 60r r ηn and, for i ∈ [r], ⌈ n r ⌉ ≥ |X i | ≥ |Y i | ≥ |X i | − |Y 0 | ≥ (1 − 1 30r η) n r . By definition, for every vertex v ∈ Y 0 there exists an index j ∈ [r] such that deg(v, X j ) ≤ (1 − 2 3 η)|X j |. Let Z j ⊂ Y 0 consists of those vertices for which j is the only such index, formally: Z j = v ∈ Y 0 : deg(v, X k ) ≤ (1 − 2 3 η)|X k | iff k = j for k ∈ [r] . We now define the final partition of V(H) as V 0 = Y 0 \ j Z j , V i = Y i ∪ Z i for i ∈ [r]. We claim that the partition V(H) = V 0 ⊔ V 1 ⊔ . . . ⊔ V r witnesses (η, r)-extremality of H. We have |V 0 | ≤ |Y 0 | ≤ 1 30r r ηn and |V i | ≥ |Y i | ≥ (1 − 1 30r η) n r for i ∈ [r]. Moreover, |V i | = |Y i | + |Z i | ≤ |Y i | + |Y 0 | ≤ ⌈ n r ⌉ + 1 30r r ηn ≤ (1 + 1 30r η) n r . That proves condition (a) of Definition 10. Next we verify condition (d). Pick any vertex v ∈ V i , i ∈ [r]. Regardless of whether v ∈ Y i or v ∈ Z i we have that deg(v, X j ) ≥ (1 − 2 3 η)|X j | for all j = i. That yields deg(v, V j ) ≥ deg(v, Y j ) ≥ deg(v, X j ) − |Y 0 | ≥ (1 − 2 3 η)|X j | − |Y 0 | ≥ (1 − 2 3 η − η 30r r−1 ) n r ≥ (1 − 2 3 η − η 30r r−1 )(1 + 1 30r η) −1 |V j | ≥ (1 − η)|V j |. To prove property (b), suppose, without loss of generality, that H[ V 1 ] contains a triangle t = {w 1 , w 2 , w 3 }. For i = 2, . . . , r let A i = N H (w 1 ) ∩ N H (w 2 ) ∩ N H (w 3 ) ∩ V i . By the already proven property (d) we have |A i | ≥ (1 − 3η)|V i | ≥ 3ηr|V i | (the last inequality uses η < 1 7r ). Since n ≥ 2rη −1 Lemma 13 now yields that the link lk H (t) contains K r−1 3 as a subgraph. This is a contradiction to Lemma 5, since lk H (t) triangulates a homology sphere of dimension 2r − 1 − 3 = 2(r − 2). Similarly, to prove (c), suppose v ∈ V 1 has three distinct neighbors w 1 , w 2 , w 3 ∈ V 1 . Applying Lemma 13 with w 1 , w 2 , w 3 and A i = N H (v) ∩ V i for i = 2, . . . , r, where |A i | ≥ (1 − η)|V i | ≥ 3ηr|V i |, we get that lk H (v) contains a K r 3 . This contradicts the fact that lk H (v) triangulates a homology 2(r − 1)-sphere. We now turn to verifying (e). Let us start with an auxiliary claim. Claim. Let v ∈ V 0 be any vertex and suppose j ∈ [r] is any index such that deg(v, X j ) ≤ (1 − 2 3 η)|X j |. Then deg(v, V j ) ≤ (1 − 1 2 η)|V j |. Proof. We have deg(v, V j ) ≤ deg(v, Y j ) + |Z j | ≤ deg(v, X j ) + |Y 0 | ≤ (1 − 2 3 η)⌈ n r ⌉ + 1 30r r ηn ≤ (1 − 2 3 η + 1 15r r−1 η) n r ≤ (1 − 2 3 η + 1 15r r−1 η)(1 − 1 30r η) −1 |V j | ≤ (1 − 1 2 η)|V j |. Now suppose that some vertex v ∈ V 0 is not of Type 2. Then, without loss of generality, deg(v, V i ) > 3ηr|V i | for i = 2, . . . , r. Suppose that deg(v, V 1 ) ≥ 3 and let w 1 , w 2, w 3 ∈ N H (v) ∩ V 1 be three distinct vertices. We already proved properties (a) and (d), so we can apply Lemma 13 with w 1 , w 2 , w 3 and A i = N H (v) ∩ V i for i = 2, . . . , r to conclude that lk H (v) contains a K r 3 , a contradiction to Lemma 5. Therefore, deg(v, V 1 ) ≤ 2. By the definition of V 0 , there exist an index j = 1 such deg(v, X j ) ≤ (1 − 2 3 η)|X j |. The above Claim then gives that deg(v, V j ) ≤ (1 − 1 2 η)|V j |. This proves that v is of Type 1. Condition (e) follows. This completes the proof of the lemma. Our last lemma says that for among (η, r)-extremal graphs, the graph J r (n) maximizes any clique function of order up to r (for sufficiently large n). Note that in this part of the proof we do not assume that H triangulates a homology manifold. Lemma 15. Let r ≥ 2 and let F be a clique function of order k, 2 ≤ k ≤ r. Set η = 1 14r r . Then there exists a number m 0 such that the following holds. If H is a graph with n ≥ m 0 vertices then F(H) ≤ F(J r (n)) and equality is attained only when H is r-radical. Proof. Let the clique function be F(G) = c k e k (G) + c k−1 e k−1 (G) + . . . + c 1 e 1 (G) + c 0 . The value of m 0 will be chosen during the proof in such a way that (3.1), (3.2), (3.3), (3.4) and (3.5) are satisfied for all n ≥ m 0 . Among all (η, r)-extremal graphs with n vertices, let us consider a graph H that maximizes F(H). We will show that H is r-radical. Claim. For each i, j ∈ [r], i = j, the bipartite graph H[V i , V j ] is complete. Proof. Suppose for a contradiction and without loss of generality that there exist vertices v 1 ∈ V 1 and v 2 ∈ V 2 that do not form an edge. Let us now add that edge to H. Observe that the modified graph H ′ is still (η, r)-extremal. We will now find a lower bound for the number of cliques in H ′ which contain the edge v 1 v 2 . By condition (d), v 1 and v 2 are both adjacent to at least (1 − 2η)|V 3 | ≥ (1 − 3η) n r vertices v 3 in V 3 . In general, given vertices v 1 ∈ V 1 , v 2 ∈ V 2 , . . . , v ℓ ∈ V ℓ there are at least (1 − ℓη)|V ℓ+1 | ≥ (1 − (ℓ + 1)η) n r vertices v ℓ+1 in V l+1 adjacent to each of v 1 , v 2 , . . . , v ℓ . This sequential extension gives at least ((1 − kη) n r ) k−2 > ( 1 2 · n r ) k−2 many k-cliques containing both v 1 and v 2 in H ′ . For each t = 2, . . . , k − 1, the number of t-cliques increased by at most n t−2 , and the number of vertices did not change. So, in total, (3.1) F(H ′ ) − F(H) ≥ c k n 2r k−2 − k−1 ∑ t=2 |c t |n t−2 > 0 , since the coefficients c t are fixed and n ≥ m 0 is large enough. This is a contradiction to the assumption that H maximizes F. Claim. The set V 0 does not contain any Type 1 vertex. Proof. Suppose that v ∈ V 0 is a Type 1 vertex. Let g and h be the two indices as in the definition of Type 1 in Definition 10. Since the average size of the sets V i , i ∈ [r], is n−|V 0 | r , there is an index j ∈ [r] so that |V j | < n r . We construct a new graph H ′ by deleting v (and its incident edges) from V 0 and introducing a new vertex w into the set V j . We make w adjacent to all the vertices in i∈[r]\j V i , and to no other. The modified graph H ′ is (η, r)-extremal. The vertex w is contained in at least ( r−1 k−1 )⌊(1 − 1 30r η) n r ⌋ k−1 many k-cliques in H ′ . Indeed, we can choose an arbitrary (k − 1)-element set {p 1 , p 2 , . . . , p k−1 } ⊂ [r] \ j, and this choice gives us at least ⌊(1 − 1 30r η) n r ⌋ k−1 choices of vertices w 1 ∈ V p 1 , . . . , w k−1 ∈ V p k−1 . By the previous Claim, for any such choice {w, w 1 , . . . , w k−1 } is a clique. Let us now upper-bound the number of k-cliques in H containing v. The number of cliques containing v and some other vertex of V 0 is at most |V 0 | · n k−2 ≤ 1 30r r ηn k−1 . The number of k-cliques through v and through a vertex from the set V g is at most 2n k−2 by the definition of Type 1. By Definition 10(c) and (d), if k ≥ 3 the number of cliques containing v and at least two vertices from a fixed V i , i ∈ [r], is at most e 2 (H[V i ]) · n k−3 ≤ |V i | · n k−3 . Therefore the number of k-cliques that touch some of the sets V i in at least two vertices is upper bounded by n k−2 . It remains to upper-bound the number of k-cliques in H through v that contain no vertex from (V 0 \ {v}) ∪ V g , and that intersect each of the sets V i in at most one vertex. Trivially, this number is at most ( r−1 k−1 )⌈(1 + 1 30r η) n r ⌉ k−1 . However, the fact that deg(v, V h ) ≤ (1 − 1 2 η)|V h | allows us to refine this upper-bound to r − 2 k − 1 ⌈(1 + 1 30r η) n r ⌉ k−1 + r − 2 k − 2 ⌈(1 + 1 30r η) n r ⌉ k−2 (1 − 1 2 η)⌈(1 + 1 30r η) n r ⌉ = ⌈(1 + 1 30r η) n r ⌉ k−1 r − 1 k − 1 − 1 2 η · r − 2 k − 2 = r − 1 k − 1 ⌈(1 + 1 30r η) n r ⌉ k−1 (1 − η 2 · k−1 r−1 ) . Putting these bounds together, we get e k (H ′ ) − e k (H) ≥ r − 1 k − 1 ⌊(1 − 1 30r η) n r ⌋ k−1 − r − 1 k − 1 ⌈(1 + 1 30r η) n r ⌉ k−1 (1 − η 2 · k−1 r−1 ) − 1 30r r ηn k−1 − 3n k−2 . Using the inequality ⌊(1 − 1 30r η) n r ⌋ > ⌈(1 + 1 30r η) n r ⌉(1 − 1 10r η) we can write e k (H ′ ) − e k (H) > r − 1 k − 1 ⌈(1 + 1 30r η) n r ⌉ k−1 [(1 − 1 10r η) k−1 − 1 + η 2 · k−1 r−1 ] − 1 30r r ηn k−1 − 3n k−2 . By Bernoulli's inequality the coefficient in the square brackets is at least 1 − k − 1 10r η − 1 + η 2 · k − 1 r − 1 > − k − 1 10r η + k − 1 2r η = 2 5 · k − 1 r η . That gives e k (H ′ ) − e k (H) > r − 1 k − 1 n k−1 r k−1 · 2 5 · k − 1 r η − 1 30r r ηn k−1 − 3n k−2 = 1 r r ηn k−1 2 5 r − 1 k − 1 (k − 1)r r−k − 1 30 − 3n k−2 ≥ 1 r r ηn k−1 ( 2 5 − 1 30 ) − 3n k−2 = 11 30r r ηn k−1 − 3n k−2 . The number of cliques of size t changed by at most n t−1 for t = 2, . . . , k − 1. That implies (3.2) F(H ′ ) − F(H) > 11 30r r ηc k n k−1 − 3c k n k−2 − k−1 ∑ t=2 |c t |n t−1 > 0 since n ≥ m 0 is sufficiently large. That contradicts the maximality of H and proves the claim. Claim. The set V 0 does not contain any Type 2 vertex. Proof. We proceed similarly as in the previous case. Suppose that v ∈ V 0 is a Type 2 vertex. Let g and h be the two indices as in Definition 10. We delete v from V 0 and introduce a new vertex w in some set V j , j ∈ [r] with |V j | < n r which we make adjacent to all the vertices in i∈[r]\{j} V i , and to no other. Let H ′ be the resulting (η, r)-extremal graph. As before, the new vertex w belongs to at least ( r−1 k−1 )⌊(1 − 1 30r η) n r ⌋ k−1 cliques of size k in H ′ . Next we upper-bound the number of cliques containing v in H. The number of k-cliques through v and through a vertex from the set V g ∪ V h ∪ V 0 \ {v} is at most (|V g ∩ N H (v)| + |V h ∩ N H (v)| + |V 0 |)n k−2 ≤ 7ηn k−1 by the definition of Type 2. The number of k-cliques through v that touch at least two vertices in some V i is at most n k−2 , as in the previous claim. Last, the number of k-cliques through v that do not intersect V g ∪ V h ∪ (V 0 \ {v}) and contain at most one vertex from each V i is upper-bounded by r − 2 k − 1 ⌈(1 + 1 30r η) n r ⌉ k−1 = r − k r − 1 r − 1 k − 1 ⌈(1 + 1 30r η) n r ⌉ k−1 (in particular it must be 0 when k = r). Proceeding as in the proof of the previous claim we get e k (H ′ ) − e k (H) ≥ r − 1 k − 1 ⌊(1 − 1 30r η) n r ⌋ k−1 − r−k r−1 r − 1 k − 1 ⌈(1 + 1 30r η) n r ⌉ k−1 − 7ηn k−1 − n k−2 ≥ r − 1 k − 1 ⌈(1 + 1 30r η) n r ⌉ k−1 [(1 − 1 10r η) k−1 − r−k r−1 ] − 7ηn k−1 − n k−2 . The expression in the square brackets is at least 1 − k − 1 10r η − 1 + k − 1 r − 1 > (k − 1) 1 r − 1 10r η > 9(k − 1) 10r ≥ 9 10r . Hence we get e k (H ′ ) − e k (H) ≥ n k−1 9 10r k r − 1 k − 1 − 7η − n k−2 > 1 3r r n k−1 − n k−2 , where we used 7η ≤ 1 2r r , and finally Thus, by the three claims above, the vertex set of H is partitioned into sets V 1 , . . . , V r , all pairs of which form complete bipartite graphs. Recall that the graphs H[V i ] are triangle-free and of maximum degree at most 2. Claim. For each i ∈ [r], we have e 2 (H[V i ]) = |V i |. Proof. The condition that the maximum degree of H[V i ] is at most 2 implies that e 2 (H[V i ]) ≤ |V i |. Suppose now that e 2 (H[V i ]) < |V i |. We replace the subgraph H[V i ] with the graph consisting of a path with e 2 (H[V i ]) edges followed by |V i | − e 2 (H[V i ]) − 1 isolated vertices. Let H ′ be the resulting graph. Note that H ′ is (η, r)extremal, and since H[V i ] was triangle-free we have e ℓ (H ′ ) = e ℓ (H) for all ℓ. Next, we create H ′′ by adding one edge to H ′ [V i ], so that we get a longer path or a cycle. We still have that H ′′ is (η, r)-extremal. The number of k-cliques increased from H ′ to H ′′ by at least ( r−1 k−2 )⌊(1 − 1 30r η) n r ⌋ k−2 ≥ ( n 2r ) k−2 . At the same time, the total number of cliques of order t = 2, . . . , k − 1 increased by at most n t−2 . Hence Claim. For each 1 ≤ i < j ≤ r, we have |V i | − 1 ≤ |V j | ≤ |V i | + 1. That leaves us with the case F(H) > (1 − 1 4 δ)F(T r (n)). Since n ≥ m 2 we get e k (H) > (1 − 1 2 δ)e k (T r (n)). By Lemma 4 we have e r+1 (H) ≤ C r n r ≤ βn r+1 . Theorem 9 now shows that we can remove at most αn 2 edges from H to obtain a K r+1 -free subgraph G with the same vertex set. The removal of one edge destroys at most n k−2 cliques of size k, therefore e k (G) ≥ e k (H) − αn k ≥ (1 − 1 2 δ)e k (T r (n)) − 1 4 δ r k 1 r k n k ≥ (1 − δ)e k (T r (n)), where in the last step we used e k (T r (n)) ≥ 1 2 ( r k ) n k r k . Theorem 8 now gives e 2 (G) ≥ (1 − δ)e 2 (T r (n)). By Theorem 7 the graph G is 1 2 ǫ-close to T r (n). Since H arises from G by adding at most αn 2 ≤ 1 2 ǫn 2 edges, we conclude that H is ǫ-close to T r (n). From Lemma 14, we have that H is (η, r)-extremal. As n ≥ m 0 , Lemma 15 now shows that F(H) ≤ F(J r (n)). That ends the proof of the inequality. If F(H) = F(J r (n)) then by Lemma 15 the graph H is r-radical. Since H triangulates a homology (2r − 1)-manifold, Lemma 12 yields that H is isomorphic to J r (n). That proves the uniqueness part. CONJECTURES First of all, it is natural to expect that the conclusion of Corollary 2 holds for flag triangulations of any size, not just sufficiently large. For the γ-vector this was conjectured in [17]. Moreover, we conjecture that the extremum is stable, in the sense that if F(M) is sufficiently close to F(J d function in dimension ℓ if it can be written asF(K) = c ℓ f ℓ (K) + c ℓ−1 f ℓ−1 (K) + . . . + c 0 f 0 (K) + c −1 with c i ∈ R and c ℓ > 0, where f i (K) is the number of i-dimensional faces of K.Theorem 1 (Main theorem). For every even number d ≥ 4 and every face function F in dimension ℓ, where 1 ≤ ℓ ≤ d 2 − 1, there exists a constant n 0 for which the following holds. If M is a flag homology manifold of dimension d − 1 with n ≥ n 0 vertices then F(M) ≤ F(J d 2 (n)) and equality holds if and only if M is isomorphic to J d 2 (n). Suppose now that d is even and let M be a homology (d − 1)-manifold. For any i the function f i (M) is clearly a face function in dimension i. For any 0 is a face function in dimension i − 1. By the Dehn-Sommerville equations if d 2 ≤ i ≤ d then h i (M) = h d−i (M) can be expressed as a face function in dimension d − i − 1. For any 0 ≤ i ≤ d − 1 we have the Dehn-Sommerville equations imply that f i is a linear combination of (h d 2 , . . . , h 0 ) with leading term ( f i (M) is equal to a face function in dimension d 2 − 1. Finally both γ i and g i are linear combinations of (h i , . . . , h 0 ) with leading term h i , hence γ i (M) and g i (M) are face functions in dimension i − 1. Using these observations the proof of Corollary 2 from Theorem 1 is immediate. (3. 3 ) 3F(H ′ ) − F(H) > 1 3r r c k n k−1 − c k n k−2 − k−1 ∑ t=2 |c t |n t−1 > 0 , because n ≥ m 0 . (H ′′ ) − F(H) = F(H ′′ ) − F(H ′ ) t |n t−2 > 0for n ≥ m 0 , a contradiction to the supposed maximality of H. is supported by VILLUM FONDEN through the network for Experimental Mathematics in Number Theory, Operator Algebras, and Topology. This research was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme (Grant Agreement Number 628974, project PaECiDM). ) . )Theorem 1 specializes to an upper bound statement for all of those simultaneously. Corollary 2. For every even number d ≥ 4 there is a constant N 0 for which the following holds. If M is a flag homology manifold of dimension d − 1 with n ≥ N 0 vertices then For any r ≥ 1 denote [r] = {1, . . . , r}. We denote by H[X] the subgraph of H induced by a set of vertices X and by H[X, Y] the bipartite sugbraph of H with parts X, Y c) H[V i ] has maximum degree at most 2, for i ∈ [r], (d) for each i, j ∈ [r], i = j, and any v∈ V i we have deg H (v, V j ) ≥ (1 − η)|V j |,(e)each vertex of V 0 is either of Type 1 or Type 2, where we say a vertex v is of Type 1 if there exist two distinct indices g, h ∈ [r] such that deg H (n)) then M is still a join of cycles of ACKNOWLEDGEMENTSThis project was carried out while JH was visiting the University of Copenhagen, and he thanks the Department of Mathematical Sciences for their hospitality.Proof. Consider the class of all (η, r)-extremal graphs G with sufficiently many vertices which are partitioned into classes V(G) = V 0 (G) ⊔ V 1 (G) ⊔ . . . ⊔ V r (G) which satisfy all the previous claims, i.e. V 0 (G) = ∅, G[V i (G), V j (G)] is complete bipartite for i, j ∈ [r], i = j, each G[V i (G)] is triangle-free and e 2 (G[V i (G)]) = |V i (G)| for i ∈ [r]. Let σ j (x 1 , . . . , x r ) = ∑ 1≤i 1 <···<i j ≤r x i 1 · · · x i j denote the j-th elementary symmetric polynomial in r variables. For a graph G in the above class we haveIt follows that there are constants c ′ k , . . . , c ′ 0 depending only on F and r such that. . , |V r (G)|) for all graphs G in the class. Now suppose, without loss of generality, that in the maximizer H we have |V 1 (H)| − |V 2 (H)| ≥ 2. Take any graph H ′ in the same class with parts of size (|V 1 | − 1, |V 2 | + 1, |V 3 |, . . . , |V r |). For any numbers x 1 , . . . , x r havefor sufficiently large n ≥ m 0 , again a contradiction to the maximality of H.The claims above clearly prove the lemma.PROOF OF THE MAIN THEOREMWe can now prove Theorem 3. Fix r ≥ k ≥ 2 and a clique functionLet η = 1 14r r and m 0 be the constants provided by Lemma 15 given r and F. Let ǫ = η 2 120r r+3 be the constant provided by Lemma 14 given r and η. Let δ be the constant from Theorem 7 for input parameters r and 1 2 ǫ. Define1 2ǫ} and let β be the constant from Theorem 9 for input r and α. Let m 1 be such that for n ≥ m1. The existence of m 1 and m 2 follows by observing that for n-vertex graphs G we have. Finally let C r be the constant from Lemma 4. We claim that Theorem 3 holds for n 0 = max{m 0 , m 1 , m 2 , C r β −1 , 2rη −1 }. Suppose H is any graph with n ≥ n 0 vertices which triangulates a homology (2r − 1)-manifold. First, suppose that F(H) ≤ (1 − 1 4 δ)F(T r (n)). Since n ≥ m 1 this implies F(H) < F(J r (n)), and the result is proved (in that case, equality is impossible). total length n and of individual lengths close to n d/2 , but not necessarily all equal (see also [2, Conjecture 5.1]).As mentioned in the introduction, Gal[10]conjectures that for flag homology spheres M the γ-vector γ(M) is non-negative. This is known to be true in a number of special cases (see[14,18,19]and the references therein). One method of showing non-negativity is to exhibit a simplicial complex of which γ(M) is the f -vector. In particular, Nevo and Petersen [18, Problem 6.4] asked if for every nvertex flag homology sphere M there exists a graph G such that the γ-vector of M is the clique vector of G. Our result γ i (M) ≤ γ i (J r (n)) supports this claim in odd dimension 2r − 1. Indeed, one checks that γ 1 (M) = n − 4r and γ i (J r (n)) = e i (T r (n − 4r)). If the conjectural graph G exists, then it is K r+1 -free, has n − 4r vertices, and thus by Zykov's theorem γ i (M) = e i (G) ≤ e i (T r (n − 4r)) = γ i (J r (n)), which is what we showed.Having proved that F(M) ≤ F(J r (n)) it is tempting to conjecture, for the classical enumeration vectors (i.e. the f -, h-, g-or γ-vector), a generalization in the spirit of Theorem 8. We pose this as an open problem.(v 1 , .. . , v r ) be any of ( f 0 , . . . , f r−1 ), (h 1 , . . . , h r ), (g 1 , . . . , g r ) or (γ 1 , . . . , γ r ). Is it true that for sufficiently large n the inequalitiesProblem 16. Lethold for any flag homology (2r − 1)-manifold (or sphere) M with n vertices?If v i = γ i and M is a homology (2r − 1)-sphere the positive answer to Problem 16 would follow directly from the conjecture of Nevo and Petersen mentioned Recall that for homology manifolds condition (ii), guaranteed by Lemma 4 and ultimately by the middle Dehn-Sommerville equation, is the weakest possible assumption which allows us to initiate the stability method for dense graphs. The first author proved in[1]that for families of flag weak (2r − 1)-pseudomanifolds which satisfy a stronger condition f r (M) ≤ Cn r for some fixed C we have f 1 (M) ≤ f 1 (J r (n)) for sufficiently large n.In even dimensions the situation seems to be more complicated. For r ≥ 1 let J * r (n) be the graph obtained from J r (n − 2) by adding two new vertices adjacent to all of J r (n − 2). Then the clique complex J * r (n) := X(J * r (n)) is a flag simplicial 2r-sphere.Conjecture 18.Fix r ≥ 2. For every flag homology 2r-sphere M with n vertices we have f i (M) ≤ f i (J * r (n)) for i = 0, . . . , 2r and γ i (M) ≤ γ i (J * r (n)) for i = 0, . . . , r. This statement is obviously true when r = 1, since ( f 0 (M), f 1 (M), f 2 (M)) = (n, 2n − 4, 3n − 6) and all the inequalities are equalities for every M. Also for r ≥ 2 the conjecture (if true) cannot be augmented by a uniqueness statement. To see this, consider the subgraph of J * r (n) induced by V i ∪ {a, b}, where V i is any part of V(J r (n − 2)) = V 1 ⊔ · · · ⊔ V r and a, b are the two additional vertices. It is a flag triangulation of S 2 as the suspension of a cycle. Upon replacing this subgraph by any other flag triangulation of S 2 with the same number of vertices one gets a flag simplicial 2r-sphere with the same face numbers as J * r (n). It is very likely that J n (r) is the maximizer of face numbers for a wider class of (2r − 1)-dimensional flag weak pseudomanifolds. A weak pseudomanifold of dimen. It is very likely that J n (r) is the maximizer of face numbers for a wider class of (2r − 1)-dimensional flag weak pseudomanifolds. A weak pseudomanifold of dimen- An upper bound theorem for a class of flag weak pseudomanifolds. M Adamaszek, arXiv:1303.5603M. Adamaszek. An upper bound theorem for a class of flag weak pseudomanifolds, 2013. arXiv:1303.5603. Dense flag triangulations of 3-manifolds via extremal graph theory. M Adamaszek, J Hladký, Trans. Amer. Math. Soc. 3674M. Adamaszek and J. Hladký. Dense flag triangulations of 3-manifolds via extremal graph theory. Trans. Amer. Math. Soc., 367(4):2743-2764, 2015. N Aisbett, Frankl-Füredi-Kalai inequalities on the γ-vectors of flag nestohedra. Discrete & Computational Geometry. 51N. Aisbett. Frankl-Füredi-Kalai inequalities on the γ-vectors of flag nestohedra. Discrete & Com- putational Geometry, 51(2):323-336, 2014. Metric geometry: connections with combinatorics. R Charney, Proceedings of FPSCA Conference. FPSCA Conference24R. Charney. Metric geometry: connections with combinatorics. Proceedings of FPSCA Conference, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 24:55-69, 1996. The Euler characteristic of a non-positively curved, piecewise Euclidean manifold. R Charney, M Davis, Pacific J. Math. 171R. Charney and M. Davis. The Euler characteristic of a non-positively curved, piecewise Euclidean manifold. Pacific J. Math., 171:117-137, 1995. On some new inequalities concerning extremal properties of graphs. P Erdős, Theory of Graphs (Proc. Colloq. Tihany; New YorkAcademic PressP. Erdős. On some new inequalities concerning extremal properties of graphs. In Theory of Graphs (Proc. Colloq., Tihany, 1966), pages 77-81. Academic Press, New York, 1968. The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. P Erdős, P Frankl, V Rödl, Graphs Combin. 22P. Erdős, P. Frankl, and V. Rödl. The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs Combin., 2(2):113-121, 1986. Über die Existenz n-dimensionaler Komplexe die nicht im den R 2n topologisch einbettar sind. A Flores, Ergeb. Math. Kolloq. 5A. Flores. Über die Existenz n-dimensionaler Komplexe die nicht im den R 2n topologisch einbettar sind. Ergeb. Math. Kolloq., 5:17-24, 1933. Face vectors of flag complexes. A Frohmader, Israel J. Math. 164A. Frohmader. Face vectors of flag complexes. Israel J. Math., 164:153-164, 2008. Real root conjecture fails for five-and higher-dimensional spheres. Ś R Gal, Discrete Comput. Geom. 342Ś. R. Gal. Real root conjecture fails for five-and higher-dimensional spheres. Discrete Comput. Geom., 34(2):269-284, 2005. Classification of simplicial triangulations of topological manifolds. D E Galewski, R Stern, Annals of Mathematics. 111D. E. Galewski and R. J Stern. Classification of simplicial triangulations of topological manifolds. Annals of Mathematics, 111:1-34, 1980. Convex hull of face vectors of colored complexes. A Goodarzi, European Journal of Combinatorics. 36A. Goodarzi. Convex hull of face vectors of colored complexes. European Journal of Combinatorics, 36:247-250, 2014. Hyperbolic groups. M Gromov, Essays in Group Theory. S. M. Gersten, M.S.R.I. Publ.Springer-VerlagM. Gromov. Hyperbolic groups. In Essays in Group Theory (S. M. Gersten, M.S.R.I. Publ. 8 eds.), pages 75-264. Springer-Verlag, 1987. The cd-index of fans and posets. K Karu, Compositio Mathematica. 1423K. Karu. The cd-index of fans and posets. Compositio Mathematica, 142(3):701-718, 2006. Topological results in combinatorics. J R Munkres, Michigan Math. J. 311J. R. Munkres. Topological results in combinatorics. Michigan Math. J., 31(1):113-128, 1984. On the cd-index and γ-vector of S*-shellable CW-spheres. S Murai, E Nevo, Mathematische Zeitschrift. 2713-4S. Murai and E. Nevo. On the cd-index and γ-vector of S*-shellable CW-spheres. Mathematische Zeitschrift, 271(3-4):1309-1319, 2012. Stellar theory for flag complexes. E Nevo, F Lutz, arxiv:1302.5197to appear in Math. ScandE. Nevo and F. Lutz. Stellar theory for flag complexes, 2013. arxiv:1302.5197, to appear in Math. Scand. On γ-vectors satisfying the Kruskal-Katona inequalities. E Nevo, T K Petersen, Discrete & Computational Geometry. 453E. Nevo and T K. Petersen. On γ-vectors satisfying the Kruskal-Katona inequalities. Discrete & Computational Geometry, 45(3):503-521, 2011. The γ-vector of a barycentric subdivision. E Nevo, T K Petersen, B E Tenner, Journal of Combinatorial Theory, Series A. 1184E. Nevo, T. K. Petersen, and B. E. Tenner. The γ-vector of a barycentric subdivision. Journal of Combinatorial Theory, Series A, 118(4):1364-1380, 2011. Upper bound theorems for homology manifolds. I Novik, Israel Journal of Mathematics. 1081I. Novik. Upper bound theorems for homology manifolds. Israel Journal of Mathematics, 108(1):45- 82, 1998. Triple systems with no six points carrying three triangles. I Z Ruzsa, E Szemerédi, Combinatorics (Proc. Fifth Hungarian Colloq. Keszthely; North-Holland, Amsterdam-New YorkIII. Z. Ruzsa and E. Szemerédi. Triple systems with no six points carrying three triangles. In Combi- natorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, volume 18 of Colloq. Math. Soc. János Bolyai, pages 939-945. North-Holland, Amsterdam-New York, 1978. A method for solving extremal problems in graph theory, stability problems. M Simonovits, Theory of Graphs (Proc. Colloq. Tihany; New YorkAcademic PressM. Simonovits. A method for solving extremal problems in graph theory, stability problems. In Theory of Graphs (Proc. Colloq., Tihany, 1966), pages 279-319. Academic Press, New York, 1968. The upper bound conjecture and cohen-macaulay rings. R Stanley, Stud. Appl. Math. 54R. P Stanley. The upper bound conjecture and cohen-macaulay rings. Stud. Appl. Math, 54:135-142, 1975. Combinatorics and Commutative Algebra. R P Stanley, Progress in Mathematics. Birkhäuser Boston. R.P. Stanley. Combinatorics and Commutative Algebra. Progress in Mathematics. Birkhäuser Boston, 2004. Komplexe in euklidischen räumen. E R Van Kampen, Abh. Math. Sem. Univ. Hamburg. 9E.R. van Kampen. Komplexe in euklidischen räumen. Abh. Math. Sem. Univ. Hamburg, 9:72-78, 1932. Minors in random and expanding hypergraphs. U Wagner, Proc. 27th Annual ACM Symposium on Computational Geometry (SoCG). 27th Annual ACM Symposium on Computational Geometry (SoCG)U. Wagner. Minors in random and expanding hypergraphs. Proc. 27th Annual ACM Symposium on Computational Geometry (SoCG), pages 351-360, 2011. On some properties of linear complexes. A A Zykov, Mat. Sbornik N.S. 2466A. A. Zykov. On some properties of linear complexes. Mat. Sbornik N.S., 24(66):163-188, 1949.
[]
[ "RELATIONSHIP BETWEEN DINUCLEAR SYSTEMS AND NUCLEI IN HIGHLY DEFORMED STATES Typeset using REVT E X 1", "RELATIONSHIP BETWEEN DINUCLEAR SYSTEMS AND NUCLEI IN HIGHLY DEFORMED STATES Typeset using REVT E X 1" ]
[ "T M Shneidman \nJoint Institute for Nuclear Research\n141980DubnaRussia\n\nInstitut für Theoretische Physik\nJustus-Liebig-Universität\nD-35392GiessenGermany\n", "G G Adamian \nJoint Institute for Nuclear Research\n141980DubnaRussia\n\nInstitut für Theoretische Physik\nJustus-Liebig-Universität\nD-35392GiessenGermany\n\nInstitute of Nuclear Physics\n702132TashkentUzbekistan\n", "N V Antonenko \nJoint Institute for Nuclear Research\n141980DubnaRussia\n\nInstitut für Theoretische Physik\nJustus-Liebig-Universität\nD-35392GiessenGermany\n", "S P Ivanova \nJoint Institute for Nuclear Research\n141980DubnaRussia\n\nInstitut für Theoretische Physik\nJustus-Liebig-Universität\nD-35392GiessenGermany\n", "W Scheid \nInstitut für Theoretische Physik\nJustus-Liebig-Universität\nD-35392GiessenGermany\n" ]
[ "Joint Institute for Nuclear Research\n141980DubnaRussia", "Institut für Theoretische Physik\nJustus-Liebig-Universität\nD-35392GiessenGermany", "Joint Institute for Nuclear Research\n141980DubnaRussia", "Institut für Theoretische Physik\nJustus-Liebig-Universität\nD-35392GiessenGermany", "Institute of Nuclear Physics\n702132TashkentUzbekistan", "Joint Institute for Nuclear Research\n141980DubnaRussia", "Institut für Theoretische Physik\nJustus-Liebig-Universität\nD-35392GiessenGermany", "Joint Institute for Nuclear Research\n141980DubnaRussia", "Institut für Theoretische Physik\nJustus-Liebig-Universität\nD-35392GiessenGermany", "Institut für Theoretische Physik\nJustus-Liebig-Universität\nD-35392GiessenGermany" ]
[]
Potential energies, moments of inertia, quadrupole and octupole moments of dinuclear systems are compared with the corresponding quantities of strongly deformed nuclei. As dinuclear system we denote two touching nuclei (clusters). It is found that the hyperdeformed states of nuclei are close to those of nearly symmetric dinuclear systems, whereas the superdeformed states are considered as states of asymmetric dinuclear systems. The superdeformed and hyperdeformed states constructed from two touching clusters have large octupole deformations. The experimental measurement of octupole deformation of the highly deformed nuclei can answer whether these nuclei have cluster configurations as described by the dinuclear model. PACS: 21.60.Ev,21.60.Gx
10.1016/s0375-9474(99)00828-3
[ "https://arxiv.org/pdf/nucl-th/9911006v1.pdf" ]
119,362,803
nucl-th/9911006
fe98770b0a706b0eac181bd027525aece5b0cdd7
RELATIONSHIP BETWEEN DINUCLEAR SYSTEMS AND NUCLEI IN HIGHLY DEFORMED STATES Typeset using REVT E X 1 2 Nov 1999 T M Shneidman Joint Institute for Nuclear Research 141980DubnaRussia Institut für Theoretische Physik Justus-Liebig-Universität D-35392GiessenGermany G G Adamian Joint Institute for Nuclear Research 141980DubnaRussia Institut für Theoretische Physik Justus-Liebig-Universität D-35392GiessenGermany Institute of Nuclear Physics 702132TashkentUzbekistan N V Antonenko Joint Institute for Nuclear Research 141980DubnaRussia Institut für Theoretische Physik Justus-Liebig-Universität D-35392GiessenGermany S P Ivanova Joint Institute for Nuclear Research 141980DubnaRussia Institut für Theoretische Physik Justus-Liebig-Universität D-35392GiessenGermany W Scheid Institut für Theoretische Physik Justus-Liebig-Universität D-35392GiessenGermany RELATIONSHIP BETWEEN DINUCLEAR SYSTEMS AND NUCLEI IN HIGHLY DEFORMED STATES Typeset using REVT E X 1 2 Nov 1999(April 19, 2022)Dinuclear systemCluster states in nucleiSuperdeformed and hyperdeformed states Potential energies, moments of inertia, quadrupole and octupole moments of dinuclear systems are compared with the corresponding quantities of strongly deformed nuclei. As dinuclear system we denote two touching nuclei (clusters). It is found that the hyperdeformed states of nuclei are close to those of nearly symmetric dinuclear systems, whereas the superdeformed states are considered as states of asymmetric dinuclear systems. The superdeformed and hyperdeformed states constructed from two touching clusters have large octupole deformations. The experimental measurement of octupole deformation of the highly deformed nuclei can answer whether these nuclei have cluster configurations as described by the dinuclear model. PACS: 21.60.Ev,21.60.Gx I. INTRODUCTION One of the most important developments in nuclear structure physics was the prediction and observation of superdeformed (SD) [1] and hyperdeformed (HD) [2] nuclear shapes. The idea that nuclei could adopt highly deformed prolate shapes at low excitation energies was originated with the discovery of deformed isomers in the actinide region [3,4]. Another group of superdeformed states with the ratio 3:2 of major to minor axis was discovered near the ground state in the A ≈ 76 ( 72 Se, 74,76 Kr) [5,6] and A ≈ 100 ( 98,100 Sr, 100 Zr) [7,8] mass regions. In these nuclei the SD ground and excited states are strongly mixed with a spherical band which coexists at low spin [9]. While in the rare earth nuclei the highly deformed shapes are stabilized by collective rotation, the highly deformed nuclei exist even at zero spin. The states with large spins are populated in heavy ion fusion reactions. By the study of rotational bands one can determine the moments of inertia of highly deformed nuclei. Based on the experimental values of the moment of inertia, it was found that the SD and HD states are related to shapes with a ratio of axes of 2:1 and 3:1, respectively. Since the intensity of γ-transitions drastically decreases with decreasing angular momentum, the experimental determination of the excitation energy of the SD band is difficult. In actinides the third potential minimum of the HD state is elucidated from a microstructure in the resonances found in the reactions (n, f ), (t, pf ) and (d, pf ) [10]. Another evidence supporting the existence of a third minimum is the observation of an asymmetric angular distribution of the light fission fragments of nuclei around 232 Th [11]. The transitions of odd multipolarity indicate a reflection-asymmetric shape of the nucleus 232 Th. Investigations of the SD and HD rotational bands in different mass regions were performed within semi-phenomenological cranked Woods-Saxon and Nilsson approaches [12,13]. Nuclear mean field theories, based on these approaches, well describe a rich variety of nuclear shapes all over the periodic table. These models allowed predictions of HD states which were recently observed experimentally [14]. The SD and HD rotational bands have been also investigated within relativistic mean field theories [15]. The nuclear shapes calculated within mean field theories are close to a rotational ellipsoid. It is known from the study of light nuclei ( 8 Be, 32 S) that the SD shape of nuclei can be considered as a symmetric di-molecular shape rather than an ellipsoid. In light α-particle nuclei the similarity between hyperdeformed and cluster-type states, i.e. quasi-molecular states, was already mentioned in [16,17]. Besides theoretical work, there are experimental evidences for the existence of the cluster-type configurations in fissioning nuclei [18]. The validity of the cluster approach for heavier nuclei has been investigated in [19][20][21][22]. An interesting observation in shell model calculations is that the nucleus in the third minimum corresponds to a dinuclear system (DNS) configuration [20]. However, in this model the clusters penetrate each other because the relative distance R between the centers of the clusters is smaller than the sum of cluster radii R 1 + R 2 . As it was shown in [23,24], the overlapping of nuclei is hindered by a repulsive potential at smaller relative distances R. Therefore, in the present paper we describe the SD and HD states by molecular-like dinuclear system configurations with a relative distance R m ≈ R 1 + R 2 , which corresponds to the minimum of the nucleus-nucleus potential [17]. In this paper we find a relationship between the DNS-type cluster configurations and highly deformed states of heavy nuclei. Consequences for assuming HD and SF states as cluster-type states are discussed. II. MULTIPOLE MOMENTS OF DNS The mass (k = m) and charge (k = c) multipole moments of the DNS shape are calculated with the expression Q (k) λµ = 16π 2λ + 1 ρ (k) (r)r λ Y λµ (Ω)dτ.(1) For small overlaps of the nuclei in the DNS when R ≥ R 1 + R 2 , where R 1 and R 2 are the radii of the nuclei and R the distance between the centers of nuclei, the nuclear mass and charge densities ρ (k) in the DNS can be written as a sum of the densities in each nucleus (frozen density approximation): ρ (k) (r) = ρ (k) 1 (r) + ρ (k) 2 (r).(2) By using Eq.(2) and assuming axial symmetry of the nuclear shapes, the multipole moments of the DNS in the center of mass of system can be expanded in the following form: Q (k) λ0 = Q (k) λ = λ λ 1 =0 λ 1 +λ 2 =λ (−1) λ λ! λ 1 !λ 2 ! (−1) λ 1 A λ 1 2 Q (k) λ 2 (1) + A λ 1 1 Q (k) λ 2 (2) R λ 1 A λ 1 ,(3) where multipole moments of the DNS nuclei Q Q (m) 1 = 0, Q (c) 1 = 2e A 2 Z 1 − A 1 Z 2 A R, Q (m) 2 = 2m 0 A 1 A 2 A R 2 + Q (m) 2 (1) + Q (m) 2 (2), Q (c) 2 = 2e A 2 2 Z 1 + A 2 1 Z 2 A 2 R 2 + Q (c) 2 (1) + Q (c) 2 (2), Q (m) 3 = 2m 0 A 1 A 2 A A 2 − A 1 A R 3 + 3 A 2 Q (m) 2 (1) − A 1 Q (m) 2 (2) A R, Q (c) 3 = 2e A 3 2 Z 1 − A 3 1 Z 2 A 3 R 3 + 3 A 2 Q (c) 2 (1) − A 1 Q (c) 2 (2) A R.(4) Here, A = A 1 +A 2 and A i , Z i (i = 1, 2) are the mass number of the system and the mass and charge numbers of the DNS nuclei, respectively; m 0 is the mass of the nucleon. Experimental values of the quadrupole moments of the DNS nuclei are taken in the calculations. We consider the DNS nuclei in pole-pole orientation which corresponds to the minimum of potential energy. Since the diffuseness in ρ i (r) in the nuclear surface does not practically influence the results for the here considered relative distances R, we disregard the effect of diffuseness in the present paper. The shape of an axially-deformed nucleus can be described by a multipole expansion of the nuclear surface R = R 0 (1 + β 0 Y 00 + β 1 Y 10 + β 2 Y 20 + β 3 Y 30 ),(5) where R 0 is the spherical equivalent radius of the nucleus and β 0 , β 1 , β 2 , β 3 are deformation parameters with respect to the center of mass. The β λ are widely used to characterize the experimental spectroscopic informations. The parameter β 0 is responsible for preserving the nuclear volume. The parameter β 1 provides the vanishing dipole moment Q (m) 10 = 0. The deformation parameters β 2 and β 3 are related to the quadrupole and octupole moments of the axially-deformed nucleus. With Eqs. (1) and (5) and assuming a constant nuclear density, we can express the mass multipole moments through the deformation parameters β λ (λ = 0, 1, 2, 3): Q (m) λ = 3 λ + 3 Am 0 R λ 0 λ+3 k 0 =0 k 0 k 1 =0 k 1 k 2 =0 k 2 k 3 =0 G λ k 0 k 1 k 2 k 3 β k 0 −k 1 0 β k 1 −k 2 1 β k 2 −k 3 2 β k 3 3(6) where G λ k 0 k 1 k 2 k 3 = 1 2 λ+k 2 λ! 3 k 1 −k 2 5 k 2 −k 3 7 k 3 (4π) k 0 λ + 3 k 0 k 0 k 1 k 1 k 2 k 2 k 3 × k 2 −k 3 i=0 k 3 j=0 (−1) i+j 3 k 2 −k 3 −i+j 5 k 3 −j k 2 − k 3 i k 3 j I ij , I ij = [λ/2] k=0 (−1) k λ k (2λ − 2k)! (λ − 2k)! 2δ k 1 +k 2 +k 3 +λ+1,odd (k 1 + k 2 + k 3 + λ + 1 − 2i − 2j − 2k) . Here, n k = n!/(k!(n − k)!). In spite of a better shape parameterization for large deformations [25] one can approximately apply Eq. (5) to the DNS. With Eq. (5) one can well describe the DNS shape for a small mass asymmetry |η| = |(A 2 − A 1 )/A| < 0.5. For larger |η|, Eq. (5) leads to smother shapes than the DNS shape. However, even asymmetric configurations can be effectively characterized by the parameters β λ . Using the experimental mass quadrupole and octupole moments, the experimentalists obtain the deformation parameters β λ (λ = 2, 3) with expressions equal or similar to (6). We do the same procedure but take the DNS multipole moments Q (m) λ from (4) instead of moments of SD and HD nuclei, extracted from experiment. Therefore, we solve the system of equations Q (m) λ = Q (m) λ(7) and find the dependences of β λ on the mass (charge) asymmetry η = (A 2 − A 1 )/A (η Z = (Z 1 − Z 2 )/Z) and the relative distance between the centers of nuclei. In Fig. 1 we show the dependence of the parameters β 1 , β 2 and β 3 on the mass asymmetry for the case that the nuclei of the DNS are assumed spherical. This dependence is independent of the total mass number which is simply demonstrated if we take A 1 = A(1 − η)/2 and A 2 = A(1 + η)/2. A dinuclear system with spherical and equal mass fragments (with radii R 0 ) in touching leads visually to the axis ratio 2:1 because it is the ratio of the DNS length (4R 0 ) to the DNS width (2R 0 ). However, the quadrupole moment Q (m) 2 for this system is equal to the quadrupole moment of ellipsoid with axis ratio 2.65:1. With deformed nuclei the ratio will be near 3:1 as for hyperdeformed nucleus. The values of quadrupole and octupole parameters of deformation become close to each other at large mass asymmetry. For very asymmetric DNS consisting of spherical nuclei, we can use simple analytical expressions for β 2 and β 3 : β 2 = 5 4π 4π 3 A 1 A 2 A 2 R 2 R 2 0 , β 3 = 7 4π 4π 3 A 1 A 2 A 2 A 2 − A 1 A R 3 R 3 0 .(8) III. MOMENT OF INERTIA OF DNS The moment of inertia is usually calculated microscopically by using the cranking-type formula and taking the residual interactions (for example, the pairing correlations) into consideration. However, one can consider simpler way to find the moment of inertia. The overlap volume of the touching nuclei in the DNS is about a few percent of the total DNS volume and the individuality of the DNS nuclei is conserved. Therefore, one can write down the potential energy U as the sum of binding energies of two nuclei and energy of their interaction (see Sect. 4) and express the multipole moments of the DNS through the multipole moments of the DNS nuclei. Due to the same reason, the DNS moment of inertia ℑ can be calculated as ℑ = ℑ 1 + ℑ 2 + m 0 A 1 A 2 A R 2 m ,(9) where the values of ℑ i (i=1,2) are the moment of inertia of the DNS clusters and in the case of small angular momentum are extracted from the experimental value of the energies [26]. For large angular momenta, the moments of inertia ℑ i (i = 1, 2) of the DNS nuclei can be with good accuracy calculated in the rigid body approximation E 2 + →0 + of 2 + → 0 + transitions: ℑ i = 3/E 2 + →0 + (h 2 /MeV)ℑ i = 1 5 m 0 A i (a 2 i + b 2 i ), a i = R 0i (1 − α 2 i 4π )(1 + 5 4π α i ), b i = R 0i (1 − α 2 i 4π )(1 − 5 16π α i )(10) because the experimental data [28] show that the moment of inertia of superdeformed states is very close to the rigid body limit (about 85 percent of last one). Here, R 0i and α i (i = 1, 2) are the spherical equivalent radii and the parameters of quadrupole deformation of the DNS nuclei, respectively. The moment of inertia is well measured for the SD and HD states. Therefore, a comparison of calculated and experimental values of the moment of inertia can prove our interpretation of the shapes of highly deformed nuclei. Of course, the complete clarification of our cluster approach and the problem how to calculate the moment of inertia could come from attempt to compute actual spectra for direct comparison with data, especially the low-lying negative parity bands symptomatic of octupole deformations. In the forthcoming paper we hope to explain the parity splitting as the effect of tunneling along mass asymmetry (by the analogy with tunneling along β 3 [29]) from the asymmetic DNS with positive η to the DNS with the same but negative η. For this purpose, we need more accurate calculations of the potential energy and, correspondingly, of the moment of inertia as a function of L. IV. POTENTIAL ENERGY OF DNS In order to check the possibility of the DNS formation in the excited compound nucleus, the potential energy of the DNS is calculated as [17] U(R, η, L) = B 1 + B 2 + V (R, η, L) − [B 12 + V ′ rot (L)].(11) Here, B 1 , B 2 , and B 12 are the realistic binding energies of the fragments and the compound nucleus, respectively. The shell effects are included in these binding energies and supply the local minima in U(R, η, L) as a function of η. The value of U(R, η, L) is normalized to the energy of the rotating compound nucleus by B 12 + V ′ rot . The nucleus-nucleus potential (11) V (R, η, L) = V coul (R, η) + V N (R, η) + V rot (R, η, L)(12) is the sum of the Coulomb [30] V coul = Z 1 Z 2 e 2 R + 3 5 Z 1 Z 2 e 2 R 3 2 i=1 R 2 0i α i Y 20 (cos θ i ) + 12 35 Z 1 Z 2 e 2 R 3 2 i=1 R 2 0i (α i Y 20 (cos θ i )) 2 ,(13) the centrifugal [17] V rot =h 2 L(L + 1) 2ℑ (14) and the nuclear V N (R, η) = ρ 1 (r 1 )ρ 2 (R − r 2 )F (r 1 − r 2 )dr 1 dr 2(15) potentials. For the nuclear part of V (R, η, L) we use a double folding formalism with the Skyrme-type effective density-dependent nucleon-nucleon interaction F (r 1 − r 2 ) = C 0 F in ρ 1 (r 1 ) + ρ 2 (R − r 2 ) ρ 0 + F ex (1 − ρ 1 (r 1 ) + ρ 2 (R − r 2 ) ρ 0 ) δ(r 1 − r 2 ), F in,ex = f in,ex + f ′ in,ex N 1 − Z 1 A 1 N 2 − Z 2 A 2 ,(16) which is well known from the theory of finite Fermi systems [17]. Here, N i (i=1,2) are neutron numbers of the nuclei. The values of C 0 =300 MeV fm 3 and the dimensionless parameters f in =0.09, f ex =-2.59, f ′ in =0 .42 and f ′ ex =0.54 are fitted to describe a large number of experimental data [31]. For the density of heavy nuclei one can use the two-parameter symmetrized Woods-Saxon function ρ i (r) = ρ 0 sinh(R i (θ i , φ i )/a i ) cosh(R i (θ i , φ i )/a i ) + cosh(r/a i ) ,(17) where ρ 0 =0.17 fm −3 is density in the center of nucleus. We use a nuclear radius parameter r 0 = 1.14 fm and a diffuseness parameter a i = (0.54−0.55 fm depending on the mass number of the isotope. For light nuclei with mass numbers A i <16, the more realistic functional dependence ρ i (r) = A i (γ 2 i /π) 3/2 exp (−γ 2 i r 2 )(18) is used, where γ i characterizes the width of the nucleon distribution in the nucleus. The value of γ i can be obtained by minimizing the nuclear binding energy in the density function [17]. The deformation effects are taken into account in the calculation of the potential energy surface [17]. The shapes of the DNS nuclei with quadrupole deformations can be written as R i (θ i , φ i ) = R 0i (1 − α 2 i 4π + α i Y 20 (θ i , φ i )).(19) With the nucleon-nucleon interaction (16) the repulsive core appears in V N (R, η) which prevents the motion to smaller distances (R < R 1 + R 2 ) and reflects the action of the Pauli principle. For R 1 + R 2 − 1 fm < R < R 1 + R 2 + 2 fm the potential V (R, η, L) as a function of the relative distance R has a pocket with small depth which is resulted by the atractive nuclear and repulsive Coulomb interactions. The DNS is localized in the minimum of this pocket at R m ≈ R 1 + R 2 = R 01 (1 + 5 4π α 1 ) + R 02 (1 + 5 4π α 2 ) for the pole-pole configuration. The relative orientation of the deformed nuclei in the DNS follows the minimum of the potential energy which yields the pole-pole orientation. At fixed mass asymmetry η, charge asymmetry η Z and deformation parameters α i (i=1,2) the minimum of V (R, η, L) of the DNS corresponds to the distance R = R m (η, η Z , α i ) when the poles of nuclei touch each other. For each η, we minimize U(R m , η, L) with respect to the charge asymmetry. Since the minimization of U with respect to α i ia very cumbersome, we took from Refs. [26,27] the parameters of the quadrupole deformation which are related to the excitation of the first 2 + states if the energies of these states are smaller than 1.5 MeV. As is known from experiments on sub-barrier fusion, these states are easily populated. For the treatment of small internal excitation of the DNS nuclei, this approximation seems to be good for deriving the minimum of U with respect to α i . So, the energy U(R m , η, L) plotted as a function of η results minimum values B 1 + B 2 + V (R, η, L) with respect to other degrees of freedom in the DNS. The potentials for all DNS were calculated with the same set of parameters and assumptions. We found that the final results are not crucial to a reasonable variation of these parameters in the calculation of the potential energy. Since the excitation energies of the DNS nuclei are relatively small, the calculation of the DNS potential energy immediately shows which DNS configuration can be related to the SD or HD nuclear systems. Among all minima in U(R m , η, L) as a function of η we selected those with the minimal values of U. These minima appear for small and large η and we analysed either almost symmetric or very asymmetric configurations. If there are several minima with close energies in some interval of η, all of them are treated. However, the calculated values of Q 2 and ℑ would be similar for these DNS. V. RESULTS AND DISCUSSIONS With the formalism given in Sections II-IV we studied various nuclei described with the dinuclear system concept. As compound nuclei we chose 152 Dy, 232 Th, 234 U, 240 Pu, 76 Kr, 148 Nd and 236 Ra which will be discussed next. a) 152 Dy: The dependences of β 2 , β 3 , Q 2 (Q 2 = Q (c) 2 ) and Q 3 (Q 3 = Q (c) 3 ) on the mass asymmetry η are presented in Fig. 2 for the DNS corresponding to the 152 Dy compound nucleus. Since the deformations of the DNS nuclei are functions of η, these dependences have some oscillations. While Q 2 and β 2 decrease with increasing η, Q 3 and β 3 have maxima. The positions of these maxima approximately correspond to maximal potential energy of the DNS as a function of η (see Fig. 3). The value of β 3 increases quickly with η from zero. For very asymmetric DNS, β 3 becomes again relatively small. Therefore, a given value of β 3 can correspond to two different DNS with different mass asymmetries. For the symmetrical DNS with spherical nuclei, the difference by a factor √ 2 between our value of β 2 and β 2 obtained in [22] is due to another definition of β 2 in [22]. The DNS potential energy as a function of the mass asymmetry is shown in Fig. 3 for the cases of spherical and deformed nuclei in the DNS forming 152 Dy. The calculation with deformed nuclei yields various minima of the potential energy, especially at η = 0.026 ( 74 Ge+ 78 Se), η = 0.16 ( 64 Ni+ 88 Sr) and η = 0.34 ( 50 Ti+ 102 Ru). For zero angular momentum, the energy of the combination 50 Ti+ 102 Ru is about 20 MeV which is close to the value estimated for the HD state in [32]. We calculated the DNS moment of inertia ℑ using the experimental moments ℑ i of the DNS nuclei (ℑ = ℑ e ) or the rigid body moments ℑ i given in eq.(10) (ℑ = ℑ r ). The first value of the moment of inertia of the system 50 Ti+ 102 Ru is obtained as ℑ e = 100h 2 MeV −1 and the second one as ℑ r = 131h 2 MeV −1 . The latter value is close to the experimental one ℑ exp = 130h 2 MeV −1 [2]. For this DNS, the obtained value β 2 = 1.3 is in agreement with the experimental estimate β 2 ≥ 0.9. Therefore, we conclude that the shape of the DNS 50 Ti+ 102 Ru is compatible with the shape of the 152 Dy nucleus in the HD state. We found that in the asymmetric DNS, for example 22 Ne+ 130 Ba (η=0.71) and 26 Mg+ 126 Xe (η=0.66) (where the potential energy has minima), the calculated moments of inertia and quadrupole moments are close to the experimental values ℑ exp = (85 ± 3)h 2 MeV −1 and Q exp 2 = (18 ± 3)e barn, respectively, known for the SD state. For the system 22 Ne+ 130 Ba, we calculated ℑ r = 100h 2 MeV −1 (and ℑ e = 63h 2 MeV −1 ), Q 2 = 20e barn and β 2 = 0.8. In the system 26 Mg+ 126 Xe we obtained ℑ r = 104h 2 MeV −1 (and ℑ e = 67h 2 MeV −1 ), Q 2 = 24e barn and β 2 = 0.9. These DNS have practically zero temperature if they are formed in the reaction 48 Ca(205 MeV) ( 108 Pd,4n) 152 Dy. At L = 0 the potential energies of these DNS with respect to the ground state of 152 Dy lie about 8 MeV higher than those estimated for the SD shapes in [32]. The energies of these two systems with respect to the energy of the compound nucleus are shown in Fig. 4 as a function of the square of the angular momentum. The SD bands in the mass region A ≈ 150 are populated with an anomalously high intensity around a spin of 55h. There is no significant population of these states below (45 − 50)h [33]. This fact can be explained by the DNS interpretation of the SD states. The dependence of the DNS potential energy on η becomes flat with decreasing angular momentum and the DNS can evolve to larger η. This takes place because the potential barrier in the direction of larger mass asymmetries decreases with L (Fig. 3). Due to the distribution of the excitation energy among many configurations at small L and due to the motion of the DNS to the compound nucleus with increasing η, the transition from the SD state to the ground state is invisible. b) 232 Th: Analysing the potential energy of the DNS as a function of η for the 232 Th compound nucleus (see Fig. 5), we found well exposed minima corresponding to the systems 100 Zr+ 132 Sn (η = 0.138) and 82 Ge+ 150 Ce (η = 0.293). Fig. 6 shows the dependence of β 2 and ℑ = ℑ r on η which is weak for small mass asymmetries. In both 232 Th systems, β 2 is about 1.5, β 3 ≈ 0.40, ℑ r about 290h 2 MeV −1 and ℑ e ≈ 240h 2 MeV −1 ; the potential energies, given in Fig. 5, are near the energy of the ground state of 232 Th. Excepting the values of β 2 , the obtained values in Table 1 are close to the corresponding ones calculated for the third minimum in [20]. In our calculation β 2 is larger in comparison to β 2 = 0.85 in [20]. The SD rotational bands in 232 Th can be interpreted as the DNS states of the configurations 28 Mg+ 204 Pt and 26 Ne+ 206 Hg (Table 1). c) 234 U and 240 Pu: For the 234 U nucleus, we found the SD and HD cluster configurations (Table 1): 26 Ne+ 208 Pb, 28 Mg+ 206 Hg, 82 Ge+ 152 Nd, 100 Zr+ 134 Te and 104 Mo+ 130 Sn. The experimentally extracted depth of the third well (which corresponds to the HD state) was determined to be (3.6 ± 0.3) MeV for the 234 U nucleus [34]. For the DNS configurations, this value is an agreement with the value of the depth of the pocket in the nucleus-nucleus potential as function of η. For example, for the 100 Zr+ 134 Te and 104 Mo+ 130 Sn configurations, the depth of the pocket is about 3.4 MeV. Fig. 5 shows also the potential energy of the 240 Pu compound nucleus where SD bands were observed in an experiment on sub-barrier fission [34]. These SD states could be considered as the DNS-type states: 32 Mg+ 208 Pb and 34 Si+ 206 Hg (see Table 1). From the results presented in Table 1 one can predict some HD cluster states for the 240 Pu nucleus. d) 74 Kr and 72 Se: The potential energy of the DNS for the 76 Kr compound nucleus is shown in Fig. 5. We find a deep minimum with an energy on the level of the energy of the compound nucleus for the cluster configuration 8 Be+ 68 Ge. For this cluster state we have Q 2 =4.9 e barn, Q 3 = 2.0 × 10 3 e fm 3 and ℑ r = 29h 2 /MeV (ℑ e = 17.2h 2 /MeV). A similar picture is observed in other nuclei in the A ≈ 76 mass region, for example, in 74 Kr and 72 Se. In the DNS configuration with the light cluster 8 Be we found Q 2 = 4.7 e barn, Q 3 = 1.9 × 10 3 e fm 3 for the 72 Se compound nucleus. For the nuclei in the mass region A ≈ 100, the energies of such cluster configurations are by 5 to 6 MeV higher than the energies of corresponding compound nuclei. Let us consider the N/Z-equilibrium. In the nuclei 72 Se, 74,76 Kr we obtained minima in the DNS potential energy for configurations with α-particle nuclei (multiple of α-clusters) at large η. The potential energies of these DNS are small because the light nuclei in them are most stable. Such behaviour of the DNS potential energy is not observed for asymmetric configurations with light clusters in the A ≈ 100 mass region. e) α-structure and dipole and octupole deformations, 148 Nd, 226 Ra and 220−228 Th: If very asymmetric DNS are energetically favorable, the wave function of the compound nucleus has components belonging to cluster-type configurations. For example, in many cases the cluster configurations A Z → (A−4) (Z − 2) + 4 He(20) have an energy which is close or even lower than the energy of the ground state of the compound nucleus. As a result, the nucleus can have an octupole deformation in the ground state. Since the shape of the nucleus is no more symmetric under space inversion, the spectrum of such a nucleus must contain states with different parity. This was experimentally found in many nuclei with Z ≈ 88 − 90 (N ≈ 86 − 90), in different isotopes of the nuclei Ra, Th and U, and with Z ≈ 60, in isotopes of Ba, Ce, Nd, Sm and Gd [35]. Another consequence of the asymmetric shape is the appearance of E1 and E3 transitions. These transitions were found in 226 Ra with Q 2 =750 e fm 2 , Q 3 =3100 e fm 3 [37] and in 148 Nd with Q 2 =400 e fm 2 , Q 3 =1500 e fm 3 [38]. The experimental dipole moments of this nuclei are found to be 0.16 e fm for 226 Ra [37] and 0.32 e fm for 148 Nd [38]. Under the assumption that the cluster configuration (20) mainly contributes to the ground state with a static octupole deformation, we can obtain the values of the multipole moments for these nuclei. For 226 Ra, we found Q 1 = Q (c) 1 /2=4 e fm, Q 2 =776 e fm 2 , Q 3 =2662 e fm 3 . For 148 Nd, we obtained Q 1 =3 e fm, Q 2 =486 e fm 2 , Q 3 =1844 e fm 3 . The calculated values of Q 2 and Q 3 are close to the experimental ones mentioned before. However, the values of Q 1 are larger by at least one order of magnitude. The same problem was observed in the cluster model [36]. These deviations of the theoretical dipole moments from the experimental ones are due to the use of a simplified treatment of the N 1 /Z 1 -ratios in the DNS nuclei. The value of Q 1 strongly depends on these ratios and vanishes (Q 1 = 0) in the limit of the same N/Z-ratio in the DNS nuclei. For very asymmetric DNS, the N/Z-ratios in the light nucleus are effectively larger than the N/Z-ratio in the α-particle (N/Z = 1)because the heavy nucleus of the DNS is strongly overlapping with the α-particle and there is at least one valence neutron supplying the coupling of the α-particle with the heavy cluster. If we take the ( 4 He+1n) cluster instead of 4 He in the DNS or slightly increase the N/Z-ratio in the contact region of the two nuclei of the DNS, then the theoretical value of Q 1 for 226 Ra and 148 Nd results in agreement with the experimental data, but with Q 2 and Q 3 practically not changed. For the nuclei with octupole deformation in the ground state, one can try to explain the parity splitting as a function of L by the tunneling in η in the DNS with α-particle. As shown in Fig. 7, the octupole (quadrupole) deformation becomes smaller (larger) with an increasing atomic number of the Th isotopes in the systems (A−4) Ra+ 4 He. This can be explained by the fact that the quadrupole deformation of the heavy cluster of the DNS gets larger in this mass region. Such a behaviour of β 2 and β 3 is in agreement with the one obtained in [39]. VI. SUMMARY The DNS potential energy as a function of mass asymmetry has a few global minima. Most of them lie higher in energy than the energy of the compound nucleus. It is possible, however, to populate these states in heavy ion induced reactions by choosing appropriate reaction partners with an appropriate bombarding energy. At high spins, these cluster states can be cold and have long lifetime. We found the energies, moments of inertia and quadrupole deformations of certain DNS close to the experimental ones of SD and HD nuclei. Since many DNS states have relatively large octupole deformations, the experimental measurement of the octupole deformation of highly deformed nuclei can answer the question whether these nuclei exist in cluster configurations. An evidence, that SD nuclei can have octupole deformations, is for example an observation of an excited SD band in 190 Hg which decays to the lowest-energy (yrast) SD-band by transitions of odd multipolarity. The E1 rate observed in [40] is three orders of magnitude larger then those rates typically observed in heavy deformed nuclei and is similar to those observed in octupole-unstable normallydeformed actinide nuclei [41]. In some nuclei with A ≈ 230 or A ≈ 76 the potential energy of the DNS has minima which lie roughly on the same energy as the energy of the compound nucleus. This means that such cluster states can exist at low spins. We attempted to describe the nuclei with static octupole deformation in the ground state like the DNS where the α -cluster is the lighter cluster. It was found that calculated quadrupole and octupole deformations are close to the experimental ones and that such DNS are energetically favorable. TABLES are calculated in their centers of mass. For example, up to λ = 3 the values of Q (k) λ are: FIG. 1 . 1Dependence of the deformation parameters β 1 , β 2 , β 3 on mass asymmetry η for a DNS with spherical nuclei. The parameters do not depend on the total mass number of the DNS. FIG. 2 . 2Dependence of Q 2 (10 −2 e fm 2 ) and Q 3 (10 −3 e fm 3 ) (upper part) and of β 2 and β 3 (lower part) on the mass asymmetry η of the DNS for the 152 Dy compound nucleus. The deformation of the DNS nuclei are taken into account. FIG. 3 .FIG. 4 .FIG. 5 .FIG. 6 .FIG. 7 . 34567Potential energy U of the DNS as a function of mass asymmetry η for the 152 Dy compound nucleus. At L = 0 the calculated results without and with deformation of the DNS nuclei are presented by solid and dotted lines, respectively (upper part). The lower part shows results for different angular momentum quantum numbers calculated with deformed DNS nuclei. Dependence of the energies of the systems 22 Ne+ 130 Ba (dotted line) and 26 Mg+ 126 Xe (solid line) on the square of the angular momentum quantum number. The energy is normalized to the energy of the rotating compound nucleus. Potential energy U of the DNS as a function of η for the compound nuclei 232 Th (upper part), 76 Kr (middle part) and 240 Pu (lower part) at L = 0. The deformation of the DNS nuclei is taken into account. Deformation parameters β 2 and β 3 (upper part) and moment of inertia ℑ = ℑ r (lower part) as a function of mass asymmetry η of the DNS corresponding to the 232 Th compound nucleus. The β 2 vs. β 3 plot for different isotopes of A Th in the cluster state (A−4) Ra+ 4 He. TABLE I . IThe values of moments of inertia ℑ r and ℑ e , charge quadrupole moments Q 2 and octupole moments Q 3 , quadrupole and octupole deformation parameters β 2 and β 3 for different DNS corresponding to the compound nuclei 232 Th, 234 U, 240 Pu (see text).Clust. Conf.ℑ r (h 2 /MeV) ℑ e (h 2 /MeV) Q 2 × 10 2 (e fm 2 ) Q 3 × 10 3 (e fm 3 ) β 2β 3 26 Ne+ 206 Hg → 232 Th 171 82 24.9 18.8 0.57 0.63 28 Mg+ 204 Pt→ 232 Th 180 94 31.0 23.9 0.65 0.68 82 Ge+ 150 Ce → 232 Th 292 249 70.9 19.8 1.53 0.47 100 Zr+ 132 Sn → 232 Th 292 235 70.1 16.2 1.48 0.34 26 Ne+ 208 Pb→ 234 U 169 79 20.9 18.0 0.47 0.61 28 Mg+ 206 Hg→ 234 U 179 91 29.9 23.7 0.61 0.68 82 Ge+ 152 Nd→ 234 U 291 257 70.4 20.5 1.49 0.49 100 Zr+ 134 Te→ 234 U 326 258 71.8 14.0 1.73 0.27 104 Mo+ 130 Sn→ 234 U 296 242 71.8 14.0 1.48 0.28 32 Mg+ 208 Pb→ 240 Pu 191 101 28.7 23.2 0.57 0.71 34 Si+ 206 Hg→ 240 Pu 197 107 33.9 25.4 0.69 0.70 82 Ge+ 158 Sm→ 240 Pu 307 267 75.3 22.1 1.53 0.50 104 Zr+ 136 Xe→ 240 Pu 305 253 74.1 14.8 1.49 0.33 106 Mo+ 134 Te→ 240 Pu 314 258 76.4 16.0 1.52 0.32 110 Ru+ 130 Sn→ 240 Pu 311 250 75.3 12.4 1.49 0.23 ACKNOWLEDGMENTSWe thank Prof. R.V.Jolos for fruitful discussions and Yu.V.Palchikov for valuable advices regarding the calculations. G.G.A. and T.M.S. are grateful the Alexander von Humboldt-Stiftung and the European Physical Society for support, respectively. This work was supported in part by DFG and RFBR. . P J Twin, Phys. Rev. Lett. 57811P.J. Twin et al., Phys. Rev. Lett. 57 (1986) 811. . A Galindo-Uribarri, Phys. Rev. Lett. 71231A. Galindo-Uribarri et al., Phys. Rev. Lett. 71 (1993) 231. . S M Polikanov, JETP. 151016S.M. Polikanov, JETP. 15 (1962) 1016. . H J Specht, Phys. Lett. 4143H.J. Specht, Phys. Lett. B41 (1972) 43. . J H Hamilton, Phys. Rev. Lett. 32239J.H. Hamilton et al., Phys. Rev. Lett. 32 (1974) 239. . R B Piercey, Phys. Rev. Lett. 471514R.B. Piercey, Phys. Rev. Lett. 47 (1981) 1514. . T A Kahn, Z. Phys. 283103T.A. Kahn et al., Z. Phys. A283 (1977) 103. . R F Azuma, Phys. Lett. 865R.F. Azuma et al., Phys. Lett. B86 (1979) 5. . J H Hamilton, Prog. Part. Nucl. Phys. 2887J.H. Hamilton, Prog. Part. Nucl. Phys. 28 (1992) 87. . B B Brick, Phys. Rev. Lett. 281707B.B. Brick et al., Phys. Rev. Lett. 28 (1972) 1707. . K Th, Brinkmann, Nucl. Phys. 502271K.Th. Brinkmann et al., Nucl. Phys. A502 (1989) 271. . S Aberg, Annu. Rev. Nucl. Part. Sci. 40439S. Aberg et al., Annu. Rev. Nucl. Part. Sci. 40 (1990) 439. . R V F Janssens, Annu. Rev. Nucl. Part. Sci. 41321R.V.F. Janssens et al., Annu. Rev. Nucl. Part. Sci. 41 (1991) 321. . J Dudek, Phys. Lett. 211252J. Dudek et al. Phys. Lett. B211 (1988) 252. . P Ring, A V Afanasjev, Prog. Part. Nucl. Phys. 38137P. Ring and A.V. Afanasjev, Prog. Part. Nucl. Phys. 38 (1997) 137. . N Cindro, J. Phys. 423N. Cindro, J. Phys. G4 (1978) L23; . N Cindro, W Greiner, J. Phys. 9175N. Cindro and W. Greiner, J. Phys. G9 (1983) L175. . G G Adamian, Int. J. Mod. Phys. 5191G.G. Adamian et al., Int. J. Mod. Phys. E5 (1996) 191. . Yu V Pyatkov, Nucl. Phys. 624140Yu.V. Pyatkov et al., Nucl. Phys. A624 (1997) 140. . S Aberg, Z. Phys. 349205S. Aberg, Z. Phys. A349 (1994) 205. . S Cwiok, Phys. Lett. 322304S. Cwiok et al., Phys. Lett. B322 (1994) 304. . B Buck, A C Merchant, S M Perez, Phys. Rev. Lett. 652975B. Buck, A.C. Merchant and S.M. Perez, Phys. Rev. Lett. 65 (1990) 2975; . Phys. Rev. Lett. 762049Phys. Rev.Phys. Rev. Lett. 76 (1996) 380; Phys. Rev. C58 (1998) 2049. . S Royer, J. Phys. 21339S. Royer, J. Phys. G21 (1995) 339. . Yu F Smirnov, Yu M , Phys.Lett. 31425Yu.F. Smirnov and Yu.M. Tchuvil'sky, Phys.Lett. B314 (1984) 25. . G G Adamian, N V Antonenko, Yu M , Phys. Lett. 451289G.G. Adamian, N.V. Antonenko and Yu.M. Tchuvil'sky, Phys. Lett. B451 (1999) 289. . V V Pashkevich, Nucl. Phys. 169275V.V. Pashkevich, Nucl. Phys. A169 (1971) 275. . S Raman, At. Data Nucl. Data Tables. 361S. Raman et al., At. Data Nucl. Data Tables 36 (1987) 1. . P Möller, R Nix, At. Data and Nucl. Tables. 39213P. Möller, R. Nix, At. Data and Nucl. Tables 39 (1988) 213; http://t2.lanl.gov/data/ astro/molnix96/massd.html. . J Dudek, Prog. Part. Nucl. Phys. 28131J. Dudek, Prog. Part. Nucl. Phys. 28 (1992) 131. . R V Jolos, P Brentano, Phys. Rev C49. 2301R.V. Jolos and P. von Brentano, Phys. Rev C49 (1994) R2301; . Nucl. Phys. 587377Nucl. Phys. A587 (1995) 377. . C Y Wong, Phys. Rev. Lett. 766C.Y. Wong, Phys. Rev. Lett. 31 (1973) 766. A B , Theory of Finite Fermi Systems and Application to Atomic Nuclei. Nauka, MoscowA. B. Migdal, Theory of Finite Fermi Systems and Application to Atomic Nuclei (Nauka, Moscow, 1982). . S Aberg, Nucl. Phys. 55717S. Aberg, Nucl. Phys. A557 (1993) 17. . M A Bentley, J. Phys. 17481M.A. Bentley et al., J. Phys. G17 (1991) 481. . D Gassmann, Verhandl. DPG (VI). 34162D. Gassmann et al., Verhandl. DPG (VI) 34 (1999) 162; . A Krasznahorkay, Phys. Rev. Lett. 4612073Phys. Lett.A. Krasznahorkay et al., Phys. Lett. B461 (1999) 15; Phys. Rev. Lett. 80 (1998) 2073. . P A Butler, W Nazarewicz, Rev. Mod. Phys. 68350P.A. Butler and W. Nazarewicz, Rev. Mod. Phys. 68 (1996) 350. . B Buck, A C Merchant, S M Perez, Phys. Rev. 59750B. Buck, A.C. Merchant and S.M. Perez, Phys. Rev. C59 (1999) 750. . H J Wollersheim, Nucl. Phys. 556261H.J. Wollersheim, Nucl. Phys. A556 (1993) 261. . R Ibbotson, Phys. Rev. Lett. 71R. Ibbotson, Phys. Rev. Lett. 71 (1993) 1990. . W Nazarewicz, Nucl. Phys. 467437W. Nazarewicz et al., Nucl. Phys. A467 (1987) 437. . B Crowell, Phys. Lett. 333320B. Crowell et al., Phys. Lett. B333 (1993) 320. . I Ahmad, P A Butler, Ann. Rev. Nucl. Part. Sci. 4371I. Ahmad and P.A. Butler, Ann. Rev. Nucl. Part. Sci. 43 (1993) 71.
[]
[ "Visual anemometry: physics-informed inference of wind for renewable energy, urban sustainability, and environmental science", "Visual anemometry: physics-informed inference of wind for renewable energy, urban sustainability, and environmental science" ]
[ "John O Dabiri [email protected] \nGraduate Aerospace Laboratories\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n\nMechanical and Civil Engineering\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n", "Michael F Howland \nCivil and Environmental Engineering\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n", "Matthew K Fu \nGraduate Aerospace Laboratories\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n", "Roni H Goldshmid [email protected] \nGraduate Aerospace Laboratories\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n" ]
[ "Graduate Aerospace Laboratories\nCalifornia Institute of Technology\n91125PasadenaCAUSA", "Mechanical and Civil Engineering\nCalifornia Institute of Technology\n91125PasadenaCAUSA", "Civil and Environmental Engineering\nMassachusetts Institute of Technology\n02139CambridgeMAUSA", "Graduate Aerospace Laboratories\nCalifornia Institute of Technology\n91125PasadenaCAUSA", "Graduate Aerospace Laboratories\nCalifornia Institute of Technology\n91125PasadenaCAUSA" ]
[]
Accurate measurements of atmospheric flows at meter-scale resolution are essential for a broad range of sustainability applications, including optimal design of wind and solar farms, safe and efficient urban air mobility, monitoring of environmental phenomena such as wildfires and air pollution dispersal, and data assimilation into weather and climate models. Measurement of the relevant microscale wind flows is inherently challenged by the optical transparency of the wind. This review explores new ways in which physics can be leveraged to "see" environmental flows non-intrusively, that is, without the need to place measurement instruments directly in the flows of interest. Specifically, while the wind itself is transparent, its effect can be visually observed in the motion of objects embedded in the environment and subjected to wind-swaying trees and flapping flags are commonly encountered examples. We describe emerging efforts to accomplish visual anemometry, the task of quantitatively inferring local wind conditions based on the physics of observed flow-structure interactions. Approaches based on first-principles physics as well as data-driven, machine learning methods will be described, and remaining obstacles to fully generalizable visual anemometry will be discussed.Key Points• Atmospheric winds near the Earth's surface mediate a variety of essential physical, chemical, and biological processes at length-scales ranging from millimeters to kilometers across the globe.• A better understanding of wind dynamics can enable more efficient renewable energy technologies, more accurate monitoring and modeling of weather and climate, and more rapid adoption of new technologies such as urban air mobility.• Visual anemometry is an emerging technique that aims to infer quantitative estimates of wind speed and direction based on visual observations of associated flow-structure interactions such as swaying trees and flapping flags.
null
[ "https://export.arxiv.org/pdf/2304.04728v2.pdf" ]
258,048,526
2304.04728
e8ec3cc64fef2fcc7178edcf642f10fc23f1b111
Visual anemometry: physics-informed inference of wind for renewable energy, urban sustainability, and environmental science John O Dabiri [email protected] Graduate Aerospace Laboratories California Institute of Technology 91125PasadenaCAUSA Mechanical and Civil Engineering California Institute of Technology 91125PasadenaCAUSA Michael F Howland Civil and Environmental Engineering Massachusetts Institute of Technology 02139CambridgeMAUSA Matthew K Fu Graduate Aerospace Laboratories California Institute of Technology 91125PasadenaCAUSA Roni H Goldshmid [email protected] Graduate Aerospace Laboratories California Institute of Technology 91125PasadenaCAUSA Visual anemometry: physics-informed inference of wind for renewable energy, urban sustainability, and environmental science 1 *Authors for correspondence: Accurate measurements of atmospheric flows at meter-scale resolution are essential for a broad range of sustainability applications, including optimal design of wind and solar farms, safe and efficient urban air mobility, monitoring of environmental phenomena such as wildfires and air pollution dispersal, and data assimilation into weather and climate models. Measurement of the relevant microscale wind flows is inherently challenged by the optical transparency of the wind. This review explores new ways in which physics can be leveraged to "see" environmental flows non-intrusively, that is, without the need to place measurement instruments directly in the flows of interest. Specifically, while the wind itself is transparent, its effect can be visually observed in the motion of objects embedded in the environment and subjected to wind-swaying trees and flapping flags are commonly encountered examples. We describe emerging efforts to accomplish visual anemometry, the task of quantitatively inferring local wind conditions based on the physics of observed flow-structure interactions. Approaches based on first-principles physics as well as data-driven, machine learning methods will be described, and remaining obstacles to fully generalizable visual anemometry will be discussed.Key Points• Atmospheric winds near the Earth's surface mediate a variety of essential physical, chemical, and biological processes at length-scales ranging from millimeters to kilometers across the globe.• A better understanding of wind dynamics can enable more efficient renewable energy technologies, more accurate monitoring and modeling of weather and climate, and more rapid adoption of new technologies such as urban air mobility.• Visual anemometry is an emerging technique that aims to infer quantitative estimates of wind speed and direction based on visual observations of associated flow-structure interactions such as swaying trees and flapping flags. • Physics-based models of flow-structure interactions and data-driven models that leverage machine learning and artificial intelligence have both demonstrated initial proofs-of-concept that site-specific visual anemometry is feasible. • Generalized visual anemometry-a technique that does not require calibration measurements or a priori collection of training data-will likely depend on the discovery of new physical principles that govern the diversity of environmental structures exposed to wind. I. Introduction The fate of life on Earth depends on macroscopic physical processes that are nonetheless imperceptible to the naked eye. Specifically, the movement of air masses at local scales mediates essential gas exchanges between the atmosphere and the terrestrial and aquatic ecosystems that lie underneath [1][2][3][4][5][6][7][8] . This flow of air is also a principal means of transportation for life ranging from bacteria 9 and seed spores [10][11][12][13][14] to animals that migrate seasonally across the globe [15][16][17] . Engineering technologies with the promise to protect those same ecosystems are also dependent on the wind. The functional reliance of technologies like wind turbines is a straightforward example [18][19][20] ; however, it may be less appreciated that the performance of solar energy farms is also determined by local wind conditions 21,22 . Text Box 1 provides further discussion of the diverse roles of wind flows in environmental sustainability applications. Given this broad and important role of the wind for current and future environmental sustainability, it is remarkable that we have relatively few tools available to quantify the wind at the length and time scales relevant to many of the applications identified above. Such measurements are inherently limited by the optical transparency of the air. To date, the most common solutions to this limitation require introducing an engineered, physical object into the flow, whose interaction with the wind can be detected visually as a qualitative indicator (e.g., a windsock or wind vane 23 ) or alternatively, by converting the physical interaction of the object and the wind into a calibrated, quantitative signal (e.g., a cup anemometer or light detection and ranging (LiDAR) system [24][25][26]. Measurements using these approaches are all fundamentally constrained by the requirement that the measurement device must be located in close proximity to the measurement domain of interest. This review explores an emerging alternative with the potential to enable multiscale, spatiotemporally resolved measurements of the wind by leveraging trillions of wind indicators already covering most of the land on Earth. These indicators include naturally occurring structures, such as the estimated three trillion trees on land 27 , as well as engineered structures such as the millions of kilometers of electrical power lines 28 . Because none of these objects is perfectly rigid, they move in response to local wind conditions in ways that could potentially be used to infer incident wind speed and direction. We call this technique visual anemometry, reflecting the opportunity to quantify local winds based solely on visual measurements at arbitrarily far, line-of-sight distances away from the region of interest. Section II begins with an introduction to the relevant physics governing flow-structure interactions of the type expected to occur in wind flows. The remainder of the section reviews current approaches toward visual anemometry, while section III highlights remaining showstoppers to successful realization of this method. We conclude the review by identifying diverse ways in which the physics community can contribute their disciplinary expertise to the development of this new field. II. Visual Anemometry II. A. Principles of flow-structure interactions II.A. 1 . Physics of vortex-induced vibration Relative motion between a solid body and a surrounding fluid (such as that illustrated in Figure 1a) will create lift and drag forces (henceforth and , respectively) that act on the body. These two forces act in the transverse and streamwise directions of the fluid, respectively, with magnitudes given by = 2 2 ,(1) and = 2 2 ,(2) where is the fluid density, is the incoming fluid speed, is the relevant body area, and and are the dimensionless lift and drag coefficients whose magnitude in an incompressible flow depends on the body's relevant Reynolds number ( ), shape, and orientation in the flow. Per convention, the Reynolds number is defined as = ⁄ , where is the relevant body length scale and is the dynamic viscosity of the fluid. While and will be nearly constant and of order 0.1 to 1 for rigid, bluff bodies at ≫ 1, these coefficients can change and often decrease considerably with increasing flow speed, as will be discussed later. Lift and drag forces are responsible for momentum exchange between the fluid (e.g., surrounding air) and the body. When the body motion is coincident with one or both forces, it will extract kinetic energy from the fluid. A simple example of this energy transference (or harvesting) occurs when a bluff body is placed in a steady (i.e., time-independent) flow. Across a wide range of flow conditions, these bodies generate an unsteady (i.e., time-varying) wake characterized by periodic vortex shedding-the formation of spatially compact regions of rotating fluid downstream of the body-at formation frequency . This vortex formation results in a spatially uneven pressure distribution on the body. For a body with a single degree of freedom, such as an elastic cantilever allowed to move in the transverse direction (see Figure 1b), this unsteady forcing will cause the structure to respond by oscillating with so-called vortex-induced vibrations (VIV). When the forcing frequency, , from the vortex shedding approaches the natural frequency, , of the structure, the dynamics of the two systems can become coupled in a state of synchronization or "lock-in." This resonant state is characterized by a significant transfer of kinetic energy from the fluid to the structure resulting in large amplitude oscillations, i.e., comparable to the characteristic length scale of the body cross-section. This mechanism is responsible for phenomena such as "singing" wires and for the notable failure of the Ferrybridge cooling tower failures 29 in England. VIVs are one of many flavors of flow-induced vibrations, along with aeroelastic flutter instabilities 30 and galloping 31,32 , which result from forcing due to unsteady pitching. Each of these response modes is a visually perceptible indication of the local wind conditions. II.A.2. Effects of flexibility and reconfiguration For relatively rigid structures, the scaling in equations (1) and (2), especially the quadratic dependence on wind speed, adequately describes the behavior of aerodynamic forces exerted by the wind. However, many naturally occurring structures, especially plants, are highly flexible and thus can deform considerably under forcing from external fluid flow ( Figure 1). Consequently, the flexural rigidity of the body plays an important role in determining the fluid forcing on many bodies 33 . Because of this flexibility, many plants will reconfigure their crosssection area and become more streamlined in higher speed flows, allowing them to experience a drag force with sub-quadratic dependency on flow speed 34 . This dependency can be expressed as ∝ 2+ , where is the Vogel exponent 33, [35][36][37][38][39][40] . Values of the Vogel exponent V < 0 capture the deviation from the canonical, inertial scaling relation due to reconfiguration. For example, V = -1 indicates a regime in which drag scales near linearly with velocity. Depending on mechanics of the reconfigurability, system-specific parameterizations have been used to quantify reconfiguration in previous studies (e.g., see a recent review 33 ). As discussed in the following sections, a goal of visual anemometry is to infer the relationship between structural response and incident wind without an a priori model of reconfiguration dynamics. II. B. Current approaches for visual anemometry II.B.1. Physics-based methods II.B.1.i Dynamics-based methods Physical objects that are both geometrically slender and mechanically stiff have proven most amenable to direct, first-principles application of flow physics to deduce a quantitative relationship between object motion and incident wind speed. In these cases, the aerodynamic force of the wind on the structure can be estimated as ≈ ̅ , where p is the dynamic pressure exerted by the wind on the windward face of the structure, the overbar indicates a spatiotemporal average, and A is the projected area of the corresponding surface. As described in Section II.A.1, the dynamic pressure is linearly proportional to the air density and quadratically proportional to the incident wind speed; hence, ∝ � 2 .(3) The structural response to small deformations can be estimated by assuming that the elastic restoring force, FE, is linearly proportional to the structure deflection: ≈ ,(4) where is the structural deflection and is the elastic constant, which depends on the structure geometry and material properties. For cantilevered, slender objects such as tree branches, plant stalks, or blades of grass, the tip deflection due to spatially uniform wind loading can be modeling using linear Euler-Bernoulli beam theory, i.e., ≈ 4 8 ,(5) where f is the applied force per unit length L, E is the elastic Young's modulus of the material comprising the structure, and I is the area moment of inertia. Comparing equations (4) and (5) above, the corresponding elastic constant is ≈ 8 3 .(6) The balance of aerodynamic and elastic forces, FW and FE, respectively, provides a relationship between observed structural deflection and incident wind speed: � ≈ � 8 3 .(7) Each of the parameters on the right-hand side of equation ( An important limitation of methods based on the preceding analysis is the necessary occurrence of a non-zero mean (i.e., time-averaged) structural deflection due to the incident wind. As described in Section II.A.1 above, the vortex-induced vibrations experienced by many environmental structures, such as plants 44 , can exhibit a mean deflection that is close to zero despite significant instantaneous deflections. The oscillatory motion of electrical power lines under wind loading is another common example; other engineered structures such as telephone poles and radio antennae can also exhibit nearly symmetric structural oscillations in a direction perpendicular to the incident wind 28 . Ref. 45 showed that the dynamic motions associated with transverse or streamwise structural oscillations can also be used to estimate wind speeds, albeit using a conceptual framework different from the quasi-steady force balance that leads to equation (7). In this case, the dynamics of the flow-structure interaction are modeled as a damped harmonic oscillator: ( ) ≈ 2 2 + + ,(8) where m and are the structure mass and damping coefficients, respectively, and the last term on the right-hand side of the equation is the elastic response analyzed previously. In principle, any time-dependent wind forcing can be represented as a superposition of harmonic forcings 46 at a spectrum of frequencies: ( ) = 0 2 + �( + ) =1(9) where an and bn are constants, and the summation includes N modes sufficient to approximate the time-dependence of the incident wind. For harmonic forcing at a single frequency , i.e., ( ) = 0 , equation (8) has the steady-state solution 45 ( ) = 0 � 1 (1 − 2 ) + (2 ) 2 � [(1 − 2 ) ( ) − 2 ( ) ],(10) where is the ratio of the forcing frequency to the natural frequency of the structure, and = √4 .(11) Inspection of this steady-state solution indicates that the amplitude of the structural oscillations is directly proportional to the amplitude of the wind forcing. Appealing to the relationship between wind speed and forcing in equation (3) above, and with additional algebraic manipulation, Ref. 45 shows that, for typical, low levels of atmospheric turbulence, i.e., if ≡ ( ) � ≪ 1,(12) then � ∝ � ( ) ,(13) where denotes the standard deviation. Field measurements demonstrated that this relationship captures the flow-structure interactions of five tree species representing a diversity of morphologies 45,47 . As with the preceding mean deflection model, visual anemometry based solely on the physics of the time-dependent structural response still requires a calibration measurement to determine the constant of proportionality in the relationship expressed by equation (13). This potentially limits visual anemometry to contexts in which one has a priori wind measurements using conventional anemometry techniques. To unlock the potential of visual anemometry for global coverage, especially in regions where conventional anemometry is inaccessible, dynamical models like those above may require augmentation with other approaches. II.B.1.ii Energy-based methods While the mechanical properties of a structure exposed to wind might be impossible to infer based on visual observation of its isolated flow-structure interactions, the presence of multiple, identical structures can potentially be leveraged to infer their common properties. Consider, for example, the wind incident on two trees aligned in the streamwise direction. The discussion in the preceding sections indicates that that the kinetic energy of each tree is derived from the kinetic energy of the incident wind, e.g., for the upstream tree: 1 ≈ ,(14) where 1 is the kinetic energy of the upstream tree, is the kinetic energy of the wind incident on the front of the canopy, and is a constant factor that quantifies the energy transfer from the wind to the trees. This factor captures the mechanical properties of the tree, e.g., its inertia, elasticity, and damping. A value = 0 would indicate no energy transfer from the incident wind to the tree, whereas a value = 1 would indicate the (unphysical) upper bound of perfect energy transfer from the wind to the tree. If the second, downwind tree is set into motion solely by remaining kinetic energy in the wake of the first tree, i.e., 1 ≡ − 1 , then we can estimate the kinetic energy of the second tree as 2 ≈ 1 ≈ (1 − )(15) To be sure, this approximation assumes that no energy was dissipated in the interaction with the upstream tree (i.e., negligible damping on the timescale of wind advection), and it assumes that the upstream wind does not also directly affect the dynamics of the second, downwind tree, e.g., via turbulent sweeps into the top or sides of the canopy 24,[48][49][50] . This approximation depends inherently on the level of turbulence in the incident wind and on the surrounding topography. We postulate that two trees (or other objects exposed to the wind) with identical structural properties are characterized by the same value of energy transfer efficiency . With this ansatz, we can eliminate the unknown structural properties by comparing the relative magnitude of the motion of the two trees: 1 2 ≈ 1 1 − ,(16) or, ≈ 1 − 2 1 .(17) The kinetic energy of each structure can be estimated as proportional to the square of its average component speeds. Hence, ≈ 1 − 2 2 � 1 2 � ,(18) where the carat denotes a spatial average of the structure motion. The model in equation (18) above could be enhanced by incorporating more realistic functional dependencies of the parameter , for example, to reflect possible sensitivity of the kinetic energy transfer efficiency to the wind speed (e.g., via structure reconfiguration as discussed in section II.A.2. above) and background turbulence levels. However, these additions would potentially re-introduce the need for local calibration measurements. Even in its current form, equation (18) illustrates the potential for canopies comprising an array of similar structures to be especially useful for visual anemometry. II.B.2. Data-driven methods The constants of proportionality needed to complete the physical relationships expressed in equations (7), (13), and (14) above depend on factors specific to the objects being visually observed, e.g., their inertia, stiffness, and damping. Non-intrusive measurement of these properties at the scale of individual environmental structures is challenging if not impossible, particularly when those objects comprise a heterogeneous composite of multiple materials. A potential way forward is to leverage the fact that the trillions of environmental objects of interest globally can be classified into a set of material categories that is several orders-of-magnitude smaller in number. For example, building codes limit the set of allowable compositions of artificial structures to a relatively small number of engineered materials 51,52 . These materials could therefore be deduced in many cases from the external appearance of the structures. As another example, high-voltage power lines are typically composed of an aluminum core and polyethylene insulation, both of standard physical dimensions [53][54][55][56] . Hence, the material properties of such an environmental structure can be deduced as soon as it is categorized. Naturally occurring structures such as vegetation present a greater challenge, given both the diversity of their physical makeup and the fact that structure inertia, stiffness, and damping depend non-trivially on factors such as age, health, moisture content, and the presence or absence of leaves, seeds, and symbiotic organisms. Notwithstanding this myriad of challenges, initial efforts toward data-driven visual anemometry have produced encouraging results [57][58][59] . For example, Ref. 57 trained a combined convolutional neural network (CNN) and long short-term memory (LSTM) network based on field observations of a magnolia tree and a cloth flag exposed to naturally occurring wind conditions over several weeks. It was postulated that the CNN learns to recognize key features of the objects exposed to the wind, e.g., tree branches and leaves, or geometric patterns on the flag. Concurrently, the LSTM was hypothesized to learn key temporal features of the object motion, e.g., recurring waving or flapping motions of the geometric patterns. The trained neural network was subsequently tested on video clips of the same tree and flag that were not included in the training data set. This purely data-driven visual anemometry achieved measurements of the mean wind speed with errors comparable with the background turbulence fluctuations at the field site of approximately 1-2 m/s ( Figure 2). Because this purely data-driven, machine learning approach has limited capacity for extrapolation beyond the training data distribution 57 , it was unable to perform similarly accurate predictions using videos of tree specimens or flag types different from those in the training data. Hence, a generalizable version of visual anemometry in this case, i.e., a method that can make accurate measurements for a diversity of vegetation and engineered structures, would likely require training on a much more comprehensive set of videos and companion anemometer data. Brute-force efforts of this type have proven successful in the past, e.g., ImageNet 60 Generalized visual anemometry-a technique that does not require calibration measurements or a priori collection of training data-will depend on the discovery of new physical principles that manifest in predictable ways across a diversity of environmental structures exposed to wind. In pursuit of fundamental concepts of this type, we have recently conducted an extensive campaign of concurrent wind and visual measurements in a large-scale wind tunnel 62 . This facility enables controlled studies of selected vegetation with a diversity of morphologies, ranging from grasses to trees (Figure 3). The incident wind speed can be described by a two-parameter Weibull probability density function 63,64 : ( ) = � 2 1 � 1 � 2 −1 −� 1 � 2 , ≥ 0 0 , < 0(19) where 1 is a positive-valued, dimensional scale factor that increases for distributions ( ) with higher variance. The dimensionless shape factor 2 typically takes values between 1 and 3 for wind distributions, with values closer to 1 indicating right-skewness of the distribution 63 . Moments of the Weibull distribution can be expressed in terms of 1 and 2 ; for example, the mean wind speed is given by, � = 1 �1 + 1 2 � (20) where Γ is the Gamma function. The motion of the vegetation can be similarly quantified using the Weibull distribution. Crosscorrelation of successive images of the moving canopy creates a displacement vector map 65 representing the spatial distribution of motion induced by the incident wind ( Figure 3). Quantilequantile 64 analysis confirmed that the time-series of the spatially averaged canopy motion can be reasonably approximated by a Weibull distribution with its own scale and shape factors, 1 and 2 , respectively. The dependence of the canopy scale and shape factor on the corresponding wind factors may provide a framework to achieve generalizable visual anemometry. For example, Figure 4 shows that the various vegetation studied to date all exhibit a similar, sigmoidal dependence of the canopy scale factor, ̃1 on the wind scale factor, ̃1 , where the tilde denotes a vegetationspecific normalization based on the width, height, and center of each sigmoid curve 62 . The physical interpretation of this possibly 'universal' curve shape can be understood by recalling the scale factor as a surrogate for the mean speed of the wind and canopy. At relatively low wind speeds, the dynamic pressure exerted by the wind on the canopy elements may be insufficient to overcome the inertia and elastic restoring force of the structures exposed to wind. In this regime, the slope of the curve in Figure 4 is expected to be nearly zero. At sufficiently high wind speeds, the resistance of the canopy to motion is overcome, and further increases in wind speed correspond to a proportional increase in canopy motion (i.e., the region of linear slope in Figure 4). At high wind speeds, further deflection of the canopy structures is limited by the fixed position of the vegetation roots in the substrate below. This constraint is reflected in the plateau of ̃1 at large values of normalized wind scale factor ̃1 in Figure 4. A key implication of the sigmoidal response curve is that its slope-a measure of the sensitivity of canopy motion to changes in the incident wind speed-exhibits regimes at both low and high winds wherein visual anemometry may be fundamentally challenged by the lack of a distinct structural response to wind dynamics. Where the response curve has zero slope, it is not possible to accomplish visual anemometry based on the curve of canopy scale versus wind scale. The location and width of each of the aforementioned regions as a function of the dimensional wind speed (e.g., in m/s) is a characteristic of each vegetation type. Placement of a given vegetation type onto the universal curve in Figure 4 required a priori knowledge of the incident wind corresponding to each canopy measurement. Generalized visual anemometry would therefore require a means to predict the placement of a given structure onto the universal curve. Additional information based on the shape factor of the motion distribution �̃2 �, the visual appearance of the structure, comparison with nearby, similar structures in a canopy (cf. section II.B.1.ii above), fine-scale changes such as leaf reconfiguration (cf. Section II.A.2), or other statistical priors could be useful for achieving this goal. We conclude with discussion of three research avenues that could accomplish the necessary model closure. III. B. New data sources A data-driven approach to generalized visual anemometry could use the discovered universal curve as a statistical prior in a physics-informed machine learning framework. This approach would anticipate that measurements collected without ground-truth wind speed measurements should exhibit a scale factor relationship between the wind and canopy distributions that is sigmoidal as in Figure 4. The relationships between the wind and canopy shape factors may provide additional physical constraints to enable a data-driven model that can extrapolate beyond its training data set. To be sure, this approach does not obviate the need for comprehensive data collection to train the neural networks or other machine-learning representations of the underlying physics. However, there exists a growing set of data sources that could be leveraged for this purpose. These include open-source, near-ground imagery 66 , e.g., from long-term ecological measurement campaigns [67][68][69][70][71][72] , hazard monitoring systems such as those deployed for wildfire detection [73][74][75][76][77][78][79][80][81] , and in the built environment, traffic and security cameras 82,83 . Emerging commercial satellite data feeds can provide a potentially transformative data source if extended to time-resolved imagery, as the wide area coverage and frequent revisits of remote locations provide data that is inaccessible by other means 78,84 . III. C. New computational tools The two primary approaches toward visual anemometry that have been explored in this reviewfirst-principles physical modeling and data-driven machine learning-have both been considered thus far from a perspective depending on empirical measurements of the relevant flow-structure interactions. Advances in high-performance computing now make it feasible to achieve physically realistic computational simulations of wind interactions with geometrically complex structures such as vegetation [85][86][87] . Hence, another promising route to generalized visual anemometry could leverage simulations to complement the aforementioned field measurement campaigns. Numerical simulations provide the added benefit of enabling complex details of environmental structures, e.g., the branches of a tree, to be tracked with high spatiotemporal fidelity. Because imagery from cameras provides only a two-dimensional projection of the threedimensional structure kinematics, the set of parameters used to describe the canopy are limited to quantities derived from that projection. The canopy motion determined from image crosscorrelation is one example (Figure 3). By contrast, numerical simulations could provide threedimensional kinematic data, from which a richer set of physical descriptors could be derived to quantify the canopy response to incident wind. That higher-dimensional description can better delineate different modes of structural response and could also be used for the task of identifying and classifying structures of interest in a machine learning context. III. D. New physics The ultimate solution to the challenge of generalized visual anemometry may lie in a combined strategy that leverages knowledge of canonical flow-structure interactions, such as those introduced in section II. A, along with libraries of representative wind interactions from empirical observations and from analogous computational models. However, the most exciting role for the physics community may lie in a third approach: the discovery and development of new physics concepts that augment our current knowledge of the nature of flow-structure interactions as well as our remote sensing capabilities. Although the flow-structure interactions to be exploited by visual anemometry are a manifestation of classical mechanics-a subfield of physics that is ostensibly mature in comparison to, say, quantum science-knowledge of those physics is still limited to a relatively small set of simplified geometries. The appeal in section II. A to objects with circular crosssections, slender or planar geometries, and moderate elasticity was by necessity, as established models for the physics of flow structure-interactions have not evolved beyond those relatively simple configurations despite intensive study for more than a century [88][89][90][91][92][93][94][95][96][97] . A historical limitation on the study of flow-induced motion of more complex structures was the inability to visualize the associated fluid-solid interactions with high spatiotemporal resolution. However, the advent of high-speed laser velocimetry 98,99 , 3D flow tomography 94,100 , and algorithms to compute the pressure field corresponding to flow velocity measurements 101,102 now make it possible for experimental physicists to revisit the classical mechanics in geometric configurations approaching the complexity of structures relevant for visual anemometry. Modern model reduction techniques from dynamical systems theory [103][104][105] have the potential to distill high-dimensional datasets such as those derived from the aforementioned new experimental measurements. Physicists familiar with the challenge of dimensionality reduction in other areas of nonlinear dynamics could apply many of the same tools here. Simplified kinematic motifs of the observed structure motion, extracted using model reduction, may prove to be robust correlates of the incident wind speed and direction. That could also provide a target for unsupervised machine learning approaches that aim to classify or even deduce material properties of objects in the wind based solely on their observed motion. While the concept of visual anemometry took initial inspiration from our human powers of visual observation, the spectrum of visible light represents a relatively small band of the electromagnetic radiation that is absorbed, reflected, and emitted by both natural and engineered objects that could be used for visual anemometry. The range of applications of the concepts introduced here could be further expanded by physicists interested in exploring the broader spectrum of electromagnetic radiation associated with objects covering the Earth's surface that are subjected to local winds. An immediate example is infrared radiation, which could enable visual anemometry measurements at night. Imaging at longer wavelengths, e.g., millimeter-wave imaging [106][107][108] , could also potentially be used to circumvent optical interference such as cloud cover, provided that the spatial resolution of those measurements still enables structural motions to be resolved. Additional optical properties, such as the polarization of reflected light, could be used to infer not only translational motion of objects but also changes in object orientation associated with flow-induced bending and torsion of reflective surfaces 109 such as leaves and blades of grass. These signatures could provide additional means to discriminate between regimes of flow speed and direction incident on the objects. Finally, it is important to recall that 70 percent of the Earth's surface is covered by water. Inference of wind fields near the ocean surface is complicated by the more complex deformations associated with the air-water interface [109][110][111][112] . This presents challenges but also opportunities. For example, the high-wind plateau in structural response observed for groundmounted structures (e.g. Figure 4) need not limit correlations between ocean surface deformation and wind speed in similarly high-wind regimes. Hence, a larger range of wind speeds may be accessible to visual anemometry over the ocean as compared to the method applied on land. In addition, ocean measurements can potential leverage not only the kinematics of the air-water interface, but also the wind-induced motion of ocean spray above the surface 113 and the waterinduced motion of submerged vegetation 92,93,114 . The aforementioned list is merely illustrative of avenues for new contributions from the physics community. If this discussion has been successful, the opportunities and challenges associated with visual anemometry that have been introduced herein will encourage the reader to pursue one or more of these research directions. Visual anemometry provides a unique opportunity for the physics community to contribute to a variety of important and far-reaching topics in global sustainability. Text Box 1 Applications of Visual Anemometry Renewable energy To achieve net-zero greenhouse gas emissions in the U.S. by 2050, it is estimated that increases in wind and solar capacity of 6-28x and 9-39x, respectively, are required 115 . Similarly, large increases in renewable energy are needed to meet global decarbonization targets. This unprecedented scale-up will require widespread proliferation of wind and solar generation in geographic regions where renewables have not previously been sited 20,116 . Diversified siting of renewable energy infrastructure creates two main challenges related to wind measurements. First, wind patterns are more uncertain in new locations that have shorter historical observation records. Second, new sites may have lower quality wind, with characteristics that are more difficult to incorporate into existing forecasting methods, such as terrain complexity [117][118][119][120] and variable land use (e.g., urban environments). Historically, two parallel approaches have been used for wind field estimation for renewable energy resource assessment. First, numerical weather forecasting models predict the winds in the atmosphere based on an approximate form of the governing equations 121 . These models require parameterizations to represent complex processes that cannot be directly resolved with available computing resources, such as turbulence, convection, and clouds. While numerical weather models provide detailed spatial and temporal coverage, the approximations in the models result in significant uncertainties in wind forecasts 122 , especially near Earth's surface where turbulence is higher than aloft. In the second approach, in situ sensors are used to observe the wind with few, if any, assumptions required regarding the nature of the wind dynamics 120,[123][124][125] . Yet these sensors are both relatively high cost and lack spatial coverage. At the intersection of these parallel approaches, data assimilation is used to combine in situ observations with numerical models [126][127][128] , but uncertainties remain in locations not covered by the measurements. Visual anemometry can provide a third, complementary approach to wind field estimation with characteristics similar to in situ observations, but with higher spatial coverage. A promising avenue may also leverage wind estimates from visual anemometry for data assimilation. Since most physical objects to be used for visual anemometry, both natural and engineered, exist tens of meters or closer to Earth's surface, measurements via visual anemometry directly quantify winds near this nominal height ( Figure 5). Small-scale wind generators, such as recently developed vertical-axis wind turbines, are designed with hub-heights on the order of ten meters 129 , making visual anemometry measurements directly applicable to the design of those systems. However, utility-scale horizontal axis wind turbines operate at hub-heights between 50-200 meters, and with rotor diameters 80-300 meters in size. For wind measurements made through visual anemometry to be used to estimate the winds incident to these utility-scale horizontal axis turbines, model-based extrapolation methods must be used 130 . Wind extrapolation to heights above a given measurement location is common, as typical weather stations also provide wind measurements 10 meters above Earth's surface and surface winds reported by typical weather and climate models are also at 10 meters. Solar power production primarily depends on spatiotemporal variations in irradiance. Irradiance is driven by known deterministic variations, such as seasonal and diurnal cycles, as well as stochastic variations of atmospheric clouds and aerosols that are challenging to predict. Therefore, physics-based irradiance forecasts rely on numerical weather prediction 131 . As noted above in the context of wind energy, visual anemometry may provide a mechanism for improved weather forecasting by increasing the availability of wind flow measurements for data assimilation. Beyond irradiance, the efficiency of solar photovoltaic (PV) cells is rated at standard test conditions (STC) of one Sun of irradiance at an air-mass ratio of 1.5 (i.e., sunlight passing obliquely through the equivalent of 1.5 times the atmospheric length at zenith) and a cell temperature of 25 degrees C. Yet PV cells typically operate at higher temperatures 132 . Solar PV efficiency decreases by approximately 0.1 to 0.5% per Kelvin above STC, but the magnitude of degradation is cell-specific 133 . To estimate efficiency in field conditions, solar cell manufacturers provide a method to approximate the cell temperature based on an empirical indicator called the nominal operating cell temperature (NOCT). Wind speed is a required input to this cell temperature approximation 134 . Improving wind estimates increases the accuracy of cell temperature and cell efficiency predictions 135 . Finally, emerging research seeks to optimally site and design solar farms to maximize passive convective cooling to reduce cell temperature 136 . Such methods require site-specific wind estimates 22 which may be provided by visual anemometry. Urban flow physics People increasingly live in urban environments. In recognition of this important trend, the United Nations Sustainable Development Goal (SDG) 11 focuses on sustainable cities and communities 137 . Air flow in urban environments affects the energy efficiency and resilience of engineered structures, pollution dispersion and air quality, and the future of urban air mobility. Given the broad impacts of urban air flow, and the limited fidelity of present observations and predictive models, urban air flow represents a grand challenge in environmental fluid mechanics 138 . Air flow affects the structural resilience and energy efficiency of buildings. Design standards incorporate site-specific wind characteristics, including extreme wind gusts 139 , which are difficult to measure or numerically model. Urban airflow also affects thermal convection in cities, which impacts the energy consumption of building heating and cooling systems, and the effectiveness of natural ventilation 140 ). Finally, the design of future aircraft and airspace for urban air mobility depends on our ability to predict the turbulent flow within and around urban environments 141 . Safe and reliable transport of people and goods requires detailed knowledge of wind gusts and turbulence, as contemporary control methods have reduced success in navigation and object avoidance in uncertain wind environments 142 . Flows in urban environments are heterogeneous and complex; these traits reduce the accuracy of numerical flow predictions and severely limit the accuracy of spatially extrapolated pointwise flow field measurements. There is an urgent need for increased spatiotemporal coverage of urban air flow measurements 138 for both validation and uncertainty quantification of numerical models 143,144 as well as for data assimilation 145 . Visual anemometry can provide a new paradigm for urban wind field sensing with wide spatial coverage and high spatiotemporal resolution ( Figure 6). Observations of the motion of naturally occurring objects such as trees and engineering objects such as flags can potentially be used in a data assimilation framework to inform wind state estimation, model parameter estimation, and uncertainty quantification, and machine learning of wind models. Conversely, developed models can inform optimal placement of additional passive sensors. Environmental and ecological processes Transport, mixing, and atmospheric conditions driven by the wind are central to numerous environmental and ecological processes (Figure 7). Wildfire prediction and mitigation are notable applications where detailed wind mapping can play a critical role. In the United States alone, wildfires have cost an average of $13.4B USD/yr 146,147 in damages over the last five years and are predicted to become even more prevalent as warmer and drier conditions driven by climate change lead to more protracted and active fire seasons 148,149 . While temperature, humidity, and stability conditions are critical contributors to the intensity of a wildfire, wind conditions play a leading role in determining the speed and direction of the wildfire spread [150][151][152] . In chaparral ecosystems such as coastal Southern California, the regions most exposed to extreme wind events (e.g., the katabatic Santa Ana winds) have been linked to increased fire danger and larger fires 153,154 . Prevailing wind conditions and their associated turbulence also drive the dispersal of critical scalar quantities (e.g., heat and mass) and particulates central to various ecological and environmental processes. Long distance dispersal of seeds 14 , spores 155 , and pollen 156 , whether by wind or organisms, is a critical yet poorly understood survival strategy 11,14,157,158 for species that adapt to changing habitats by overcoming geographic isolation 10,159,160 . Many of the world's most important agricultural grains, including wheat, barley, corn, and rice, are grasses that are pollinated primarily through the wind due to their lack of flowering structures 161 . Though such a mechanism is most prevalent for plants located in close proximity, e.g., within a range of 1 km 162 , airborne transport and mating of taller plants, e.g., trees, has been documented at distances exceeding 10 km 12,156,163 . Understanding these dynamics is critical for ensuring or minimizing cross-pollination between various crops; the latter is especially critical to limit contamination from genetically modified crops 162 in the natural environment. Even propagated material that is transported by organisms depends in part on wind dynamics for its dispersal. Natural dispersal, convection, and mixing by wind has similarly been leveraged by humans for various engineering purposes, including for pollution dispersal 164 and natural ventilation 165,166 . Proper ventilation is necessary to ensure a healthy indoor environment, but it comes with a tangible energy cost. Ventilation comprises approximately 11% of the nearly 7 quadrillion BTU (2 million GWh) and $141 B USB spent by commercial buildings in the US alone 167 . This expenditure does not include the energy spent on space heating or cooling, which are both also substantial (i.e., 32% and 8% of energy consumption by commercial buildings in the US, respectively). Natural ventilation presents an efficient and largely passive alternative to conventional ventilation approaches. This approach uses naturally-occurring forcing from wind and/or buoyancy to exchange air between the indoors and outdoors through openings in a building structure. While this resource is freely available in appropriate climates, it can be challenging to control and predict 168,169 due to the inherent complexity and variability of the airflow inside connected rooms and around the structure. Designing a system to adequately take advantage of wind-driven forcing requires detailed knowledge of the turbulent wind patterns in and around the building across diurnal and seasonal variations 170 . This represents another potentially transformative application of visual anemometry. Figure 1 . 1Physics of flow-structure interactions. Slender objects, such as the branches of a plant, can undergo vortex-induced vibrations in a direction transverse to the incoming flow (a), due to vortex shedding and associated lift and drag forces (b). In high-speed flows, flexible structures such as vegetation can undergo reconfiguration (c), which reduces proportional increases in aerodynamic forces from the wind. 7) can be estimated from visual observation of the structure, with the exception of the Young's modulus of the material. Ref. 41 demonstrated the use of single-point calibration to determine the unknown material property. Alternatively, computer vision techniques can potentially be used to deduce the likely material properties based on libraries of environmentally observed structures and their known material properties 42,43 . Figure 2 . 2Data-driven implementation of visual anemometry based on measurements collected at a research field site and in a laboratory wind tunnel. A combined CNN-LSTM network successfully predicted the wind speed corresponding to new videos of the same structures included in the training data set (left panel). However, significantly lower sensitivity to wind speed was observed for videos of structures not included in the training data set (right panel).Figure adapted from Ref. 57. Figure 3 . 3Schematic of large-scale wind tunnel measurements of vegetation under controlled wind conditions. A 3-m x 3-m array of 1296 individually addressable fans generates wind conditions with user-defined spatiotemporal profiles and mean speeds up to 20 m/s (right). Vegetation exposed to the wind is recorded from above using a high-speed camera. Spatial crosscorrelation of successive images reveals the local, instantaneous displacement of the vegetation, illustrated in the planar vector field. Adjacent to the vegetation, a sonic anemometer (white) measures the local wind speed for comparison to visual anemometry. Figure 4 . 4Compilation of visual anemometry measurements of eight vegetation species, plotted in terms of the normalized canopy scale factor, ̃1 versus normalized wind scale factor, ̃1 . Black curve corresponds to sigmoid equation. Data derived from Ref. 62. Figure 5 . 5Traditional tools for wind measurement (left) include meteorological towers carrying sensors for wind speed and direction, as well as light detection and ranging (LiDAR) systems. Visual anemometry (center) measures near-ground wind conditions relevant to solar photovoltaic (PV) and solar thermal energy infrastructure, as well as smaller wind turbines (e.g., vertical-axis wind turbines, VAWT). The measurements can also be extrapolated to higher altitudes relevant to traditional horizontal-axis wind turbines (HAWT). [Tree graphic adapted from: http://clipart-library.com/free/forest-silhouette-clip-art.html] Figure 6 . 6Schematic of visual anemometry for urban flow physics. Figure 7 . 7A diverse range of ecological processes are mediated by near-ground winds, including (a) wildfires, (b) pollen and seed dispersal, and (c) natural ventilation. and COCO61 . However, to develop the equivalent data set for visual anemometry would likely require leveraging a combination of existing open-source data and new, dedicated measurement campaigns. We revisit this possibility in the conclusion of this review.III. Outlook: Unsolved Challenges, Untapped Opportunities III. A. Theoretical constraints on visual anemometry AcknowledgementsThe authors gratefully acknowledge seminal contributions from Jennifer L. Cardona in development of several of the concepts presented in this review, as well as discussions with Katie Bouman, Jennifer Sun, Yisong Yue, and Pietro Perona at Caltech. Additional helpful discussions occurred in the CV4Ecology Summer Workshop, supported by the Caltech Resnick Sustainability Institute. Funding was generously provided by the National Science Foundation (Grant CBET-2019712) and the Center for Autonomous Systems and Technologies at Caltech. Additional support from Heliogen is gratefully acknowledged.The authors declare no competing interests. A cubic relationship between air-sea CO2 exchange and wind speed. R Wanninkhof, W R Mcgillis, Geophys Res Lett. 26Wanninkhof, R. & McGillis, W. R. A cubic relationship between air-sea CO2 exchange and wind speed. Geophys Res Lett 26, 1889-1892 (1999). North America's net terrestrial CO2 exchange with the atmosphere. A W King, Biogeosciences. 12King, A. W. et al. North America's net terrestrial CO2 exchange with the atmosphere 1990-2009. Biogeosciences 12, 399-414 (2015). Integrating terrestrial and aquatic ecosystems to constrain estimates of land-atmosphere carbon exchange. J P Casas-Ruiz, Nat Commun. 141571Casas-Ruiz, J. P. et al. Integrating terrestrial and aquatic ecosystems to constrain estimates of land-atmosphere carbon exchange. Nat Commun 14, 1571 (2023). Relationship between wind speed and gas exchange over the ocean. R Wanninkhof, J Geophys Res. 977373Wanninkhof, R. Relationship between wind speed and gas exchange over the ocean. J Geophys Res 97, 7373 (1992). Reduced air-sea CO2 exchange in the Atlantic Ocean due to biological surfactants. R Pereira, I Ashton, B Sabbaghzadeh, J D Shutler, R C Upstill-Goddard, Nat Geosci. 11Pereira, R., Ashton, I., Sabbaghzadeh, B., Shutler, J. D. & Upstill-Goddard, R. C. Reduced air-sea CO2 exchange in the Atlantic Ocean due to biological surfactants. Nat Geosci 11, 492-496 (2018). Air-sea gas exchange in the coastal zone. R C Upstill-Goddard, Estuar Coast Shelf Sci. 70Upstill-Goddard, R. C. Air-sea gas exchange in the coastal zone. Estuar Coast Shelf Sci 70, 388-404 (2006). Role of air-mass transformations in exchange between the Arctic and mid-latitudes. F Pithan, Nat Geosci. 11Pithan, F. et al. Role of air-mass transformations in exchange between the Arctic and mid-latitudes. Nat Geosci 11, 805-812 (2018). Atmospheric carbon dioxide and the ocean. U Siegenthaler, J L Sarmiento, Nature. 365Siegenthaler, U. & Sarmiento, J. L. Atmospheric carbon dioxide and the ocean. Nature 365, 119-125 (1993). High concentrations of biological aerosol particles and ice nuclei during and after rain. J A Huffman, Atmos Chem Phys. 13Huffman, J. A. et al. High concentrations of biological aerosol particles and ice nuclei during and after rain. Atmos Chem Phys 13, 6151-6164 (2013). Wind-borne seed and fruit movement. F M Burrows, New Phytologist. 75Burrows, F. M. Wind-borne seed and fruit movement. New Phytologist 75, 405- 418 (1975). Long-distance dispersal of tree seeds by wind. H S Horn, R A N Nathan, S R Kaplan, Ecol Res. 16Horn, H. S., Nathan, R. A. N. & Kaplan, S. R. Long-distance dispersal of tree seeds by wind. Ecol Res 16, 877-885 (2001). A theoretical framework for data analysis of wind dispersal of seeds and pollen. A Okubo, S A Levin, Ecology. 70Okubo, A. & Levin, S. A. A theoretical framework for data analysis of wind dispersal of seeds and pollen. Ecology 70, 329-338 (1989). Effects of wind on plants. E De Langre, Annu Rev Fluid Mech. 40de Langre, E. Effects of wind on plants. Annu Rev Fluid Mech 40, 141-168 (2008). Mechanisms of long-distance dispersal of seeds by wind. R Nathan, Nature. 418Nathan, R. et al. Mechanisms of long-distance dispersal of seeds by wind. Nature 418, 409-413 (2002). Wind assistance: a requirement for migration of shorebirds. R W Butler, T D Williams, N Warnock, M A Bishop, Auk. 114Butler, R. W., Williams, T. D., Warnock, N. & Bishop, M. A. Wind assistance: a requirement for migration of shorebirds? Auk 114, 456-466 (1997). Sandpipers go with the flow: correlations between estuarine conditions and shorebird abundance at an important stopover on the Pacific Flyway. R Canham, S A Flemming, D D Hope, M C Drever, Ecol Evol. 11Canham, R., Flemming, S. A., Hope, D. D. & Drever, M. C. Sandpipers go with the flow: correlations between estuarine conditions and shorebird abundance at an important stopover on the Pacific Flyway. Ecol Evol 11, 2828-2841 (2021). The influence of atmospheric structure and motions on insect migration. V A Drake, R A Farrow, Annu Rev Entomol. 33Drake, V. A. & Farrow, R. A. The influence of atmospheric structure and motions on insect migration. Annu Rev Entomol 33, 183-210 (1988). . T Burton, N Jenkins, D Sharpe, E Bossanyi, Wind energy handbook. Burton, T., Jenkins, N., Sharpe, D. & Bossanyi, E. Wind energy handbook. (2011). Grand challenges in the science of wind energy. P Veers, Science. 366Veers, P. et al. Grand challenges in the science of wind energy. Science (1979) 366, (2019). Research challenges and needs for the deployment of wind energy in hilly and mountainous regions. A Clifton, S Barber, A Stökl, H Frank, T Karlsson, Wind Energy Science. 7Clifton, A., Barber, S., Stökl, A., Frank, H. & Karlsson, T. Research challenges and needs for the deployment of wind energy in hilly and mountainous regions. Wind Energy Science 7, 2231-2254 (2022). Row spacing as a controller of solar module temperature and power output in solar farms. B J Stanislawski, T Harman, T J Silverman, R B Cal, M Calaf, Journal of Renewable and Sustainable Energy. 1463702Stanislawski, B. J., Harman, T., Silverman, T. J., Cal, R. B. & Calaf, M. Row spacing as a controller of solar module temperature and power output in solar farms. Journal of Renewable and Sustainable Energy 14, 063702 (2022). Viewing convection as a solar farm phenomenon broadens modern power predictions for solar photovoltaics. S E Smith, Journal of Renewable and Sustainable Energy. 1463502Smith, S. E. et al. Viewing convection as a solar farm phenomenon broadens modern power predictions for solar photovoltaics. Journal of Renewable and Sustainable Energy 14, 063502 (2022). Mediaeval kites and Windsocks. C Hart, The Aeronautical Journal. 73Hart, C. Mediaeval kites and Windsocks. The Aeronautical Journal 73, 1019-1026 (1969). Atmospheric boundary layer flows: their structure and measurement. J C Kaimal, J J Finnigan, 72Kaimal, J. C. & Finnigan, J. J. Atmospheric boundary layer flows: their structure and measurement. Book vol. 72 (1994). Sub-meter wind detection with pulsed coherent Doppler lidar. Y Zhang, J Yuan, Y Wu, J Dong, H Xia, Phys Rev Fluids. 822701Zhang, Y., Yuan, J., Wu, Y., Dong, J. & Xia, H. Sub-meter wind detection with pulsed coherent Doppler lidar. Phys Rev Fluids 8, L022701 (2023). Flow measurement handbook: industrial designs, operating principles, performance, and applications. R C Baker, 10.1017/CBO9780511471100Flow Measurement Handbook. Cambridge University PressBaker, R. C. Flow measurement handbook: industrial designs, operating principles, performance, and applications. Flow Measurement Handbook (Cambridge University Press, 2000). doi:10.1017/CBO9780511471100. A trillion trees. P Goymer, Nat Ecol Evol. 2Goymer, P. A trillion trees. Nat Ecol Evol 2, 208-209 (2018). Wind-induced instability of structures. G V Parkinson, Philosophical Transactions of the Royal Society of London. Series A. 269Mathematical and Physical SciencesParkinson GV. Wind-induced instability of structures. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 269, 395-413 (1971). Wind forces and the proximity of cooling towers to each other. D J Gunn, A A Malik, Nature. 210Gunn, D. J. & Malik, A. A. Wind forces and the proximity of cooling towers to each other. Nature 210, 1142-1143 (1966). . W J Mccroskey, Unsteady Airfoils, Annu Rev Fluid Mech. 14McCroskey, W. J. Unsteady Airfoils. Annu Rev Fluid Mech 14, 285-311 (1982). Transmission line vibration due to sleet. J P Hartog, Den, Transactions of the American Institute of Electrical Engineers. 51Hartog, J. P. Den. Transmission line vibration due to sleet. Transactions of the American Institute of Electrical Engineers 51, 1074-1076 (1932). Misconceptions and generalizations of the Den Hartog galloping criterion. N Nikitas, J H Macdonald, J Eng Mech. 140Nikitas, N. & Macdonald, J. H. G. Misconceptions and generalizations of the Den Hartog galloping criterion. J Eng Mech 140, (2014). Mechanics of a plant in fluid flow. F P Gosselin, J Exp Bot. 70Gosselin, F. P. Mechanics of a plant in fluid flow. J Exp Bot 70, 3533-3548 (2019). Drag reduction through self-similar bending of a flexible body. S Alben, M Shelley, J Zhang, Nature. 420Alben, S., Shelley, M. & Zhang, J. Drag reduction through self-similar bending of a flexible body. Nature 420, 479-481 (2002). Drag reduction of flexible plates by reconfiguration. F Gosselin, E De Langre, B A Machado-Almeida, J Fluid Mech. 650Gosselin, F., de Langre, E. & Machado-Almeida, B. A. Drag reduction of flexible plates by reconfiguration. J Fluid Mech 650, 319-341 (2010). Drag and reconfiguration of broad leaves in high winds. S Vogel, J Exp Bot. 40Vogel, S. Drag and reconfiguration of broad leaves in high winds. J Exp Bot 40, 941-948 (1989). Life in moving fluids: the physical biology of flow. S Vogel, Princeton University PressVogel, S. Life in moving fluids: the physical biology of flow. (Princeton University Press, 1994). Foliage motion under wind, from leaf flutter to branch buffeting. L Tadrist, J R Soc Interface. 1520180010Tadrist, L. et al. Foliage motion under wind, from leaf flutter to branch buffeting. J R Soc Interface 15, 20180010 (2018). Effects of sea anemones on the flow forces they encounter. M A R Koehl, Journal of Experimental Biology. 69Koehl, M. A. R. Effects of sea anemones on the flow forces they encounter. Journal of Experimental Biology 69, 87-105 (1977). Drag and flexibility in sessile organisms. S Vogel, Am Zool. 24Vogel, S. Drag and flexibility in sessile organisms. Am Zool 24, 37-44 (1984). Wind speed inference from environmental flow-structure interactions. J L Cardona, K L Bouman, J O Dabiri, Flow. 14Cardona, J. L., Bouman, K. L. & Dabiri, J. O. Wind speed inference from environmental flow-structure interactions. Flow 1, E4 (2021). Estimating the material properties of fabric from video. K L Bouman, B Xiao, P Battaglia, W T Freeman, 10.1109/ICCV.2013.4552013 IEEE International Conference on Computer Vision. IEEEBouman, K. L., Xiao, B., Battaglia, P. & Freeman, W. T. Estimating the material properties of fabric from video. in 2013 IEEE International Conference on Computer Vision 1984-1991 (IEEE, 2013). doi:10.1109/ICCV.2013.455. Cloth in the wind: a case study of physical measurement through simulation. T F H Runia, K Gavrilyuk, C G M Snoek, A W Smeulders, Runia, T. F. H., Gavrilyuk, K., Snoek, C. G. M. & Smeulders, A. W. M. Cloth in the wind: a case study of physical measurement through simulation. 10498-10507 (2020). Plant vibrations at all scales: a review. E De Langre, J Exp Bot. 70de Langre, E. Plant vibrations at all scales: a review. J Exp Bot 70, 3521-3531 (2019). Wind speed inference from environmental flowstructure interactions, part 2: leveraging unsteady kinematics. J L Cardona, J O Dabiri, Cardona, J. L. & Dabiri, J. O. Wind speed inference from environmental flow- structure interactions, part 2: leveraging unsteady kinematics. Flow 2, (2021). Wind induced deformation and vibration of a Platanus acerifolia leaf. C.-P Shao, Y.-J Chen, J.-Z Lin, Acta Mechanica Sinica. 28Shao, C.-P., Chen, Y.-J. & Lin, J.-Z. Wind induced deformation and vibration of a Platanus acerifolia leaf. Acta Mechanica Sinica 28, 583-594 (2012). Self-supervised keypoint discovery in behavioral videos. J J Sun, 10.1109/CVPR52688.2022.002212022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). IEEESun, J. J. et al. Self-supervised keypoint discovery in behavioral videos. in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) 2161-2170 (IEEE, 2022). doi:10.1109/CVPR52688.2022.00221. Turbulence in plant canopies. J Finnigan, Annu Rev Fluid Mech. 32Finnigan, J. Turbulence in plant canopies. Annu Rev Fluid Mech 32, 519-571 (2000). Measurements of the structure of the Reynolds stress in a turbulent boundary layer. S S Lu, W W Willmarth, J Fluid Mech. 60Lu, S. S. & Willmarth, W. W. Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J Fluid Mech 60, 481-511 (1973). An introduction to boundary layer meteorology. R B Stull, Springer13NetherlandsStull, R. B. An introduction to boundary layer meteorology. Book vol. 13 (Springer Netherlands, 1988). . California Building Standards Commission. California Building Standards Commission. https://www.dgs.ca.gov/BSC/Codes. Building codes illustrated: a guide to understanding the 2018 International Building Code. F D K Ching, S R Winkel, International Code CouncilChing, F. D. K. & Winkel, S. R. Building codes illustrated: a guide to understanding the 2018 International Building Code. (International Code Council). Electrical power transmission system engineering: analysis and design. Turan Gönen, Gönen, Turan. Electrical power transmission system engineering: analysis and design. 1066. High-voltage power cables plug into the future. I Metwally, IEEE Potentials. 27Metwally, I. High-voltage power cables plug into the future. IEEE Potentials 27, 18-25 (2008). . 10.1007/978-981-32-9938-2SpringerSingaporeSpringer handbook of power systems.Springer handbook of power systems. (Springer Singapore, 2021). doi:10.1007/978- 981-32-9938-2. High voltage engineering. E Gockenbach, 10.1007/978-981-32-9938-2_3Springer Handbooks. Springer Science and Business Media Deutschland GmbHGockenbach, E. High voltage engineering. in Springer Handbooks 131-182 (Springer Science and Business Media Deutschland GmbH, 2021). doi:10.1007/978-981-32-9938-2_3. Seeing the wind: visual wind speed prediction with a coupled convolutional and recurrent neural network. J L Cardona, M F Howland, J O Dabiri, NeurIPS. Cardona, J. L., Howland, M. F. & Dabiri, J. O. Seeing the wind: visual wind speed prediction with a coupled convolutional and recurrent neural network. NeurIPS (2019). See the wind: wind scale estimation with optical flow and VisualWind dataset. Q Zhang, J Xu, M Crane, C Luo, Science of The Total Environment. 846157204Zhang, Q., Xu, J., Crane, M. & Luo, C. See the wind: wind scale estimation with optical flow and VisualWind dataset. Science of The Total Environment 846, 157204 (2022). Visualwind: a novel video dataset for cameras to sense the wind. Q Zhang, J Xu, M Crane, C Luo, IGARSS 2022 -2022 IEEE International Geoscience and Remote Sensing Symposium vols 2022. IEEEZhang, Q., Xu, J., Crane, M. & Luo, C. Visualwind: a novel video dataset for cameras to sense the wind. in IGARSS 2022 -2022 IEEE International Geoscience and Remote Sensing Symposium vols 2022-July 1924-1927 (IEEE, 2022). ImageNet: A large-scale hierarchical image database. J Deng, 10.1109/CVPR.2009.52068482009 IEEE Conference on Computer Vision and Pattern Recognition. IEEEDeng, J. et al. ImageNet: A large-scale hierarchical image database. in 2009 IEEE Conference on Computer Vision and Pattern Recognition 248-255 (IEEE, 2009). doi:10.1109/CVPR.2009.5206848. Microsoft COCO: common objects in context. T.-Y Lin, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics. 8693Springer VerlagLNCSLin, T.-Y. et al. Microsoft COCO: common objects in context. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) vol. 8693 LNCS 740-755 (Springer Verlag, 2014). Visual anemometry measurements of eight vegetation species. R H Goldshmid, J O Dabiri, 10.22002/crb7k-1gj48CaltechDATA. Goldshmid, R. H. & Dabiri, J. O. Visual anemometry measurements of eight vegetation species. CaltechDATA (2023) doi:https://doi.org/10.22002/crb7k-1gj48. The characteristics of wind velocity that favor the fitting of a Weibull distribution in wind speed analysis. S E Tuller, A C Brett, Journal of Climate and Applied Meteorology. 23Tuller, S. E. & Brett, A. C. The characteristics of wind velocity that favor the fitting of a Weibull distribution in wind speed analysis. Journal of Climate and Applied Meteorology 23, 124-134 (1984). Probability plotting methods for the analysis of data. M B Wilk, R Gnanadesikan, Biometrika. 551Wilk, M. B. & Gnanadesikan, R. Probability plotting methods for the analysis of data. Biometrika 55, 1 (1968). Particle image velocimetry for MATLAB: accuracy and enhanced algorithms in PIVlab. W Thielicke, R Sonntag, J Open Res Softw. 912Thielicke, W. & Sonntag, R. Particle image velocimetry for MATLAB: accuracy and enhanced algorithms in PIVlab. J Open Res Softw 9, 12 (2021). The Ames Stereo Pipeline: NASA's open source software for deriving and processing terrain data. R A Beyer, O Alexandrov, S Mcmichael, Earth and Space Science. 5Beyer, R. A., Alexandrov, O. & McMichael, S. The Ames Stereo Pipeline: NASA's open source software for deriving and processing terrain data. Earth and Space Science 5, 537-548 (2018). The Auto Arborist Dataset: a large-scale benchmark for multiview urban forest monitoring under domain shift. S Beery, 10.1109/CVPR52688.2022.020612022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). IEEEBeery, S. et al. The Auto Arborist Dataset: a large-scale benchmark for multiview urban forest monitoring under domain shift. in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) 21262-21275 (IEEE, 2022). doi:10.1109/CVPR52688.2022.02061. Automatic storage and analysis of camera trap data. G Harris, R Thompson, J L Childs, J G Sanderson, Bulletin of the Ecological Society of America. 91Harris, G., Thompson, R., Childs, J. L. & Sanderson, J. G. Automatic storage and analysis of camera trap data. Bulletin of the Ecological Society of America 91, 352- 360 (2010). Limitations of recreational camera traps for wildlife management and conservation research: a practitioner's perspective. S Newey, Ambio. 44Newey, S. et al. Limitations of recreational camera traps for wildlife management and conservation research: a practitioner's perspective. Ambio 44, 624-635 (2015). Towards good practice guidance in using camera-traps in ecology: influence of sampling design on validity of ecological inferences. S Hamel, Methods Ecol Evol. 4Hamel, S. et al. Towards good practice guidance in using camera-traps in ecology: influence of sampling design on validity of ecological inferences. Methods Ecol Evol 4, 105-113 (2013). Seabird surveillance: combining CCTV and artificial intelligence for monitoring and research. J Hentati-Sundberg, 10.1002/rse2.329Remote Sens Ecol Conserv. Hentati-Sundberg, J. et al. Seabird surveillance: combining CCTV and artificial intelligence for monitoring and research. Remote Sens Ecol Conserv (2023) doi:10.1002/rse2.329. Wildlife surveillance using deep learning methods. R Chen, R Little, L Mihaylova, R Delahay, R Cox, Ecol Evol. 9Chen, R., Little, R., Mihaylova, L., Delahay, R. & Cox, R. Wildlife surveillance using deep learning methods. Ecol Evol 9, 9453-9466 (2019). Drone assisted deep learning based wildfire detection system. V Gupta, S Roy, V Jaiswal, K Bhardwaj, P S Rana, 10.1109/PDGC56933.2022.100531232022 Seventh International Conference on Parallel, Distributed and Grid Computing (PDGC). IEEEGupta, V., Roy, S., Jaiswal, V., Bhardwaj, K. & Rana, P. S. Drone assisted deep learning based wildfire detection system. in 2022 Seventh International Conference on Parallel, Distributed and Grid Computing (PDGC) 162-166 (IEEE, 2022). doi:10.1109/PDGC56933.2022.10053123. Video fire detection -review. A E Çetin, Digit Signal Process. 23Çetin, A. E. et al. Video fire detection -review. Digit Signal Process 23, 1827- 1843 (2013). BLSTM based night-time wildfire detection from video. A K Agirman, K Tasdemir, PLoS One. 17269161Agirman, A. K. & Tasdemir, K. BLSTM based night-time wildfire detection from video. PLoS One 17, e0269161 (2022). Aerial imagery pile burn detection using deep learning: The FLAME dataset. A Shamsoshoara, Computer Networks. 193108001Shamsoshoara, A. et al. Aerial imagery pile burn detection using deep learning: The FLAME dataset. Computer Networks 193, 108001 (2021). Real-time video fire/smoke detection based on CNN in antifire surveillance systems. S Saponara, A Elhanashi, A Gagliardi, J Real Time Image Process. 18Saponara, S., Elhanashi, A. & Gagliardi, A. Real-time video fire/smoke detection based on CNN in antifire surveillance systems. J Real Time Image Process 18, 889-900 (2021). Active fire detection in Landsat-8 imagery: a large-scale dataset and a deep-learning study. G H De Almeida Pereira, A M Fusioka, B T Nassu, R Minetto, ISPRS Journal of Photogrammetry and Remote Sensing. 178de Almeida Pereira, G. H., Fusioka, A. M., Nassu, B. T. & Minetto, R. Active fire detection in Landsat-8 imagery: a large-scale dataset and a deep-learning study. ISPRS Journal of Photogrammetry and Remote Sensing 178, 171-186 (2021). Forest fire susceptibility modeling using a convolutional neural network for Yunnan Province of China. G Zhang, M Wang, K Liu, International Journal of Disaster Risk Science. 10Zhang, G., Wang, M. & Liu, K. Forest fire susceptibility modeling using a convolutional neural network for Yunnan Province of China. International Journal of Disaster Risk Science 10, 386-403 (2019). Deep convolutional neural networks for forest fire detection. Q Zhang, J Xu, L Xu, H Guo, 10.2991/ifmeita-16.2016.105Proceedings of the 2016 International Forum on Management, Education and Information Technology Application. the 2016 International Forum on Management, Education and Information Technology ApplicationAtlantis PressZhang, Q., Xu, J., Xu, L. & Guo, H. Deep convolutional neural networks for forest fire detection. in Proceedings of the 2016 International Forum on Management, Education and Information Technology Application (Atlantis Press, 2016). doi:10.2991/ifmeita-16.2016.105. Analysis of deep learning methods for early wildfire detection systems: review. A S Mahdi, S A Mahmood, 10.1109/IICETA54559.2022.98885152022 5th International Conference on Engineering Technology and its Applications (IICETA). IEEEMahdi, A. S. & Mahmood, S. A. Analysis of deep learning methods for early wildfire detection systems: review. in 2022 5th International Conference on Engineering Technology and its Applications (IICETA) 271-276 (IEEE, 2022). doi:10.1109/IICETA54559.2022.9888515. The traffic congestion investigating system by image processing from CCTV camera. B Eamthanakul, M Ketcham, N Chumuang, 10.1109/ICDAMT.2017.79049692017 International Conference on Digital Arts, Media and Technology (ICDAMT). IEEEEamthanakul, B., Ketcham, M. & Chumuang, N. The traffic congestion investigating system by image processing from CCTV camera. in 2017 International Conference on Digital Arts, Media and Technology (ICDAMT) 240- 245 (IEEE, 2017). doi:10.1109/ICDAMT.2017.7904969. CADP: a novel dataset for CCTV traffic camera based accident analysis. A P Shah, J.-B Lamare, T Nguyen-Anh, A Hauptmann, 10.1109/AVSS.2018.863916015th IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS). IEEEShah, A. P., Lamare, J.-B., Nguyen-Anh, T. & Hauptmann, A. CADP: a novel dataset for CCTV traffic camera based accident analysis. in 2018 15th IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS) 1-9 (IEEE, 2018). doi:10.1109/AVSS.2018.8639160. A review on forest fire detection techniques. A A A Alkhatib, Int J Distrib Sens Netw. 10597368Alkhatib, A. A. A. A review on forest fire detection techniques. Int J Distrib Sens Netw 10, 597368 (2014). 3D modelling of forest canopy structure for remote sensing simulations in the optical and microwave domains. M Disney, P Lewis, P Saich, Remote Sens Environ. 100Disney, M., Lewis, P. & Saich, P. 3D modelling of forest canopy structure for remote sensing simulations in the optical and microwave domains. Remote Sens Environ 100, 114-132 (2006). Real-Time interactive tree animation. E Quigley, Y Yu, J Huang, W Lin, R Fedkiw, IEEE Trans Vis Comput Graph. 24Quigley, E., Yu, Y., Huang, J., Lin, W. & Fedkiw, R. Real-Time interactive tree animation. IEEE Trans Vis Comput Graph 24, 1717-1727 (2018). Windy trees. S Pirk, T Niese, T Hädrich, B Benes, O Deussen, ACM Trans Graph. 33Pirk, S., Niese, T., Hädrich, T., Benes, B. & Deussen, O. Windy trees. ACM Trans Graph 33, 1-11 (2014). Modeling of fluid-structure interaction. E H Dowell, K C Hall, 10.1146/annurev.fluid.33.1.44533Dowell, E. H. & Hall, K. C. Modeling of fluid-structure interaction. http://dx.doi.org/10.1146/annurev.fluid.33.1.445 33, 445-490 (2003). Fluid-structure interactions and flow induced vibrations: a review. R Parameshwaran, S J Dhulipalla, D R Yendluri, Procedia Eng. 144Parameshwaran, R., Dhulipalla, S. J. & Yendluri, D. R. Fluid-structure interactions and flow induced vibrations: a review. Procedia Eng 144, 1286-1293 (2016). Quasi-geostrophic jet-like flow with obstructions. M Mossa, J Fluid Mech. 92112Mossa, M. et al. Quasi-geostrophic jet-like flow with obstructions. J Fluid Mech 921, A12 (2021). Turbulent jet through porous obstructions under Coriolis effect: an experimental investigation. F De Serio, Exp Fluids. 62218De Serio, F. et al. Turbulent jet through porous obstructions under Coriolis effect: an experimental investigation. Exp Fluids 62, 218 (2021). Oscillatory flow through submerged canopies: 1. Velocity structure. R J Lowe, J R Koseff, S G Monismith, J Geophys Res Oceans. 110Lowe, R. J., Koseff, J. R. & Monismith, S. G. Oscillatory flow through submerged canopies: 1. Velocity structure. J Geophys Res Oceans 110, 1-17 (2005). Modeling flow in coral communities with and without waves: A synthesis of porous media and canopy flow approaches. R J Lowe, U Shavit, J L Falter, J R Koseff, S G Monismith, Limnol Oceanogr. 53Lowe, R. J., Shavit, U., Falter, J. L., Koseff, J. R. & Monismith, S. G. Modeling flow in coral communities with and without waves: A synthesis of porous media and canopy flow approaches. Limnol Oceanogr 53, 2668-2680 (2008). Combined threedimensional flow field measurements and motion tracking of freely moving spheres in a turbulent boundary layer. R Van Hout, A Hershkovitz, G E Elsinga, J Westerweel, J Fluid Mech. 94412van Hout, R., Hershkovitz, A., Elsinga, G. E. & Westerweel, J. Combined three- dimensional flow field measurements and motion tracking of freely moving spheres in a turbulent boundary layer. J Fluid Mech 944, A12 (2022). Vortex shedding from oscillating bluff bodies. P W Bearman, Annu Rev Fluid Mech. 16Bearman, P. W. Vortex shedding from oscillating bluff bodies. Annu Rev Fluid Mech 16, 195-222 (1984). Aerodynamics: constituting the first volume of a complete work on aerial flight. F W Lanchester, Lanchester, F. W. Aerodynamics: constituting the first volume of a complete work on aerial flight. (1907). The aeroplane: a concise scientific study. A Fage, Fage, A. The aeroplane: a concise scientific study. (1917). A high-order time-accurate interrogation method for timeresolved PIV. K Lynch, F Scarano, Meas Sci Technol. 2435305Lynch, K. & Scarano, F. A high-order time-accurate interrogation method for time- resolved PIV. Meas Sci Technol 24, 035305 (2013). Long-duration time-resolved PIV to study unsteady aerodynamics. Z J Taylor, R Gurka, G A Kopp, A Liberzon, IEEE Trans Instrum Meas. 59Taylor, Z. J., Gurka, R., Kopp, G. A. & Liberzon, A. Long-duration time-resolved PIV to study unsteady aerodynamics. IEEE Trans Instrum Meas 59, 3262-3269 (2010). Tomographic particle image velocimetry. G E Elsinga, F Scarano, B Wieneke, B W Van Oudheusden, Exp Fluids. 41Elsinga, G. E., Scarano, F., Wieneke, B. & van Oudheusden, B. W. Tomographic particle image velocimetry. Exp Fluids 41, 933-947 (2006). Instantaneous pressure and material acceleration measurements using a four-exposure PIV system. X Liu, J Katz, Exp Fluids. 41Liu, X. & Katz, J. Instantaneous pressure and material acceleration measurements using a four-exposure PIV system. Exp Fluids 41, 227-240 (2006). An algorithm to estimate unsteady and quasi-steady pressure fields from velocity field measurements. J O Dabiri, S Bose, B J Gemmell, S P Colin, J H Costello, Journal of Experimental Biology. 217Dabiri, J. O., Bose, S., Gemmell, B. J., Colin, S. P. & Costello, J. H. An algorithm to estimate unsteady and quasi-steady pressure fields from velocity field measurements. Journal of Experimental Biology 217, 331-336 (2014). SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics. K Kaheman, J N Kutz, S L Brunton, Proceedings of the Royal Society A. 476Kaheman, K., Kutz, J. N. & Brunton, S. L. SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics. Proceedings of the Royal Society A 476, (2020). Physics-informed dynamic mode decomposition. P J Baddoo, B Herrmann, B J Mckeon, J Nathan Kutz, S L Brunton, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 47920220576Baddoo, P. J., Herrmann, B., McKeon, B. J., Nathan Kutz, J. & Brunton, S. L. Physics-informed dynamic mode decomposition. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 479, 20220576 (2023). Kernel learning for robust dynamic mode decomposition: linear and nonlinear disambiguation optimization. P J Baddoo, B Herrmann, B J Mckeon, S L Brunton, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 478Baddoo, P. J., Herrmann, B., McKeon, B. J. & Brunton, S. L. Kernel learning for robust dynamic mode decomposition: linear and nonlinear disambiguation optimization. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 478, (2022). An airborne millimeter-wave imaging radiometer for cloud, precipitation, and atmospheric water vapor studies. P Racette, J Atmos Ocean Technol. 13Racette, P. et al. An airborne millimeter-wave imaging radiometer for cloud, precipitation, and atmospheric water vapor studies. J Atmos Ocean Technol 13, 610-619 (1996). Real-time highresolution millimeter-wave imaging for in-vivo skin cancer diagnosis. A Mirbeik, R Ashinoff, T Jong, A Aued, N Tavassolian, Scientific Reports. 121Mirbeik, A., Ashinoff, R., Jong, T., Aued, A. & Tavassolian, N. Real-time high- resolution millimeter-wave imaging for in-vivo skin cancer diagnosis. Scientific Reports 2022 12:1 12, 1-10 (2022). Passive millimeter wave imaging. L Yujiri, M Shoucri, P Moffa, IEEE Microw Mag. 4Yujiri, L., Shoucri, M. & Moffa, P. Passive millimeter wave imaging. IEEE Microw Mag 4, 39-50 (2003). Efficient machine learning method for spatio-temporal water surface waves reconstruction from polarimetric images. N Ginio, D Liberzon, M Lindenbaum, B Fishbain, Meas Sci Technol. 3455801Ginio, N., Liberzon, D., Lindenbaum, M. & Fishbain, B. Efficient machine learning method for spatio-temporal water surface waves reconstruction from polarimetric images. Meas Sci Technol 34, 055801 (2023). Experimental study of the initial stages of wind waves' spatial evolution. D Liberzon, L Shemer, J Fluid Mech. 681Liberzon, D. & Shemer, L. Experimental study of the initial stages of wind waves' spatial evolution. J Fluid Mech 681, 462-498 (2011). Structure of the airflow above surface waves. M P Buckley, F Veron, J Phys Oceanogr. 46Buckley, M. P. & Veron, F. Structure of the airflow above surface waves. J Phys Oceanogr 46, 1377-1397 (2016). The turbulent airflow over wind generated surface waves. M P Buckley, F Veron, European Journal of Mechanics -B/Fluids. 73Buckley, M. P. & Veron, F. The turbulent airflow over wind generated surface waves. European Journal of Mechanics -B/Fluids 73, 132-143 (2019). Ocean spray. F Veron, Annu Rev Fluid Mech. 47Veron, F. Ocean spray. Annu Rev Fluid Mech 47, 507-538 (2015). Hydrodynamics of coral reefs. S G Monismith, 10.1146/annurev.fluid.38.050304.09212539Monismith, S. G. Hydrodynamics of coral reefs. https://doi.org/10.1146/annurev.fluid.38.050304.092125 39, 37-55 (2006). Net-zero America: potential pathways, infrastructure, and impacts. E Larson, Princeton UniversityLarson, E. et al. Net-zero America: potential pathways, infrastructure, and impacts. (Princeton University, 2020). The need for continued innovation in solar, wind, and energy storage. V Sivaram, J O Dabiri, D M Hart, Joule. 2Sivaram, V., Dabiri, J. O. & Hart, D. M. The need for continued innovation in solar, wind, and energy storage. Joule 2, 1639-1642 (2018). Crossing multiple gray zones in the transition from mesoscale to microscale simulation over complex terrain. F Chow, Atmosphere (Basel). 10274Chow, F. et al. Crossing multiple gray zones in the transition from mesoscale to microscale simulation over complex terrain. Atmosphere (Basel) 10, 274 (2019). Obtaining turbulence statistics of thermally driven anabatic flow by sonic-hot-film combo anemometer. R Hilel Goldshmid, D Liberzon, Environmental Fluid Mechanics. 20Hilel Goldshmid, R. & Liberzon, D. Obtaining turbulence statistics of thermally driven anabatic flow by sonic-hot-film combo anemometer. Environmental Fluid Mechanics 20, 1221-1249 (2018). Separation of upslope flow over a plateau. R Hilel Goldshmid, Atmosphere (Basel). 9165Hilel Goldshmid, R. et al. Separation of upslope flow over a plateau. Atmosphere (Basel) 9, 165 (2018). The MATERHORN: unraveling the intricacies of mountain weather. H J S Fernando, Bull Am Meteorol Soc. 96Fernando, H. J. S. et al. The MATERHORN: unraveling the intricacies of mountain weather. Bull Am Meteorol Soc 96, 1945-1967 (2015). A description of the advanced research WRF model version 4. W C Skamarock, National Center for Atmospheric Research. Skamarock, W. C. et al. A description of the advanced research WRF model version 4. National Center for Atmospheric Research: Boulder, CO, USA https://opensky.ucar.edu/islandora/object/technotes:576/datastream/PDF/download/ citation.pdf (2019). Assessing boundary condition and parametric uncertainty in numerical-weather-prediction-modeled, long-term offshore wind speed through machine learning and analog ensemble. N Bodini, W Hu, M Optis, G Cervone, S Alessandrini, Wind Energy Science. 6Bodini, N., Hu, W., Optis, M., Cervone, G. & Alessandrini, S. Assessing boundary condition and parametric uncertainty in numerical-weather-prediction-modeled, long-term offshore wind speed through machine learning and analog ensemble. Wind Energy Science 6, 1363-1377 (2021). Springer handbook of experimental fluid mechanics. C Tropea, A L Yarin, J F Foss, Springer Science+Business MediaTropea, C., Yarin, A. L. & Foss, J. F. Springer handbook of experimental fluid mechanics. (Springer Science+Business Media, 2007). In situ calibration of hot-film probes using a collocated sonic anemometer: implementation of a neural network. E Kit, A Cherkassky, T Sant, H J S Fernando, J Atmos Ocean Technol. 27Kit, E., Cherkassky, A., Sant, T. & Fernando, H. J. S. In situ calibration of hot-film probes using a collocated sonic anemometer: implementation of a neural network. J Atmos Ocean Technol 27, 23-41 (2010). Next generation combined sonichotfilm anemometer: wind alignment and automated calibration procedure using deep learning. R H Goldshmid, E Winiarska, D Liberzon, Exp Fluids. 6330Goldshmid, R. H., Winiarska, E. & Liberzon, D. Next generation combined sonic- hotfilm anemometer: wind alignment and automated calibration procedure using deep learning. Exp Fluids 63, 30 (2022). Atmospheric modeling, data assimilation and predictability. E Kalnay, Kalnay, E. Atmospheric modeling, data assimilation and predictability. (2003). The quiet revolution of numerical weather prediction. P Bauer, A Thorpe, G Brunet, Nature. 525Bauer, P., Thorpe, A. & Brunet, G. The quiet revolution of numerical weather prediction. Nature 525, 47-55 (2015). Short-term wind forecast of a data assimilation/weather forecasting system with wind turbine anemometer measurement assimilation. W Y Y Cheng, Y Liu, A J Bourgeois, Y Wu, S E Haupt, Renew Energy. 107Cheng, W. Y. Y., Liu, Y., Bourgeois, A. J., Wu, Y. & Haupt, S. E. Short-term wind forecast of a data assimilation/weather forecasting system with wind turbine anemometer measurement assimilation. Renew Energy 107, 340-351 (2017). Nearwake structure of full-scale vertical-axis wind turbines. N J Wei, I D Brownstein, J L Cardona, M F Howland, J O Dabiri, J Fluid Mech. 91417Wei, N. J., Brownstein, I. D., Cardona, J. L., Howland, M. F. & Dabiri, J. O. Near- wake structure of full-scale vertical-axis wind turbines. J Fluid Mech 914, A17 (2021). New methods to improve the vertical extrapolation of near-surface offshore wind speeds. M Optis, N Bodini, M Debnath, P Doubrawa, Wind Energy Science. 6Optis, M., Bodini, N., Debnath, M. & Doubrawa, P. New methods to improve the vertical extrapolation of near-surface offshore wind speeds. Wind Energy Science 6, 935-948 (2021). Solar energy forecasting and resource assessment. J Kleissl, Kleissl, J. Solar energy forecasting and resource assessment. (2013). Renewable and efficient electric power systems. G Masters, Masters, G. Renewable and efficient electric power systems. (2013). Thermal behavior of photovoltaic devices. O Dupré, R Vaillon, M A Green, 10.1007/978-3-319-49457-9Springer International PublishingDupré, O., Vaillon, R. & Green, M. A. Thermal behavior of photovoltaic devices. Springer (Springer International Publishing, 2017). doi:10.1007/978-3-319-49457- 9. Modeling of the nominal operating cell temperature based on outdoor weathering. M Koehl, M Heck, S Wiesmeier, J Wirth, Solar Energy Materials and Solar Cells. 95Koehl, M., Heck, M., Wiesmeier, S. & Wirth, J. Modeling of the nominal operating cell temperature based on outdoor weathering. Solar Energy Materials and Solar Cells 95, 1638-1646 (2011). Wind Effect on PV module temperature: analysis of different techniques for an accurate estimation. C Schwingshackl, Energy Procedia. 40Schwingshackl, C. et al. Wind Effect on PV module temperature: analysis of different techniques for an accurate estimation. Energy Procedia 40, 77-86 (2013). Pathways for mitigating thermal losses in solar photovoltaics. R Vaillon, O Dupré, R B Cal, M Calaf, Sci Rep. 813163Vaillon, R., Dupré, O., Cal, R. B. & Calaf, M. Pathways for mitigating thermal losses in solar photovoltaics. Sci Rep 8, 13163 (2018). United Nations sustainable development goals. United Nations sustainable development goals. https://www.un.org/sustainabledevelopment/. Confronting grand challenges in environmental fluid mechanics. T Dauxois, Phys Rev Fluids. 620501Dauxois, T. et al. Confronting grand challenges in environmental fluid mechanics. Phys Rev Fluids 6, 020501 (2021). Gust loading factor-past, present and future. A Kareem, Y Zhou, Journal of Wind Engineering and Industrial Aerodynamics. 91Kareem, A. & Zhou, Y. Gust loading factor-past, present and future. Journal of Wind Engineering and Industrial Aerodynamics 91, 1301-1328 (2003). Natural ventilation in cities: the implications of fluid mechanics. J Song, Building Research & Information. 46Song, J. et al. Natural ventilation in cities: the implications of fluid mechanics. Building Research & Information 46, 809-828 (2018). Designing airspace for urban air mobility: a review of concepts and approaches. A Bauranov, J Rakas, Progress in Aerospace Sciences. 125100726Bauranov, A. & Rakas, J. Designing airspace for urban air mobility: a review of concepts and approaches. Progress in Aerospace Sciences 125, 100726 (2021). Reactive trajectory generation and formation control for groups of UAVs in windy environments. K Cole, George Washington UniversityCole, K. Reactive trajectory generation and formation control for groups of UAVs in windy environments. (George Washington University, 2018). Predictive large eddy simulations for urban flows: challenges and opportunities. C García-Sánchez, J Van Beeck, C Gorlé, Build Environ. 139García-Sánchez, C., van Beeck, J. & Gorlé, C. Predictive large eddy simulations for urban flows: challenges and opportunities. Build Environ 139, 146-156 (2018). Parameter uncertainty quantification in an idealized GCM with a seasonal cycle. M F Howland, O R A Dunbar, T Schneider, J Adv Model Earth Syst. 14Howland, M. F., Dunbar, O. R. A. & Schneider, T. Parameter uncertainty quantification in an idealized GCM with a seasonal cycle. J Adv Model Earth Syst 14, e2021MS002735 (2022). Improving urban flow predictions through data assimilation. J Sousa, C García-Sánchez, C Gorlé, Build Environ. 132Sousa, J., García-Sánchez, C. & Gorlé, C. Improving urban flow predictions through data assimilation. Build Environ 132, 282-290 (2018). Economic footprint of California wildfires. D Wang, Nat Sustain. 4Wang, D. et al. Economic footprint of California wildfires in 2018. Nat Sustain 4, 252-260 (2020). NOAA National Centers for Environmental Information (NCEI). U.S. billion-dollar weather and climate disasters. 10.25921/STKW-7W73National Centers for Environmental Information. NOAA National Centers for Environmental Information (NCEI). U.S. billion-dollar weather and climate disasters. National Centers for Environmental Information https://accession.nodc.noaa.gov/0209268 (2023) doi:10.25921/STKW-7W73. Exacerbated fires in Mediterranean Europe due to anthropogenic warming projected with non-stationary climate-fire models. M Turco, Nat Commun. 93821Turco, M. et al. Exacerbated fires in Mediterranean Europe due to anthropogenic warming projected with non-stationary climate-fire models. Nat Commun 9, 3821 (2018). Warming and earlier spring increase western U.S. forest wildfire activity. A L Westerling, H G Hidalgo, D R Cayan, T W Swetnam, Science. 313Westerling, A. L., Hidalgo, H. G., Cayan, D. R. & Swetnam, T. W. Warming and earlier spring increase western U.S. forest wildfire activity. Science (1979) 313, 940-943 (2006). Representativeness of wind measurements in fire experiments: lessons learned from large-eddy simulations in a homogeneous forest. F Pimont, J.-L Dupuy, R R Linn, R Parsons, N Martin-Stpaul, Agric For Meteorol. 232Pimont, F., Dupuy, J.-L., Linn, R. R., Parsons, R. & Martin-StPaul, N. Representativeness of wind measurements in fire experiments: lessons learned from large-eddy simulations in a homogeneous forest. Agric For Meteorol 232, 479-488 (2017). Prediction of fire spread in grasslands. N Cheney, J Gould, W Catchpole, Int J Wildland Fire. 81Cheney, N., Gould, J. & Catchpole, W. Prediction of fire spread in grasslands. Int J Wildland Fire 8, 1 (1998). The interaction of wind and fire. T Beer, Boundary Layer Meteorol. 54Beer, T. The interaction of wind and fire. Boundary Layer Meteorol 54, 287-308 (1991). Spatial variation in extreme winds predicts large wildfire locations in chaparral ecosystems. M A Moritz, T J Moody, M A Krawchuk, M Hughes, A Hall, Geophys Res Lett. 374801Moritz, M. A., Moody, T. J., Krawchuk, M. A., Hughes, M. & Hall, A. Spatial variation in extreme winds predicts large wildfire locations in chaparral ecosystems. Geophys Res Lett 37, 4801 (2010). The Santa Ana wildfire threat index: methodology and operational implementation. T Rolinski, Weather Forecast. 31Rolinski, T. et al. The Santa Ana wildfire threat index: methodology and operational implementation. Weather Forecast 31, 1881-1897 (2016). The role of intermittent wind in the dispersal of fungal pathogens. D E Aylor, Annu Rev Phytopathol. 28Aylor, D. E. The role of intermittent wind in the dispersal of fungal pathogens. Annu Rev Phytopathol 28, 73-92 (1990). Wind pollination over mesoscale distances: an investigation with Scots pine. J J Robledo-Arnuncio, New Phytologist. 190Robledo-Arnuncio, J. J. Wind pollination over mesoscale distances: an investigation with Scots pine. New Phytologist 190, 222-233 (2011). Long-distance dispersal of plants. R Nathan, Science. 313Nathan, R. Long-distance dispersal of plants. Science (1979) 313, 786-788 (2006). Aerial dispersal of pathogens on the global and continental scales and its impact on plant disease. J K M Brown, M S Hovmøller, Science. 297Brown, J. K. M. & Hovmøller, M. S. Aerial dispersal of pathogens on the global and continental scales and its impact on plant disease. Science (1979) 297, 537-541 (2002). Wind-borne insects mediate directional pollen transfer between desert fig trees 160 kilometers apart. S Ahmed, S G Compton, R K Butlin, P M Gilmartin, Proceedings of the National Academy of Sciences. 106Ahmed, S., Compton, S. G., Butlin, R. K. & Gilmartin, P. M. Wind-borne insects mediate directional pollen transfer between desert fig trees 160 kilometers apart. Proceedings of the National Academy of Sciences 106, 20342-20347 (2009). Wind as a long-distance dispersal vehicle in the southern hemisphere. J Muñoz, A M Felicísimo, F Cabezas, A R Burgaz, I Martínez, Science. 304Muñoz, J., Felicísimo, A. M., Cabezas, F., Burgaz, A. R. & Martínez, I. Wind as a long-distance dispersal vehicle in the southern hemisphere. Science (1979) 304, 1144-1147 (2004). Pollination of crops in Australia and New Zealand. M Goodwin, Goodwin, M. Pollination of crops in Australia and New Zealand. (2012). Evidence for landscape-level, pollen-mediated gene flow from genetically modified creeping bentgrass with CP4 EPSPS as a marker. L S Watrud, Proceedings of the National Academy of Sciences. 101Watrud, L. S. et al. Evidence for landscape-level, pollen-mediated gene flow from genetically modified creeping bentgrass with CP4 EPSPS as a marker. Proceedings of the National Academy of Sciences 101, 14533-14538 (2004). Some evolutionary consequences of being a tree. R J Petit, A Hampe, Annu Rev Ecol Evol Syst. 37Petit, R. J. & Hampe, A. Some evolutionary consequences of being a tree. Annu Rev Ecol Evol Syst 37, 187-214 (2006). Role of temperature, wind, and precipitation in heavy metal contamination at copper mines: a review. A Punia, Environmental Science and Pollution Research. 28Punia, A. Role of temperature, wind, and precipitation in heavy metal contamination at copper mines: a review. Environmental Science and Pollution Research 28, 4056-4072 (2021). Study of natural ventilation in buildings by large eddy simulation. Y Jiang, Q Chen, Journal of Wind Engineering and Industrial Aerodynamics. 89Jiang, Y. & Chen, Q. Study of natural ventilation in buildings by large eddy simulation. Journal of Wind Engineering and Industrial Aerodynamics 89, 1155- 1178 (2001). Improving thermal model predictions for naturally ventilated buildings using large eddy simulations. L W Chew, C Chen, C Gorlé, Build Environ. 220109241Chew, L. W., Chen, C. & Gorlé, C. Improving thermal model predictions for naturally ventilated buildings using large eddy simulations. Build Environ 220, 109241 (2022). . US Energy Information Administration. Commercial buildings energy consumption survey, consumption and expenditures highlights. US Energy Information Administration. Commercial buildings energy consumption survey, consumption and expenditures highlights. (2018). The fluid mechanics of natural ventilation. P F Linden, Annu Rev Fluid Mech. 31Linden, P. F. The fluid mechanics of natural ventilation. Annu Rev Fluid Mech 31, 201-238 (1999). Improving predictions of the urban wind environment using data. C Gorlé, Technology|Architecture + Design. 3Gorlé, C. Improving predictions of the urban wind environment using data. Technology|Architecture + Design 3, 137-141 (2019). Single-sided natural ventilation in buildings: a critical literature review. H.-Y Zhong, Build Environ. 212108797Zhong, H.-Y. et al. Single-sided natural ventilation in buildings: a critical literature review. Build Environ 212, 108797 (2022).
[]
[ "The Markov approximation revisited: inconsistency of the standard quantum Brownian motion model", "The Markov approximation revisited: inconsistency of the standard quantum Brownian motion model" ]
[ "Andrea Rocco *[email protected]†[email protected] \nCenter for Nonlinear Science\nUniversity of North Texas\nP.O. Box 30537076203-305370DentonTexas\n\nDipartimento di Fisica dell'Universita' di Pisa\nPiazza Torricelli 256100PisaItaly\n\nIstituto di Biofisica del Consiglio Nazionale delle Ricerche\nVia S. Lorenzo 2656127PisaItaly\n", "Paolo Grigolini \nCenter for Nonlinear Science\nUniversity of North Texas\nP.O. Box 30537076203-305370DentonTexas\n" ]
[ "Center for Nonlinear Science\nUniversity of North Texas\nP.O. Box 30537076203-305370DentonTexas", "Dipartimento di Fisica dell'Universita' di Pisa\nPiazza Torricelli 256100PisaItaly", "Istituto di Biofisica del Consiglio Nazionale delle Ricerche\nVia S. Lorenzo 2656127PisaItaly", "Center for Nonlinear Science\nUniversity of North Texas\nP.O. Box 30537076203-305370DentonTexas" ]
[]
We revisit the Markov approximation necessary to derive ordinary Brownian motion from a model widely adopted in literature for this specific purpose. We show that this leads to internal inconsistencies, thereby implying that further search for a more satisfactory model is required.
10.1016/s0375-9601(98)00940-2
[ "https://arxiv.org/pdf/quant-ph/9909051v1.pdf" ]
18,456,611
quant-ph/9909051
5dbf45eef0638800054b8728af8b947660841bfd
The Markov approximation revisited: inconsistency of the standard quantum Brownian motion model 16 Sep 1999 Andrea Rocco *[email protected]†[email protected] Center for Nonlinear Science University of North Texas P.O. Box 30537076203-305370DentonTexas Dipartimento di Fisica dell'Universita' di Pisa Piazza Torricelli 256100PisaItaly Istituto di Biofisica del Consiglio Nazionale delle Ricerche Via S. Lorenzo 2656127PisaItaly Paolo Grigolini Center for Nonlinear Science University of North Texas P.O. Box 30537076203-305370DentonTexas The Markov approximation revisited: inconsistency of the standard quantum Brownian motion model 16 Sep 1999(July 30, 2018) We revisit the Markov approximation necessary to derive ordinary Brownian motion from a model widely adopted in literature for this specific purpose. We show that this leads to internal inconsistencies, thereby implying that further search for a more satisfactory model is required. I. INTRODUCTION According to Penrose [1], the time evolution of a quantum system should be thought of as the combination of two processes: the U-process, i.e., the unitary time evolution prescribed by the Schrödinger equation, and the R-process, namely, the genuine randomness associated to the collapse of the wave function. This perspective is very attractive, and many attempts are currently being made to set the R-process on the same dynamical basis as the U-process. This purpose is realized by some authors [2][3][4] by supplementing the Schrödinger equation with stochastic corrections, which are then interpreted as a manifestation of the environment influence. As well known [5], the path-integral formalism is quite equivalent to ordinary quantum mechanics. Within this formalism the joint action of the U-process and the R-process is described by the theory of the path integral with constraints developed by Mensky [6]. We think that a possible physical meaning of the Mensky approach is made especially transparent by the interesting recent work of Presilla, Onofrio and Tambini [7,8]. These authors prove that the well known method of the influence functional introduced by Feynmann and Vernon [9] can be used to derive the Mensky path integral with constraint as an effect of the influence of the environment. This is a very interesting result which might reflect significant elements of truth [10]. The conviction that the wave-function collapse is generated by an environment and that no such systems exist in nature as isolated systems is accepted by the majority of physicists. In spite of this general consensus, we find some element of inconsistency in this view. First, we observe that no claim of a genuine derivation of stochastic properties can be made without involving the Markov approximation. Adelman [11] built up a sort of Fokker-Planck equation equivalent to the generalized Langevin equation and this, in turn, can be derived from a dynamic picture not involving any Markov approximation [12,13], thereby generating the impression that the Markov assumption is not indispensable to the foundation of a stochastic process. However, as later proved by Fox [14], the non-Markovian Fokker-Planck equation of Adelman is not a stochastic process. Within a quantum mechanical context the close connection between the Markov assumption and a stochastic picture is pointed out by the recent work of Kleinert and Shabanov [15]. On the same token we think that a stochastic picture not implying the Markov approximation is not genuinely stochastic, in spite of a recent claim to the contrary [16]. The celebrated generalized Langevin equation of Mori [12,13] is an illuminating example showing that the so called stochastic force is actually a deterministic function of the initial conditions and can be considered as stochastic only if additional assumptions are made, such as that of coarse graining or of an incomplete information. On the other hand, these arbitrary assumptions might have essential effects also on the dynamics: for instance, that of abolishing slow tails of the velocity correlation function, conflicting with the exponential nature of the relaxation of macroscopic variables, a basic tenet of stochastic physics [17]. This paper is devoted to the discussion of the physical conditions necessary to establish from within a quantum-mechanical picture a standard fluctuation-dissipation process for a macroscopic variable. This property implies the exponential nature of the relaxation process of the macroscopic variable, and this is incompatible with both classical [18] and quantum mechanics [19]. The dynamical approach to the fluctuation-dissipation process [17] leads us to conclude that there are problems even if we disregard the case where the microscopic time scale is infinite, and the Markov approximation is impossible. In the case where the Markov approximation seems to be legitimate due to the existence of a finite time scale, it has the effect of disregarding a weak but persistent slow tail of the correlation function of the macroscopic variable. Thus, the Markovian approximation turns out to be equivalent to the influence of real physical processes, the so called spontaneous fluctuations [10], which are in fact shown to kill these slow tails [20]. The change of perspective is radical, even if it does not affect the resulting transport equations, and throughout this letter we shall refer to it as the transition from the subjective to the objective Markov approximation. Another way of expressing our aim is as follows. We plan to establish the intensity of the corrections to ordinary quantum mechanics necessary to make valid the Markov assumption, which is incompatible with ordinary quantum mechanics. In literature a scarce attention is devoted to the physical meaning of the error associated to the Markov approximation. To properly establish the intensity of this error we interpret the Markov approximation as a property made genuine by proper corrections to quantum mechanics. Then we make a balance on the intensity of these corrections. If they turn out to be too large, we shall consequently judge the error associated with the Markov approximation to be unacceptable. To realize this purpose we adopt the Caldeira and Leggett model [21] and thus our departure point is close to that established by the results of Presilla, Onofrio and Tambini [7,8], who, as earlier pointed out, showed that the contraction over the irrelevant bath variables yields the Mensky path integral. Similar results have been more recently obtained by Mensky himself, adopting as a bath the internal atomic structure [22]. To trigger the transition from these subjective Markov assumptions to an objective Markov approximation we adopt the same perspective as that used in [10] to address the problem of the wave-function collapse, through the environment-induced enhancement of spontaneous fluctuations. Furthermore, rather than using the model of Ghirardi, Pearle and Rimini [23] as a source of spontaneous fluctuations, we adopt here a Mensky constraint conceived as a correction to ordinary quantum mechanics. II. MARKOV APPROXIMATION TO FRICTION IN THE ZERO-TEMPERATURE LIMIT Notice that the high-temperature assumption of [7,8,21] with the Mensky constraint as a source of spontaneous fluctuations would make the calculations exceedingly difficult. Thus, we do not address the calculation of fluctuations and we limit ourselves to studying dissipation in the zero-temperature limit. As proved by Caldeira and Leggett [21], in the case of harmonic baths the friction term derived from the subjective Markov approximation is independent of temperature. Thus, we make the plausible assumption that even in the presence of the Mensky constraint, the high-temperature friction is the same as that derived by us in the zero-temperature limit (see Section III) 1 . Let us consider as in [21], an oscillator of interest interacting with an environment of N oscillators, with N ≫ 1. The Lagrangians describing this system are given by the following expressions: L 0 (q) = 1 2 m 0q 2 − 1 2 m 0 ω 2 0 q 2 (1) L B (Q i ) = N j=1 1 2 m BQ 2 j − 1 2 m B ω 2 j Q 2 j (2) L I (q, Q j ) = q N j=1 g j Q j .(3) Thus the resulting quantum mechanical motion of the whole system is described by the following quantum mechanical propagator: q F , Q F 1 , ..., Q F N , T |q I , Q I 1 , ..., Q I N , 0 = [dq] exp ī h T 0 dt 1 2 m 0q 2 − 1 2 m 0 ω 2 0 q 2 × N j=1 [dQ j ] exp ī h T 0 dt 1 2 m BQ 2 j − 1 2 m B ω 2 j Q 2 j + g j qQ j . (4) Let us use the contracted description already adopted by Feynman and co-workers [5,9]. This means the reduced density matrix ρ A [21] for the system of interest defined by: ρ A (q F , q I , T ) = dq ′ F , dq ′ I K(q F , q I , T ; q ′ F , q ′ I , 0)ρ A (q ′ F , q ′ I , 0).(5) Here K denotes the superpropagator K(q F , q I , T ; q ′ F , q ′ I , 0) = [dq 1 ][dq 2 ] exp ī h [S 0 (q 1 ) − S * 0 (q 2 )] F [q 1 , q 2 ],(6) where S 0 is the action related to L 0 and F is the influence functional [5,9] whose explicit expression is F [q 1 , q 2 ] = exp − 1 h T 0 dt[q 1 (t) − q 2 (t)]I R (t) − ī h T 0 dt[q 1 (t) − q 2 (t)]I I (t) ,(7) with the two convolution integrals I R (t) and I I (t) given by: I R (t) = t 0 dt ′ α R (t − t ′ )[q 1 (t ′ ) − q 2 (t ′ )],(8)I I (t) = t 0 dt ′ α I (t − t ′ )[q 1 (t ′ ) + q 2 (t ′ )].(9) The two memory kernels α R (t−t ′ ) and α I (t−t ′ ) present in (8) and (9) in the zero-temperature limit are given by the expressions so as to recover, in the proper limit, an ordinary fluctuation-dissipation process. α R (t − t ′ ) = η π Ω 0 dωω cos ω(t − t ′ ),(10)α I (t − t ′ ) = − η π Ω 0 dωω sin ω(t − t ′ ),(11) where η is the friction and Ω is a cut-off frequency introduced assuming ohmic environment [24]. We are now in a position to discuss the problems raised by the subjective Markov approximation on the two kernels (10) and (11). First, we observe that the Markov approximation is expressed by: I(t) = t 0 dt ′ f (t ′ )K(t − t ′ ) = t 0 dτ f (t − τ )K(τ ) ≃ f (t) ∞ 0 dτ K(τ ).(12) According to the traditional wisdom, the closer K(τ ) to δ(τ ), the more accurate this approximation is expected to be. From within the perspective adopted in this paper, however, we have to establish a clearcut separation between the case Ω = ∞ and the case Ω < ∞, regardless of how large Ω might be in this latter case. This is so because we know [17] that, in accordance with the work of Ref. [18,19], the relaxation process of a variable made unstable by a bath with a bounded spectrum, cannot be exactly exponential. At long times, after the completion of the exponential decay of the macroscopic variable, non-exponential tails appear [17]. As weak as these tails are made by increasing the value of Ω, their existence is incompatible with the claim that a fluctuation-dissipation relation has been derived. The ideal case of Ω = ∞, on the contrary, would make the property of (12) exact, and there would be no conflict with the requirements of stochastic theory. However, we think that the recourse to the property Ω = ∞ is a sort of subterfuge and that this choice and the thermodynamical properties of crystals are at odds. Let us discuss now in this light the memory kernels of the case under study. Let us consider first the real part of the memory kernel α R (t − t ′ ). This corresponds to fluctuations, and in accordance with the known low-temperature predictions [25] we expect a strong non Markovian behavior to emerge out of it. In fact, renaming τ = t − t ′ , and integrating (10), we get: α R (τ ) = η π Ω sin Ωτ τ − 2 sin 2 (Ωτ /2) τ 2 .(13) This is integrable, and the adoption of the Markovian approximation, which is apparently possible, would lead to the puzzling result: ∞ 0 α R (τ ) = ηΩ 2 − ηΩ 2 = 0.(14) Actually, in accordance with [25,26], this is a case of infinite memory strikingly departing from the δ-function condition which is reached with Ω → ∞ by the dissipation part. Not even the subjective Markov approximation is admitted in this case. Then let us consider α I (τ ). In this case, on the basis of [21] we expect a Markovian behavior, temperature independent, to be legitimate at a subjective level. The following calculations confirm our expectation. From Eq. (11), we get α I (τ ) = η d dτ 1 π sin Ωτ τ .(15) By using lim Ω→∞ 1 π sin Ωτ τ = δ(τ ),(16) we get the expression: α I (τ ) = ηδ ′ (τ ).(17) Therefore the convolution integral (9) becomes: I I (t) = −ηδ(0)[q 1 (t) + q 2 (t)] + η 2 [q 1 (t) +q 2 (t)].(18) The adoption of this approach makes it possible to evidentiate that, as earlier pointed out, the case Ω = ∞ yields an exact result. In the case Ω < ∞ the same result can be obtained using the approximation of (12): I I (t) = η t 0 dt ′ d d(t − t ′ ) 1 π sin Ω(t − t ′ ) t − t ′ ) [q 1 (t ′ ) + q 2 (t ′ )] ≃ −η 1 π sin Ω(t − t ′ ) t − t ′ [q 1 (t ′ ) + q 2 (t ′ )] t 0 + η π [q 1 (t) +q 2 (t)] ∞ 0 dτ sin Ωτ τ = −η Ω π [q 1 (t) + q 2 (t)] + η 2 [q 1 (t) +q 2 (t)].(19) This result coincides with that resting on the condition Ω = ∞. However, as earlier remarked, the Markov approximation is equivalent to disregarding weak but very slow tails [17]. In both cases, the dissipative part of the influence functional is therefore written as [25,26]: F (diss) [q 1 , q 2 ] = exp ī h T 0 dtηδ(0)[q 2 1 (t) − q 2 2 (t)] − ī h T 0 dt η 2 [q 1 (t) − q 2 (t)][q 1 (t) +q 2 (t)] .(20) Notice that the divergence associated with δ(0) can be settled either by assigning to the Lagrangian a proper counter-term [24,27] or assuming a correlated initial condition for the total system [28]. III. FROM THE SUBJECTIVE TO THE OBJECTIVE MARKOV APPROXIMATION We want to explore here the possibility that the problems with the choice of a finite value of Ω might be solved assuming that the bath oscillators are subjected to a Mensky measurement process [6,7]. Adopting the perspective of Onofrio, Presilla and Tambini [7,8], these measurement processes should be traced back to the interaction between each bath oscillator and its own bath. This would not be yet satisfactory because the finite Ω problem would now affect the baths of the bath: This would be the beginning of an endless chain of baths of baths. We truncate this endless chain by assuming that the measurement process on the bath oscillator is an expression of a generalized version of quantum mechanics, an expression of spontaneous fluctuations, in the spirit of the theory of Ghirardi, Pearle and Rimini [23]. Of course, also the oscillator of interest should undergo the direct influence of this process. However, we are also making the assumption that the corrections to ordinary quantum mechanics are extremely weak, thereby implying that the direct influence of the Mensky measurement process on the oscillator of interest can be safely neglected. On these premises we are naturally led to adopt the following quantum mechanical propagator: q F , Q F 1 , ..., Q F N , T |q I , Q I 1 , ..., Q I N , 0 Γ = [dq] exp ī h T 0 dt 1 2 m 0q 2 − 1 2 m 0 ω 2 0 q 2 × N j=1 [dQ j ] exp ī h T 0 dt 1 2 m BQ 2 j − 1 2 m B ω 2 j Q 2 j + ihk(ω j )(Q j − a j ) 2 + g j qQ j ,(21) where a j = a j (t) is a function which expresses the result of the measurement process taking place on the j-th bath oscillator and k(ω j ) is the strenght of such a process. For the sake of generality, we assume the strenght of this measurement process to depend on the bathoscillator frequency. After several trials we established that the most convenient form to adopt for our purposes is the following linear dependence: k(ω j ) = Γm B h ω j − iΓ 2 .(22) The main tenet of all theoretical derivations of fluctuation-dissipation processes [29] is that the fluctuation-dissipation processes are perceived at a contracted level of description, and that these processes are nothing but a manifestation of the interaction with a bath, whose elementary constituents are not directly observed. This makes tempting the adoption of Markov approximation, as an expression of this lack of knowledge. However, in the same way as the information approach to statistical mechanics by Jaynes [30] implies probabilistic ingredients which might be foreign to classical mechanics, the Markov approximation is equivalent to an elementary randomness, which would be foreign to quantum mechanics, if this is not supplemented by the action of R-processes [1]. The purpose of this letter is to establish the amount of this elementary randomness with the help of the Mensky formalism. Note that the assumption that the bath is found in a condition of equilibrium, expressed by a canonical distribution at temperature T is a very strong assumption, already implying the inclusion of thermodynamical arguments. As discussed in Section IV, we have in mind a perspective, based on chaotic dynamics, where thermodynamics can be really reduced to dynamics, and this should prevent us from adopting this strong assumption. At the same time, this assumption, based on the fact that dynamics and statistics are decoupled the ones from the others, makes it possible to adopt a harmonic thermal bath, whose regular dynamics has the effects of rendering excessively large, as we shall see, the corrections to quantum mechanics necessary to make the Markov assumption possible. In conclusion, using again the contracted description already adopted in the previous Section, we get for the influence functional the expression: F Γ [q 1 , q 2 ] = exp − 1 h ∞ 0 dω g 2 (ω) 2m B ω dN dω T 0 dt t 0 dt ′ q 1 (t)q 1 (t ′ )e −iω(t−t ′ )−Γ(t−t ′ ) + T 0 dt t 0 dt ′ q 2 (t)q 2 (t ′ )e iω(t−t ′ )−Γ(t−t ′ ) − T 0 dt t 0 dt ′ q 1 (t)q 2 (t ′ )e iω(t−t ′ )−Γ(2T −t−t ′ ) − T 0 dt t 0 dt ′ q 2 (t)q 1 (t ′ )e −iω(t−t ′ )−Γ(2T −t−t ′ ) ,(23) where we have neglected terms of order Γ or higher related with the definition (22). Notice the presence of the two memory kernels depending on the sum of the two times t and t ′ rather than on their difference. Eq. (23) can be rewritten under a form similar to (7): F Γ [q 1 , q 2 ] = exp − 1 h T 0 dt[q 1 (t)I (Γ) R,1 (t) + q 2 (t)I (Γ) R,2 (t) − q 1 (t)J (Γ) R,2 (t) − q 2 (t)J (Γ) R,1 (t)] − ī h T 0 dt[q 1 (t)I (Γ) I,1 (t) − q 2 (t)I (Γ) I,2 (t) + q 1 (t)J (Γ) I,2 (t) − q 2 (t)J (Γ) I,1 (t)] ,(24) where I (Γ) λ,k (t) = t 0 dt ′ q k (t ′ )α (Γ) λ (t − t ′ )(25) and J (Γ) λ,k (t) = e −2Γ(T −t) I (Γ) λ,k (t),(26) with k = 1, 2 and λ = R, I. The modified kernels α I (τ ) = α I (τ )e −Γτ = − η π Ω 0 dωω sin ωτ e −Γτ ,(28) that is, neglecting terms of order Γ: α (Γ) R (τ ) = η π Ω sin Ωτ τ − 2 sin 2 (Ωτ /2) τ 2 e −Γτ(29) α (Γ) I (τ ) = η π d dτ sin Ωτ τ e −Γτ .(30) Both memory kernels decay exponentially in time, a fact that according to the earlier remarks fully legitimates the Markov behavior of the system. Let us see it in some more detail. As far as α (Γ) R is concerned, its scale becomes now τ R = ∞ 0 α (Γ) R (τ )dτ = η 2π Γ ln Γ 2 + Ω 2 Γ 2 = η 2π O(Γ),(31) that is, it is different from zero for O(Γ). In the limit Γ → 0, τ R → 0, restating the already discussed non-Markovian properties. More interesting is the case of α (Γ) I . For Γ small but finite, the Markov approximation becomes now rigorous because of the presence of the e −Γτ term. We get, for the convolution integral (25): I (Γ) I,k (t) = η π t 0 dτ d dτ sin Ωτ τ e −Γτ q k (t − τ ) = − η π Ωq k (t) + η π ∞ 0 dτ sin Ωτ τ e −Γτ q k (t) = − η π Ωq k (t) + η π arctan Ω Γ q k (t)(32) Notice that we could get this result without carrying out the limit Ω → ∞. In the limit Γ → 0, I (Γ) I,k (t) → − η π Ωq k (t) + η 2q k (t) (33) J (Γ) I,k (t) → − η π Ωq k (t) + η 2q k (t)(34) and from (24), Eq. (19) and (20) are recovered, with Ω replacing δ(0). IV. INTENSITY OF THE CORRECTIONS TO ORDINARY QUANTUM MECHANICS NECESSARY TO THE OBJECTIVE MARKOV APPROXIMATION This Section is devoted to a balance of the results obtained in this paper. To make this balance crystal clear it is convenient to warn the reader from mistaking our main purpose with that of many papers on the subject of master equation and the foundation of stochastic Schrödinger equation. To establish this difference of perspective and purposes, we find it convenient to make some comments on a representative group of papers. These are those of Refs. [31][32][33][34][35][36][37]. We have to point out that it is possible, in principle, to derive exact master equations to describe the time evolution of an open system [31]. However, in the specific case of the model of Caldeira and Leggett [21], which is the specific model studied in this letter, it is well known that the Markov approximation is responsible for the birth of unphysical effects. This difficulty has been addressed by different authors with different methods. For instance, Diosi [32] has recovered the Lindblad form [38] by adding two additional damping terms to the result of the Markov approximation. Munro and Gardiner [33] made the interesting observation that the unphysical effects might be the consequence of a transient process produced by the regression to equilibrium from the initial factorization of the system and bath's density operator. However, the reduced density matrix fails being positive semidefinite in a short-time region which is probably beyond experimental observation due to the coarse-grain time-scale approach used in its derivation. This explicit admission of resting on a coarse-graining procedure is of significant importance to stress the main aim of this letter, as we shall see in the remainder part of this Section. We have to mention finally that the important problem of unravelling the master equation so as to build up an equivalent Schrödinger equation is now being currently extended with success to the case of non-Markov master equations [34][35][36][37]. The main purpose of this paper is totally different from that of these papers, but it is much more closely related to the conceptual issues of [1] and [23]. It is probably not so easy to appreciate these differences especially because the formal structures of the stochastic Schrödinger equation of [23] might generate the false impression that these authors, and we with them, adopt a phenomenological approach rather than a more attractive derivation from ordinary quantum mechanics. First of all, we have to stress that our main thrust is on the connection between the unification of quantum and classical physics and the unification of mechanics and thermodynamics. The structure of the stochastic Schrödinger equation of Ghirardi, Pearle and Rimini [23] has been determined by the need of establishing a unified perspective embracing macroscopic classical physics. The spontaneous wave-function collapses are described by a correction to the ordinary Schrödinger equation, compatible with the Lindblad structure [38], but conceived as real correction to ordinary quantum mechanics rather than as expression of the influence of an environment. We are convinced that to consider these corrections as manifestations of the environment influence we should prove that the Lindblad structure [38] can be derived from the quantum mechanical picture of a system interacting with an environment with no conflict with ordinary quantum mechanics. This cannot be done because the Lindbland structure [38] implies a rigorously exponential decay. When it is used, as we did in this paper, as a seed of stochasticity, it certainly kills the long-time deviations from the exponential decay, in conflict with the well-known fact that the exponential-like decay regime is only admitted in an intermediate time region [19]. The conviction that stochastic processes are compatible with ordinary statistical mechanics is questionable from a conceptual perspective, even if it is especially attractive under the proviso of adopting the for all practical purposes point of view. We know that the prototype of stochastic processes is given by Brownian motion, and that the formal structure encompassing it within the usual statistical treatments is the ordinary Fokker-Plank equation. Therefore a convincing demonstration of the compatibility between ordinary quantum mechanics and stochastic processes would be given by a derivation of the Fokker-Planck equation with no statistical or coarse-graining assumption whatsoever. The ordinary approach to the Fokker-Planck equation is not only affected by statistical assumptions such as averaging on the initial conditions: If the Markoffian assumption is not made, the resulting equation of motion cannot be identified with a bona fide Fokker-Planck equation [14]. This does not rule out, of course, the possibility that a stochastic process with infinite memory might exist. However, it can be shown [39] that in this case this stochastic process turns out to be equivalent to transport equations expressed in terms of fractional derivatives, thereby departing from conventional statistical mechanics to which we would like to limit the discussion of this final Section. The problem with advocating our point of view is that of the experimental verification of the so called spontaneous fluctuations [40]. Bonci et al. [40] have recently shown that the influence of the environment might not result, in principle, in a wave-function collapse, but only in a blurring of the wave function, even if from a statistical point of view wave-function collapses turn out to be indistinguishable from decoherence processes with no collapse. A wave-function collapse is a single-system property which can be provoked by the environmental fluctuations only if these happen to be genuinely stochastic. Unfortunately, to experimentally prove this perspective it would be necessary to find cases where the effect of spontaneous fluctuations is statistically more significant than the environmental-induced fluctuations. We feel unconfortable in accepting a view where a wave-function collapse, requiring the action of a genuine stochastic process, is actually caused by our ignorance of the microscopic details, or more in general by the limitations of the human observer. However, to change a philosophical debate into a scientific issue it would be necessary to single out experimental effects, and in this sense the results of [40] are not encouraging. In fact, the toy model adopted to study quantum jumps show that the rate of the spontaneous wave-function collapses is much lower than that of the environment-induced decoherence, thereby making them virtually invisible to statistical observation: A nice price to pay to leave quantum statistical mechanics essentially unchanged. We can show, however, that in spite of the discouraging conditions concerning the experimental settlement of this problem, the adoption of our perspective can lead to the definition of new interesting theoretical problems, concerning the triggering action of stochastic processes, regardless the philosophical view of the investigator. A nice example of this kind is provided by Zurek and Paz [41]. These authors address indeed the problem of the foundation of the second principle of thermodynamics by means of the same theoretical arguments as those adopted by Zurek to address the intriguing problem of the wave-function collapse [42]. Although we do not share Zurek's view, since we think it to be impossible to derive white noise from the contraction over the environmental degrees of freedom, we find attractive the ensuing picture of how the influence of this randomness, which in our perspective would be a spontaneous fluctuations, is enhanced by the chaotic properties of the system under study. It has to be pointed out that much of the confidence of the advocates of the foundation of thermodynamics on the basis of ordinary physics rests on the identification of the correlation time τ M with the predictability time τ S . Both times are defined in the paper of Ref. [43], which has been devoted to shed light on this intriguing issue, and where it was proved that the conventional view is reinforced by the key role of the following inequality τ M ≪ τ S ≪ 1 η .(35) This is Eq. (24) of Ref. [43] with the oscillator friction denoted by η to fit the notation of this letter. If this inequality is fulfilled, then the resulting macroscopic stochastic dynamics become indistinguishable from those predicted by ordinary physics supplemented by the arbitrary assumption of identifying τ M with the predictability time τ S . Actually τ M is the lifetime of a correlation function, and this is not necessarily identical to the time at which the system under study stops being predictable. An illuminating example is given by the superposition of many harmonic oscillations with slight different frequencies. The resulting process might be characterized by a relaxation-like behavior, but it would mistifying to identify the resulting decorrelation time with the beginning of an unpredictable regime [43]. In classical physics this unpredictability time is proved to be inversely proportional to the Lyapunov coefficient. Apparently there might be a conflict between this interpretation and the fact that other investigators might identify this time with the inverse of the correlation time 1/τ M . This apparent conflict is settled by adopting the perspective of Zurek and Paz [41]: The key action of the environmental seed of irreversibility makes it possible to identify the two times, since in the case studied by Zurek and Paz τ M is slightly shorter than τ S . This is in a perfect accordance with the point of view that we are trying to illustrate here, even if are inclined to identify the seed of irreversibility with spontaneous fluctuations either of the form of those of Ghirardi, Pearle and Rimini, or of the Mensky constrained path integral studied in this letter. The main aim of this paper has been that of assessing whether or not the adoption of the Mensky path integral with constraint, as an expression of correction to the ordinary quantum dynamics of the bath oscillators, makes it possible to realize the basic condition (35). This condition, fully expressed in terms of the notations used in this paper, would read η < Γ < Ω.(36) In fact, 1/Ω plays the same role as ordinary correlation time, and the parameter Γ can be adopted to denote the rate of the Mensky process on a single oscillator. Note that the Mensky measurement process is the R-process of the perspective established by Penrose [1], and consequently the time τ S must be identified with 1/Γ, since the measurement process is by definition a genuinely stochastic process and the calculations of Section III have shown that a single bath oscillator is made unpredictable in time with the rate Γ. Note that establishing this inequality, as simple as it is, cannot be easily done without the extended calculations of Section III, and especially without the complex process of repeated trials yielding Eq. (22). Following [10], we adopt for the dissipation parameter of the oscillator, η, the value 10 9 sec −1 . This plausible choice and the inequality (36) set a minimum value for the parameter Γ, which turns out too large to be compatible with the Taylor series expansion adopted in our theoretical treatment. Furthermore, if we fix Γ = 10 10 sec −1 , m B = 10 −24 gr, Ω = 10 12 sec −1 , we find that the real part of k is equal to the value 10 25 sec −1 cm −2 . This means that the corrections made are unacceptable from the physical as well as from the mathematical point of view. In other words, the results are not selfconsistent and imply corrections to ordinary physics too large to be seriously taken into account. The spontaneous fluctuations of Ghirardi, Pearle and Rimini [23] lead to much smaller corrections, and consequently might be regarded as being in principle an acceptable corrections. However, as established in [10], in spite of the enhancement produced by the interplay with the ordinary fluctuation-dissipation process, these fluctuations fail producing the objective wave-function collapse in a reasonably short time, thereby sharing the weakness of the Mensky constrained path integral studied in this letter. According to the perspective here adopted, that the Markov assumption corresponds to tacitly correcting quantum mechanics, this conclusion seems to be equivalent to establishing that the Markov approximation made in literature are unacceptable. However, suggesting that all master equations based on this assumption, widely used in literature, cannot be trusted, would be a dramatic conclusion. We are convinced that the results of this letter lead to a rather different conclusion that will be illustrated at the end as the main result of this paper. Before reaching this conclusion, we want to caution the reader from mistaking this conclusion for the discovery that the Markov assumption can break the condition that the density matrix is positive definite [32,33]. We cannot rule out completely the possibility that the perspective of this paper can be used to shed light on some issues raised, for instance, by Munro and Gardiner [33] 2 . However, our thrust focuses on the different issue widely illustrated in this Section. We are convinced that the results of this letter leads to a conclusion different from stating the breakdown of the master equations. The master equations, supplemented in case with the improvements established by [32] and [33], might lead to correct result, in spite of the fact that Eq. (36) seems to be unphysical. It has to be remarked that the approach to statistical mechanics here adopted is not genuinously dynamic: the bath of oscillators is linear and it is arbitrarily forced to stay in a canonical equilibrium condition. For this reason the conventional assumption that the bath oscillators are in a state of canonical equilibrium has to be replaced by dynamical ingredients (see, for instance, the discussion made in recent literature [44,45]). This means that each bath oscillator must be assigned its own bath, and this bath must be non-linear so as to result, in the classical limit, in deterministic chaos. This is expected to lead to the fulfillment of Eq. (36) without internal inconsistencies, thereby explaining why, after all, the linear baths of literature can be profitably used with no conflict with reality. This is a plausible conjecture that has to be proved by a direct use of a nonlinear bath. There are already outstanding examples of dynamic approaches to quantum statistical mechanics, based on the so called quantum chaos [46,47]. We note that within this context it has already been noticed [41] that a chaotic dynamic system leads to a significant increase of the Gibbs entropy in times of the order of the inverse of the Lyapunov coefficients, provided that a seed of genuine irreversibility is introduced as an effect of environmental fluctuations. As earlier said, this fits very well the perspective established by the present paper. The dependence on these environmental fluctuations is through the logarithm of their intensity, thereby implying no significant change to ordinary quantum mechanics. If we interpret these environmental fluctuations as weak corrections to ordinary quantum mechanics, of the same type as that here studied, we might reach the satisfactory conclusion that a non arbitrary Markov assumption can be realized with extremely weak corrections to ordinary quantum mechanics. The results of this paper therefore must be interpreted as an incentive to study directly, from now on, the influence that white fluctuations have on dynamic systems which would be strongly chaotic in the classical limit. 2 We refer here to some key equations of Ref. [33]. It seems to us that the short-time expansion of the density matrix terms of Eqs. (3.3) and (3.5) with the joint use of (3.11) might lead to a breakdown of the property of the density matrix to be positive definite in a time region more extended than the correlation time of Eq. (2.20). This seems to support our view that the correlation time τ M , identified by us with the time of Eq. (2.20), cannot be regarded as the true coarse-graining time. Averages should be made on times more extended, as indicated by Eq. (36) of this letter. Of course, after completing the calculation of Section III, yielding a Markov dissipation, we make the implicit assumption that the fluctuations are properly decorrelated by the Mensky constraint R Penrose, Shadows of the Mind -A Search for the Missing Science of Consciousness. OxfordOxford University PressR. Penrose, Shadows of the Mind -A Search for the Missing Science of Consciousness, Oxford University Press, Oxford, 1994. . A Barchielli, G Lupieri, J. Math. Phys. 262222A. Barchielli and G. Lupieri, J. Math. Phys. 26 (1985) 2222. . A Barchielli, Phys. Rev. A. 341642A. Barchielli, Phys. Rev. A 34 (1986) 1642. . V P Belavkin, Phys. Lett. A. 140355V.P. Belavkin, Phys. Lett. A 140 (1989) 355; . J. Phys. A: Math. and Gen. 221109J. Phys. A: Math. and Gen. 22 (1989) L1109. R P Feynman, A R Hibbs, Quantum Mechanics and Path Integrals. McGraw-Hill, NYR.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, NY, 1965. . M Mensky, Phys. Lett. 196159M. Mensky, Phys. Lett. A196 (1994) 159. . C Presilla, R Onofrio, U Tambini, Ann. Phys. 24895C. Presilla, R. Onofrio and U.Tambini, Ann. Phys. 248 (1996) 95. . C Presilla, R Onofrio, M Patriarca, J. Phys. 307385C. Presilla, R. Onofrio and M. Patriarca, J. Phys. A30 (1997) 7385. . R P Feynman, F L Vernon, Ann, Phys. 24118R.P. Feynman and F.L. Vernon, Ann, Phys. 24 (1963) 118. . L Tessieri, D Vitali, P Grigolini, Phys. Rev. A. 514404L. Tessieri, D. Vitali, P. Grigolini, Phys. Rev. A 51 (1995) 4404. . S A Adelman, J. Chem. Phys. 64124S.A. Adelman, J. Chem. Phys. 64 (1976) 124. . H Mori, Prog. Theor. Phys. 33423H. Mori, Prog. Theor. Phys. 33 (1965) 423. . H Mori, Prog. Theor. Phys. 34399H. Mori, Prog. Theor. Phys. 34 (1965) 399. . R F Fox, J. Math. Phys. 182331R.F. Fox, J. Math. Phys. 18 (1977) 2331. . H Kleinert, S V Shabanov, Phys. Lett. A. 200224H. Kleinert, S.V. Shabanov, Phys. Lett. A 200 (1995) 224. . L Diósi, W T Strunz, quant-ph/9706050L. Diósi and W.T. Strunz, quant-ph/9706050. . D Vitali, P Grigolini, Phys. Rev. 391486D. Vitali and P. Grigolini, Phys. Rev. A39 (1989) 1486. . M H Lee, Phys. Rev. Lett. 511227M.H. Lee, Phys. Rev. Lett. 51 (1983) 1227. . L Fonda, G C Ghirardi, A Rimini, Rep. Prog. Phys. 41587L. Fonda, G.C. Ghirardi and A. Rimini, Rep. Prog. Phys. 41 (1978) 587. . A Mazza, P Grigolini, Phys. Lett. A. 238169A. Mazza, P. Grigolini, Phys. Lett. A 238 (1998) 169. . A O Caldeira, A J Leggett, Physica A. 121587A.O. Caldeira and A.J. Leggett, Physica A 121 (1983) 587. . M B Mensky, Phys. Lett. A. 231M. B. Mensky, Phys. Lett. A 231 (1997). . G C Ghirardi, P Pearle, A Rimini, Phys. Rev. A. 4278G.C. Ghirardi, P. Pearle, and A. Rimini, Phys. Rev. A 42 (1990) 78. . A O Caldeira, A J Leggett, Ann. of Phys. 149374A.O. Caldeira and A.J. Leggett, Ann. of Phys. 149 (1983) 374. . K Lindenberg, B J West, Phys. Rev. 30568K. Lindenberg, B.J. West, Phys. Rev. A30 (1984) 568. . B L Hu, J P Paz, Y Zhang, Phys. Rev. D. 452843B.L. Hu, J.P. Paz and Y. Zhang, Phys. Rev. D 45 (1992) 2843. . H Grabert, P Schramm, G.-L Ingold, Phys. Rep. 168115H. Grabert, P. Schramm and G.-L. Ingold, Phys. Rep. 168 (1988) 115. . M Patriarca, Nuovo Cimento B. 11161M. Patriarca, Nuovo Cimento B 111 (1996) 61. . P Grigolini, Adv. Chem. Phys. 621P. Grigolini, Adv. Chem. Phys. 62 (1985) 1. . E T Jaynes, Phys. Rev. 106620E.T. Jaynes, Phys. Rev. 106 (1957) 620; . Phys. Rev. 108171Phys. Rev. 108 (1957) 171. . R R Puri, S V Lawande, Phys. Lett. A. 62143R.R. Puri and S.V. Lawande, Phys. Lett. A 62 (1977) 143. . L Diosi, Physica A. 199517L. Diosi, Physica A 199 (1993) 517. . W J Munro, C W Gardiner, Phys. Rev A. 532633W.J. Munro and C.W. Gardiner, Phys. Rev A 53 (1996) 2633. . W T Strunz, Phys. Lett. A. 22425W.T. Strunz, Phys. Lett. A 224 (1996) 25. . L Diosi, N Gisin, W T Strunz, Phys. Rev. A. 581699L. Diosi, N. Gisin, and W.T. Strunz, Phys. Rev. A 58 (1998) 1699. . H P Breuer, B Kappler, F Petruccione, quant-ph/9806026H.P. Breuer, B. Kappler, and F. Petruccione, quant-ph/9806026. . M W Jack, M J Collett, D F Walls, quant-ph/9807028M.W. Jack, M.J. Collett, and D.F. Walls, quant-ph/9807028. . G Lindblad, Comm. Math. Phys. 48119G. Lindblad, Comm. Math. Phys. 48 (1976) 119. . P Grigolini, A Rocco, B J West, cond-mat/9809075P. Grigolini, A. Rocco, B.J. West, cond-mat/9809075. . L Bonci, P Grigolini, G Morabito, L Tessieri, D Vitali, Phys. Lett. A. 209129L. Bonci, P. Grigolini, G. Morabito, L. Tessieri, and D. Vitali, Phys. Lett. A 209 (1995) 129. . W H Zurek, J P Paz, Phys. Rev. Lett. 722508W.H. Zurek and J.P. Paz, Phys. Rev. Lett. 72 (1994) 2508. . W H Zurek, Physics Today. 441087W.H. Zurek, Physics Today 44 (10) (1984) 87. . V Giovannetti, P Grigolini, G Tesi, D Vitali, Phys. Lett. A. 22431V. Giovannetti, P. Grigolini, G. Tesi and D. Vitali, Phys. Lett. A 224 (1996) 31. . M Bianucci, R Mannella, B J West, P Grigolini, Phys. Rev. E. 513002M. Bianucci, R. Mannella, B.J. West, P. Grigolini, Phys. Rev. E 51, (1995) 3002. . Wm G Hoover, Computational Statistical Mechanics. ElsevierWm.G. Hoover, Computational Statistical Mechanics, Elsevier, Amsterdam, 1991. . M Srednicki, Phys. Rev. E. 50888M. Srednicki, Phys. Rev. E 50 (1994) 888. . V V Flambaun, F M Izrailev, Phys. Rev. E. 565144V.V. Flambaun and F.M. Izrailev, Phys. Rev. E 56 (1997) 5144.
[]
[ "Neutralino, axion and axino cold dark matter in minimal, hypercharged and gaugino AMSB", "Neutralino, axion and axino cold dark matter in minimal, hypercharged and gaugino AMSB" ]
[ "Howard Baer [email protected] \nDept. of Physics and Astronomy\nUniversity of Oklahoma\n73019NormanOKUSA\n\nIntroduction\n\n", "Radovan Dermíšek \nDept. of Physics\nIndiana University\n47405BloomingtonINUSA\n", "† ", "Shibi Rajagopalan \nDept. of Physics and Astronomy\nUniversity of Oklahoma\n73019NormanOKUSA\n\nIntroduction\n\n", "‡ ", "Heaya Summy \nDept. of Physics and Astronomy\nUniversity of Oklahoma\n73019NormanOKUSA\n\nIntroduction\n\n" ]
[ "Dept. of Physics and Astronomy\nUniversity of Oklahoma\n73019NormanOKUSA", "Introduction\n", "Dept. of Physics\nIndiana University\n47405BloomingtonINUSA", "Dept. of Physics and Astronomy\nUniversity of Oklahoma\n73019NormanOKUSA", "Introduction\n", "Dept. of Physics and Astronomy\nUniversity of Oklahoma\n73019NormanOKUSA", "Introduction\n" ]
[]
Supersymmetric models based on anomaly-mediated SUSY breaking (AMSB) generally give rise to a neutral wino as a WIMP cold dark matter (CDM) candidate, whose thermal abundance is well below measured values. Here, we investigate four scenarios to reconcile AMSB dark matter with the measured abundance: 1. non-thermal wino production due to decays of scalar fields (e.g. moduli), 2. non-thermal wino production due to decays of gravitinos, 3. non-thermal wino production due to heavy axino decays, and 4. the case of an axino LSP, where the bulk of CDM is made up of axions and thermally produced axinos. In cases 1 and 2, we expect wino CDM to constitute the entire measured DM abundance, and we investigate wino-like WIMP direct and indirect detection rates. Wino direct detection rates can be large, and more importantly, are bounded from below, so that ton-scale noble liquid detectors should access all of parameter space for m Z 1 < ∼ 500 GeV. Indirect wino detection rates via neutrino telescopes and space-based cosmic ray detectors can also be large. In case 3, the DM would consist of an axion plus wino admixture, whose exact proportions are very model dependent. In this case, it is possible that both an axion and a wino-like WIMP could be detected experimentally. In case 4., we calculate the re-heat temperature of the universe after inflation. In this case, no direct or indirect WIMP signals should be seen, although direct detection of relic axions may be possible. For each DM scenario, we show results for the minimal AMSB model, as well as for the hypercharged and gaugino AMSB models. *
10.1088/1475-7516/2010/07/014
[ "https://arxiv.org/pdf/1004.3297v2.pdf" ]
55,757,475
1004.3297
8d56ffcba905e2f6005290c42b55a9017a4581df
Neutralino, axion and axino cold dark matter in minimal, hypercharged and gaugino AMSB 18 Jun 2010 Howard Baer [email protected] Dept. of Physics and Astronomy University of Oklahoma 73019NormanOKUSA Introduction Radovan Dermíšek Dept. of Physics Indiana University 47405BloomingtonINUSA † Shibi Rajagopalan Dept. of Physics and Astronomy University of Oklahoma 73019NormanOKUSA Introduction ‡ Heaya Summy Dept. of Physics and Astronomy University of Oklahoma 73019NormanOKUSA Introduction Neutralino, axion and axino cold dark matter in minimal, hypercharged and gaugino AMSB 18 Jun 2010 Supersymmetric models based on anomaly-mediated SUSY breaking (AMSB) generally give rise to a neutral wino as a WIMP cold dark matter (CDM) candidate, whose thermal abundance is well below measured values. Here, we investigate four scenarios to reconcile AMSB dark matter with the measured abundance: 1. non-thermal wino production due to decays of scalar fields (e.g. moduli), 2. non-thermal wino production due to decays of gravitinos, 3. non-thermal wino production due to heavy axino decays, and 4. the case of an axino LSP, where the bulk of CDM is made up of axions and thermally produced axinos. In cases 1 and 2, we expect wino CDM to constitute the entire measured DM abundance, and we investigate wino-like WIMP direct and indirect detection rates. Wino direct detection rates can be large, and more importantly, are bounded from below, so that ton-scale noble liquid detectors should access all of parameter space for m Z 1 < ∼ 500 GeV. Indirect wino detection rates via neutrino telescopes and space-based cosmic ray detectors can also be large. In case 3, the DM would consist of an axion plus wino admixture, whose exact proportions are very model dependent. In this case, it is possible that both an axion and a wino-like WIMP could be detected experimentally. In case 4., we calculate the re-heat temperature of the universe after inflation. In this case, no direct or indirect WIMP signals should be seen, although direct detection of relic axions may be possible. For each DM scenario, we show results for the minimal AMSB model, as well as for the hypercharged and gaugino AMSB models. * Introduction Supersymmetric (SUSY) models of particle physics are very attractive in that they stabilize the gauge hierarchy problem, and provide an avenue for the incorporation of gravity via local SUSY, or supergravity [1]. They also receive some indirect experimental support via the unification of gauge couplings under Minimal Supersymmetric Standard Model (MSSM) renormalization group evolution (RGE) [2], and they provide several different candidates (neutralinos, gravitinos, axions/axinos, · · ·) which can serve as cold dark matter (CDM) in the universe. If evidence for SUSY is found at LHC, then a paramount question will be: what is the mechanism of SUSY breaking, and how is it communicated to the visible sector? Some of the possibilities proposed in the literature include: gravity-mediation (SUGRA) with a gravitino mass m 3/2 ∼ 1 TeV [4], gauge-mediation (GMSB) with m 3/2 ≪ 1 TeV [5], and anomaly mediation (AMSB) with m 3/2 ∼ 100 TeV [6,7,8]. Anomaly-mediated supersymmetry breaking (AMSB) models have received much attention in the literature due to their attractive properties [6,7]: 1. the soft supersymmetry (SUSY) breaking terms are completely calculable in terms of just one free parameter (the gravitino mass, m 3/2 ), 2. the soft terms are real and flavor invariant, thus solving the SUSY flavor and CP problems and 3. the soft terms are actually renormalization group invariant [9], and can be calculated at any convenient scale choice. In order to realize the AMSB set-up, it was proposed that the hidden sector be "sequestered" on a separate brane from the observable sector in an extra-dimensional universe, so that tree-level supergravity breaking terms do not dominate the soft term contributions. Such a set-up can be realized in brane-worlds, where SUSY breaking takes place on one brane, with the visible sector residing on a separate brane. A further attractive feature of AMSB models arises due to the scale of their gravitino mass. SUGRA-type models with m 3/2 ∼ 1 TeV suffer from the cosmological gravitino problem. There are two parts to the gravitino problem [10]. 1. If the re-heat temperature after inflation T R > ∼ 10 10 GeV, then the high rate of thermal gravitino production leads to an overabundance of neutralino dark matter [11]. 2. Even for lower values of T R ∼ 10 5 −10 10 GeV, thermal production ofG followed by late decays to particle + sparticle pairs injects high energy particles into the cosmic soup during or after BBN, thus disrupting one of the pillars of Big-Bang theory [11]. If m 3/2 > ∼ 5 TeV, then the lifetime τG drops below 0.1 − 1 sec, and gravitino decay occurs before or at the onset of BBN. In AMSB models where m 3/2 ∼ 100 TeV, the gravitino is much too short-lived to be afflicted by the BBN bounds. In spite of their attractive features, AMSB models suffer from the well-known problem that slepton mass-squared parameters are found to be negative, giving rise to tachyonic states. The original "solution" to this problem was to posit that scalars acquire as well a universal mass m 0 , which when added to the AMSB SSB terms, renders them positive [6,7]. The derived form of soft SUSY breaking terms, supplemented by a universal scalar mass m 0 and implemented at the GUT scale, constitutes what is usually called the minimal AMSB, or mAMSB model. In mAMSB and the additional models described below, it is assumed that electroweak symmetry is broken radiatively due to the large top quark mass, so that the magnitude of the µ parameter is determined to gain the correct value of M Z , and the bilinear soft term B is traded for the ratio of Higgs field vevs, tan β. An alternative set-up for AMSB has been advocated in Ref. [12], known as hypercharged anomaly-mediation (HCAMSB). It is a string-motivated scenario which uses a similar setup as the one envisioned for AMSB. In HCAMSB, the MSSM resides on a D-brane, and the hypercharge gaugino mass is generated in a geometrically separated hidden sector. An additional contribution to the U(1) Y gaugino mass M 1 is generated, and its magnitude is parametrized by an additional parameter α. The large value of M 1 feeds into slepton mass evolution through the MSSM RGE, and acts to lift the weak-scale slepton soft masses beyond tachyonic values. Thus, the HCAMSB model naturally solves the tachyonic slepton mass problem which is endemic to pure AMSB scenarios. A third scenario has recently been proposed in Ref. [13], under the name gaugino AMSB, or inoAMSB. The inoAMSB model is suggested by recent work on the phenomenology of flux compactified type IIB string theory [14], which reduces to N = 1 supergravity below the compactification scale. The essential features of this scenario are that the gaugino masses are of the anomaly-mediated SUSY breaking (AMSB) form, while scalar and trilinear soft SUSY breaking terms are highly suppressed: they are taken as m 0 = A 0 ≃ 0 at energy scale Q ∼ M GU T , at first approximation. The normally large value of M 1 as generated in AMSB models feeds into the scalar soft term evolution, lifting slepton soft masses to generate an allowable sparticle mass spectrum, while at the same time avoiding tachyonic sleptons or charged LSPs (lightest SUSY particles). Charged LSPs are common in models with negligible soft scalar masses, such as no-scale [15] or gaugino mediation models [16]. Since scalar and trilinear soft terms are highly suppressed, the SUSY induced flavor and CP -violating processes are also suppressed in inoAMSB. All three of these models-mAMSB, HCAMSB and inoAMSB-share the common feature that the lightest MSSM particle is a neutral wino, while the lightest chargino is wino-like with a mass m W 1 ∼ m Z 1 . The W 1 -Z 1 mass gap is of order ∼ 200 MeV [17], so that dominantly W ± 1 → Z 1 π ± , with the decay-produced pion(s) being very soft. The small mass gap makes the W 1 rather long lived (τ W 1 ∼ 10 −9 sec), and it may yield observable highly ionizing tracks (HITs) of order cm in length at LHC detectors [32]. An important consequence of wino-like neutralinos is that the thermal abundance of neutralino cold dark matter falls generally an order of magnitude or so below the measured abundance: Ω CDM h 2 = 0.1123 ± 0.0035 68% CL(1) according to the WMAP7 data analysis [18]. This latter fact has led many to consider AMSBlike models as perhaps less interesting than SUGRA-type models, wherein the bino-like or mixed bino-higgsino neutralino can more easily yield the measured relic abundance. In this paper, we address the question of the dark matter abundance in AMSB models. While the calculated thermal abundance of wino-like neutralinos is found to be below measured values (for m Z 1 < ∼ 800 GeV), we find that there exists a variety of attractive methods to augment the wino abundance, thus bringing the calculated abundance into accord with experiment. These include: 1. Decay of scalar (e.g. moduli) fields into sparticles, ultimately terminating in Z 1 production [19]. In this case, the LSP is expected to be a relic wino-like neutralino, which would constitute all of the CDM. 2. Thermal production [20,21] of gravitinosG and also possibly gravitino production via moduli [22] or inflaton [23] decay, followed byG → particle + sparticle → Z 1 + X (where X = assorted SM debris). Here also, the LSP would be a relic wino-like neutralino, which would constitute all of the CDM. 3. Thermal production of heavy axinos [25] followed byã → particle + sparticle → Z 1 + X [26]. Here, the LSP is again a relic wino-like neutralino, but the CDM would consist of a wino-like WIMP plus axion mixture. 4. A scenario where mã < m Z 1 , so theã is instead the stable LSP [24]. In this case, a combination of thermally produced axinos plus vacuum mis-alignment produced axions would constitute the CDM [28,29]. In Sec. 2, we present some details of the three AMSB models which we investigate. In Sec. 3, we present four methods of reconciling the AMSB CDM relic abundance with the measured value. Given cases 1 and 2, and possibly 3, where we expect all (or some fraction) of the measured abundance to consist of relic winos, in Sec. 4 we present rates for direct and indirect detection of wino-like neutralinos. Unlike SUGRA models, the wino CDM direct detection rate is bounded from below. We find the current experiments like Xenon-100 should be able to explore the parameter space of AMSB-like models with a wino-like neutralino up to m Z 1 < ∼ 200 GeV. Next generation detectors such as ton-scale noble liquids or SuperCDMS should be able to push to m Z 1 ∼ 500 GeV. This would correspond to a reach in mg ∼ 3850 GeV, i.e. well beyond the projected reach of LHC. We also find excellent prospects for indirect detection of wino-like CDM via detection of wino annihilation into γs, e + s,ps orDs in the galactic halo. In fact, Kane et al. have already proposed wino CDM as an explanation for the recent anomalies seen by Pamela, ATIC, Fermi and others [31]. Neutrino telescopes such as IceCube will also have a reach for wino-like neutralinos, especially for large tan β. In case 4, we would expect the axinoã to be the LSP, and so here no direct or indirect WIMP detection signals are expected. However, it may be the case that large amounts of axions a are produced in the early universe, in which case direct detection of axions may be possible at ADMX [53]. We present parameter expectations in Sec. 3.4 for the scenario of mixed axion/axino CDM to occur. In Sec. 5, we present a summary and our conclusions. 2 Overview of mAMSB, HCAMSB and inoAMSB models Minimal AMSB The AMSB contribution to the gaugino mass is given by, M i = β g i g i m3 2 = b i g 2 i 16π 2 m 3/2 ,(2) where β i is the corresponding beta function, defined by β g i ≡ dg i /d ln µ. The constants b i = (33/5, 1, −3) for i = 1 − 3. The AMSB contribution to soft SUSY breaking scalar masses is given by, m 2 f = − 1 4 dγ dg β g + dγ df β f m 2 3 2(3) where β f is the β-function for the corresponding superpotential Yukawa coupling, and γ = ∂ ln Z/∂ ln µ, with Z the wave function renormalization constant. Complete expressions for the MSSM can be found e.g. in Ref's [1,32]. Since these give rise to tachyonic slepton masses, each term is supplemented by +m 2 0 , where m 0 is some additional universal contribution to scalar masses. Finally, the anomaly-mediated contribution to the trilinear SUSY breaking scalar coupling is given by, A f = β f f m3 2 ,(4) where f labels the appropriate Yukawa coupling (e.g. f t is the top-quark Yukawa coupling). Thus, the parameter space of the "minimal" AMSB model (mAMSB) is given by [32] m 0 , m 3/2 , tan β, sign(µ) (mAMSB). In the mAMSB model, we assume as usual that electroweak symmetry is broken radiatively by the large top-quark Yukawa coupling. Then the SSB B term and the superpotential µ term are given as usual by the scalar potential minimization conditions which emerge from requiring an appropriate breakdown of electroweak symmetry. The above expressions for the soft SUSY breaking terms are usually imposed as GUT-scale boundary conditions, and weak scale values are calculated via renormalization group evolution. Hypercharged AMSB In HCAMSB, SUSY breaking is localized at the bottom of a strongly warped hidden region, geometrically separated from the visible region where the MSSM resides. The warping suppresses contributions due to tree-level gravity mediation [33] and the anomaly mediation can become the dominant source of SUSY breaking in the visible sector. Possible exceptions to this sequestering mechanism are gaugino masses of U(1) gauge symmetries [34]. Thus, in the MSSM, the mass of the bino-the gaugino of U(1) Y -can be the only soft SUSY breaking parameter not determined by anomaly mediation [12]. Depending on its size, the bino mass M 1 can lead to a small perturbation to the spectrum of anomaly mediation, or it can be the largest soft SUSY breaking parameter in the visible sector: as a result of RG evolution, its effect on other soft SUSY breaking parameters can dominate the contribution from anomaly mediation. We parametrize the HCAMSB SSB contributionM 1 using a dimensionless quantity α such thatM 1 = αm 3/2 ; then, α governs the size of the hypercharge contribution to soft terms relative to the AMSB contribution [35]. The soft SUSY breaking terms are then exactly the same as in mAMSB, except there is no m 2 0 contribution to scalar masses, and the U(1) Y gaugino mass is given by M 1 = α + b 1 g 2 1 16π 2 m 3/2 ,(6) so that α = 0 takes us back to pure AMSB soft terms, with their concommitant tachyonic sleptons. Then the parameter space of HCAMSB models is given by α, m 3/2 , tan β, sign(µ) (HCAMSB),(7) where the dimensionless α typically ranges between 0.01 − 0.2 for allowable spectra [35]. While the lightest neutralino is mainly wino-like in mAMSB and HCAMSB models, it is important phenomenologically to realize that for certain regions of parameter space, the Z 1 picks up a substantial higgsino component, thus becoming a mixed wino/higgsino particle. This occurs in mAMSB at large m 0 values, and in HCAMSB at large α values. The situation is shown in Fig. 1 for m 3/2 = 50 TeV, tan β = 10 and µ > 0, where we show RW ≡ v (1) 3 (the wino component of Z 1 in the notation of Ref. [1]) and Rh ≡ v (1)2 1 + v (1)2 2 (the higgsino component of Z 1 ) as a function of a). m 0 in mAMSB and b). α for HCAMSB. We see indeed that in both these cases, Rh is increasing as the relevant parameter increases. Gaugino AMSB Phenomenologically viable versions of string theory require the stabilization of all moduli fields as well as weak to intermediate scale supersymmetry breaking. Models satisfying these criteria were first developed in the context of type IIB string theory using flux compactifications and non-perturbative effects on Calabi-Yau orientifolds (CYO's) [36]. The low energy limit of type-IIB string theory after compactification on a CYO is expected to be N = 1 supergravity (SUGRA). Two classes of the above models which yield an interesting supersymmetry breaking scenario have been studied: a) Those with only a single Kähler modulus (SKM models). These are essentially of the KKLT type [37] but with uplift coming from one-loop quantum effects. b) Large Volume Scenario (LVS) [38] models which require at least two moduli. In both of these types of models, the moduli fields are stabilized using a combination of fluxes and non-perturbative effects. Additionally, supersymmetry is broken by the moduli fields acquiring non-zero F-terms and interacting gravitationally with the MSSM. For both models, the gauginos acquire mass predominately through the Weyl anomaly while the classical contribution to the scalar masses and trilinear coupling constants are naturally suppressed. Here, we take the limit where scalar and trilinear soft breaking parameters are exactly zero at the GUT scale: m 0 = A 0 = 0, while gaugino masses are of the AMSB form. The parameter space of inoAMSB models is then given by m 3/2 , tan β, sign(µ) (inoAMSB).(8) As shown in Ref. [13], the inoAMSB model solves the problem of tachyonic scalars in AMSB, since now the GUT scale scalar masses vanish. It also solves the problem of charged LSPs which is endemic to no-scale SUGRA or gaugino-mediated SUSY breaking (inoMSB) models, which have m 0 = A 0 = 0 but with universal gaugino masses equal to m 1/2 . For inoAMSB models-with scalar masses m 0 = 0 at the GUT scale-the large GUT-scale U(1) Y gaugino mass M 1 pulls all scalar masses to large values, leaving no tachyons and a wino-like neutralino as the lightest MSSM particle. Thermally produced wino CDM in AMSB models The above mAMSB and HCAMSB models have been included into the Isasugra subprogram of the event generator Isajet [39]. In addition, sparticle mass spectra for the inoAMSB model can easily be generated using usual mSUGRA input parameters with m 0 = A 0 = 0, but with non-universal gaugino masses as specified by AMSB models. After input of mAMSB, HCAMSB or inoAMSB parameters, Isasugra then implements an iterative procedure of solving the MSSM RGEs for the 26 coupled renormalization group equations, taking the weak scale measured gauge couplings and third generation Yukawa couplings as inputs, as well as the above-listed GUT scale SSB terms. Isasugra implements full 2-loop RG running in the DR scheme, and minimizes the RG-improved 1-loop effective potential at an optimized scale choice Q = √ mt L mt R [40] to determine the magnitude of µ and m A . All physical sparticle masses are computed with complete 1-loop corrections, and 1-loop weak scale threshold corrections are implemented for the t, b and τ Yukawa couplings [41]. The off-set of the weak scale boundary conditions due to threshold corrections (which depend on the entire superparticle mass spectrum), necessitates an iterative up-down RG running solution. The resulting superparticle mass spectrum is typically in close accord with other sparticle spectrum generators [42]. Once the weak scale sparticle mass spectrum is known, then sparticle annihilation cross sections may be computed. To evaluate the thermally produced neutralino relic density, we adopt the IsaReD program [43], which is based on CalcHEP [44] to compute the several thousands of neutralino annihilation and co-annihilation Feynman diagrams. Relativistic thermal averaging of the cross section times velocity is performed [45]. As an example, in Fig. 2, we show the thermally produced neutralino relic density Ω Z 1 h 2 versus m 3/2 for all three models: mAMSB, HCAMSB and inoAMSB. We take tan β = 10 and µ > 0. For mAMSB, we also take m 0 = 0.01m 3/2 , and for HCAMSB, we take α = 0.02. On the upper axis, we also indicate the corresponding values of mg. We see that the relic abundance is typically well below WMAP7-measured levels, until m 3/2 > ∼ 450 TeV, corresponding to mg > ∼ 8 TeV, and m Z 1 > ∼ 1.3 TeV: well beyond any conceivable LHC reach. The well-known tiny relic abundance arises due to the large Z 1 Z 1 → W + W − annihilation and also Z 1 W 1 and W 1 W 1 co-annihilation processes. GeV. For mAMSB, we also take m 0 = 0.01m 3/2 GeV, and for HCAMSB, we take α = 0.02. 3 Dark matter scenarios for AMSB models Neutralino production via moduli decay Shortly after the introduction of AMSB models, Moroi and Randall proposed a solution to the AMSB dark matter problem based on augmented neutralino production via the decays of moduli fields in the early universe [19]. The idea here is that string theory is replete with additional moduli fields: neutral scalar fields with gravitational couplings to matter. In generic supergravity theories, the moduli fields are expected to have masses comparable to m 3/2 . When the Hubble expansion rate becomes comparable to the moduli mass m φ , then an effective potential will turn on, and the moduli field(s) will oscillate about their minima, producing massive excitations, which will then decay to all allowed modes: e.g. gauge boson pairs, higgs boson pairs, gravitino pairs, · · ·. The neutralino production rate via moduli decay has been estimated in Ref. [19]. It is noted in Ref. [46] that the abundance-given by Ω Z 1 h 2 ∼ 0.1 × m Z 1 100 GeV 10.75 g * 1/4 σ 0 σv 100 TeV m φ 3/2(9) with σ 0 = 3 × 10 −24 cm 3 /sec-yields nearly the measured dark matter abundance for winolike neutralino annihilation cross sections and m φ ∼ 100 TeV. 1 These authors dub this the "non-thermal WIMP miracle". A necessary condition for augmented neutralino production via scalar field decay is that the re-heat temperature of radiation T R induced by moduli decays is bounded by T R > ∼ 5 MeV (in order to sustain Big Bang Nucleosynthesis (BBN) as we know it), and T R < T f o , where T f o is the freeze-out temperature for thermal neutralino production T f o ∼ m Z 1 /20. If T R exceeds T f o , then the decay-produced neutralinos will thermalize, and the abundance will be given by the thermal calculation as usual. This "low re-heat" neutralino production mechanism has been investigated extensively by Gondolo and Gelmini [47]. The low re-heat neutralino abundance calculation depends on the input value of T R and the ratio b/m φ , where b is the average number of neutralinos produced in moduli decay, and m φ is the scalar field mass. They note that theories with an underabundance of thermally produced neutralino CDM with Ω T P Z 1 > ∼ 10 −5 100 GeV m Z 1 can always be brought into accord with the measured DM abundance for at least one and sometimes two values of T R . 2 While the low T R ∼ 10 − 1000 MeV scenario with DM generation via scalar field decay is compelling, we note here that it is also consistent with some baryogenesis mechisms: e.g. Affleck-Dine baryogenesis wherein a large baryon asymmetry is generated early on, only to be diluted to observable levels via moduli decay [48], or a scenario wherein the baryon asymmetry is actually generated by the moduli decay [49]. 1 In inoAMSB models, we expect moduli with SUSY breaking scale masses, m φ ∼ m 3/2 / √ V ≪ m 3/2 , where V is the (large) volume of the compactified manifold: V ∼ 10 5 in Planck units. In this case, the mechanism would not so easily apply. 2 Ref. [47] also shows that an overabundance of thermally produced neutralino CDM can also be brought into accord with the measured abundance via dilution of the neutralino number density by entropy injection from the φ field decay. Since this case doesn't attain in AMSB models (unless m Z1 > ∼ 1300 GeV), we will neglect it here. Neutralino production via gravitino decay An alternative possibility for augmenting the production of wino-like neutralinos in AMSB models is via gravitino production and decay in the early universe. While gravitinos would not be in thermal equilibrium during or after re-heat, they still can be produced thermally via radiation off ordinary sparticle scattering reactions in the early universe. The relic density of thermally produced gravitinos as calculated in Ref's [50,21] is given by Ω T P G h 2 = 3 i=1 ω i g 2 i   1 + M 2 i 3m 2 3/2   log k i g i m 3/2 100 GeV T R 10 10 GeV ,(10) where g i and M i are the gauge couplings and gaugino masses evaluated at scale Q = T R , and ω i = (0.018, 0.044, 0.117) and k i = (1.266, 1.312, 1.271). Each gravitino ultimately cascade decays down to the wino-like Z 1 state, so the neutralino relic density is given by Ω Z 1 h 2 = Ω T P Z 1 h 2 + m Z 1 m 3/2 Ω T P G h 2 .(11) A plot of the value of T R and m 3/2 which is required to yield Ω Z 1 h 2 = 0.11 from Eq'n 11 is shown in Fig. 3 for mAMSB (m 0 = 0.01m 3/2 ), HCAMSB (α = 0.02) and inoAMSB using tan β = 10 and µ > 0. The region above the Ω Z 1 h 2 = 0.11 curves would yield too much dark matter, while the region below the curves yields too little. We should consider the curves shown in Fig. 3 as only indicative of the simplest scenario for wino production via gravitino decay. Three other effects can substantially change the above picture from what is presented in Eq. 11. • On the one hand, if moduli fields φ m exist with mass m φm > 2m 3/2 , then gravitinos can also be produced via moduli production and decay [22]. The exact abundance of these moduli-produced gravitinos is very model dependent, and depends on the moduli and gravitino mass and branching fractions. • A second case arises if we consider gravitino production via inflaton decay at the end of inflation [23]. This production mechanism depends on unknown properties of the inflaton: e.g. its mass and branching fractions, and the re-heat temperature generated by inflaton decay. These latter quantities are very model dependent. • Additional entropy production generated via the inflaton, moduli and gravitino decays may also dilute the above relic abundance in Eq. 11. We will bear in mind that these possibilities permit much lower or much higher values of T R and m 3/2 than those shown by the Ω Z 1 h 2 = 0.1 contour of Fig. 3. Neutralino production via heavy axino decay A third mechanism for increasing the wino-like relic abundance is presented in Ref. [26], in the context of the PQMSSM. If we adopt the Peccei-Quinn (PQ) solution to the strong CP problem within the context of supersymmetric models, then it is appropriate to work with the PQ-augmented MSSM, which contains in addition to the usual MSSM states, the axion a, the R-parity even saxion field s, and the spin-1 2 R-parity odd axinoã. The axino can serve as the lightest SUSY particle if it is lighter than the lightest R-odd MSSM particle. The a andã have couplings to matter which are suppressed by the value of the PQ breaking scale f a , usually considered to be in the range 10 9 GeV < ∼ f a < ∼ 10 12 GeV [27]. In Ref. [26], it is assumed that mã > m Z 1 , where Z 1 is the LSP. In the AMSB scenarios considered here, we will assume T R < ∼ 10 10 GeV, so as to avoid overproduction of dark matter via gravitinos. With these low values of T R , we are also below the axino decoupling temperature Tã −dcp = 10 11 GeV fa 10 12 GeV 2 0.1 αs 3 , so the axinos are never considered at thermal equilibrium [30]. However, axinos can still be produced thermally via radiation off usual MSSM scattering processes at high temperatures. The calculation of the thermally produced axino abundance, from the hard thermal loop approximation, yields [25] Ω T P a = h 2 ≃ 5.5g 6 s ln 1.211 g s 10 11 GeV f a /N 2 mã 0.1 GeV T R 10 4 GeV(12) where g s is the strong coupling evaluated at Q = T R and N is the model dependent color anomaly of the PQ symmetry, of order 1. Since these axinos are assumed quite heavy, they will decay to gg or Z i γ modes, which further decay until the stable LSP state, assumed here to be the neutral wino, is reached. If the temperature of radiation due to axino decay (T D ) exceeds the neutralino freeze-out temperature T f o , then the thermal wino abundance is unaffected by axino decay. If T D < T f o , then the axino decay will add to the neutralino abundance. However, this situation breaks up into two possibilities: a). a case wherein the axinos can dominate the energy density of the universe, wherein extra entropy production from heavy axino decay may dilute the thermal abundance of the wino-like LSPs, and b). a case where they don't. In addition, if the yield of winos from axino decay is high enough, then additional annihilation of winos after axino decay may occur; this case is handled by explicit solution of the Boltzmann equation for the wino number density. Along with a component of wino-like neutralino CDM, there will of course be some component of vacuum mis-alignment produced axion CDM: thus, in this scenario, we expect a WIMP/axion mixture of CDM. Mixed axion/axino CDM in AMSB models In this case, we again consider the PQMSSM, as in Subsec. 3.3. But now, we consider a light axino with mã < m Z 1 , so thatã is the stable LSP [24]. Here, the thermally produced wino-like neutralinos will decay via Z 1 →ãγ, so we will obtain a very slight dark matter abundance from neutralino decay: Ω N T P a = mã m Z 1 Ω Z 1 h 2 , since each thermally produced neutralino gives rise to one non-thermally produced (NTP) axino. We will also produce axinos thermally via Eq'n 12. Finally, we will also produce axion CDM via the vacuum mis-alignment mechanism [51]: Ω a h 2 ≃ 1 4 fa/N 10 12 GeV 7/6 θ 2 i (we will take here the initial mis-alignment angle θ i ≃ 1). The entire CDM abundance is then the sum Ω aã h 2 = Ω N T P a h 2 + Ω T P a h 2 + Ω a h 2 .(13) In this case, the TP axinos constitute CDM as long as mã > ∼ 0.1 MeV. The NTP axinos constitute warm DM for mã < ∼ 1 GeV [52], but since their abundance is tiny, this fact is largely irrelevant. The entire CDM abundance then depends on the parameters f a , mã and T R ; it also depends extremely weakly on Ω Z 1 h 2 , since this is usually small in AMSB models. As an example, we plot in Fig. 4 the three components of mixed axion/axino DM abundance from HCAMSB benchmark point 1 in Ref. [35]: α = 0.025, m 3/2 = 50 TeV, tan β = 10 and µ > 0. The neutralino thermal DM abundance would be Ω Z 1 h 2 = 0.0015 if the Z 1 was stable. We require instead Ω aã h 2 = 0.11, and plot the three components of Ω aã h 2 versus f a /N, for three values of T R = 10 6 , 10 7 and 10 8 GeV. The value of mã is determined by the constraint Ω aã h 2 = 0.11. We see that at low values of f a /N, the NTP axino abundance is indeed tiny. Also the axion abundance is tiny since the assumed initial axion field strength is low. The TP axino abundance dominates. As f a /N increases, the axion abundance increases, taking an ever greater share of the measured DM abundance. The TP axino abundance drops with increasing f a /N, since the effective axino coupling constant is decreasing. Around f a /N ∼ 3 × 10 11 GeV, the axion abundance becomes dominant. It is in this range that ADMX [53] would stand a good chance of measuring an axion signal using their microwave cavity experiment. In Fig. 5, we again require Ω aã h 2 = 0.11 for HCAMSB benchmark point 1, but this time plot the value of T R which is needed versus mã, for various values of f a /N. The plots terminate at high T R in order to avoid reaching the axion decoupling temperature T a−dcp . Dashed curves indicate regions where over 50% of the DM is warm, instead of cold. Solid curves yield the bulk of DM as being cold. We see that for very light axino masses, and large values of f a , the value of T R easily reaches beyond 10 6 GeV, while maintaining the bulk of dark matter as cold . Such high values of T R are good enough to sustain baryogenesis via non-thermal leptogenesis [54], although thermal leptogenesis requires T R > ∼ 10 10 GeV [55]. Since f a is quite large, we would expect that the dominant portion of DM is composed of relic axions, rather than axinos; as such, detection of the relic axions may be possible at ADMX [53]. While Fig's 4 and 5 were created for the HCAMSB model, quite similar results are obtained for the mAMSB or inoAMSB models. detection of relic axions is more likely: in a nearly equal mixture of WIMPs and axions, possibly detection of both could occur! In case 4 above, we would expect no WIMP signals to occur in either direct or indirect detection experiments. f a = 1 E 1 1 G e V f a = 5 E 1 0 G e V f a = 1 E 1 0 G e V f a = 5 E 9 G e V f a = 1 E 9 G e V f a = 4 E 1 1 G e V Direct wino detection rates in AMSB models Direct detection of WIMPs depends on the WIMP-nucleon scattering cross section, but also on assumptions about the local WIMP density (usually assumed to be ρ local ≃ 0.3 GeV/cm 3 ), and the velocity distribution of the relic WIMPs (usually assumed to follow a Maxwellian distribution f (v) ∼ v 2 e −v 2 /v 2 0 where v 0 ∼ 220 km/sec, the sun's velocity about the galactic center). In our case, where WIMPs are mainly produced non-thermally via moduli, gravitino or axino decay, the original velocity distribution due to decays will be red-shifted away and the current distribution will arise mainly from gravitational infall, as is the case with thermal WIMP production. The direct detection reach plots are usually presented in terms of the WIMP-nucleon scattering cross section. Then, the experimental reach depends on factors like the mass and spin of the nuclear target, and the assumed local WIMP density and velocity profiles. Direct detection of WIMPs is usually broken down into two components: detection via spin-independent (SI) interactions , and detection via spin-dependent interactions (SD). For SI interactions, it may be best to use heavy target nuclei, since the SI nucleon-WIMP interactions sum coherently over the nuclear mass. For the SI WIMP-nucleon cross section, we use the Isatools subroutine IsaReS [57]. Our results for SI direct detection of wino-WIMPs is shown in Fig. 6, in the σ( Z 1 p) vs. m Z 1 plane. Here, we scan over m 3/2 for all models, and m 0 (for mAMSB) and α (for HCAMSB). We show results for tan β = 10 and 40, while taking µ > 0. The inoAMSB results occur as lines, since there is no m 0 or α dependence. We keep only solutions that obey the LEP2 limit on a wino-like chargino: m W 1 > 91.9 GeV [58]. Several crucial features emerge from the plot. First, we note that for a given value of m Z 1 , the value of σ( Z 1 p) is bounded from below, unlike the case of the mSUGRA model. That means that wino-WIMP dark matter can be either detected or excluded for a given m Z 1 value. Second, we note that the cross section values generally fall in the range that is detectable at present or future DD experiments. The purple contour, for instance, exhibits the CDMS reach based on 2004-2009 data, and already excludes some points, especially those at large tan β. We also show the reach of Xenon-100, LUX, Xenon-100 upgrade, and Xenon 1 ton[59]. These experiments should be able to either discover or exclude AMSB models with m Z 1 values below ∼ 90, 100, 200 and 500 GeV respectively. These WIMP masses correspond to values of mg ∼ 690, 770, 1540 and 3850 GeV, respectively! The latter reach far exceeds the 100 fb −1 of integrated luminosity reach of LHC for mg. 3 For inoAMSB models, where the minimal value of σ SI ( Z 1 p) exceeds that of mAMSB or HCAMSB for a given m Z 1 value, the Xenon 1 ton reach is to m Z 1 ∼ 800 GeV, corresponding to a reach in mg of 6200 GeV! In Fig. 7, we show the SD direct detection cross section σ SD ( Z 1 p) versus m Z 1 for mAMSB, : Spin-independent Z 1 − p scattering cross section versus m Z 1 for mAMSB, HCAMSB and inoAMSB models for tan β = 10 and 40 and µ > 0. The parameters m 3/2 and also m 0 (for mAMSB) and α (for HCAMSB) have been scanned over. We also show the CDMS limit and projected Xenon and LUX sensitivities. HCAMSB and inoAMSB models with tan β = 10 and 40. We also show a recent limit on this cross section from the COUPP experiment, which is above the theory expectation by two orders of magnitude. We also show two limits from IceCube in 2009, which do approach the theory region, but only for rather large values of m Z 1 . The IceCube SD reach is quite significant, because the rate for WIMP annihilation in the core of the sun mainly depends on the sun's ability to sweep up neutralinos as it passes along its orbit. The target here is the solar hydrogen, where the SD cross section usually dominates the SI one, since the atomic mass is minimal (an enhancement by number of nucleons per nucleus is usually necessary to make the SI cross section competetive with the SD one). Since IceCube is mainly sensitive to very high energy muons with E µ > 50 GeV, it can access mainly higher values of m Z 1 . The IceCube DeepCore reach is also shown. The DeepCore project will allow IceCube to access much lower energy muons, and thus make it more useful for generic WIMP searches. While DeepCore will access a portion of parameter space, it will not reach the lower limit on SD cross sections as predicted by AMSB models. Indirect wino detection rates in mAMSB Next, we present rates for indirect detection (ID) of wino-like DM via neutrino telescopes, and via detection of gamma rays and anti-matter from WIMP annihilation in the galactic halo. The ID detection rates depend (quadratically [60]) on the assumed galactic DM density (halo) profile. We will show results using two profiles: isothermal and Navarro-Frenk-White (NFW) [61] (see e.g. [62] for plots of several recent halo profiles). Most halo models are in near accord at the earth's position at ∼ 8 kpc from the galactic center. However, predictions for the DM density near the galactic center differ wildly, which translates to large uncertainties for DM annihilation rates near the galactic core. The corresponding uncertainty will be smaller for anti-protons, and smaller still for positrons; since these particles gradually lose energy while propagating through the galaxy, they can reach us only from limited distances over which the halo density is relatively well-known. Possible clumping of DM yields an additional source of uncertainty in ID detection rates. In Fig. 8a.), we show for comparison the SI direct detection scattering cross section versus m 0 in the mAMSB model for m 3/2 = 50 TeV and tan β = 10. For these parameters, the winolike neutralino has mass m Z 1 ≃ 144 GeV. We also indicate an approximate reach of Xenon-10 and Xenon-100. While the SI direct detection cross section is just below Xenon-100 reach for low m 0 , as m 0 increases, the value of µ drops, much as it does in mSUGRA as we approach the focus point region. For large m 0 , the Z 1 becomes mixed wino-higgsino, and its direct detection cross section increases into the range which is accessible to Xenon-100. In Fig. 8b.), we show the flux of muons from ν µ → µ conversions at earth coming from neutralino annihilation to SM particles within the solar core. Here, we use the Isajet/DarkSUSY interface for our calculations [63], and require E µ > 50 GeV. The predicted rate depends, in this case, mainly on the sun's ability to sweep up and capture neutralinos, which depends mainly on the spin-dependent neutralino-nucleon scattering cross section (since in this case, the neutralinos mainly scatter from solar Hydrogen, and there is no mass number enhancement), which is mostly sensitive to Z * exchange. The rates are again low for low m 0 with wino-like neutralinos. They nearly reach the IceCube detectability level at large m 0 where the neutralinos, while remaining mainly wino-like, have picked up an increasing higgsino component, so that the neutralino couplings to Z become large. In Fig. 8c.), we show the expected flux of gamma rays with E γ > 1 GeV, as required for the Fermi Gamma-ray Space Telescope (FGST), arising from DM annihilations in the galactic core. In this case, we see a signal rate which is flat with respect to m 0 . Here, the rate depends mainly on the Z 1 Z 1 → W + W − annihilation cross section, which occurs via chargino exchange; since the Z 1 s remain mainly wino-like, and the chargino mass hardly varies, the annihilation rate hardly varies with m 0 . The predictions for two halo profiles differ by over an order of magnitude, reflecting the large uncertainty in our knowledge of the DM density at the center of our Galaxy. Both projections are above the approximate reach of the FGST. In Fig. 8d.)-f.), we show the expected flux of positrons e + , antiprotonsp and antideuterons D from neutralino halo annihilations. Each of these frames show detectable rates by Pamela [64] (for e + s andps) and by GAPS [65] (for anti-deuterons). These elevated IDD rates (compared to mSUGRA [66] for similar tan β values) for anti-matter detection reflect the elevated rate for the wino − wino annihilation into W + W − cross section. The halo model uncertainty for antimatter detection is much smaller than in the γ-ray case, since for charged particle detection, it is necessary that the anti-matter is generated relatively close to earth, where the DM density profile is much better known. In Fig. 9, we show rates for direct and indirect detection of wino-like WIMPs in mAMSB versus m 0 with m 3/2 = 50 TeV, tan β = 40 and µ > 0. The SI direct detection rate shown in Fig. 9a.) shows a notable enhancement at low m 0 , and the usual enhancement at large m 0 due to the increasing higgsino component of Z 1 . The low m 0 enhancement arises because the mass of the heavy Higgs scalar H has dropped with increasing tan β, and is now quite light: m H ∼ 152 GeV for m 0 = 600 GeV (compared to m H = 1019 GeV for the same m 0 with tan β = 10. The g Z 1 → g Z 1 loop diagram via H exchange is enhanced, resulting in a huge direct detection cross section. This range of m 0 is already excluded by DD WIMP searches! In Fig. 9b.), we show the muon flux from mAMSB models versus m 0 for m 3/2 = 50 TeV and tan β = 40. In this case, we again see a huge enhancement at low m 0 . While normally the SD Z 1 p cross section dominates the solar accretion rate for WIMPs, in this case, the 3 orderof-magnitude increase in SI cross section shown in frame a.) contributes and greatly increases the solar capture rate, and hence the muon flux from the sun. Of course, this region would already be excluded by present DD limits. We also see a curious "anti-resonance" effect in Φ µ around m 0 ∼ 850 GeV. In this case, m A ∼ 2m Z 1 , and neutralino annihilation is enhanced by the A resonance. Normally, for AMSB models, Z 1 Z 1 → V V (V = W ± or Z) is the dominant annihilation mechanism. But on the Higgs resonance, Z 1 Z 1 → bb instead dominates. The energy distribution of neutrinos from b decay is far softer than that from W or Z decay, leading to ν µ → µ conversions to lower energy muons. Since we require E µ > 50 GeV for IceCube, fewer muons are detected, and hence the anti-resonance effect. At large m 0 and tan β = 40, the muon flux is again enhanced by the increased WIMP scattering rate via its increasing higgsino component. In Fig. 9c-f.), we see the flux of gamma rays and anti-matter versus m 0 at large tan β. Here, the rate versus m 0 is again flat, reflecting the usually constant Z 1 Z 1 annihilation rate into vector bosons. The exception occurs at m 0 ∼ 800 GeV, where annihilation through the A-resonance enhances the halo annihilation rate [66]. At large m 0 and tan β = 40, the e + andp detection rates drop. This is due to the changing final state from Z 1 Z 1 annihilation: at low m 0 it is mainly to vector bosons, leading to a hard e + andp distribution. At large m 0 , annihilations to bb increase and become prominent, but the energy distribution of e + andp softens, and since we require E e + ,p = 20 GeV, the detection rate drops. In frame f.), showing theD rate, the rate actually increases at large m 0 , since here we already require quite low energyDs for detection, and the distribution only reflects the increased annihilation rate. Indirect wino detection rates in HCAMSB In this subsection, we present wino-like WIMP DD and ID rates in the HCAMSB model for m 3/2 = 50 TeV, versus varying α. As shown in Ref. [35], a low value of α ∼ 0 corresponds to pure anomaly-mediation, while large α gives an increasing mass M 1 to the hypercharge gaugino at the GUT scale. The large value of M 1 pulls sparticle masses to larger values via RG evolution, with the pull increasing in accord with the matter state's hypercharge quantum number: thus-at large α-we expect relatively heavyẽ R states, but comparatively lightũ L and d L states. In fact, the U(1) RGE effect-coupled with the large t-quark Yukawa coupling[1]leads to relatively light, and dominantly left-, top squark statest 1 at large α. For very high values of α ∼ 0.15 − 0.2, the value of |µ| diminishes until radiative EWSB no longer occurs (much as in the focus point region of mSUGRA). In Fig. 10a.), we show the SI neutralino direct detection rate versus α for tan β = 10. The cross section σ SI ( Z 1 p) is of order 10 −9 pb at low α, consistent with pure wino-like neutralinos. As α increases, the increasing sparticle masses feed into m 2 Hu , diminishing the term X t = m 2 Q 3 + m 2 t R + m 2 Hu + A 2 t and leading to a lessened downward push by the top Yukawa coupling. Since µ 2 ∼ −m 2 Hu at the weak scale, the |µ| term is also diminished, leading to an increasing higgsino component of Z 1 . The increased higgsino component yields an enhanced σ SI ( Z 1 p) via Higgs exchange at large α. In Fig. 10b.), we plot the muon flux in the HCAMSB model vs. α. The muon flux is quite small at low α, but at high α, the increasing higgsino component of Z 1 leads to an increased σ SD ( Z 1 p) via Z-exchange. In Fig's 10c-f.), we plot the γ-ray, e + ,p andD fluxes versus α. In these cases, the rates are large due to the large wino − wino → V V annihilation cross section and is relatively flat with α. At the largest α values, annihilation to bb states is enhanced, leading to diminished rates for e + andp (due to softened energy distributions) but to a slightly increased rate forDs. In Fig. 11, we show the same rates vs. α in the HCAMSB model, except now for tan β = 40. The SI direct detection rate in Fig. 11a.) is enhanced relative to the tan β = 10 case due to the much lighter Higgs mass m H and the increased b-quark Yukawa coupling. The rate diminishes as α increases due to increasing squark and Higgs masses, until very high α is reached, and the rate is enhanced by the growing higgsino component of Z 1 . The parameter space terminates above α ∼ 0.15 due to lack of REWSB. In Fig. 11b.), we see that the muon flux due to solar core annihilations is also enhanced. In this case, the increase is again due to the large enhancement in SI scattering cross section, which feeds into the solar accretion rate. As in Fig. 9b.), we find an anti-resonance dip at the α value where 2m Z 1 ∼ m A , and WIMP annihilations occur instead mainly into bb rather than V V states. Fig's 11c-f.) show the halo annihilation rates for HCAMSB at tan β = 40. These rates are generally flat with changing α, and do not suffer an increase compared with low tan β results, since wino − wino → V V still dominates the annihilation rate. The exception occurs at α ∼ 0.035, where 2m Z 1 ∼ m A , and halo annihilation is enhanced by the pseudoscalar Higgs resonance. Indirect detection rates vs. m 3/2 for mAMSB, HCAMSB and inoAMSB In Fig. 12, we show direct and indirect wino DM detection rates versus m 3/2 for tan β = 10 and µ > 0. For mAMSB, we take m 0 = 1 TeV, while for HCAMSB, we take α = 0.1. The associated mass spectra versus m 3/2 can be found in Ref. [35] for mAMSB and HCAMSB, and in Ref. [13] for the inoAMSB model. Spectra for all three models are shown in Table 1 of Ref. [13] for the case of m 3/2 = 50 TeV. In Fig. 12a.), the SI direct detection cross section is shown for all three models. In this case, we see for a given value of m 3/2 , the inoAMSB model gives the highest cross section, while mAMSB gives the lowest. The larger inoAMSB cross section is due in part because inoAMSB models have a smaller µ value for a given value of m 3/2 , and so SI scattering via Higgs exchange (which involves a product of higgsino and gaugino components) is enhanced. Figure 12: Direct and indirect detection rates of wino CDM in mAMSB, HCAMSB and inoAMSB models vs. m 3/2 , for tan β = 10 and µ > 0. For mAMSB, we take m 0 = 1 TeV,while for HCAMSB, we take α = 0.1. In these plots, we adopt the NFW DM halo profile. In Fig. 12b.), we show the relative rates for indirect wino detection due to WIMP annihilation into ν µ states in the solar core, with subsequent muon detection from ν µ → µ conversions in Antarctic ice, as might be seen by IceCube. We require E µ > 50 GeV. The muon flux is mainly related to the spin-dependent direct detection rate, which enters the sun's ability to capture WIMPs. Here again, inoAMSB yields the highest rates, and mAMSB the lowest. This follows the relative values of µ in the three models: low µ in inoAMSB leads to a larger higgsino component of Z 1 , and an increased SD scattering rate via Z * exchange. A rough reach of the IceCube detector is shown, and indicates that the low m 3/2 portion of parameter space of inoAMSB and HCAMSB may be accessible to ν µ → µ searches. In Fig. 12c-f.), we show the ID rates for detection of γ, e + s,ps andDs, for the energy ranges indicated on the plots. All these plots adopt the NFW halo profile. In all these cases, all three models yield almost exactly the same detection rates for a given value of m 3/2 . This is due to the dominance of Z 1 Z 1 → V V halo annihilations, which mainly depend on the gaugino component of Z 1 , which is nearly all wino-like. The rough reach of Fermi-LAT, Pamela and GAPS is shown for reference. The high rates for wino halo annihilations should yield observable signals. As mentioned previously, Kane et al. promote wino-like WIMPs as a source of the Pamela anomaly [31]. In this case, a largep signal should be seen as well, although the Pamelā p rate seems to agree with SM background projections. In Fig. 13, we show direct and indirect wino detection rates versus m 3/2 as in Fig. 12, except now for a large value of tan β = 40. In frame a.), we see that the SI direct detection rates are all elevated with respect to the tan β = 10 case. The well-known large tan β enhancement [67,68] occurs due to enhanced Higgs exchange contributions, where now the value of m H is lower and the b-quark Yukawa coupling is larger. For low m 3/2 , the inoAMSB model has the smallest value of m H and the lowest value of µ for a given m 3/2 value, and thus the highest value of σ SI ( Z 1 p). As m 3/2 increases, the value of m H increases for inoAMSB and HCAMSB, while it actually decreases for mAMSB. Thus, for m 3/2 > ∼ 75 TeV, the mAMSB model yields the highest value of σ SI ( Z 1 p). Figure 13: Direct and indirect detection rates of wino CDM in mAMSB, HCAMSB and inoAMSB vs. m 3/2 , for tan β = 40 and µ > 0. For mAMSB, we take m 0 = 1 TeV,while for HCAMSB, we take α = 0.1. In these plots, we adopt the NFW DM halo profile. In Fig. 13b.), we show the muon flux for IceCube due to wino annihilation in the solar core. Here again, the rates are elevated compared to the tan β = 10 case. The inoAMSB model yields the highest flux at low m 3/2 , since it has the lowest µ value, and the highest higgsino component, which enters into the Z * exchange diagram for q Z 1 scattering. At m 3/2 ∼ 60 TeV in the mAMSB model, we obtain 2m Z 1 ∼ m A , and the solar core annihilations mainly proceed to bb states instead of V V , which diminishes the muon energy distribution and hence the detection rate for µs with E µ > 50 GeV. At higher m 3/2 values, the resonance is passed, and annihilation once again proceeds dominantly into V V . For high m 3/2 , the mAMSB model yields the highest muon flux, due to its elevated value of σ SI ( Z 1 p). In Fig. 13c-f.), we find relatively little change in halo annihilation rates due to an increase in tan β, since the annihilations mainly proceed via wino − wino → V V , which depends mainly on gauge couplings. The exception occurs in the mAMSB model, where we do get the resonance enhancement of halo annihilations when 2m Z 1 ∼ m A . We also obtain some tan β enhancement of theD detection rate for inoAMSB and mAMSB at large m 3/2 because in these cases the Z 1 Z 1 → bb annihilation rate, which does receive tan β enhancement, contributes to the detection of rather low energyDs. Discussion and conclusions In this paper, we have investigated aspects of cold dark matter in three models of anomaly mediation: mAMSB, HCAMSB and inoAMSB. Typically, each gives rise to a wino-like lightest neutralino, unless very high values of m 0 (for mAMSB) or α (for HCAMSB) are used, in which case the Z 1 becomes a mixed wino-higgsino state. In this class of models with a wino-like Z 1 , the thermal abundance of neutralino CDM is well below measured values, unless m Z 1 > ∼ 1300 GeV. We discuss four ways to reconcile the predicted abundance of CDM with experiment: 1. enhanced neutralino production via scalar field (e.g. moduli) decay, 2. enhanced neutralino production via gravitino decay, where gravitinos may arise thermally, or by moduli or inflaton decay, 3. enhanced neutralino production via heavy axino decay, and 4. neutralino decay to axinos, where the bulk of CDM comes from a mixture of vacuum mis-alignment produced axions and thermally produced axinos. Cases 1 and 2 should lead to a situation where all of CDM is comprised of wino-like WIMPs; they will be very hard, perhaps impossible, to tell apart. Case 3 would contain a mixture of axion and wino-like WIMP CDM. It is a scenario where it is possible that both a WIMP and an axion could be detected. Case 4 predicts no direct or indirect detection of WIMPs, but a possible detection of relic axions. It is important to note that more than one of these mechanisms may occur at once: for instance, we may gain additional neutralino production in the early universe from moduli, gravitino and axino decay all together. In Sec. 4, we presented rates for direct and indirect detection of relic wino-like WIMPs. The SI direct detection cross sections are bounded from below. Ultimately, ton-scale noble liquid or SuperCDMS experiments should probe out to m Z 1 ∼ 500 GeV, which would exceed the 100 fb −1 reach of LHC; a non-observation of signal would put enormous stress on AMSB-like models as new physics. We also evaluated SD direct detection: current experiments have little reach for AMSB-like models, although IceCube DeepCore and possibly COUPP upgrades may probe more deeply. We also presented indirect WIMP detection rates for all three AMSB models. The IceCube experiment has some reach for WIMPs from AMSB models, especially at high tan β or when the Z 1 picks up a higgsino component. We noted an interesting inverse resonance effect in the muon flux detection rate, caused by transition from solar core annihilations to V V states, to annihilations to mainly bb states. The detection of γs, e + s,ps andDs are all elevated in AMSB-like models compared to mSUGRA, due to the high rate for Z 1 Z 1 → V V annihilation in the galactic halo. The results do depend on the assumed halo profile, especially for γ-ray detection in the direction of the galactic core. Generally, if a signal is seen in the e + channel, then one ought to be seen in thep channel, and ultimately in the γ,D (if/when GAPS flies) or direct detection channel. In addition, a sparticle production signal should ultimately be seen at LHC, at least for mg < ∼ 2400 GeV, once 100 fb −1 of integrated luminosity is accrued. As a final remark, we note here that the dark matter detection signals all provide complementary information to that which will be provided by the CERN LHC. At LHC, each model-mAMSB, HCAMSB and inoAMSB-will provide a rich assortment of gluino and squark cascade decay signals which will include multi-jet plus multi-lepton plus missing E T events. In all cases, the wino-like lightest neutralino state will be signaled by the well-known presence of highly ionizing tracks (HITs) from quasi-stable charginos with track length of order cms, before they decay to soft pions plus a Z 1 . It is noted in Ref's [35] and [13] that the three models should be distinguishable at LHC by the differing opposite-sign/same flavor dilepton invariant mass distributions. In the case of mAMSB, with ml L ≃ ml R , we expect a single mass edge from Z 2 → ℓl L,R → ℓ + ℓ − Z 1 decay. In HCAMSB, the sleptons are rather heavy, and instead Z 2 → Z 1 Z occurs at a large rate, leading to a bump in m(ℓ + ℓ − ) ∼ M Z , upon a continuum distribution. In inoAMSB, with m Z 2 > ml L,R , but withl L andl R split in mass (due to different U(1) Y quantum numbers), a characteristic double mass edge is expected in the m(ℓ + ℓ − ) invariant mass distribution. Figure 1 : 1The wino and higgsino content of the neutralino Z 1 in a). mAMSB versus m 0 and b). HCAMSB versus α for m 3/2 = 50 TeV, tan β = 10 and µ > 0. Figure 2 : 2Thermally produced relic abundance of wino-like neutralino cold dark matter in mAMSB, HCAMSB and inoAMSB versus m 3/2 for tan β = 10, with µ > 0 and m t = 172.6 Figure 3 : 3Plot of allowed region of T R vs. m 3/2 plane allowed for wino-like neutralino DM from thermal production plus thermally produced gravitino decay. Figure 4 : 4Abundance of TP and NTP axino DM and vacuum-misalignment production of axion CDM versus f a /N, for various values of T R . Figure 5 : 5Plot of T R needed to ensure Ω aã h 2 = 0.1 for HCAMSB benchmark Pt. 1, versus mã for various values of the PQ breaking scale f a . The dashed curves yield mainly warm axino DM, while solid curves yield mainly cold mixed axion/axino DM. Figure 6 6Figure 6: Spin-independent Z 1 − p scattering cross section versus m Z 1 for mAMSB, HCAMSB and inoAMSB models for tan β = 10 and 40 and µ > 0. The parameters m 3/2 and also m 0 (for mAMSB) and α (for HCAMSB) have been scanned over. We also show the CDMS limit and projected Xenon and LUX sensitivities. Figure 7 : 7Spin-dependent Z 1 − p scattering cross section versus m Z 1 for mAMSB, HCAMSB and inoAMSB models for tan β = 10 and 40 and µ > 0. The parameters m 3/2 and also m 0 (for mAMSB) and α (for HCAMSB) have been scanned over. We also show the COUPP and IceCube limits in σ SD ( Z 1 p). Figure 8 : 8Direct and indirect detection rates of neutralino CDM in mAMSB vs. m 0 , for m 3/2 = 50 TeV, tan β = 10 and µ > 0. Figure 9 : 9Direct and indirect detection rates of neutralino CDM in mAMSB vs. m 0 , for m 3/2 = 50 TeV, tan β = 40 and µ > 0. Figure 10 : 10Direct and indirect detection rates of neutralino CDM in HCAMSB vs. α, for m 3/2 = 50 TeV, tan β = 10 and µ > 0. Figure 11 : 11Direct and indirect detection rates of neutralino CDM in HCAMSB vs. α, for m 3/2 = 50 TeV, tan β = 40 and µ > 0. Direct and indirect detection of wino CDM in AMSB modelsFor AMSB dark matter cases 1 and 2 above, it is expected that the thermal wino abundance will be supplemented by either moduli or gravitino decay in the early universe, thus increasing the wino abundance into accord with measured values. In these cases, it may be possible to detect relic wino-like WIMPs with either direct or indirect detection experiments[56]. Also, in case 3 above, it is expected that the DM abundance is comprised of an axion/wino mixture. If the wino component of this mixture is substantial, then again direct or indirect WIMP detection may be possible, while if axions are dominant, then a WIMP signal is less likely, but direct In Ref.[35], the 100 fb −1 reach of LHC for HCAMSB is found to be mg ∼ 2.2 − 2.4 TeV. In Ref.[13], the 100 fb −1 reach of LHC for inoAMSB was found to be mg < 2.6 TeV. AcknowledgmentsWe thank Shanta de Alwis for comments on the manuscript. This work was supported in part by the U.S. Department of Energy. H Baer, X Tata, Weak Scale Supersymmetry: From Superfields to Scattering Events. Cambridge University PressH. Baer and X. Tata, Weak Scale Supersymmetry: From Superfields to Scattering Events, (Cam- bridge University Press, 2006) . S Dimopoulos, S Raby, F Wilczek, Phys. Rev. D. 241681S. Dimopoulos, S. Raby and F. Wilczek, Phys. Rev. D 24 (1981) 1681; . S Dimopoulos, H Georgi, Nucl. Phys. B. 193150S. Dimopoulos and H. Georgi, Nucl. Phys. B 193 (1981) 150; . L Ibanez, G Ross, Phys. Lett. B. N. Sakai105153Z. Phys.L. Ibanez and G. Ross, Phys. Lett. B 105 (1981) 439; N. Sakai, Z. Phys. C11 (1981) 153; . M Einhorn, D R T Jones, Nucl. Phys. B. 196475M. Einhorn and D. R. T. Jones, Nucl. Phys. B 196 (1982) 475; . W Marciano, G Senjanovic, Phys. Rev. D. 253092W. Marciano and G. Senjanovic, Phys. Rev. D 25 (1982) 3092. . U Amaldi, W Boer, H Furstenau, Phys. Lett. B. 260447U. Amaldi, W. de Boer and H. Furstenau, Phys. Lett. B 260 (1991) 447; . J Ellis, S Kelley, D V Nanopoulos, Phys. Lett. B. 260131J. Ellis, S. Kelley and D. V. Nanopoulos, Phys. Lett. B 260 (1991) 131; . P Langacker, Luo , Phys. Rev. D. 44817P. Langacker and Luo, Phys. Rev. D 44 (1991) 817. . D Freedman, S Ferrara, P Van Nieuwenhuizen, Phys. Rev. D. 133214D. Freedman, S. Ferrara and P. van Nieuwenhuizen, Phys. Rev. D 13 (1976) 3214; . S Deser, B Zumino, Phys. Lett. B. 62335S. Deser and B. Zumino, Phys. Lett. B 62 (1976) 335; . S Ferrara, D Freedman, P Van Nieuwenhuizen, P Breitenlohner, G Gliozzi, J Scherk, Phys. Rev. D. 151013S. Ferrara, D. Freedman, P. van Nieuwenhuizen, P. Breitenlohner, G. Gliozzi and J. Scherk, Phys. Rev. D 15 (1977) 1013; . E Cremmer, S Ferrara, L Girardello, A Van Proeyen, Nucl. Phys. B. 212413E. Cremmer, S. Ferrara, L. Girardello and A. van Proeyen, Nucl. Phys. B 212 (1983) 413 . M Dine, W Fischler, M Srednicki, Nucl. Phys. B. 189575M. Dine, W. Fischler and M. Srednicki, Nucl. Phys. B 189 (1981) 575; . S Dimopoulos, S Raby, Nucl. Phys. B. 192353S. Dimopoulos and S. Raby, Nucl. Phys. B 192 (1982) 353; . M Dine, W Fischler, Phys. Lett. B. 110227M. Dine and W. Fischler, Phys. Lett. B 110 (1982) 227 . C Nappi, B Ovrut, Nucl. Phys. B. 204175Phys. Lett. Band Nucl. Phys. B 204 (1982) 346; C. Nappi and B. Ovrut, Phys. Lett. B 113 (1982) 175; . M Alvarez-Gaume, M Claudson, ; S Wise ; 96, S Dimopoulos, Raby, Nucl. Phys. B. 207479Nucl. Phys. BAlvarez-Gaume, M. Claudson and M. Wise, Nucl. Phys. B 207 (1982) 96; S. Dimopoulos and S. Raby, Nucl. Phys. B 219 (1983) 479; ) 2658; for a review, see G. Giudice and R. Rattazzi. M Dine, A Nelson, Y Nir, Y Shirman, Phys. Rev. D. 53419Phys. Rept.M. Dine, A. Nelson, Y. Nir and Y. Shirman, Phys. Rev. D 53 (1996) 2658; for a review, see G. Giudice and R. Rattazzi, Phys. Rept. 322 (1999) 419. . L Randall, R Sundrum, Nucl. Phys. B. 55779L. Randall and R. Sundrum, Nucl. Phys. B 557 (1999) 79. . G Giudice, M Luty, H Murayama, R Rattazzi, J. High Energy Phys. 1227G. Giudice, M. Luty, H. Murayama and R. Rattazzi, J. High Energy Phys. 12 (1998) 027. For recent discussion, see M. Dine and N. Seiberg. J. High Energy Phys. 070340For recent discussion, see M. Dine and N. Seiberg, J. High Energy Phys. 0703 (2007) 040; . S De Alwis, Phys. Rev. D. 77105020S. de Alwis, Phys. Rev. D 77 (2008) 105020. . I Jack, D R T Jones, Phys. Lett. B. 465148I. Jack and D. R. T. Jones, Phys. Lett. B 465 (1999) 148. . S Weinberg, Phys. Rev. Lett. 481303S. Weinberg, Phys. Rev. Lett. 48 (1982) 1303; . R H Cyburt, J Ellis, B D Fields, K A Olive, Phys. Rev. D. 67103521R. H. Cyburt, J. Ellis, B. D. Fields and K. A. Olive, Phys. Rev. D 67 (2003) 103521; . K Jedamzik, Phys. Rev. D. 7063524K. Jedamzik, Phys. Rev. D 70 (2004) 063524; . M Kawasaki, K Kohri, T Moroi, Phys. Lett. B. 6257M. Kawasaki, K. Kohri and T. Moroi, Phys. Lett. B 625 (2005) 7. . M Kawasaki, K Kohri, T Moroi, Phys. Rev. D. 7183502M. Kawasaki, K. Kohri and T. Moroi, Phys. Rev. D 71 (2005) 083502. . R Dermisek, H Verlinde, L-T Wang, Phys. Rev. Lett. 100131804R. Dermisek, H. Verlinde and L-T. Wang, Phys. Rev. Lett. 100 (2008) 131804. . H Baer, S De Alwis, K Givens, S Rajagopalan, H Summy, J. High Energy Phys. 100569H. Baer, S. de Alwis, K. Givens, S. Rajagopalan and H. Summy, J. High Energy Phys. 1005 (2010) 069. . S P De Alwis, J. High Energy Phys. 100378S. P. de Alwis, J. High Energy Phys. 1003 (2010) 078. . J R Ellis, C Kounnas, D V Nanopoulos, Nucl. Phys. B. 247373J. R. Ellis, C. Kounnas and D. V. Nanopoulos, Nucl. Phys. B 247 (1984) 373; . J R Ellis, A B Lahanas, D V Nanopoulos, K Tamvakis, Phys. Lett. B. 134429J. R. Ellis, A. B. Lahanas, D. V. Nanopoulos and K. Tamvakis, Phys. Lett. B 134 (1984) 429; ) 303; For a review, see A. Lahanas and D. V. Nanopoulos. G A Diamandis, J R Ellis, A B Lahanas, D V Nanopoulos, Phys. Lett. B. 1731Phys. Rept.G. A. Dia- mandis, J. R. Ellis, A. B. Lahanas and D. V. Nanopoulos, Phys. Lett. B 173 (1986) 303; For a review, see A. Lahanas and D. V. Nanopoulos, Phys. Rept. 145 (1987) 1. . D E Kaplan, G D Kribs, M Schmaltz, Phys. Rev. D. 6235010D. E. Kaplan, G. D. Kribs and M. Schmaltz, Phys. Rev. D 62 (2000) 035010; . Z Chacko, M A Luty, A E Nelson, E Ponton, J. High Energy Phys. 013Z. Chacko, M. A. Luty, A. E. Nelson and E. Ponton, J. High Energy Phys. 01 (2000) 003. . H C Cheng, B Dobrescu, K Matchev, Nucl. Phys. B. 54347H. C. Cheng, B. Dobrescu and K. Matchev, Nucl. Phys. B 543 (1999) 47. . E Komatsu, WMAP collaborationarXiv:1001.4538E. Komatsu et al. (WMAP collaboration), arXiv:1001.4538 (2010). . T Moroi, L Randall, Nucl. Phys. B. 570455T. Moroi and L. Randall, Nucl. Phys. B 570 (2000) 455. . M Bolz, A Brandenburg, W Buckmuller, Nucl. Phys. B. 606518M. Bolz, A. Brandenburg and W. Buckmuller, Nucl. Phys. B 606 (2001) 518. . J Pradler, F Steffen, Phys. Rev. D. 7523509J. Pradler and F. Steffen, Phys. Rev. D 75 (2007) 023509. . K Kohri, M Yamaguchi, J Yokoyama, Phys. Rev. D. 7283510K. Kohri, M. Yamaguchi and J. Yokoyama, Phys. Rev. D 72 (2005) 083510; . T Asaka, S Nakamura, M Yamaguchi, Phys. Rev. D. 7423520T. Asaka, S. Naka- mura and M. Yamaguchi, Phys. Rev. D 74 (2006) 023520. . M Endo, F Takahashi, T T Yanagida, Phys. Rev. D. 7683509M. Endo, F. Takahashi and T. T. Yanagida, Phys. Rev. D 76 (2007) 083509. . L Covi, J E Kim, L Roszkowski, Phys. Rev. Lett. 824180L. Covi, J. E. Kim and L. Roszkowski, Phys. Rev. Lett. 82 (1999) 4180; . L Covi, H B Kim, J E Kim, L Roszkowski, J. High Energy Phys. 010533L. Covi, H. B. Kim, J. E. Kim and L. Roszkowski, J. High Energy Phys. 0105 (2001) 033. . A Brandenburg, F Steffen, 04088A. Brandenburg and F. Steffen, JCAP0408 (2004) 008. . K.-Y Choi, J E Kim, H M Lee, O Seto, Phys. Rev. D. 77123501K.-Y. Choi, J. E. Kim, H. M. Lee and O. Seto, Phys. Rev. D 77 (2008) 123501. For a recent review, see P. Sikivie. hep-ph/0509198For a recent review, see P. Sikivie, hep-ph/0509198; . M Turner, Phys. Rept. 19767M. Turner, Phys. Rept. 197 (1990) 67; . J E Kim, G Carosi, arXiv:0807.3125J. E. Kim and G. Carosi, arXiv:0807.3125 (2008). . H Baer, H Summy, Phys. Lett. B. 6665H. Baer and H. Summy, Phys. Lett. B 666 (2008) 5. . H Baer, A Box, H Summy, J. High Energy Phys. 090880H. Baer, A. Box and H. Summy, J. High Energy Phys. 0908 (2009) 080. . K Rajagopal, M Turner, F Wilczek ; See, L Covi, J E Kim, ; F Steffen, arXiv:0902.0769Eur. Phys. J. C. 358557Nucl. Phys. BK. Rajagopal, M. Turner and F. Wilczek, Nucl. Phys. B 358 (1991) 447; for recent reviews, see L. Covi and J. E. Kim, arXiv:0902.0769 and F. Steffen, Eur. Phys. J. C 59 (2009) 557. . P Grajek, G Kane, D Phalen, A Pierce, S Watson, Phys. Rev. D. 7943506P. Grajek, G. Kane, D. Phalen, A. Pierce and S. Watson, Phys. Rev. D 79 (2009) 043506; . G Kane, R Lu, S Watson ; D. Feldman, G Kane, R Lu, B Nelson, Phys. Lett. B. 681363Phys. Lett. BG. Kane, R. Lu and S. Watson, Phys. Lett. B 681 (2009) 151. D. Feldman, G. Kane, R. Lu and B. Nelson, Phys. Lett. B 687 (2010) 363. . T Gherghetta, G Giudice, J Wells, Nucl. Phys. B. 55927T. Gherghetta, G. Giudice and J. Wells, Nucl. Phys. B 559 (1999) 27; . J L Feng, T Moroi, L Randall, M Strassler, S Su, Phys. Rev. Lett. 831731J. L. Feng, T. Moroi, L. Randall, M. Strassler and S. Su, Phys. Rev. Lett. 83 (1999) 1731; . J L Feng, T Moroi, Phys. Rev. D. 6195004J. L. Feng and T. Moroi, Phys. Rev. D 61 (2000) 095004; hep-ph/0001249. F Paige, J Wells, ; A Datta, K Huitu, Phys. Rev. D. 67115006F. Paige and J. Wells, hep-ph/0001249; A. Datta and K. Huitu, Phys. Rev. D 67 (2003) 115006; . S Asai, T Moroi, K Nishihara, T T Yanagida, Phys. Lett. B. 65381S. Asai, T. Moroi, K. Nishihara and T. T. Yanagida, Phys. Lett. B 653 (2007) 81; . S Asai, T Moroi, T T Yanagida, Phys. Lett. B. 664185S. Asai, T. Moroi and T. T. Yanagida, Phys. Lett. B 664 (2008) 185; . H Baer, J K Mizukoshi, X Tata, Phys. Lett. B. 488367H. Baer, J. K. Mizukoshi and X. Tata, Phys. Lett. B 488 (2000) 367; . A J Barr, C Lester, M Parker, B Allanach, P Richardson, J. High Energy Phys. 030345A. J. Barr, C. Lester, M. Parker, B. Allanach and P. Richardson, J. High Energy Phys. 0303 (2003) 045. . S Kachru, L Mcallister, R Sundrum, J. High Energy Phys. 071013S. Kachru, L. McAllister and R. Sundrum, J. High Energy Phys. 0710 (2007) 013. . H Verlinde, L-T Wang, M Wijnholt, I Yavin, J. High Energy Phys. 080282H. Verlinde, L-T. Wang, M. Wijnholt and I. Yavin, J. High Energy Phys. 0802 (2008) 082; . M Buican, D Malyshev, D R Morrison, H Verlinde, M Wijnholt, J. High Energy Phys. 0701107M. Buican, D. Malyshev, D. R. Morrison, H. Verlinde and M. Wijnholt, J. High Energy Phys. 0701 (2007) 107. . H Baer, R Dermisek, S Rajagopalan, H Summy, J. High Energy Phys. 091078H. Baer, R. Dermisek, S. Rajagopalan and H. Summy, J. High Energy Phys. 0910 (2009) 078. . M R Douglas, S Kachru, arXiv:hep-th/0610102Rev. Mod. Phys. 79733M. R. Douglas and S. Kachru, Rev. Mod. Phys. 79, 733 (2007) [arXiv:hep-th/0610102]. . S Kachru, R Kallosh, A Linde, S Trivedi, Phys. Rev. D. 6846005S. Kachru, R. Kallosh, A. Linde and S. Trivedi, Phys. Rev. D 68 (2003) 046005. . V Balasubramanian, P Berglund, J P Conlon, F Quevedo, J. High Energy Phys. 037V. Balasubramanian, P. Berglund, J. P. Conlon and F. Quevedo, J. High Energy Phys. 03 (2005) 007. . F Paige, S Protopopescu, H Baer, X Tata, hep-ph/0312045F. Paige, S. Protopopescu, H. Baer and X. Tata, hep-ph/0312045. . H E Haber, R Hempfling, A Hoang, Z. Phys. 75539H. E. Haber, R. Hempfling and A. Hoang, Z. Phys. C75 (1997) 539. . D Pierce, J Bagger, K Matchev, R Zhang, Nucl. Phys. B. 4913D. Pierce, J. Bagger, K. Matchev and R. Zhang, Nucl. Phys. B 491 (1997) 3. . G Belanger, S Kraml, A Pukhov, Phys. Rev. D. 7215003G. Belanger, S. Kraml and A. Pukhov, Phys. Rev. D 72 (2005) 015003. . By H Isared, C Baer, A Balazs, Belyaev, J. High Energy Phys. 020342IsaRED, by H. Baer, C. Balazs and A. Belyaev, J. High Energy Phys. 0203 (2002) 042. . A Pukhov, hep-ph/9908288CompHEP v.33.23CompHEP v.33.23, by A. Pukhov et al., hep-ph/9908288 (1999). . P Gondolo, G Gelmini, Nucl. Phys. 360145P. Gondolo and G. Gelmini, Nucl. Phys. B360, 145 (1991); . J Edsjö, P Gondolo, Phys. Rev. 561879J. Edsjö and P. Gondolo, Phys. Rev. D56, 1879 (1997). . B Acharya, G Kane, S Watson, P Kumar, Phys. Rev. D. 8083529B. Acharya, G. Kane, S. Watson and P. Kumar, Phys. Rev. D 80 (2009) 083529. . G Gelmini, P Gondolo, Phys. Rev. D. 7423510G. Gelmini and P. Gondolo, Phys. Rev. D 74 (2006) 023510; . G Gelmini, P Gondolo, A Soldatenko, C Yaguna, Phys. Rev. D. 7483514G. Gelmini, P. Gondolo, A. Sol- datenko and C. Yaguna, Phys. Rev. D 74 (2006) 083514; . G Gelmini, P Gondolo, A Soldatenko, C Yaguna, Phys. Rev. D. 7615010G. Gelmini, P. Gondolo, A. Soldatenko and C. Yaguna, Phys. Rev. D 76 (2007) 015010. . M Kawasaki, K Hakayama, Phys. Rev. D. 7643502M. Kawasaki and K. Hakayama, Phys. Rev. D 76 (2007) 043502. . R Kitano, H Murayama, M Ratz, Phys. Lett. B. 669145R. Kitano, H. Murayama and M. Ratz, Phys. Lett. B 669 (2008) 145. . M Bolz, A Brandenburg, W Buchmuller, Nucl. Phys. B. 606518M. Bolz, A. Brandenburg and W. Buchmuller, Nucl. Phys. B 606 (2001) 518 . L F Abbott, P Sikivie, Phys. Lett. B. 120133L. F. Abbott and P. Sikivie, Phys. Lett. B 120 (1983) 133; . J Preskill, M Wise, F Wilczek, Phys. Lett. B. 120127J. Preskill, M. Wise and F. Wilczek, Phys. Lett. B 120 (1983) 127; . M Dine, W Fischler, Phys. Lett. B. 120137M. Dine and W. Fischler, Phys. Lett. B 120 (1983) 137; . M Turner, Phys. Rev. D. 33889M. Turner, Phys. Rev. D 33 (1986) 889; . L Visinelli, P Gondolo, Phys. Rev. D. 8035024L. Visinelli and P. Gondolo, Phys. Rev. D 80 (2009) 035024. . K Jedamzik, M Lemoine, G Moultaka, 060710K. Jedamzik, M. LeMoine and G. Moultaka, JCAP0607 (2006) 010. . L Duffy, Ann. Rev. Nucl. Part. Sci. 95293Phys. Rev. Lett.L. Duffy et al., Phys. Rev. Lett. 95 (2005) 091304 and Phys. Rev. D 74 (2006) 012006; for a review, see S. Asztalos, L. Rosenberg, K. van Bibber, P. Sikivie and K. Zioutas, Ann. Rev. Nucl. Part. Sci.56 (2006) 293. . G Lazarides, Q Shafi, Phys. Lett. B. 258305G. Lazarides and Q. Shafi, Phys. Lett. B 258 (1991) 305; . K Kumekawa, T Moroi, T Yanagida, Prog. Theor. Phys. 92437K. Kumekawa, T. Moroi and T. Yanagida, Prog. Theor. Phys. 92 (1994) 437; . T Asaka, K Hamaguchi, M Kawasaki, T Yanagida, Phys. Lett. B. 46412T. Asaka, K. Hamaguchi, M. Kawasaki and T. Yanagida, Phys. Lett. B 464 (1999) 12. . W Buchmüller, P Di Bari, M Plumacher, New Jou. Phys. 643105Nucl. Phys. BW. Buchmüller, P. Di Bari and M. Plumacher, Nucl. Phys. B 643 (2002) 367 and Erratum- ibid,B793 (2008) 362; Annal. Phys. 315 (2005) 305 and New Jou. Phys.6 (2004) 105. . P Grajek, G Kane, D Phalen, A Pierce, S Watson, arXiv:0807.1508P. Ullio, J. High Energy Phys. 010653P. Grajek, G. Kane, D. Phalen, A. Pierce and S. Watson, arXiv:0807.1508; P. Ullio, J. High Energy Phys. 0106 (2001) 053. . H Baer, C Balazs, A Belyaev, J , JCAP. 03097H. Baer, C. Balazs, A. Belyaev and J. O'Farrill, JCAP 0309, (2003) 007. . Lepsusywg, note LEPSUSYWG/02-04.1LEPSUSYWG, note LEPSUSYWG/02-04.1. G Jungman, M Kamionkowski, K Griest, For reviews. 267195For reviews, see e.g. G. Jungman, M. Kamionkowski and K. Griest,Phys. Rept. 267 (1996) 195; . A Lahanas, N Mavromatos, D Nanopoulos, Int. J. Mod. Phys. D. 121529A. Lahanas, N. Mavromatos and D. Nanopoulos, Int. J. Mod. Phys. D 12 (2003) 1529; . M Drees, hep-ph/0410113M. Drees, hep-ph/0410113; . K Olive, astro-ph/0503065Tasi Lectures on Astroparticle Physics. K. Olive, "Tasi Lectures on Astroparticle Physics", astro-ph/0503065; For a recent review of axion/axino dark matter, see. G Bertone, D Hooper, J Silk, ; F Steffen, arXiv:0811.3347Phys. Rept. 405279G. Bertone, D. Hooper and J. Silk, Phys. Rept. 405 (2005) 279. For a recent review of axion/axino dark matter, see F. Steffen, arXiv:0811.3347 (2008). . J Navarro, C S Frenk, S D M White, Astrophys. J. 462563J. Navarro, C. S. Frenk and S. D. M. White, Astrophys. J. 462 (1996) 563. . H Baer, E K Park, X Tata, New Jou. Phys. 11H. Baer, E. K. Park and X. Tata, New Jou. Phys.11 (105024) 2009. ) 008; for a review, see F. Boudjema. P Gondolo, J Edsjo, P Ullio, L Bergstrom, M Schelke, E A Baltz, ; P Gondolo, arXiv:1003.4748JCAP. 0407P. Gondolo, J. Edsjo, P. Ullio, L. Bergstrom, M. Schelke and E. A. Baltz, JCAP 0407 (2004) 008; for a review, see F. Boudjema, J. Edsjo and P. Gondolo, arXiv:1003.4748. M Pearce, Pamela CollaborationProc. Suppl. Suppl113314M. Pearce (Pamela Collaboration), Nucl. Phys. 113 (Proc. Suppl.) (2002) 314. . K Mori, C J Hailey, E A Baltz, W W Craig, M Kamionkowski, W T Serber, P Ullio, Astrophys. J. 566604K. Mori, C. J. Hailey, E. A. Baltz, W. W. Craig, M. Kamionkowski, W. T. Serber and P. Ullio, Astrophys. J. 566 (2002) 604; . C J Hailey, JCAP. 06017C. J. Hailey et al., JCAP 0601 (2006) 007; . H Baer, S Profumo, JCAP. 05128H. Baer and S. Profumo, JCAP 0512 (2005) 008. . H Baer, J , JCAP. 04045H. Baer and J. O'Farrill, JCAP 0404 (2004) 005; . H Baer, A Belyaev, T Krupovnickas, J , JCAP. 04085H. Baer, A. Belyaev, T. Krupovnickas and J. O'Farrill, JCAP 0408 (2004) 005. . M Drees, M Nojiri, Phys. Rev. D. 47376M. Drees and M. Nojiri, Phys. Rev. D 47 (1993) 376; . H Baer, M Brhlik, Phys. Rev. D. 57567H. Baer and M. Brhlik, Phys. Rev. D 57 (1998) 567
[]
[ "Bayesian Nonparametric Multilevel Clustering with Group-Level Contexts", "Bayesian Nonparametric Multilevel Clustering with Group-Level Contexts" ]
[ "Vu Nguyen \nCenter for Pattern Recognition and Data Analytics (PRaDA)\nDeakin University\nAustralia\n", "Dinh Phung [email protected] \nCenter for Pattern Recognition and Data Analytics (PRaDA)\nDeakin University\nAustralia\n", "Xuanlong Nguyen [email protected] \nDepartment of Statistics\nUniversity of Michigan\nAnn ArborUSA\n", "Svetha Venkatesh [email protected] \nCenter for Pattern Recognition and Data Analytics (PRaDA)\nDeakin University\nAustralia\n", "Hung Hai Bui \nLaboratory for Natural Language Understanding\nNuance Communications\nSunnyvaleUSA\n", "Bui H Hung@gmail Com " ]
[ "Center for Pattern Recognition and Data Analytics (PRaDA)\nDeakin University\nAustralia", "Center for Pattern Recognition and Data Analytics (PRaDA)\nDeakin University\nAustralia", "Department of Statistics\nUniversity of Michigan\nAnn ArborUSA", "Center for Pattern Recognition and Data Analytics (PRaDA)\nDeakin University\nAustralia", "Laboratory for Natural Language Understanding\nNuance Communications\nSunnyvaleUSA" ]
[]
We present a Bayesian nonparametric framework for multilevel clustering which utilizes grouplevel context information to simultaneously discover low-dimensional structures of the group contents and partitions groups into clusters. Using the Dirichlet process as the building block, our model constructs a product base-measure with a nested structure to accommodate content and context observations at multiple levels. The proposed model possesses properties that link the nested Dirichlet processes (nDP) and the Dirichlet process mixture models (DPM) in an interesting way: integrating out all contents results in the DPM over contexts, whereas integrating out group-specific contexts results in the nDP mixture over content variables. We provide a Polyaurn view of the model and an efficient collapsed Gibbs inference procedure. Extensive experiments on real-world datasets demonstrate the advantage of utilizing context information via our model in both text and image domains.
null
[ "https://arxiv.org/pdf/1401.1974v4.pdf" ]
438,986
1401.1974
9531ba78c41c82e7a6c24773031c01ad45fea87e
Bayesian Nonparametric Multilevel Clustering with Group-Level Contexts 29 Jan 2014 Vu Nguyen Center for Pattern Recognition and Data Analytics (PRaDA) Deakin University Australia Dinh Phung [email protected] Center for Pattern Recognition and Data Analytics (PRaDA) Deakin University Australia Xuanlong Nguyen [email protected] Department of Statistics University of Michigan Ann ArborUSA Svetha Venkatesh [email protected] Center for Pattern Recognition and Data Analytics (PRaDA) Deakin University Australia Hung Hai Bui Laboratory for Natural Language Understanding Nuance Communications SunnyvaleUSA Bui H Hung@gmail Com Bayesian Nonparametric Multilevel Clustering with Group-Level Contexts 29 Jan 2014 We present a Bayesian nonparametric framework for multilevel clustering which utilizes grouplevel context information to simultaneously discover low-dimensional structures of the group contents and partitions groups into clusters. Using the Dirichlet process as the building block, our model constructs a product base-measure with a nested structure to accommodate content and context observations at multiple levels. The proposed model possesses properties that link the nested Dirichlet processes (nDP) and the Dirichlet process mixture models (DPM) in an interesting way: integrating out all contents results in the DPM over contexts, whereas integrating out group-specific contexts results in the nDP mixture over content variables. We provide a Polyaurn view of the model and an efficient collapsed Gibbs inference procedure. Extensive experiments on real-world datasets demonstrate the advantage of utilizing context information via our model in both text and image domains. Introduction In many situations, content data naturally present themselves in groups, e.g., students are grouped into classes, classes grouped into schools, words grouped into documents, etc. Furthermore, each content group can be associated with additional context information (teachers of the class, authors of the document, time and location stamps). Dealing with grouped data, a setting known as multilevel analysis (Hox, 2010;Diez-Roux, 2000), has diverse application domains ranging from document modeling (Blei et al., 2003) to public health (Leyland & Goldstein, 2001). This paper considers specifically the multilevel clustering problem in multilevel analysis: to jointly cluster both the content data and their groups when there is group-level context information. By context, we mean a secondary data source attached to the group of primary content data. An example is the problem of clustering documents, where each document is a group of words associated with grouplevel context information such as time-stamps, list of authors, etc. Another example is image clustering where visual image features (e.g. SIFT) are the content and image tags are the context. To cluster groups together, it is often necessary to perform dimensionality reduction of the content data by forming content topics, effectively performing clustering of the content as well. For example, in document clustering, using bag-of-words directly as features is often problematic due to the large vocabulary size and the sparsity of the in-document word occurrences. Thus, a typical approach is to first apply dimensionality reduction techniques such as LDA (Blei et al., 2003) or HDP (Teh et al., 2006b) to find word topics (i.e., distributions on words), then perform document clustering using the word topics and the document-level context information as features. In such a cascaded approach, the dimensionality reduction step (e.g., topic modeling) is not able to utilize the context information. This limitation suggests that a better alternative is to perform context-aware document clustering and topic modeling jointly. With a joint model, one can expect to obtain improved document clusters as well as context-guided content topics that are more predictive of the data. Recent work has attempted to jointly capture word topics and document clusters. Parametric approaches (Xie & Xing, 2013) are extensions of the LDA (Blei et al., 2003) and require specifying the number of topics and clusters in advance. Bayesian nonparametric approaches including the nested Dirichlet process (nDP) (Rodriguez et al., 2008) and the multi-level clustering hierarchical Dirichlet Process (MLC-HDP) (Wulsin et al., 2012) can automatically adjust the number of clusters. We note that none of these methods can utilize context data. This paper propose the Multilevel Clustering with Context (MC 2 ), a Bayesian nonparametric model to jointly cluster both content and groups while fully utilizing grouplevel context. Using the Dirichlet process as the building block, our model constructs a product base-measure with a nested structure to accommodate both content and context observations. The MC 2 model possesses properties that link the nested Dirichlet process (nDP) and the Dirichlet process mixture model (DPM) in an interesting way: integrating out all contents results in the DPM over contexts, whereas integrating out group-level context results in the nDP mixture over content variables. For inference, we provide an efficient collapsed Gibbs sampling procedure for the model. The advantages of our model are: (1) the model automatically discovers the (unspecified) number of groups clusters and the number of topics while fully utilizing the context information; (2) content topic modeling is informed by group-level context information, leading to more predictive content topics; (3) the model is robust to partially missing context information. In our experiments, we demonstrate that our proposed model achieves better document clustering performances and more predictive word topics in realworld datasets in both text and image domains. Related Background There have been extensive works on clustering documents in the literature. Due to limited scope of the paper, we only describe works closely related to probabilistic topic models. We note that standard topic models such as LDA (Blei et al., 2003) or its nonparametric Bayesian counter part, HDP (Teh et al., 2006b) exploits the group structure for word clustering. However these models do not cluster documents. An approach to document clustering is to employ a twostage process. First, topic models (e.g. LDA or HDP) are applied to extract the topics and their mixture proportion for each document. Then, this is used as feature input to another clustering algorithm. Some examples of this approach include the use of LDA+Kmeans for image clustering (Xuan et al., 2011;Elango & Jayaraman, 2005) and HDP+Affinity Propagation for clustering human activities (Nguyen et al., 2013). A more elegant approach is to simultaneously cluster documents and discover topics. The first Bayesian nonparametric model proposed for this task is the nested Dirichlet Process (nDP) (Rodriguez et al., 2008) where documents in a cluster share the same distribution over topic atoms. Although the original nDP does not force the topic atoms to be shared across document clusters, this can be achieved by simply introducing a DP prior for the nDP base measure. The same observation was also made by (Wulsin et al., 2012) who introduced the MLC-HDP, a 3-level extension to the nDP. This model thus can cluster words, documents and document-corpora with shared topic atoms throughout the group hierarchy. Xie et al (Xie & Xing, 2013) recently introduced the Multi-Grain Clustering Topic Model which allows mixing between global topics and document-cluster topics. However, this is a parametric model which requires fixing the number of topics in advance. More crucially, all of these existing models do not attempt to utilize grouplevel context information. Modelling with Dirichlet Process We provide a brief account of the Dirichlet process and its variants. The literature on DP is vast and we refer to (Hjort et al., 2010) for a comprehensive account. Here we focus on DPM, HDP and nDP which are related to our work. Dirichlet process (Ferguson, 1973) is a basic building block in Bayesian nonparametrics. Let (Θ, B, H) be a probability measure space, and γ is a positive number, a Dirichlet process DP (γ, H) is a distribution over discrete random probability measure G on (Θ, B). Sethuraman (Sethuraman, 1994) provides an alternative constructive definition which makes the discreteness property of a draw from a Dirichlet process explicit via the stick-breaking representation: G = ∞ k=1 β k δ φ k where φ k iid ∼ H, k = 1, . . . , ∞ and β = (β k ) ∞ k=1 are the weights constructed through a 'stick- breaking' process β k = v k s<k (1 − v s ) with v k iid ∼ Beta (1, γ). It can be shown that ∞ k=1 β k = 1 with probability one, and as a convention (Pitman, 2002), we hereafter write β ∼ GEM (γ). Due to its discrete nature, Dirichlet process has been widely used in Bayesian mixture models as the prior distribution on the mixing measures, each is associated with an atom φ k in the stick-breaking representation of G above. A likelihood kernel F (·) is used to generate data x i | φ k iid ∼ F (· | φ k ), resulting in a model known as the Dirichlet process mixture model (DPM), pioneered by the work of (Antoniak, 1974) and subsequently developed by many others. In section 3 we provide a precise definition for DPM. While DPM models exchangeable data within a single group, the Dirichlet process can also be constructed hierarchically to provide prior distributions over multiple exchangeable groups. Under this setting, each group is modelled as a DPM and these models are 'linked' together to reflect the dependency among them -a formalism which is generally known as dependent Dirichlet processes (MacEachern, 1999). One particular attractive approach is the hierarchical Dirichlet processes (Teh et al., 2006b) which posits the dependency among the group-level DPM by another Dirichlet process, i.e., G j | α, G 0 ∼ DP (α, G 0 ) and G 0 | γ, H ∼ DP (γ, H) where G j is the prior for the j-th group, linked together via a discrete measure G 0 whose distribution is another DP. Yet another way of using DP to model multiple groups is to construct random measure in a nested structure in which the DP base measure is itself another DP. This formalism is the nested Dirichlet Process (Rodriguez et al., 2008), specifically G j iid ∼ U where U ∼ DP (α × DP (γH)). Modeling G j (s) hierarchically as in HDP and nestedly as in nDP yields different effects. HDP focuses on exploiting statistical strength across groups via sharing atoms φ k (s), but it does not partition groups into clusters. This statement is made precisely by noting that P (G j = G j ′ ) = 0 in HDP. Whereas, nDP emphasizes on inducing clusters on both observations and distributions, hence it partitions groups into clusters. To be precise, the prior probability of two groups being clustered together is P (G j = G j ′ ) = 1 a+1 . Finally we note that this original definition of nDP in (Rodriguez et al., 2008) does not force the atoms to be shared across clusters of groups, but this can be achieved by simply introducing a DP prior for the nDP base measure, a modification that we use in this paper. This is made clearly in our definition for nDP mixture in section 3. Multilevel Clustering with Contexts Model description and stick-breaking Consider data presented in a two-level group structure as follows. Denote by J the number of groups; each group j contains N j exchangeable data points, represented by w j = w j1 , w j2 , . . . , w jNj . For each group j, the groupspecific context data is denoted by x j . Assuming that the groups are exchangeable, the overall data is {(x j , w j )} J j=1 . The collection {w 1 , . . . , w J } represents observations of the group contents, and {x 1 , . . . , x J } represents observations of the group-level contexts. We now describe the generative process of MC 2 that generates a two-level clustering of this data. We use a grouplevel DP mixture to generate an infinite cluster model for groups. Each group cluster k is associated with an atom having the form of a pair (φ k , Q * k ) where φ k is a parameter that generates the group-level contexts within the cluster and Q * k is a measure that generates the group contents within the same cluster. To generate atomic pairs of context parameter and measurevalued content parameter, we introduce a product basemeasure of the form H × DP(vQ 0 ) for the group-level DP mixture. Drawing from a DP mixture with this base measure, each realization is a pair (θ j , Q j ); θ j is then used to generate the context x j and Q j is used to repeatedly produce the set of content observations w ji within the group j. Specifically, U ∼ DP (α(H × DP(vQ 0 ))) where Q 0 ∼ DP (ηS) (θ j , Q j ) iid ∼ U for each group j (1) x j ∼ F (.|θ j ), ϕ ji iid ∼ Q j , w ji ∼ Y (.|ϕ ji ) In the above, H and S are respectively base measures for context and content parameters θ j and ϕ ji . The context and content observations are then generated via the likelihood kernels F (· | θ j ) and Y (· | ϕ ji ). To simplify inference, H and S are assumed to be conjugate to F and Y respectively. The generative process is illustrated in Figure 1. STICK-BREAKING REPRESENTATION We now derive the stick-breaking construction for MC 2 where all the random discrete measures are specified by a distribution over integers and a countable set of atoms. The random measure U in Eq. (7) has the stick-breaking form: U = ∞ k=1 π k δ (φk,Q * k )(2) where π ∼ GEM (α) and (φ k , Q * k ) iid ∼ H × DP (vQ 0 ). Equivalently, this means φ k is drawn i.i.d. from H and Q * k drawn i.i.d. from DP (vQ 0 ). Since Q 0 ∼ DP (ηS), Q 0 and Q * k have the standard HDP (Teh et al., 2006b) stick-breaking forms: Q 0 = ∞ m=1 ǫ m δ ψm where ǫ ∼ GEM(η), ψ m iid ∼ S; Q * k = ∞ m=1 τ k,m δ ψm where τ k = (τ k1 , τ k2 , . . .) ∼ DP (v, ǫ). For each group j we sample the parameter pair (θ j , Q j ) iid ∼ U ; equivalently, this means drawing z j iid ∼ π and letting θ j = φ zj and Q j = Q * zj . For the i-th content data within the group j, the content parameter ϕ ji is drawn iid ∼ Q j = Q * zj ; equivalently, this means drawing l ji iid ∼ τ zj and letting ϕ ji = ψ lji . Figure 1 presents the graphical model of this stick-breaking representation. Inference and Polya Urn View We use collapsed Gibbs sampling, integrating out φ k (s), ψ m (s), π and τ k (s). Latent variables z, l, ǫ and the hyperparameters α, v, η will be resampled. We only describe the key inference steps in sampling z, l and ǫ here and refer to Appendix A.2 for the rest of the details (including how to sample the hyper-parameters). Sampling z. The required conditional distribution is p(z j = k | z −j , l, x, α, H) ∝ p (z j = k|z −j , α) p (x j |z j = k, z −j , x −j , H) × p (l j * |z j = k, l −j * , z −j , ǫ, v) The first term can be recognized as a form of the Chinese restaurant process (CRP). The second term is the predictive likelihood for the context observations under the component φ k after integrating out φ k . This can be evaluated analytically due to conjugacy of F and H. The last term is the predictive likelihood for the group content-index l j * = {l ji |i = 1 . . . N j }. Since l ji | z j = k iid ∼ Mult (τ k ) where τ k ∼ Dir (vǫ 1 , . . . , vǫ M , ǫ new ), the last term can also be evaluated analytically by integrating out τ k using the Multinomial-Dirichlet conjugacy property. Sampling l. Let w −ji be the same set as w exclud- ing w ji , let w −ji (m) = {w j ′ i ′ |(j ′ i ′ ) = (ji) ∧ l j ′ i ′ = m} and l −ij (k) = {l j ′ i ′ |(j ′ i ′ ) = (ji) ∧ z j ′ = k}. Then p (l ji = m | l −ji , z j = k, z −j , v, w, ǫ, S) ∝ p(w ji |l, w −ji , S) p(l ji = m|l −ji , z j = k, z −j , ǫ, v) =p (w ji | w −ji (m), S) p (l ji = m | l −ji (k), ǫ, v) The first term is the predictive likelihood under mixture component ψ m after integrating out ψ m , which can be evaluated analytically due to the conjugacy of Y and S. The second term is in the form of a CRP similar to the one that arises during inference for HDP (Teh et al., 2006b). Sampling ǫ. Sampling ǫ requires information from both z and l. p (ǫ | l, z, v, η) ∝ p (l | ǫ, v, z, η) × p (ǫ | η)(3) Using a similar strategy in HDP, we introduce auxiliary variables (o km ), then alternatively sample together with ǫ: p (o km = h | ·) ∝ Stirl (h, n km ) (vǫ m ) h , h = 0, 1, . . . , n km p (ǫ | ·) ∝ ǫ η−1 new M m=1 ǫ k o km −1 m where Stirl (h, n km ) is the Stirling number of the first kind, n km is the count of seeing the pair (z j = k, l ji = m) : ∀i, j, and finally M is the current number of active content topics. It clear that o km can be sampled from a Multinomial distribution and ǫ from an (M + 1)-dim Dirichlet distribution. POLYA URN VIEW Our model exhibits a Polya-urn view using the analogy of a fleet of buses, driving customers to restaurants. Each bus represents a group and customers on the bus are data points within the group. For each bus j, z j acts as the index to the restaurant for its destination. Thus, buses form clusters at their destination restaurants according to a CRP: a new bus drives to an existing restaurant with the probability proportional to the number of other buses that have arrived at that restaurant, and with probability proportional to α, it goes to a completely new restaurant. Once all the buses have delivered customers to the restaurants, all customers at the restaurants start to behave in the same manner as in a Chinese restaurant franchise (CRF) process: customers are assigned tables according to a restaurant-specific CRP; tables are assigned with dishes ψ m (representing the content topic atoms) according to a global franchise CRP. In addition to the usual CRF, at restaurant k, a single dessert φ k (which represents the context-generating atom, drawing iid ∼ from H) will be served to all the customers at that restaurant. Thus, every customer on the same bus j will be served the same dessert φ zj . We observe three sub-CRPs, corresponding to the three DP(s) in our model: the CRP at the dish level is due to the DP (ηS), the CRP forming tables inside each restaurant is due to the DP(vQ 0 ), and the CRP aggregating buses to restaurants is due to the DP (α(H × DP(vQ 0 ))). Marginalization property We study marginalization property for our model when either the content topics ϕ ji (s) or context topics θ j (s) are marginalized out. Our main result is established in Theorem 9 where we show an interesting link to nested DP and DPM via our model. Let H be a measure over some measurable spaces (Θ, Σ). Let P be the set of all measures over (Θ, Σ), suitably endowed with some σ-algebra. Let G ∼ DP(αH) and θ i iid ∼ G. The collection (θ i ) then follows the DP mixture distribution which is defined formally below. Definition 1. (DPM) A DPM is a probability measure over Θ n ∋ (θ 1 , . . . , θ n ) with the usual product sigma algebra Σ n such that for every collection of measurable sets {(S 1 , . . . , S n ) : S i ∈ Σ, i = 1, . . . , n}: DPM(θ 1 ∈ S 1 , . . . , θ n ∈ S n |α, H) = n i=1 G (S i ) DP (dG | αH) We now state a result regarding marginalization of draws from a DP mixture with a joint base measure. Consider two measurable spaces (Θ 1 , Σ 1 ) and (Θ 2 , Σ 2 ) and let (Θ, Σ) be their product space where Θ = Θ 1 × Θ 2 and Σ = Σ 1 × Σ 2 . Let H * be a measure over the product space Θ = Θ 1 × Θ 2 and let H 1 be the marginal of H * over Θ 1 in the sense that for any measurable set A ∈ Σ 1 , H 1 (A) = H * (A × Θ 2 ). Then drawing (θ (1) i , θ(2) i ) from a DP mixture with base measure αH and marginalizing out (θ (2) i ) is the same as drawing (θ (1) i ) from a DP mixture with base measure H 1 . Formally Proposition 2. Denote by θ i the pair θ (1) i , θ (2) i , there holds DPM θ (1) 1 ∈ S 1 , . . . , θ (1) n ∈ S n | αH 1 = DPM (θ 1 ∈ S 1 × Θ 2 , . . . , θ n ∈ S n × Θ 2 | αH * ) for every collection of measurable sets {(S 1 , . . . , S n ) : S i ∈ Σ 1 , i = 1, . . . , n}. Proof. see Appendix 7. Next we give a formal definition for the nDP mixture: ϕ ji iid ∼ Q j , Q j iid ∼ U , U ∼ DP(αDP(vQ 0 )), Q 0 ∼ DP (ηS). Definition 3. (nested DP Mixture) An nDPM is a probabil- ity measure over Θ J j=1 Nj ∋ (ϕ 11 , . . . , ϕ 1N1 , . . . , ϕ JNJ ) equipped with the usual product sigma algebra Σ N1 × . . . × Σ NJ such that for every collection of measurable sets {(S ji ) : S ji ∈ Σ, j = 1, . . . , J, i = 1 . . . , N j }: nDPM(ϕ ji ∈ S ji , ∀i, j|α, v, η, S) =    J j=1 Nj i=1 Q j (S ji ) U (dQ j )    × DP (dU | αDP (vQ 0 )) DP (dQ 0 | η, S) We now have the sufficient formalism to state the marginalization result for our model. Theorem 4. Given α, H and α, v, η, S, let θ = (θ j : ∀j) and ϕ = (ϕ ji : ∀j, i) be generated as in Eq (7). Then, marginalizing out ϕ results in DPM (θ | α, H), whereas marginalizing out θ results in nDPM (ϕ|α, v, η, S). Proof. We sketch the main steps, Appendix 9 provides more detail. Let H * = H 1 × H 2 , we note that when either H 1 or H 2 are random, a result similar to Proposition 7 still holds by taking the expectation on both sides of the equality. Now let H 1 = H and H 2 = DP (vQ 0 ) where Q 0 ∼ DP(ηS) yields the proof for the marginalization of ϕ; let H 1 = DP (vQ 0 ) and H 2 = H yields the proof for the marginalization of θ. Experiments We first evaluate the model via simulation studies, then demonstrate its applications on text and image modeling using three real-world datasets. Throughout this section, unless explicitly stated, discrete data is modeled by Multinomial with Dirichlet prior, while continuous data is modeled by Gaussian (unknown mean and unknown variance) with Gaussian-Gamma prior. Simulation studies The main goal is to investigate the posterior consistency of the model, i.e., its ability to recover the true group clusters, context distribution and content topics. To synthesize the data, we use M = 13 topics which are the 13 unique letters in the ICML string "INTERNA-TIONAL CONFERENCE MACHINE LEARNING". Similar to (Griffiths & Steyvers, 2004), each topic ψ m is a distribution over 35 words (pixels) and visualized as a 7 × 5 binary image. We generate K = 4 clusters of 100 documents each. For each cluster, we choose a set of topics corresponding to letters in the each of 4 words in the ICML string. The topic mixing distribution τ k is an uniform distribution over the chosen topic letters. Each cluster is also assigned a context-generating univariate Gaussian distribution. These generating parameters are shown in Figure 2 (left). Altogether we have J = 400 documents; for each document we sample N j = 50 words and a context variable x j drawing from the cluster-specific Gaussian. We model the word w ji with Multinomial and Gaussian for context x j . After 100 Gibbs iterations, the number of context and content topics (K = 4, M = 13) are recovered correctly: the learned context atoms φ k and topic ψ m are almost identical to the ground truth (Figure 2, right) and the model successfully identifies the 4 clusters of documents with topics corresponding to the 4 words in the ICML string. To demonstrate the importance of context observation, we then run LDA and HDP with only the word observations (ignoring context) where the number of topic of LDA is set to 13. As can be seen from Figure 2 (right), LDA and HDP have problems in recovering the true topics. They cannot distinguish small differences between the overlapping character topics (e.g M vs N, or I vs T). Further analysis of the role of context in MC 2 is provided in Appendix A.3. Experiments with Real-World Datasets We use two standard NIPS and PNAS text datasets, and the NUS-WIDE image dataset. NIPS contains 1,740 documents with vocabulary size 13,649 (excluding stop words); timestamps (1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999), authors (2,037) and title information are available and used as group-level context. PNAS contains 79,800 documents, vocab size = 36,782 with publication timestamp . For NUS-WIDE we use a subset of the 13-class animals 1 comprising of 3,411 images (2,054 images for training and 1357 images for testing) with off-the-shelf features including 500-dim bag-of-word SIFT vector and 1000-dim bag-of-tag annotation vector. Text Modeling with Document-Level Contexts We use NIPS and PNAS datasets with 90% for training and 10% for held-out perplexity evaluation. We compare the perplexity with HDP (Teh et al., 2006b) where no grouplevel context can be used, and npTOT (Dubey et al., 2012) where only timestamp information can be used. We note that unlike our model, npTOT requires replication of document timestamp for every word in the document, which is somewhat unnatural. We use perplexity score (Blei et al., 2003) on 1 downloaded from http://www.ml-thu.net/~jun/data/ held-out data as performance metric, defined as 2 exp − J j=1 log p w test j | x train , w train / j N test j . To ensure fairness and comparable evaluation, only words in held-out data is used to compute the perplexity. We use univariate Gaussian for timestamp and Multinomial distributions for words, tags and authors. We ran collapsed Gibbs for 500 iterations after 100 burn-in samples. Table 1 shows the results where MC 2 achieves significant better performance. This shows that group-level context information during training provide useful guidance for the modelling tasks. Regarding the informative aspect of group-level context, we achieve better perplexity with timestamp information than with titles and authors. This may be explained by the fact that 1361 authors (among 2037) show up only once in the data while title provides little additional information than what already in that abstracts. Interestingly, without the group-level context information, our model still predicts the held-out words better than HDP. This suggests that inducing partitions over documents simultaneously with topic modelling is beneficial. Beyond the capacity of HDP and npTOT, our model can induce clusters over documents (value of K in Table 1). Figure 3 shows an example of one such document cluster discovered from NIPS data with authors as context. Our proposed model also allows flexibility in deriving useful understanding into the data and to evaluate on its predictive capacity (e.g., who most likely wrote this article, which authors work in the same research topic and so on). Another possible usage is to obtain conditional distributions among context topics φ k (s) and content topics ψ m (s). For example if the context information is timestamp, the model immediately yields the distribution over time for Figure 4 illustrates an example of a distribution over time for a content topic discovered from PNAS dataset where timestamp was used as context. This topic appears to capture a congenital disorder known as Albinism. This distribution illustrates research attention to this condition over the past 100 years from PNAS data. To seek evidence for this result, we search the term "Albinism" in Google Scholar, using the top 50 searching results and plot the histogram over time in the same figure. Surprisingly, we obtain a very close match between our results and the results from Google Scholar as evidenced in the figure. Image Clustering with Image-Level Tags We evaluate the clustering capacity of MC 2 using contexts on an image clustering task. Our dataset is NUS-WIDE described earlier. We use bag-of-word SIFT features from each image for its content. Since each image in this dataset comes with a set of tags, we exploit them as context information, hence each context observation x j is a bag-of-tag annotation vector. First we perform the perplexity evaluation for this dataset using a similar setting as in the previous section. Table 2 presents the results where our model again outperforms HDP even when no context (tags) is used for training. Next we evaluate the clustering quality of the model using the provided 13 classes as ground truth. We report performance on four well-known clustering evaluation metrics: Purity, Normalized Mutual Information (NMI), Rand-Index (RI), and Fscore (detailed in (Rand, 1971;Cai et al., 2011)). We use the following baselines for comparison: • Kmeans and Non-negative Matrix Factorization (NMF) (Lee & Seung, 1999). For these methods, we need to specify the number of clusters in advance, hence we vary this number from 10 to 40. We then report the min, max, mean and standard deviation. requires a similarity score between two documents and we use the Euclidean distance for this purpose. • Hierarchical Dirichlet Process (HDP) + AP: we first run HDP using content observations, and then apply Affinity Propagation with similarity score derived from the symmetric KL divergence between the mixture proportions from two documents. Figure 5 shows the result in which our model consistently delivers highest performance across all four metrics. For purity and NMI, our model beats all by a wide margin. To gain some understanding on the clusters of images induced by our model, we run t-SNE (Van der Maaten & Hinton, 2008), projecting the feature vectors (both content and context) onto a 2D space. For visual clarity, we randomly select 7 out of 28 clusters and display in Figure 6 where it can be seen that they are reasonably well separated. Effect of partially observed and missing data Missing and unlabelled data is commonly encountered in practical applications. Here we examine the effect of context observability on document clustering performance. To do so, we again use the NUS-WIDE 13-animal subset as described previously, then vary the amount of observing context observation x j with missing proportion ranges from 0% to 100%. Table 3 reports the result. We make two observations: a) utilizing context results in a big performance gain as evidenced in the difference between the top and bottom row of the table, and b) as the proportion of missing context starts to increase, the performance degrades gracefully up to 50% missing. This demonstrates the robustness of model against the possibility of missing context data. Missing (%) Purity Conclusion We have introduced an approach for multilevel clustering when there are group-level context information. Our MC 2 provides a single joint model for utilizing group-level contexts to form group clusters while discovering the shared topics of the group contents at the same time. We provide a collapsed Gibbs sampling procedure and perform extensive experiments on three real-world datasets in both text and image domains. The experimental results using our model demonstrate the importance of utilizing context information in clustering both at the content and at the group level. Since similar types of contexts (time, tags, locations, ages, genres) are commonly encountered in many real-world data sources, we expect that our model will also be further applicable in other domains. Our model contains a novel ingredient in DP-based Bayesian nonparametric modeling: we propose to use a base measure in the form of a product between a context-generating prior H and a content-generating prior DP(vQ 0 ). Doing this results in a new model with one marginal being the DPM and another marginal being the nDP mixture, thus establishing an interesting bridge between the DPM and the nDP. Our product base measure construction can be generalized to yield new models suitable for data presenting in more complicated nested group structures (e.g., more than 2-level deep). A. Appendix This note provides supplementary information for the main paper. It has three parts: a) the proof for the marginalization property of our proposed model, b) detailed derivations for our inference, and c) equations to show how the perplexity in the experiment was computed. A.1. Proof for Marginalization Property (Theorem 4) We start with a proposition on the marginalization result for DPM with the product measure then move on the final proof for our proposed model. A.1.1. MARGINALIZATION OF DPM WITH PRODUCT MEASURE Let H be a measure over some measurable space (Θ, Σ). Let P be the set of all measures over (Θ, Σ), suitably endowed with some σ-algebra. Let G ∼ DP(αH) be a draw from a Dirichlet process. Lemma 5. Let S 1 . . . S n be n measurable sets in Σ. We form a measurable partition of Θ, a collection of disjoint measurable sets, that generate S 1 , . . . , S n as follows. If S is a set, let S 1 = S and S −1 = Θ\S. Then S * = { n i=1 S ci i |c i ∈ {1, −1}} is a partition of Θ into a finite collection of disjoint measurable sets with the property that any S i can be written as a union of some sets in S * . Let the element of S * be A 1 . . . A n * (note n * ≤ 2 n ). Then the expectation E G [G (S 1 ) , . . . , G (S n )] = (4) n i=1 G (S i ) DP (dG | αH) (5) depends only on α and H(A i ). In other words, the above expectation can be written as a function E n (α, H(A 1 ), . . . H(A n * )). It is easy to see that since S i can always be expressed as the sum of some disjoints A i , G (S i ) can respectively be written as the sum of some G (A i ). Furthermore, by definition of a Dirichlet process, the vector (G (A 1 ) , . . . , G (A n * )) distributed according to a finite Dirichlet distribution (αH (A 1 ) , . . . , αH (A n * )), therefore the expectation E Definition 6. (DPM) A DPM is a probability measure over Θ n ∋ (θ 1 , . . . , θ n ) with the usual product sigma algebra Σ n such that for every collection of measurable sets {(S 1 , . . . , S n ) : S i ∈ Σ, i = 1, . . . , n}: DPM(θ 1 ∈ S 1 , . . . , θ n ∈ S n |α, H) = (6) G n i=1 G (S i ) DP (dG | αH) Consider two measurable spaces (Θ 1 , Σ 1 ) and (Θ 2 , Σ 2 ) and let (Θ, Σ) be their product space where Θ = Θ 1 × Θ 2 and Σ = Σ 1 × Σ 2 . We present the general theorem that states the marginal result from a product base measure. Proposition 7. Let H * be a measure over the product space Θ = Θ 1 ×Θ 2 . Let H 1 be the marginal of H * over Θ 1 in the sense that for any measurable set A ∈ Σ 1 , H 1 (A) = H * (A × Θ 2 ). Denote by θ i the pair θ (1) i , θ (2) i , then: DPM θ (1) 1 ∈ S 1 , . . . , θ (1) n ∈ S n | αH 1 = DPM (θ 1 ∈ S 1 × Θ 2 , . . . , θ n ∈ S n × Θ 2 | αH * ) for every collection of measurable sets {(S 1 , . . . , S n ) : S i ∈ Σ 1 , i = 1, . . . , n}. Proof. Since {(S 1 , . . . , S n ) : S i ∈ Σ 1 , i = 1, . . . , n} are rectangles, expanding the RHS using Definition 6 gives: RHS = G (S 1 × Θ 2 ) . . . G (S n × Θ 2 ) dDP(dG|α, H * ) Let T i = S i × Θ 2 , the above expression is the expectation of i G(T i ) when G ∼ DP (αH * ). Forming collection of the disjoint measurable sets T * = (B 1 . . . B n * ) that generates T i , then note that B i = A i × Θ 2 , and S * = (A 1 . . . A n * ) generates S i . By definition of H 1 , H 1 (A i ) = H * (A i × Θ 2 ) = H * (B i ). Using the Lemma 5 above, RHS = E n (α, H * (B 1 ) . . . H * (B n * )), while LHS = E n (α, H 1 (A 1 ) . . . H 1 (A n * )) and they are indeed the same. We note that H * can be any arbitrary measure on Θ and, in general, we do not require H * to factorize as product measure. A.1.2. MARGINALIZATION RESULT FOR OUR PROPOSED MODEL Recall that we are considering a product base-measure of the form H * = H × DP(vQ 0 ) for the group-level DP mixture. Drawing from a DP mixture with this base measure, each realization is a pair (θ j , Q j ); θ j is then used to generate the context x j and Q j is used to repeatedly generate the set of content observations w ji within the group j. Specifically, U ∼ DP (α(H × DP(vQ 0 ))) where Q 0 ∼ DP (ηS) (θ j , Q j ) iid ∼ U for j = 1, . . . , J(7) ϕ ji iid ∼ Q j , for each j and i = 1, . . . , N j In the above, H and S are respectively base measures for context and content parameters θ j and ϕ ji . We start with a definition for nested Dirichlet Process Mixture (nDPM) to proceed further. Definition 8. (nested DP Mixture) An nDPM is a probability measure over Θ J j=1 Nj ∋ (ϕ 11 , . . . , ϕ 1N1 , . . . , ϕ JNJ ) equipped with the usual product sigma algebra Σ N1 × . . . × Σ NJ such that for every collection of measurable sets {(S ji ) : S ji ∈ Σ, j = 1, . . . , J, i = 1 . . . , N j }: nDPM(ϕ ji ∈ S ji , ∀i, j|α, v, η, S) =    J j=1 Nj i=1 Q j (S ji ) U (dQ j )    × DP (dU | αDP (vQ 0 )) DP (dQ 0 | η, S) We now state the main marginalization result for our proposed model. Theorem 9. Given α, H and α, v, η, S, let θ = (θ j : ∀j) and ϕ = (ϕ ji : ∀j, i) be generated as in Eq (7). Then, marginalizing out ϕ results in DPM (θ | α, H), whereas marginalizing out θ results in nDPM (ϕ|α, v, η, S). Proof. First we make observation that if we can show Proposition 7 still holds when H 1 is random with H 2 is fixed and vice versa, then the proof required is an immediate corollary of Proposition 7 by letting H * = H 1 × H 2 where we first let H 1 = H, H 2 = DP (vQ 0 ) to obtain the proof for the first result, and then swap the order H 1 = DP (vQ 0 ) , H 2 = H to get the second result. To see that Proposition 7 still holds when H 2 is a random measure and H 1 is fixed, we let the product base measure H * = H 1 × H 2 and further let µ be a prior probability measure for H 2 , i.e, H 2 ∼ µ (·). Denote by θ i the pair θ i , θ (2) i , consider the marginalization over H 2 : H2 DPM (θ 1 ∈ S 1 × Θ 2 , . . . , θ n ∈ S n × Θ 2 | α, H * ) µ (H 2 ) = Σ2 DPM θ (1) 1 ∈ S 1 , . . . , θ (1) n ∈ S n | α, H 1 constant w.r.t H2 µ (H 2 ) = DPM θ(1)1 ∈ S 1 , . . . , θ (1) n ∈ S n | α, H 1 Σ2 µ (H 2 ) = DPM θ(1) 1 ∈ S 1 , . . . , θ (1) n ∈ S n | α, H 1 When H 1 is random and H 2 is fixed. Let λ (·) be a prior probability measure for H 1 , ie., H 1 ∼ λ (·). It is clear that Proposition 7 holds for each draw H 1 from λ (·). This complete our proof. A.1.3. ADDITIONAL RESULT FOR CORRELATION ANALYSIS IN NDPM We now consider the correlation between ϕ ik and ϕ jk ′ for arbitrary i, j, k and k ′ , i.e., we need to evaluate: P (ϕ ik ∈ A 1 , ϕ jk ′ ∈ A 2 | α, η, v, S) for two measurable sets A 1 , A 2 ∈ Σ by integrating out over all immediate random measures. We use an explicit stick-breaking representation for U where U ∼ DP (αDP (vQ 0 )) as follows U = ∞ k=1 π k δ Q * k(8) where π ∼ GEM (α) and Q * k iid ∼ DP (vQ 0 ). We use the notation δ Q * k to denote the atomic measure on measure, placing its mass at measure Q * k . For i = j, we have: P (ϕ ik ∈ A 1 , ϕ jk ′ ∈ A 2 | Q 1 , . . . , Q J ) = Q i (A 1 ) Q i (A 2 ) Sequentially take expectation over Q i and U : Qi Q i (A 1 ) Q i (A 2 ) dU (Q i ) = Qi Q i (A 1 ) Q i (A 2 ) d ∞ k=1 π k δ Q * k = k π k [Q * k (A 1 ) Q * k (A 2 )] U ∞ k=1 π k [Q * k (A 1 ) Q * k (A 2 )] dDP (U | αDP (vQ 0 )) = E k π k [Q * k (A 1 ) Q * k (A 2 )] = k E [π k ] E [Q * k (A 1 ) Q * k (A 2 )] = Q 0 (A 1 ∩ A 2 ) + Q 0 (A 1 ) Q 0 (A 2 ) v (v + 1) k E [π k ] = Q 0 (A 1 ∩ A 2 ) + Q 0 (A 1 ) Q 0 (A 2 ) v (v + 1) Integrating out Q 0 ∼ DP (vS) we get: P (ϕ ik ∈ A 1 , ϕ jk ′ ∈ A 2 | α, v, η, S) = E Q0|η,S Q 0 (A 1 ∩ A 2 ) + Q 0 (A 1 ) Q 0 (A 2 ) v (v + 1) = 1 v (v + 1) S (A 1 ∩ A 2 ) + S (A 1 ∩ A 2 ) + S (A 1 ) S (A 2 ) η (η + 1) = S (A 1 ∩ A 2 ) v (v + 1) + S (A 1 ∩ A 2 ) + S (A 1 ) S (A 2 ) v (v + 1) η (η + 1) For i = j, since Q i and Q j are conditionally independent given U , we get: P (ϕ ik ∈ A 1 , ϕ jk ′ ∈ A 2 | Q 1 , . . . , Q J ) = Q i (A 1 ) Q j (A 2 ) Let a k = Q * k (A 1 ) , b k = Q * k (A 2 ) and using Definition (8), integrating out U conditional on Q 0 with the stick-breaking representation in Eq (8): P (ϕ ik ∈ A 1 , ϕ jk ′ ∈ A 2 | vQ 0 ) = U Q i (A 1 ) dU U Q j (A 2 ) dU = E k π k Q * k (A 1 ) k ′ π k ′ Q * k ′ (A 2 ) = E (π 1 a 1 + π 2 a 2 + . . .) (π 1 b 1 + π 2 b 2 + . . .) = E k π 2 k a k b k + E   k =k ′ π k π k ′ a k b k ′   = AE k π 2 k + BE   k =k ′ π k π k ′   = A k E π 2 k + B 1 − k E π 2 k where A = E [a k b k ] = E [Q * k (A 1 ) Q * k (A 2 )] = Q 0 (A 1 ∩ A 2 ) + Q 0 (A 1 ) Q 0 (A 2 ) v (v + 1) and since Q * k (s) are iid draw from DP (vQ 0 ) we have: B = E [a k b k ′ ] = E [Q * k (A 1 ) Q * k ′ (A 2 )] = E [Q * k (A 1 )] E [Q * k ′ (A 2 )] = Q 0 (A 1 ) Q 0 (A 2 ) Lastly, since (π 1 , π 2 , . . .) ∼ GEM (α), using the property of its stick-breaking representation k E π 2 k = 1 1+α . Put things together we obtain the expression for the correlation of ϕ ik and ϕ jk ′ for i = j conditional on Q 0 as: P (ϕ ik ∈ A 1 , ϕ jk ′ ∈ A 2 | vQ 0 ) = Q 0 (A 1 ∩ A 2 ) + Q 0 (A 1 ) Q 0 (A 2 ) (1 + α) v (v + 1) + α 1 + α Q 0 (A 1 ) Q 0 (A 2 ) = Q 0 (A 1 ∩ A 2 ) (1 + α) v (v + 1) + αv (v + 1) + 1 (1 + α) v (v + 1) Q 0 (A 1 ) Q 0 (A 2 ) Next, integrating out Q 0 ∼ DP (vS) we get: P (ϕ ik ∈ A 1 , ϕ jk ′ ∈ A 2 | α, v, η, S) = αv (v + 1) + 1 (1 + α) v (v + 1) E [Q 0 (A 1 ) Q 0 (A 2 )] + E [Q 0 (A 1 ∩ A 2 )] (1 + α) v (v + 1) = αv (v + 1) + 1 (1 + α) v (v + 1) S (A 1 ∩ A 2 ) + S (A 1 ) S (A 2 ) η (η + 1) + S (A 1 ∩ A 2 ) (1 + α) v (v + 1) A.2. Model Inference Derivations We provide detailed derivations for model inference with the graphical model displayed in Fig 1. The variables φ k , ψ m , π, τ k are integrated out due to conjugacy property. We need to sample these latent variables z, l, ǫ and hyper parameters α, v, η. For convenience of notation, we denote z −j is a set of latent context variable z in all documents excluding document j, l j * is all of hidden variables l ji in document j, and l −j * is all of l in other documents rather than document j-th. SAMPLING z Sampling context index z j needs to take into account the influence of the corresponding context topics: p(z j = k | z −j , l, x, α, H) ∝ p (z j = k | z −j , α) CRP for context topic (9) × p (x j | z j = k, z −j , x −j , H) context predictive likelihood × p (l j * | z j = k, l −j * , z −j , ǫ, v) content latent marginal likelihood As R → ∞, we have p (ǫ | o, l, z, v, η) ∞ = ǫ η−1 new M m=1 ǫ K o km −1 m Finally, we sample ǫ jointly with the auxiliary variable o km by: p (o km = h | ·) ∝ Stirl (h, n km ) (vǫ m ) h , h = 0, 1, . . . , n km p(ǫ) ∝ ǫ η−1 new M m=1 ǫ K o km −1 m Sampling hyperparameters In the proposed model, there are three hyper-parameters which need to be sampled : α, v and η. SAMPLING η Using similar strategy and using technique from Escobar and West (Escobar & West, 1995), we have p (M | η, u) = Stirl (M, u) η M Γ (η) Γ (η + u) where u = m u m with u m = K o km is in the previous sampling ǫ and M is the number of active content atoms. Let η ∼ Gamma (η 1 , η 2 ). Recall that: Γ (η) Γ (η + u) = 1 0 t η (1 − t) u−1 1 + u η dt that we have just introduced an auxiliary variable t p (t | η) ∝ t η (1 − t) u−1 = Beta (η + 1, u) Therefore, p (η | t) ∝ η η1−1+M exp {−ηη 2 } × t η (1 − t) u−1 1 + u η = η η1−1+M × exp {−η(η 2 − log t)} × (1 − t) u−1 + η η1−1+M−1 exp {−η(η 2 − log t)} × (1 − t) u−1 u ∝ η η1−1+M exp {−η(η 2 − log t)} + uη η1−1+M−1 exp {−η(η 2 − log t)} = π t Gamma (η 1 + M, η 2 − log t) (16) + (1 − π t ) Gamma (η 1 + M − 1, η 2 − log t) where π t satisfies this following equation to make the above expression a proper mixture density: π t 1 − π t = η 1 + M − 1 u (η 2 − log t)(17) To re-sample η, we first sample t ∼ Beta (η + 1, u), compute π t as in equation 17, and then use π t to select the correct Gamma distribution to sample η as in Eq. 16. SAMPLING α Again sampling α is similar to Escobar et al (Escobar & West, 1995). Assuming α ∼ Gamma (α 1 , α 2 ) with the auxiliary variable t: p (t | α, K) ∝t α1 (1 − t) J−1 p (t | α, K) ∝Beta (α 1 + 1, J) J: number of document p (η | t, K) ∼π t Gamma (α 1 + K, α 2 − log(t)) + (1 − π t )Gamma (α 1 + K − 1, α 2 − log(t)) where c, d are prior parameter for sampling η following Gamma distribution and πt 1−πt = α1+K−1 J(α2−log t) SAMPLING v Sampling v is similar to sampling concentration parameter in HDP (Teh et al., 2006b). Denote o k * = m o km , where o km is defined previously during the sampling step for ǫ, n k * = m n km , where n km is the count of |{l ji | z ji = k, l ji = m}|. Using similar technique in (Teh et al., 2006b), we write: p (o 1 * , o 2 * .., o K * | v, n 1 * , ...n K * ) = K k=1 Stirl(n k * , o k * )α o k * 0 × Γ(v) Γ (v + n k * ) where the last term can be expressed as Γ(v) Γ (v + n k * ) = 1 Γ(n k * ) 1 0 b v k (1 − b k ) n k * −1 1 + n k * v db k Assuming v ∼ Gamma (v 1 , v 2 ), define the auxiliary variables b = (b k | k = 1, . . . , K) , b k ∈ [0, 1] and t = (t k | k = 1, . . . , K) , t k ∈ {0, 1} we have q (v, b, t) ∝ v v1−1+ k M k exp {−vv 1 } × K k=1 b v k (1 − b k ) M k −1 M k v t k We will sample the auxiliary variables b k , t k in accordance with v that are defined below: q(b k | v) =Beta (v + 1, o k * ) q (t k | .) =Bernoulli o k * /v 1 + o k * /v q(v | .) =Gamma v 1 + k (o k * − t k ) , v 2 − k log b k A.3. Relative Roles of Context and Content Data Regarding the inference of the cluster index z j (Eq. 9), to obtain the marginal likelihood (the third term in Eq. 9) one has to integrate out the words' topic labels l ji . In doing so, it can be shown that the sufficient statistics coming from the content data toward the inference of the topic frequencies and the clustering labels will just be the empirical word frequency from each document. As each document becomes sufficiently long, the empirical word frequency quickly concentrates around its mean by the central limit theorem (CLT), so as soon as the effect of CLT kicks in, increasing document length further will do very little in improving this sufficient statistics. Increasing the document length will probably not hurt, of course. But to what extent it contributes relative to the number of documents awaits a longer and richer story to be told. We confirm this argument by varying the document length and the number of documents in the synthetic document and see how they affect the posterior of the clustering labels. Each experiment is repeated 20 times. We record the mean and standard deviation of the clustering performance by NMI score. As can be seen from Fig 7, using context observation makes the model more robust in recovering the true document clusters. A.4. Perplexity Evaluation The standard perplexity proposed by Blei et al (Blei et al., 2003), used to evaluate the proposed model as following: perplexity w Test = exp − JTest j=1 log p w Test j JTest j=1 N Test j During individual sampling iteration t, we utilize the important sampling approach (Teh et al., 2006a) to compute p (w Test ). The posterior estimation of ψ m in a Multinomial-Dirichlet case is defined below, note that it can be in other types of conjugacies (Gelman et al., 2003) (e.g. Gaussian-Wishart, Binomial-Poisson): ψ t m,v = n t m,v + smooth V u=1 n t m,v + V × smooth τ t k,m = c k,m + vv × ǫ m M m=1 (c k,m + vv × ǫ m ) where n t m,v is number of times a word v, v ∈ {1, ..., V } is assigned to context topic ψ m in iteration t, and c k,m is the count of the set {w ji | z j = k, l ji = m, 0 ≤ j ≤ J, 0 ≤ i ≤ N j }. There is a constant smooth parameter (Asuncion et al., 2009) that influence on the count, roughly set as 0.1. Supposed that we estimate z Test j = k and l Test ji = m, then the probability p w Test j is computed as: Note: Document clustering performance is evaluated on the estimated document cluster z_j vs their groundtruth. p w Test j = N Test j i=1 1 T T t=1 τ t k, Proceedings of the 31 st International Conference on Machine Learning, Beijing, China, 2014. JMLR: W&CP volume 32. Copyright 2014 by the author(s). Figure 1 . 1Graphical model representation for the proposed model. Right figure illustrates a stick breaking representation. Figure 2 . 2Results from simulation study. Left: illustration of data generation with ground truth for context atoms are 4 univariate Gaussians centered at 2, 4, 6 and 8 respectively (different variances). Right: Our model recovers the correct 4 group clusters, their context distributions and the set of shared topics. LDA and HDP are unable to recover the true content topics without using contexts. Figure 3 . 3An example of document cluster from NIPS. Top: distribution over authors. Middle: examples of paper titles. Bottom: examples of word topics in this cluster. a topic, showing when the topic rises and falls. Figure 4 . 4Topic Albinism discovered from PNAS dataset and its conditional distribution over time using our model; plotted together with results independently searched from Google Scholar using the top 50 hits. •Figure 5 .Figure 6 . 56Affinity Propagation (AP)(Frey & Dueck, 2007): AP Clustering performance measured in purity, NMI, Rand-Index and F-score using NUS-WIDE dataset. Projecting 7 discovered clusters (among 28) on 2D using t-SNE (Van der Maaten & Hinton, 2008). G [G (S i )] depends only on α and H (A i ) (s). Figure 7 . 7Document clustering performance with different numbers of observed words and documents. m ψ t m,w Test ji where T is the number of collected Gibbs samples. MC2 on Synthetic Data J: number of document. NJ: number of word per document. NMI: normalized mutual information.Without Context Observation With Context Observation J=10 J=20 J=50 Appendix A.4 provides further details on how to derive this score from our model The first term can easily be recognized as a form of Chinese Restaurant Process (CRP):where n k −j is the number of data z j = k excluding z j , and n * −j is the count of all z, except z j . The second expression is the predictive likelihood from the context observations under the context component φ k .Specifically, let f (· | φ) and h (·) be respectively the density function for F (φ) and H, the conjugacy between F and H allows us to integrate out the mixture component parameter φ k , leaving us the conditional density of x j under the mixture component k given all the context data items exclude x j :Finally, the last term is the contribution from the multiple latent variables of corresponding topics to that context. Since l ji | z j = k iid ∼ Mult (τ k ) where τ k ∼ Dir (vǫ 1 , . . . , vǫ M , ǫ new ), we shall attempt to integrate out τ k . Using the Multinomial-Dirichlet conjugacy property we proceed to compute the last term in Eq (9) as following:is the count of topic m being assigned to context k excluding document j. Using this result, p (l j * | τ k ) is a predictive likelihood for l j * under the posterior Dirichlet parameters τ k in Eq 11 and therefore can be evaluated to be:, when sampling z j we only use M active components from the previous iteration. In summary, the conditional distribution to sample z j is given as:Implementation note: to evaluate A and B, we make use of the marginal likelihood resulted from a Multinomial-Dirichlet conjugacy.SAMPLING lLet w −ji be the same set as w excluding w ji , i.e w −ji = {w uv : u = j ∩ v = i}, then we can writeCRF for content topicThe first argument is computed as log likelihood predictive of the content with the component ψ mAnd the second term is inspired by Chinese Restaurant Franchise (CRF) as:where c k,m is the number of data point |{l ji |l ji = m, z j = k, 1 ≤ j ≤ J, 1 ≤ i ≤ N j }|. The final form to sample l ji is given as:Sampling ǫNote that sampling ǫ require both z and l.Isolating the content variables l k ji generated by the same context z j = k into one group l k j = {l ji : 1 ≤ i ≤ N j , z j = k} the first term of 15 can be expressed following:where n k * = |{w ji | z j = k, i = 1, ...N j }| and n km = |{w ji | z j = k, l ji = m, 1 ≤ j ≤ J, 1 ≤ i ≤ N j , }|.Let η r = η R , η new = R−M R η and recall that ǫ ∼ Dir (η r , . . . , η r , η new ), the last term of Eq 15 is a Dirichlet density:Using the result:Thus, Eq 15 becomes: Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. C E Antoniak, The Annals of Statistics. 26Antoniak, C.E. Mixtures of Dirichlet processes with appli- cations to Bayesian nonparametric problems. The Annals of Statistics, 2(6):1152-1174, 1974.
[]
[ "Photonic Lantern", "Photonic Lantern" ]
[ "Sergio G Leon-Saval [email protected] \nBland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia\n", "Alexander Argyros \nBland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia\n", "Joss Bland-Hawthorn \nBland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia\n", "Sergio G Leon-Saval \nBland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia\n", "Sergio G Leon-Saval \nBland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia\n", "Alexander Argyros \nBland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia\n", "Joss \nBland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia\n" ]
[ "Bland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia", "Bland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia", "Bland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia", "Bland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia", "Bland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia", "Bland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia", "Bland-Hawthorn: Institute of Photonics and Optical Sci-ence\nSchool of Physics\nUniversity of Sydney\nSydneyAustralia" ]
[]
Multimode optical fibers have been primarily (and almost solely) used as "light pipes" in short distance telecommunications and in remote and astronomical spectroscopy. The modal properties of the multimode waveguides are rarely exploited and mostly discussed in the context of guiding light. Until recently, most photonic applications in the applied sciences have arisen from developments in telecommunications. However, the photonic lantern is one of several devices that arose to solve problems in astrophotonics and space photonics. Interestingly, these devices are now being explored for use in telecommunications and are likely to find commercial use in the next few years, particularly in the development of compact spectrographs. Photonic lanterns allow for a low-loss transformation of a multimode waveguide into a discrete number of single-mode waveguides and vice versa, thus, enabling the use of single-mode photonic technologies in multimode systems. In this review, we will discuss the theory and function of the photonic lantern, along with several different variants of the technology. We will also discuss some of its applications in more detail. Furthermore, we foreshadow future applications of this technology to the field of nanophotonics.
10.1515/nanoph-2013-0035
[ "https://export.arxiv.org/pdf/1503.03269v1.pdf" ]
13,575,491
1503.03269
86ae434760388e6abddb851acd89b351f1825e23
Photonic Lantern Sergio G Leon-Saval [email protected] Bland-Hawthorn: Institute of Photonics and Optical Sci-ence School of Physics University of Sydney SydneyAustralia Alexander Argyros Bland-Hawthorn: Institute of Photonics and Optical Sci-ence School of Physics University of Sydney SydneyAustralia Joss Bland-Hawthorn Bland-Hawthorn: Institute of Photonics and Optical Sci-ence School of Physics University of Sydney SydneyAustralia Sergio G Leon-Saval Bland-Hawthorn: Institute of Photonics and Optical Sci-ence School of Physics University of Sydney SydneyAustralia Sergio G Leon-Saval Bland-Hawthorn: Institute of Photonics and Optical Sci-ence School of Physics University of Sydney SydneyAustralia Alexander Argyros Bland-Hawthorn: Institute of Photonics and Optical Sci-ence School of Physics University of Sydney SydneyAustralia Joss Bland-Hawthorn: Institute of Photonics and Optical Sci-ence School of Physics University of Sydney SydneyAustralia Photonic Lantern *Corresponding author:photonic lanternmultimode opti- cal fibersoptical fiber tapersastrophotonicsmulticore fibersspatial division multiplexing (SDM)mode-multiplexerspectroscopyspec- trographssurface plasmons Multimode optical fibers have been primarily (and almost solely) used as "light pipes" in short distance telecommunications and in remote and astronomical spectroscopy. The modal properties of the multimode waveguides are rarely exploited and mostly discussed in the context of guiding light. Until recently, most photonic applications in the applied sciences have arisen from developments in telecommunications. However, the photonic lantern is one of several devices that arose to solve problems in astrophotonics and space photonics. Interestingly, these devices are now being explored for use in telecommunications and are likely to find commercial use in the next few years, particularly in the development of compact spectrographs. Photonic lanterns allow for a low-loss transformation of a multimode waveguide into a discrete number of single-mode waveguides and vice versa, thus, enabling the use of single-mode photonic technologies in multimode systems. In this review, we will discuss the theory and function of the photonic lantern, along with several different variants of the technology. We will also discuss some of its applications in more detail. Furthermore, we foreshadow future applications of this technology to the field of nanophotonics. Introduction Photonics has been described as "molding the flow of light" [1,2] and, as such, it offers extensive options to manipulate light propagating along a fiber or within a waveguide. Examples include fiber Bragg gratings, array waveguide gratings, optical circulators, fiber couplers, ring resonators, beam shapers, switches, and many more. For the most part, these devices were developed in the context of telecommunications, and their function is restricted to single-mode propagation. Hence multimode systems have not been investigated to the same extent and have been limited in their functionality. Astrophotonics, a field that lies at the interface of photonics and astronomy, emerged in about 2001 to investigate new enabling technologies for astronomical instrumentation. While optical fibers have been in use in astronomy since the 1980s, they were restricted to merely transporting light between the telescope focus and the instrument. Large-core multimode fibers were used in order to maximize the light captured at the telescope focal plane while matching the field of view of the investigated objects. Subsequently, in the 1990s, single-mode fibers found important uses in astronomical interferometers as a way of filtering the noisy input signal and combining the signal of multiple telescopes or apertures [3][4][5]. However, for multimode fibers, more complex forms of light manipulation were not considered until only a decade ago in the context of complex filters that were being developed to suppress noise in the form of hundreds of narrow-frequency emission lines from the Earth's atmosphere. The first devices were based on fiber Bragg grating technology and thus could only operate efficiently in single-mode fibers [6]. It was thus recognized that a new photonic device was needed that would transfer properties of single-mode waveguides to a multimode system, interfacing efficiently between single-and multimode fibers. The device that achieved this we now refer to as a photonic lantern and began to emerge soon after [7]. The photonic lantern forms the interface between a multimode waveguide and a set of single-mode waveguides, and thus allows a lowloss transition from one to other as required by the function of the optical system under consideration. Most generally, it consists of a collection of single-mode waveguides (the singlemode (SM) end) that are interfaced to a multimode waveguide (multimode end (MM)) through a physical waveguide transition. In a standard fiber-based photonic lantern, an array of single-mode fibers is placed inside a secondary cladding, of lower index than both the cores and cladding of each of the optical fibers. The transition involves the cores of the SM fibers reducing in size and losing their ability to confine the light. The light thus spreads to the cladding, and becomes confined by the secondary lower index cladding, which have now become core and cladding of the final MM fiber waveguide, respectively. In this paper we outline the principles that allow photonic lanterns to operate, and discuss the mode transitions that take place, modeling approaches and design considerations. We also present an overview of different implementations of photonic lanterns using different fabrication methods. Although originally developed for Astrophotonics, photonic lanterns are finding uses in other fields that require single-to multimode interfacing, and as such we outline their use in astronomy and space photonics, and telecommunications. Theory and modeling The single-mode end of a photonic lantern comprises an array of isolated identical single-mode waveguides. The array of single-mode waveguides is a multimode system whose spatial modes are the supermodes of the array, which are all degenerate. The number of degenerate supermodes is equal to the number of waveguides [8]. Light can couple between this array of single-mode waveguides and a multimode waveguide via a gradual transition. If the transition is adiabatic, then the supermodes of the single-mode array evolve into the modes of the multimode waveguide, and vice versa. The second law of thermodynamics (brightness theorem) does not allow lossless coupling of light from an arbitrarily excited multimode system into one single-mode system. However if the two systems have the same number of degrees of freedom (i.e. same number of single-mode waveguides as modes in the multimode waveguide), then lossless coupling becomes possible by conserving the entropy of the system. This is a necessary but not sufficient condition. The parameters and properties of ideal low-loss photonic lantern are discussed in more details in the following sections. Quantum analogy We consider a lantern with M uncoupled SM waveguides, that transition to a MM fiber with M modes. How the uncoupled SM waveguide modes evolve through an adiabatic transition to become the modes of the MM waveguide can be described by analogy with the Kronig-Penney model [9,10] for the interaction of electrons in a periodic potential well. At an atomic scale, electrons begin to display wavelike properties, including interference and non-localization. This information is contained within the wavefunction which obeys the wave-equation-like Schrödinger equation. A free electron may be represented by a plane wave, whereas for conduction electrons inside a periodic crystal, as in the Kronig-Penney model, the electron wavefunction is described by Bloch functions. We can compare the photonic transition that occurs in a photonic lantern between an isolated array of M single-mode waveguides and a multimode one with M modes to a quantum mechanical (QM) system which evolves from M isolated potential wells, each with a single discrete allowed energy level, to a single broader potential with M discrete energy levels. This analogy is depicted in Fig. 1. Figure 1(A) shows the one dimensional refractive index profile (n) vs radius widely used to represent optical fibers. However, 1/n instead of n is used in the vertical axis as this allows the refractive index profile of the photonic lantern to be represented in the same fashion as the potential (V) in the 1D Kronig-Penney model. Thus we compare optical fiber cores with quantum wells as shown in Fig. 1(B). In the QM case, energy is used to define the wavefunction corresponding to discrete energy levels; however this is not applicable to the electromagnetic (EM) case of spatial modes propagating along a fiber core, as the energy of each photon is fixed (E = hf). These spatial modes in the EM case are defined by their propagation constant β = Kn eff (K being the wavenumber and n eff the effective index of the mode) and a transverse wavevector K T (see Fig. 1(A) top right). In this analogy K T (EM) and E (QM) behave qualitatively the same. We compare the change in energy of the standing wave solutions of the electron inside the quantum wells, with the change in K T of the spatial modes in the waveguide. A fiber core can be designed to have only one spatial mode (i.e. the fundamental mode) by choosing the right refractive index (n) profile. This mode has its electric field concentrated in the region of high n and hence has the highest modal effective index (n eff ) and the highest propagation constant β, hence the lowest transverse wavevector (K T ) (see Fig. 1(A)). In the QM case of an isolated potential well, only dis-crete energies for the electron wavefunction are allowed, and the electron wavefunction takes the form of standing waves. With the right potential and geometry, a potential well can be designed to allow only one discrete energy level (i.e. the ground state). These standing wave solutions of the independent quantum wells have the lowest energy (E) and typically their amplitude is concentrated in the regions of low potential (V). Hence, the ground state of the potential well becomes analogous to the fundamental mode of the waveguides. At the start of the transition ( Fig.1 (A)), each quantum well allows only one electron in its ground state (fundamental mode). The taper transition renders the quantum wells progressively narrower such that each electron's wavefunction begins to be increasingly less localized, and the energy of each state increases. With the wells closer together, the leaky "conduction" electrons behave as if confined to a periodic crystal. At the point where the taper ends, the wells have essentially vanished, and the collective behaviour of the electrons is described by M states (cf. supermodes) confined to a single broad potential well (cf. multimode core). Mode evolution and modeling In order to understand how a set of identical modes evolve into an equal number of nondegenerate modes, the entire taper transition of the photonic lantern must be modeled. The most effective approach is to model the transition at discrete points along its length, such that at each point the 2D waveguide geometry is considered. The modes at each point may be found by any number of available mode solvers, such as CUDOS MOF Utilities [11], RSoft [12] or COMSOL [13]. The geometry may be constructed and scaled in size, or more practically, the geometry may remain fixed and the wavelength changed by the same scaling factor -increasing the wavelength being equivalent to having a smaller waveguide. This approach makes the implicit assumption that the lantern only varies in scale along the tapered transition, but any deviations from this are unlikely to affect the modes of interest in most cases, and certainly do not affect the mode evolution overall. For example, air gaps between the single-mode fibers at the single-mode end of the transition (which are not present at the multimode end) will not affect the modes which are highly concentrated in the single-mode cores at that stage. A second approach is to model the entire three-dimensional transition using methods such as beam propagation (e.g. RSoft). The results of such modeling will depend on the initial conditions used in the model, and thus do not directly give information about the modes and the mode evolution but do allow an exploration of the effect of launch conditions themselves. However, such beam propagation methods may be sensitive to the exact geometry of the structure being modeled, thus the insight derived may be more qualitative than quantitative. Full threedimensional simulations using other approaches such as finite-element methods may be impractical due to the multi-scale nature of the problem, having a small wavelength and waveguide features compared to the transition length. Insight into the mode evolution can be gained by the first approach of solving for the modes along the length of the transition. An example of such a study from Ref. [9,14] is shown in Fig. 2. A 7×1 lantern with a 437 µm diameter at the single-mode end and a 90 µm diameter at the multimode end was modeled using CUDOS MOF Utilities [11] to calculate the effective indices of the modes along the photonic lantern transition. The structure was defined to be equivalent to the multimode end of the transition, with all features in the cross-section approximated as circles as required by the software. The material parameters were set to those corresponding to the chosen wavelength, and the transition was modeled by varying the wavelength rather than the geometry. The mode profile of the final modes of the structure shown in Fig. 2 were calculated using RSoft. Figure 2 shows the mode evolution for the 7×1 photonic lantern in Ref [14]. At large lantern diameters (not shown) the modes of the single-mode cores are strongly confined, and thus do not couple and remain near-degenerate. In addition, there is a near-continuum of cladding modes. As the lantern diameter decreases, the V-parameter of the single-mode cores also decreases and the modal fields expand, such that interaction between the single-mode cores increases. The resulting coupling leads the formation of non-degenerate supermodes that form from linear superposition of these original 14 modes. In Fig. 2, it can be seen that this coupling becomes significant at diameters below 190 µm. These modes continue to separate in effective index and eventually become the modes of the multimode core at the end of the transition. At the same time, the previous nearcontinuum of cladding modes is eliminated as all the modes reach cut-off; in Fig. 2 this occurs at an outer diameter of 100 µm. The number of single-mode cores determines the number of modes of interest, which must be conserved across the transition in a functioning lantern. In the example of Fig. 2, the 7 singlemode cores and the multimode core both supports 14 modes (including degeneracies). The symmetry of the structure also plays a role and in this case turns the otherwise 4-fold degenerate LP 31 mode into two 2-fold degenerate pairs LP 31a and LP 31b , one aligned with the cores, and the other with the regions between them. The former remains, but the latter is cutoff to give the desired number of modes in the multimode end of the transition. One final remark is that the fundamental mode (LP 01 ) is ordinarily expected to asymptote at the cladding index (1.4440), as it has no cut-off. In this case, the presence of the external cladding results in its effective index reducing below this value. Apart from revealing the evolution of the modes, Fig. 2 also contains information about the wavelength response of the lantern. Since it depicts the modes along the transition at a single wavelength, the modes at another wavelength would behave identically, once the appropriate scaling was accounted for. For example, a larger wavelength would exhibit the same evolution at correspondingly larger lantern diameters. This neglects only material dispersion. The mode transition seen in Fig. 2 is in some sense ideal, in that (i) all the cladding modes have been cut-off by the end of the transition, meaning the multimode end of the lantern has the correct number of modes to match the single-mode end, and (ii) there are no crossings between the core modes coming from the single-mode end and the cladding modes, which would create opportunities for loss through power transfer between the desired and undesired modes. However, the above conditions are necessary, rather than sufficient and losses can still occur. The total length of the transition must also be taken into account when designing a lantern. In every device in which crosssectional geometry changes along the propagation length, the change in the mode field of a propagating mode must be gradual for low-loss transmission. As such, it has been demonstrated numerically by the Beam Propagation Method that the losses of a photonic lantern depend on the tapering angle along the transition length [15,16]. The optimal transition length, i.e. tapering angle, will also depend on the number of modes in the photonic lantern. As in the case of conventional optical fiber tapers, the adiabaticity criterion is more severe for higher order modes [17,18]. This can be seen quantitatively in the published results in which a 3-mode lantern was shown to required a tapering angle of < 0.69º [16]; whilst a 85-mode lantern required a tapering angle of < 0.21º [15] to achieve low transmission losses. Optimum waveguide geometry So far in published results, photonic lantern core geometries have consisted of hexagonal lattices, rounded array or square lattices with the number of cores approximately equal to the number of final waveguide modes as required [14,15,[19][20][21]. However, it has been theoretically demonstrated that the best starting point for a low loss lantern design is an uncoupled core geometry that best approximates or samples the geometry of the modes of the final MM fiber [16,22,23]. Hence, the ideal arrangements and number of single-mode cores which minimize losses are unique for each photonic lantern and depend on the number of modes required in each case. When the cores are isolated or weakly coupled, their supermodes (or superposition of supermodes) should resemble the final multimode waveguide modes. In a step-index MMF that supports the LP lm mode, l is the maximum azimuthal node number, and m is the radial node number. To best approximate this mode shape, the geometry of the uncoupled single-mode waveguides must have m concentric rings, with each ring having an odd number of waveguides so as to differentiate between the cos(lφ) and sin(lφ) azimuthal dependence of the (degenerate) LP lm modes. The number of waveguides in each ring is required to be 2p+1 where p is the largest l for each radial mode number, m. For example, Fig. 3(A) shows the 15 lowest order LP modes for a standard step-index MMF, and each panel groups the modes with almost identical cut-off frequency. Fig. 3(B) shows the waveguide patterns that best approximate a MMF supporting 3, 6, 8, 10, 12, and 15 spatial modes in total, and Fig. 3(C) shows the corresponding supermodes of the 15 core array. The exact coupling from the modes in the single-mode end to the those of the multimode end and vice versa is in practice extremely complicated, and will depend on the amplitude and phase of the light in each mode or waveguide, the transition, and any perturbations or imperfections in the transition. In an adiabatic photonic lantern with no perturbations, light launched in a particular supermode as in Fig. 3 (C); i.e. launched into the corresponding set of single-mode waveguides with the correct intensity and phase, will end up in the matching MM fiber mode. Under normal operating conditions, light launched into the multimode end will couple into a set of MM fiber guided modes and then into a set of different SM waveguides with different intensities (through the corresponding superposition of supermodes). Thus standard photonic lanterns are not "true" mode (de)multiplexers and do not couple each mode of the MM waveguide into just one of the SM waveguides. However, with a careful waveguide design and geometry a photonic lantern functioning as a "true" mode (de)multiplexer (i.e. mode sorter) into the different SM waveguides could be achieved. Wavelength dependence The properties and efficiencies of these multimode to single-mode converters are indeed wavelength dependant. A essential condition for a low loss device is that the number of modes in the multimode section M m and the number of modes in the single-mode section M sm must be the same: M m =M sm . M sm is fixed by the number of single-mode cores, and does not change with wavelength so long as the cores remain singlemoded, however M m is wavelength dependent. The approximate number of unpolarized spatial modes guided by a MMF with a step-index profile is given by ! ≈ !" ! ! ! = ! ! ! ! ! !" ! ! !! !(1) where Vp m is the V-parameter from waveguide theory [14]; d m is the MM fiber core diameter; and NA m is the numerical aperture of the waveguide for a given wavelength λ. A mismatch in the number of modes (M m ≠ M sm ) will increase the transmission losses of the lantern. Light could couple to modes that become cutoff in the SM to MM direction when M m < M sm and similarly, when M m >M sm in the MM to SM direction. Thus, a perfect lantern device has an intrinsically limited bandwidth, since the number of modes could only be perfectly matched over a restricted wavelength ranges determined by the modal cut-off of the multimode waveguide at those wavelengths and the number of single-mode waveguides. Devices with a 300 nm bandwidth have been already demonstrated with a minimum 85% transmission at the edges of the usable wavelength span [15,24]. Different approaches The only way that these mode converters can be made is by making a physical transition in which the single-mode waveguides either stop acting as such, and/or cease behaving as independent uncoupled waveguides. The final aim of this physical transition is to adiabatically form a multimode waveguide in which the single-mode waveguides either vanish or form a composite waveguide formed by strong cou-pling between them. Photonic lanterns to date have been manufactured and demonstrated by three different techniques; two using optical fibers [14,15] and the third by using ultrafast laser writing techniques [20,25]. The very first photonic lantern, reported in 2005 [7], was fabricated by using a drawing tower fabrication method for photonic crystal fibers devices. This method, called the 'ferrule technique' [26], was the precursor of the photonic lanterns and was used to interface different photonic crystal fibers to single-mode standard step-index fibers. Nowadays, an all solid optical fiber splitter/combiner fabrication technique [14] is most commonly used. A bundle of single-mode fibers is inserted into a low index glass capillary tube which is then fused and tapered down in a glass processing machine to form an all solid multimode fiber at the other end ( Fig. 4 (B) and (D)). The second approach using optical fibers for the realization of photonic lanterns is by using a multicore fiber with an array of identical single-mode cores [15]. A photonic lantern can be made by tapering such a multicore fiber whilst, again, placing a low refractive index jacket around the fiber to form the cladding of the multimode fiber (Fig. 4 A). Each method has its advantages and disadvantages. The multicore fiber approach has the potential to produce photonic lanterns with a much larger number of modes in the system in a more straight forward manner. A multicore fiber with hundreds of cores can be fabricated in the same fashion as an all-solid photonic bandgap fiber [27], for example. Furthermore, the fabrication procedure for a multicore photonic lantern -one single fiber inside the low index jacket -is simpler and less prone to fabrication imperfections during the tapering process. However, when input/output are considered and/or a different photonic function is required for each single-mode core, then multicore photonic lanterns are not the appropriate choice. Photonic lanterns produced by the fiber fused bundle approach with large number of fibers are cumbersome to fabricate. Nevertheless, this approach produces lanterns that are easily interfaced and connectorized to any standard single-mode fiber system, opening a broad range of possible applications. An entirely different approach for the fabrication of photonic lanterns is based on ultrafast laser inscription (ULI) techniques. Here, laser writing is used to create waveguides in a piece of bulk glass (Fig. 4 C). The laser illumination causes an increase in the refractive index, which forms the waveguide core, and the technique allows for positioning of these cores in 3D within the bulk glass [28,29]. In this case, the isolated single-mode waveguide cores are created and gradually brought together such as they couple strongly. This creates the adiabatic optical transition required for the single-mode to multimode conversion, and the strongly coupled cores form the final multimode composite waveguide [20,21,25]. ULI photonic lanterns, also called integrated photonic lanterns, are a very versatile approach. Almost any singlemode waveguide geometry can be produced at the same time as offering the possibility to scale to very large numbers of waveguides. Their compact footprint and versatility makes them a very attractive approach for the implementation of photonic lanterns, however, interfacing to standard single-mode systems and the transmission losses need to be addressed. Applications Astronomy and Spectroscopy Multimode fibers are widely used in optical and infrared astronomy to allow many celestial sources to be observed simultaneously. These large-core fibers (50-300 µm core diameter) are used to transport light from the telescope focal plane to a remote spectrograph. Mainstream astronomy has generally avoided single-mode fibers because of the difficulty of coupling light into these efficiently. Even the best performing adaptive optics systems, which attempt to deliver a diffraction-limited beam, are unable to couple light efficiently into the Gaussian-like mode of single-mode fibers below 2500 nm [3,30,31]. Consequently, astronomers have been unable to exploit numerous technological advances in photonics over the past three decades, as these have almost entirely been based on singlemode telecommunications fiber. There are two major drivers in astronomy for the photonic lantern technology: one is the use of single-mode photonic technologies such as Fiber Bragg Gratings (FBG) [32] and Arrayed Waveguide Gratings [33,34], and the second is a push towards more compact, stable and precise next generation astronomical spectrographs working in the diffraction limited regime [35,36] and hence requiring single-mode fiber inputs. The latter also has direct applications to small satellite instrumentation and exoplanets research. An early successful application of photonic lanterns has been to suppress unwanted noise sources in astronomical observations. An OHsky suppression instrument called GNOSIS [24], installed at the 3.9 meters Anglo Australian Telescope (AAT) in Australia (Fig. 5 (A)), was made possible by photonic lantern technology. This instrument efficiently transfers the light captured and coupled into multimode fibers by the telescope into single-mode fibers through a photonic lantern. These are spliced to FBGs which suppress the OH emission lines emitted by the night sky, followed by another photonic lantern transforming the filtered light back into a multimode fiber before entering the telescope spectrograph. This improves the signal to noise ratio, having removed the noise on the way to the spectrograph and thus the quality of the scientific observations. It also highlights the fundamental premise behind the photonic lantern itself, in efficiently interfacing a multimode requirement (for light capturing efficiency) and a single-mode requirement (for the FBG light filtering). In order to exploit the full capabilities of this OH suppression concept new exciting approaches are being pursued recently with a follow up astronomical instrument called PRAXIS. Multicore fibers with 37, 55 and 120 cores are being produced for the fabrication of OH sky filters. There, the standard fiber fused photonic lanterns will be replaced with multicore photonic lanterns. The advantage will be in the ability to inscribe the gratings in all the single-mode cores (the multicore fiber) simultaneously, rather than in each single-mode fiber separately. However, the major technological challenge in this concept is the realization of high quality fiber Bragg gratings in multicore fibers with such large numbers of cores, but recent pub-lished results on the fabrication of fiber Bragg gratings in multicore fibers [15,37] have been promising. A further application relates to the spectrographs themselves. One of the fundamental design characteristics of a spectrograph is the amount of light collected, dispersed and reimaged onto a detector. Spectrographs operating at high spectral resolution (R=λ/Δλ > 20,000) are the most challenging. In low light applications there is a strong tension between the need to broaden the spectrograph input (i.e. a larger slit entrance, to allow for more light) and preserving the spectroscopic performance of the instrument (which requires a narrower slit). In 2010, the Photonic Integrated Multi-Mode Spectrograph (PIMMS) was proposed; this promised a fundamental shift in fiber-fed spectrograph design [35,36]. In this approach, the photonic lantern conversion from MM to SM is used to completely decouple the spectrograph design and performance from the light source at the MM input. Photonic lanterns can achieve throughputs equivalent to a MM fiber design with a spectral resolution of a diffractionlimited slit width, which is provided by the SM fibers. The spectrograph is designed to match the output of the array of SM fibers (whose output remains fundamentally unchanged regardless of the source at the MM input), reducing its complexity and size. This reduction in size and optical components has opened a new path towards extremely high resolution, high accuracy, and portable spectrographs. Two areas where this approach is being proposed and studied currently are in high accuracy spectrographs for exoplanet research [38] and space satellite spectroscopy [39]. Extremely precise Doppler spectroscopy is an indispensable tool to find and characterize earth-like planets; however, nearly one order of magnitude better radial velocity (RV) precision over the best current spectrographs [40,41] is required. The two major limiting factors for achieving this extremely high precision is the multimode nature of the fiber feeding the spectrograph, and the stability of the system. Transforming the multimode input into a single-mode diffraction limited slit by means of a photonic lantern eliminates and reduces, respectively, both of those problems. Firstly, by eliminating the MM fiber input at the spectrographs slit the Point Spread Function instability due to fiber illumination, guiding or scrambling will be removed [42]. Secondly, the diffraction limited input to the spectrograph from the SM fiber allows for a very compact instrument design (as shown in Fig. 6 B) providing a much easier system to control and deliver high optomechanical stability. The applications in space photonics arise from limitations in payload technology of small satellites due to their restricted mass and sizes. Earth observation and scientific missions are still a relatively minor application of small satellite technology, and the use of such satellites is primarily dominated by telecommunications. However, in recent years there has been a definite trend towards putting small things together to achieve big accomplishments. Photonic lanterns have been proposed for space-borne spectroscopy instrumentation as a solution to the mass and size limitations. By using photonic lanterns, medium to high-resolution spectrographs can be designed to fit in smaller spaces and with less optical components compared to their standard multimode fiber-fed counterparts. Small prototypes spectrographs have already been developed and tested in balloon launches. Fig. 6 (B) left panel, shows a < 90 g diffraction limited spectrograph fed by 8 single-mode fibers with a resolution of 0.45 nm and an operating wavelength range of 400 to 700 nm. Photonic lanterns are also been recently studied in astronomy for their modal mixing and scrambling properties [15,43]. These multimode/single-mode hybrid systems show interesting modal scrambling properties for use in exoplanets research and astronomical instrumentation were low NA fiber delivery is required. Furthermore, photonic lanterns can be used as NA convertors, i.e. focal ratio convertors; asymmetric photonic lanterns can be fabricated such as both ends of a multimode/singlemode/multimode device could in principle have different core sizes and input/output NAs, while conserving the etendue of the system. Telecommunications The data capacity carried by a single-mode fiber is rapidly approaching its limits [44,45]. Multiplexing has been used to increase the capacity of a single SM fiber, with various approaches including wavelength division multiplexing (WDM) and polarization. Spatial Division Multiplexing (SDM) [46] is another approach investigated that has recently being advanced by the research community, albeit being proposed many years ago. This approach can further address this capacity crunch by using a MM (few moded) fiber instead of a SM fiber, and using its spatial modes [47][48][49][50] and even high order orbital angular momentum modes in specialty fiber [51] as an additional degree of freedom to increase the number of data channels. Another SDM approach is the use of multicore fibers [52][53][54], here the spatial multiplexing is done by using the uncoupled cores of a multicore fiber as independent data channels. Recently, photonic lanterns have been studied as adiabatic mode converters (i.e. as modemultiplexers and mode-demultiplexers) for their use in SDM systems for coherent multiple-input multiple-output (MIMO) networks [55,56]. A MIMO system consists of a spatial multiplexer to couple T channels to M fiber modes, a MM fiber with M supported modes, an M modedemultiplexer to couple to an array of R coherent receivers (Coh. Rx), and electrical MIMO processing (Fig. 7 (A)). Maximal capacity in systems with T = M = R can occur only when the mode-dependent losses (MDL) are negligible [55]. Some spatial multiplexers directly excite the spatial modes and others excite an orthogonal combination of modes. In free beam systems due to the use of passive beamcombiners, phase masks, and spatial-light modulators the coupling losses (CPL) increase proportionally with the number of modes in the system. Thus, spatial multiplexers supporting large number of modes with negligible MDL and CPL are highly desirable. Photonic lanterns can be used as both the spatial multiplexer and spatial demultiplexer in this context. Optimizing the geometrical arrangement of the single-mode waveguides, as discussed in Section 2.3 along the taper transition could reduce the MDL and CPL to zero. Launching into the isolated singlemode core modes distributes the information across the MM fiber modes, this could mitigate the effect of MDL along the transmission. This approach will excite an orthogonal combination of modes such that all channels experience similar modal dependencies reducing the outage probability [57][58][59]. A further approach currently under development is the use of photonic lanterns to produce mode sorters, i.e. true mode multiplexers. It is challenging to build spatial multiplexers for few-mode fiber that excites each mode without loss since the M spatial modes of the few-mode fiber are spatially overlapping and cannot be simply separated. For instance, phase-mask spatial multiplexers convert Gaussian beams into the different spatial modes, then overlap the shaped beams onto the few-mode fiber using passive beam combining [58]. These multiplexers can have in excess of 20-dB mode selectivity, but suffer from large insertion losses from the passive splitting that increases proportionally to M. In the contrary, photonic lanterns can be nearly lossless, can scale to many modes, and they can be spliced directly o the few-mode fiber as well as easily integrated through the SM fiber pigtails. Standard photonic lanterns have no mode-selectivity since they scramble each signal launched on the SM fiber end onto a combination of the few-mode fiber modes. However, it is possible to add mode-selectivity to the photonic lantern spatial multiplexer by building the lantern with M dissimilar SM fibers rather than M identical fibers. During the adiabatic taper, the dissimilarity can control the coupling between the cores and force certain cores to evolve into specific mode groups (e.g., LP 01 or LP 11 ) [60]. Multimode Plasmonics A very fast-moving area on Nanophotonics is the study of surface plasmons (SPs) [61], a field known as Plasmonics. SPs are extremely sensitive to any change in the material parameters and/or surface topology. In addition, the exponential decay is usually so strong that most SPs exhibit transverse, sub-wavelength confinement. Due to these properties, SPs have been extensively studied in recent years and are considered especially attractive for enhancing optical phenomena, detecting biochemical events [62] and controlling light at the nanoscale [61]. The field of Plasmonics offers the potential towards higher integration densities for optical circuits by combining the high capacity in photonics and the miniaturization technologies in electronics. Photonic lantern technology and concepts could be exploited to produce for example beam splitters which are an important functionality for integrated optical circuits. Several concepts in plasmonics have already been developed, e.g., Y-shaped splitters [63], and splitters related to the multimode interference (MMI) [64], howev-er analog photonic functionalities are still far from developed. For instance, the concept of multimode SPs existence and excitation is still an open area of research. It has been shown recently that a metal surface strip can support SP polariton modes, leaky modes with phase constants which are close to those of a SP polariton travelling along an extended thin film. This study demonstrated that the propagation of light along surface plasmon waveguides is mediated by a discrete number of guided polariton modes as well as a continuum of radiation modes [65]. At a given frequency and for a sufficiently wide SP waveguide (with several SP polariton eigenmodes), one could achieve a multimode SP polariton excitation. In fact a strong indication for this are the multimode interference studies [64], which provide direct evidence for multiple guided modes. A SP photonic lantern device could indeed help to study this exciting new direction for plasmonics and nanophotonics. Furthermore, this multimode to single-mode convertors could be used for clean excitation and/or splitting of SP waveguides enhancing even further the SP capabilities for controlling light at the nanoscale. Conclusions The photonic lantern is a versatile and powerful concept, allowing the transformation of an optical multimode system into a single-mode one and enabling the use of single-mode-based photonic technologies in multimode systems for the first time. Photonic lanterns increase the functionality and possible applications of fewmoded devices and systems. We have presented the operating principle of these devices and how they can be fabricated with current and standard waveguide technologies. Furthermore, the key current applications for this multimode to single-mode convertors have been presented. Being a relatively recent development, photonic lanterns are likely to find uses in other emerging fields. These mode convertors offer the possibility of improving the light collecting ability while keeping and opening new photonic functionalities. Applications such as gas or Raman spectroscopy as well as coherent detection sys-tems like for example light detection and ranging (LIDAR), are areas in which photonic lanterns could in principle revolutionize existing instrumentation and applications. Fig. 1 . 1(A) Schematics of the Kronig-Penney model analogy for the photonic lantern. (B) Schematics of a one dimensional quantum well and a one dimensional waveguide. Fig. 2 . 2The evolution of modes throughout a 7 fiber photonic lantern. The 7 core modes are degenerate at large diameters, but become nondegenerate at smaller diameters and fill the range of neff available in the multimode core at the end of the transition. The red horizontal dashed lines indicate the core and cladding index of the final multimode core (nco = 1.444; ncl = 1.4431). (right panel) Detail of the calculated modes supported by the photonic lantern at the multimode end. Fig. 4 . 4Schematics of the three different photonic lantern fabrication approaches. (A) Multicore fiber; (B) Fiber bundle; and (C) Ultrafast laser inscription (integrated photonic lantern). (D) Optical photograph of a 7×1 fiber fused photonic lantern and (bottom left panel) an integrated photonic lantern fabricated by Dr R. Thomson (Ref [20]). Fig. 5 . 5(A) Optical photographs of the built GNOSIS instrument: (top) FBG filter tray with two 19×1 photonic lanterns spliced back to back with the OH FBG filters between them; (middle) Fully assembled GNOSIS FBG unit which consists of 7 trays like the one in the top panel image; and (bottom) the 3.9 m Anglo-Australian Telescope in Australia where the GNOSIS instrument was installed and tested. (B) On sky results of the GNOSIS instrument. Bottom plot shows a comparison of the filtered and unfiltered OH sky background. Fig. 6 . 6(A) Schematics of a single-mode fiber fed diffraction-limited spectrograph constructed with a photonic lantern. (B) (left panel) IR spectrograph with a optical resolution of R~30,000 (~0.045nm) at 1550 nm. (right panel) Extremely hand-palm compact size diffraction limited VIS spectrograph with an optical resolution of R~1,100 (~0.44 nm) at 550 nm. Fig. 7 . 7(A) Coherent SDM with electronic MIMO processing (T<=M<=R). (B) 3-and 6-moded photonic lanterns mode multiplexers for low-loss SDM coupling fabricated at the University of Sydney, optical photographs and calculated modes. Fig.3. (A) 15 lowest order step index fiber spatial modes. (B) Coupled waveguide arrays whose supermodes closely match the fiber modes. (C) 15 lowest order modes of the near optimal 15 core arrangement. AcknowledgementsThe authors acknowledge the contribution of many people to the development and implementation of this technology over its relatively short life span. A special acknowledgement has to go to Tim Birks, another one of the pioneers of this technology; Danny Noordegraaf, for his pioneering demonstration of standard fiber fused lanterns; and to Nick Fontaine, for opening the door to the telecom applications. J D Joannopoulos, S G Johnson, J N Winn, R D Meade, Photonic Crystals: Molding the Flow of Light. Princeton University Presssecond editionJ.D. Joannopoulos, S.G. Johnson, J.N. Winn, and R.D. Meade. Photonic Crystals: Molding the Flow of Light (second edition). Princeton University Press 2008. Molding the flow of light: photonics in astronomy. J Bland-Hawthorn, P Kern, Physics Today. J.Bland-Hawthorn and P.Kern. Molding the flow of light: photonics in astronomy. Physics Today May 2012; 31-37. Coupling Star light into Single-mode fiber Optics. S Shaklan, F Roddier, Appl Opt. 27S.Shaklan, F.Roddier. Coupling Star light into Sin- gle-mode fiber Optics. Appl Opt 1988;27;2334-38. P Connes, F Reynaud, Fiber tests on a radiotelescope. MerkleF. ESOP.Connes, F.Reynaud. Fiber tests on a radiotele- scope. MerkleF. (ed.) ESO, Garching 1988;1117-29. Application of optical fibers and integrated optics to astronomical interferometry. P Connes, proceedings of Integrated Optics for Astronomical Interferometry. Integrated Optics for Astronomical InterferometryGrenoble, FranceP.Connes. Application of optical fibers and integrat- ed optics to astronomical interferometry. In proceed- ings of Integrated Optics for Astronomical Interfer- ometry, Grenoble, France, October 15-16 1996;122- 127. New approach to atmospheric OH suppression using an aperiodic fiber Bragg grating. J Bland-Hawthorn, M Englund, G Edvell, Opt Express. 12J.Bland-Hawthorn, M.Englund, and G.Edvell. New approach to atmospheric OH suppression using an aperiodic fiber Bragg grating. Opt Express 2004;12;5902-09. Multimode fiber devices with singlemode performance. S G Leon-Saval, T A Birks, J Bland-Hawthorn, M Englund, Opt Letters. 30S.G.Leon-Saval, T.A.Birks, J.Bland-Hawthorn, and M.Englund. Multimode fiber devices with single- mode performance. Opt Letters 2005;30;2545-47. Multiport single-mode fiber splitters. F Ladouceur, J Love, Opt. Quantum Electron. 22F.Ladouceur and J.Love. Multiport single-mode fiber splitters. Opt. Quantum Electron 1990;22;453- 6. Photonic lanterns: a study of light propagation in multimode to single-mode converters. S G Leon-Saval, A Argyros, J Bland-Hawthorn, Opt Express. 18S.G.Leon-Saval, A.Argyros, J.Bland-Hawthorn. Photonic lanterns: a study of light propagation in multimode to single-mode converters. Opt Express 2010;18;8430-39. Introduction to Solid State Physics. C Kittel, John Wiley & SonsC.Kittel. Introduction to Solid State Physics. John Wiley & Sons 2005. Multipole method for microstructured optical fibers. II. Implementation and results. B T Kuhlmey, T P White, G Renversez, D Maystre, L C Botten, C M De Sterke, R C Mcphedran, JOSA B. 19B.T. Kuhlmey, T.P. White, G. Renversez, D. Maystre, L.C. Botten, C.M. de Sterke, R.C. McPhedran. Multipole method for microstructured optical fibers. II. Implementation and results. JOSA B 2002;19; 2331-40. Efficient multi-mode to single-mode coupling in a Photonic Lantern. D Noordegraaf, P M W Skovgaard, M D Nielsen, J Bland-Hawthorn, Opt Express. 17D.Noordegraaf, P.M.W. Skovgaard, M.D.Nielsen, and J.Bland-Hawthorn. Efficient multi-mode to sin- gle-mode coupling in a Photonic Lantern. Opt Ex- press 2009;17;1988-94. Photonic lantern" spectral filters in multi-core fiber. T A Birks, B J Mangan, A Díez, J L Cruz, D F Murphy, Opt Express. 20T.A.Birks, B.J.Mangan, A.Díez, J.L.Cruz, and D.F.Murphy. "Photonic lantern" spectral filters in multi-core fiber. Opt Express 2012;20;13996-14008. Geometric requirements for photonic lanterns in space division multiplexing. N K Fontaine, R Ryf, J Bland-Hawthorn, S G Leon-Saval, Opt Express. 20N.K.Fontaine, R.Ryf, J.Bland-Hawthorn, and S.G.Leon-Saval. Geometric requirements for pho- tonic lanterns in space division multiplexing. Opt Express 2012;20;27123-32. Optical Waveguide Theory. W Snyder, J D Love, Chapman & HallW.Snyder and J.D.Love. Optical Waveguide Theo- ry. Chapman & Hall 1983. Tapered single-mode fibers and devices part 1: adiabaticity criteria. J D Love, W M Henry, W J Stewart, R J Black, S Lacroix, F Gonthier, IEEE Proceedings PT J. 138J.D.Love, W.M.Henry, W.J.Stewart, R.J.Black, S.Lacroix, and F.Gonthier. Tapered single-mode fi- bers and devices part 1: adiabaticity criteria. IEEE Proceedings PT J 1991;138;343-354. Multi-mode to single-mode conversion in a 61 port Photonic Lantern. D Noordegraaf, P M W Skovgaard, M D Maack, J Bland-Hawthorn, R Haynes, J Laegsgaard, Opt Express. 18D.Noordegraaf, P.M.W.Skovgaard, M.D.Maack, J.Bland-Hawthorn, R.Haynes, and J.Laegsgaard. Multi-mode to single-mode conversion in a 61 port Photonic Lantern. Opt Express 2010;18; 4673-78. Ultrafast laser inscription of an integrated photonic lantern. R R Thomson, T A Birks, S G Leon-Saval, A K Kar, J Bland-Hawthorn, Opt Express. 19R.R.Thomson, T.A.Birks, S.G.Leon-Saval, A.K.Kar, and J.Bland-Hawthorn. Ultrafast laser inscription of an integrated photonic lantern. Opt Express 2011;19;5698-5705. Integrated photonic building blocks for next-generation astronomical instrumentation I: the multimode waveguide. N Jovanovic, I Spaleniak, S Gross, M V Ireland, J S Lawrence, C Miese, A Fuerbach, M J Withford, Opt Express. 20N.Jovanovic, I.Spaleniak, S.Gross, M.V.Ireland, J.S.Lawrence, C.Miese, A.Fuerbach, and M.J.Withford. Integrated photonic building blocks for next-generation astronomical instrumentation I: the multimode waveguide. Opt Express 2012;20;17029-43. Spotbased mode coupler for mode-multiplexed transmission in few-mode fiber. R Ryf, N K Fontaine, R.-J Essiambre, IEEE Summer Topicals. paper TuC3.2R.Ryf, N.K.Fontaine and R.-J.Essiambre. Spot- based mode coupler for mode-multiplexed transmis- sion in few-mode fiber. IEEE Summer Topicals 2012;paper TuC3.2. Optical-mode demultiplexing by optical MIMO filtering of spatial samples. H Bülow, IEEE Photon Technol Lett. H.Bülow. Optical-mode demultiplexing by optical MIMO filtering of spatial samples. IEEE Photon Technol Lett 2012;1045-47. GNOSIS: The First Instrument to Use Fiber Bragg Gratings for OH Suppression. C Q Trinh, S C Ellis, J Bland-Hawthorn, J S Lawrence, A J Horton, S G Leon-Saval, K Shortridge, J Bryant, S Case, M Colless, W Couch, K Freeman, H.-G Löhmannsröben, L Gers, K Glazebrook, R Haynes, S Lee, J O&apos;byrne, S Miziarski, M M Roth, B Schmidt, C G Tinney, J Zheng, The Astronomical Journal. 145C.Q. Trinh, S.C. Ellis, J. Bland-Hawthorn, J.S. Law- rence, A.J.Horton, S.G.Leon-Saval, K.Shortridge, J.Bryant, S.Case, M.Colless, W.Couch, K.Freeman, H.-G. Löhmannsröben, L. Gers, K. Glazebrook, R. Haynes, S.Lee, J.O'Byrne, S.Miziarski, M.M.Roth, B.Schmidt, C.G.Tinney, and J.Zheng. GNOSIS: The First Instrument to Use Fiber Bragg Gratings for OH Suppression. The Astronomical Journal 2013;145; 51-62. Enabling photonic technologies for seeing-limited telescopes: fabrication of integrated photonic lanterns on a chip. I Spaleniak, N Jovanovic, S Gross, M V Ireland, J Lawrence, M Withford, Proceedings of Modern Technologies in Space-and Ground-based Telescopes and Instrumentation II. Modern Technologies in Space-and Ground-based Telescopes and Instrumentation II8450I.Spaleniak, N.Jovanovic, S.Gross, M.V.Ireland, J.Lawrence, and M.Withford. Enabling photonic technologies for seeing-limited telescopes: fabrica- tion of integrated photonic lanterns on a chip. In Proceedings of Modern Technologies in Space-and Ground-based Telescopes and Instrumentation II, SPIE 2012;8450;845015-1. Splice-free interfacing of photonic crystal fibers. S G Leon-Saval, T A Birks, N Y Joly, A K George, W J Wadsworth, G Kakarantzas, P J St, Russell, Opt Letters. 30S.G.Leon-Saval, T.A.Birks, N.Y.Joly, A.K.George, W.J.Wadsworth, G.Kakarantzas, and P.St.J.Russell. Splice-free interfacing of photonic crystal fibers. Opt Letters 2005;30;1629-31. All-solid photonic bandgap fiber. F Luan, A K George, T D Hedley, G J Pearce, D M Bird, J C Knight, P J St, Russell, Opt Letters. 29F.Luan, A.K.George, T.D.Hedley, G.J.Pearce, D.M.Bird, J.C.Knight, and P.St.J.Russell. All-solid photonic bandgap fiber. Opt Letters 2004;29;2369- 71. Writing waveguides in glass with a femtosecond laser. K M Davis, K Miura, N Sugimoto, K Hirao, Opt Letters. 21K.M.Davis, K.Miura, N.Sugimoto, and K.Hirao. Writing waveguides in glass with a femtosecond la- ser. Opt Letters 1996;21;1729-31. Writing optical waveguides in bulk fused silica using 1kHz femtosecond infrared pulses. A Saliminia, N T Nguyen, M C Nadeau, S Petit, S L Chin, R Vallée, Journal Appl Phys. 93A.Saliminia, N.T.Nguyen, M.C.Nadeau, S.Petit, S.L.Chin, and R.Vallée. Writing optical waveguides in bulk fused silica using 1kHz femtosecond infrared pulses. Journal Appl Phys 2003;93;3724-31. Coupling light into few-mode optical fibers I: The diffraction limit. A J Horton, J Bland-Hawthorn, Opt Express. 15A.J.Horton and J.Bland-Hawthorn. Coupling light into few-mode optical fibers I: The diffraction limit. Opt Express 2007;15;1443-53. Using single-mode fibers to monitor fast Strehl ratio fluctuations. V Coudé Du Foresto, M Faucherre, N Hubin, P Gitton, Astronomy and Astrophysics Supplement Series. 145V.Coudé du Foresto, M.Faucherre, N.Hubin, and P.Gitton. Using single-mode fibers to monitor fast Strehl ratio fluctuations. Astronomy and Astrophys- ics Supplement Series 2000;145;305-310. . J Bland-Hawthorn, S C Ellis, S G Leon-Saval, R Haynes, M M Roth, H G Löhmannsröben, A J , J. Bland-Hawthorn, S.C. Ellis, S.G. Leon-Saval, R. Haynes, M.M. Roth, H.G. Löhmannsröben, A.J. A complex multi-notch astronomical filter to suppress the bright infrared sky. J G Horton, T A Cuby, J S Birks, P Lawrence, S D Gillingham, C Ryder, Trinh, Nat Comms. 2;581-90Horton, J.G. Cuby, T.A. Birks, J.S. Lawrence, P. Gillingham, S.D.Ryder, and C.Trinh. A complex multi-notch astronomical filter to suppress the bright infrared sky. Nat Comms 2011;2;581-90. Developing arrayed waveguide grating spectrographs for multi-object astronomical spectroscopy. N Cvetojevic, N Jovanovic, J Lawrence, M Withford, J Bland-Hawthorn, Opt Express. 20N.Cvetojevic, N.Jovanovic, J.Lawrence, M. With- ford, and J.Bland-Hawthorn. Developing arrayed waveguide grating spectrographs for multi-object as- tronomical spectroscopy. Opt Express 2012;20; 2062-71. First starlight spectrum captured using an integrated photonic micro-spectrograph. N Cvetojevic, N Jovanovic, C H Betters, J S Lawrence, S C Ellis, G Robertson, J Bland-Hawthorn, Astronomy and Astrophysics. 5441N. Cvetojevic, N. Jovanovic, C.H. Betters, J.S. Law- rence, S.C. Ellis, G. Robertson, and J. Bland- Hawthorn. First starlight spectrum captured using an integrated photonic micro-spectrograph. Astronomy and Astrophysics 2012;544;L1. Demonstration and design of a compact diffraction limited spectrograph. C H Betters, S G Leon-Saval, J Bland-Hawthorn, G Robertson, Proceedings of Modern Technologies in Space-and Ground-based Telescopes and Instrumentation II. Modern Technologies in Space-and Ground-based Telescopes and Instrumentation II8446C.H.Betters, S.G.Leon-Saval, J.Bland-Hawthorn, and G.Robertson. Demonstration and design of a compact diffraction limited spectrograph. In Pro- ceedings of Modern Technologies in Space-and Ground-based Telescopes and Instrumentation II, SPIE 2012;8446; 84463H-84463H-9. PIMMS: photonic integrated multimode microspectrograph. J Bland-Hawthorn, J S Lawrence, G Robertson, S Campbell, B Pope, C H Betters, S G Leon-Saval, T A Birks, R Haynes, N Cvetojevic, N Jovanovic, Proceedings of Ground-based and Airborne Instrumentation. Ground-based and Airborne InstrumentationSPIE. 7735;77350N.J.Bland-Hawthorn, J.S.Lawrence, G.Robertson, S.Campbell, B.Pope, C.H.Betters, S.G.Leon-Saval, T.A.Birks, R.Haynes, N.Cvetojevic, and N. Jo- vanovic. PIMMS: photonic integrated multimode microspectrograph. In Proceedings of Ground-based and Airborne Instrumentation for Astronomy III, SPIE 2012;7735;77350N. Multicore fiber with integrated fiber Bragg gratings for background-free Raman sensing. S Dochow, I Latka, M Becker, R Spittel, J Kobelke, K Schuster, A Graf, S Brückner, S Unger, M Rothhardt, B Dietzek, C Krafft, J Popp, Opt Express. 20S.Dochow, I.Latka, M.Becker, R.Spittel, J.Kobelke, K.Schuster, A.Graf, S.Brückner, S.Unger, M.Rothhardt, B.Dietzek, C.Krafft, and J.Popp. Mul- ticore fiber with integrated fiber Bragg gratings for background-free Raman sensing. Opt Express 2012;20;20156-69. Single mode, extreme precision Doppler spectrographs. C Schwab, S G Leon-Saval, C H Betters, J Bland-Hawthorn, S Mahadevan, arXiv:1212.4867Proceedings of IAU Proceedings. IAU ProceedingsC.Schwab, S.G.Leon-Saval, C.H.Betters, J.Bland- Hawthorn, S.Mahadevan. Single mode, extreme pre- cision Doppler spectrographs. In Proceedings of IAU Proceedings, General Assembly 2012;arXiv: 1212.4867. NanoSpec: a diffraction limited micro-spectrograph for pico-and nano-satellites. C H Betters, S G Leon-Saval, J Bland-Hawthorn, Proceedings of Space Telescopes and Instrumentation. Space Telescopes and Instrumentation844284424C.H.Betters, S.G.Leon-Saval and J.Bland-Hawthorn. NanoSpec: a diffraction limited micro-spectrograph for pico-and nano-satellites. In Proceedings of Space Telescopes and Instrumentation, SPIE 2012;8442;84424B. . M Mayor, F Pepe, D Queloz, F Bouchy, G Rupprecht, G Lo Curto, G Avila, W Benz, J.-L Bertaux, X Bonfils, Th, H Dall, B Dekker, M.Mayor, F.Pepe, D.Queloz, F.Bouchy, G.Rupprecht, G.Lo Curto, G.Avila, W.Benz, J.- L.Bertaux, X.Bonfils, Th.Dall, H.Dekker, B. W Delabre, M Eckert, A Fleury, D Gilliotte, J Gojak, D Guzman, J.-L Kohler, A Lizon, C Longinotti, D Lovis, L Megevand, J Pasquini, J.-P Reyes, D Sivan, R Sosnowska, S Soto, A Udry, L Van Kesteren, U Weber, Weilenmann, Setting New Standards with HARPS. The Messenger. 114Delabre, W.Eckert, M.Fleury, A.Gilliotte, D. Gojak, J.Guzman, D.Kohler, J.-L.Lizon, A.Longinotti, C.Lovis, D.Megevand, L.Pasquini, J. Reyes, J.- P.Sivan, D.Sosnowska, R.Soto, S.Udry, A.van Kes- teren, L.Weber, U. Weilenmann. Setting New Standards with HARPS. The Messenger (ISSN0722- 6691) 2003;114;20-24. . F A Pepe, S Cristiani, R Lopez, N C Santos, A Amorim, G Avila, W Benz, F.A.Pepe, S.Cristiani, R.Rebolo Lopez, N.C.Santos, A.Amorim, G.Avila, W.Benz; . P Bonifacio, P.Bonifacio; . A Cabral; P.Carvas; R.Cirami, ; J Coelho, A.Cabral; P.Carvas; R.Cirami; J.Coelho; . M Comari, M.Comari; . I Coretti; V.De, Caprio, I.Coretti; V.De Caprio; . H Dekker, H.Dekker; . B Delabre, B.Delabre; . P , Di Marcantonio, P.Di Marcantonio; . V Odorico, V.D'Odorico; . M Fleury; R.García, ; J. Herreros Linares, M.Fleury; R.García; J. Herreros Linares; . I Hughes; O.Iwert, ; J Lima, ; J Lizon, I.Hughes; O.Iwert; J.Lima; J. Li- zon; . G Lo Curto, ; C Lovis, ; A Manescau, ; C Martins, G.Lo Curto; C.Lovis; A.Manescau; C. Martins; . D Mégevand; A.Moitinho, ; P Molaro, D.Mégevand; A.Moitinho; P.Molaro; . M Monteiro, M. Monteiro; . L Pasquini; C.Mordasini, ; D Queloz, ; J Rasilla, L.Pasquini; C.Mordasini; D.Queloz; J.Rasilla; . J M S Rebordão; S, ; P Tschudi, ; D Santin, Sosnowska, J.M.Rebordão; S.S.Tschudi; P.Santin; D.Sosnowska; . P Spanò, ; F Tenegi, P.Spanò; F.Tenegi; . S Udry, ; E Vanzella, ; M Viel, S.Udry; E.Vanzella; M.Viel; ESPRESSO: the Echelle spectrograph for rocky exoplanets and stable spectroscopic observations. M Osorio, ; F Zerbi, Proceedings of Ground-based and Airborne Instrumentation for Astronomy III. Ground-based and Airborne Instrumentation for Astronomy III77350M.Zapatero Osorio; F.Zerbi. ESPRESSO: the Echelle spectrograph for rocky exoplanets and stable spectroscopic observations. In Proceedings of Ground-based and Airborne Instrumentation for As- tronomy III, SPIE 2010;7735;77350F. Use and Limitations of Single-and Multi-Mode Optical Fibers for Exoplanet Detection. Recent Progress in Optical Fiber Research. J F P Spronck, D A Fischer, Z A Kaplan, Moh. YasinJ.F.P.Spronck, D.A.Fischer and Z.A. Kaplan. Use and Limitations of Single-and Multi-Mode Optical Fibers for Exoplanet Detection. Recent Progress in Optical Fiber Research, Dr Moh. Yasin (Ed.), ISBN: 978-953-307-823-6. 1:61 photonic lanterns for astrophotometry: a performance study. J.-C Olaya, S G Leon-Saval, D Schirdewahn, K Ehrlich, D M Haynes, R Haynes, Monthly Notices of the Royal Astronomical Society. 427J.-C.Olaya, S.G.Leon-Saval, D. Schirdewahn, K. Ehrlich, D.M.Haynes, and R.Haynes. 1:61 photonic lanterns for astrophotometry: a performance study. Monthly Notices of the Royal Astronomical Society 2012;427;1194-1208. Plenary paper: The coming capacity crunch. A Chraplyvy, European Conference on Optical Communication. 1A.Chraplyvy. Plenary paper: The coming capacity crunch. In European Conference on Optical Com- munication, ECOC 2009;1. Capacity limits of optical fiber networks. R.-J Essiambre, G Kramer, P J Winzer, G J Foschini, B Goebel, Journal of Lightwave Technol. 28R.-J.Essiambre, G.Kramer, P.J.Winzer, G.J. Foschi- ni, and B.Goebel. Capacity limits of optical fiber networks. Journal of Lightwave Technol 2010;28;662-701. Space Division Multiplexing in Optical Fibers. D J Richardson, J M Fini, L E Nelson, Nature Photonics. 7D.J.Richardson, J.M.Fini and L.E.Nelson. Space Division Multiplexing in Optical Fibers. Nature Photonics 2013;7;354-362. Mode division multiplexing in optical fibers. S Berdague, P Facq, App Optics. 21S.Berdague, P.Facq. Mode division multiplexing in optical fibers. App Optics 1982;21;1950-55. 6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization. S Randel, R Ryf, A Sierra, P J Winzer, A H Gnauck, C A Bolle, R.-J Essiambre, D W Peckham, A Mccurdy, R Lingle, Opt. Express. 19S.Randel, R.Ryf, A.Sierra, P.J.Winzer, A.H.Gnauck, C.A. Bolle, R.-J.Essiambre, D.W. Peckham, A. McCurdy, and R.Lingle. 6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fi- ber enabled by 6×6 MIMO equalization. Opt. Ex- press 2011;19;16697-707. . R Ryf, S Randel, A H Gnauck, C A Bolle, A Sierra, S Mumtaz, M Esmaeelpour, E C Burrows, R.-J , R.Ryf, S.Randel, A.H.Gnauck, C.A.Bolle, A.Sierra, S.Mumtaz, M.Esmaeelpour, E.C. Burrows, R.-J. Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing. P J Essiambre, D W Winzer, A Peckham, R Mccurdy, Lingle, Journal of Lightwave Technol. 30Essiambre, P.J.Winzer, D.W. Peckham, A.McCurdy, and R.Lingle. Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing. Journal of Lightwave Technol 2012;30; 521-31. Modedivision multiplexed transmission with inline fewmode fiber amplifier. N Bai, E Ip, Y Huang, E Mateo, F Yaman, M.-J Li, S Bickham, S Ten, J Liñares, C Montero, V Moreno, X Prieto, V Tse, K M Chung, A P T Lau, H.-Y Tam, C Lu, Y Luo, G.-D Peng, G Li, T Wang, Opt Express. 20N.Bai, E.Ip, Y.Huang, E.Mateo, F.Yaman, M.-J. Li, S.Bickham, S.Ten, J.Liñares, C.Montero, V.Moreno, X.Prieto, V.Tse, K.M.Chung, A.P.T.Lau, H.-Y.Tam, C.Lu, Y.Luo, G.-D.Peng, G.Li, and T.Wang. Mode- division multiplexed transmission with inline few- mode fiber amplifier. Opt Express 2012;20;2668-80. Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers. N Bozinovic, Y Yue, Y Ren, M Tur, P Kristensen, H Huang, A E Willner, S Ramachandran, Science. N.Bozinovic, Y.Yue, Y.Ren, M.Tur, P.Kristensen, H.Huang, A.E. Willner, S. Ramachandran. Terabit- Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers. Science 2013;1545-1548. Multicore opticl fiber. S Inao, T Sato, S Sentsui, T Kuroha, Y Nishimura, Proceedings of Optical Fiber Communications Conference. Optical Fiber Communications Conference1S.Inao, T.Sato, S.Sentsui, T.Kuroha, and Y. Nishi- mura. Multicore opticl fiber. In Proceedings of Opti- cal Fiber Communications Conference, OFC 1979;WB1;46-8. 12-core fiber with one ring structure for extremely large capacity transmission. S Matsuo, Y Sasaki, T Akamatsu, I Ishida, K Takenaga, K Okuyama, K Saitoh, M Kosihba, Opt Express. 20S.Matsuo, Y.Sasaki, T.Akamatsu, I.Ishida, K. Takenaga, K.Okuyama, K.Saitoh, and M.Kosihba. 12-core fiber with one ring structure for extremely large capacity transmission. Opt Express 2012;20;28398-408. High-Capacity Space-Division-Multiplexed DWDM Transmissions Using Multicore Fiber. B Zhu, J M Fini, M F Yan, X Liu, S Chandrasekhar, T F Taunay, M Fishteyn, E M Monberg, F V Dimarcello, Journal of Lightwave Technol. 30B.Zhu, J.M. Fini, M.F. Yan, X. Liu, S. Chandrasek- har, T. F. Taunay, M. Fishteyn, E.M. Monberg, and F.V.Dimarcello. High-Capacity Space-Division- Multiplexed DWDM Transmissions Using Multi- core Fiber. Journal of Lightwave Technol 2012;30;486-492. MIMO capacities and outage probabilities in spatially multiplexed optical transport systems. P J Winzer, G J Foschini, Opt Express. 19P.J.Winzer and G.J.Foschini. MIMO capacities and outage probabilities in spatially multiplexed optical transport systems. Opt Express 2011;19;16680-96. Dual-LP 11 mode 4x4 MIMO-OFDM transmission over a two-mode fiber. A Amin, A Li, S Chen, X Chen, G Gao, W Shieh, Opt Express. 19A.Al Amin, A.Li, S.Chen, X.Chen, G.Gao, and W.Shieh. Dual-LP 11 mode 4x4 MIMO-OFDM transmission over a two-mode fiber. Opt Express 2011;19;16672-79. Statistics of group delays in multimode fiber with strong mode coupling. K Ho, J M Kahn, Journal of Lightwave Technol. 29K.Ho and J.M.Kahn. Statistics of group delays in multimode fiber with strong mode coupling. Journal of Lightwave Technol 2011;29;3119-28. Low-Loss mode coupler for Mode-Multiplexed transmission in Few-Mode fiber. R Ryf, M A Mestre, A Gnauck, S Randel, C Schmidt, R Essiambre, P Winzer, R Delbue, P Pupalaikis, A Sureka, Y Sun, X Jiang, D Peckham, A H Mccurdy, R Lingle, Proceedings of Optical Fiber Communications Conference. Optical Fiber Communications ConferencePDP5B.5R.Ryf, M.A.Mestre, A.Gnauck, S.Randel, C. Schmidt, R.Essiambre, P.Winzer, R.Delbue, P. Pu- palaikis, A.Sureka, Y.Sun, X.Jiang, D.Peckham, A.H. McCurdy, and R.Lingle. Low-Loss mode cou- pler for Mode-Multiplexed transmission in Few- Mode fiber. In Proceedings of Optical Fiber Com- munications Conference, OFC 2012; PDP5B.5. Coherent Multimode-Fiber MIMO transmission with spatial constellation modulation. H Bülow, H Al-Hashimi, B Schmauss, European Conference on Optical Communications. Tu.5.B.3H.Bülow, H.Al-Hashimi, and B.Schmauss. Coherent Multimode-Fiber MIMO transmission with spatial constellation modulation. In European Conference on Optical Communications, ECOC 2011; Tu.5.B.3. Fontaine et al in preparation for. S G Leon-Saval, N K , Optics Letters. S.G.Leon-Saval, N.K.Fontaine et al in preparation for Optics Letters (2013). Surface plasmon subwavelength optics. W L Barnes, A Dereux, T W Ebbesen, Nature. 4246950W.L.Barnes, A.Dereux, and T.W.Ebbesen. Surface plasmon subwavelength optics. Nature 2003; 424(6950); 824-830. Surface Plasmon Resonance Sensors for Detection of Chemical and Biological Species. J Homola, Chem. Rev. 1082J.Homola. Surface Plasmon Resonance Sensors for Detection of Chemical and Biological Species. Chem. Rev., 2008;108(2);462-493. Wideband Ysplitter and aperture-assisted coupler based on subdiffraction confined plasmonic slot waveguides. Z Han, A Y Elezzabi, V Van, Appl. Phys. Lett. 131106Z.Han, A.Y.Elezzabi and V.Van. Wideband Y- splitter and aperture-assisted coupler based on sub- diffraction confined plasmonic slot waveguides. Appl. Phys. Lett. 2010;96;131106:1-3. Plasmonic multi-mode interference couplers. Y Tsai, A Degiron, N M Jokerst, D R Smith, Opt. Express. 17Y.Tsai, A.Degiron, N.M.Jokerst and D.R.Smith. Plasmonic multi-mode interference couplers. Opt. Express 2009;17;17471-17482. Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides. R Zia, J A Schuller, M L Brongersma, Phys. Rev. B. 165415R.Zia, J.A.Schuller and M.L.Brongersma. Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides. Phys. Rev. B 2006;74;165415.
[]
[ "The bipartite unconstrained 0-1 quadratic programming problem: polynomially solvable cases *", "The bipartite unconstrained 0-1 quadratic programming problem: polynomially solvable cases *" ]
[ "Abraham P Punnen \nDepartment of Mathematics\nSimon Fraser University Surrey\n250-13450 102nd AVV3T 0A3Central City, SurreyBritish ColumbiaCanada\n", "Piyashat Sripratak \nDepartment of Mathematics\nSimon Fraser University Surrey\n250-13450 102nd AVV3T 0A3Central City, SurreyBritish ColumbiaCanada\n", "Daniel Karapetyan \nDepartment of Mathematics\nSimon Fraser University Surrey\n250-13450 102nd AVV3T 0A3Central City, SurreyBritish ColumbiaCanada\n" ]
[ "Department of Mathematics\nSimon Fraser University Surrey\n250-13450 102nd AVV3T 0A3Central City, SurreyBritish ColumbiaCanada", "Department of Mathematics\nSimon Fraser University Surrey\n250-13450 102nd AVV3T 0A3Central City, SurreyBritish ColumbiaCanada", "Department of Mathematics\nSimon Fraser University Surrey\n250-13450 102nd AVV3T 0A3Central City, SurreyBritish ColumbiaCanada" ]
[]
We consider the bipartite unconstrained 0-1 quadratic programming problem (BQP01) which is a generalization of the well studied unconstrained 0-1 quadratic programming problem (QP01). BQP01 has numerous applications and the problem is known to be MAX SNP hard. We show that if the rank of an associated m × n cost matrix Q = (q ij ) is fixed, then BQP01 can be solved in polynomial time. When Q is of rank one, we provide an O(n log n) algorithm and this complexity reduces to O(n) with additional assumptions. Further, if q ij = a i + b j for some a i and b j , then BQP01 is shown to be solvable in O(mn log n) time. By restricting m = O(log n), we obtain yet another polynomially solvable case of BQP01 but the problem remains MAX SNP hard if m = O( k √ n) for a fixed k. Finally, if the minimum number of rows and columns to be deleted from Q to make the remaining matrix non-negative is O(log n) then we show that BQP01 polynomially solvable but it is NP-hard if this number is O( k √ n) for any fixed k.
10.1016/j.dam.2015.04.004
[ "https://arxiv.org/pdf/1212.3736v3.pdf" ]
14,741,213
1212.3736
b00b0f1d083358f50a7b24e2d553fadc5a85063b
The bipartite unconstrained 0-1 quadratic programming problem: polynomially solvable cases * 28 Apr 2014 Abraham P Punnen Department of Mathematics Simon Fraser University Surrey 250-13450 102nd AVV3T 0A3Central City, SurreyBritish ColumbiaCanada Piyashat Sripratak Department of Mathematics Simon Fraser University Surrey 250-13450 102nd AVV3T 0A3Central City, SurreyBritish ColumbiaCanada Daniel Karapetyan Department of Mathematics Simon Fraser University Surrey 250-13450 102nd AVV3T 0A3Central City, SurreyBritish ColumbiaCanada The bipartite unconstrained 0-1 quadratic programming problem: polynomially solvable cases * 28 Apr 2014arXiv:1212.3736v3 [math.OC]quadratic programming0-1 variablespolynomial algorithmscomplexitypseudo-Boolean programming We consider the bipartite unconstrained 0-1 quadratic programming problem (BQP01) which is a generalization of the well studied unconstrained 0-1 quadratic programming problem (QP01). BQP01 has numerous applications and the problem is known to be MAX SNP hard. We show that if the rank of an associated m × n cost matrix Q = (q ij ) is fixed, then BQP01 can be solved in polynomial time. When Q is of rank one, we provide an O(n log n) algorithm and this complexity reduces to O(n) with additional assumptions. Further, if q ij = a i + b j for some a i and b j , then BQP01 is shown to be solvable in O(mn log n) time. By restricting m = O(log n), we obtain yet another polynomially solvable case of BQP01 but the problem remains MAX SNP hard if m = O( k √ n) for a fixed k. Finally, if the minimum number of rows and columns to be deleted from Q to make the remaining matrix non-negative is O(log n) then we show that BQP01 polynomially solvable but it is NP-hard if this number is O( k √ n) for any fixed k. Introduction The unconstrained 0-1 quadratic programming problem (QP01) is to Maximize f (x) = x T Q ′ x + c ′ x + c ′ 0 Subject to x ∈ {0, 1} n , where Q ′ is an n × n matrix, c ′ is a row vector in R n , and c ′ 0 is a constant. The problem QP01 is formulated and studied in various alternative formats. For example, without loss of generality, one can assume that c 0 is zero, c ′ is the zero vector and Q is symmetric. If c ′ is allowed to be arbitrary, then the diagonal elements of Q can be assumed to be zero, without loss of generality. In another variation, x is assumed to be in {−1, 1} n instead of {0, 1} n . However, this variation can be reduced to the 0-1 version using a linear transformation. QP01 has been studied extensively in literature because of its various applications and rich theoretical structure. For further details and additional references on QP01 we refer to [5,6,7,10,11,17,25,37]. The focus of this paper is on a problem closely related to QP01 which we call the bipartite unconstrained 0-1 quadratic programming problem (BQP01). The problem BQP01 can be defined as follows. Maximize f (x, y) = x T Qy + cx + dy + c 0 Subject to x ∈ {0, 1} m , y ∈ {0, 1} n , where Q = (q ij ) is an m × n matrix, c = (c 1 , c 2 , . . . , c m ) is a row vector in R m , d = (d 1 , d 2 , . . . , d n ) is a row vector in R n and c 0 is a constant. Without loss of generality, we assume that m ≤ n. Let Q = O m×m Q O n×m O n×n ,c = c d and w T = x y ,(1) where O n×n , O n×m and O m×m are zero matrices. Then BQP01 can be formulated as the QP01 Maximize f (w) = w TQ w +cw + c 0 (2) Subject to w ∈ {0, 1} m+n . ( Thus, the exact and heuristic algorithms available to solve QP01 can be used directly to solve BQP01. However, such a transformation increases the problem size and could limit our ability in handling large scale problems. Further, BQP01 has additional structure that can be exploited to obtain efficient algorithms and derive interesting theoretical properties. We can also show that QP01 is a special case of BQP01. Choose Q = Q ′ + 2MI, c = 1 2 c ′ − Mu and d = 1 2 c ′ − Mu,(4) where I is an n × n identity matrix, u ∈ R n is an all one row vector and M is a very large number. Then the resulting BQP01 solves QP01 since the penalty parameter M forces x i = y i in an optimal solution of this modified BQP01. The transformation discussed in equation (4) is important since it provides additional flexibility in developing algorithms for QP01 through BQP01 formulations. This provides additional motivation to study BQP01. Consider the continuous relaxation BQPC of BQP01 given by: BQPC: Maximize x T Qy + cx + dy + c 0 Subject to x ∈ [0, 1] m , y ∈ [0, 1] n , Note that multilinear polynomials in binary variables attain their maximum at a vertex of the unit cube [6,33]. Thus, BQPC and BQP01 are equivalent. BQPC is a special case of the bilinear programming problem (BLP) [3,20,38] and hence BQP01 can also be solved using any algorithm for solving the BLP. However, by exploiting the special structure of BQPC, more efficient algorithms may be obtained. To the best of our knowledge, BQP01 has not been studied in literature from the point of view of bilinear programs by exploiting the inherent structure of the problem. A graph theoretic interpretation of BQP01 can be given as follows. Let V 1 = {1, 2, . . . , m} and V 2 = {1, 2, . . . , n}. Consider the bipartite graph G = (V 1 , V 2 , E). For each node i ∈ V 1 and j ∈ V 2 , respective costs c i and d j are prescribed. Further, for each (i, j) ∈ E, a cost q ij is given. Then the maximum weight induced subgraph problem on G is to find U 1 ⊆ V 1 , U 2 ⊆ V 2 such that i∈U 1 c i + j∈U 2 d j + (i,j)∈E 1,2 q ij is maximized, where E 1,2 is the edge set of the subgraph of G induced U 1 ∪ U 2 . The maximum weight induced subgraph problem on G is precisely a BQP01, where q ij = 0 if (i, j) / ∈ E. Many well known combinatorial optimization problems can be modeled as BQP01. Consider the bipartite graph G = (V 1 , V 2 , E) with w ij > 0 being the weight of the edge (i, j) ∈ E. Then the maximum weight biclique problem (MWBP) [4,35] is to find a biclique in G of maximum total edge-weight. Define q ij = w ij if (i, j) ∈ E −M otherwise , where M is a large positive number. Set c and d as zero vectors and also set c 0 = 0. Then the BQP01 with this choice of Q, c, d and c 0 solves the MWBP. This immediately shows that BQP01 is NP-hard and one can also establish some approximation hardness results with appropriate assumptions [4,35]. MWBP has applications in data mining, clustering and bioinformatics [9,36] which in turn become applications of BQP01. Another application of BQP01 arises in approximating a matrix by a rank-one binary matrix [13,22,23,26,34]. For example, let H = (h ij ) be a given m × n matrix and we want to find an m × n matrix A = (a ij ), where a ij = u i v j and u i , v j ∈ {0, 1}, such that m i=1 n j=1 (h ij − u i v j ) 2 is minimized. The matrix A is called a rank one approximation of H and can be identified by solving the BQP01 with q ij = 1 − 2h ij , c i = 0, d j = 0 and c 0 = 0 for all i ∈ I and j ∈ J. Binary matrix factorization is an important topic in mining discrete patterns in binary data [26,34]. If u i and v j are required to be in {−1, 1} then also the resulting approximation problem can be formulated as a BQP01. The maximum cut problem on a bipartite graph can be formulated as a BQP01 and this gives yet another application of the model. BQP01 can also be used to find approximations to the cut-norm of a matrix [2,31]. When the matrix Q ′ is positive semidefinite, QP01 can be solved directly as a BQP01. This follows from a corresponding result from BLP [20] and the equivalence between BQPC and BQP01. To the best of our knowledge, BQP01 has not been thoroughly investigated in literature, especially from the point of view of polynomially solvable special cases. Some recent references on the problem considers theoretical analysis of approximation algorithms [30] and experimental analysis of heuristics [14,18]. The primary focus of this work is to identify polynomially solvable special cases of BQP01. We show that BQP01 can be solved in polynomial time if the rank of Q is fixed. It may be noted that the corresponding version of QP01 is NP-hard even if Q ′ is of rank one [16]. However, when rank of Q ′ is fixed and Q ′ is positive semidefinite with c ′ = 0, QP01 can be solved in polynomial time [1,11,19,27]. When Q is of rank one, we show that BQP 01 can be solved in O(n log n) time. In addition, we obtain an O(n) algorithm for a special case of this rank-one problem. When q ij = a i + b j , we present an O(mn log n) algorithm. 2 Equivalent formulations QP01 is often presented in literature in various equivalent forms. Similar equivalent formulations can be obtained for BQP01 as well which are summarized in this section. Although many of the transformations discussed here follows almost directly from the corresponding transformation for QP01, we present them here for completness, comparison, and for notational convenience. Further, these transformations, albeit simple, preserve the bipartite structure of the problem which allows us establishing interesting complexity results. When the linear and constant terms from BQP01 are dropped, we get the homogeneous version of the problem defined as, BQP01(H): Maximize f (x, y) = x T Qy Subject to x ∈ {0, 1} m , y ∈ {0, 1} n . In fact, the general BQP01 can be represented in the homogeneous form by introducing two additional variables. LetQ = Q c T d c 0 .(5) Then BQP01 is equivalent to BQP01(H): Maximize f (x,ȳ) =x TQȳ Subject tox ∈ {0, 1} m+1 ,ȳ ∈ {0, 1} n+1 ,x m+1 = 1,ȳ n+1 = 1, wherex m+1 andȳ n+1 are respectively the (m + 1)th and (n + 1)th components ofx andȳ. The restrictionsx m+1 = 1,ȳ n+1 = 1 in BQP01(H) can be dropped by replacing c 0 with a large number M inQ. Thus, we have Remark 1. A BQP01 can always be represented in an equivalent homogeneous form in the sense that the two problems have identical optimal solutions set. Another important variation of BQP01 is obtained by restricting the variables to 1 or −1 instead of 0 or 1. We call this the cut version of BQP01 and denote it by BQP-11. The problem BQP-11 can be stated as BQP-11: Maximize φ(x, y) = x T Qy + cx + dy + c 0 Subject to x ∈ {−1, 1} m , y ∈ {−1, 1} n . By dropping the linear and constant terms from BQP-11, we get the corresponding homogeneous version BQP-11(H): Maximize φ(x, y) = x T Qy Subject to x ∈ {−1, 1} m , y ∈ {−1, 1} n . As in the case of BQP01, its cut version BQP-11 can also be represented in the corresponding homogeneous form by introducing two additional variables. LetQ be defined as in equation (5) andx ∈ {−1, 1} m+1 ,ȳ ∈ {−1, 1} n+1 . Then BQP-11 is equivalent to BQP-11(H): Maximize φ(x,ȳ) =x TQȳ Subject tox ∈ {−1, 1} m+1 ,ȳ ∈ {−1, 1} n+1 ,x m+1 = 1,ȳ n+1 = 1. Note that the restrictionx m+1 = 1,ȳ n+1 = 1 in BQP-11(H) can be replaced byx m+1 =ȳ n+1 without affecting the optimal solution of BQP-11(H). This is possible because, for any solutionx,ȳ withx m+1 = −1,ȳ n+1 = −1, the solution −x, −ȳ satisfies −x m+1 = 1, −ȳ n+1 = 1 and φ(x,ȳ) = φ(−x, −ȳ). Further, the conditionx m+1 =ȳ n+1 can be dropped by replacing c 0 with a large number M inQ resulting in a homogeneous version of BQP-11. Thus, we have Remark 2. A BQP-11 can always be represented in an equivalent homogeneous version in the sense that the two problems have identical optimal solutions set. Consider the linear transformation x = 2w − e m and y = 2z − e n ,(6) where e m and e n are all-one vectors in R m and R n , respectively. Using (6), BQP-11 can be reduced to the BQP01 Maximize f (w, z) = w TQ z +cw +dz +c 0 Subject to w ∈ {0, 1} m , z ∈ {0, 1} n , whereQ = 4Q,c = 2(c − (Qe n ) T ),d = 2(d − e T m Q) andc 0 = e T m Qe n − ce m − de n + c 0 . Similarly, using the linear transformation x = 1 2 (w + e m ) and y = 1 2 (z + e n )(7) BQP01 can be reduced to the BQP-11 Maximize φ(w, z) = w TQ z +ĉw +dz +ĉ 0 Subject to w ∈ {−1, 1} m , z ∈ {−1, 1} n whereQ = 1 4 Q,ĉ = 1 4 (Qe n ) T + 1 2 c,d = 1 4 e T m Q + 1 2 d andĉ 0 = 1 4 e T m Qe n + 1 2 ce m + 1 2 de n + c 0 . It is well known that the maximum weight cut problem (MaxCut) on a general graph G is equivalent to QP01 [17]. In the maximum weight cut problem, if we restrict the graph G to be bipartite, we get an instance of the bipartite maximum weight cut problem which is denoted by B-MaxCut. Indeed, viewing B-MaxCut as a general MaxCut problem yields an equivalent QP01. Many approximation algorithms for MaxCut assume that the associated edge weights are nonnegative [15]. However, for nonnegative edge weights, B-MaxCut is a trivial problem since the generic bipartition of the underlying graph gives the optimal cut. Thus, the topic that is more interesting is when the edge weights take positive as well as negative values. Developing approximation algorithms for BQP01 or for B-MaxCut is not the focus of this paper. However, we do want to indicate the equivalence between B-MaxCut and BQP01 to further understand the inherent complexity of the problems. Let G = (I, J, E) be a bipartite graph. Two vectors x ∈ {−1, 1} m and y ∈ {−1, 1} n determine a cut (U 1 ∪ U 2 , H 1 ∪ H 2 ) in G if and only if U 1 = {i ∈ I : x i = 1}, H 1 = {i ∈ I : x i = −1}, U 2 = {j ∈ J : y j = 1}, and H 2 = {j ∈ J : y j = −1}. We call (x, y) the incidence vector of the cut (U 1 ∪ U 2 , H 1 ∪ H 2 ). Let q ij be the weight of the edge (i, j) in G. Then the value of the cut (U 1 ∪ U 2 , H 1 ∪ H 2 ) is given by i∈U 1 ,j∈H 2 q ij + i∈H 1 ,j∈U 2 q ij = x i =−y j q ij (8) Theorem 1. The problems BQP01 and B-MaxCut are equivalent in the sense that: 1. For any instance of BQP01, it is possible to construct a complete bipartite graph G such that an optimal solution to the B-MaxCut problem on G gives an optimal solution to BQP01. For any instance of B-MaxCut on a bipartite graph G = (I, J, E) , it is possible to construct an instance of BQP01 with an m × n cost matrix Q such that an optimal solution to the BQP01 gives an optimal solution to the B-MaxCut problem on G. Proof. Since BQP01 is equivalent to a BQP-11(H) (see remarks 1 and 2), without loss of generality we assume that BQP01 is given in the equivalent homogeneous cut form BQP-11(H). Now, φ(x, y) = ij q ij x i y j = x i =y j q ij − x i =−y j q ij = ij q ij − 2 x i =−y j q ij Since ij q ij is a constant, maximizing φ(x, y) is equivalent to maximizing −2 x i =−ȳ j q ij . Thus, by solving the B-MaxCut problem on a complete bipartite graph K m,n with −2q ij chosen as the weight of the edge (i, j), we can solve BQP-11(H). To establish the second part of the theorem, we show that the B-MaxCut problem on G with edge weights c ij for (i, j) ∈ E can be solved as a BQP-11(H). Let (U 1 ∪ U 2 , H 1 ∪ H 2 ) be a cut in G and (x, y) be the corresponding incidence vector. Let g(U, H) = x i =−y j c ij be the value of the cut (U 1 ∪ U 2 , H 1 ∪ H 2 ). Then it can be verified that g(U, H) = 1 2 ij c ij − 1 2 ij c ij x i y j and, hence, maximizing g(U, H) is equivalent to solving a BQP-11(H) with cost matrix q ij = − 1 2 c ij . Polynomially solvable cases In BQP01 if we restrict the variables to 1 or −1 instead of 0 or 1, we get the cut version of the problem. BQP01 and its cut version can be reduced to each other using appropriate linear transformations. Alon and Naor [2] showed that the cut version of BQP01 with no linear or constant terms is MAX SNP hard. As a consequence, BQP01 is also MAX SNP hard. Thus, let us focus on polynomially solvable special cases of BQP01. Polynomially solvable special cases of QP01 have been investigated by many authors [1,8,11,16,19,24,27,29]. Since BQP01 can be formulated as a QP01 using a linear transformation, whenever the transformed data satisfies the known conditions for polynomial solvability of QP01, we can solve the corresponding BQP01 also in polynomial time. Thus, it is interesting to focus on new polynomially solvable cases that exploit the special structure of BQP01. Fixed parameter problems Let us first consider a simple case where the number of rows of G is fixed. Theorem 2. BQP01 can be solved in polynomial time if m = O(log n) but remains MAX SNP hard if m = O( k √ n) for any constant k. Proof. If x is fixed to all possible values, the resulting objective function linear in y and have a closed form solution of the type: y 0 j = 1 if m i=1 q ij x i + d j > 0 0 otherwise .(9) Since there are only 2 m choices for x, the polynomial solvability for m = O(log n) follows. To establish that BQP01 is MAX SNP hard when m = O( k √ n), consider an instance of BQP01 with data Q, c, d and c 0 . Let Q = Q O m×n k O (n−m)×(n) O (n−m)×n k ,c = cc , andd = dd , where O m×n k , O (n−m)×(n) and O (n−m)×n k are zero matrices,c is a zero row vector of dimension n − m andd is a zero row vector of dimension n k . Now consider the instance of BQP01 with dataQ,c,d and c 0 . It can be verified that this new instance of BQP01 satisfies the condition of the theorem. Further, for every solution to the original BQP01, we can find a solution to the constructed instance of BQP01, where the corresponding objective function values are identical and viceversa. Since BQP01 is MAX SNP hard, the result follows. If m is fixed, it can be verified that BQP01 can be solved in O(n) time. Further, if m = O(log k n) for fixed k, the complexity becomes quasi-polynomial. Let I ⊆ V 1 = {1, 2, . . . , m} and J ⊆ V 2 = {1, 2, . . . , n}. We say that I ∪ J is a negative eliminator of Q if the matrix Q * obtained from Q by deleting rows corresponding to I and columns corresponding J has only non-negative entries. A negative eliminator of smallest cardinality is called a minimum negative eliminator. Construct a bipartite graph G B = (V 1 , V 2 , E) where (i, j) ∈ E if and only if q ij < 0. Then a minimum negative eliminator of Q is precisely a minimum vertex cover of G B . Since the vertex cover problem on a bipartite graph can be solved in polynomial time, the minimum negative eliminator of Q can be identified in polynomial time. Let S − = I − ∪ J − be a minimum negative eliminator of Q. Theorem 3. BQP01 can be solved in polynomial time if |S − | is O(log n) and the problem is NP-hard if |S − | is O( k √ n) for some fixed k. Proof. Suppose |S − | is O(log n). Fixing the variables x i for i ∈ I − and y j for j ∈ J − at 0-1 values results in a reduced BQP01 with cost matrix have non-negative elements. Such a BQP01 can be solved as a minimum cut problem [32]. Since there are at most 2 |S − | ways to fix the variables associated with I − and J − , BQP01 can be solved in polynomial time if |S − | = O(log n). The second part of the theorem can be proved by reducing a general BQP01 to a BQP01 satisfying the conditions of the theorem. This can be achieved by increasing the number of columns (or rows) of Q to a sufficiently large number, yet polynomial for fixed k and filling these columns (rows) with entries 0. It may be noted that Theorem 3 allows arbitrary c and d. Fixed rank cost matrix QP01 is polynomially solvable if Q ′ is a symmetric positive semidefinite matrix [1,11,19,27] with fixed rank and c ′ = 0. If c ′ is allowed to have arbitrary elements, the problem is NP-hard even if the rank of Q ′ is one [16]. We now show that BQP01 is solvable in polynomial time if rank of Q is fixed. No assumption is made on any other property of Q such as positive semidefiniteness and no restrictions on c and d are imposed. Our algorithm was inspired by [21]. First, let us prove some preliminary results. Consider the multiparametric linear programming problem (MLP) f (λ) = max cx Subject to: Ax = λ x ∈ [0, 1] m , where A is a p × m matrix of full row rank and λ T = (λ 1 , λ 2 , . . . , λ p ) ∈ R p . The j th column of the matrix A is denoted by C B B −1 A j − c j ≥0 for j ∈ L (10) C B B −1 A j − c j ≤0 for j ∈ U.(11) Conditions (10) and (11) are also known as reduced cost optimality conditions [28]. Let (B, L, U) be a dual feasible basis structure for MLP. Then the basic solution of MLP corresponding to this basis structure is optimal for all λ ∈ R p satisfying 0 ≤ B −1 λ − B −1 A u v ≤ 1,(12) where A u is the submatrix formed by columns of A corresponding to the indices of U, v is a vector of size |U| with all entry equal to 1, 0 is the zero vector in R p and 1 is the vector in R p with all entries equal to 1. The polyhedral set represented by (12) is called the characteristic region of the basis structure (B, L, U). A dual feasible basis structure (B, L, U) is dual nondegenerate if (10) and (11) are satisfied as strict inequalities. If any of these inequalities is satisfied as an equality, then (B, L, U) is a dual degenerate basis structure. Let S be the collection of all the extreme points of the characteristic regions associated with all dual feasible basis structures of MLP. Lemma 1. |S| ≤ m C p 2 p . Proof. For simplicity, we assume that all dual feasible basis structures of MLP are dual nondegenerate. This is not a restriction since we can achieve this by an ǫ-perturbation of the cost vector c in MLP by an appropriately small ǫ and this change will not underestimate |S|. Thus, the inequalities (10) and (11) are satisfied as strict inequalities. Consequently, given B, the choice of L and U is unique for any dual feasible basis structure, i.e., if (B, L 1 , U 1 ) and (B, L 2 , U 2 ) are two dual non-degenerate basis structures for MLP then at most one of them can be dual feasible. Thus, there exist at most m C p dual feasible and dual non-degenerate basis structures for MLP. The characteristic region of such a basis structure (B, L, U) is defined by the inequalities (12). Each extreme point of this polyhedron is determined by the unique solution of p tight inequalities from (12) that are satisfied as equalities. Since there are exactly 2 p choices for these tight inequalities, the result follows. Note that f (λ) is a piecewise linear concave function when λ ∈ R p [12]. It is linear when λ is restricted to a characteristic region associated with any dual feasible basis structure (B, L, U). These extreme points are called breakpoints of f (λ). Thus, f (λ) will have at most m C p 2 p breakpoints. Let S(B, L, U) be the collection of all extreme points of the characteristic region associated with (B, L, U). Then, using inequalities (12), it can be verified that S(B, L, U) = {λ τ : λ τ = Bτ + A u v, τ ∈ {0, 1} p }(13) where (13), λ = Bτ + A u v for some τ ∈ {0, 1} p and hence x λ B = B −1 λ − B −1 A u v = τ.(14) The non-basic variables of Let us now consider BQP01 where rank of Q is p, a fixed number. We assume that Q is given in the rank-p factorized form. i.e. Q = AB where A is an m×p matrix and B is a p×n matrix. Such a factorization can easily be constructed from the reduced row echelon form Q R of Q by choosing A as the matrix obtained from Q by deleting non-pivot columns and B as the matrix obtained from Q R by removing the zero rows. Let a k = (a k 1 , a k 2 , . . . , a k m ) be the k th column of A and b k = (b k 1 , b k 2 , . . . , b k n ) be the k th row of B. Since BQP01 is equivalent to BQPC, this problem can be stated as BQPC(p): Maximize p k=1 a k xb k y + cx + dy Subject to: x ∈ [0, 1] m , y ∈ [0, 1] n . Consider the multiparametric linear program [12] MLP1: h 1 (λ) = max cx Subject to: a k x = λ k for k = 1, 2, . . . , p x ∈ [0, 1] m , where λ = (λ 1 , λ 2 , . . . , λ p ). Then h 1 (λ) is a piecewise linear concave function [12]. Let S 1 be the set of breakpoints of h 1 (λ) and x(λ) be an optimal basic feasible solution of MLP1 at λ ∈ S 1 . By Theorem 4, x(λ) ∈ {0, 1} m . Let y(λ) = (y 1 (λ), y 2 (λ), . . . , y n (λ)) be an optimal solution to BQPC(p) when x is restricted to x(λ). Then it can be verified that y(λ) satisfies Let h(λ) be the optimal objective function value of BLP1 where λ is fixed. Now, h(λ) can be decomposed into h 1 (λ) + h 2 (λ), where y j (λ) = 1 if d j + p k=1 b k j m i=1 a k i x i (λ) > 0 0 otherwise .(15h 2 (λ) = max p k=1 λ k b k y + dy Subject to: y ∈ [0, 1] n , and h 1 (λ) is as defined in MLP1. Then the optimal objective function value of BQPC(p) can be identified by finding the global maximum of h(λ) over all λ ∈ R p for which BLP1 is feasible. As a function of λ, h 2 (λ) is piecewise linear convex [12] and h 1 (λ) is piecewise linear concave [12]. Thus, h(λ) is piecewise linear but need not be convex or concave. But h 1 (λ) is linear when λ is restricted to any characteristic region of MLP1. Thus, h(λ) is convex when λ is restricted to any characteristic region associated with h 1 (λ). Hence, the maximum of h(λ) is attained at a breakpoint of h 1 (λ). Let λ = λ 0 be such a breakpoint and x(λ 0 ) be an optimal basic feasible solution of MLP1 at λ = λ 0 . By Theorem 4, x(λ 0 ) is a 0-1 solution and it yields the optimal value y(λ) as given by (15) for the y variables for BQPC(p) subject to the condition that x = x 0 (λ). Since one of the x(λ) for λ ∈ S 1 gives an optimal x-value, the result follows. Based on Theorem 5, BQPC(p) and hence BQP01 can be solved by generating the set S 1 , computing the set S 2 = {(x(λ), y(λ)) : λ ∈ S 1 } and choosing the best solution in S 2 . Note that we do not need to compute explicitly the set S 1 . The solution set {x(λ) : λ ∈ S 1 } can be identified without computing λ. Let (B, L, U) be a dual feasible and dual non-degenerate basis structure. Let B = {B1, B2, . . . , Bp} and B be the associated basis matrix. For each τ ∈ {0, 1} p , we get an extreme point λ τ = Bw + A u v of its characteristic region. This follows from the inequality (12). But then, from equation (14), the corresponding basic variables x(λ τ ) B is precisely τ . The non-basic variables are fixed at 0/1 values guided by L and U. Thus, for each basis structure (B, L, U) we can generate 2 p basic feasible solutions corresponding to its extreme points by varying τ ∈ {0, 1, } p (see proof of Theorem 4). For each such basic feasible solution x we can compute the corresponding y value easily (see equation (15)). Below we present a high-level description of our algorithm for solving BQPC(p): Step 1: Let A be the coefficient matrix of MLP1 and Γ be the collection of all dual feasible basis structures associated A. Step 2: For each basis structure (B, L, U) ∈ Γ, construct the set of optimal basic feasible solutions corresponding to the extreme points of its characteristic region. LetS be the collection of all such solutions obtained. Step 3: For each each x ∈S compute the best y ∈ R n , say y x . Let S 2 = {(x, y x ) : x ∈S}. Step 4: Output the best solution in S 2 . There are m C p choices for B and for each such choice, there is a unique allocation of non-basic variables to L and U at zero or one values. (The uniqueness follows from dual non-degeneracy assumption which can be achieved by appropriate perturbation of the cost vector.) The basis inverse can be obtained in O(p 3 ) time and given this inverse, L and U can be identified in O(mp 2 ) time so that (B, L, U) is dual feasible. Thus |Γ| ≤ m C p and Γ can be identified in O( m C p (p 3 + mp 2 ) time. The characteristic region associated with each (B, L, U) ∈ Γ has at most 2 p extreme points and the optimal solution to MLP1 when λ is fixed at these extreme points can be identified as discussed in the proof of Theorem 4 without explicitly computing λ. Thus, given Γ,S in Step 2 can be identified in O( m C p 2 p m) time. For each x ∈S we can compute the corresponding optimal y x in O(mnp) time using equation (15). The best solution in S 2 can be identified in O( m C p 2 p (mnp)) time. Summarizing the foregoing discussions, we have, Theorem 6. BQP01 can be solved in O( m C p 2 p mnp) time when rank of Q = p and Q is given in the rank factored form. Note that for a fixed p the above bound is polynomial and for p = O(log n) it is quasipolynomial. For specific choices of p, the complexity of the procedure discussed above may be improved. This is illustrated in the next subsection for the case when p = 1. This is a special case of MLP1 for p = 1. Let h 1 (λ) be the optimal objective function value of PKP(λ) for a given λ. Then for λ ≤ λ ≤ λ, h 1 (λ) is a piecewise linear concave function [28]. Let λ = λ 1 < λ 2 < · · · < λ s = λ be the breakpoints of h 1 (λ), x k be an optimal basic feasible solution of PKP(λ) for λ ∈ [λ k , λ k+1 ], 1 ≤ k ≤ s − 1, and x s be an optimal basic feasible solution to PKP(λ s ). Let y k be an optimal solution to BQPC(1) when x is restricted to x k , the vector y k can be identified by appropriate modification of the equation (15). By Theorem 5, there exists an optimal solution to BQPC(1) amongst the solutions x k , y k : k = 1, 2, . . . , s. From Lemma 1, the number of breakpoints of h 1 (λ) is at most 2m. We now observe that the number of breakpoints of h 1 (λ) cannot be more than m+ 1 and obtain closed form values of these breakpoints. Rank one cost matrix Let T = c i a i : i = 1, 2, . . . , m, a i = 0 and consider a descending arrangement c π 1 a π 1 > c π 2 a π 2 > · · · > c πr a πr(16) of all distinct elements of T . Let T (k) = i : c π k a π k = c i a i . Then the breakpoints of h 1 (λ) are given by λ 1 = λ and λ k+1 = λ k + i∈T (k) |a i | for k = 1, 2, . . . , r. An optimal solution to PKP(λ) at λ = λ k for k = 1, 2, . . . , r + 1 can be identified recursively as x 1 i = 1 if a i = 0 and c i > 0 or a i < 0 0 otherwise and x k+1 i =      x k i if i / ∈ T (k) 1 if i ∈ T (k) and a i > 0 0 otherwise. Thus, it can be verified that given h(λ k ) and x k , h(λ k+1 ) and x k+1 can be identified in O(|T (k)|) time. The complexity for generating these solutions and breakpoints are dominated by that of constructing the descending arrangement (16) which is O(m log m). Note that h 1 (λ) has at most m + 1 breakpoints and given x k , a corresponding solution y k can be computed in O(n) time. This leads to a complexity of O(mn). The bottleneck operation here is the computation of y k for k = 1, 2, . . . , r. We now show that these points can be identified in O(n log n) time. Consider the parametric unconstrained linear optimization problem ULP(µ): Maximize dy + µby Subject to: y ∈ [0, 1] n and λ ≤ µ ≤ λ. Let h 2 (µ) be the optimal objective function value of ULP(µ). Then h 2 (µ) is a piecewise linear convex function. Let S + = {j : d j + λb j ≥ 0}, S − = {j : d j + λb j < 0}, B + = {j : b j > 0}, and B − = {j : b j < 0}. Also, let T 2 = { d j b j : j ∈ B + ∪ B − } and consider an ascending arrangement λ < d τ 1 b τ 1 < d τ 2 b τ 2 · · · < d τt b τt(17) of all distinct elements of T 2 greater than λ. Let µ 1 , µ 2 , . . . , µ t be the breakpoints of h 2 (µ). Then µ ℓ = dτ ℓ bτ ℓ for ℓ = 1, 2, . . . t. Let ∆ i = {j : d j b j = dτ i bτ i }. Then the optimal solution y ℓ for ULP(µ) corresponding to the breakpoint µ ℓ for ℓ = 1, 2, . . . , t is given recursively by Then the optimal objective function value at µ ℓ is given by h 2 (µ ℓ ) = D ℓ + µ ℓ B ℓ . Given y ℓ−1 , D ℓ−1 and B ℓ−1 , we can compute y ℓ , D ℓ , and B ℓ in O(|∆ ℓ |) time and, hence, h 2 (µ ℓ ) and y ℓ can be identified in O(|∆ ℓ |) time. Since ∆ ℓ ∩ ∆ k = ∅ for ℓ = k, y ℓ and h 2 (µ ℓ ) for ℓ = 1, 2, . . . , t can be identified in O(n log n) time. Now the algorithm for solving BQPC(1) can be described as follows. First, compute x 1 , y 1 , h 1 (λ) and h 2 (λ). Set f (x 1 , y 1 ) = h 1 (λ) + h 2 (λ). Sort all breakpoints of h 1 (λ) and h 2 (µ) for λ ≤ λ, µ ≤ λ (see Fig: 1) and scan these breakpoints starting from λ in the increasing order. As we pass breakpoints of h 2 (µ) keep updating the solution y of ULP(µ) corresponding to this breakpoint and the objective function value of this solution until we hit a breakpoint λ k of h 1 (λ). At this point compute the solution x k and h 1 (λ k ). The most recent solution y identified is selected as y k and compute h 2 (λ k ). Note that h 2 (λ k ) can be obtained in O(1) time using slope of h 2 (µ) for the interval containing λ k . Update f (x k , y k ) and the process is continued until all breakpoints of h 1 (λ) including λ are examined and the overall best solution is selected. It is not difficult to verify that the complexity of this procedure is O(n log n). y ℓ j =      y ℓ−1 j if j / ∈ ∆ ℓ 1 if j ∈ ∆ ℓ and y ℓ−1 j = 0 0 if j ∈ ∆ ℓ and y ℓ−1 j = 1, where y 0 j = 1 if j ∈ S + 0 otherwise. Define D 0 = j∈S + d j , B 0 = j∈S + b j , D ℓ = D ℓ−1 − j∈∆ ℓ ,y ℓ−1 j =1 d j + j∈∆ ℓ ,y ℓ−1 j =0 d j and B ℓ = B ℓ−1 − j∈∆ ℓ ,y ℓ−1 j =1 b j + j∈∆ ℓ ,y ℓ−1 j =0 b j . λ 1 µ 1 µ 2 µ 3 λ 2 µ 4 λ 3 µ 5 λ 4 λ 5 h 1 (λ) h 2 (µ) Let us now consider a special case of BQP01 when Q is of rank one (equivalently a special case of BQPC(1)) of the form BQPC(1, c ∨ d = 0): Maximize (a 0 + ax)(b 0 + by) + cx + dy Subject to: x ∈ [0, 1] m , y ∈ [0, 1] n , where either c = 0 or d = 0. We now show that this problem can be solved more efficiently. Convexity of L(λ) guarantees that its maximum is attained when λ = λ 0 or λ * , i.e., when y = y 0 or y = y * . But when y is fixed, an optimal x can be obtained in linear time. Since y 0 and y * can be identified in O(n) time, the result follows. The case when c = 0 can be established analogously. Additively decomposable cost matrix Let us now examine the case when q ij = a i + b j for i = 1, 2, . . . , m and j = 1, 2, . . . , n. Note that when q ij = a i + b j , rank of Q is at most 2 and hence can be solved in polynomial time. We now present an O(mn log n) algorithm to solve this problem. x i = L For K = 0, 1, . . . , n, let α K be a permutation of size m such that Ka α K (i) + c α K (i) ≥ Ka α K (i+1) + c α K (i+1) , i = 1, . . . , m − 1. For L = 0, 1, . . . , m, let β L be a permutation of size n such that Lb β L (j) + d β L (j) ≥ Lb β L (j+1) + d β L (j+1) , j = 1, 2, . . . , n − 1. Observe that, for fixed K and L, the optimal x = x K,L can be obtained by setting x i = 1 for i = α K (1), α K (2), . . . , α K (L) and x i = 0 for the rest of indices. Similarly, the optimal y = y K,L can be obtained by setting y j = 1 for j = β L (1), β L (2), . . . , β L (K) and y j = 0 for the rest of indices. Observe also that f K,0 1 = 0. Hence, for a fixed K, we can calculate f K,L . Then the optimal solution of the BQP01 is (x K 0 ,L 0 , y K 0 ,L 0 ), and it can be obtained in O(nm log m + mn log n + mn) = O(mn log n) time. Conclusion In this paper we studied the problem BQP01 which generalizes QP01, a well studied combinatorial optimization problem. BQP01 is known to be MAX SNP hard. Several interesting polynomially solvable special cases of the problem are identified. In particular, we showed that when rank of the matrix Q is fixed, BQP01 can be solved in polynomial time and improved complexity results are provided for the rank one case and a special case when rank of Q is at most two. By restricting m = O(log n) or by restricting the size of minimum negative eliminator of Q to be O(log n), we obtained additional polynomially solvable cases. If m = k √ n BQP01 is MAX SNP hard and a similar result is obtained if the size of a minimum negative eliminator of Q is O( k √ n). It would be interesting to explore the complexity of the problem when the problem size or data restrictions fall in between these extreme cases. Exploiting the algorithms for our polynomially solvable special cases, efficient exact and heuristic algorithms may be obtained to solve BQP01. This is a topic for further investigation. Further, we observe that if m = O(log n) then BQP01 can be solved in polynomial time but if m = O( k √ n) for a fixed k, the problem remains MAX SNP hard. Similarly, if the number of negative entries in Q are O(log n) then BQP01 is solvable in polynomial time but if the number of negative entries are O( k √ n) for a fixed k, the problem becomes NP-hard. A j . A partition (B, L, U) of I = {1, 2, . . . , m} with |B| = p is referred to as a basis structure for MLP. For each basic feasible solution of MLP, a basis structure (B, L, U) is associated, where L is the index set of nonbasic variables at the lower bound 0, U is the index set of non-basic variables at the upper bound 1 and B = {B1, B2, . . . , Bp} is the index set of basic variables. Let B be the p × p matrix with its i th column A Bi . Then the set B is called a basis set or simply a basis and B is the associated basis matrix. A basis provides an implicit ordering of its elements. Thus, x Bi is called the i th basic variable with respect to B. Let C B = (c B1 , c B2 , . . . , c Bp ). Then the basis structure (B, L, U) is dual feasible if and only if[28] v is the all-one vector of size |U|. LetS(B, L, U) be the collection of all basic feasible solutions of MLP associated with the extreme points in S(B, L, U). Theorem 4 . 4S(B, L, U) ⊆ {0, 1} m and |S(B, L, U)| = 2 p . Proof. Let x(λ) ∈S(B, L, U) be a basic feasible solution for MLP corresponding to the extreme point λ ∈ S(B, L, U) and B be the basis matrix associate with (B, L, U). Let x λ B be the vector of basic variables of x(λ). From x(λ) by definition take 0-1 values and hence x(λ) ∈ {0, 1} m . Thus x(λ) ⊆S(B, L, U). Since there are 2 p choices for τ in equation (14) and nonbasic variables in L and U are fixed for a given (B, L, U) (independent of the choice of τ ), we have |S(B, L, U)| = 2 p . ) Theorem 5 . 5There exists an optimal solution to BQPC(p) amongst the solutions {(x(λ), y(λ)) :λ ∈ S 1 }.Proof. BQPC(p) is equivalent to the bilinear b k y + cx + dy Subject to: a k x = λ k for k = 1, 2, . . . , px ∈ [0, 1] m , y ∈ [0, 1] n , λ = (λ 1 , λ 2 , . . . , λ p ) T ∈ R p . if rank of Q is 1, BQP01 can be solved in O(m 2 n) time. We now show that the problem can be solved in O(n log n) time by careful organization of our computations. Recall that m ≤ n. As in the general case, let us consider the bilinear equivalent version:BQPC(1): Maximize axby + cx + dy Subject to: x ∈ [0, 1] m , y ∈ [0, 1] n , where a = (a 1 , a 2 , . . . , a m ), c = (c 1 , c 2 , . . . , c m ) ∈ R m and b = (b 1 , b 2 , . . . , b n ), d = (d 1 , d 2 , . . . , d n ) ∈ R n . Let A − = {i : a i < 0} and A + = {i : a i > 0}. Define λ = i∈A − a i and λ = i∈A + a i ,where summation over the empty set is taken as zero. Note that λ and λ are respectively the smallest and the largest values of ax when x ∈ [0, 1] m . Consider the parametric continuous knapsack problem (PKP(λ)) given below.PKP(λ):Maximize cx Subject to: ax = λx ∈ [0, 1] m , and λ ≤ λ ≤ λ. Figure 1 : 1An example of h 1 (λ) and h 2 (µ) when a = [ Theorem 7 . 7An optimal solution to BQPC(1, c ∨ d = 0) can be identified in O(n) time. Proof. Suppose d = 0. Let L(λ) = Maximum x∈{0,1} m {(a 0 + ax) λ + cx}. Clearly, L(λ) is is a piecewise linear convex function of λ. Suppose b 0 + by maximizes at y 0 and minimizes at y * with respective objective function values, say, λ 0 and λ * . As λ varies in the interval [λ * , λ 0 ], L(λ) traces the best objective function values for BQPC(1, c ∨ d = 0) for all possible solution vectors y. Thus, BQPC(1, c ∨ d = 0) reduces to maximizing L(λ) for λ ∈ [λ * , λ 0 ]. Theorem 8 .( 8If q ij = a i + b j for i = 1, 2, . . . , m and j = 1, 2, . . . , n, then BQP01 can be solved in O(mn log n).Proof. For any feasible solution x, y, f (x, y) = x T Qy + cx + dy + c y j + c 0 .Let n j=1 y j = K and m i=1 x i = L, where K and L are two parameters. Then, Lb j + d j )y j + c 0 . + Ka α K (L+1) + c α K (L+1) . 1, . . . , m} in O(m) time. Similarly, for a fixed L, we can calculate f K,L 2 for each K = 0, 1, . . . , n in O(n) time. Let (K 0 , L 0 ) be the values of K and L that maximize f K Acknowledgement:We are thankful to the referees for their insightful comments which improved the presentation of the paper. A polynomial case of unconstrained zero-one quadratic optimization. K Allemand, K Fukuda, . M Th, E Liebling, Steiner, Mathematical Programming, Ser. A. 91K. Allemand, K. Fukuda, Th. M. Liebling, and E. Steiner. A polynomial case of uncon- strained zero-one quadratic optimization. Mathematical Programming, Ser. A, 91 (2001) 49-52. Approximating the cut-norm via Grothendieck's inequality. N Alon, A Naor, SIAM Journal of Computing. 35N. Alon and A. Naor, Approximating the cut-norm via Grothendieck's inequality, SIAM Journal of Computing 35 (2006) 787-803. Bulletin de l' Académie Polonaise des. M Altman, Sciences. 16Bilinear programmingM. Altman, Bilinear programming, Bulletin de l' Académie Polonaise des Sciences 16 (1968) 741-746. Inapproximability results for maximum edge biclique, minimum linear arrangement, and sparsest cut. C Ambühl, M Mastrolilli, O Svensson, SIAM Journal of Computing. 40C. Ambühl, M. Mastrolilli, and O. Svensson, Inapproximability results for maximum edge biclique, minimum linear arrangement, and sparsest cut, SIAM Journal of Computing 40 (2011) 567-596. The max-cut problem and quadratic 01 optimization; polyhedral aspects, relaxations and bounds. E Boros, P L Hammer, Annals of Operations Research. 33E. Boros and P. L. Hammer, The max-cut problem and quadratic 01 optimization; poly- hedral aspects, relaxations and bounds, Annals of Operations Research 33 (1991) 151-180. Pseudo-Boolean optimization. E Boros, P L Hammer, Discrete Applied Mathematics. 123E. Boros and P. L. Hammer, Pseudo-Boolean optimization, Discrete Applied Mathematics 123 (2002) 155-225. Local search heuristics for Quadratic Unconstrained Binary Optimization. E Boros, P L Hammer, G Tavares, Journal of Heuristics. 13E. Boros, P.L. Hammer, and G. Tavares. Local search heuristics for Quadratic Uncon- strained Binary Optimization. Journal of Heuristics 13 (2007) 99-132. Polynomially solvable cases of the constant rank unconstrained quadratic 0-1 programming problem. E Cela, B Klinz, C Meyer, Journal of Combinatorial Optimization. 12E. Cela, B. Klinz, and C. Meyer, Polynomially solvable cases of the constant rank uncon- strained quadratic 0-1 programming problem, Journal of Combinatorial Optimization 12 (2006) 187-215. Exploring biological interaction networks with tailored weighted quasi-bicliques. W.-C Chang, S Vakati1, R Krause, O Eulenstein, BMC Bioinformatics. 131016SupplW.-C. Chang, S. Vakati1, R. Krause, and O. Eulenstein, Exploring biological interaction networks with tailored weighted quasi-bicliques, BMC Bioinformatics 13(Suppl 10):S16, 2012. Maximizing quadratic programs: extending Grothendieck's inequality. M Charikar, A Writh, M. Charikar and A. Writh, Maximizing quadratic programs: extending Grothendieck's inequality, FOCS 2004, 54-60. Solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm. J.-A Ferrez, K Fukuda, Th M Liebling, European Journal of Operational Research. 166J.-A. Ferrez, K. Fukuda, and Th.M. Liebling, Solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm. European Journal of Operational Research, 166 (2005) 35-50. Postoptimal analysis, parametric programming, and related topics. T , Walter de GruyterBerlinT. Gal, Postoptimal analysis, parametric programming, and related topics, Walter de Gruyter, Berlin, 1995. Low-rank matrix approximation with weights or missing data is NP-hard. N Gillis, F Glineur, SIAM Journal of Matrix Analysis and Applications. 32N. Gillis and F. Glineur, Low-rank matrix approximation with weights or missing data is NP-hard, SIAM Journal of Matrix Analysis and Applications 32 (2011) 1149-1165. Integrating tabu search and VLSN search to develop enhanced algorithms: A case study using bipartite boolean quadratic programs. F Glover, T Ye, A P Punnen, G Kochenberger, arXiv:1305.5610cs.AIF. Glover, T. Ye, A. P. Punnen, and G. Kochenberger, Integrating tabu search and VLSN search to develop enhanced algorithms: A case study using bipartite boolean quadratic programs, arXiv:1305.5610 [cs.AI], 2013. Improved approximation algorithms for maximum cut and satisfieability problems using semidefinite programming. M X Goemans, D P Williamson, Journal of the Association for Computing Machinery. 42M. X. Goemans and D. P. Williamson, Improved approximation algorithms for max- imum cut and satisfieability problems using semidefinite programming, Journal of the Association for Computing Machinery 42 (1995) 1115-1145. JR, Maximizing the product of two linear functions in 0-1 variables. P L Hammer, P Hansen, P Pardalos, D J Rader, Optimization. 51P. L. Hammer, P. Hansen, P. Pardalos, and D. J. Rader, JR, Maximizing the product of two linear functions in 0-1 variables, Optimization 51 (2002) 511-537. Some network flow problems solved using pseudo-boolean programming. P L Ivȃnescu, Operations Research. 13P. L. Ivȃnescu, Some network flow problems solved using pseudo-boolean programming, Operations Research 13 (1965) 388-399. D Karapetyan, A P Punnen, arXiv:1210.3684Heuristic algorithms for the bipartite unconstrained 0-1 quadratic programming problem. cs.DMD. Karapetyan and A. P. Punnen, Heuristic algorithms for the bipartite unconstrained 0-1 quadratic programming problem, arXiv:1210.3684 [cs.DM], 2013. Efficient computation of the binary vector that maximizes a rank-deficient quadratic form. G N Karystinos, A P Liavas, IEEE Transactions on information theory. 56G. N. Karystinos and A. P. Liavas, Efficient computation of the binary vector that maximizes a rank-deficient quadratic form, IEEE Transactions on information theory, 56 (2010) 3581 -3593. Maximization of a convex quadratic function under linear constraints. H Konno, Mathematical programming. 11H. Konno, Maximization of a convex quadratic function under linear constraints, Math- ematical programming 11 (1976) 117-127. Parametric simplex algorithms for a class of NPcomplete problems whose average number of steps is polynomial. H Konno, T Kuno, Y Yajima, Computational optimization and applications. 1H. Konno, T. Kuno, and Y. Yajima, Parametric simplex algorithms for a class of NP- complete problems whose average number of steps is polynomial, Computational opti- mization and applications 1 (1992) 227-239. Compression, clustering, and pattern discovery in very digh-dimensional discrete-attribute data sets. M Koyutürk, A Grama, N Ramakrishnan, IEEE Transactions on Knowledge and Data Engineering. 17M. Koyutürk, A. Grama, and N. Ramakrishnan, Compression, clustering, and pattern discovery in very digh-dimensional discrete-attribute data sets, IEEE Transactions on Knowledge and Data Engineering, 17 (2005) 447-461. Nonorthogonal Decomposition of Binary Matrices for Bounded-Error Data Compression and Analysis. M Koyutürk, A Grama, N Ramakrishnan, ACM Transactions on Mathematical Software. 32M. Koyutürk, A. Grama, and N. Ramakrishnan, Nonorthogonal Decomposition of Bi- nary Matrices for Bounded-Error Data Compression and Analysis, ACM Transactions on Mathematical Software 32 (2006) 33-69. Polynomially solvable cases of binary quadratic programs, in Optimization and Optimal Control. D Li, X Sun, S Gu, J Gao, C Liu, Springer Optimization and its Applications 39. A. Chinchuluun et al.D. Li, X. Sun, S. Gu, J. Gao, and C. Liu, Polynomially solvable cases of binary quadratic programs, in Optimization and Optimal Control, A. Chinchuluun et al. (eds), Springer Optimization and its Applications 39, 2010. A hybrid metaheuristic approach to solving the UBQP problem. Z Lü, F Glover, J.-K Hao, European Journal of Operational Research. 207Z. Lü, F. Glover, and J.-K. Hao, A hybrid metaheuristic approach to solving the UBQP problem, European Journal of Operational Research 207 (2010) 1254-1262. Weighted rank-one binary matrix factorization. H Lu, J Vaidya, V Atluri, H Shin, L Jiang, H. Lu, J. Vaidya, V. Atluri, H. Shin, and L. Jiang, Weighted rank-one binary matrix factorization Application of computational geometry to multiuser detection in CDMA. G Manglani, A K Chaturvedi, IEEE Transactions on Commun. 54G. Manglani and A. K. Chaturvedi, Application of computational geometry to multiuser detection in CDMA, IEEE Transactions on Commun., 54 (2006) 204-207. Linear programming. K G Murty, John Wiley & SonsNew YorkK. G. Murty, Linear programming, John Wiley & Sons, New York, 1983. Minimum cuts and related problems. J C Picard, H D Ratliff, Networks. 5J.C.Picard and H.D.Ratliff, Minimum cuts and related problems, Networks 5 (1974)357- 370. Domination analysis of algorithms for the bipartite boolean quadratic programs. A P Punnen, P Sripratak, D Karapetyan, LNCS. 8070SpringerA. P. Punnen, P. Sripratak, and D. Karapetyan, Domination analysis of algorithms for the bipartite boolean quadratic programs, Springer LNCS 8070, 271-282, 2013. Towards computing the Grothendieck constant. P Raghavendra, D Steurer, Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09). the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09)Philadelphia, PA, USAP. Raghavendra and D. Steurer, Towards computing the Grothendieck constant. In Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 525-534, 2009. A selection problem of shared fixed costs and network flows. J Rhys, Management Science. 17J. Rhys, A selection problem of shared fixed costs and network flows, Management Science 17 (1970) 200-207. 0-1 optimization and non-linear programming, Revue Française d'Automatique, d'Informatique et de. I G Rosenberg, Recherche Operationnelle. 2I. G. Rosenberg, 0-1 optimization and non-linear programming, Revue Française d'Automatique, d'Informatique et de Recherche Operationnelle 2 (1972) 95-97. Mining discrete patterns via binary matrix factorization. B.-H Shen, S Ji, J Ye, KDD'09: Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining. New York, USAB.-H. Shen, S. Ji, and J. Ye, Mining discrete patterns via binary matrix factorization, In KDD'09: Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, New York, USA, (2009) 757-766. Inapproximability of maximum weighted edge biclique and its applications. J Tan, Proceedings of the 5th international conference on Theory and applications of models of computation (TAMC'08). Dingzhu Du, Zhenhua Duan, and Angsheng Lithe 5th international conference on Theory and applications of models of computation (TAMC'08)Berlin, HeidelbergSpringer-VerlagManindra AgrawalJ. Tan. Inapproximability of maximum weighted edge biclique and its applications. In Proceedings of the 5th international conference on Theory and applications of models of computation (TAMC'08), Manindra Agrawal, Dingzhu Du, Zhenhua Duan, and Ang- sheng Li (Eds.). Springer-Verlag, Berlin, Heidelberg, 282-293, 2008. Discovering statistically significant biclusters in gene expression data. A Tanay, R Sharan, R Shamir, Bioinformatics. 18Suppl. 1, S136-S144A. Tanay, R. Sharan, and R. Shamir, Discovering statistically significant biclusters in gene expression data, Bioinformatics 18 (2002) Suppl. 1, S136-S144. Path relinking for unconstrained binary quadratic programming. Y Wang, Z Lu, F Glover, J-K Hao, European Journal of Operational Research. 223Y. Wang, Z. Lu, F. Glover, J-K. Hao, Path relinking for unconstrained binary quadratic programming, European Journal of Operational Research 223(2012) 595-604. Efficient algorithms for solving rank two and rank three bilinear programming problems. Y Yajima, H Konno, Journal of global optimization. 1Y. Yajima and H. Konno, Efficient algorithms for solving rank two and rank three bilinear programming problems, Journal of global optimization 1 (1991) 155-171.
[]
[ "A Model Theoretic Proof of Szemerédi's Theorem", "A Model Theoretic Proof of Szemerédi's Theorem" ]
[ "Henry Towsner " ]
[]
[]
We present a short proof of Szemerédi's Theorem using a dynamical system enriched by ideas from model theory. The resulting proof contains features reminiscent of proofs based on both ergodic theory and on hypergraph regularity.
null
[ "https://arxiv.org/pdf/1002.4456v3.pdf" ]
115,177,693
1002.4456
43c9b3de4be1377c330e5c66fc68c7a3156b70b6
A Model Theoretic Proof of Szemerédi's Theorem 26 Jan 2011 January 27, 2011 Henry Towsner A Model Theoretic Proof of Szemerédi's Theorem 26 Jan 2011 January 27, 2011 We present a short proof of Szemerédi's Theorem using a dynamical system enriched by ideas from model theory. The resulting proof contains features reminiscent of proofs based on both ergodic theory and on hypergraph regularity. Introduction Szemerédi's Theorem states: Theorem 1. For any δ > 0 and any k, there is an n such that whenever N ≥ n, A ⊆ [1, N ], and |A| ≥ δN , there exists an a and a d such that a, a + d, a + 2d, . . . , a + (k − 1)d ∈ A. Szemerédi's original proof [14] used graph theoretic methods, in particular the Szemerédi Regularity Lemma [15]. Shortly after, Furstenberg gave a different proof [5,4], based on a correspondence argument which translates the problem into one in ergodic theory. Beginning with a new proof given by Gowers [7], a number of new proofs have been developed in the last decade. (Tao counts a total of roughly sixteen different proofs [19]. ) Hrushovski has recently used a stronger correspondence-type argument [9] to make progress on a similar combinatorial problem (the so-called non-commutative Freiman conjecture). In this paper, we use Hrushovski's method to give a short proof of Szemerédi's theorem. The proof here bears a similarity to proofs based on hypergraph regularity, such as [8,12,13,16]; in particular the proof is very similar to the infinitary regularity-like arguments introduced by Tao [17] and used by Austin to prove both Szemerédi's Theorem [2] and generalizations [1]. Indeed, this proof was inspired by noticing that the use of "wide types" (countable intersections of definable sets of positive measure) in Hrushovski's arguments was analogous to the use of the regularity lemma in finitary arguments. (In fact, Hrushovski essentially sketches a proof of the k = 3 case of Szemerédi's Theorem in [9]; however his arguments depend on stability theoretic methods which don't seem to generalize to higher k. This seems related to the fact that stability implies 3-amalgamation, but not 4-amalgamation.) The methods here are also reminiscent of those used by Tao to prove the "diagonal ergodic theorem" [18], and especially to our infinitary reformulation of that proof [20]. This paper might shed light on the connection between that method and the technique of "pleasant extensions" used by Austin [3]. We are grateful to the members of UCLA's reading seminar on [9]: Matthias Aschenbrenner, Isaac Goldbring, Greg Hjorth, Terence Tao, and Anush Tserunyan. A Correspondence Principal Suppose that Szemerédi's Theorem fails; that is, for some δ > 0 and every n, there is an A n ⊆ [1, n] with |A n | ≥ δn such that A n contains no k-term arithmetic progression. We must first make a technical adjustment: we view the sets A n as subsets of the group [1, 2n + 1]. Then |A n | ≥ δn/2 − ǫ where ǫ → 0 as n → ∞, and A n contains no k-term arithmetic progressions in this group. 1 We extend the language of groups with a predicate symbol A and the following additional class of formulas: • Whenever α 1 , . . . , α k , γ is a sequence of rationals, x a tuple, and B 1 ( x, y 1 ), . . . , B k ( x, y k ) a sequence of formulas, i≤k α i · B i dm( x) > γ is a formula with free variables y 1 , . . . , y k We let ([1, 2n + 1], A n ) be models, interpreting the symbol A by A n and   i≤k α i · φ i dm( x) > γ   ( p 1 , . . . , p k ) ⇔ i≤k α i |{ x | φ i ( x, p i )}| n | x| > γ We write m x (φ) as an abbreviation for 1 · φdm( x), and sometimes omit x when it is clear from context. Form an ultraproduct of those groups [1, 2n + 1] such that 2n + 1 is prime. We obtain a model (G, +, A). By transfer, the formula ∃a, d(a ∈ A ∧ a + d ∈ A ∧ · · · a + d + · · · + d ∈ A) is false. Observe that for any countable set M and any n, the sets of n-tuples definable with parameters from M form an algebra of internal sets, and using the Loeb measure construction, we may extend the internal counting measure on this model to a measure µ n on the σ-algebra of Borel sets generated from these definable sets (for basic facts about this construction, see [6]). The measures µ n satisfy Fubini's Theorem [10,11]; that is, f dµ n = f dµ n0 dµ n1 where n 0 + n 1 = n. Lemma 2. If (G, +, A) m(φ) > γ then µ | x| ({ x | φ( x)}) ≥ γ. If (G, +, A) ¬m(φ) > γ then µ | x| ({ x | φ( x)}) ≤ γ. Proof. For the first part, since (G, +, A) m(φ) > γ, for almost every n, ([1, 2n + 1], +, A 2n+1 ) m(φ) > γ, and therefore |{ x | φ( x)}| (2n + 1) | x| > γ holds in ([1, 2n + 1], A 2n+1 ). (Here φ may contain parameters.) But since this holds for almost every n, by transfer |{ x | φ( x)}| (2n + 1) | x| > γ holds in (G, +, A) where the inequality is between nonstandard rational numbers. This means that µ | x| ({ x | φ( x)}) = st( |{ x | φ( x)}| (2n + 1) | x| ) ≥ γ. For the second part, suppose (G, +, A) ¬m(φ) > γ. Then for almost every n, ([1, 2n + 1], +, A 2n+1 ) ¬m(φ) > γ, and therefore |{ x | φ( x)}| (2n + 1) | x| ≤ γ holds in ([1, 2n + 1], A 2n+1 ) . But since this holds for almost every n, by transfer, |{ x | φ( x)}| (2n + 1) | x| ≤ γ holds in (G, +, A), so also µ | x| ({ x | φ( x)}) = st( |{ x | φ( x)}| (2n + 1) | x| ) ≤ γ. Since both m and µ are additive, this extends immediately to the corresponding integrals. As a notational convenience, let us write x for a sequence x 1 , . . . , x k , and x i for the sequence x 1 , . . . , x i−1 , x i+1 , . . . , x k and x i,k for the sequence x 1 , . . . , x i−1 , x i+1 , . . . , x k−1 . Observe that for any b, the function a → a+b is a definable bijection, as is a → k·a for any integer k. For r ≤ k, we define formulas A r on x r . When r < k, we define A r (x r ) :⇔ i<k,i =r i · x i + r ·   x k − i<k,i =r x i   ∈ A. We define A k (x k ) :⇔ i<k i · x i ∈ A. Note that for any x k−1,k , µ({x k−1 | x k ∈ A k }) = µ(A), so by the Fubini property of these measures, µ k−1 (A k ) = µ(A). Define σ( x k ) := i<k x i ; it is easy to see that for any a i,k , the function a i → σ( a k ) is a bijection. Whenever A k ( a k ), also A i ( a i,k , σ( a k )) for each i < k. In particular, if we define := { a k | A k ( a k )∧∀i < kA i ( a i,k , σ( a k ))}, we have = A k , and so µ k−1 (Â) = µ(A) > 0. In the next section, we will show that, under these conditions, µ k ( i≤k A i ) > 0. First, however, we show that this is enough to prove Szemerédi's Theorem. If µ k ( i≤k A i ) > 0, we may find a ∈ i≤k A i such that a k = i<k a i . Setting a := i<k i · a i and d := a k − i<k a i , we have a + id ∈ A for i ∈ [0, k − 1]. Therefore a, a + d, . . . , a + (k − 1)d is a k-term arithmetic progression in A. This contradicts the construction of the model, which in turn means that the initial assumption that the sets A N ⊆ [1, N ] exist must fail. Therefore for every δ, there is an N such that for every A N ⊆ [1, N ] with |A N | ≥ δn, A N contains an arithmetic progression of length k. {(x 1 , . . . , x n ) | φ({x i } i∈I )} where φ is a formula with parameters from M . We write S k for the collection of subsets of S with cardinality k. When k ≤ n, we define B n,k (M ) = I∈( [1,n] k ) B n,I (M ). We also define B n,<I (M ) = J∈( I |I|−1 ) (M ). If B is any Boolean algebra, we write B σ for the σ-algebra generated by B. We equate formulas with the sets they define, so we will also speak of B as being a Boolean algebra of formulas. Then for any Proof. Suppose not. Then setting ǫ := ||f − E(f | B σ )|| L 2 and δ := ||f − E(f | D σ 0 )|| L 2 , we must have δ < ǫ. For some β 1 , . . . , β m and A 1 ( x, b 1 f ∈ L 2 (B σ 1 ), ||E(f | B σ ) − E(f | B σ 0 )|| = 0.), . . . , A m ( x, b m ) ∈ B 1 with each b i ∈ M , we have ||f − i≤m β i χ Ai ( x, b i )|| L 2 < (ǫ − δ)/4. Since ||f −E(f | B σ 0 )|| = δ, there are α 1 , . . . , α n and D 1 ( x, a 1 ), . . . , D n ( x, α n ) ∈ D 0 with ||f − i≤n α i χ Di ( x, b i )|| L 2 < ǫ − (ǫ − δ)/2, and therefore || i≤m β i χ Ai ( x, b i ) − i≤n α i χ Di ( x, b i )|| L 2 < ǫ − 3(ǫ − δ)/4. This means the formula ∃ y 1 , . . . , y m ¬(|| i≤m β i χ Ai ( x, b i ) − i≤n α i χ Di ( x, y i )|| 2 L 2 > [ǫ − 3(ǫ − δ)/4] 2 ) is satisfied (where, to view this as a formula, we expand the norm into an integral of sums of definable formulas). This is a formula with parameters from M , so by the elementarity of M , there are witnesses a ′ 1 , . . . , a ′ n in M satisfying: || i≤m β i χ Ai ( x, b i ) − i≤n α i χ Di ( x, a ′ i )|| L 2 ≤ ǫ − 3(ǫ − δ)/4. It follows that ||f − i≤n α i χ Di ( x, a ′ i )|| L 2 < ǫ, and since i≤n α i χ Di ( x, a ′ i ) is measurable with respect to D σ , this contradicts the assumption that ||f − E(f | D σ )|| L 2 = ǫ. Theorem 5. Let n ≤ k and suppose that for each each I ⊆ [1, n] with |I| = k, we have a set A I ∈ B σ n,I (M ), and suppose there is a δ > 0 such that whenever B I ∈ B n,I (M ) and µ n (A I \ B I ) < δ for all I ∈ [1,n] k , B I is non-empty. Then µ n ( I∈( [1,n] k ) A I ) > 0. Proof. We proceed by main induction on k. When k = 1, the claim is trivial: we must have µ(A I ) > 0 for all I, since otherwise we could take B I = ∅; then µ( A I ) = µ(A I ) > 0. So we assume that k > 1 and that whenever B I ∈ B n,I (M ) and µ n (A I \ B I ) < δ for all I, I∈( [1,n] k ) B I is non-empty. Throughout this proof, the variable I ranges over [1,n] k . Claim 1. For any I 0 , (χ AI 0 − E(χ AI 0 | B σ n,<I0 (M ))) I =I0 χ AI dµ n = 0. Proof. When k = n, this is trivial since I =I0 χ AI is an empty product, and therefore equal to 1. If k < n, we have (χ AI 0 − E(χ AI 0 | B σ n,<I0 (M ))) I =I0 χ AI dµ n = (χ AI 0 − E(χ AI 0 | B σ n,<I0 (M ))) I =I0 χ AI dµ k ({x i } i∈I0 )dµ n−k ({x i } i ∈I0 ). Observe that for any choice of {a i } i ∈I0 , B n,<I0 (M ), B n,<I0 (M ∪{a i } i ∈I0 ), B n,I0 (M ) satisfy the preceding lemma, so ||E(χ AI 0 | B σ n,<I0 (M )) − E(χ AI 0 | B σ n,<I0 (M ∪ {a i } i ∈I0 ))|| L 2 = 0. The function I =I0 χ Ai ({x i } i∈I0 , {a i } i ∈I0 ) is measurable with respect to B σ n,<I0 (M ∪ { a i }). Combining these two facts, we have (χ AI 0 − E(χ AI 0 | B σ n,<I0 (M ))) I =I0 χ AI dµ k ({x i } i∈I0 ) = (χ AI 0 − E(χ AI 0 | B σ n,<I0 (M ∪ {a i } i ∈I0 ))) I =I0 χ Ai dµ k ({x i })µ n (A ′ I0 \ B I0 ) < δ, I∈( [1,n] k ) B I is non-empty, and • If µ n (A ′ I0 ∩ I =I0 A I ) > 0, µ n ( A I ) > 0. Proof. Define A ′ I0 := { x i | E(χ AI 0 | B σ n,<I0 (M ))( x i ) > 0}. If µ n (A ′ I0 ∩ I =I0 A I ) > 0 then we have E(χ AI 0 | B σ n,<I0 (M )) I =I0 χ AI dµ n > 0 and by the previous claim, this implies that µ( A I ) > 0. Suppose that for each I, B I ∈ B n,I (M ) with µ n (A I \ B I ) < δ for I = I 0 and µ n (A ′ I0 \ B I ) < δ. Since µ n (A I0 \ A ′ I0 ) = χ AI 0 (1 − χ A ′ I 0 )dµ n = E(χ AI 0 | B σ n,<I0 )(1 − χ A ′ I 0 )dµ n = 0, we have µ n (A I0 \ B I0 ) < δ as well, and therefore B I is non-empty. ⊣ By applying the previous claim to each I ∈ [1,n] k , we may assume for the rest of the proof that for each I, A I ∈ B σ n,<I (M ). Fix some finite algebra B ⊆ B n,k−1 (M ) so that for every I, ||χ AI − E(χ AI | B)|| L 2 < √ δ √ 2(( n k )+1) (such a B exists because there are finitely many I and each A I is B σ n,k−1 (M )-measurable). For each I, set A * I = { a i | E(χ AI | B)( a) > ( n k ) ( n k )+1 }. Claim 3. For each I, µ(A I \ A * I ) ≤ δ/2 Proof. A I \ A * I is the set of points such that χ AI − E(χ AI | B)( a) > 1 ( n k )+1 . By Chebyshev's inequality, the measure of this set is at most In particular, x ∈ B i,J ∪ I ′ ⊇J,I ′ =I (A * I ′ ,i I ′ ,J ) for the particular i we have chosen. Since x ∈ A * I,i I ′ ,J for each I ′ ⊃ J, it must be that x ∈ B i,J . This holds for any J, so x ∈ J B i,J . ⊣ From our assumption, I B * I is non-empty, and therefore there is some i such that J B i,J . But this leads to a contradiction, so it must be that µ( I A * I ) > 0, and therefore, as we have shown, µ( I A I ) ≥ 1 ( n k )+1 µ( I A * I ) > 0. ( n k + 1) 2 (χ AI − E(χ AI | B)) 2 dµ = ( n k + 1) 2 ||χ AI − E(χ AI | B)|| 2 L 2 < δ 2 . ⊣ Claim 4. µ( I A I ) ≥ µ( I A * I )/ n k + 1 . Proof. For each I 0 , µ((A * I0 \ A I0 ) ∩ I =I0 A * I ) = χ A * I 0 (1 − χ AI 0 ) I =I0 χ A * I dµ n = χ A * I 0 (1 − E(χ AI 0 | B)) I =I0 χ A * I dµ n . Let M ⊆ G be a set, let n be a positive integer, and let I ⊆ [1, n] be given. We define B n,I (M ) to be the Boolean algebra of subsets of G n of the form Lemma 4 . 4Let B, B 0 , B 1 be Boolean algebras with B ⊆ B 0 ∩ B 1 . Suppose there is an elementary submodel M such that: • Every parameter in every formula in B 1 belongs to M , • If φ( x, a) ∈ B 0 , b ∈ M , and | a| = | b|, then φ( x, b) ∈ B. An illustrating case is when B = B n,k (M ), B 0 = B n,k (M ∪ N ), and B 1 = B n,k+1 (M ). =0 Since this holds for any {a i } i ∈I0 , the claim follows by integrating over all choices of {a i }. ⊣ Claim 2. For any I 0 , there is an A ′ I0 ∈ B σ n,<I0 (M ) such that: • Whenever B I ∈ B n,I (M ) for each I, µ n (A I \ B I ) < δ for each I = I 0 , and ⊣ Each A * I may be written in the form i≤rI A *I,i where A * I,i = J∈( I k−1 ) A * I,i,J and A * I,i,J is an element of B n,J (M ). We may assume that if i = i ′ then A * I,i ∩ A * I,i ′ = ∅. ForI each i ∈ I [1, r I ], let D i = I J∈( I k−1 ) A * iI ,J,I . Each A * I,iI ,J is an element of B n,J (M ), so we may group the components and write D i = J∈( n k−1 ) D i,J where D i,J = I⊃J A * I,iI ,J . Suppose, for a contradiction, that µ( I A * I ) = 0. Then for every i ∈ I [1, r I ], µ(D i ) = µ( J D i,J ) = 0. By the inductive hypothesis, for each γ > 0, there is a collection B i,J ∈ B n,J (M ) such that µ(D i,J \ B i,J ) < γ and J B i,J = ∅. In particular, this holds with γ = δ 2( k k−1 )( I rI )(maxI rI ) . For each I, i ≤ r I , J ⊂ I, ′ ⊇J,I ′ =I Proof. Suppose x ∈ I B * I = I i≤rI J B * I,i,J . Then for each I, there is an i I ≤ r I such that x ∈ J B * I,iI ,J . Since B * I,iI ,J ⊆ A * I,iI ,J , for each I and J ⊂ I, x ∈ A * I,iI ,J . For any J, let I ⊃ J. Then x ∈ B * I,iI ,J = A * I,iI ,J ∩ Without this modification, An might contain no arithmetic progressions as a set of integers, but contain arithmetic progressions as a subset of the group, since the group contains progressions which "wrap around": for instance, 3, 10, 4 is an arithmetic progression in[1,13], but does not correspond to an arithmetic progression in the integers. Expanding the group by adding a "dead zone" disjoint from An is a standard way of avoiding this problem. Proof. Since µ(A I \ A * I ) < δ/2, it suffices to show that µ(A * I \ B * I ) < δ/2. Deducing the density Hales-Jewett Theorem from an infinitary removal lemma. Tim Austin, Tim Austin. Deducing the density Hales-Jewett Theorem from an infinitary re- moval lemma. http://arxiv.org/abs/0903.1633, March 2009. Deducing the multidimensional Szemerédi Theorem from an infinitary removal lemma. Tim Austin, Tim Austin. Deducing the multidimensional Szemerédi Theorem from an infini- tary removal lemma. http://www.arxiv.org/abs/0808.2267, March 2009. Title: On the norm convergence of nonconventional ergodic averages. Tim Austin, Tim Austin. Title: On the norm convergence of nonconventional ergodic aver- ages. http://www.arxiv.org/abs/0805.0320, February 2009. The ergodic theoretical proof of Szemerédi's theorem. H Furstenberg, Y Katznelson, D Ornstein, Bull. Amer. Math. Soc. (N.S.). 73H. Furstenberg, Y. Katznelson, and D. Ornstein. The ergodic theoretical proof of Szemerédi's theorem. Bull. Amer. Math. Soc. (N.S.), 7(3):527-552, 1982. Ergodic behavior of diagonal measures and a theorem of szemerdi on arithmetic progressions. Harry Furstenberg, 1977.10.1007/BF02813304Journal d'Analyse Mathmatique. 31Harry Furstenberg. Ergodic behavior of diagonal measures and a theorem of szemerdi on arithmetic progressions. Journal d'Analyse Mathmatique, 31:204- 256, 1977. 10.1007/BF02813304. Robert Goldblatt, Lectures on the hyperreals. New YorkSpringer-Verlag188An introduction to nonstandard analysisRobert Goldblatt. Lectures on the hyperreals, volume 188 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1998. An introduction to nonstandard analysis. A new proof of Szemerédi's theorem. W T Gowers, Geom. Funct. Anal. 113W. T. Gowers. A new proof of Szemerédi's theorem. Geom. Funct. Anal., 11(3):465-588, 2001. Hypergraph regularity and the multidimensional Szemerédi theorem. W T Gowers, Ann. of Math. 1662W. T. Gowers. Hypergraph regularity and the multidimensional Szemerédi theo- rem. Ann. of Math. (2), 166(3):897-946, 2007. Stable group theory and approximate subgroups. Ehud Hrushovski, Ehud Hrushovski. Stable group theory and approximate subgroups. Hyperfinite model theory. H , Jerome Keisler, Logic Colloquium 76. Oxford; AmsterdamNorth-Holland87H. Jerome Keisler. Hyperfinite model theory. In Logic Colloquium 76 (Oxford, 1976), pages 5-110. Studies in Logic and Found. Math., Vol. 87. North-Holland, Amsterdam, 1977. An infinitesimal approach to stochastic analysis. H , Jerome Keisler, Mem. Amer. Math. Soc. 48297184H. Jerome Keisler. An infinitesimal approach to stochastic analysis. Mem. Amer. Math. Soc., 48(297):x+184, 1984. The counting lemma for regular k-uniform hypergraphs. Brendan Nagle, Vojtěch Rödl, Mathias Schacht, Random Structures Algorithms. 282Brendan Nagle, Vojtěch Rödl, and Mathias Schacht. The counting lemma for regular k-uniform hypergraphs. Random Structures Algorithms, 28(2):113-179, 2006. Regularity lemma for k-uniform hypergraphs. Vojtěch Rödl, Jozef Skokan, Random Structures Algorithms. 251Vojtěch Rödl and Jozef Skokan. Regularity lemma for k-uniform hypergraphs. Random Structures Algorithms, 25(1):1-42, 2004. On sets of integers containing no k elements in arithmetic progression. E Szemerédi, Collection of articles in memory of Juriȋ Vladimirovič Linnik. 27E. Szemerédi. On sets of integers containing no k elements in arithmetic progres- sion. Acta Arith., 27:199-245, 1975. Collection of articles in memory of Juriȋ Vladimirovič Linnik. Regular partitions of graphs. Endre Szemerédi, Problèmes combinatoires et théorie des graphes. Orsay; ParisCNRS260CNRS, Univ. OrsayColloq. Internat.Endre Szemerédi. Regular partitions of graphs. In Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 of Colloq. Internat. CNRS, pages 399-401. CNRS, Paris, 1978. A variant of the hypergraph removal lemma. Terence Tao, J. Combin. Theory Ser. A. 1137Terence Tao. A variant of the hypergraph removal lemma. J. Combin. Theory Ser. A, 113(7):1257-1280, 2006. A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma. Terence Tao, 10.1007/s11854-008-0001-0Journal d'Analyse Mathmatique. 103Terence Tao. A correspondence principle between (hyper)graph theory and prob- ability theory, and the (hyper)graph removal lemma. Journal d'Analyse Mathma- tique, 103:1-45, 2007. 10.1007/s11854-008-0001-0. Norm convergence of multiple ergodic averages for commuting transformations. Ergodic Theory Dynam. Terence Tao, Systems. 282Terence Tao. Norm convergence of multiple ergodic averages for commuting transformations. Ergodic Theory Dynam. Systems, 28(2):657-688, 2008. Yet another proof of Szemerédi's Theorem. Terence Tao, Terence Tao. Yet another proof of Szemerédi's Theorem. Convergence of diagonal ergodic averages. Henry Towsner, 10.1017/S0143385708000722Ergodic Theory Dynam. Systems. 294Henry Towsner. Convergence of diagonal ergodic averages. Ergodic Theory Dynam. Systems, 29(4):1309-1326, 2009. 10.1017/S0143385708000722.
[]
[ "Robust and Fast Measure of Information via Low-rank Representation", "Robust and Fast Measure of Information via Low-rank Representation" ]
[ "Yuxin Dong [email protected] \nSchool of Computer Science and Technology\nXi'an Jiaotong University\n710049Xi'anChina\n\nMinistry of Education\nShaanxi Provincial Key Laboratory of Big Data Knowledge Engineering\nXi'an 710049China\n", "Tieliang Gong \nSchool of Computer Science and Technology\nXi'an Jiaotong University\n710049Xi'anChina\n\nMinistry of Education\nShaanxi Provincial Key Laboratory of Big Data Knowledge Engineering\nXi'an 710049China\n", "Shujian Yu \nMachine Learning Group\nUiT\nThe Arctic University of Norway\n\n", "Hong Chen [email protected] \nCollege of Science\nHuazhong Agriculture University\n430070WuhanChina\n\nEngineering Research Center of Intelligent Technology for Agriculture\nMinistry of Education\n430070WuhanChina\n", "Chen Li \nSchool of Computer Science and Technology\nXi'an Jiaotong University\n710049Xi'anChina\n\nMinistry of Education\nShaanxi Provincial Key Laboratory of Big Data Knowledge Engineering\nXi'an 710049China\n" ]
[ "School of Computer Science and Technology\nXi'an Jiaotong University\n710049Xi'anChina", "Ministry of Education\nShaanxi Provincial Key Laboratory of Big Data Knowledge Engineering\nXi'an 710049China", "School of Computer Science and Technology\nXi'an Jiaotong University\n710049Xi'anChina", "Ministry of Education\nShaanxi Provincial Key Laboratory of Big Data Knowledge Engineering\nXi'an 710049China", "Machine Learning Group\nUiT\nThe Arctic University of Norway\n", "College of Science\nHuazhong Agriculture University\n430070WuhanChina", "Engineering Research Center of Intelligent Technology for Agriculture\nMinistry of Education\n430070WuhanChina", "School of Computer Science and Technology\nXi'an Jiaotong University\n710049Xi'anChina", "Ministry of Education\nShaanxi Provincial Key Laboratory of Big Data Knowledge Engineering\nXi'an 710049China" ]
[]
The matrix-based Rényi's entropy allows us to directly quantify information measures from given data, without explicit estimation of the underlying probability distribution. This intriguing property makes it widely applied in statistical inference and machine learning tasks. However, this information theoretical quantity is not robust against noise in the data, and is computationally prohibitive in large-scale applications. To address these issues, we propose a novel measure of information, termed low-rank matrix-based Rényi's entropy, based on low-rank representations of infinitely divisible kernel matrices. The proposed entropy functional inherits the specialty of of the original definition to directly quantify information from data, but enjoys additional advantages including robustness and effective calculation. Specifically, our low-rank variant is more sensitive to informative perturbations induced by changes in underlying distributions, while being insensitive to uninformative ones caused by noises. Moreover, low-rank Rényi's entropy can be efficiently approximated by random projection and Lanczos iteration techniques, reducing the overall complexity from O(n 3 ) to O(n 2 s) or even O(ns 2 ), where n is the number of data samples and s n. We conduct large-scale experiments to evaluate the effectiveness of this new information measure, demonstrating superior results compared to matrix-based Rényi's entropy in terms of both performance and computational efficiency.
10.48550/arxiv.2211.16784
[ "https://export.arxiv.org/pdf/2211.16784v1.pdf" ]
254,095,976
2211.16784
83838422effa238682d462396a5a251fabb079b3
Robust and Fast Measure of Information via Low-rank Representation Yuxin Dong [email protected] School of Computer Science and Technology Xi'an Jiaotong University 710049Xi'anChina Ministry of Education Shaanxi Provincial Key Laboratory of Big Data Knowledge Engineering Xi'an 710049China Tieliang Gong School of Computer Science and Technology Xi'an Jiaotong University 710049Xi'anChina Ministry of Education Shaanxi Provincial Key Laboratory of Big Data Knowledge Engineering Xi'an 710049China Shujian Yu Machine Learning Group UiT The Arctic University of Norway Hong Chen [email protected] College of Science Huazhong Agriculture University 430070WuhanChina Engineering Research Center of Intelligent Technology for Agriculture Ministry of Education 430070WuhanChina Chen Li School of Computer Science and Technology Xi'an Jiaotong University 710049Xi'anChina Ministry of Education Shaanxi Provincial Key Laboratory of Big Data Knowledge Engineering Xi'an 710049China Robust and Fast Measure of Information via Low-rank Representation The matrix-based Rényi's entropy allows us to directly quantify information measures from given data, without explicit estimation of the underlying probability distribution. This intriguing property makes it widely applied in statistical inference and machine learning tasks. However, this information theoretical quantity is not robust against noise in the data, and is computationally prohibitive in large-scale applications. To address these issues, we propose a novel measure of information, termed low-rank matrix-based Rényi's entropy, based on low-rank representations of infinitely divisible kernel matrices. The proposed entropy functional inherits the specialty of of the original definition to directly quantify information from data, but enjoys additional advantages including robustness and effective calculation. Specifically, our low-rank variant is more sensitive to informative perturbations induced by changes in underlying distributions, while being insensitive to uninformative ones caused by noises. Moreover, low-rank Rényi's entropy can be efficiently approximated by random projection and Lanczos iteration techniques, reducing the overall complexity from O(n 3 ) to O(n 2 s) or even O(ns 2 ), where n is the number of data samples and s n. We conduct large-scale experiments to evaluate the effectiveness of this new information measure, demonstrating superior results compared to matrix-based Rényi's entropy in terms of both performance and computational efficiency. Introduction The practical applications of traditional entropy measures e.g. Shannon's entropy (Shannon 1948) and Rényi's entropy (Rényi 1961) have long been hindered by their heavy reliance on the underlying data distributions, which are extremely hard to estimate or even intractable in highdimensional spaces (Fan and Li 2006). Alternatively, the matrix-based Rényi's entropy proposed by (Sanchez Giraldo, Rao, and Principe 2014) treats the entire eigenspectrum of a normalized kernel matrix as a probability distribution, thus allows direct quantification from given data samples by projecting them in reproducing kernel Hilbert spaces (RKHS) without the exhausting density estimation. This intriguing property makes matrix-based Rényi's entropy and its multivariate extensions (Yu et al. 2019) successfully applied in various data science applications, ranging from classical dimensionality reduction and feature selection (Brockmeier et al. 2017;Álvarez-Meza et al. 2017) problems to advanced deep learning problems such as network pruning (Sarvani et al. 2021) and knowledge distillation (Miles, Rodríguez, and Mikolajczyk 2021). Despite the empirical success of matrix-based Rényi's entropy, it has been shown to be not robust against noises in the data (Yu et al. 2019), because it cannot distinguish them from linear combinations of informative features in highdimensional scenarios. Moreover, the exact calculation requires O(n 3 ) time complexity with traditional eigenvalue decomposition techniques e.g. CUR decomposition and QR factorization (Mahoney and Drineas 2009;Watkins 2008), greatly hampering its application in large scale tasks due to the unacceptable computational cost. Inspired by the success of min-entropy which uses the largest outcome solely as a measure of information (Wan et al. 2018;Konig, Renner, and Schaffner 2009), we seek for a robust information quantity by utilizing low-rank representations of kernel matrices. Our new definition, termed lowrank matrix-based Rényi's entropy (abbreviated as low-rank Rényi's entropy), fulfills the entire set of axioms provided by Rényi (Rényi 1961) that a function must satisfy to be considered a measure of information. Compared to the original matrix-based Rényi's entropy, our low-rank variant is more sensitive to informative perturbations caused by variation of the underlying probability distribution, while being more robust to uninformative ones caused by noises in the data samples. Moreover, our low-rank Rényi's entropy can be efficiently approximated by random projection and Lanczos iteration techniques, achieving substantially lower time complexity than the trivial eigenvalue decomposition approach. We theoretically analyze the quality of approximation results, and conduct large-scale experiments to evaluate the effectiveness of low-rank Rényi's entropy as well as the approximation algorithms. The main contributions of this work are summarized as follows: • We extend Giraldo et al.'s definition and show that a measure of entropy can be built upon the low-rank representation of the kernel matrix. Our low-rank definition can be naturally extended to measure the interactions between multiple random variables, including joint entropy, arXiv:2211.16784v1 [cs.LG] 30 Nov 2022 conditional entropy, and mutual information. • Theoretically, we show that low-rank Rényi's entropy is more insensitive to random perturbations of the data samples under mild assumptions. We also give empirical examples of low-rank Rényi's entropy achieving higher discriminability for different eigenspectrum distributions through a proper choice of the hyper-parameter k. • We develop efficient algorithms to approximate low-rank Rényi's entropy through random projection and Lanczos iteration techniques, enabling fast and accurate estimations respectively. The overall complexity is reduced from O(n 3 ) to O(n 2 s) or even O(ns 2 ) for some s n, leading to a significant speedup compared to the original matrix-based Rényi's entropy. • We evaluate the effectiveness of low-rank Rényi's entropy on large-scale synthetic and real-world datasets, demonstrating superior performance compared to the original matrix-based Rényi's entropy while bringing tremendous improvements in computational efficiency. Related Work Matrix-based Rényi's Entropy Given random variable X with probability density function (PDF) p(x) defined in a finite set X , the α-order Rényi's entropy (α > 0, α = 1) H α (X) is defined as H α (X) = 1 1−α log 2 X p α (x) dx, where the limit case α → 1 yields the well-known Shannon's entropy. It is easy to see that Rényi's entropy relies heavily on the distribution of the underlying variable X, preventing its further adoption in data-driven science, especially for high-dimensional scenarios. To alleviate this issue, an alternative measure namely matrix-based Rényi's entropy was proposed (Sanchez Giraldo, Rao, and Principe 2014): Definition 1. Let κ : X × X → R be an infinitely divisible positive kernel (Bhatia 2006). Given {x i } n i=1 ⊂ X , each x i being a real-valued scalar or vector, and the Gram matrix K obtained from K ij = κ(x i , x j ), a matrix-based analogue to Rényi's α-entropy can be defined as: S α (A) = 1 1−α log 2 n i=1 λ α i (A) , where A ij = 1 n Kij √ KiiKjj is a normalized kernel matrix and λ i (A) is the i-th largest eigenvalue of A. The kernel matrix A is positive semi-definite (PSD) and satisfies tr(A) = 1, therefore λ i ∈ [0, 1] for all i ∈ [1, n]. With this setting, one can similarly define matrix notion of Rényi's conditional entropy S α (A|B), mutual information I α (A; B), and their multivariate extensions (Yu et al. 2019). Approximating Matrix-based Rényi's Entropy Exactly calculating S α (A) requires O(n 3 ) time complexity in general with traditional eigenvalue decomposition techniques. Recently, several attempts have been made towards accelerating the computation of S α (A) from the perspective of randomized numerical linear algebra (Gong et al. 2021;Dong et al. 2022). Although we also develop fast approximations, the motivation and technical solutions are totally different: we aim to propose a new measure of information that is robust to noise in data and also enjoys fast computation, whereas Gong and Dong et al. only accelerate the original matrix-based Rényi's entropy. Moreover, in terms of adopted mathematical tools, we mainly focus on random projection and Lanczos iteration algorithms, rather than stochastic trace estimation and polynomial approximation techniques used in their works. As a result, the corresponding theoretical error bounds are also different. A Low-rank Definition of Rényi's Entropy Our motivations root in two observations. Recall that the min-entropy (Konig, Renner, and Schaffner 2009), defined by H min (X) = − log 2 max x∈X p(x), measures the amount of information using solely the largest probability outcome. In terms of quantum statistical mechanics, it is the largest eigenvalue of the quantum state ρ which is PSD and has unit trace (Ohya and Petz 2004). On the other hand, the eigenvalues with the maximum magnitude characterize the main properties of a PSD matrix. Inspired by these observations, we develop a robust information theoretical quantity by exploiting the low-rank representation: Definition 2. Let κ : X × X → R be an infinitely divisible kernel. Given {x i } n i=1 ⊂ X and integer k ∈ [1, n − 1], the low-rank Rényi's α-order entropy is defined as: S k α (A) = 1 1−α log 2 k i=1 λ α i (A) + (n − k)λ α r (A) , where A is the normalized kernel matrix constructed from {x i } n i=1 and κ, λ i (A) is the i-th largest eigenvalues of A and λ r (A) = 1 n−k 1 − k i=1 λ i (A) . Let A k be the best rank-k approximation of A and L k (A) be the matrix constructed by replacing the smaller n − k eigenvalues in A to λ r (A). It is easy to verify that S k α (A) = S k α (A k ) = S k α (L k (A)) = S α (L k (A)). Definition 2 complements the smaller eigenvalues through a uniform distribution, which is the unique method that fulfills all axioms below (the uniqueness is discussed in the appendix 1 ). Proposition 1. Let A, B ∈ R n×n be arbitrary normalized kernel matrices, then (a) S k α (PAP ) = S k α (A) for any orthogonal matrix P. (b) S k α (pA) is a continuous function for 0 < p ≤ 1. (c) 0 ≤ S k α (A) ≤ S k α ( 1 n I) = log 2 (n). (d) S 2nk−k 2 α L k (A) ⊗ L k (B) = S k α (A) + S k α (B). (e) If AB = BA = 0 and tr(A k ) = tr(B k ) = 1, then for g(x) = 2 (1−α)x and t ∈ [0, 1], we have S 2k α tA + (1 − t)B = g −1 tg(S k α (A)) + (1 − t)g(S k α (B)) . (f) S k α A•B tr(A•B) ≥ max S k α (A), S k α (B) . (g) S k α A•B tr(A•B) ≤ S k α (A) + S k α (B). i PDF Entropy rank (k) CDF Figure 1: Left: PDF (solid) and CDF (dashed) of the altered eigenspectrum for different ranks k. Right: The convergence behavior of S k α (A) (solid) to S α (A) (dashed) with the increase of rank k for different EDR (r). Remark 1. Proposition 1 characterizes the basic properties of low-rank Rényi's entropy, in which (a)-(e) are the set of axioms provided by Rényi (Rényi 1961) that a function must satisfy to be a measure of information. Additionally, (f) and (g) together imply a definition of joint entropy which is also compatible with the individual entropy measures: S k α (A, B) = S k α A•B tr(A•B) . This further allows us to define the low-rank conditional entropy S k α (A|B) and mutual information I k α (A; B), whose positiveness is guaranteed by (f) and (g) respectively: S k α (A|B) = S k α (A, B) − S k α (B), I k α (A; B) = S k α (A) + S k α (B) − S k α (A, B ). An intuitive overview of the comparative behavior between S α (A) and S k α (A) for n = 1000 is reported in Figure 1 and 2, where we evaluate the impact of k, α and eigenspectrum decay rate (EDR) r respectively. The eigenvalues are initialized by λ i = e −ri/n and then normalized. It can be observed from Figure 1 that S k α (A) is always larger than S α (A) since the uncertainty of the latter n − k outcomes are maximized. Moreover, S k α (A) quickly converges to S α (A) with the increase of k, especially in extreme cases when the eigenspectrum of A is flat or steep. From Figure 2, we can see that for small k, S k α (A) decreases slow with the increase of α when α < 1 and fast otherwise. This behavior is the opposite when k becomes large. Furthermore, we can see that EDR directly influences the value of entropy, as a flat eigenspectrum indicates higher uncertainty and steep the opposite. As can be seen, S k α (A) monotonically decreases with the increase of r, and decreases faster than S α (A) in a certain range which varies according to the choice of k, indicating higher sensitivity to informative distribution changes when the hyper-parameter k is selected properly. Moreover, consider the case that the data samples {x i } n i=1 are randomly perturbed, i.e. y i = x i + εp i , where p i are random vectors comprised of i.i.d. entries with zero expectation and unit variance. Let A and B be kernel matrices constructed from {x i } n i=1 and {y i } n i=1 respectively, and let {λ i } n i=1 , {µ i } n i=1 be their eigenvalues. Then it satisfies that (Ngo 2005), where u i is the corresponding eigenvector of λ i . When ε is small, the entries as Entropy order (α) well as the eigenvalues of A are nearly independently perturbed. The following theorem shows that S k α (A) is more robust against small noises in data compared to S α (A): µ i ≈ λ i + u i (B − A)u iS k α (A) EDR (r) S k α (A) Sα(A) k = 1 k = 3 k = 10 k = 30 k = 100Theorem 1. Let {ν i } n i=1 be independent random variables with zero mean and variance {σ 2 i } n i=1 . Let A and B be PSD matrices with eigenvalues λ i and µ i = λ i + ν i re- spectively. If k i=1 σ 2 i ≤ n i=k+1 σ 2 i or α > 1, there exists > 0 such that when all |ν i | ≤ , we have Var[IP k α (B)] ≤ Var[IP α (B)] , where IP is the information potential (Gokcay and Principe 2000) defined as IP α (B) = 2 (1−α)Sα(B) and IP k α (B) = 2 (1−α)S k α (B) . Remark 2. Theorem 1 indicates that IP k α (B) enables lower variance than IP α (B) against random perturbation of the eigenvalues under mild conditions, which is easy to be satisfied since in most cases we have k n. Combining with our discussion above, the low-rank Rényi's entropy is more sensitive to informative variations in probability distributions which will surely induce an increase or decrease in entropy, while being insensitive to uninformative perturbations caused by noises in the data samples. Extending to Multivariate Scenarios Following Definition 2 and Proposition 1, the low-rank variant of multivariate Rényi's joint entropy, in virtue of the Venn diagram relation for Shannon's entropy (Yeung 1991), could be naturally derived: Definition 3. Let {κ i } L i=1 : X i × X i → R be positive in- finitely divisible kernels and {x 1 i , · · · , x L i } n i=1 ⊂ X 1 ×· · ·× X L , the low-rank Rényi's joint entropy is defined as: S k α (A 1 , · · · , A L ) = S k α A1•···•A L tr(A1•···•A L ) , where A 1 , · · · , A L are normalized kernel matrices con- structed from {x 1 i } n i=1 , · · · , {x L i } n i=1 respectively and • denotes the Hadamard product. This joint entropy definition enables further extension to multivariate conditional entropy and mutual information: S k α (A 1 , · · · , A k |B) = S k α (A 1 , · · · , A k , B) − S k α (B), I k α (A 1 , · · · , A k ; B) = S k α (A 1 , · · · , A k ) + S k α (B) − S k α (A 1 , · · · , A k , B), Algorithm 1: Approximation via Random Projection 1: Input: Integers n, k ∈ [1, n/2], s ≥ k, kernel matrix A ∈ R n×n , order α > 0. 2: Output: Approximation to S k α (A); 3: Construct a random projection matrix P ∈ R n×s . 4: Calculate = AP ∈ R n×s . 5: Calculate the largest k singular valuesλ i , i ∈ [1, k] of A through singular value decomposition. 6: Calculateλ r = 1 n−k 1 − k i=1λ i . 7: Return:Ŝ k α (A) = 1 1−α log 2 k i=1λ α i + (n − k)λ α r . where A 1 , · · · , A L and B are normalized kernel matrices constructed from the variables {x 1 i } n i=1 , · · · , {x L i } n i=1 and the target label {y i } n i=1 respectively. Their positiveness can be guaranteed through a reduction to axiom (f) and (g). These multivariate information quantities enable much more widespread applications e.g. feature selection, dimension reduction and information-based clustering. Approximating Low-rank Rényi's Entropy Although only the largest eigenvalues are accessed by our entropy definition, one still needs to calculate the full eigenspectrum of the PSD matrix A through eigenvalue decomposition algorithms, resulting in O(n 3 ) overall time cost. To alleviate the computational burden, we design fast approximations by leveraging random projection and Lanczos iteration techniques for low-rank Rényi's entropy. Random Projection Approach Random projection offers a natural way to approximate the low-rank representation of kernel matrices. The core idea is to project the n×n PSD matrix A into a n×s subspace, and then use the largest k singular value of the projected matrix as approximations of the largest k eigenvalues, as summarized in Algorithm 1. In this way, the main computation cost is reduced to O(n 2 s) or even O(ns 2 ), (s n), substantially lower than the original O(n 3 ) approach. Based on this fact, we develop efficient approximation algorithms by exploring different random projection techniques, in which the construction of P varies depending on the practical applications, ranging from simple but effective Gaussian distributions to advanced random orthogonal projections. Gaussian Random Projection As one of the most widely used random projection techniques, Gaussian random projection (GRP) admits a simple but elegant solution for eigenvalue approximation: P = n/s · G, where the columns of G ∈ R n×s are initialized by i.i.d random standard Gaussian variables and then orthogonalized. The time complexity of GRP is O(n 2 s). Subsampled Randomized Hadamard Transform SRHT (Lu et al. 2012;Tropp 2011) is a simplification of the fast Johnson-Lindenstrauss transform (Ailon and Chazelle 2009) which preserves the geometry of an entire subspace of vectors compared to GRP. In our settings, the n × s SRHT matrix is constructed by P = 1/s · DHS, where D ∈ R n×n is a diagonal matrix with random {±1} entries, H ∈ R n×n is a Walsh-Hadamard matrix, S ∈ R n×s is a subsampling matrix whose columns are a uniformly chosen subset of the standard basis of R n . Two key ingredients make SRHT an efficient approximation strategy: first, it takes only O(n 2 min(log(n), s)) time complexity to calculate the projected matrixÂ; second, the orthonormality between the columns of A can be preserved after projection, thus is more likely to achieve lower approximation error compared to GRP. Input-Sparsity Transform Similar to SRHT, input-sparsity transform (IST) (Mahoney 2011;Woodruff and Zandieh 2020) utilizes the fast John-Lindenstrauss transform to reduce time complexity for leastsquare regression and low-rank approximation: P = n/s · DS, where D and S are constructed in the same way as SRHT. The complexity of calculating using IST is O(nnz(A)), where nnz denotes the number of non-zero entries, resulting in a total complexity of O(min(nnz(A), ns 2 )). Sparse Graph Sketching The idea of using sparse graphs as sketching matrices is proposed in (Hu et al. 2021). It is shown that the generated bipartite graphs by uniformly adding edges enjoy elegant theoretical properties known as the Expander Graph or Magical Graph with high probability, and thus serve as an effective random projection strategy: P = 1/p · G, where p ∈ N is the hyper-parameter that controls the sparsity, and each column g of G is constructed independently by uniformly sampling c ⊂ [n] with |c| = p, and then setting g i = {±1} randomly for i ∈ c and g i = 0 for i / ∈ c. Similar to IST, sparse graph sketching (SGS) also utilizes the sparsity of input matrices and achieves O(nnz(A)p) computational complexity to calculate the projected matrix. Theoretical Results Next, we provide the main theorem on characterizing the quality-of-approximation for low-rank Rényi's entropy: Theorem 2. Let A be positive definite and s =        O(k + log(1/δ)/ 2 0 ), for GRP O((k + log n) log k/ 2 0 ), for SRHT O(k 2 / 2 0 ), for IST O(k log(k/δ 0 )/ 2 0 ), for SGS p = O(log(k/δ 0 )/ 0 ), for SGS where 0 = λ k λ r , then for k ≤ n/2, with confidence at least 1 − δ, the output of Algorithm 1 satisfies |λ 2 i −λ 2 i | ≤ Algorithm 2: Approximation via Lanczos Iteration 1: Input: Integers n, k ∈ [1, n/2], s ≥ k, kernel matrix A ∈ R n×n , order α > 0, initial vector q. 2: Output: Approximation to S k α (G). 3: Set q 0 = 0, β 0 = 0, q 1 = q/ q . 4: for j = 1, 2, · · · , s do 5:q j+1 = Aq j − β j−1 q j−1 , γ j = q j+1 , q j . 6:q j+1 =q j+1 − γ j q j . 7: Orthogonalizeq j+1 against q 1 , · · · , q j−1 . 8: β j = q j+1 , q j+1 =q j+1 /β j . 9: end for 10: Calculate the largest k eigenvaluesλ i , i ∈ [1, k] of T =     γ 1 β 1 0 β 1 γ 2 . . . β s−1 0 β s−1 γ s     . 11: Calculateλ r = 1 n−k 1 − k i=1λ i . 12: Return:Ŝ k α (A) = 1 1−α log 2 k i=1λ α i + (n − k)λ α r . for all i ∈ [1, k] eigenvalues of A and |S k α (A) −Ŝ k α (A)| ≤ | α 1−α log 2 (1 − )|. Remark 3. Theorem 2 provides the accuracy guarantees for low-rank Rényi's entropy approximation via random projections. It can be observed that the approximation error grows with the increase of α when α is small. Note that although the error bound is additive in nature, it can be further reduced to a relative error bound under mild condition S k α (G) ≥ √ . In general, Theorem 2 requires s = O(k + 1/ 2 ) to achieve 1 ± absolute accuracy, which is consistent with the complexity results of least squares and low rank approximations (Mahoney 2011). Lanczos Iteration Approach Besides random projection, the Lanczos algorithm is also widely adopted to find the k extreme (largest or smallest in magnitude) eigenvalues and the corresponding eigenvectors of an n × n Hermitian matrix A. Given an initial vector q, the Lanczos algorithm utilizes the Krylov subspace spanned by {q, Aq, · · · , A s q} to construct an tridiagonalization of A whose eigenvalues converge to those of A along with the increase of s, and are satisfactorily accurate even for s n. As shown in Algorithm 2, the main computation cost is the O(n 2 s) matrix-vector multiplications in the Lanczos process, which could be further reduced to O(nnz(A)s) when A is sparse. The computational cost of reorthogonalization can be further alleviated by explicit or implicit restarting Lanczos methods. The following theorem establishes the accuracy guarantee of Algorithm 2: Theorem 3. Let A be positive definite, q be the initial vector, {φ i } k i=1 be the corresponding eigenvectors and s = k + 1 2 log R log 4θ 2 K 2 λ1 λr , where R = γ + γ 2 − 1, γ = 1 + 2 min i∈[1,k] λi−λi+1 λi+1−λn , θ = max i∈[1,k] tan φ i , q , K = k−1 j=1λ j −λn λj −λ k , then for k ≤ n/2, the output of Algorithm 2 satisfies 0 ≤ λ i −λ i ≤ λ i for all i ∈ [1, k] eigenvalues of A and |S k α (A) −Ŝ k α (A)| ≤ | α 1−α log 2 (1 − )|. Remark 4. Theorem 3 provides the accuracy guarantee for the Lanczos algorithm. The relationship between approximation error and α is similar to those in Theorem 2. Algorithm 2 achieves a much faster convergence rate compared to Algorithm 1 while achieving the same level of absolute precision. When is small, R, θ and K can be regarded as constants that depends only on the eigenspectrum of A and the initial vector q, so that s = O(k + log(1/ )) is enough to guarantee a 1 ± accuracy. In practice, q is suggested to be generated by random Gaussian in order to avoid a large θ with high probability (Urschel 2021). Experimental Results In this section, we evaluate the proposed low-rank Rényi's entropy and the approximation algorithms under large-scale experiments. Our experiments are conducted on an Intel i7-10700 (2.90GHz) CPU and an RTX 2080Ti GPU with 64GB of RAM. The algorithms are implemented in C++ with the Eigen library and in Python with the Pytorch library. Simulation studies We first test the robustness of S k α (A) against noises in the data. As indicated by Theorem 1, low-rank Rényi's entropy achieves lower variance under mild conditions in terms of the information potential. We consider the case that the input data points are randomly perturbed, i.e. y i = x i +εp i for i ∈ [1, n], where p i is comprised of i.i.d. random variables. Let {λ i } n i=1 , {µ i } n i=1 denote the eigenvalues of normalized kernel matrices constructed from {x i } n i=1 and {y i } n i=1 respectively. We test the following noise distributions: Standard Gaussian N (0, 1), Uniform U (− √ 3, √ 3), Student-t t(3)/ √ 3 and Rademacher {±1} with n = 100 (detailed settings are given in the appendix). The examples of variation in eigenvalues (µ i − λ i ) and the standard deviation (multiplied by n) of entropy values after 100 trials are reported in Figure 3. It verifies our analysis that when ε is small, the eigenvalues µ i are nearly independently perturbed. Moreover, our low-rank definition achieves lower variance than matrix-based Rényi's entropy under different choices of α, in which smaller k corresponds to higher robustness. Real Data Examples In this section, we demonstrate the great potential of applying our low-rank Rényi's entropy functional and its multivariate extensions in two representative real-world information-related applications, which utilize the mutual information (information bottleneck) and multivariate mutual information (feature selection) respectively. Application to Information Bottleneck The Information Bottleneck (IB) methods recently achieve great success in compressing redundant or irrelevant information in the inputs and preventing overfitting in deep neural networks. Formally, given network input X and target label Y, the IB approach tries to extract a compressed intermediate representation Z from X that maintains minimal yet meaningful information to predict the task Y by optimizing the following IB Lagrangian: L IB = I(Y, Z) − β · I(X, Z), where β is the hyper-parameter that balances the trade-off between sufficiency (predictive performance of Z on task Y, quantified by I(Y, Z)) and minimality (the complexity of Z, quantified by I(X, Z)). In practice, optimizing I(Y, Z) is equivalent to the cross-entropy (CE) loss for classification tasks, so our target remains to optimize the latter term I(X, Z). However, mutual information estimation is extremely hard or even intractable for high-dimension distributions, which is usually the case in deep learning. To address this issue, there have been efforts on using variational approximations to optimize a lower bound of I(X, Z), e.g. Variational IB (VIB) (Alemi et al. 2017) and Nonlinear IB (NIB) (Kolchinsky, Tracey, and Wolpert 2019). We show that with low-rank Rényi's entropy, I(X, Z) can be directly optimized by approximating the largest k eigenvalues of the kernel matrix A constructed by X and Z. Recall that the Lanczos method constructs an approximation A ≈ QTQ , where Q ∈ R n×s has orthogonal columns and T ∈ R s×s is tridiagonal, we haveλ i = λ i (Q AQ) for all i ∈ [1, s]. Let s i=1λ i u i u i be the eigenvalue decomposition of Q AQ, we can approximate the gradient of S k α (A) as: ∂S k α (A) ∂A ≈ k i=1 ∂Ŝ k α (A) ∂λ i · Qu i u i Q . In this experiment, we test the performance of matrix-based Rényi's IB (MRIB) and our lowrank variant (LRIB) with variational approximation-based objectives using VGG16 as the backbone and CIFAR10 as the classification task. All models are trained for 300 epochs with 100 batch size and 0.1 initial learning rate which is divided by 10 every 100 epochs. Following the settings in (Yu, Yu, and Principe 2021), we select α = 1.01, β = 0.01, k = 10 and s = 20. The final results are reported in Table 1. It can be seen that the matrix-based approaches MRIB and LRIB outperform other methods, while our LRIB achieves the highest performance with significantly less training time. Application to Feature Selection In practical regression or classification machine learning tasks, many features can be completely irrelevant to the learning target or redundant in the context of others. Given a set of features S = {X 1 , · · · , X L } and the target label Y, we aim to find a subset S sub ⊂ S which leverage the expressiveness and the complexity simultaneously. In the field of information theoretic learning, this target is equivalent to maximizing the multivariate mutual information I(S sub ; Y), which is computationally prohibitive due to the curse of high dimensionality. As a result, there have been tremendous efforts on approximation techniques that retain only the first or second order interactions and build mutual information estimators upon low-dimensional probability distributions, including Mutual Information-based Feature Selection (MIFS) (R. Battiti 1994), First-Order Util- (Fleuret 2004) and Double Input Symmetrical Relevance (DISR) (Meyer and Bontempi 2006) which achieve state-of-the-art performance in information-based feature selection tasks. We evaluate the performance of matrix-based Rényi's mutual information (MRMI) and our low-rank variant (LRMI) with these methods on 8 widely-used classification datasets as shown in Table 3, which is chosen to cover a broad variety of instance-feature ratios, number of classes and discreteness. Notice that non-Rényi methods can only handle discrete features, so we discretize them into 5 bins under equal-width criterion as adopted in (Brown et al. 2012). In this experiment, we choose the Support Vector Machine (SVM) algorithm with RBF kernel (σ = 1) as the classifier for continuous datasets and a 3-NN classifier for discrete datasets. Following the settings of (Yu et al. 2019), we select α ∈ {0.6, 1.01, 2}, k ∈ {100, 200, 400} via crossvalidation, s = k + 50 and use the Gaussian kernel of width σ = 1 for matrix-based entropy measures. Considering that it is NP-hard to evaluate each subset of S, we adopt a greedy strategy to incrementally select 10 features that maximize our target I(S sub ; Y). That is, in each step, we fix the current subset S sub = {X i1 , · · · , X i l−1 } and add a new feature X i l ∈ S/S sub to S sub . The average rank of each method across different number of features and the running time of MRMI and LRMI are reported in Table 2 and Table 3. As we can see, both MRMI and LRMI significantly outperform other Shannon entropy based methods. Compared to MRMI, LRMI achieves 6 to 27 times speedup, 15 times on average via Lanczos approximation. Furthermore, LRMI outperforms MRMI on 4 datasets in our test benchmark, which verifies our theoretical analysis that low-rank Rényi's entropy enables higher robustness against noises in the data. This demonstrates the great potential of our lowrank Rényi's entropy on information-related tasks. Conclusion In this paper, we investigate an alternative entropy measure built upon the largest k eigenvalues of the data kernel matrix. Compared to the original matrix-based Rényi's entropy, our definition enables higher robustness to noises in the data and sensitivity to informative changes in eigenspectrum distribution with a proper choice of hyper-parameter k. Moreover, low-rank Rényi's entropy can be efficiently approximated with O(ns 2 ) random projection and O(n 2 s) Lanczos iteration techniques, substantially lower than the O(n 3 ) complexity required to compute matrix-based Rényi's entropy. We conduct large-scale simulation and real-world experiments on information bottleneck and feature selection tasks to validate the effectiveness of low-rank Rényi's entropy, demonstrating elegant performance with significant improvements in computational efficiency. Supplementary Experimental Results Parameter Settings of Robustness Experiment In the first simulation study, we set n = 100, d = 400, ε = 1/n = 0.01 and use the linear kernel κ(x i , x j ) = x i x j to generate the kernel matrices. The data samples {x i } n i=1 are generated by i.i.d Gaussian distribution N (0, 0.1 2 ). The noise distributions E are selected following the criterion that E[E] = 0 and Var[E] = 1. It can be seen that we get similar results under different noise settings, which further verifies our analysis that when ε is small, the variance of S k α (A) mainly depends on the variance of random perturbations of data samples. Additional Results of Approximation Algorithms Additionally, we evaluate the impact of α and c on approximation accuracy. We keep the previous n = 8192 parameter settings and set k = 64. The results of MRE curves with c = 1.0 and varying α are reported in Figure 4. For SGS, we set the sparsity hyper-parameter p = 2. For Lanczos, we randomly select the initial vector q from standard Gaussian. It can be seen that GRP achieves the lowest MRE amongst all random projection algorithms. When α is small, all MRE curves exhibit similar behavior and grow with the increase of α. This behavior starts to differ when α gets larger. For random projection algorithms, MRE keeps at the same level; for Lanczos algorithm, MRE starts to decrease when α > 2. Recall that the larger eigenvalues of A take the main role in the calculation of S k α (A) for large α, this phenomenon indicates that Lanczos algorithm achieves higher precision for larger eigenvalues than smaller ones (as shown in our proof, it requires only O(i + log(1/ )) steps to approximate λ i to relative error 1 ± ), while random projection achieves similar level of precision for all of the k eigenvalues. We then evaluate the impact of EDR (c) on approximation accuracy. In Figure 5, we report the MRE curves for α = 2 while c varies from 0 (flat) to 2 (steep). It is interesting that the behavior of the two types of methods is entirely different. For random projections, the MRE curves grow slowly (GRP) or keep unchanged (SRHT, IST and SGS) when c is small, and start to increase at a constant rate when c gets large. This is because S k α (A) is decreasing along with the increase of c, whose slope is low at first and high when c gets large (see Figure 2). This results in the slow to fast increasing behavior in relative error since the absolute error is upper bounded (Theorem 2). For Lanczos method, the ratio λ 1 /λ r in Theorem 3 increases fast when c is small, resulting in the increase of MRE. When c gets large, λ 1 gradually reaches its upper bound λ 1 ≤ 1, while the intervals between adjacent eigenvalues also increase and result in a higher R (Theorem 3). Moreover, recall that Lanczos algorithm approximates larger eigenvalues to higher precision, these reasons together explain the increase and then decrease MRE of the Lanczos approach. Next, we conduct large-scale experiments to evaluate the approximation algorithms. The kernel matrices are generated by A = ΦΣΦ , where Φ ∈ R n×n is a random orthogonal matrix, Σ ∈ R n×n is a diagonal matrix such that Σ ii = i −c for i ∈ [1, n], and c is a constant that con- trols the EDR. We set size of the kernel matrix n = 8192. For random projection methods, we make s vary from 100 to 1000; while for Lanczos algorithm, s varies from 64 to 110. The mean relative error (MRE) and ± 1 4 standard deviation (SD) are reported in Figure 6 for each test after 100 trials with α = 1.5 and c ∈ {1.5, 1.0, 0.5}, which correspond to high, medium and low EDR respectively. For comparison, the trivial eigenvalue decomposition approach requires 134 seconds. It can be seen that all random projection methods yield similar approximation accuracy, in which GRP achieves slightly lower MRE when c = 0.5 while IST & SGS bring the highest speedup. The Lanczos method achieves the highest accuracy with significantly lower s values but requires longer running time. Generally, we recommend IST or SGS for medium precision approximation, and Lanczos when high precision is required. These methods achieve more than 25 times speedup compared to the trivial approach for an 8192 × 8192 kernel matrix. Additional Results of Feature Selection The hyper-parameter selection result of α and k for matrixbased Rényi's entropy and low-rank Rényi's entropy in feature selection experiment via cross-validation are shown in Table 4. As can be seen, k = 100 is already suitable for most circumstances. We perform a Nemenyi's post-hoc test (Demvsar 2006) to give the significant level, in which the confidence that method i significantly outperforms method j is calculated as: p ij = Φ (R j − R i ) M (M + 1) 6N , where Φ is the CDF of standard normal distribution, R i is the average rank of method i, M is the number of methods and N is the number of datasets. For our case, we have M = 8, N = 8 and the value of R i are given in the last column of table 2. The confidence level of different methods is shown in Figure 7. It can be seen that under significance level p = 0.05, LRMI significantly outperforms all Shannon's entropy-based methods, while the confidence of MRMI outperforming CMIM is not significant enough. In Figure 8, we report the classification accuracy achieved by different feature selection methods for the first 10 features. It can be seen that classification error tends to stabilize after selecting the 10 most informative features. For (c): Notice that S k α (A) = S α (L k (A)), where tr(L k (A)) = 1 and λ i (L k (A)) ∈ [0, 1] for all i ∈ [1, n]. Then we have tr(L α k (A)) ≥ 1 when α ∈ (0, 1) and tr(L α k (A)) ≤ 1 when α > 1, which further implies that S k α (A) ≥ 0. Let f (x) = x α , it is obvious that f is concave when α ∈ (0, 1) and convex when α > 1. Then by Jensen's inequality, tr(f (L k (A))) ≤ tr(f ( 1 n I)) when α ∈ (0, 1) and otherwise the opposite, which further implies that S k α (A) ≤ S k α ( 1 n I). Moreover, it is straightforward to show that S k α ( 1 n I) = S α ( 1 n I) = log 2 (n). t i=1 λ i (A • B) ≤ 1 n t i=1 λ i (B), where t is any integer in [1, n]. Therefore t i=1 λ i (L k (A • B)) ≤ 1 n t i=1 λ i (L k (B)), ∀t ∈ [1, k], n i=1 λ i (L k (A • B)) = 1 n n i=1 λ i (L k (B)) = 1 n , From the case t = k we know that λ r (A • B) ≥ λ r (B)/n, therefore for any t ∈ [k + 1, n], we have = 1 n t i=1 λ i (L k (B)). Then we can prove that S k α A • B tr(A • B) = S α L k A • B tr(A • B) ≥ S α (L k (B)) = S k α (B) following the proof in (Sanchez Giraldo, Rao, and Principe 2014). For (g): From the proof of Proposition 4.1 in (Sanchez Giraldo, Rao, and Principe 2014), when A = 1 n 11 and A = 1 n I, we have t i=1 λ i (A • B) ≤ 1 n t i=1 λ i (B) and 1 n t i=1 λ i (A • B) ≤ 1 n t i=1 λ i (B) respectively, where t is any integer in [1, n]. Similar with the proof of (f), for these two extreme cases we can prove that t i=1 λ i (L k (A • B)) ≤ 1 n t i=1 λ i (L k (B)) and 1 n t i=1 λ i (L k (A • B)) ≤ 1 n t i=1 λ i (L k (B)) respectively. These inequalities imply that S α L k A • B tr(A • B) ≤ S α (L k (A)) + S α (L k (B)) following the proof in (Sanchez Giraldo, Rao, and Principe 2014). Proof of Theorem 1 Proof. Without loss of generality, we assume µ 1 ≥ µ 2 ≥ · · · ≥ µ n . Note that λ i , i ∈ [1, n] may not be monotonically decreasing. By the definition of information potential, we have IP α (B) = n i=1 µ α i , IP k α (B) = k i=1 µ α i + (n − k)µ α r , µ r = 1 n − k 1 − k i=1 µ i . When ν i is small, we have the following first-order approximation: µ α i = (λ i + ν i ) α = λ α i + αλ α−1 i ν i + α(α − 1) 2 λ α−2 i ν 2 i + · · · = λ α i + αλ α−1 i ν i + o(ν i ). Therefore Var[IP α (B)] = Var[IP α (B) − IP α (A)] = Var n i=1 µ α i − λ α i = Var n i=1 αλ α−1 i ν i + o(ν i ) ≈ α 2 n i=1 Var λ α−1 i ν i = α 2 n i=1 σ 2 i λ 2(α−1) i . Similarly, we have Var[IP k α (B)] = Var[IP k α (B) − IP k α (A)] = Var k i=1 (µ α i − λ α i ) + (n − k)(µ α r − λ α r ) = Var k i=1 αλ α−1 i ν i + o(ν i ) − α(n − k)λ α−1 r · 1 n − k k i=1 ν i . When α ∈ (0, 1), i.e. α − 1 < 0, we have λ α−1 r ≥ λ α−1 i for i ∈ [1, k]. Therefore Var[IP k α (B)] ≤ Var αλ α−1 r k i=1 ν i = α 2 λ 2(α−1) r k i=1 σ 2 i ≤ α 2 n i=k+1 σ 2 i λ 2(α−1) i k i=1 σ 2 i n i=k+1 σ 2 i (1) ≤ Var[IP α (B)]. (1) follows by Jensen's inequality using the fact that λ r = 1 n−k n i=k+1 λ i and σ i are non-negative, since the function B)]. This completes the proof. f (x) = x 2(α−1) is convex. Otherwise when α > 1, we have λ α−1 r ≤ λ α−1 i for i ∈ [1, k]. Therefore Var[IP k α (B)] ≤ Var α k i=1 λ α−1 i ν i = α 2 k i=1 σ 2 i λ 2(α−1) i ≤ Var[IP α ( Uniqueness of Low-rank Rényi's Entropy Let S k α (A) be a measure of entropy defined on the largest k eigenvalues of A. Then S k α (A) must adopt some strategy to build a probability distribution upon known eigenvalues, i.e. let the summation of all eigenvalues be exactly 1, otherwise S k α (A) will not be continuous at α = 1. One choice is to adopt some strategy to complement the missing eigenvalues. Let L k (A) be the complemented matrix, we have λ i (L k (A)) = λ i (A), ∀i ∈ [1, k], λ n (L k (A)) ≤ · · · ≤ λ k+1 (L k (A)) ≤ λ k (A) and tr(L k (A)) = 1. Let F A (t) be the CDF of A: F A (t) = t i=1 λ i (A), and let λ r (A) = 1 n−k (1 − k i=1 λ i (A)). Then we have F L k (A) (t) ≥ k i=1 λ i (A) + (t − k)λ r (A)(2) for all t ∈ [k+1, n] since the function F A is always concave. Let B ∈ R n×n be a PSD matrix satisfying t i=1 λ i (A) ≤ t i=1 λ i (B) for all t ∈ [1, n], then in order to maintain the triangle inequality (axiom (f) and (g)), The function F A must satisfy F L k (A) (t) ≤ F L k (B) (t), ∀t ∈ [k + 1, n]. Construct B by letting λ 1 (B) = 1 − (n − 1)λ r (A) and λ 2 (B) = · · · = λ n (B) = λ r (A), then combining with the fact that λ i (L k (B)) ≤ λ k (B), ∀i ∈ [k + 1, n], we have F L k (A) (t) ≤ F L k (B) (t) = k i=1 λ i (A) + (t − k)λ r (A). (3) Eq. (2) and (3) together imply that taking λ i (L k (A)) = λ r (A) for all i ∈ [k + 1, n] is the only choice that fulfills all axioms in Proposition 1. Another reasonable choice to normalize the probability distribution is to scale the known largest k eigenvalues: S k α (A) = 1 1 − α log 2 k i=1 λ i (A) n i=1 λ i (A) α , or S k α (A) = 1 1 − α log 2 k i=1 λ α i (A) n i=1 λ i (A) . However, these methods do not fulfill the triangle inequality, i.e. we cannot infer S k α (A) ≥ S k α (B) from the condition that F A (t) ≤ F B (t), ∀t ∈ [1, k]. This results in violations of axioms (f) and (g). Proof of Theorem 2 We first present the 2 embedding results for RGP, SRHT, IST and SGS in Lemma 1, 2, 3 and 4 respectively, where the dimension of embedding subspace is given to guarantee the error. Lemma 5 presents the permutation bound for symmetric positive definite matrix. All these theoretical results are helpful to our proof. Lemma 1. (Foucart and Rauhut 2013) Let U ∈ R n×k such that U U = I k and P ∈ R n×s constructed by GRP. Then, with probability at least 1 − δ, U PP U − I k 2 ≤ , by setting s = O k + log(1/δ)/ 2 . Lemma 2. (Drineas et al. 2012) Let U ∈ R n×k such that U U = I k and P ∈ R n×s constructed by SRHT. Then, with probability at least 0.9, n k U PP U − I k 2 ≤ , by setting s = O (k + log n) log k 2 . Lemma 3. (Woodruff 2014) Let U ∈ R n×k such that U U = I k and P ∈ R n×s constructed by IST. Then, with probability at least 0.9, U PP U − I k 2 ≤ , by setting s = O k 2 / 2 . Lemma 4. (Hu et al. 2021) Let U ∈ R n×k such that U U = I k and P ∈ R n×s constructed by SGS. Then, with probability at least 1 − δ, U PP U − I k 2 ≤ , by setting s = O k log(k/δ )/ 2 , p = O (log(k/δ )/ ) . Lemma 5. (Demmel and Veselić 1992) Let DGD be a symmetric positive definite matrix such that D is a diagonal matrix and G ii = 1 for all i. Let DED be a permutation matrix such that E 2 < λ min (G). Let λ i be the i-th eigenvalue of DGD andλ i be the i-th eigenvalue of D(G + E)D. Then, for all i, |λ i −λ i | ≤ E 2 λ min (G) . The following proposition shows that if we can bound each eigenvalue of A to absolute error , we have an absolute bound for S α (A). Proposition 2. Let A and be positive definite matrices with eigenvalues λ i andλ i , i ∈ [1, n] respectively, such that for each i ∈ [1, n], |λ i −λ i | ≤ , then |S α (A) − S α (Â)| ≤ α 1 − α log 2 1 − λ n . Proof. Let λ n > 0 be the smallest eigenvalue of A and let 0 = /λ n , then we have |λ i −λ i | ≤ 0 λ i for each i ∈ [1, n]. Observe that when α < 1, S α (Â) = 1 1 − α log 2 n i=1λ α i ≥ 1 1 − α log 2 (1 − 0 ) α n i=1 λ α i = 1 1 − α log 2 n i=1 λ α i + α 1 − α log 2 (1 − 0 ) = S α (A) + α 1 − α log 2 (1 − 0 ). Similarly, we have S α (Â) = 1 1 − α log 2 n i=1λ α i ≤ 1 1 − α log 2 (1 + 0 ) α n i=1 λ α i = 1 1 − α log 2 n i=1 λ α i + α 1 − α log 2 (1 + 0 ) = S α (A) + α 1 − α log 2 (1 + 0 ). We can get the same results for the other case when α > 1, which finishes the proof. Proof of Theorem 2. Note that λ min (G) in the Lemma 5 is a real, strictly positive number since G is positive definite and the fact 0 ≤ E 2 λ min (G). Now consider the matrix APP A , we will show that the singular values of APP A are sufficient approximation to that of AA by the permutation theory presented in Lemma 5. Let λ i , i ∈ [1, n] be the eigenvalues of the positive definite kernel matrix A,λ i be their approximations and A = ΦΣΦ be the eigenvalue decomposition A. Since Φ is an orthogonal matrix, we have that the eigenvalues of ΦΣΦ PP ΦΣΦ are equal to the eigenvalues of ΣΦ PP ΦΣ. Let Σ k be the k × k diagonal matrix containing the k largest eigenvalues of A and Φ k be the matrix containing the corresponding eigenvectors, then λ 2 i , i ∈ [1, k] are the eigenvalues of matrix Σ k I k Σ k , andλ 2 i , i ∈ [1, k] are the eigenvalues of matrix Σ k Φ k PP Φ k Σ k (since the first k singular values of Σ k Φ k P are equal to those of ΦΣΦ P = AP). Let E = Φ k PP Φ k − I k , we know from Lemma 1 (or Lemma 2, 3 and 4) that E 2 ≤ 0 with high probability. It meets the condition of Lemma 5 since λ min (I k ) = 1. Hence, we have |λ 2 i −λ 2 i | ≤ 0 , ∀i ∈ [1, k], which then implies that λ i − λ 2 i − 0 ≤ |λ i − λ i | ≤ λ 2 i + 0 − λ i . Since λ k is the smallest eigenvalue amongst λ i , i ∈ [1, k], we have |λ i − λ i | ≤ λ k − λ 2 k − 0 = λ k 1 − 1 − 0 λ 2 k ≤ λ k 0 λ 2 k = 0 λ k . Combining with k ≤ n/2, we have |λ r − λ r | = k i=1λ i − k i=1 λ i (n − k) ≤ 0 λ k · k n − k ≤ 0 λ k . Let 0 = λ k λ r and B be a positive definite matrix with the first k eigenvalues equal toλ i , i ∈ [1, k] and the other n − k eigenvalues equal toλ r . Recall that λ r is the smallest eigenvalue of L k (A), by applying Proposition 2, we have |S k α (A) −Ŝ k α (A)| = |S α (L k (A)) − S α (B)| ≤ α 1 − α log 2 (1 − ) . A Potential Improvement The upper bound of s in Theorem 2 relies on λ r , which grows large if the kernel matrix A is ill-posed and λ r is small. Alternatively, we derive an upper bound for s in terms of n and tr(A α ), which is tighter for such kernel matrices. if α < 1. Proof. S α (Ã) − S α (A) = 1 1 − α log 1 − n i=1λ α i − n i=1 λ α i n i=1λ α i ≤ 1 1 − α log(1 − β) , where β = n i=1 (λ i + ) α − n i=1 λ α i n i=1 (λ i + ) α ≤ n i=1 (λ i + ) α − n i=1 λ α i n i=1 λ α i . When α > 1, we have β ≤ α n i=1 (λ i + ) α−1 n i=1 (λ i + ) α ≤ nα 1 + n , where the last step takes equality if and only if λ 1 = · · · = λ n = 1 n . Otherwise when α < 1, β ≤ n i=1 α n i=1 λ α i = n α tr(A α ) . One can upper bound S α (A) − S α (Ã) through the same strategy, which finishes the proof. Proof of Theorem 3 The following lemma gives the convergence rate of the Lanczos algorithm: Lemma 6. (Saad 1980) Let q be the initial vector, λ i be the i-th largest eigenvalue of A with associated eigenvector φ i such that φ i , q = 0,λ i be the corresponding approximation of λ i after s steps of Lanczos iteration, and assume that λ i−1 > λ i . Let γ i = 1 + 2 λ i − λ i+1 λ i+1 − λ n , K i = i−1 j=1λ j −λn λj −λi , i > 1 1, i = 1 , then 0 ≤ λ i −λ i ≤ (λ i − λ n ) · K i T s−i (γ i ) tan φ i , q 2 , where T i (x) = 1 2 (x + √ x 2 − 1) i + (x − √ x 2 − 1) i is the Chebyshev polynomial of the first kind of degree i. Proof of Theorem 3. It is easy to see that K i is monotonically increasing with the increase of i. Let γ = min i∈ [1,k] γ i , θ = max i∈ [1,k] tan φ i , q , R = γ + γ 2 − 1, then ∀i ∈ [1, k], λ i −λ i ≤ λ i · 2θK i R s−i + R −(s−i) 2 ≤ λ i · 4θ 2 K 2 i R −2(s−i) ≤ λ i · 4θ 2 K 2 k R −2(s−k) . By selecting s = k + log(4θ 2 K 2 k / 0) 2 log R , we have that ∀i ∈ [1, k], |λ i −λ i | ≤ 0 λ i . Similarly, by combining with k ≤ n/2 we have |λ r − λ r | ≤ 0 λ 1 . Let 0 = λ r /λ 1 , by applying Proposition 2, we have |S k α (A) −Ŝ k α (A)| ≤ α 1 − α log 2 (1 − ) . Figure 2 : 2Left: The behavior of S k α (A) when the entropy order α varies from 0 to 2. Right: The behavior of S k α (A) when the EDR of A varies from flat to steep. Figure 3 : 3Upper: perturbation of the eigenvalues, i.e. µ i − λ i . Lower: standard deviation of matrix-based Rényi's entropy and low-rank Rényi's entropy against random perturbations of the data samples for different values of α. Figure 4 : 4α versus MRE curves for entropy approximation. Figure 5 : 5c versus MRE curves for entropy approximation. Figure 6 : 6s versus MRE curves for entropy approximation. The first three sub-figure correspond to different c values, while the last sub-figure show the running time. For (a): Let A = UΛU be the eigenvalue decomposition of A, then PU is a unitary matrix and λ i (A) = λ i (PAP ) for all i ∈ [1, k].For (b): When p > 0, tr((pL k (A)) α ) = p α · tr(L α k (A)) > 0, then (b) follows by the continuity of the logarithm function. For (d): From Proposition 4.1 in (Sanchez Giraldo, Rao, and Principe 2014) we have that S α(A ⊗ B) = S α (A) + S α (B), therefore S α (L k (A) ⊗ L k (B)) = S α (L k (A)) + S α (L k (B)). Notice that the smaller (n − k) 2 eigenvalues of L k (A) ⊗ L k (B) are equal to λ r (A)λ r (B), we have S n 2 −(n−k) 2 α (L k (A)⊗L k (B)) = S k α (L k (A))+S k α (L k (B)). For (e): From Proposition 4.1 in (Sanchez Giraldo, Rao, and Principe 2014) we have that S α (tA k + (1 − t)B k ) = Figure 7 : 7Confidence of significant outperforming (%) for different feature selection methods.g −1 tg(S α (A)) + (1 − t)g(S α (B)) . Notice that A k = A when tr(A k ) = 1, we have S k α (tA + (1 − t)B) = g −1 tg(S k α (A)) + (1 − t)g(S k α (B)). For (f): From the proof of Proposition 4.1 in (Sanchez Giraldo, Rao, and Principe 2014) we have that Figure 8 : 8(L k (A • B)) = 1 n − (n − t)λ r (A • B) Number of Features (l) versus Classification Error (%) curves for different feature selection methods. Table 1 : 1Classification accuracy and training time of different IB objectives. Left is the time spent on IB calculation and right is the total training time. Table 2 : 2Information theoretic feature selection methods and their average rank over different number of features in each dataset. The first and second best performances are marked as bold and underlined respectively.ity (FOU) (Brown 2009), Maximum-Relevance Minimum- Redundancy (MRMR) (Peng, Long, and Ding 2005), Joint Mutual Information (JMI) (Yang and Moody 1999), Condi- tional Mutual Information Maximization (CMIM) Table 3 : 3Number of instances (#I), features (#F), classes (#C) and discreteness of classification datasets used in fea- ture selection experiments, running time comparison (min- utes) of MRMI (left) and LRMI (right), and speedup ratios. https://github.com/Gamepiaynmo/LRMI Acknowledgments The fast Johnson-Lindenstrauss transform and approximate nearest neighbors. N Ailon, B Chazelle, SIAM Journal on computing. 391Ailon, N.; and Chazelle, B. 2009. The fast Johnson- Lindenstrauss transform and approximate nearest neighbors. SIAM Journal on computing, 39(1): 302-322. A A Alemi, I Fischer, J V Dillon, K Murphy, Deep variational information bottleneck. ICLR. Alemi, A. A.; Fischer, I.; Dillon, J. V.; and Murphy, K. 2017. Deep variational information bottleneck. ICLR, 1-19. Kernel-based dimensionality reduction using Rényi's α-entropy measures of similarity. A M Alvarez-Meza, J A Lee, M Verleysen, G Castellanos-Dominguez, Neurocomputing. 222Alvarez-Meza, A. M.; Lee, J. A.; Verleysen, M.; and Castellanos-Dominguez, G. 2017. Kernel-based dimension- ality reduction using Rényi's α-entropy measures of similar- ity. Neurocomputing, 222: 36-46. Infinitely divisible matrices. R Bhatia, The American Mathematical Monthly. 1133Bhatia, R. 2006. Infinitely divisible matrices. The American Mathematical Monthly, 113(3): 221-235. Quantifying the informativeness of similarity measurements. A J Brockmeier, T Mu, S Ananiadou, J Y Goulermas, JMLR. 18Brockmeier, A. J.; Mu, T.; Ananiadou, S.; and Goulermas, J. Y. 2017. Quantifying the informativeness of similarity measurements. JMLR, 18: 1-61. A new perspective for information theoretic feature selection. G Brown, Journal of Machine Learning Research. 51Brown, G. 2009. A new perspective for information the- oretic feature selection. Journal of Machine Learning Re- search, 5(1): 49-56. Conditional likelihood maximisation: a unifying framework for information theoretic feature selection. G Brown, A Pocock, M.-J Zhao, M Luján, JMLR. 13Brown, G.; Pocock, A.; Zhao, M.-J.; and Luján, M. 2012. Conditional likelihood maximisation: a unifying framework for information theoretic feature selection. JMLR, 13: 27- 66. Jacobi's method is more accurate than QR. J Demmel, K Veselić, SIAM Journal on Matrix Analysis and Applications. 134Demmel, J.; and Veselić, K. 1992. Jacobi's method is more accurate than QR. SIAM Journal on Matrix Analysis and Applications, 13(4): 1204-1245. Statistical comparisons of classifiers over multiple data sets. The Journal of Machine learning research. J Demvsar, 7Demvsar, J. 2006. Statistical comparisons of classifiers over multiple data sets. The Journal of Machine learning re- search, 7: 1-30. Y Dong, T Gong, S Yu, C Li, arXiv:2205.07426Optimal Randomized Approximations for Matrix based Renyi's Entropy. arXiv preprintDong, Y.; Gong, T.; Yu, S.; and Li, C. 2022. Optimal Ran- domized Approximations for Matrix based Renyi's Entropy. arXiv preprint arXiv:2205.07426. Fast approximation of matrix coherence and statistical leverage. P Drineas, M Magdon-Ismail, M W Mahoney, D P Woodruff, The Journal of Machine Learning Research. 131Drineas, P.; Magdon-Ismail, M.; Mahoney, M. W.; and Woodruff, D. P. 2012. Fast approximation of matrix co- herence and statistical leverage. The Journal of Machine Learning Research, 13(1): 3475-3506. Statistical challenges with high dimensionality. J Fan, R Li, Proceedings of the international Congress of Mathematicians. the international Congress of MathematiciansFan, J.; and Li, R. 2006. Statistical challenges with high di- mensionality. In Proceedings of the international Congress of Mathematicians. Fast binary feature selection with conditional mutual information. F Fleuret, Journal of Machine learning research. 59Fleuret, F. 2004. Fast binary feature selection with condi- tional mutual information. Journal of Machine learning re- search, 5(9). A Mathematical Introduction to Compressive Sensing. S Foucart, H Rauhut, Birkhäuser BaselFoucart, S.; and Rauhut, H. 2013. A Mathematical Introduc- tion to Compressive Sensing. Birkhäuser Basel. A new clustering evaluation function using Renyi's information potential. E Gokcay, J C Principe, Y Dong, S Yu, H Chen, B Dong, C Li, Q Zheng, arXiv:2112.137202000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No. 00CH37100). 6arXiv preprintComputationally Efficient Approximations for Matrix-based Renyi's EntropyGokcay, E.; and Principe, J. C. 2000. A new clustering eval- uation function using Renyi's information potential. In 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No. 00CH37100), vol- ume 6, 3490-3493. IEEE. Gong, T.; Dong, Y.; Yu, S.; Chen, H.; Dong, B.; Li, C.; and Zheng, Q. 2021. Computationally Efficient Approxi- mations for Matrix-based Renyi's Entropy. arXiv preprint arXiv:2112.13720. Sparse graph based sketching for fast numerical linear algebra. D Hu, S Ubaru, A Gittens, K L Clarkson, L Horesh, V Kalantzis, ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing. IEEEHu, D.; Ubaru, S.; Gittens, A.; Clarkson, K. L.; Horesh, L.; and Kalantzis, V. 2021. Sparse graph based sketching for fast numerical linear algebra. In ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 3255-3259. IEEE. A Kolchinsky, B D Tracey, D H Wolpert, Nonlinear information bottleneck. 211181Kolchinsky, A.; Tracey, B. D.; and Wolpert, D. H. 2019. Nonlinear information bottleneck. Entropy, 21(12): 1181. The operational meaning of min-and max-entropy. R Konig, R Renner, C Schaffner, IEEE Transactions on Information theory. 559Konig, R.; Renner, R.; and Schaffner, C. 2009. The opera- tional meaning of min-and max-entropy. IEEE Transactions on Information theory, 55(9): 4337-4347. Faster Ridge Regression via the Subsampled Randomized Hadamard Transform. Y Lu, P S Dhillon, D P Foster, NIPS. Lu, Y.; Dhillon, P. S.; Foster, D. P.; et al. 2012. Faster Ridge Regression via the Subsampled Randomized Hadamard Transform. In NIPS. Randomized algorithms for matrices and data. Foundations and Trends in Machine Learning. M W Mahoney, 3Mahoney, M. W. 2011. Randomized algorithms for matri- ces and data. Foundations and Trends in Machine Learning, 3(2). CUR matrix decompositions for improved data analysis. M W Mahoney, P Drineas, Proceedings of the National Academy of Sciences. 1063Mahoney, M. W.; and Drineas, P. 2009. CUR matrix de- compositions for improved data analysis. Proceedings of the National Academy of Sciences, 106(3): 697-702. On the use of variable complementarity for feature selection in cancer classification. P E Meyer, G Bontempi, Workshops on applications of evolutionary computation. SpringerMeyer, P. E.; and Bontempi, G. 2006. On the use of variable complementarity for feature selection in cancer classifica- tion. In Workshops on applications of evolutionary compu- tation, 91-102. Springer. . R Miles, A L Rodríguez, K Mikolajczyk, arXiv:2112.00459Information Theoretic Representation Distillation. arXiv preprintMiles, R.; Rodríguez, A. L.; and Mikolajczyk, K. 2021. Information Theoretic Representation Distillation. arXiv preprint arXiv:2112.00459. An approach of eigenvalue perturbation theory. K V Ngo, Applied Numerical Analysis & Computational Mathematics. 21Ngo, K. V. 2005. An approach of eigenvalue perturbation theory. Applied Numerical Analysis & Computational Math- ematics, 2(1): 108-125. Quantum entropy and its use. M Ohya, D Petz, Springer Science & Business MediaOhya, M.; and Petz, D. 2004. Quantum entropy and its use. Springer Science & Business Media. Feature selection based on mutual information criteria of max-dependency, max-relevance, and min-redundancy. H Peng, F Long, C Ding, IEEE Transactions. 278Peng, H.; Long, F.; and Ding, C. 2005. Feature selection based on mutual information criteria of max-dependency, max-relevance, and min-redundancy. IEEE Transactions on pattern analysis and machine intelligence, 27(8): 1226- 1238. Using mutual information for selecting features in supervised neural net learning. R Battiti, IEEE Transactions on Neural Networks. 54R. Battiti. 1994. Using mutual information for selecting fea- tures in supervised neural net learning. IEEE Transactions on Neural Networks, 5(4)(4): 537-550. On measures of entropy and information. A Rényi, Contributions to the Theory of Statistics. University of California Press1Rényi, A. 1961. On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Math- ematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, 547-561. University of California Press. On the rates of convergence of the Lanczos and the block-Lanczos methods. Y Saad, SIAM Journal on Numerical Analysis. 175Saad, Y. 1980. On the rates of convergence of the Lanczos and the block-Lanczos methods. SIAM Journal on Numeri- cal Analysis, 17(5): 687-706. Measures of entropy from data using infinitely divisible kernels. L G Sanchez Giraldo, M Rao, J C Principe, IEEE TIT. 611Sanchez Giraldo, L. G.; Rao, M.; and Principe, J. C. 2014. Measures of entropy from data using infinitely divisible ker- nels. IEEE TIT, 61(1): 535-548. HRel: Filter pruning based on high relevance between activation maps and class labels. C Sarvani, M Ghorai, S R Dubey, S S Basha, Neural Networks. Sarvani, C.; Ghorai, M.; Dubey, S. R.; and Basha, S. S. 2021. HRel: Filter pruning based on high relevance between acti- vation maps and class labels. Neural Networks. A mathematical theory of communication. The Bell system technical journal. C E Shannon, 27Shannon, C. E. 1948. A mathematical theory of communi- cation. The Bell system technical journal, 27(3): 379-423. Improved analysis of the subsampled randomized Hadamard transform. J A Tropp, Advances in Adaptive Data Analysis. 301n02Tropp, J. A. 2011. Improved analysis of the subsampled ran- domized Hadamard transform. Advances in Adaptive Data Analysis, 3(01n02): 115-126. Uniform error estimates for the Lanczos method. J C Urschel, SIAM Journal on Matrix Analysis and Applications. 423Urschel, J. C. 2021. Uniform error estimates for the Lanczos method. SIAM Journal on Matrix Analysis and Applications, 42(3): 1423-1450. Minentropy latent model for weakly supervised object detection. F Wan, P Wei, J Jiao, Z Han, Q Ye, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionWan, F.; Wei, P.; Jiao, J.; Han, Z.; and Ye, Q. 2018. Min- entropy latent model for weakly supervised object detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 1297-1306. The QR algorithm revisited. D S Watkins, SIAM review. 501Watkins, D. S. 2008. The QR algorithm revisited. SIAM review, 50(1): 133-145. Near input sparsity time kernel embeddings via adaptive sampling. D Woodruff, A Zandieh, ICML. Woodruff, D.; and Zandieh, A. 2020. Near input sparsity time kernel embeddings via adaptive sampling. In ICML, 10324-10333. Sketching as a Tool for Numerical Linear Algebra. D P Woodruff, Foundations and Trends in Theoretical Computer Science. 101-2Woodruff, D. P. 2014. Sketching as a Tool for Numeri- cal Linear Algebra. Foundations and Trends in Theoretical Computer Science, 10(1-2): 1-157. Advances in neural information processing systems, 12. Yeung, R. W. 1991. A new outlook on Shannon's information measures. H Yang, J Moody, IEEE transactions on information theory. 373Data visualization and feature selection: New algorithms for nongaussian dataYang, H.; and Moody, J. 1999. Data visualization and fea- ture selection: New algorithms for nongaussian data. Ad- vances in neural information processing systems, 12. Yeung, R. W. 1991. A new outlook on Shannon's informa- tion measures. IEEE transactions on information theory, 37(3): 466-474. Multivariate Extension of Matrix-Based Rényi alpha-Order Entropy Functional. S Yu, L G Sanchez Giraldo, R Jenssen, J C Principe, IEEE TPAMI. 4211Yu, S.; Sanchez Giraldo, L. G.; Jenssen, R.; and Principe, J. C. 2019. Multivariate Extension of Matrix-Based Rényi alpha-Order Entropy Functional. IEEE TPAMI, 42(11): 2960-2966. Deep Deterministic Information Bottleneck with Matrix-Based Entropy Functional. X Yu, S Yu, J C Principe, ICASSP. Yu, X.; Yu, S.; and Principe, J. C. 2021. Deep Deterministic Information Bottleneck with Matrix-Based Entropy Func- tional. ICASSP, 3160-3164.
[ "https://github.com/Gamepiaynmo/LRMI" ]
[ "Single qubit gates in frequency-crowded transmon systems", "Single qubit gates in frequency-crowded transmon systems" ]
[ "R Schutjens \nQuantum Transport\nDelft University of Technology\n2628 CJDelftThe Netherlands\n\nTheoretical Physics\nUniversität des Saarlandes\n66123SaarbrückenGermany\n", "F Abu Dagga \nTheoretical Physics\nUniversität des Saarlandes\n66123SaarbrückenGermany\n", "D J Egger \nTheoretical Physics\nUniversität des Saarlandes\n66123SaarbrückenGermany\n", "F K Wilhelm \nTheoretical Physics\nUniversität des Saarlandes\n66123SaarbrückenGermany\n\nIQC and Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1OntarioCanada\n" ]
[ "Quantum Transport\nDelft University of Technology\n2628 CJDelftThe Netherlands", "Theoretical Physics\nUniversität des Saarlandes\n66123SaarbrückenGermany", "Theoretical Physics\nUniversität des Saarlandes\n66123SaarbrückenGermany", "Theoretical Physics\nUniversität des Saarlandes\n66123SaarbrückenGermany", "Theoretical Physics\nUniversität des Saarlandes\n66123SaarbrückenGermany", "IQC and Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1OntarioCanada" ]
[]
Recent experimental work on superconducting transmon qubits in 3D cavities show that their coherence times are increased by an order of magnitude compared to their 2D cavity counterparts. However to take advantage of these coherence times while scaling up the number of qubits it is advantageous to address individual qubits which are all coupled to the same 3D cavity fields. The challenge in controlling this system comes from spectral crowding, where leakage transition of qubits are close to computational transitions in other. Here it is shown that fast pulses are possible which address single qubits using two quadrature control of the pulse envelope while the DRAG method of Refs. [1, 2] alone only gives marginal improvements over the conventional Gaussian pulse shape. On the other hand, a first order result using the Magnus expansion gives a fast analytical pulse shape which gives a high fidelity gate for a specific gate time, up to a phase factor on the second qubit. Further numerical analysis corroborates these results and yields to even faster gates, showing that leakage state anharmonicity does not provide a fundamental quantum speed limit.
10.1103/physreva.88.052330
[ "https://arxiv.org/pdf/1306.2279v1.pdf" ]
4,571,739
1306.2279
b3be54f0e4e61becedaac97d3d867e38828f0937
Single qubit gates in frequency-crowded transmon systems 10 Jun 2013 R Schutjens Quantum Transport Delft University of Technology 2628 CJDelftThe Netherlands Theoretical Physics Universität des Saarlandes 66123SaarbrückenGermany F Abu Dagga Theoretical Physics Universität des Saarlandes 66123SaarbrückenGermany D J Egger Theoretical Physics Universität des Saarlandes 66123SaarbrückenGermany F K Wilhelm Theoretical Physics Universität des Saarlandes 66123SaarbrückenGermany IQC and Department of Physics and Astronomy University of Waterloo N2L 3G1OntarioCanada Single qubit gates in frequency-crowded transmon systems 10 Jun 2013(Dated: May 7, 2014)arXiv:1306.2279v1 [quant-ph] Recent experimental work on superconducting transmon qubits in 3D cavities show that their coherence times are increased by an order of magnitude compared to their 2D cavity counterparts. However to take advantage of these coherence times while scaling up the number of qubits it is advantageous to address individual qubits which are all coupled to the same 3D cavity fields. The challenge in controlling this system comes from spectral crowding, where leakage transition of qubits are close to computational transitions in other. Here it is shown that fast pulses are possible which address single qubits using two quadrature control of the pulse envelope while the DRAG method of Refs. [1, 2] alone only gives marginal improvements over the conventional Gaussian pulse shape. On the other hand, a first order result using the Magnus expansion gives a fast analytical pulse shape which gives a high fidelity gate for a specific gate time, up to a phase factor on the second qubit. Further numerical analysis corroborates these results and yields to even faster gates, showing that leakage state anharmonicity does not provide a fundamental quantum speed limit. I. INTRODUCTION Superconducting qubits are a promising candidate for the realization of a quantum computer [3][4][5][6][7], owing in large parts to the success of circuit QED (CQED), where those qubits are coupled to microwave resonators [8][9][10]. There is a multitude of designs of such qubits [4]. A key challenge for implementing quantum computing in the solid state is decoherence from uncontrolled degrees of freedom. Decoherence sources range from the electromagnetic environment [11] to sources inherent to the material [12]. Remarkably, many of the material sources could be mitigated by changes in the circuit layout such as the optimum working point first embodied in the Quantronium [13][14][15][16] and later in the Transmon [11] and the 3D-Transmon [17,18]. Coherence times have been improved by going from the two dimensional implementation of a qubit interacting with a stripline resonator [8] to a three dimensional system [17,18]. In the latter, a single Josephson junction transmon qubit [11,19] is placed inside a 3D cavity and addressed with the surrounding microwave field. What is common to these approaches is the trade-off of coherence against control flexibility and ultimately operation speed. While this has been studied in single Quantronium [16] the precise trade-off is not fully understood in samples containing multiple qubits let alone multiple 3D transmons. The gain in coherence times comes at a cost in controllability. This is strongly felt when more than one qubit is in the cavity. To create single qubit operations each qubit must be addressed individually requiring them to have significantly different energy splitting between the ground and first excited state. Spectral crowding refers to transitions coming too close to address them individually. Now with the limited control, even if the logical transitions are well-spaced, crowding can occur between logical and leakage transition, e.g., if the logical tran-sition of first qubit is close in frequency to the leakage transition, the transition between a computational and a non-computational state, of the second qubit. Thus when performing, e.g., anX gate on first qubit leakage to second qubit's |2 state will occure. Although high fidelity gates have been demonstrated with single junction transmsons in the 2D architecture [20] spectral crowding will limit the gate fidelity in 3D architectures. In order to mitigate spectral overlap, the Derivative Removal by Adiabatic Gate (DRAG) technique has been developed [1,2]. We will apply this technique to the problem at hand and show that on its own it is of limited success. We will then combine DRAG with sideband drive to show a possibility to do these single-qubit gates fast. In this work we thus address the issue of spectral crowding with optimal control theory methods. To better illustrate the problem and show the effectiveness of the analytical pulses we introduce specific gate fidelity functions in section III. In section IV we demonstrate the limitations of the DRAG technique alone for this problem. We then present an analytical pulse, found through the Magnus expansion [21], capable of minimizing leakage out of the computational subspace of both qubits in section V. We then, in section VI, show pulses obtained numerically that show similar characteristic but, with additional ingredients, improved fidelities. II. SYSTEM Optimized superconducting qubits such as 3D transmons are well described by weakly anharmonic oscillators [1,22]. A realistic model of the qubit has to take at least one extra non-computational level (a leakage level) into account [23][24][25]. This is reflected in the following Hamiltonian for two superconducting transmon qubits in a common 3D cavity H (t) =Ĥ 0 +Ĥ C (t) = 2 k=1 ω knk + ∆ kΠ (k) 2 + Ω (t) 2 j=1 λ (1) jσ x(1) j,j−1 + λ (2) jσ x(2) j,j−1 . (1) The 0 ↔ 1 transition frequency and number operator of qubit k are, respectively, ω k andn k = j j |j j| (k) . We call the transition from the excited state |1 to the extra state |2 the leakage transition. It is detuned from ω k by the anharmonicity ∆ k . In the reminder of this work we assume ∆ 1 = ∆ 2 = ∆. The projectors on the energy levels of transmon k areΠ (k) k = |j j| (k) . The terms coupling adjacent energy levels of qubit k arê σ x(k) j,j−1 = |j j − 1| (k) + |j − 1 j| (k) andσ y(k) j,j−1 = i |j j − 1| (k) − i |j − 1 j| (k) . Ω(t) is the drive field and is applied simultaneously to both qubits. The strength at which Ω(t) drives the 1 ↔ 2 transition relative to the 0 ↔ 1 is given by λ 1 1 λ (k) 2 √ 2 √ 2 Qubits are usually addressed by frequency selection through pulses tuned to the respective qubit level splitting. This is necessary whenever the control field cannot be selectively focused on individual qubits as is the case for multiple 3D transmons in the same cavity. An eventual implementation of a quantum computer will consist of many such qubits, probably a whole register in one cavity. The problem to distinguish different qubits can thus be seen as a problem of spectral crowding. In transmon systems this can lead to the 0 ↔ 1 transition of the first qubit being very close to the 1 ↔ 2 transition of the second qubit. The frequency difference of these two transitions is named δ. With δ/2π = 45MHz, the leakage 1 These values were suggested to describe an experiment by Leo DiCarlo transition of qubit two is closer to the driving fields frequency than the leakage transition of qubit one detuned by ∆/2π = −350 MHz. The situation is depicted in Fig. 1. Qubit 2 ω 2 ω 1 + δ ∆ Qubit 1 ω d ω 1 ∆ FIG. 1. Level diagram of the two qubits. The driving field is set to have the same frequency as the 0 ↔ 1 transition of first qubit which we wish to drive. Requiring that the same transition of the second qubit be far detuned results in its leakage transition being only slightly detuned by δ with 0 ↔ 1 of first qubit. The second term in equation (1) is the control Hamiltonian, described as a semiclassical dipolar interaction between the qubits and the classical cavity field Ω (t) = Ω X (t) cos (ω d t) + Ω Y (t) sin (ω d t) .(2) Both quadrature envelopes can be modulated separately. In the reminder of this work, we assume resonance between the drive and qubit 1, i.e. ω d = ω 1 . Single quadrature pulses employ Gaussian shapes Ω g due to their limited bandwidth [2]. To remove fast oscillating terms we move to another reference frame and invoke the rotating wave approximation (RWA). The transformation into an appropriate frame is accomplished by the time-dependent unitaryR that acts on the Hamiltonian aŝ H R =RĤR † + iṘR † .(3) Here, R (t) = j e −iω (1) j tΠ(1) j ⊗ j e −iω (2) j tΠ(2) j . Transformations into this type of frame can lead to either the rotating frame with respect to the drive ω d or the interaction frame by choosing ω (l) j = jω d , ω (l) j = jω (l) +∆ (l) j respectively. Here, we choose the former. In the rotating frame, we use the RWA to neglect the fast oscillating terms such as ±2ω d , the system's original Hamiltonian given by (1), iŝ H R = ∆Π (1) 2 + (δ − ∆)Π (2) 1 + δΠ (2) 2 + Ω X (t) 2 2 j=1 λ (1) jσ x(1) j,j−1 + λ (2) jσ x(2) j,j−1 + Ω Y (t) 2 2 j=1 λ (1) jσ y(1) j,j−1 + λ (2) jσ y(2) j,j−1 .(4) III. SINGLE QUBIT GATES We aim at applying, up to a global phase φ, a gate on the first qubit without affecting the second onê U F = e iφÛ (1) ⊗ 1.(5) Unless otherwise specifiedÛ (1) is anX-gate. A specific control pulse of duration t g results in a final gate given byÛ (t g ). The fidelity with which a control pulse meets the target gate is measured by Φ = 1 d 2 Tr Û F †Û (t g ) 2 ,(6) where d is the dimension of the Hilbert space of the system. The trace is taken over the computational subspace consisting of {|00 , |01 , |10 , |11 }. This takes leakage into account since leaving this subspace diminishes the matrix elements of the projected unitary [2,26]. We will also investigate single-qubit gates that shift the phase of the second qubit. Such gates can be made more efficiently and we later show how to correct the phase. Such gates can be studies using the reduced fidelity functions Φ | * ,i = 1 2 2 Tr {|0,i ,|1,i } Û F †Û (t g ) 2 .(7) The trace is taken over states where the second qubit is exclusively in the |0 or |1 . A gate producing a good Φ | * ,i has qubit 2 starting and ending in state |i . The average of the Φ | * ,i 's gives a fidelity function insensitive to the phase of the second qubit Φ avg = 1 2 Φ | * ,0 + Φ | * ,1 .(8) In other words, Φ avg is maximal ifÛ (t g ) (in the computational subspace of the two qubits) has the form U (t g ) = e iα 0 1 1 0 ⊗ 1 0 0 e i(γ−α) .(9) For a given gate time the phase error can be calculated and subsequently corrected as this gate is not entangling. In fact, an entangling gate would be detected by deteriorating Φ avg and given that the qubit controls are local and the two qubits are uncoupled, no entanglement is generated. IV. APPLYING DRAG The DRAG method [1,2,27] negates leakage to the |2 state with a two quadrature drive. Here we show that this method does not provide a sizable improvement over a single Gaussian envelope. We transformĤ R a second time along the lines of eq. (3) using the transformation matrix V (t) = exp   −i Ω X 2β 2 j=1 λ (1) jσ y(1) j,j−1 + λ (2) jσ y(2) j,j−1   . (10) This is the two-qubit version of the DRAG transformation [2,27]. The parameter β selects which transition is suppressed. A first order expansion in η = Ω X (t) /β ≪ 1 givesĤ V =Ĥ diag +Ĥ Y +Ĥ (1) X +Ĥ (2) X(11) The diagonal terms are of O(η 2 ), henceĤ diag is neglected on our level of approximation.Ĥ Y contains a term generated by the time-derivative in eq. (3) as well as the Y drivê H Y = Ω Y (t) 2 +Ω X (t) 2β 2 j=1 λ (1) jσ y(1) j,j−1 + λ (2) jσ y(2) j,j−1 . (12) H Y can be suppressed by choosing Ω Y (t) = −Ω X (t) /β. This is the essence of the DRAG method [1]. The last two terms respectively drive the first and second qubit according tô H (1) X (t) = Ω X (t)σ x(1) 10 + λ β − ∆ 2β Ω X (t)σ x(1) 21 + λ∆ 8β 2 Ω X (t) 2σ x(1) 20 , H(2)X (t) = η β − δ + ∆ 2β Ω X (t)σ x(2) 10 + ηλ β − δ 2β Ω X (t)σ x(2) 21 + η 2 λ∆ 8β 2 Ω 2 X (t)σ x(2) 20 . Depending on the value of β a specific off resonant transition can be suppressed. If β = δ the second qubit leakage transition is removed. However, since δ < ∆ (by a factor > 7 for the numbers in table I) the compensation field Ω Y becomes large and strongly drives the other leakage transitions, i.e., introduces errors of a size comparable to what it is suppressing. Note, that for fast pulses with β = δ the perturbation expansion in [1,2,27]. naturally breaks down. Selecting β = ∆ suppresses the leakage transition of the first qubit, but does not solve the leading spectral crowding issue based on the smallness of δ. We are explicitly highlighting this in figure 2. It shows the fidelity, as a function of gate time, for the single quadrature Gaussian (thin lines) and DRAG (thick lines) solutions with β = ∆. The difference between the fidelity function Φ, eq. (6), and the special fidelity functions Φ | * ,i and Φ avg , eq. (7) and (8), show that while it is difficult to perform an X gate on qubit 1 without affecting qubit 2, we can implement a high fidelityX gate with an additional phase shift on the other qubit for t g > 42ns. This marks a time limitation that for DRAG alone to produce a high-fidelity gate the time needs to be at least on the boundaries of the adiabatic regime. Φ Φ |*,0> Φ |*,1> Φ avg FIG. 2. Error for a single control with a Gaussian pulse shape as a function of gate time and a single quadrature (thin lines) and for the DRAG method with β = ∆ (thick lines). The DRAG method gives only marginal improvements over the single quadrature Gaussian pulse shape for Φavg which is slightly lower at the dip around 42 ns. The DRAG solution shown here is the optimal from picking β ǫ {∆, δ, δ − ∆}. V. MAGNUS EXPANSION Here we show how to find an improved pulse capable of performing the desired gate faster and with better fidelity. The full effect of system and Hamiltonian is described by the time evolution operator U (t g ) = T exp    −i tg 0 dtĤ(t)   (13) where T is the time-ordering operator. This can in general not be computed in closed form even for driven twostate systems with notable exceptions [28]. Still being unitary, the solution of equation (13) can be written as the exponential of an Hermitian matrix [21]. An expansion in this effective Hamiltonian gives the Magnus expansion U (t g ) = e −i kΘ k (tg ) .(14) The equation above still requires exponentiating a matrix. However the absence of time ordering considerably simplifies the derivation of an explicit expression forÛ . The Magnus expansion is asymptotic. Here, it converges quickly as nested integrals lead to cancellations of fast oscillating terms. The constraints on the controls set by the zeroth order in the expansion will thus be most important. The first terms in the expansion are given by [21] Θ 0 (t g ) = tg 0 dtĤ(t), Θ 1 (t g ) = − i 2 tg 0 dt 2 t2 0 dt 1 Ĥ (t 2 ) ,Ĥ (t 1 ) .(15) Here Ĥ (t 2 ) ,Ĥ (t 1 ) is the commutator of the Hamiltonian at different times. Higher order terms in the expansion can be worked out as nested commutators similar as those shown above. We start with the system in the interaction frame (the transformation is given in section II) H I = Ω C 2 2 j=1 λ (1) j e −iδ (1) j t |j − 1 j| (1) + λ (2) j e −iδ (2) j t |j − 1 j| (2) + h.c.(16) Here we have combined Ω C = Ω X + iΩ Y and set δ 1 = 0, δ (1) 2 = ∆, δ (2) 1 = δ − ∆, and δ (2) 2 = δ.(1) In the interaction frame, the Hamiltonian is purely off-diagonal and the desired gate is changed by a phase on the |1 state of the second qubit. This phase is known since any unitary transformationV (t), transforms the time evolution followingÛ V (t g ) =V (t g )Û (t g )V † (0). In equation (17) U F transforms in this way. If the zeroth order term is to implement the gate, the control problem becomeŝ U F = e −iΘ0 = e −i tg 0 dtĤ I (t) .(17) As an aside, this highlights why Θ 0 /t g is often called the average Hamiltonian and kΘ k (t g )/t g the effective Hamiltonian in NMR [21]. This and the formĤ I imposes restrictions on the control Ω C 1 2 tg 0 dt Ω C = π(18)1 2 tg 0 dt e −i∆t Ω C = 0(19)1 2 tg 0 dt e −iδt Ω C = 0(20)1 2 tg 0 dt e −i(δ−∆)t Ω C = 0(21) These constraints are the Fourier transforms of the control evaluated at the different detunings in the system as is familiar from spectroscopy at weak drive [21, 29-31]-but here derived under intermediate-to-strong drive conditions. They state that the control should contain no power at the off resonant frequencies. If Ω C is palindromic the complex conjugated equations are also satisfied. If equations (18)(19)(20)(21) are met, the final unitary evolution will be e iφσ x ⊗ 1. So that the zeroth order implements the gate, higher order terms have to be zero. Here is an example of the first order termΘ 1 . It only gives extra terms on the diagonal and the 0 ↔ 2 transition. This calculation is quite involved and here is an example of the term involving |0, 1 0, 1| (neglecting terms oscillating faster than δ) 01|Θ 1 (t g ) |01 = 1 4 tg 0 dt 2 t2 0 dt 1 Ω(t 1 , t 2 ) [1 + cos (δ (t 1 − t 2 )) − sin (δ (t 1 − t 2 ))] ,(22) with Ω(t 1 , t 2 ) = Ω X (t 2 ) Ω Y (t 1 ) − Ω X (t 1 ) Ω Y (t 2 ) . In the spirit of the Magnus expansion, all slow oscillating terms have the form above and are negligible if their integral is small. This suggest a control pulse where Ω X is modulated with a sinusoidal function Ω X =A π e − 1 2σ 2 t− tg 2 2 1 − A cos ω x t − t g 2 , Ω Y = − 1 βΩ X .(23) This is a Gaussian with added sideband modulation on the in-phase part Ω x supplemented by DRAG on the quadrature Ω y . A frequency modulation with cos(ω x t) for a bandwidth of Ω g < 2ω x can be seen as adding an effective drive at ω x proportional to Ω g . This added drive can be used to counteract the population transfer of a specific transition. The absolute errors of eqs. (19)(20)(21) are minimized by varying A, ω x , β yielding a pulse with a sideband modulation of δ/2 Ω X =A π e − 18 t 2 g t− tg 2 2 1 − cos δ 2 t − t g 2 , Ω Y = − 1 2∆Ω X .(24) Here we chose σ = t g /6. The factor of 2 in the denominator of Ω Y comes from the absence of control over the qubit frequency [2]. This is shown experimentally in ref. [32,33]. The pulse is shown in figure 3 for t g = 17ns and other parameters given by the values in table I. In order for the pulse to produce the X gate A π should be chosen so that relation (18) is satisfied. A. Sideband modulation The black line in figure 4 shows the error of pulse (24) as function of gate time. Compared to the Gaussian and DRAG results, the error has a minimum ( 4%) at a shorter gate time, around 20ns. The reduced fidelity functions Φ | * ,i (red and blue lines) and Φ avg (gray line) give additional insight by allowing a phase shift on qubit 2. Comparing to figure 2, it is seen that the sideband modulated pulse attains a high fidelity (> 99.9%) in less (23) for tg = 17ns. The amplitude of Ωx is somewhat smaller than for a Gaussian only pulse (which has been used in figure 2). than half the time (17 ns compared to 42 ns) of the Gaussian or DRAG solutions. The 1 ↔ 2 transition of the second qubit is still the limiting factor since the reduced error 1 − Φ | * ,1 is always the biggest. Nonetheless for a specific gate time a high fidelity is possible. The state populations during the pulse reveal the underlying mechanism. Figure 5 shows the populations for gate times 17 and 20 ns. In the latter there is still a net population in the |2 state of the qubit 2 after the gate. For the former, there is no net change to the second qubit at the end. This suggest that the the drive on the second qubit makes it perform a closed transition cycle in the (|1 , |2 ) subspace, thus acquiring a local phase. Finally we note in this section that the method worked out here is not the only way to determine new analytical results for pulse shapes. In general, the different terms of equation (14) need to combine into the correct gate in some manner, whereas we have enforced that this combination consists of all terms beyond the lowest one to vanish. Our approach has the advantage that it produces an intuitive result, providing frequency selectivity criteria eqs. (18,19,20,21) in the form of the Fourier transform of the driving pulse. B. Phase correction The average reduced fidelity (8) is insensitive to the phase of the second qubit and leads to a gate of the form of eq. (9). This phase error does not influence population measurements after the gate; only the X and Y component have different contributions. The global phase α and the phase error γ for specific gate times are plotted in fig. 6. One can correct for this error in multiple ways. If there is a Z control available on the separate qubits [17] one can simply compensate the phase following π 2 = Z 1 (t) dt α (t g ) = Z 2 (t) dt(25) Instead of compensating the qubit phase, one can adjust the phase of the next gate in the XY -plane accordingly. This is possible because the phase error is constant given a set gate time, as shown in figure 6. In essence this is the same as changing the frame in the XY plane according to X ′ = cos (α (t g )) X + sin (α (t g )) Y Y ′ = − sin (α (t g )) X + cos (α (t g )) Y.(26) This technique is analogous to phase ramping as described in Refs. [1,2] The phases in the leakage states are irrelevant, it is thus sufficient to correct the computational subspaces of the qubits individually. (9) of the gate with the control sequence from equation (24). It is by these phases that the qubits or the subsequent gates need to be corrected. C. Experimental protocol The procedure to implement the pulse on an actual experiment is • Use spectroscopy to determine the qubit frequencies, yielding δ and ∆. • Equation (24) gives the shape of the pulses for all possible gate times t g . The normalization parameter A π is chosen so that the area theorem, equation (18), is satisfied, which in general requires numerical root finding. • The gate time t g is chosen so that the pulse sequence optimizes the reduced average fidelity defined by equation (8). • With the gate time known, the phase offset α(t g ) is computed, so that it can be corrected according to the procedures given in section V B. VI. NUMERICAL OPTIMIZED CONTROLS By using numerical methods one can go beyond the analytic methods discussed in the last sections. Here is discussed how further improvements can be made with the GRAPE algorithm. A. GRAPE To handle our system numerically we use the GRadient Ascent Pulse Engineering (GRAPE) algorithm [34]. GRAPE maximizes the fidelity eq. (6) by changing the control amplitudes at discrete times. In discrete time the evolution operator is given byÛ (t g ) = jÛ j , witĥ U j = exp[−iĤ (j∆t) ∆t]. The fidelity is increased by updating the controls in the direction of the gradient Ω l (j) = Ω j l → Ω j l + ǫ∂Φ/∂Ω j l . An analytic expression for the gradient is given in ref. [35]. B. Numerical results The system of equation (4) is numerically optimized using the parameters in table I. Figure 7 is an example of a short gate (4 ns) high fidelity (99.999%) GRAPE pulse. This pulse has t g ≪ π/δ and therefore the smallest spectral crowding frequency scale δ does not impose a quantum speed limit. The limit rather seems to be set by the number of control parameters available. E.g., we have verified that if the size of a time step is 1 ns as in current experimental equipment, the shortest possible time is 8ns. From numerical results we have not observed a quantum speed limit. By decreasing the gate time the pulse can be shortened at the expense of higher amplitudes. The pulse in figure 7 has large amplitudes at t = 0 and t = t g . These can be removed by adding penalties to the fidelity used by GRAPE [36]. Only a small increase in gate time is usually needed to enforce pulse sequences to start and end at zero amplitude. The numerical results show that no speed limit is set by the overlap of the control field in the frequency domain with different qubit transitions. Additionally, numerical pulse sequences don't leave a phase error on the second qubit, eliminating the need for post-processing. To get insight for the shape of the solutions we run the GRAPE algorithm for short time steps and longer gate times to increase the resolution of the discrete time Fourier transform (DTFT). These solutions show rapid oscillations, figure 8. The DTFT of the pulse sequence shows that both quadrature components have contributions at the energy splittings δ, δ − ∆, ∆, 2δ − ∆. This shows that the numerical solution augments the one When one goes to shorter gate times however Fourier analysis shows that the contribution of the higher frequency components increases, making the Fourier transform less useful due to the lower frequency resolution. For faster pulses one could suggest that adding more sideband modulations could improve the results further. VII. CONCLUSION We have found numerical as well as analytical pulse shapes implementing single qubit gates in a 3D cavity coupled to two single junction Transmons. Such qubits are typically hindered by spectral crowding whereby leakage transitions lie close in frequency to main qubit 0 ↔ 1 transitions. We combine average Hamiltonian theory for arbitrary waveforms with the DRAG methodology, shows that it is possible to find better controls using a sideband modulation. Numerically optimized pulses support this conclusion and provide greater improvements in fidelity. They show that qubits can still be addressed individually with short gate times. Faster control pulses require more bandwidth and amplitude, therefore the limiting factor is the capabilities of the arbitrary waveform generator. No speed limit has been observed in numerically optimized pulses, which is contrary to the believe that spectral crowding limits the scalability of the 3D cavity architecture in cQED. We would like to thank Leo DiCarlo for suggesting this problem as well as Felix Motzoi for useful discussions. This work was supported through IARPA within the MQCO program and the European Union within SCALEQIT. FIG. 3 . 3Example of the control functions of equation FIG. 4 . 4Error as a function of gate time for the pulse with sideband modulation. The target gate isσx ⊗ 1. At tg ∼ 17ns, Φavg reaches a maximum. The gate fidelity functions are defined in equations (6),(7)and(8)respectively. FIG. 5 . 5Populations of the states during the pulse sequence of equation(24) for gate time of 17ns (a), and 20ns (b). At 20ns the pulse sequence clearly leaves part of the excitation in the {|1 , |2 } subspace of qubit two, while at 17ns the trajectory is optimal in the sense that no net population transfer is present on qubit two. FIG. 6 . 6Phases as defined in equation FIG. 7 .FIG. 8 . 78Example of a numerically optimized pulse for gate time tg = 4ns and ∆t = 0.01ns. The pulses for shorter gate time are highly oscillating. The ΩY control is usually not proportional to the derivative of ΩX .based on the Magnus expansion by adding small further sideband drives. Solution found by GRAPE for a long gate time, here ∆t = 0.01ns and tg = 130ns. The dotted line shows a rescaled version of the derivative of the ΩX control. FIG. 9 . 9Fourier transform of the pulse shown in 8 found by GRAPE. Table I TABLE I . IIshow the variables and numerical values used in simulations.1 System parameters as shown in equation(1).Qubit 1 Qubit 2 ω k /2π 5.508 5.5903 GHz ∆/2π −350 −350 MHz λ (k) 1 . F Motzoi, J M Gambetta, P Rebentrost, F K Wilhelm, Phys. Rev. Lett. 103110501F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm, Phys. Rev. Lett. 103, 110501 (2009). . J M Gambetta, F Motzoi, S T Merkel, F K Wilhelm, Physical Review A. 8312308J. M. Gambetta, F. Motzoi, S. T. Merkel, , and F. K. Wilhelm, Physical Review A 83, 012308 (2011). . Y Makhlin, G Schön, A Shnirman, Rev. Mod. Phys. 73357Y. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). . J Clarke, F K Wilhelm, Nature. 4531031J. Clarke and F. K. Wilhelm, Nature 453, 1031 (2008). . J You, F Nori, Phys. Today. 5842J. You and F. Nori, Phys. Today 58, 42 (2005). . M D Reed, L Dicarlo, S E Nigg, L Sun, L Frunzio, S M Girvin, R J Schoelkopf, Nature Physics. 482382M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Nature Physics 482, 382 (2012). . E Lucero, R Barends, Y Chen, J Kelly, M Mariantoni, A Megrant, P O&apos;malley, D Sank, A Vainsencher, J Wenner, Nature Physics. 8719E. Lucero, R. Barends, Y. Chen, J. Kelly, M. Mariantoni, A. Megrant, P. O'Malley, D. Sank, A. Vainsencher, J.Wenner, et al., Nature Physics 8, 719 (2012). . A Blais, R.-S Huang, A Wallraff, S M Girvin, R J Schoelkopf, Phys. Rev. A. 6962320A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004). . R Schoelkopf, S Girvin, Nature. 451664R. Schoelkopf and S. Girvin, Nature 451, 664 (2008). . J You, F Nori, Nature. 474589J. You and F. Nori, Nature 474, 589 (2011). . J Koch, T M Yu, J Gambetta, A A Houck, D I Schuster, J Majer, A Blais, M H Devoret, S M Girvin, R J Schoelkopf, Phys. Rev. A. 7642319J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 76, 042319 (2007). . J M Martinis, K B Cooper, R Mcdermott, M Steffen, M Ansmann, K D Osborn, K Cicak, S Oh, D P Pappas, R W Simmonds, Phys. Rev. Lett. 95210503J. M. Martinis, K. B. Cooper, R. McDermott, M. Stef- fen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, et al., Phys. Rev. Lett. 95, 210503 (2005). . D Vion, A Aassime, A Cottet, P Joyez, H Pothier, C Urbina, D Esteve, M Devoret, Science. 296866D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, and M. Devoret, Science 296, 866 (2002). . Y Makhlin, A Shnirman, Phys. Rev. Lett. 92178301Y. Makhlin and A. Shnirman, Phys. Rev. Lett. 92, 178301 (2004). . P Rebentrost, I Serban, T Schulte-Herbrüggen, F Wilhelm, Phys. Rev. Lett. 10290401P. Rebentrost, I. Serban, T. Schulte-Herbrüggen, and F. Wilhelm, Phys. Rev. Lett. 102, 090401 (2009). . M Metcalfe, E Boaknin, V Manucharyan, R Vijay, I Siddiqi, C Rigetti, L Frunzio, R J Schoelkopf, M H Devoret, Phys. Rev. B. 76174516M. Metcalfe, E. Boaknin, V. Manucharyan, R. Vijay, I. Siddiqi, C. Rigetti, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Phys. Rev. B 76, 174516 (2007). . H Paik, D I Schuster, L S Bishop, G Kirchmair, G Catelani, A P Sears, B R Johnson, M J Reagor, L Frunzio, L I Glazman, Phys. Rev. Lett. 107240501H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, et al., Phys. Rev. Lett. 107, 240501 (2011). . C Rigetti, J M Gambetta, S Poletto, B L T Plourde, J M Chow, A D Corcoles, J A Smolin, S T Merkel, J R Rozen, G A Keefe, Phys. Rev. B. 86100506C. Rigetti, J. M. Gambetta, S. Poletto, B. L. T. Plourde, J. M. Chow, A. D. Corcoles, J. A. Smolin, S. T. Merkel, J. R. Rozen, G. A. Keefe, et al., Phys. Rev. B 86, 100506 (2012). . D I Schuster, A A Houck, J A Schreier, A Wallraff, J M Gambetta, A Blais, L Frunzio, J Majer, B Johnson, M H Devoret, Nature. 445515D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. John- son, M. H. Devoret, et al., Nature 445, 515 (2007). . J M Chow, J M Gambetta, A D Córcoles, S T Merkel, J A Smolin, C Rigetti, S Poletto, G A Keefe, M B Rothwell, J R Rozen, Phys. Rev. Lett. 10960501J. M. Chow, J. M. Gambetta, A. D. Córcoles, S. T. Merkel, J. A. Smolin, C. Rigetti, S. Poletto, G. A. Keefe, M. B. Rothwell, J. R. Rozen, et al., Phys. Rev. Lett. 109, 060501 (2012). . W Warren, J. Chem. Phys. 815437W. Warren, J. Chem. Phys. 81, 5437 (1984). . J M Gambetta, A A Houck, A Blais, Phys. Rev. Lett. 10630502J. M. Gambetta, A. A. Houck, and A. Blais, Phys. Rev. Lett. 106, 030502 (2011). . J Chow, J Gambetta, L Tornberg, J Koch, L Bishop, A Houck, B Johnson, L Frunzio, S Girvin, R Schoelkopf, Phys. Rev. Lett. 10290502J. Chow, J. Gambetta, L. Tornberg, J. Koch, L. Bishop, A. Houck, B. Johnson, L. Frunzio, S. Girvin, and R. Schoelkopf, Phys. Rev. Lett. 102, 090502 (2009). . M Steffen, J Martinis, I Chuang, Phys. Rev. B. 68224518M. Steffen, J. Martinis, and I. Chuang, Phys. Rev. B 68, 224518 (2003). . B Khani, J Gambetta, F Motzoi, F Wilhelm, Phys. Scr. 13714021B. Khani, J. Gambetta, F. Motzoi, and F. Wilhelm, Phys. Scr. T137, 014021 (2009). . P , F Wilhelm, arXiv:0808.2680Phys. Rev. B. 7960507P. and F. Wilhelm, Phys. Rev. B 79, 060507(R) (2009), arXiv:0808.2680. . F Motzoi, J Gambetta, F Wilhelm, in preparationF. Motzoi, J. Gambetta, and F. Wilhelm, in preparation. . A Gangopadhyay, M Dzero, V Galitski, Phys. Rev. B. 8224303A. Gangopadhyay, M. Dzero, and V. Galitski, Phys. Rev. B 82, 024303 (2010). . R Vold, J Waugh, M Klein, D Phelps, J. Chem. Phys. 433831R. Vold, J. Waugh, M. Klein, and D. Phelps, J. Chem. Phys. 43, 3831 (1968). . D Hoult, J. Magn. Res. 3569D. Hoult, J. Magn. Res. 35, 69 (1979). R Freeman, Spin Choreography, Basic Steps in High Resolution NMR. New YorkOxford University PressR. Freeman, Spin Choreography: Basic Steps in High Resolution NMR (Oxford University Press, New York, 1998). . J Chow, L Dicarlo, J Gambetta, F Motzoi, L Frunzio, S Girvin, R Schoelkopf, Phys. Rev. A. 8240305J. Chow, L. DiCarlo, J. Gambetta, F. Motzoi, L. Frun- zio, S. Girvin, and R. Schoelkopf, Phys. Rev. A 82, 040305(R) (2010). . E Lucero, J Kelly, R C Bialczak, M Lenander, M Mariantoni, M Neeley, A D O&apos;connell, D Sank, H Wang, M Weides, Phys. Rev. A. 8242339E. Lucero, J. Kelly, R. C. Bialczak, M. Lenander, M. Mariantoni, M. Neeley, A. D. O'Connell, D. Sank, H. Wang, M. Weides, et al., Phys. Rev. A 82, 042339 (2010). . N Khaneja, T Reiss, C Kehlet, T Schulte-Herbrüggen, S Glaser, J. Magn. Reson. 172296N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. Glaser, J. Magn. Reson. 172, 296 (2005). . S Machnes, U Sander, S J Glaser, P De Fouquières, A Gruslys, S Schirmer, T Schulte-Herbrüggen, Phys. Rev. A. 8422305S. Machnes, U. Sander, S. J. Glaser, P. de Fouquières, A. Gruslys, S. Schirmer, and T. Schulte-Herbrüggen, Phys. Rev. A 84, 022305 (2011). . P Rebentrost, F K Wilhelm, Phys. Rev. B. 7960507P. Rebentrost and F. K. Wilhelm, Phys. Rev. B 79, 060507 (2009).
[]
[ "It's Like Python But: Towards Supporting Transfer of Programming Language Knowledge", "It's Like Python But: Towards Supporting Transfer of Programming Language Knowledge" ]
[ "Nischal Shrestha \nNC State University Raleigh\nNCUSA\n", "Titus Barik [email protected] \nMicrosoft Redmond\nWAUSA\n", "Chris Parnin [email protected] \nNC State University Raleigh\nNCUSA\n" ]
[ "NC State University Raleigh\nNCUSA", "Microsoft Redmond\nWAUSA", "NC State University Raleigh\nNCUSA" ]
[]
Expertise in programming traditionally assumes a binary novice-expert divide. Learning resources typically target programmers who are learning programming for the first time, or expert programmers for that language. An underrepresented, yet important group of programmers are those that are experienced in one programming language, but desire to author code in a different language. For this scenario, we postulate that an effective form of feedback is presented as a transfer from concepts in the first language to the second. Current programming environments do not support this form of feedback.In this study, we apply the theory of learning transfer to teach a language that programmers are less familiar with--such as R--in terms of a programming language they already know--such as Python. We investigate learning transfer using a new tool called Transfer Tutor that presents explanations for R code in terms of the equivalent Python code. Our study found that participants leveraged learning transfer as a cognitive strategy, even when unprompted. Participants found Transfer Tutor to be useful across a number of affordances like stepping through and highlighting facts that may have been missed or misunderstood. However, participants were reluctant to accept facts without code execution or sometimes had difficulty reading explanations that are verbose or complex. These results provide guidance for future designs and research directions that can support learning transfer when learning new programming languages.
10.1109/vlhcc.2018.8506508
[ "https://arxiv.org/pdf/1808.09008v1.pdf" ]
52,109,765
1808.09008
e716fac140c7b61bcb3c86f06e6d5c03d1b3b643
It's Like Python But: Towards Supporting Transfer of Programming Language Knowledge Nischal Shrestha NC State University Raleigh NCUSA Titus Barik [email protected] Microsoft Redmond WAUSA Chris Parnin [email protected] NC State University Raleigh NCUSA It's Like Python But: Towards Supporting Transfer of Programming Language Knowledge Expertise in programming traditionally assumes a binary novice-expert divide. Learning resources typically target programmers who are learning programming for the first time, or expert programmers for that language. An underrepresented, yet important group of programmers are those that are experienced in one programming language, but desire to author code in a different language. For this scenario, we postulate that an effective form of feedback is presented as a transfer from concepts in the first language to the second. Current programming environments do not support this form of feedback.In this study, we apply the theory of learning transfer to teach a language that programmers are less familiar with--such as R--in terms of a programming language they already know--such as Python. We investigate learning transfer using a new tool called Transfer Tutor that presents explanations for R code in terms of the equivalent Python code. Our study found that participants leveraged learning transfer as a cognitive strategy, even when unprompted. Participants found Transfer Tutor to be useful across a number of affordances like stepping through and highlighting facts that may have been missed or misunderstood. However, participants were reluctant to accept facts without code execution or sometimes had difficulty reading explanations that are verbose or complex. These results provide guidance for future designs and research directions that can support learning transfer when learning new programming languages. I. INTRODUCTION Programmers are expected to be fluent in multiple programming languages. When a programmer switches to a new project or job, there is a ramp-up problem where they need to become proficient in a new language [1]. For example, if a programmer was proficient in Python, but needed to learn R, they would need to consult numerous learning resources such as documentation, code examples, and training lessons. Unfortunately, current learning resources typically do not take advantage of a programmer's existing knowledge and instead present material as if they were a novice programmer [2]. This style of presentation does not support experienced programmers [3] who are already proficient in one or more languages and harms their ability to learn effectively and efficiently [4]. Furthermore, the new language may contain many inconsistencies and differences to previous languages which actively inhibit learning. For example, several blogs and books [5] have been written for those who have become frustrated or confused with the R programming language. In an online document [6], Smith lists numerous differences of R from other high-level languages which can confuse programmers such as the following: Sequence indexing is base-one. Accessing the zeroth element does not give an error but is never useful. In this paper, we explore supporting learning of programming languages through the lens of learning transfer, which occurs when learning in one context either enhances (positive transfer) or undermines (negative transfer) a related performance in another context [7]. Past research has explored transfer of cognitive skills across programming tasks like comprehension, coding and debugging [8], [9], [10]. There has also been research exploring the various difficulties of learning new programming languages [11], [12] and identifying programming misconceptions held by novices [13]. However, limited research has focused on the difficulties of learning languages for experienced programmers and the interactions and tools necessary to support transfer. To learn how to support transfer, we built a new training tool called Transfer Tutor that guides programmers through code snippets of two programming languages and highlights reusable concepts from a familiar language to learn a new language. Transfer Tutor also warns programmers about potential misconceptions carried over from the previous language [14]. We conducted a user study of Transfer Tutor with 20 participants from a graduate Computer Science course at North Carolina State University. A qualitative analysis on think-aloud protocols revealed that participants made use of learning transfer even without explicit guidance. According to the responses to a user satisfaction survey, participants found several features useful when learning R, such as making analogies to Python syntax and semantics. However, participants also pointed out that Transfer Tutor lacks code executability and brevity. Despite these limitations, we believe a learning transfer tool can be successful in supporting expert learning of programming languages, as well as other idioms within the same language. We discuss future applications of learning transfer in other software engineering contexts, such as assisting in code translation tasks and generating documentation for programming languages. II. MOTIVATING EXAMPLE Consider Trevor, a Python programmer who needs to switch to R for his new job as a data analyst. Trevor takes an online course on R, but quickly becomes frustrated as the course presents material as if he is a novice programmer and does not make use of his programming experience with Python and Pandas, a data analysis library. Now, Trevor finds himself illequipped to get started on his first task at his job, tidying data on popular questions retrieved from Stack Overflow (SO), a question-and-answer (Q&A) community [15]. Even though he is able to map some concepts over from Python, he experiences difficulty understanding the new syntax due to his Python habits and the inconsistencies of R. Trevor asks help from Julie, a seasoned R programmer, by asking her to review his R script (see Fig. 1) so he can learn proper R syntax and semantics. Trevor's task is to conduct a typical data analysis activity, tidying data. He is tasked with the following: 1) read in a comma-separated value (csv) file containing Stack Overflow questions 2) filter the data according to positive scores and 3) select the top five rows. Julie walks him through his Python code and explains how they relate to the equivalent code she wrote in R. Julie teaches Trevor that R has several assignment operators that he can use to assign values to variables but tells him that the <syntax is commonly used by the R community. However, she tells him that the = operator can also be used in R just like Python. To read a csv file, Julie instructs Trevor to use a built-in function called read.csv() which is quite similar to Python's read csv() function. Moving on to the next line, Julie explains that selecting rows and columns in R is very similar to Python with some subtle differences. The first subtle difference that she points out is that when subsetting (selecting) rows or columns from a data frame in Python, using the [ syntax selects rows. However, using the same operator in R will select columns. Julie explains that the equivalent effect of selecting rows works if a comma is inserted after the row selection and the right side of the comma is left empty (Figure 1b). Julie tells him that since the right side is for selecting columns, leaving it empty tells R to select all the columns. To reference a column of a data frame in R, Julie explains that it works almost the same way as in Python, except the . (dot) must be replaced with a $ instead. Finally, Julie points out that R's indexing is 1-based, so the range for selecting the five rows must start with 1, and unlike Python, the end index is inclusive. Trevor now has some basic knowledge of R. Could tools help Trevor in the same way Julie was able to? III. TRANSFER TUTOR A. Design Rationale We created a new training tool called Transfer Tutor that takes the place of an expert like Julie and makes use of learning transfer to teach a new programming language. Transfer Tutor teaches R syntax and semantics in terms of Python to help provide scaffolding [16] so programmers can start learning from a familiar context and reuse their existing knowledge. Our approach is to illustrate similarities and differences between code snippets in Python and R with the use of highlights on syntax elements and different types of explanations. We designed Transfer Tutor as an interactive tool to promote "learnable programming" [17] so that users can focus on a single syntax element at a time and be able to step through the code snippets on their own pace. We made the following design decisions to teach data frame manipulations in R: 1) highlighting similarities between syntax elements in the two languages 2) explicit tutoring on potential misconceptions and 3) stepping through and highlighting elements incrementally. B. Learning Transfer Transfer Tutor supports learning transfer through these feedback mechanisms in the interface: • Negative Transfer: 'Gotchas' warn programmers about a syntax or concept that either does not work in the new language or carries a different meaning and therefore should be avoided. • Positive Transfer: 'Transfer' explanations describe a syntax or concept that maps over to the new language. • New Fact: 'New facts' describe a syntax or concept that has little to no mapping to the previous language. Each type of feedback consists of a highlighted portion of the code in the associated language (Python or R) with its respective explanation, which serves as affordances for transfer [18]. Furthermore, we support deliberate connections between elements, by allowing participants to step through the code, which helps them make a mindful abstraction of the concepts [19]. Finally, we focus on transferring declarative knowledge [20], such as syntax rules, rather than procedural knowledge, such as general problem-solving strategies. C. User Experience This section presents screenshots of Transfer Tutor and a use case scenario. The user experience of Transfer Tutor is presented from the perspective of Trevor who decides to use the tool to learn how to select columns of a data frame in R, a 2D rectangular data structure which is also used in Python/Pandas. The arrows and text labels are used to annotate the various features of the tool and are not actually presented to the users. 1) Code Snippets and Highlighting: Trevor opens up Transfer Tutor and notices that the tool displays two lines of code, where the top line is Python, the language that he is already familiar with and on the bottom is the language to learn which is R. Trevor examines the stepper buttons below the snippets and clicks 3 which begins the lesson and highlights some syntax elements: Start Over Finish Lesson Highlight Previous Element Begin / Highlight Next Element Current Element R Transfer Element Python Transfer Element Travis notices 1 points to the current syntax element in Python and R indicated by 2a and 2b . Trevor looks over to the right at the explanation box: 2) Explanation Box: Trevor sees 1 which refers to a Python 'transfer' with 2 showing the transfer icon. He reads 3 and learns that the [ operator can be used in R. Transfer Tutor treats this syntax as a positive transfer since it can be reused. Trevor moves on to the next element: Trevor looks at 1 which is a red highlight on the Python code. He reads 2 in the explanation box for clarification. Trevor learns about a Python 'gotcha': the [[ syntax from Python can't be used in R. Trevor then reads 3 which explains an R 'gotcha' about how the [[ syntax is legal in R, but semantically different from the Python syntax as it only selects a single column. In this case, Transfer Tutor warns him about a subtle difference, a negative transfer that could cause him issues in R. Trevor moves on to the next element and examines the elements that are highlighted blue: Trevor looks at 1 then 2 and realizes he's looking at a 'new fact' about R. Transfer Tutor describes the c() function used to create a vector in R, which doesn't have a direct mapping to a Python syntax. 3) Code Output Box: Finally, Trevor steps through the code to the end, and the code output box now appear at the bottom which displays the state of the data frame: Trevor reads 1 and inspects 2 to understand the contents of the data frame in R and how it differs from Python's data frame: 1) NaNs from Python are represented as NAs and 2) Row indices start from 1 as opposed to 0. Transfer Tutor makes it clear that selecting columns of a data frame in R is similar to Python with some minor but important differences. IV. METHODOLOGY A. Research Questions We investigated three research questions using Transfer Tutor to: 1) determine face validity of teaching a new language using an interactive tool 2) examine how programmers use Transfer Tutor and 3) determine which affordances they found to being useful for learning a new language. RQ1: Are programmers learning R through Transfer Tutor? To identify if training through learning transfer is an effective approach in the context of programming, this question is used to determine the face validity of Transfer Tutor's ability to teach R. RQ2: How do programmers use Transfer Tutor? Investigating how programmers use Transfer Tutor can identify when it supports learning transfer, and whether the affordances in the tool align with the way programmers reason about the problem. RQ3: How satisfied are programmers with learning R when using the Transfer Tutor? We want to learn what features of Transfer Tutor programmers felt were useful to them. If programmers are satisfied with the tool and find it useful, it is more likely to be used. B. Study Protocol 1) Participants: We recruited 20 participants from a graduate Computer Science course at our University, purposely sampling for participants with experience in Python, but not R. We chose to teach R for Python programmers because both languages are used for data science programming tasks, yet have have subtle differences that are known to perplex novice R programmers with a background in Python [5], [6], [21]. Through an initial screening questionnaire, participants reported programming experience and demographics. Participants reported their experience with Python programming with a median of "1-3 years" (7), on a 4-point Likert-type item scale ranging from "Less than 6 months", "1-3 years", "3-5 years", and "5 years or more" ( ). Participants reported a median of "Less than 6 months" (19) of experience with R programming ( ), and reported a medium of "1-3 years" with data analysis activities ( ). 16 participants reported their gender as male, and four as female; the average age of participants was 25 years (sd = 5). All participants conducted the experiment in a controlled lab environment on campus, within a 1-hour time block. The first author of the paper conducted the study. 2) Onboarding: Participants consented before participating in the study. They were presented with a general instructions screen which described the format of the study and familiarized them with the interface. The participants then completed a pre-test consisting of seven multiple choice or multiple answer questions, to assess prior knowledge on R programming constructs for tasks relating to indexing, slicing, and subsetting of data frames. The questions were drawn from our own expertise in the language and quizzes from an online text. 1 The presentation of questions was randomized to mitigate ordering effects. We also asked participants to thinkaloud during the study, and recorded these think-aloud remarks as memos. 3) Study Materials: The authors designed four lessons on the topic of data frame manipulation, where each lesson consists of a one line code snippet in both languages and explanations associated with the relevant syntax elements. The authors also designed questions for the pre-test and post-test (see Table I). Finally, the authors designed a user satisfaction survey of Transfer Tutor. The study materials are available online. 2 4) Tasks: Participants completed the following lessons on R: 1) assignment and reading data, 2) selecting columns, 3) filtering, and 4) selecting rows and sorting. Participants stepped through each lesson as described in Section III. Within each lesson, participants interacted with 5-8 highlights and corresponding explanation boxes. 5) Wrap-up: At the end of the study, participants completed a post-test containing the same questions as the pre-test. Participants completed a user satisfaction survey asking the participants for additional feedback on the tool. The survey asked them to rate statements about the usefulness of the tool using a 5-point Likert scale. These statements targeted different features of the tool such as whether or not highlighting syntax elements was useful for learning R. The survey also contained free-form questions for feedback regarding the tool such as the most positive and negative aspects, how they could benefit from using the tool and what features they would add to make it more useful. Finally, participants were given the opportunity to debrief for any general questions they may have had about the study. C. Analysis RQ1: Are programmers learning R through Transfer Tutor? We used differences in pre-test and post-test performance as a proxy measure for learning. We assigned equal weight to each question, with each question being marked as incorrect (0 points) or correct (1 point), allowing us to treat them as ordinal values. For the multiple answer questions, the participants received credit if they choose all the correct answers. A Wilcoxon signed-rank test between the participants' pre-test and post-test scores was computed to identify if the score differences were significant (α = 0.05). RQ2: How do programmers use Transfer Tutor? All authors of the paper jointly conducted an open card sort-a qualitative technique for discovering structure from an unsorted list of statements [22]. Our card sorting process consisted of two phases: preparation and execution. In the preparation phase, we extracted the think-aloud and observational data from the written memos into individual cards, with each card containing Given a data frame df with column indices 1, 2, and 3, which one of these will cause an error? In the execution phase, we sorted the cards into meaningful themes. The card sort is open because the themes were not pre-defined before the sort. The result of a card sort is not to a ground truth, but rather, one of many possible organizations that help synthesize and explain how programmers interact with tool. RQ3: How satisfied are programmers with learning R when using Transfer Tutor? We summarized the Likert responses for each of the statements in the user satisfaction survey using basic descriptive statistics. We also report on suggestions provided by participants in the free-form responses for questions, which include suggestions for future tool improvements. V. RESULTS In this section we present the results of the study, organized by research question. A. RQ1: Are programmers learning R after using Transfer Tutor? All participants had a positive increase in overall score (n = 20). The Wilcoxon signed rank test identified the posttest scores to be significantly higher than the pre-test scores (S = 105, p < .0001), and these differences are presented in Table I. Questions 1, 3, 5 and 6 provide strong support for learning transfer. In Question 2 and Question 4, most participants already supplied the correct answer with the pretest: thus, there was a limited increase in learning transfer. The result of Question 7, however, was unexpected: no participants answered the pre-test question correctly, and there was essentially no learning transfer. We posit potential explanations for this in Limitations (Section VI). Based on these results, using test performance has face validity in demonstrating Transfer Tutor's effectiveness in supporting learning transfer from Python to R. B. RQ2: How do programmers use Transfer Tutor? The card sorting results of the observational and thinkaloud memos are presented in this section, organized into four findings. Evidence of using transfer: We collected 398 utterances from our participants during their think-aloud during card sorting. All participants' think-aloud contained utterances related to learning transfer. 35.9% of the total utterances related to transfer, revealing positive (18.9%) and negative transfers (66.4%). They also verbalized or showed behavior to indicate that they were encountering something that was new and didn't map to something they already knew (14%). Other utterances not related to transfer involved verbatim reading of text or reflection on the task or tool. Participants identified several positive transfers from Python, often without explicit guidance from Transfer Tutor. P4 guessed that the range for selecting a column in the Python code was equivalent to the one in R without Transfer Tutor explicitly mentioning this fact: "both are the same, 2 colon in Python means 3 in R." Another participant correctly related Python's dot notation to reference a data frame's column to R's use of dollar sign: "Oh looks like $ sign is like the dot." [P17]. This is evidence that programmers are naturally using learning transfer and Transfer Tutor helps support this strategy. Participants also encountered several negative transfers from either Python or their previous languages. P15 thought the dot in the read.csv() function signified a method call and verbalized that the "read has a csv function" and later realized the mistake: "read is not an Object here which I thought it was!" P5 expressed the same negative transfer, thinking that "R has a module called read.". This indicates a negative transfer from object-oriented languages where the dot notation is typically used for a method call. Participants would also verbalize or show signs of behavior indicating that they have encountered a new fact, or a nontransfer, in R. This behavior occurred before progressing to the element with its associated explanation. P7 encountered the subsetting syntax in R and wondered, "Why is the left side Tool highlighted facts participants may have misunderstood or missed: The highlighting of the syntax elements and stepping through the code incrementally helped participants focus on the important parts of the code snippets. For additional feedback, one participant said "I was rarely confused by the descriptions, and the colorized highlighting helped me keep track of my thoughts and reference what exactly it was I was reading about with a specific example" [P17]. P13 had a similar feedback remarking that the "highlighting was good since most people just try to summarize the whole code at once." However, a few participants found the stepper to progress the lesson too slowly. P17 read the entire line of code on the 'Selecting rows and sorting' lesson and said that they "didn't understand drop=FALSE, hasn't been mentioned" before Transfer Tutor had the opportunity to highlight it. Reluctance of accepting facts without execution or examples: Participants were reluctant to accept certain facts without confirming for themselves through code execution, or without seeing additional examples. One participant was "not too sold on the explanation" [P2] for why parentheses aren't required around conditions when subsetting data frames. Another participant expressed doubt and confusion when reading about an alternate [ syntax that doesn't require specifying both rows and columns: "Ok but then it says you can use an alternate syntax without using the comma" [P20]. Regarding the code output, one participant suggested that "it would've been more useful if I could change [the code] live and observe the output" [P18]. There were a few participants who wanted more examples. For example, P17 was unclear on how to use the [[ syntax in R and suggested that "maybe if there was a specific example here for the [[ that would help". Information overload: Although several participants reported that Transfer Tutor is "interactive and easy to use" [P13], there were a few who thought that there was "information overload in the textual explanations" [P1]. Some syntax elements had lengthy explanations and one participant felt that "sometimes too many new things were introduced at once" [P18] and P5 expressed that "complex language is used" to describe a syntax or concept in R. Participants also expressed that they wanted "more visual examples" [P5]. C. RQ3: How satisfied are programmers with learning R when using Transfer Tutor? Table II shows the distribution of responses for each statement from the user satisfaction survey, with each statement targeting a feature of Transfer Tutor. Overall, participants indicated that features of Transfer Tutor were useful in learning R. However, a few participants strongly disagreed about the usefulness of explanations relating back to R, and the output boxes. The free-form responses from participants offers additional insight into the Likert responses which will be discussed next. The highlighting feature had no negative ratings and all participants indicated that it was useful to them in some way. One participant thought that "the highlighting drew [their] attention" [P2] while another commented that "it showed the differences visually and addressed almost all my queries" [P1]. The stepper received some neutral (3) ratings and one participant disagreed on its usefulness. Nevertheless, most participants did find the stepper useful and expressed that they "like how it focuses on things part by part" [P20]. Participants generally found the explanations relating R to Python was useful in learning R. One of the participants "liked the attempt to introduce R syntax based on Python syntax" [P18] and P14 thought that "comparing it with Python makes it even more easy to understand R language". All participants thought this feature was useful except for one. This participant did not provide any feedback for why. The 'new facts' explanations also had no negative ratings and was useful to all participants. Although participants didn't speak explicitly about the feature, P8 expressed that there was "detailed explanation for each element" and P16 said that "Every aspect of the syntax changes has been explained very well". Most participants also found 'gotchas' to be useful. P7 for example said that "Gotchas! were interesting to learn and to avoid errors while coding." For the explanation box, some participants suggested that this affordance would need to "reduce the need for scrolling and (sadly) reading" [P2]. Still other participants wanted deeper explanations for some concepts, perhaps with "links to more detailed explanations" [P12]. For the output boxes, participants who disagreed with its usefulness suggested that the output boxes would be more useful if the output code be dynamically adjusted by changing the code [P6, P9, P12], and P17 suggested that the output boxes were "a little difficult to read" because of the small font. VI. LIMITATIONS A. Construct Validity We used pre-test and post-test questions as a proxy to assess the participants' understanding of R concepts as covered by Transfer Tutor. Because of time constraints in the study, we could only ask a limited number of questions. Consequently, these questions are only approximations of participants' understanding. For instance, Question 7 illustrates several reasons why questions may be problematic for programmers. First, the question may be confusingly-worded, because of the use of except in the question statement. Second, the response may be correct, but incomplete-due to our scoring strategy, responses must be completely correct to receive credit. Third, questions are only approximations of the participants' understanding. A comparative study is necessary to properly measure learning from using Transfer Tutor to other traditional methods of learning languages by measuring performance on programming tasks. B. Internal Validity Participants in the study overwhelmingly found the features of Transfer Tutor to be positive (Section V-C). It's possible, however, that this positivity is artificially high due to social desirability bias [23]-a phenomenon in which participants tend to respond more positively in the presence of the experimenters than they would otherwise. Given the novelty of Transfer Tutor, it is likely that they assumed that the investigator was also the developer of the tool. Thus, we should be conservative about how we interpret user satisfaction with Transfer Tutor and its features. A second threat to internal validity is that we expected Transfer Tutor to be used by experts in Python, and novices in R. Although all of our participants have limited knowledge with R, very few participants were also experts with Python or the Pandas library (Section IV). On one hand, this could suggest that learning transfer would be even more effective with expert Python/Pandas participants. On the other hand, this could also suggest that there is a confounding factor that explains the increase in learning that is not directly due to the tool. For instance, it may be that explanations in general are useful to participants, whether or not they are phrased in terms of transfer [24], [25]. C. External Validity We recruited graduate students with varying knowledge of Python and R, so the results of the study may not generalize to other populations, such as industry experts. The choice of Python and R, despite some notable differences, are both primarily intended to be used as scripting languages. How effective language transfer can be when language differences are more drastic is still an open question; for example, consider if we had instead used R and Rust-languages with very different memory models and programming idioms. VII. DESIGN IMPLICATIONS This section presents the design implications of the results and future applications for learning transfer. A. Affordances for supporting learning transfer Stepping through each line incrementally with corresponding highlighting updates allows programmers to focus on the relevant syntax elements for source code. This helps novice programmers pinpoint misconceptions that could be easily overlooked otherwise, but prevents more advanced programmers from easily skipping explanations from Transfer Tutor. Despite the usefulness of always-on visualizations in nice environments [26], [27], an alternative implementation approach to always-on may be to interactively allow the programmer to activate explanations on-demand. We found that live code execution is an an important factor for programmers as they can test new syntax rules or confirm a concept. We envision future iterations of Transfer Tutor that could allow code execution and adapt explanations in the context of the programmers' custom code. Reducing the amount of text and allowing live code execution were two improvements suggested by the participants. This suggests that Transfer Tutor needs to reduce information overload and balance the volume of explanation against the amount of code to be explained. One solution is to externalize additional explanation to documentation outside of Transfer Tutor, such as web resources. Breaking up lessons into smaller segments could also reduce the amount of reading required. B. Expert learning can benefit from learning transfer To prevent negative consequences for experienced learners, we intentionally mitigated the expertise reversal effect [4] by presenting explanations in terms of language transfer-in the context of a language that the programmer is already an expert at. Participants in our study tried to guess positive transfers on their own, which could lead to negative transfers from their previous languages. This cognitive strategy is better supported by a tool like Transfer Tutor as it guides programmers on the correct positive transfers and warns them about potential negative transfers. We think that tools such as ours serves as a type of intervention design: like training wheels, programmers new to the language can use our tool to familiarize themselves with the language. As they become experts, they would reduce and eventually eliminate use of Transfer Tutor. C. Learning transfer within programming languages Our study explored learning transfer between programming languages, but learning transfer issues can be found within programming languages as well, due to different programming idioms within the same language. For example, in the R community, a collection of packages called tidyverse encourage an opinionated programming style that focuses on consistency and readability, through the use of a fluent design pattern. In contrast to 'base' R-which is usually structured as a sequence of data transformation instructions on data frames-the fluent pattern uses 'verbs' that pipe together to modify data frames. D. Applications of learning transfer beyond tutorials Learning transfer could be applied in other contexts, such as within code review tools, and within integrated development environments such as Eclipse and Visual Studio. For example, consider a scenario in which a software engineer needs to translate code from one programming language to another: this activity is an instance in which learning transfer is required. Tools could assist programmers by providing explanations in terms of their expert language through existing affordances in development environments. Learning transfer tools can be beneficial even when the language conversion is automatic. For example, SMOP (Small Matlab and Octave to Python compiler) is one example of a transpiler-the system takes in Matlab/Octave code and outputs Python code. 3 The generated code could embed explanations of the translation that took place so that programmers can better understand why the translation occurred the way that it did. Another potential avenue for supporting learning transfer with tools can be found in the domain of documentation generation for programming languages. Since static documentation can't support all types of readers, authors make deliberate design choices to focus their documentation for certain audiences. For example, the canonical Rust book 4 makes the assumption that programmers new to Rust have experience with some other language-though it tries not to assume any particular one. Automatically generating documentation for programmers tailored for prior expertise in a different language might be an interesting application for language transfer. VIII. RELATED WORK There are many studies on transfer between tools [28], [29], [30], [31] but fewer studies examining transfer in programming. Transfer of declarative knowledge between programming languages has been studied by Harvey and Anderson [20], which showed strong effects of transfer between Lisp and Prolog. Scholtz and Wiedenbeck [11] conducted a thinkaloud protocol where programmers who were experienced in Pascal or C tried implementing code in Icon. They demonstrated that programmers could suffer from negative transfer of programming languages. Wu and Anderson conducted a similar study on problem-solving transfer, where programmers who had experience in Lisp, Pascal and Prolog wrote solutions to programming problems [12]. The authors found positive transfer between the languages which could improve programmer productivity. Bower [32] used a new teaching approach called Continual And Explicit Comparison (CAEC) to teach Java to students who have knowledge of C++. They found that students benefited from the continual comparison of C++ concepts to Java. However, none of these studies investigated tool support. Fix and Wiedenbeck [14] developed and evaluated a tool called ADAPT that teaches Ada to programmers who know Pascal and C. Their tool helps programmers avoid high level plans with negative transfer from Pascal and C, but is targeted at the planning level. Our tool teaches programmers about negative transfers from Python, emphasizing both syntax and semantic issues by highlighting differences between the syntax elements in the code snippets of the two languages. Transfer Tutor also covers pitfalls in R that doesn't relate to Python. We leverage existing techniques used in two interactive learning tools for programming, namely Python Tutor [33] and Tutorons [34]. Python Tutor is an interactive tool for computer science education which allows the visualization and execution of Python code. We borrowed the idea of Python Tutor's ability to step through the code and pointing to the current line the program is executing to help the programmer stay focused. Head et al. designed a new technique of generating explanations or Tutorons that helps programmers learn about code snippets on the web browser by providing pop-ups with brief explanations of user highlighted code [34]. Although our tool does not automatically generate explanations for highlighted code, it uses the idea of providing details about syntax elements as the programmer steps through the syntax elements which are already highlighted for them. IX. CONCLUSION In this paper, we evaluated the effectiveness of using learning transfer through a training tool for expert Python developers who are new to R. We found that participants were able to learn basic concepts in R and they found Transfer Tutor to be useful in learning R across a number of affordances. Observations made in the think-aloud study revealed that Transfer Tutor highlighted facts that were easy to miss or misunderstand and participants were reluctant to accept certain facts without code execution. The results of this study suggest opportunities for incorporating learning transfer feedback in programming environments. code for reading data, filtering for positive scores and selecting 5 rows. (b) The equivalent code in R. one of these correctly subsets the first five rows and the first column of a data frame df and returns the result as a data framenumber of participants who answered correctly in pre-test.2 Difference in the number of participants who answered correctly in pre-test and post-test. a statement or participant observation. We labeled each of the cards as either being an indicator of positive transfer, negative transfer, or non-transfer. To do so, we used the following rubric to guide the labeling process:1) Statements should not be labeled if it includes verbatim or very close reading of the text provided by Transfer Tutor. 2) The statement can be labeled as positive if it demonstrates the participant learning a syntax or concept from Python that can be used in R. 3) The statement can be labeled as negative if it demonstrates the participant learning a syntax or concept in R that is different from Python or breaks their expectation. 4) The statement can be labeled as a non-transfer if it demonstrates the participant encountering a new fact in R for which there is no connection to Python. responses: Strongly Disagree (SD), Disagree (D), Neutral (N), Agree (A), Strongly Agree (SA). 2 Net stacked distribution removes the Neutral option and shows the skew between positive (more useful) and negative (less useful) responses. Strongly Disagree, Disagree, Agree; Strongly Agree. of the comma blank?" Another participant wondered about the meaning of a negative sign in front of R's order function by expressing they "don't get why the minus sign is there." [P8]. TABLE I IPRE-TEST AND POST-TEST QUESTIONSID Question Text Tot. 1 ∆ 2 1 Select all the valid ways of assigning a 1 to a variable 'x' in R. 0 18 2 Select all the valid vector types that can be used to subset a data frame. 13 2 3 How would one check if 'x' is the value NA? 0 20 4 TABLE II FOLLOW II-UP SURVEY RESPONSES Likert Resp. Counts 1 % Agree SD D N A SA Distribution 2 The code output box helped me understand new concepts in R.50% 50% 0% The highlighting feature was useful in learning about R. 95% 0 0 1 5 14 Stepping through the syntax was useful in learning about R. 79% 0 1 3 2 14 The explanations that related R back to another language like Python was useful. 89% 1 0 1 6 12 The 'new facts' in the information box helped me learn new syntax and concepts. 95% 0 0 1 6 13 The 'gotchas' in the information box were helpful in learning about potential pitfalls. 93% 0 2 0 6 12 The code output box helped me understand new syntax in R. 79% 3 0 1 8 8 http://adv-r.had.co.nz, chapters "Data Structures" and "Subsetting." 2 https://github.com/alt-code/Research/tree/master/TransferTutor https://github.com/victorlei/smop 4 https://doc.rust-lang.org/book/second-edition/ ACKNOWLEDGEMENTSThis material is based in part upon work supported by the National Science Foundation under Grant Nos. 1559593 and 1755762. The ramp-up problem in software projects: A case study of how software immigrants naturalize. S E Sim, R C Holt, International Conference on Software Engineering (ICSE). S. E. Sim and R. C. Holt, "The ramp-up problem in software projects: A case study of how software immigrants naturalize," in International Conference on Software Engineering (ICSE), 1998, pp. 361-270. Programming, problem solving, and self-awareness: Effects of explicit guidance. D Loksa, A J Ko, W Jernigan, A Oleson, C J Mendez, M M Burnett, Human Factors in Computing Systems (CHI). D. Loksa, A. J. Ko, W. Jernigan, A. Oleson, C. J. Mendez, and M. M. Burnett, "Programming, problem solving, and self-awareness: Effects of explicit guidance," in Human Factors in Computing Systems (CHI), 2016, pp. 1449-1461. Beyond program understanding: A look at programming expertise in industry. L M Berlin, Empirical Studies of Programmers (ESP). 93744L. M. Berlin, "Beyond program understanding: A look at programming expertise in industry," Empirical Studies of Programmers (ESP), vol. 93, no. 744, pp. 6-25, 1993. The expertise reversal effect. S Kalyuga, P Ayres, P Chandler, J Sweller, Educational Psychologist. 381S. Kalyuga, P. Ayres, P. Chandler, and J. Sweller, "The expertise reversal effect," Educational Psychologist, vol. 38, no. 1, pp. 23-31, 2003. The R Inferno. P Burns, P. Burns. (2012) The R Inferno. [Online]. Available: http://www. burns-stat.com/documents/books/the-r-inferno/ aRrgh: a newcomer's (angry) guide to R. T Smith, K Ushey, T. Smith and K. Ushey, "aRrgh: a newcomer's (angry) guide to R," http://arrgh.tim-smith.us. Transfer of learning. D N Perkins, G Salomon, P Press, International Encyclopedia of Education. Pergamon PressD. N. Perkins, G. Salomon, and P. Press, "Transfer of learning," in International Encyclopedia of Education. Pergamon Press, 1992. Learning strategies and transfer in the domain of programming. P Pirolli, M Recker, Cognition and Instruction. 123P. Pirolli and M. Recker, "Learning strategies and transfer in the domain of programming," Cognition and Instruction, vol. 12, no. 3, pp. 235-275, 1994. Transfer of programming skills in novice LISP learners. C M Kessler, Carnegie Mellon UniversityPh.D. dissertationC. M. Kessler, "Transfer of programming skills in novice LISP learners," Ph.D. dissertation, Carnegie Mellon University, 1988. Transfer of training between cognitive subskills: Is knowledge use specific?. N Pennington, R Nicolich, J Rahm, Cognitive Psychology. 282N. Pennington, R. Nicolich, and J. Rahm, "Transfer of training between cognitive subskills: Is knowledge use specific?" Cognitive Psychology, vol. 28, no. 2, pp. 175-224, 1995. Learning second and subsequent programming languages: A problem of transfer. J Scholtz, S Wiedenbeck, International Journal of Human-Computer Interaction. 21J. Scholtz and S. Wiedenbeck, "Learning second and subsequent pro- gramming languages: A problem of transfer," International Journal of Human-Computer Interaction, vol. 2, no. 1, pp. 51-72, 1990. Problem-solving transfer among programming languages. Q Wu, J R Anderson, Carnegie Mellon University, Tech. Rep.Q. Wu and J. R. Anderson, "Problem-solving transfer among program- ming languages," Carnegie Mellon University, Tech. Rep., 1990. Identifying student misconceptions of programming. L C Kaczmarczyk, E R Petrick, J P East, G L Herman, Computer Science Education (SIGCSE). L. C. Kaczmarczyk, E. R. Petrick, J. P. East, and G. L. Herman, "Identifying student misconceptions of programming," in Computer Science Education (SIGCSE), 2010, pp. 107-111. An intelligent tool to aid students in learning second and subsequent programming languages. V Fix, S Wiedenbeck, Computers & Education. 272V. Fix and S. Wiedenbeck, "An intelligent tool to aid students in learning second and subsequent programming languages," Computers & Education, vol. 27, no. 2, pp. 71 -83, 1996. Stack Overflow. "Stack Overflow," https://stackoverflow.com. The Cambridge Handbook of the Learning Sciences. R K Sawyer, Cambridge University PressR. K. Sawyer, The Cambridge Handbook of the Learning Sciences. Cambridge University Press, 2005. Learnable programming. B Victor, B. Victor. (2012) Learnable programming. [Online]. Available: http://worrydream.com/LearnableProgramming/ Transfer of situated learning. J G Greeno, J L Moore, D R Smith, Transfer on trial: Intelligence, cognition, and instruction. Westport, CT, USAblex PublishingJ. G. Greeno, J. L. Moore, and D. R. Smith, "Transfer of situated learning." in Transfer on trial: Intelligence, cognition, and instruction. Westport, CT, US: Ablex Publishing, 1993, pp. 99-167. D H Schunk, Learning Theories: An Educational Perspective. 6th ed. PearsonD. H. Schunk, Learning Theories: An Educational Perspective, 6th ed. Pearson, 2012. Transfer of declarative knowledge in complex information-processing domains. L Harvey, J Anderson, Human-Computer Interaction. 111L. Harvey and J. Anderson, "Transfer of declarative knowledge in com- plex information-processing domains," Human-Computer Interaction, vol. 11, no. 1, pp. 69-96, 1996. Python for R Users: A Data Science Approach. A Ohri, John Wiley & SonsA. Ohri, Python for R Users: A Data Science Approach. John Wiley & Sons, 2017. D Spencer, Card Sorting: Designing Usable Categories. Rosenfeld. D. Spencer, Card Sorting: Designing Usable Categories. Rosenfeld, 2009. Yours is better!': Participant response bias in HCI. N Dell, V Vaidyanathan, I Medhi, E Cutrell, W Thies, Human Factors in Computing Systems (CHI). N. Dell, V. Vaidyanathan, I. Medhi, E. Cutrell, and W. Thies, "'Yours is better!': Participant response bias in HCI," in Human Factors in Computing Systems (CHI), 2012, pp. 1321-1330. Too much, too little, or just right? Ways explanations impact end users' mental models. T Kulesza, S Stumpf, M Burnett, S Yang, I Kwan, W.-K Wong, Visual Languages and Human-Centric Computing (VL/HCC). T. Kulesza, S. Stumpf, M. Burnett, S. Yang, I. Kwan, and W.-K. Wong, "Too much, too little, or just right? Ways explanations impact end users' mental models," in Visual Languages and Human-Centric Computing (VL/HCC), 2013, pp. 3-10. Are explanations always important?. A Bunt, M Lount, C Lauzon, in Intelligent User Interfaces (IUI). A. Bunt, M. Lount, and C. Lauzon, "Are explanations always impor- tant?" in Intelligent User Interfaces (IUI), 2012, pp. 169-178. Omnicode: A novice-oriented live programming environment with always-on run-time value visualizations. H Kang, P J Guo, User Interface Software and Technology (UIST. H. Kang and P. J. Guo, "Omnicode: A novice-oriented live programming environment with always-on run-time value visualizations," in User Interface Software and Technology (UIST), 2017, pp. 737-745. Augmenting code with in situ visualizations to aid program understanding. J Hoffswell, A Satyanarayan, J Heer, Human Factors in Computing Systems (CHI). 53212J. Hoffswell, A. Satyanarayan, and J. Heer, "Augmenting code with in situ visualizations to aid program understanding," in Human Factors in Computing Systems (CHI), 2018, pp. 532:1-532:12. A quantitative theory of human-computer interaction. P G Polson, Interfacing Thought: Cognitive Aspects of Human-Computer Interaction. P. G. Polson, "A quantitative theory of human-computer interaction," in Interfacing Thought: Cognitive Aspects of Human-Computer Interaction, 1987, pp. 184-235. Transfer between text editors. P G Polson, S Bovair, D Kieras, Human Factors in Computing Systems and Graphics Interface (CHI/GI). 17P. G. Polson, S. Bovair, and D. Kieras, "Transfer between text editors," in Human Factors in Computing Systems and Graphics Interface (CHI/GI), vol. 17, no. SI, 1986, pp. 27-32. A test of a common elements theory of transfer. P G Polson, E Muncher, G Engelbeck, Human Factors in Computing Systems (CHI). 17P. G. Polson, E. Muncher, and G. Engelbeck, "A test of a common elements theory of transfer," in Human Factors in Computing Systems (CHI), vol. 17, no. 4, 1986, pp. 78-83. A keystroke analysis of learning and transfer in text editing. M K Singley, J R Anderson, Human-Computer Interaction. 33M. K. Singley and J. R. Anderson, "A keystroke analysis of learning and transfer in text editing," Human-Computer Interaction, vol. 3, no. 3, pp. 223-274, 1987. Continual and explicit comparison to promote proactive facilitation during second computer language learning. M Bower, A Mciver, Innovation and Technology in Computer Science Education (ITiCSE). M. Bower and A. McIver, "Continual and explicit comparison to pro- mote proactive facilitation during second computer language learning," in Innovation and Technology in Computer Science Education (ITiCSE), 2011, pp. 218-222. Online Python Tutor: Embeddable web-based program visualization for cs education. P J Guo, Computer Science Education (SIGCSE). P. J. Guo, "Online Python Tutor: Embeddable web-based program visu- alization for cs education," in Computer Science Education (SIGCSE), 2013, pp. 579-584. Tutorons: Generating context-relevant, on-demand explanations and demonstrations of online code. A Head, C Appachu, M A Hearst, B Hartmann, Visual Languages and Human-Centric Computing (VL/HCC). A. Head, C. Appachu, M. A. Hearst, and B. Hartmann, "Tutorons: Gen- erating context-relevant, on-demand explanations and demonstrations of online code," in Visual Languages and Human-Centric Computing (VL/HCC), 2015, pp. 3-12.
[ "https://github.com/alt-code/Research/tree/master/TransferTutor", "https://github.com/victorlei/smop" ]
[ "A Pattern Logic for Automata with Outputs ⋆", "A Pattern Logic for Automata with Outputs ⋆" ]
[ "Emmanuel Filiot \nUniversité libre de Bruxelles\n\n", "Nicolas Mazzocchi \nUniversité libre de Bruxelles\n\n", "Jean-François Raskin \nUniversité libre de Bruxelles\n\n" ]
[ "Université libre de Bruxelles\n", "Université libre de Bruxelles\n", "Université libre de Bruxelles\n" ]
[]
We introduce a logic to express structural properties of automata with string inputs and, possibly, outputs in some monoid. In this logic, the set of predicates talking about the output values is parametric, and we provide sufficient conditions on the predicates under which the model-checking problem is decidable. We then consider three particular automata models (finite automata, transducers and automata weighted by integers -sum-automata -) and instantiate the generic logic for each of them. We give tight complexity results for the three logics and the model-checking problem, depending on whether the formula is fixed or not. We study the expressiveness of our logics by expressing classical structural patterns characterising for instance finite ambiguity and polynomial ambiguity in the case of finite automata, determinisability and finite-valuedness in the case of transducers and sum-automata. Consequently to our complexity results, we directly obtain that these classical properties can be decided in PTIME.IntroductionMotivations An important aspect of automata theory is the definition of automata subclasses with particular properties, of algorithmic interest for instance. As an example, the inclusion problem for non-deterministic finite automata is PSPACE-C but becomes PTIME if the automata are k-ambiguous for a fixed k [21].By automata theory, we mean automata in the general sense of finite state machines processing finite words. This includes what we call automata with outputs, which may also produce output values in a fixed monoid M = (D, ⊕, 0). In such an automaton, the transitions are extended with an (output) value in D, and the value of an accepting path is the sum (for ⊕) of all the values occurring along its transitions. Automata over finite words in Λ * and with outputs in M define subsets of Λ * × D as follows: to any input word w ∈ Λ * , we associate the set of values of all the accepting paths on w. For example, transducers are automata with outputs in a free monoid: they process input words and produce output words and therefore define binary relations of finite words[15].The many decidability properties of finite automata do not carry over to transducers, and many restrictions have been defined in the literature to recover decidability, or just to define subclasses relevant to particular applications. The inclusion problem for transducer is undecidable [13], but decidable for finitevalued transducers[23]. Another well-known subclass is that of the determinisable transducers [5], defining sequential functions of words. Finite-valuedness and determinisability are two properties decidable in PTIME, i.e., it is decidable in PTIME, given a transducer, whether it is finite-valued (resp. determinisable). As a second example of automata with outputs, we also consider sum-automata, i.e. automata with outputs in (Z, +, 0), which defines relations from words to Z. Properties such as functionality, determinisability, and k-valuedness (for a fixed k) are decidable in PTIME for sum-automata[11,10].In our experience, it is quite often the case that deciding a subclass goes in two steps: (1) define a characterisation of the subclass through a "simple" pattern, (2) show how to decide the existence of a such a pattern. For instance, the determinisable transducers have been characterised via the so called twinning property[6,24,4], which, said briefly, asks that the output words produced by any two different paths on input words of the form uv n cannot differ unboundedly when n grows, with a suitable definition of "differ". ⋆ We warmly thank the anonymous reviewers for their helpful comments, and Ismaël Jecker for spotting a bug in a preliminary version of the paper. E.
10.1142/s0129054120410038
[ "https://arxiv.org/pdf/1810.03515v1.pdf" ]
52,151,179
1810.03515
c467bc34b6f755d435fd7d1fc5008c8b96bc0d0d
A Pattern Logic for Automata with Outputs ⋆ 8 Oct 2018 Emmanuel Filiot Université libre de Bruxelles Nicolas Mazzocchi Université libre de Bruxelles Jean-François Raskin Université libre de Bruxelles A Pattern Logic for Automata with Outputs ⋆ 8 Oct 2018 We introduce a logic to express structural properties of automata with string inputs and, possibly, outputs in some monoid. In this logic, the set of predicates talking about the output values is parametric, and we provide sufficient conditions on the predicates under which the model-checking problem is decidable. We then consider three particular automata models (finite automata, transducers and automata weighted by integers -sum-automata -) and instantiate the generic logic for each of them. We give tight complexity results for the three logics and the model-checking problem, depending on whether the formula is fixed or not. We study the expressiveness of our logics by expressing classical structural patterns characterising for instance finite ambiguity and polynomial ambiguity in the case of finite automata, determinisability and finite-valuedness in the case of transducers and sum-automata. Consequently to our complexity results, we directly obtain that these classical properties can be decided in PTIME.IntroductionMotivations An important aspect of automata theory is the definition of automata subclasses with particular properties, of algorithmic interest for instance. As an example, the inclusion problem for non-deterministic finite automata is PSPACE-C but becomes PTIME if the automata are k-ambiguous for a fixed k [21].By automata theory, we mean automata in the general sense of finite state machines processing finite words. This includes what we call automata with outputs, which may also produce output values in a fixed monoid M = (D, ⊕, 0). In such an automaton, the transitions are extended with an (output) value in D, and the value of an accepting path is the sum (for ⊕) of all the values occurring along its transitions. Automata over finite words in Λ * and with outputs in M define subsets of Λ * × D as follows: to any input word w ∈ Λ * , we associate the set of values of all the accepting paths on w. For example, transducers are automata with outputs in a free monoid: they process input words and produce output words and therefore define binary relations of finite words[15].The many decidability properties of finite automata do not carry over to transducers, and many restrictions have been defined in the literature to recover decidability, or just to define subclasses relevant to particular applications. The inclusion problem for transducer is undecidable [13], but decidable for finitevalued transducers[23]. Another well-known subclass is that of the determinisable transducers [5], defining sequential functions of words. Finite-valuedness and determinisability are two properties decidable in PTIME, i.e., it is decidable in PTIME, given a transducer, whether it is finite-valued (resp. determinisable). As a second example of automata with outputs, we also consider sum-automata, i.e. automata with outputs in (Z, +, 0), which defines relations from words to Z. Properties such as functionality, determinisability, and k-valuedness (for a fixed k) are decidable in PTIME for sum-automata[11,10].In our experience, it is quite often the case that deciding a subclass goes in two steps: (1) define a characterisation of the subclass through a "simple" pattern, (2) show how to decide the existence of a such a pattern. For instance, the determinisable transducers have been characterised via the so called twinning property[6,24,4], which, said briefly, asks that the output words produced by any two different paths on input words of the form uv n cannot differ unboundedly when n grows, with a suitable definition of "differ". ⋆ We warmly thank the anonymous reviewers for their helpful comments, and Ismaël Jecker for spotting a bug in a preliminary version of the paper. E. Quite often, the most difficult part is step (1) and step (2) is technical but less difficult to achieve, as long as we do not seek for optimal complexity bounds (by this we mean that PTIME is good enough, and obtaining the best polynomial degree is not the objective). We even noticed that in transducer theory, even though step (2) share common techniques (reduction to emptiness of reversal-bounded counter machines for instance), the algorithms are often ad-hoc to the particular subclass considered. Here is a non-exhaustive list of subclasses of transducers which are decidable in PTIME: determinisable transducers [6,24,5,4,1,7], functional transducers [5,4], k-sequential transducers (for a fixed k) [8], multi-sequential transducers [16,7], k-valued transducers (for a fixed k) [14], finite-valued transducers [18,23]. Our goal in this paper is to define a common tool for step (2), i.e., define a generic way of deciding a subclass characterised through a structural pattern. More precisely, we want to define logics, tailored to particular monoids M, able to express properties of automata with outputs in M, such that model-checking these properties on given automata can be done in PTIME. Contributions We define a general logic, denoted PL[Ø] for "pattern logic", to express properties of automata with outputs in a fixed monoid M = (D, ⊕, 0). This logic is parameterised by a set of predicates Ø interpreted on D. We first give sufficient conditions under which the problem of model-checking an automaton with outputs in M against a formula in this logic is decidable. Briefly, these conditions require the existence of a machine model accepting tuples of runs which satisfy the atomic predicates of the logic, is closed under union and intersection, and has decidable emptiness problem. Then, we study three particular classes of automata with outputs: finite automata (which can be seen as automata with outputs in a trivial monoid with a single element), transducers (automata with outputs in a free monoid), and sum-automata (automata with outputs in (Z, +, 0)). For each of them, we define particular logics, called PL NFA , PL Trans and PL Sum to express properties of automata with outputs in these particular monoids. Formulas in these logics have the following form: where the π i are path variables, the p i , q i are state variables, the u i are (input) word variables and the v i are output value variables (interpreted in D). The subformula C is a quantifier free Boolean combinations of constraints talking about states, paths, input words and output values. Such a formula expresses the fact that there exists a path π 1 from some state p 1 to some state q 1 , over some input word u 1 , producing some value v 1 , some path π 2 etc. such that they all satisfy the constraints in C. In the three logics, paths can be tested for equality. Input words can be compared with the prefix relation, w.r.t. their length, and their membership to a regular language be tested. States can be compared for equality, and it can be expressed whether they are initial or final. The predicates we take for the output values depends on the monoids. For transducers, output words can be compared with the non-prefix relation (and by derivation =), a predicate which cannot be negated (otherwise model-checking becomes undecidable), and can also be compared with respect to their length, and membership to a regular language can be tested. For sum-automata, the output values can be compared with < (and by derivation =, =, ≤). As an example, a transducer (resp. sum-automaton) is not (n − 1)valued iff it satisfies the following PL Trans -formula (resp. PL Sum -formula): ∃π 1 : p 1 u|v1 − −− → q 1 , . . . , ∃π n : p n u|vn − −− → q n , n i=1 init(p i ) ∧ final(q i ) ∧ 1≤i<j≤n v i = v j . For the three logics, we show that deciding whether a given automaton satisfies a given formula is PSPACE-C. When the formula is fixed, the model-checking problem becomes NLOGSPACE-C for PL NFA and PL Trans , and NP-C for PL Sum . If output values can only be compared via disequality = (which cannot be negated), then PL Sum admits PTIME model-checking. We show that many of the properties from the literature, including all the properties mentioned before, can be expressed in these logics. As a consequence, we show that most of the PTIME upper-bounds obtained for deciding subclasses of finite automata in [25,2], of transducers in [6,14,24,22,16,7,5,18,8] and sum-automata in [11,10,8,3], can be directly obtained by expressing in our logics the structural patterns given in these papers, which characterise these subclasses. Related works In addition to the results already mentioned, we point out that the syntax of our logic is close to a logic, defined in [9] by Figueira and Libkin, to express path queries in graph databases (finite graphs with edges labelled by a symbol). In this work, there is no disjunction nor negation, and no distinction between input and output values. By making such a distinction, and by adding negation and disjunction, we were able to tailor our logics to particular automata models and add enough power to be able to directly express classical structural automata properties. Finite Automata with Outputs In this section, we define a general model of finite automata defining functions from the free monoid Λ * (where Λ is a finite input alphabet) to any monoids M = (D, ⊕, 0). More precisely, they are parametrised by a monoid of output values, read input words over some alphabet and output elements of the output monoid, obtained by summing the output values met along accepting paths. Formally, a monoid M is a tuple (D, ⊕ M , 0 M ) where D is a set of elements which we call here values or sometimes outputs, ⊕ M is an associative binary operation on D, for which 0 M ∈ D is neutral. Monoids of interest in this paper are the free monoid (Λ * , ·, ε) for some finite alphabet of symbols Λ (where · denotes the concatenation), and the monoid (Z, +, 0). We also let Λ ε = Λ ∪ {ε}. For w ∈ Λ * , |w| denotes its length, in particular |ε| = 0. The set of positions of w is {1, . . . , |w|} (and empty if w = ǫ). We let w[i] be the ith symbol of w. Given w 1 , w 2 , we write w 1 ⊑ w 2 whenever w 1 is a prefix of w 2 . All over this paper, the input alphabet is denoted by the letter Λ. We write #(A) to refer to the number of states of A. A path in A is a sequence π = q 0 a 1 d 1 q 1 . . . a n d n q n ∈ Q(Λ ε DQ) * , for n ≥ 0, such that for all 1 ≤ i ≤ n we have (q i−1 , a i , q i ) ∈ ∆ and γ(q i−1 , a i , q i ) = d i . The input of π is defined as the word in(π) = a 1 . . . a n (and ε if π ∈ Q), the output of π as the element out(π) = d 1 ⊕ M · · · ⊕ M d n (and 0 M if π ∈ Q), and the size of π as |π| = n. We may write π : q 0 in(π)|out(π) − −−−−−− → q n to denote that π is a path from q 0 to q n on input in(π) and output out(π). For convenience we write π ⊳ , π ⊲ to denote respectively the starting state q 0 and the ending state q n of the path π. The set of all paths of A is written Paths(A). A path π : q 0 u|v − − → q n is initial if q 0 ∈ I, final if q n ∈ F and accepting if it is both initial and final. The set of accepting paths of A is denoted by Paths acc (A). The input/output relation (or just relation) defined by A is the set of pairs R(A) ⊆ Λ * × D defined by R(A) = {(u, v) | ∃π ∈ Paths acc (A) · in(π) = u ∧ out(π) = v} Finite automata, transducers and sum-automata In this paper, we consider three instances of automata with outputs. First, finite automata (over Λ), are seen as automata with outputs in a trivial monoid (and which is therefore ignored). Transducers are automata with outputs in the free monoid Γ * . They define relations from Λ * to Γ * . Finally, sum-automata are automata with outputs in the monoid (Z, +, 0). A Pattern Logic for Automata with Outputs In this section, we introduce a generic pattern logic. It is built over four kind of variables, namely path, state, input and output variables. More precisely, we let X P = {π, π 1 , . . . }, X Q = {q, q 1 , p . . . , }, X I = {u, u 1 , . . . } and X O = {v, v 1 , . . . } be disjoint and countable sets of resp. path, state, input and output variables. We define T erms(X O , ⊕, 0) as the set of terms built over variables of X O , a binary function symbol ⊕ (representing the monoid operation) and constant symbol 0 (neutral element). The logic syntax is parametrised by a set of output predicates Ø. Output predicates of arity 0 are called constant symbols, and we denote by Ø| n the predicates of arity n. Predicates talking about states, paths and input words are however fixed in the logic. Definition 2. A pattern formula ϕ over a set of output predicates Ø is of the form ϕ = ∃π 1 : p 1 u1|v1 −−−→ q 1 , . . . , ∃π n : p n un|vn − −−− → q n , C where for all 1 ≤ i ≤ n, π i ∈ X P and they are all pairwise different, p i , q i ∈ X Q , u i ∈ X I , v i ∈ X O , and C is a Boolean combination of atoms amongst Input constraints : u ⊑ u ′ | u ∈ L | |u| ≤ |u ′ | u, u ′ ∈ X I Output constraints : p(t 1 , . . . , t n ) p ∈ Ø| n , t i ∈ T erms(X O , ⊕, 0) State constraints : init(q) | final(q) | q = q ′ q, q ′ ∈ X Q Path constraints : π = π ′ π, π ′ ∈ X P where L is a regular language of words over Λ (assumed to be represented as an NFA). The sequence of existential quantifiers before C in ϕ is called the prefix of ϕ. We denote by PL(Ø) the set of pattern formulas over Ø, and by PL + (Ø) the fragment where output predicates does not occur under an odd number of negations. The size of a formula is the number of its symbols plus the number of states of all NFA representing the membership constraints. We denote by Var(ϕ) the variables occurring in any pattern formula ϕ, and by Var P (ϕ) (resp. Var Q (ϕ), Var I (ϕ), Var O (ϕ)) its restriction to path (resp. state, input, output) variables. We finally let (u = u ′ ) def = u ⊑ u ′ ∧ u ′ ⊑ u, (|u| = |u ′ |) def = (|u| ≤ |u ′ |) ∧ (|u ′ | ≤ |u|), (|u| < |u ′ |) def = ¬(|u ′ | ≤ |u|).u ⊑ u ′ if ν(u) is a prefix of ν(u ′ ), u ∈ L if ν(u) ∈ L, |u| ≤ |u ′ | if |ν(u)| ≤ |ν(u ′ )|. Given a predicate p ∈ Ø of arity α(p), an atom p(t 1 , . . . , t α(p) ) is satisfied by ν if (t ν,M 1 , . . . , t ν,M α(p) ) ∈ p M . Finally, ν satisfies init(q) (resp. final(q)) if ν(q) is initial (resp. ν(q) is final). The satisfiability relation is naturally extended to Boolean combinations of atoms. Finally, assume that ϕ is of the form ∃π 1 : p 1 u1|v1 −−−→ q 1 , . . . , ∃π n : p n un|vn − −−− → q n , C, we say that A satisfies ϕ, denoted by A |= ϕ, if there exists a valuation ν of Var(ϕ) such that for all i ∈ {1, . . . , n}, ν(π i ) : ν(p i ) ν(ui)|ν(vi) − −−−−−− → ν(q i ) and ν satisfies C (ν |= C) . Given a pattern formula ϕ and an automaton with outputs A, the model-checking problem consists in deciding whether A satisfies ϕ, i.e. A |= ϕ. Example 1. Given k ∈ N, the k-valuedness property has been already expressed in Introduction (assuming = ∈ Ø). The formula ∃π 0 : p 0 u|v0 − −− → q 0 , . . . , ∃π k : p k u|v k − −− → q k , C 0 where C 0 = 0≤i<j≤k π i = π j ∧ k i=0 init(p i ) ∧ final(q i ) expresses the fact that an automaton is not (k − 1)-ambiguous (has at least k accepting paths for some input). Model-Checking Problem In this section, we give sufficient conditions on the output monoid M and the set of output predicates Ø by which the model-checking of automata with outputs in M against pattern formulas over the output predicates Ø is decidable. In the next sections, we study the precise complexity of the model-checking problem for particular monoids M. Tuple acceptors Since automata with outputs can get their output values in arbitrary monoids, to get an effective model-checking algorithm, we will assume the existence of machines, called tuple acceptors, that can recognise sets of word tuples. These machines will be required to satisfy some key properties, forming the notion of good class of tuple acceptors. First, what we call a tuple acceptor is a machine M whose semantics is a set of tuples of words [[M ]] ⊆ (Σ * ) n , for some alphabet Σ and some arity n ≥ 1. The notion of good class, formally defined later, require (i) that any regular set of tuples is recognised by some machine, for a regularity notion that we will make clear (roughly, by seeing tuples of words as words resulting from the overlapping of all components), (ii) all output predicates (and their negation) are recognised by some machine, (iii) the class is closed under union and intersection. Regular sets of word tuples Let Σ be some alphabet containing some symbol ⊥, π ∈ Σ * and m ≥ |π|. The padding of π with respect to m is the word π ′ = π⊥ m−|π| . Let π 1 , π 2 ∈ Σ * and let m = max(|π 1 |, |π 2 |). For j = 1, 2, let π ′ j the padding of π j with respect to m. Note that |π ′ ⊥). The convolution can be naturally extended to multiple words as follows: 1 | = |π ′ 2 | = m. The convolution π 1 ⊗π 2 is the word of length m defined for all 1 ≤ i ≤ n by (π 1 ⊗ π 2 )[i] = (π ′ 1 [i], π ′ 2 [i]). E.g. q 1 λ 1 d 1 q 2 ⊗ p 1 = (q 1 , p 1 )(λ 1 , ⊥)(d 1 , ⊥)(q 2 ,n i=1 π i = π 1 ⊗ (π 2 ⊗ · · · ⊗ π n ). Definition 3. A set of n-ary word tuples P ⊆ (Σ * ) n is regular if L = { n i=1 π i | (π 1 , . . . , π n ) ∈ P } is a regular language over Σ n . We often identify L and P . Good class of tuple acceptors First, any valuation ν of a set of path variables X into paths of some automaton with values in some monoid M gives a way to interpret terms t ∈ Terms(X, ⊕, 0) as follows: for π ∈ X, π ν,M = out(ν(π)), 0 ν, M = 0 M and (t 1 ⊕ t 2 ) ν,M = t ν,M 1 ⊕ M t ν,M 2 . Then, for a class C (i.e. a set) of tuple acceptors, we denote by C| n its restriction to acceptors of arity n. Effectiveness of a good class gives effective model-checking, as announced. Theorem 1. Let M be a monoid and Ø be a set of output predicates, interpreted over M. If there exists an effective good class C (resp. effective weakly good class) of tuple acceptors for M and Ø, then the modelchecking problem of automata with outputs in M against pattern formulas ψ ∈ PL[Ø] (resp. ψ ∈ PL + [Ø]) is decidable. Proof (sketch). First, the formula is put in negation normal form: negation is pushed down to the atoms. Then, given an automaton with outputs in M, we show that any tuple of paths which satisfy state, input and path predicates and their negations is a regular set of path tuples (this is doable even for input equality as well as input length comparison thanks to the way paths are overlapped by the definition of convolution). By condition 1a, these sets of tuples are accepted by acceptors of C. By conditions 1(b)i and 1(b)ii, tuples of paths satisfying output predicates and their negations are also accepted by acceptors of C. Then, the closure properties (condition 2) allows us to construct an acceptor for the tuples of paths satisfying the whole formula inductively. A pattern logic for finite automata Finite automata can be seen as automata with outputs in a trivial monoid (with a single element). As the monoid is trivial, there is no need for predicates over it and so we specialize our pattern logic into PL NFA = PL[∅]. = ¬C | C ∨ C | u ⊑ u ′ | u ∈ L | |u| ≤ |u ′ | | init(q) | final(q) | q = q ′ | π = π ′ where for all i = j, π i = π j , L is a regular language over Λ (assumed to be represented as an NFA), u, u ′ ∈ {u 1 , . . . , u n }, q, q ′ ∈ {q 1 , . . . , q n } and π, π ′ ∈ {π 1 , . . . , π n }. As a yardstick to measure the expressiveness of PL NFA , we have considered the structural properties of NFA studied in two classical papers: [25] by Weber and Seidl and in [2] by Allauzen et al. The authors of these two papers give PTIME membership algorithms for k-ambiguity, finite ambiguity, polynomial ambiguity and exponential ambiguity (with as applications the approximation of the entropy of probabilistic automata for example). We refer the interested readers to these papers for the formal definitions of those classes. The solutions to these membership problems follow a recurrent schema: one defines (1) a pattern that identifies the members of the class and (2) an algorithm to decide if an automaton satisfies the pattern. The next theorem states that all these membership problems can be reduced to the model-checking problem of PL NFA using a constant space reduction. The proof of this theorem is obtained by showing how the patterns identified in [25], can be succinctly and naturally encoded into (fixed) PL NFA formulas. As a corollary, we get that all the class membership problems are in NLOGSPACE, using a model-checking algorithm that we defined below for PL NFA . Proof. For each membership problem, our reduction copies (in constant space) the NFA and considers the model-checking for this NFA against a fixed PL NFA (one for each class). As illustration, k-ambiguity has already been expressed in Example 1. As a second example, an automaton is not polynomially ambiguous iff there exists a state p which is reachable from an initial state, and the source of two different cycles labelled identically by a word v. With PL NFA this gives: ∃π 0 : q 0 u1 −→ p, ∃π 1 : p u2 −→ p, ∃π 2 : p u2 −→ p, ∃π 3 : p u3 −→ q, init(q 0 )∧π 1 = π 2 ∧final(q) The model-checking problem asks if a given NFA A satisfies a given PL NFA -formula ϕ. Theorem 3. The model-checking problem of NFA against formulas in PL NFA is PSPACE-C. It is in NLOGSPACE- C if the formula is fixed. Proof (sketch). We use NFA as acceptors for tuples of paths. The algorithm presented in the proof of Theorem 1 yields an exponentially large NFA (and polynomial if the formula is fixed). We show that it does not need to be constructed explicitly and that a short non-emptiness witness can be searched nondeterministically on-the-fly. For PSPACE-hardness, we notice that the non-emptiness of the intersection of n DFA can be easily expressed in PL NFA , by seeing the n DFA as a disjoint union, and by asking for the existence of n different accepting paths over the same input in this union. 2 and 3). The membership problem to the classes of k-ambiguous, finitely ambiguous, polynomially ambiguous and exponentially ambiguous NFA is in NLOGSPACE. Corollary 1 (of Theorems A pattern logic for transducers Transducers are automata with outputs in a free monoid M T rans = (Γ * , ·, ε) and therefore define subsets of Λ * × Γ * . Since our general pattern logic can test for output equalities (by repeating twice an output variable in the prefix), the model-checking is easily shown to be undecidable by encoding PCP: To obtain a decidable logic for transducers, we need to exclude equality tests on the output words in the logic. However, as we will see, we can instead have inequality test = as long as it is not under an odd number of negations in the formula. We also allow to test (non) membership of output word concatenations to a regular language, as well as comparison of output word concatenations wrt their length. Formally: = ¬C | C ∨ C | u ⊑ u ′ | u ∈ L | |u| ≤ |u ′ | | init(q) | final(q) | q = q ′ | π = π ′ | t ⊑ t ′ | t ∈ N | |t| ≤ |t ′ | where for all 1 ≤ i < j ≤ n, π i = π j and v i = v j (no implicit output equality tests), L (resp. N ) is a regular language over Λ (resp. Γ ), assumed to be represented as an NFA, u, u ′ ∈ {u 1 , . . . , u n }, q, q ′ ∈ {q 1 , . . . , q n }, t, t ′ ∈ T erms({v 1 , . . . , v n }, ·, ǫ), π, π ′ ∈ {π 1 , . . . , π n }, and t ⊑ t ′ does not occur under an odd number of negations. We define the macros t = t ′ def = t ⊑ t ′ ∨ t ′ ⊑ t, mismatch(t, t ′ ) def = t ⊑ t ′ ∧ t ′ ⊑ t and SDel = (t 1 , t ′ 1 , t 2 , t ′ 2 ) def = (|t ′ 1 | = |t ′ 2 |) ∨ [ t ′ 1 t ′ 2 = ǫ ∧ mismatch(t 1 , t 2 )] Let us explain the latter macro. Many properties of transducers are based on the notion of output delays, by which to compare output words. Formally, for any two words v 1 , v 2 , delay(v 1 , v 2 ) = (α 1 , α 2 ) such that v 1 = ℓα 1 and v 2 = ℓα 2 where ℓ is the longest common prefix of v 1 and v 2 . It can be seen that for any words v 1 , v ′ 1 , v 2 , v ′ 2 , if we have SDel = (v 1 , v ′ 1 , v 2 , v ′ 2 ), then delay(v 1 , v 2 ) = delay(v 1 v ′ 1 , v 2 v ′ 2 ), but the converse does not hold. But, if delay(v 1 , v 2 ) = delay(v 1 v ′ 1 , v 2 v ′ 2 ), then SDel = (v 1 (v ′ 1 ) i , v ′ 1 , v 2 (v ′ 2 ) i , v ′ 2 ) holds for some i ≥ 0. These two facts allows us to express all the known transducer properties from the literature relying on the notion of delays. We leave however as open whether our logic can express a constraint such as delay(v 1 , v 2 ) = delay(v 3 , v 4 ). We review here some of the main transducer subclasses studied in the literature. We refer the reader to the mentioned references for the formal definitions. As for the NFA subclasses of the previous section, deciding them usually goes in two steps: (1) identify a structural pattern characterising the property, (2) decide whether such as pattern is satisfied by a given transducer. The class of determinisable transducers are the transducers which define sequential functions [6,5,24]. The k-sequential transducers are the transducers defining unions of (graphs) of k sequential functions [8]. The multi-sequential ones are the union of all ksequential transducers for all k [16,7]. Finally, the k-valued transducers are the transducers for which any input word has at most k output words [14,19], and the finite-valued ones are all the k-valued transducers for all k [22,23,18]. All these classes, according to the given references, are decidable in PTIME. Proof. Without going through all the properties, let us remind the reader that the formula for k-valuedness has been given in the introduction. We also give the PL Trans formulas for the class of determinisable transducer. It is known that a transducer is determinisable iff it satisfies the twinning property, which is literally the negation of: ∃π 1 : q 1 u|v1 − −− → p 1 , ∃π ′ 1 : p 1 u ′ |v ′ 1 − −− → p 1 , ∃π ′′ 1 : p 1 u ′′ |v ′′ 1 − −−− → r 1 ,∃π 2 : q 2 u|v2 − −− → p 2 , ∃π ′ 2 : p 2 u ′ |v ′ 2 − −− → p 2 , ∃π ′′ 2 : p 2 u ′′ |v ′′ 2 − −−− → r 2 , init(q 1 ) ∧ init(q 2 ) ∧ final(r 1 ) ∧ final(r 2 ) ∧ SDel = (v 1 , v ′ 1 , v 2 , v ′ 2 ) Theorem 6. The model checking of transducers against formulas in PL T rans is PSPACE-C. It is in NLOGSPACE-C if the formula is fixed. Proof (sketch). We use Parikh automata as acceptors for tuples of paths. They extend automata with counters that can only be incremented and never tested for zero. The acceptance condition is given by a semilinear set (represented for instance by an existential Presburger formula). The formal definition can be found e.g. in [9]. The counters allow us to compare the output length of paths, or to identify some output position of two paths with different labels (to test v ⊑ v ′ ). The counters are needed because this position may not occur at the same location in the convolution encoding of path tuples. Corollary 2 (of Theorems 5 and 6). The membership problem of transducers to the classes of determinisable, functional, k-sequential, multi-sequential, k-valued, and finite-valued transducers (for fixed k) is decidable in NLOGSPACE. A pattern logic for sum-automata We remind the reader that sum-automata are automata with outputs in the monoid M Sum = (Z, +, 0) (assumed to be encoded in binary) and therefore define subsets of Λ * × Z. We consider in this section two logics for expressing structural properties of sum-automata: the logic PL Sum which is obtained as PL [{≤}] where the output predicate ≤ is interpreted by the natural total order over integers, and a subset of this logic PL = Sum obtained as PL + [{ =}] where the predicate = never appears in the scope of an odd number of negations (to avoid the expressibility of the equality predicate). We show that the fragment PL = Sum enjoys better complexity results. Formally, those two logics are defined as follows: C ∨ C | u ⊑ u ′ | u ∈ L | |u| ≤ |u ′ | | init(q) | final(q) | q = q ′ | π = π ′ | t ≤ t ′ where for all 1 ≤ i < j ≤ n, π i = π j , L is a regular language over Λ assumed to be represented as an NFA, u, u ′ ∈ {u 1 , . . . , u n }, q, q ′ ∈ {q 1 , . . . , q n }, t, t ′ ∈ T erms({v 1 , . . . , v n }, ·, ǫ) and π, π ′ ∈ {π 1 , . . . , π n }. The logic PL = Sum is defined as above but the constraint t ≤ t ′ is replaced by t = t ′ and this constraint does not occur under an odd number of negations, and moreover v i = v j for all 1 ≤ i < j ≤ n (no implicit output equality tests). We review here some of the main sum-automata subclasses decidable in PTIME studied in the literature. We refer the reader to the mentioned references for the formal definitions. The class of functional sumautomata [11] are those such that all accepting paths associated with a given word return the same value. The classes of k-valued [10] and k-sequential sum-automata [8] are defined similarly as for transducers. Proof. We have already shown in the introduction that functionality [11] and more generally k-valuedness [10] are expressible in PL = Sum . The twinning property [11,1] is as well expressible in PL = Sum , just by replacing in the formula expressing it for transducers (proof of Thm. 5) the atom SDel = (v 1 , v ′ 1 , v 2 , v ′ 2 ) by v ′ 1 = v ′ 2 . In [8], a generalization of the twinning property is shown to be complete for testing k-sequentiality. The proof of the results below for PL Sum follows arguments that are similar to those developed for transducers in the proof of Theorem 6, and for the PTIME result for PL = Sum , we use a reduction to the k-valuedness problem of sum-automata [10]. Theorem 8. The model checking of sum-automata against formulas in PL Sum is PSPACE-C, NP-C when the formula is fixed, and NLOGSPACE-C if in addition the values of the automaton are encoded in unary. The model checking of sum-automata against formulas in PL = Sum is PSPACE-C, and in PTIME when the formula is fixed (even if the values of the automaton are encoded in binary). Corollary 3 (of Theorems 7 and 8). The membership problem of sum-automata in the class of functional, k-valued, and k-sequential automata is decidable in PTIME. Note that we have shown that the k-valuedness property is expressible in PL = Sum , and so the k-valuedness property is reducible to the model-checking problem of PL = Sum . Nevertheless, this result does not provide a new algorithm for k-valuedness as our model-checking algorithm is based on a reduction to kvaluedness [10]. Extensions and Future Work The logics we have presented can be extended in two ways by keeping the same complexity results, no matter what the output monoid is. The first extension allows to express properties of automata whose states can be coloured by an arbitrary (but fixed) set of colours. This is useful for instance to express properties of disjoint unions of automata, the colours allowing to identify the subautomata. The second extension is adding a bunch of universal state quantifiers before the formula. This does not change the complexity, and allow for instance to express properties such as whether an automaton is trim (all its states are accessible and co-accessible). As future work, we would like to investigate other monoids (discounted sum group for instance [11]), and other data structures for which transducers and weighted automata have been defined: nested words, infinite words and trees are the main structures we want to work on. q 1 , . . . , ∃π n : p n un|vn − −−− → q n , C Definition 1 ( 1Automata with outputs). An automaton A with outputs over an (output) monoid M = (D, ⊕ M , 0 M ) is a tuple Q, I, F, ∆, γ where Q is a non-empty finite set of states, I ⊆ Q the set of initial states, F ⊆ Q the set of final states, ∆ ⊆ Q × Λ ε × Q the set of transitions labelled with some element of Λ ε , and γ : ∆ → D a mapping from transitions to output values 1 . The set of automata over M is written A out (M). Semantics To define the semantics of a pattern formula ϕ, we first fix some monoid M = (D, ⊕ M , 0 M ) together with an interpretation p M of each output predicates p ∈ Ø of arity α(p), such that p M ∈ D if p is a constant and p M ⊆ D α(p) otherwise. Given a valuation ν : X O → D, the interpretation . M can be inductively extended to terms t by letting 0 ν,M = 0 M , (t 1 ⊕ t 2 ) ν,M = t ν,M 1 ⊕ M t ν,M 2 and x ν,M = ν(x). Then, a formula ϕ ∈ PL(Ø) is interpreted in an automaton with outputs A ∈ A out (M) as a set of valuations [[ϕ]] A of Var(ϕ) which we now define. Each valuation ν ∈ [[ϕ]] A maps state variables to states of A, path variables to paths of A, etc. Such a valuation ν satisfies an atom Definition 4 ( 4Good class). A class of tuple acceptors C is said to be good for an output monoid M = (D, ⊕ M , 0 M ), a set of output predicates Ø and an interpretation p M ⊆ D α(p) for all p ∈ Ø of arity α(p), if the following conditions are satisfied: 1. for all automata with outputs A ∈ A out (M) with a set of states Q we have: (a) ∀n ≥ 1,∀R ⊆ Paths(A) n regular, R = [[M ]] for some M ∈ C| n . (b) all p ∈ Ø of arity α(p), all X = {π 1 , . . . , π n } finite sets of path variables and all t 1 , . . . , t α(p) ∈ Terms(X, ⊕, 0), there exist M, M ′ ∈ C| n such that i. [[M ]] = {(ν(π 1 ), . . . , ν(π n )) |ν : X → Paths(A) ∧ (t ν,M 1 , . . . , t ν,M α(p) ) ∈ p M } ii. [[M ′ ]] = Paths(A) n \ [[M ]]. 2. ∀n ≥ 1, ∀M 1 , M 2 ∈ C| n , there exist M, M ′ ∈ C| n such that [[M ]] = [[M 1 ]] ∩ [[M 2 ]] and [[M ′ ]] = [[M 1 ]] ∪ [[M 2 ]].We say that C is effective if all properties are effective and moreover it is decidable whether [[M ]] = ∅ for any (effectively represented) M ∈ C. We say that C is weakly good if all properties hold except 1(b)ii. Definition 5 ( 5Pattern logic for NFA). The logic PL NFA is the set of formulas ϕ ::= ∃π 1 : p 1 u1 −→ q 1 , . . . , ∃π n : p n un − − → q n , C C :: Theorem 2 . 2The membership problem to the subclasses of k-ambiguous, finitely ambiguous, polynomially ambiguous and exponentially ambiguous NFA can be reduced to the model-checking problem of PL NFA with constant space reduction. The obtained formulas are constant (for fixed k). Theorem 4 . 4The model-checking problem of transducers against formulas in PL[∅] is undecidable. Definition 6 ( 6Pattern logic for transducers). The logic PL Trans is the set of formulas of the form ϕ ::= ∃π 1 : p 1 u1|v1 −−−→ q 1 , . . . , ∃π n : p n un|vn − −−− → q n , C C :: Theorem 5 . 5The membership problem of transducers to the classes of determinisable, functional, k-sequential, multi-sequential, k-valued, and finite-valued transducers can be reduced to the model-checking problem of PL Trans with a constant space reduction. The obtained formulas are constant (as long as k is fixed). Definition 7 ( 7Two pattern logics for sum-automata). The logic PL Sum is the set of formulas of the form ϕ ::= ∃π 1 : p 1 u1|v1 −−−→ q 1 , . . . , ∃π n : p n un|vn − −−− → q n , C C ::= ¬C | Theorem 7 . 7The membership problem of sum-automata in the class of functional, k-valued, and k-sequential automata can be reduced to the model-checking problem of PL = Sum . Moreover, the obtained PL = Sum formulas are constant (as long as k is fixed). Often in the literature, output values are directly given in the transitions, i.e. the transition relation is a (finite) subset of Q × Λε × D × Q. Our definition is then equivalent modulo PTIME transformation, and allows for a clearer distinction between input and output mechanisms. Efficient algorithms for testing the twins property. C Allauzen, M Mohri, Journal of Automata, Languages and Combinatorics. 82C. Allauzen and M. Mohri. Efficient algorithms for testing the twins property. Journal of Automata, Languages and Combinatorics, 8(2):117-144, 2003. General algorithms for testing the ambiguity of finite automata and the double-tape ambiguity of finite-state transducers. C Allauzen, M Mohri, A Rastogi, Int. J. Found. Comput. Sci. 224C. Allauzen, M. Mohri, and A. Rastogi. General algorithms for testing the ambiguity of finite automata and the double-tape ambiguity of finite-state transducers. Int. J. Found. Comput. Sci., 22(4):883-904, 2011. Unambiguous automata denoting finitely sequential functions. S Bala, A Koninski, LATA. 7810S. Bala and A. Koninski. Unambiguous automata denoting finitely sequential functions. In LATA, LNCS 7810, 2013. Determinization of transducers over finite and infinite words. M.-P Béal, O Carton, Theoretical Computer Science. 2891M.-P. Béal and O. Carton. Determinization of transducers over finite and infinite words. Theoretical Computer Science, 289(1):225-251, 2002. Squaring transducers: an efficient procedure for deciding functionality and sequentiality. M.-P Béal, O Carton, C Prieur, J Sakarovitch, TCS. 2921M.-P. Béal, O. Carton, C. Prieur, and J. Sakarovitch. Squaring transducers: an efficient procedure for deciding functionality and sequentiality. TCS, 292(1), 2003. Une caracterisation des fonctions sequentielles et des fonctions sous-sequentielles en tant que relations rationnelles. C Choffrut, Theor. Comput. Sci. 53C. Choffrut. Une caracterisation des fonctions sequentielles et des fonctions sous-sequentielles en tant que rela- tions rationnelles. Theor. Comput. Sci., 5(3), 1977. Decomposition de Fonctions Rationnelles. C Choffrut, M P Schutzenberger, STACS. C. Choffrut and M. P. Schutzenberger. Decomposition de Fonctions Rationnelles. In STACS, 213-226, 1986. Degree of sequentiality of weighted automata. L Daviaud, I Jecker, P.-A Reynier, D Villevalois, FOS-SACS. L. Daviaud, I. Jecker, P.-A. Reynier, and D. Villevalois. Degree of sequentiality of weighted automata. In FOS- SACS, 2017. Path logics for querying graphs: Combining expressiveness and efficiency. D Figueira, L Libkin, LICS. D. Figueira and L. Libkin. Path logics for querying graphs: Combining expressiveness and efficiency. In LICS, pages 329-340, 2015. Finite-valued weighted automata. E Filiot, R Gentilini, J.-F Raskin, FSTTCS. E. Filiot, R. Gentilini, and J.-F. Raskin. Finite-valued weighted automata. In FSTTCS, pages 133-145, 2014. Quantitative languages defined by functional automata. E Filiot, R Gentilini, J.-F Raskin, LMCS. 113E. Filiot, R. Gentilini, and J.-F. Raskin. Quantitative languages defined by functional automata. LMCS, 11(3), 2015. Hierarchies of complete problems. Z Galil, Acta Informatica. 61Z. Galil. Hierarchies of complete problems. Acta Informatica, 6(1):77-88, 1976. The unsolvability of the equivalence problem for lambda-free nondeterministic generalized machines. T V Griffiths, Journal of the ACM. 153T. V. Griffiths. The unsolvability of the equivalence problem for lambda-free nondeterministic generalized ma- chines. Journal of the ACM, 15(3):409-413, 1968. A note on finite-valued and finitely ambiguous transducers. Theory of Computing Systems. E M Gurari, O H Ibarra, 16E. M. Gurari and O. H. Ibarra. A note on finite-valued and finitely ambiguous transducers. Theory of Computing Systems, 16(1):61-66, 1983. Transductions and Context-Free Languages. J Berstel, TeubnerStuttgartJ. Berstel. Transductions and Context-Free Languages. Teubner, Stuttgart, 1979. Multi-sequential word relations. I Jecker, E Filiot, IJFCS. 292I. Jecker and E. Filiot. Multi-sequential word relations. IJFCS, 29(2), 2018. Monadic second-order logics with cardinalities. F Klaedtke, H Rueß, ICALP. F. Klaedtke and H. Rueß. Monadic second-order logics with cardinalities. In ICALP, 2003. On the decidability of bounded valuedness for transducers. J Sakarovitch, R Souza, MFCS. J. Sakarovitch and R. de Souza. On the decidability of bounded valuedness for transducers. In MFCS, pages 588-600, 2008. Lexicographic decomposition of k -valued transducers. Theory of Computing Systems. J Sakarovitch, R Souza, 47J. Sakarovitch and R. de Souza. Lexicographic decomposition of k -valued transducers. Theory of Computing Systems, 47(3), 2010. Complexity of subcases of presburger arithmetic. B Scarpellini, In Transactions of the American Mathematical Society. 284203218B. Scarpellini. Complexity of subcases of presburger arithmetic. In Transactions of the American Mathematical Society 284, page 203218, 1984. On the equivalence and containment problems for unambiguous regular expressions, regular grammars and finite automata. R E Stearns, H B Hunt, Iii , SIAM Journal on Computing. 143R. E. Stearns and H. B. Hunt III. On the equivalence and containment problems for unambiguous regular expres- sions, regular grammars and finite automata. SIAM Journal on Computing, 14(3):598-611, 1985. On the valuedness of finite transducers. A Weber, Acta Inf. 278A. Weber. On the valuedness of finite transducers. Acta Inf., 27(8), 1990. Decomposing finite-valued transducers and deciding their equivalence. A Weber, SIAM Journal on Computing. 221A. Weber. Decomposing finite-valued transducers and deciding their equivalence. SIAM Journal on Computing, 22(1):175-202, 1993. Economy of description for single-valued transducers. A Weber, R Klemm, Information and Computation. 1182A. Weber and R. Klemm. Economy of description for single-valued transducers. Information and Computation, 118(2):327-340, 1995. On the degree of ambiguity of finite automata. A Weber, H Seidl, Theor. Comput. Sci. 882A. Weber and H. Seidl. On the degree of ambiguity of finite automata. Theor. Comput. Sci., 88(2):325-349, 1991.
[]
[ "Dephasing in a quantum pump", "Dephasing in a quantum pump" ]
[ "J N H J Cremers \nLyman Laboratory of Physics\nHarvard University\n02138CambridgeMA\n", "P W Brouwer \nLaboratory of Atomic and Solid State Physics\nCornell University\n14853-2501IthacaNY\n" ]
[ "Lyman Laboratory of Physics\nHarvard University\n02138CambridgeMA", "Laboratory of Atomic and Solid State Physics\nCornell University\n14853-2501IthacaNY" ]
[]
We study how dephasing affects the distribution of the dc current pumped through a chaotic quantum dot. We introduce dephasing by the addition of a voltage probe to the quantum dot, treating both the case of controlled dephasing (when the voltage probe is coupled to the dot via a ballistic point contact with a conductance that is increased stepwise) and intrinsic dephasing (for which the voltage probe serves as a phenomelogical model). While dephasing eventually suppresses the dc current through the dot, we also find that, for a quantum dot with single-mode point contacts, a small amount of dephasing actually decreases the likelihood of a zero pumped current.
10.1103/physrevb.65.115333
[ "https://export.arxiv.org/pdf/cond-mat/0110437v1.pdf" ]
119,100,575
cond-mat/0110437
47a815983219db10a522a8c06a655e20484c46c9
Dephasing in a quantum pump 20 Oct 2001 J N H J Cremers Lyman Laboratory of Physics Harvard University 02138CambridgeMA P W Brouwer Laboratory of Atomic and Solid State Physics Cornell University 14853-2501IthacaNY Dephasing in a quantum pump 20 Oct 2001(March 22, 2022)PACS numbers: 7210Bg, 7323-b We study how dephasing affects the distribution of the dc current pumped through a chaotic quantum dot. We introduce dephasing by the addition of a voltage probe to the quantum dot, treating both the case of controlled dephasing (when the voltage probe is coupled to the dot via a ballistic point contact with a conductance that is increased stepwise) and intrinsic dephasing (for which the voltage probe serves as a phenomelogical model). While dephasing eventually suppresses the dc current through the dot, we also find that, for a quantum dot with single-mode point contacts, a small amount of dephasing actually decreases the likelihood of a zero pumped current. We study how dephasing affects the distribution of the dc current pumped through a chaotic quantum dot. We introduce dephasing by the addition of a voltage probe to the quantum dot, treating both the case of controlled dephasing (when the voltage probe is coupled to the dot via a ballistic point contact with a conductance that is increased stepwise) and intrinsic dephasing (for which the voltage probe serves as a phenomelogical model). While dephasing eventually suppresses the dc current through the dot, we also find that, for a quantum dot with single-mode point contacts, a small amount of dephasing actually decreases the likelihood of a zero pumped current. Quantum mechanical interference is a key player determining sample-specific properties of small metal or semiconductor particles. For closed systems, quantum interference fixes the precise positions of energy levels and the microscopic details of wavefunctions; for open systems it governs, e.g., the fluctuations of the conductance. 1,2 In all these cases, interference provides a (sometimes large) correction added to a nonzero background. Such a background is absent in a so-called "quantum electron pump", which has received considerable experimental 3 and theoretical 4-12 attention recently. This requirement of phase coherence for the observation of a pumped current makes a quantum pump a sensitive instrument to study quantum interference in quantum dots. In the experimental proposal of Ref. 3, a quantum electron pump is made of a semiconductor quantum dot connected to two electron reservoirs. A dc current is generated by periodically varying two gate voltages that characterize shape of the dot. (Alternatively, one may pump electrons by periodic variation of the Fermi energy, magnetic flux, etc.) The electron pump is called "quantum", because the variation pertains to the quantum interference only, not to the classical dynamics of the dot: In a trajectory-based picture, the variations affect phases of the trajectories, not their weights. For this reason, this "quantum" electron pump is fundamentally different from electron pumps or turnstiles that rely on Coulomb charging physics and do not require phase coherence. 13,14 In the quantum pump, the precise value of the pumped current and its direction depend on microscopic details and vary from sample to sample. Therefore, one needs to address the probability distribution P (I) of pumped currents for an ensemble of quantum pumps. Theoretical work has focused on electron pumps in a chaotic quantum dot, for which this distribution can be calculated using random matrix theory. 5,6,8,10 For a pump with single-channel point contacts, it was found that P (I) has a cusp at I = 0 and algebraic tails, 5 while for many-channel point contacts, P (I) is Gaussian. 5,6,8 In this paper, we study how the probability distribution of the pumped current is affected if phase coherence in the quantum pump is gradually destroyed. As in Refs. 3,5-10, we consider the case of a quantum pump that consists of a chaotic quantum dot. A controlled way to destroy phase coherence is to couple the dot to a voltage probe. 15 While not drawing a net current, the voltage probe absorbs and reinjects electrons without any phaserelationship, thereby destroying phase coherence. The dephasing rate γ φ can be tuned by varying the conductance of the point contact connecting the quantum dot and the voltage probe. While a voltage probe can be used as an external source to controllably increase the dephasing rate in the quantum pump, it can also be used as a model for intrinsic dephasing processes in the pump itself, such as electron electron interactions or two-level systems in the dot. [15][16][17] A similar approach was taken in a recent paper by Moskalets and Büttiker, who studied the effect of a voltage probe on the current pumped through an electron pump consisting of a one-dimensional wire with two tunable tunnel barriers of oscillating strength. 18 However, unlike the quantum dot geometry studied here, the pump of Ref. 18 can also operate as an electron pump in the absence of phase coherence. The variance of the pumped current in the case of many-channel point contacts was previously obtained by Shutenko et al. using a different model. 8 In Sec. II we use the scattering approach to derive a formula for the pumped current in the presence of the voltage probe. In Sec. III we then study the distribution of the pumped current for the cases of controlled dephasing and intrinsic dephasing. We focus on the cases of a quantum dot with single channel point contacts and with multi-channel point contacts. We conclude in Sec. IV. II. PUMPED CURRENT IN THE PRESENCE OF DEPHASING A schematic picture of the system is shown in Fig.1. It consists of a quantum dot connected by ballistic point contacts to three electron reservoirs, labeled 1, 2, and 3. The point contacts allow N 1 , N 2 and N 3 propagating channels at the Fermi level, respectively. The total number of channels in all point contacts will be denoted as N = N 1 + N 2 + N 3 . The reservoirs 1 and 2 are held at the same voltage V 1 = V 2 = 0, while reservoir 3 serves as a voltage probe: the time-dependent voltage V 3 (t) is adjusted such that the current I 3 (t) = 0 at all times. Two external parameters X 1 (t) and X 2 (t) that determine the shape (or other characteristics) of the quantum dot are varied periodically with frequency ω. As a result of this periodic variation, a dc current I = I 1 = −I 2 will flow between reservoirs 1 and 2. We will now evaluate this pumped current in the presence of the voltage probe. Starting point of our evaluation is the relation 5 between the pumped current and the emissivity edn(m)/dX j , the charge that leaves the dot through contact m = 1, 2 as the parameter X j is varied by an amount dX j , I = ωe 2π A dX 1 dX 2 ∂ ∂X 1 dn(1) dX 2 − ∂ ∂X 2 dn(1) dX 1 .(1) Here A is the area in (X 1 , X 2 )-space enclosed by the parameters [X 1 (t), X 2 (t)] in one cycle. Equation (1) is valid for the adiabatic regime, ωτ d ≪ 1, where τ d ∼h/N ∆ is the dwell time of the quantum dot, ∆ being the mean single-particle level spacing. In the absence of the voltage probe, and for ballistic point contacts between the dot and reservoirs 1 and 2, so that effects of charge quantization in the dot can be ignored, 19 the emissivity edn(m)/dX j is related to the N × N scattering matrix S of the dot as 20 dn(m) dX = 1 2π β α∈m Im ∂S αβ ∂X S * αβ .(2) With voltage probe, charge can either leave the quantum dot directly through the contacts 1 or 2, or via inelastic scattering in reservoir 3. Hence, we can write the emissivity dn(m)/dX as a sum of an elastic and an inelastic contribution, dn(m) dX = dn(m) dX el + dn(m) dX in ,(3) where the elastic contribution (dn(m)/dX) el is still given by Eq. (2) and the inelastic contribution reads dn(m) dX in = G m3 2π(G 13 + G 23 ) Im β α∈3 ∂S αβ ∂X S * αβ .(4) Here G ij = α∈i β∈j |S αβ | 2 is an element of the 3 × 3 condutance matrix of the dot. Substitution of Eqs. (2)-(4) into Eq. (1) then yields the current formula I = ωe 2π A dX 1 dX 2 (i dir + i rect ) ,(5a)i dir = G 23 π(G 13 + G 23 ) Im β α∈1 ∂S αβ ∂X 2 ∂S * αβ ∂X 1 − G 13 π(G 13 + G 23 ) Im β α∈2 ∂S αβ ∂X 2 ∂S * αβ ∂X 1 , (5b) i rect = 1 4π Im β α∈3 S * αβ ∂S αβ ∂X 2 ∂ ∂X 1 G 13 − G 23 G 13 + G 23 − 1 4π Im β α∈3 S * αβ ∂S αβ ∂X 1 ∂ ∂X 2 G 13 − G 23 G 13 + G 23 . (5c) The first contribution I dir represents the charge that exits the dot either directly or indirectly via the reservoir, while the second contribution I rect is an additional contribution to the dc current that arises from rectification of the voltage V 3 (t). 18 One verifies that in the limit where the third reservoir is decoupled from the quantum dot, I rect vanishes, while the expression for I dir approaches that for current in the absence of dephasing (see Ref. 5). In the next section we proceed to calculate the statistical distribution of the current I for an ensemble of chaotic quantum dots. We restrict ourselves to the case of small pumping amplitudes X 1 = δX 1 sin(ωt), X 2 = δX 2 sin(ωt + φ), for which the current I is bilinear in δX 1 and δX 2 , I = 1 2 eωi sin φδX 1 δX 2 , i = i dir + i rect .(6) We will consider both the case of "controlled dephasing", corresponding to a real voltage probe coupled to the quantum dot via a ballistic point contact, and that of intrinsic dephasing, which is modeled by a voltage probe coupled to the quantum dot via a wide tunneling contact. 17 In the absence of dephasing, the pumped current is not symmetric under reversal of a magnetic field through the quantum dot. 8,11 One easily verifies that in the presence of dephasing the current I remains asymmetric under reversal of the magnetic field, even in the presence of strong dephasing (N 3 large). III. DISTRIBUTION OF THE PUMPED CURRENT IN THE PRESENCE OF DEPHASING For an ensemble of chaotic quantum dots the distribution of the scattering matrix and its derivatives is known. 21 It is most conveniently expressed through the parameterization S = U U ′ , ∂S ∂X j = U Q j U ′ ,(7) where U and U ′ are N × N unitary matrices and Q j is an N ×N hermitian matrix. In the presence of time-reversal symmetry, U ′ = U T and Q j = Q T j . We also introduce the matrix Q ε that parameterizes the energy-derivative ∂S/∂ε, ∂S ∂ε = 2π ∆ U Q ε U ′ . For ballistic point contacts to reservoirs 1, 2, and 3, the matrices U and U ′ are uniformly distributed in the unitary group, independently of Q 1 , Q 2 , and Q ε , while the joint distribution of the matrices Q 1 , Q 2 , and Q ε is given by P ∝ (det Q ε ) −N/2−2(βN +2−β) Θ(Q) (8) × exp   − β 2 tr   Q −1 ε + 1 8 2 j=1 (Q −1 ε Q j ) 2     , where Θ(Q) = 1 if all eigenvalues of Q are positive and Θ(Q) = 0 otherwise and β = 1 (2) in the presence (absence) of time-reversal symmetry. Note that at fixed Q ε the distribution of Q 1 and Q 2 is Gaussian with a width set by Q ε . We will now study the distribution of the pumped dc current for controlled dephasing and instrinsic dephasing. In each case we will first study the variance and then the full distribution. A. Controlled Dephasing The effect of dephasing on the distribution of the pumped current can be studied in a controlled setting by increasing the number N 3 of open channels in the voltage probe one by one. The third reservoir serves as a true voltage probe if the charge relaxation rate of that reservoir is much larger than the frequency ω, so that the voltage V 3 (t) can adjust essentially instantaneously to balance any current I 3 (t) flowing into or out of that reservoir as a result of the pumping action on the quantum dot. A voltage probe connected to the dot via a ballistic point contact with N 3 channels gives rise to a dephasing rate γ φ = 1 τ φ = N 3 ∆/h.(9) We now fix N 1 and N 2 and calculate the distribution of the pumped current as a function of N 3 . We consider the cases N 1 = N 2 = 1 of single channel current-carrying leads and N 1 , N 2 ≫ 1 of many-channel current-carrying leads. In the former case, we have calculated the first two moments of the distribution P (i) of the dimensionless current i analytically, using the technique of Ref. 22 to perform the integrations over the matrices U and U ′ . The average current i was thus found to be zero, while the variance i 2 decreases with increasing N 3 ; i 2 is divergent for N 3 = 0 for β = 2 and for N 3 < 6 for β = 1. The results of this calculation are shown in Fig. 2. For large N 3 , we find i 2 = i 2 dir = 16N 1 N 2 π 2 (N 1 + N 2 )(N 1 + N 2 + N 3 ) 3 . (10) Details of the calculation can be found in Appendix A. [Equation (10) is valid for arbitrary N 1 and N 2 , as long as N 3 ≫ 1.] For N 3 ≫ 1, the contribution of the rectification current i rect to i 2 is proportional to N −4 3 and hence negligible. (A similar conclusion for the rectification contribution was reached in Ref. 12 for the comparison of measurements of pumped current and pumped voltage.) As shown in Fig. 2, in general, i 2 is larger with time-reversal symmetry than without, though the difference between the two variances vanishes for large dephasing rates. We obtained the full distribution P (i) for N 1 = N 2 = 1 using Monte-Carlo integration with the distribution (8). The result is shown in Fig. 3 for various values of N 3 . Details of the numerical procedure are outlined in App. B. Note that the distributions are symmetric around i = 0. In the absence of dephasing (N 3 = 0), P (i) has a cusp at i = 0, which is smoothed out for N 3 > 0. The distribution P (i) has algebraic tails with powers that increase with increasing dephasing rate. In the presence of timereversal symmetry the tails fall off with a power smaller than 3 as long as N 3 < 6, corresponding to a divergent second moment i 2 (cf. Fig. 2). Interestingly, while dephasing decreases the second moment i 2 for singlechannel point contacts N 1 = N 2 = 1, a small amount of dephasing also decreases the probability P (i = 0) of finding no pumped current at all. This can be understood physically, since the cusp in the distribution P (i) at zero pumped current arises from a the occurence of destructive interference, which is as much suppressed by dephasing as the constructive interference responsible for large values of i and the tails of the distribution. For N 3 ≫ 1, the distribution P (i) approaches a Gaussian. For multichannel point contacts in the current-carrying leads, N 1 , N 2 ≫ 1, the distribution P (i) is a Gaussian centered around i = 0. To calculate the second moment i 2 , we parameterize the scattering matrix S and its derivatives as in Eq. (7), perform the Gaussian integration over Q 1 and Q 2 with the distribution (8) at fixed Q ε , integrate over the unitary matrices U and U ′ using the diagrammatic technique of Ref. 22 and finally integrate over the eigenvalues τ j , j = 1, . . . , N , of Q ε , see appendix A for details. The result is given by Eq. (10) above. B. Intrinsic dephasing The voltage probe can also be used as a model for intrinsic dephasing in the quantum dot. Sources of dephasing may be, e.g., electron-electron interactions or interactions with an external bath of photons and/or phonons. In such a case the source of decoherence is delocalized throughout the dot. This situation is well modeled by a voltage probe with a tunnel barrier with transmission probability Γ 3 ≪ 1 and many channels N 3 ≫ 1, such that the product N 3 Γ 3 = hγ φ ∆(11) is kept fixed. 17 In order to find the distribution of the pumped current in this case, we need the distribution of the scattering matrix S and its derivatives when the contact to the third reservoir contains a tunnel barrier. This problem can be solved by a statistical mapping which connects the scattering matrix S to a scattering matrix S 0 that is taken from an ensemble appropriate for a quantum dot with ballistic point contacts, 23,24 S = √ 1 − Γ − √ Γ 1 1 − S 0 √ 1 − Γ S 0 √ Γ,(12) where Γ is a diagonal N × N matrix with Γ jj = 0 for index j corresponding to the current-carrying leads 1 and 2 and Γ jj = Γ 3 for index j corresponding to the voltage probe. The distribution of S 0 and its derivatives is as described in Eqs. (7)-(8) at the beginning of this section. We can then find the distribution of S from Eq. (12) and the distribution of its derivatives upon differentiating Eq. (12). We have calculated the variance i 2 and the full distribution of the current for N 1 = N 2 = 1 using Monte-Carlo integration with the above distribution. The results are shown in Figs. 4 and 5. We note that intrinsic dephasing cuts off the tails of the distribution P (i). This is in contrast to the case of controlled dephasing by a few-channel voltage probe with ballistic point contacts, which merely replaces the algebraic tail of the distribution at zero dephasing by another, faster decaying, algebraic tail. As for controlled dephasing, the variance of the pumped current is larger with time-reversal symmetry than without, though for intrinsic dephasing the difference is smaller than in the case of controlled dephasing (cf. Fig. 2). At large dephasing rates, the dependence of i 2 on the presence or absence of time-reversal symmetry vanishes. Also note that the probability to find zero pumped current initially decreases when the dephasing rate is increased, although the effect is not as strong as in the case of controlled dephasing. For large dephasing rates, both intrinsic and controlled dephasing yield the same distribution P (i). For large N 1 and N 2 the distribution P (i) is again Gaussian, with zero mean and with variance given by Eq. IV. CONCLUSION In summary, we have derived the current distribution of electrons pumped adiabatically through a chaotic quantum dot, in the presence of a voltage probe. The voltage can either serve as a controlled source of dephasing, or as an effective description of intrinsic dephasing processes inside the quantum dot. For the case of a quantum dot with two single-mode current-carrying point contacts, the conductance distribution is non-Gaussian with algebraic tails. Remarkably, while dephasing shifts the weight of the probability distribution P (I) of the pumped current towards zero current, a small amount of dephasing actually reduces the probability P (I = 0) to find no pumped current at all. This can be understood if the probability to find zero pumped current is enhanced by destructive interference -note the cusp in P (I) at I = 0 -, which is then suppressed by dephasing. For a quantum dot with many-channel point contacts (as well as for a quantum dot with single-channel point contacts and a large dephasing rate), the current distribution is Gaussian. The width of the distribution decreases monotonically with increasing dephasing rate. For dephasing rates γ φ much larger than the escape rate γ ∼ (N 1 +N 2 )∆ to the reservoirs, the r.m.s. current decays ∝ (γ/γ φ ) 3/2 (see also Ref. 8). Not only dephasing, but also thermal smearing can reduce the size of the pumped current. Thermal smearing has been considered by Shutenko et al., 8 who found that, without dephasing, the r.m.s. current decays ∝ (γ/T ) −1/2 for temperature T ≫ γ. Experimentally, the dephasing rate γ φ has been found to increase ∝ T (or faster), which implies that, for such a dephasing rate, dephasing is the more effective mechanism to reduce the current pumped in a quantum pump. It is interesting to compare the quantum pump to a (quantum) rectifier. The latter device generates a dc current in response to an a.c. bias voltage. For a setup similar to the quantum dot pump considered here, the variance of the rectified current is proportional to (∂G/∂X) 2 . 25 For large dephasing rates, typically (∂G/∂X) 2 ∼ (γ/γ φ ) 3 , 26 so that the r.m.s. rectified current decays ∝ (γ/γ φ ) 3/2 . As in an experimental realization a true quantum pump may coexist with a rectifier (the a.c. voltages arising from displacement currents and parasitic capacitive coupling), 27 we conclude that dephasing does not change the relative importance of one mechanism over the other. In this appendix we outline the calculation of the moments i 2 in the presence of controlled dephasing. A detailed report of the calculations will appear in Ref. 29. In order to calculate the average i 2 , we need to perform an average over the N × N matrices U , U ′ , Q 1 , and Q 2 defined in Eq. (7). The matrices U and U ′ are uniformly distributed in the unitary group [U ′ = U T in the presence of time-reversal symmetry (TRS)], while the distribution of the matrices Q 1 and Q 2 is given by Eq. (8). First we perform the average over Q 1 and Q 2 . Hereto, we parameterize Q 1 and Q 2 as Q i = Ψ †−1 H i Ψ −1 , i = 1, 2,(A1) where H 1 and H 2 are hermitian N × N matrices (real symmetric in the presence of time-reversal symmetry) and Ψ is a complex (real) N × N matrix such that 28 Q ε = Ψ †−1 Ψ −1 .(A2) Substitution of Eqs. (A1) and (A2) into the distribution (8) shows that the elements of H 1 and H 2 have a Gaussian distribution with zero mean and with variance H ij H kl = 4 (δ il δ jk + δ ik δ jl ) TRS, 4δ il δ jk no TRS.(A3) We first perform the Gaussian average over the matrices H 1 and H 2 . The resulting expression contains the matrix Ψ in the combination Q ε = Ψ †−1 Ψ −1 only. We decompose Q ε in terms of its eigenvalue matrixτ and a random unitary (orthogonal) matrix V , Q ε = Vτ V † .(A4) and average over V using the diagrammatic method of Ref. 22 (or its straightforward generalization to orthogonal matrices V in the case when time-reversal symmetry is present). The result of the average is an expression that depends on the scattering matrix S and the matrix of eigenvaluesτ only. In the absence of time-reversal symmetry, the result of this calculation is i 2 dir = 8 π 2 N (N 2 − 1) trτ 2 (trτ ) 2 − trτ 4 × N 1 N 2 + N 3 N 1 G 2 23 + N 2 G 2 13 (G 13 + G 23 ) 2 (A5) while in the presence of time-reversal symmetry, we find i 2 dir = 8 π 2 N (N − 1) G 12 + G 13 G 23 G 13 + G 23 trτ 2 (trτ ) 2 − (trτ 2 ) 2 + 2trτ (trτ 3 ) − 2trτ 4 .(A6) The contribution from i rect is a factor N smaller than i dir for large N . Now, the large-N result, Eq. (10), follows easily employing the known density of dimensionless delay times, ρ(τ ) = j δ(τ − τ j ) = N 2πτ 2 (τ + − τ )(τ − τ − ),(A7) where τ ± = (3 ± √ 8)/N , to integrate overτ . For the case N 1 = N 2 = 1, we were able to obtain exact expressions for i dir i rect and i 2 rect for arbitrary N 3 and time-reversal symmetry is broken. These expressions, which contain up to a product of four traces involving the matrixτ , were too lengthy to be reported here, and will be published in Ref. 29. The results for i 2 are shown in Fig. 2. We summarize the main steps of the calculation below. Starting from Eq. (A5) and similar expressions for the variance of i rect , 29 we integrate over the eigenvalues τ j of the dimensionless time-delay matrix Q ε and the scattering matrix S. To average over the τ j , j = 1, . . . , N , we introduce the dimensionless escape rates x n = 1/τ n which are distributed according to the generalized Laguerre ensemble, 21 P (x 1 , . . . , x N ) ∝ n<m (x n − x m ) 2 n x N n exp(x n ). (A8) Since the τ n appear only in products of up to four traces, we need the marginal n-point distributions R n (x 1 , . . . , x n ) for n = 1, . . . , 4 only, where R n (x 1 , . . . , x n ) = N ! (N − n)! × ∞ 0 dx n+1 . . . dx N P (x 1 , . . . , x N ). We can find exact expressions of R 1 , . . . , R 4 using the associated Laguerre polynomials L N n (x) which are orthogonal with respect to the weight function x N e −x . in particular, for the case of broken time-reversal symmetry, R n (x 1 , . . . , x n ) = det [K(x i , x j )] i,j=1,...,n , where K(x i , x j ) = N l=1 L N l (x i )L N l (x j )e (xi+xj )/2 (x i x j ) N/2 . The average over S can be done using the polar decomposition of S, S = u 0 0 v √ 1 − t † t it † it √ 1 − t † t u ′ 0 0 v ′ , where u and u ′ (v and v ′ ) are 2 × 2 (N 3 × N 3 ) unitary matrices and t is an N 3 ×2 matrix with all elements equal to zero except t nn = √ T n , n = 1, 2. In the presence of time-reversal symmetry, u ′ = u T and v ′ = v T . The parameters T 1 and T 2 govern the escape rate into the voltage probe. The uniform distribution of U and U ′ in the unitary group yields the integration measure dS = |T 1 − T 2 | 2 (T 1 T 2 ) N3−2 dudu ′ dvdv ′ dT 1 dT 2 . The average over S then reduces to an integral over T 1 , T 2 and one angle φ uniformly distributed in the interval 0 < φ < π/2. In terms of these variables we can write G 13 = T 1 cos 2 φ + T 2 sin 2 φ G 23 = T 1 sin 2 φ + T 2 cos 2 φ tr S 33 S † 33 S 33 S † 33 = N − 4 + (1 − T 1 ) 2 + (1 − T 2 ) 2 tr S † 13 S 13 S † 33 S 33 = (1 − T 1 )T 1 cos 2 φ + (1 − T 2 )T 2 sin 2 φ, where S ij denotes the ij block in the scattering matrix S. (The last two terms appear in the expression for i rect . 29 ) APPENDIX B: In order to obtain the full distribution of the pumped current i, we numerically generated matrices U , U ′ , Q 1 , and Q 2 according to the appropriate distributions. The matrices U and U ′ are uniformly distributed in the unitary group (U ′ = U T in the presence of time-reversal symmetry). Using a parameterization in terms of Euler angles, 30 their generation is relatively straightforward. The numerical generation of matrices Q 1 and Q 2 according to the distribution (8) makes use of a trick that was inspired by Ref. 31, which would like to explain below. We parameterize the N × N matrices Q 1 , Q 2 , and Q ε as Q ε = C T−1 C −1 , Q i = C T−1 H i C −1 , i = 1, 2 (B1) where C is a complex N × 2N matrix [real N × (2N + 1) matrix in the presence of time-reversal symmetry] and H i (i = 1, 2) is a hermitian 2N × 2N matrix [real symmetric (2N + 1) × (2N + 1) matrix in the presence of time-reversal symmetry]. In Eq. (B1), the inverse C −1 is defined as the right-inverse, C −1 = C T (CC T ) −1 . In Ref. 31 it was shown that the matrix Q ε has the distribution (8) if the elements of C are all chosen from a Gaussian distribution with unit variance. From there, one can show by substitution of Eq. (B1) into Eq. (8), that the parameterization (B1) reproduces the correct distribution (8) for all three matrices Q ε , Q 1 , and Q 2 if the elements of the matrices H 1 and H 2 are all chosen from a Gaussian distribution with variance given by Eq. (A3) above. 1. A quantum dot with two parameters X1 and X2 which describe a deformation of its shape. As X1 and X2 are varied periodically a dc current I = I1 = −I2 flows from the right to the left reservoir. The lead on the bottom, through which no current flows (I3 = 0), has a variable number of channels N3. FIG. 2 . 2Variance of the pumped current with one open channel in each of the current carrying leads (N1 = N2 = 1) as a function of the number of open channels in the third lead (N3). The presence (absence) of time-reversal symmetry is denoted by open circles (closed circles). The dashed line is the variance in the asymptotic limit N1 = N2 = 1, N3 → ∞. The variance in the presence of time-reversal symmetry is infinite for N3 < 6. FIG. 3 . 3The probability distribution of the pumped current without (a) and with (b) time-reversal symmetry, for the case N1 = N2 = 1 of single-channel point contacts in the current-carrying leads and as a function of the number of channels N3 in the voltage probe. The dashed line shows the distribution without dephasing (N3 = 0). The solid lines show the distribution with N3 = 1, (lowest curve at i = 0) 2, ... , N3 = 10 (highest curve). FIG. 4 .FIG. 5 . 45(10) above with N 3 replaced by γ. This result agrees with what was previously obtained by Shutenko et al. using a different method. The variance of the pumped current in the case of intrinsic dephasing as a function of dimensionless dephasing rate γ in the presence (dashed curve) and absence (solid curve) of time-reversal symmetry. The asymptotic result (dotted curve) is also shown. There is one open channel in each lead (N1 = N2 = 1The probability distribution of the pumped current with N1 = N2 = 1 in the presence of time-reversal symmetry and intrinsic dephasing. The dashed curve is the probability distribution in the absence of dephasing. The two solid curves have dimensionless dephasing rates γ = .1 (highest) and γ = 1 (lowest). The dot-dashed curves have dephasing rates γ = 8 (lowest) and γ = 15 (highest). The inset shows the probability distribution of the pumped current with N1 = N2 = 1 in the absence of time-reversal symmetry with the same parameters. The distribution with γ = .1 is indistinguishable from the case without dephasing (γ = 0). ACKNOWLEDGMENTS We thank B.I. Halperin and C. Marcus for stimulating discussions. Shortly before completion of our manuscript, we learned of Ref.18, where the effect of dephasing in an electron pump is studied with an approach similar to ours. This work was supported by the NSF under grant Mesoscopic Quantum Physics. E. Akkermans, G. Montambaux, J.-L. Pichard, and J. Zinn-JustinNorth-Holland, AmsterdamMesoscopic Quantum Physics, edited by E. Akkermans, G. Montambaux, J.-L. Pichard, and J. Zinn-Justin (North- Holland, Amsterdam, 1995). L P Kouwenhoven, C M Marcus, P L Mceuen, S Tarucha, R M Westervelt, N S Wingreen, Mesoscopic Electron Transport. L. L. Sohn, L. P. Kouwenhoven, and G. SchönDordrechtKluwerL. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R. M. Westervelt, and N. S. Wingreen, in Meso- scopic Electron Transport, edited by L. L. Sohn, L. P. Kouwenhoven, and G. Schön (Kluwer, Dordrecht, 1997). . M Switkes, C M Marcus, K Campman, A C Gossard, Science. 2831905M. Switkes, C. M. Marcus, K. Campman, and A. C. Gos- sard, Science 283, 1905 (1999). . B Spivak, F Zhou, M T Beal Monod, Phys. Rev. B. 5113226B. Spivak, F. Zhou, and M. T. Beal Monod, Phys. Rev. B 51, 13226 (1995). . P W Brouwer, Phys. Rev. B. 5810135P. W. Brouwer, Phys. Rev. B 58, 10135 (1998). . F Zhou, B Spivak, B L Altshuler, Phys. Rev. Lett. 82608F. Zhou, B. Spivak, and B. L. Altshuler, Phys. Rev. Lett. 82, 608 (1999). . B L Altshuler, L I Glazman, Science. 2831864B. L. Altshuler and L. I. Glazman, Science 283, 1864 (1999). . T A Shutenko, I L Aleiner, B L Altshuler, Phys. Rev. B. 6110366T. A. Shutenko, I. L. Aleiner, and B. L. Altshuler, Phys. Rev. B 61, 10366 (2000). . J E Avron, A Elgart, G M Graf, L Sadun, Phys. Rev. B. 6210618J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, Phys. Rev. B 62, 10618 (2000). . M G Vavilov, V Ambegaokar, I L Aleiner, Phys. Rev. B. 63195313M. G. Vavilov, V. Ambegaokar and I. L. Aleiner, Phys. Rev. B 63, 195313 (2001). . I L Aleiner, B L Altshuler, A Kamenev, Phys. Rev. B. 6210373I. L. Aleiner, B. L. Altshuler, and A. Kamenev, Phys. Rev. B 62, 10373 (2000). . M L Polianski, P W Brouwer, Phys. Rev. B. 6475304M. L. Polianski and P. W. Brouwer, Phys. Rev. B 64, 75304 (2001). . L P Kouwenhoven, A T Johnson, N C Van Der, C J P M Vaart, C T Harmans, Foxon, Phys. Rev. Lett. 671626L. P. Kouwenhoven, A. T. Johnson, N. C. van der Vaart, C. J. P. M. Harmans, and C. T. Foxon, Phys. Rev. Lett. 67, 1626 (1991). . H Pothier, P Lafarge, C Urbina, D Esteve, M H Devoret, Europhys. Lett. 17249H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M. H. Devoret, Europhys. Lett. 17, 249 (1992). . M Büttiker, Phys. Rev. B. 333020M. Büttiker, Phys. Rev. B 33, 3020 (1986); . IBM J. Res. Dev. 3263IBM J. Res. Dev. 32, 63 (1988). . H U Baranger, P A Mello, Phys. Rev. B. 514703H. U. Baranger and P. A. Mello, Phys. Rev. B 51, 4703 (1995). . P W Brouwer, C W J Beenakker, Phys. Rev. B. 554695P. W. Brouwer and C. W. J. Beenakker, Phys. Rev. B 55, 4695 (1997). . M Moskalets, M Büttiker, cond-mat/0108061M. Moskalets and M. Büttiker, cond-mat/0108061. . I L Aleiner, P W Brouwer, L I Glazman, cond- mat/0103008I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, cond- mat/0103008. . M Büttiker, J. Phys. Condens. Matter. 59361M. Büttiker, J. Phys. Condens. Matter 5, 9361 (1993); . M Büttiker, H Thomas, A Prêtre, Z. Phys. B. 94133M. Büttiker, H. Thomas, and A. Prêtre, Z. Phys. B 94, 133 (1994). P W Brouwer, K M Frahm, C W J Beenakker, Waves in Random Media. 7891P. W. Brouwer, K. M. Frahm, and C. W. J. Beenakker, Phys. Rev. Lett. 78, 4737 (1997); Waves in Random Media 9, 91 (1999). . P W Brouwer, C W J Beenakker, J. Math. Phys. 374904P. W. Brouwer and C. W. J. Beenakker, J. Math. Phys. 37, 4904 (1996). . P A Mello, P Pereyra, T H Seligman, Ann. Phys. 161254P. A. Mello, P. Pereyra and T. H. Seligman, Ann. Phys. 161, 254 (1985). . P W Brouwer, Phys. Rev. B. 5116878P. W. Brouwer, Phys. Rev. B 51, 16878 (1995). . P W Brouwer, Phys. Rev. B. 63121303P. W. Brouwer, Phys. Rev. B 63, 121303 (2001). . C W J Beenakker, Rev. Mod. Phys. 69731C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). . M Switkes, StanfordPh. D. thesisM. Switkes, Ph. D. thesis (Stanford, 1999). A2) only fixes Ψ up to right-multiplication with a unitary matrix (orthogonal matrix in the presence of time-reversal symmetry). However, this unitary (orthogonal) matrix can be absorbed into H1 and H2, the distribution of which is. Note that Eq.invariant under unitary (orthogonal) transformationsNote that Eq. (A2) only fixes Ψ up to right-multiplication with a unitary matrix (orthogonal matrix in the presence of time-reversal symmetry). However, this unitary (orthog- onal) matrix can be absorbed into H1 and H2, the dis- tribution of which is invariant under unitary (orthogonal) transformations. . J N H J Cremers, Harvard UniversityPh.D. Thesisin preparationJ.N.H.J. Cremers, Ph.D. Thesis, Harvard University, in preparation. . K Zyczkowski, M Kus, Phys. Rev. E. 53319K. Zyczkowski and M. Kus, Phys. Rev. E 53, 319 (1996). . E Brézin, S Hikami, A Zee, Nucl. Phys. B. 464411E. Brézin, S. Hikami, and A. Zee, Nucl. Phys. B 464, 411 (1996).
[]
[ "The inverse problem beyond two-body interaction: the cubic mean-field Ising model", "The inverse problem beyond two-body interaction: the cubic mean-field Ising model" ]
[ "Pierluigi Contucci ", "Godwin Osabutey ", "Cecilia Vernia ", "\nDipartimento di Matematica\nDipartimento di Scienze Fisiche Informatiche e Matematiche\nUniversità di Bologna\nItaly\n", "\nUniversità di Modena e Reggio Emilia\nModenaItaly\n" ]
[ "Dipartimento di Matematica\nDipartimento di Scienze Fisiche Informatiche e Matematiche\nUniversità di Bologna\nItaly", "Università di Modena e Reggio Emilia\nModenaItaly" ]
[]
In this paper we solve the inverse problem for the cubic mean-field Ising model. Starting from configuration data generated according to the distribution of the model we reconstruct the free parameters of the system. We test the robustness of this inversion procedure both in the region of uniqueness of the solutions and in the region where multiple thermodynamics phases are present.
10.1103/physreve.107.054124
[ "https://export.arxiv.org/pdf/2210.03260v1.pdf" ]
252,762,607
2210.03260
3f37da123c19a4cb57a2157004140a482b9c9405
The inverse problem beyond two-body interaction: the cubic mean-field Ising model 6 Oct 2022 (Dated: October 10, 2022) Pierluigi Contucci Godwin Osabutey Cecilia Vernia Dipartimento di Matematica Dipartimento di Scienze Fisiche Informatiche e Matematiche Università di Bologna Italy Università di Modena e Reggio Emilia ModenaItaly The inverse problem beyond two-body interaction: the cubic mean-field Ising model 6 Oct 2022 (Dated: October 10, 2022) In this paper we solve the inverse problem for the cubic mean-field Ising model. Starting from configuration data generated according to the distribution of the model we reconstruct the free parameters of the system. We test the robustness of this inversion procedure both in the region of uniqueness of the solutions and in the region where multiple thermodynamics phases are present. In this paper we solve the inverse problem for the cubic mean-field Ising model. Starting from configuration data generated according to the distribution of the model we reconstruct the free parameters of the system. We test the robustness of this inversion procedure both in the region of uniqueness of the solutions and in the region where multiple thermodynamics phases are present. I. INTRODUCTION In this paper we study the inverse problem for a class of mean-field models in statistical mechanics with cubic interaction. The direct problem of statistical mechanics is to compute macroscopic variables (i.e. the average values of magnetizations and correlations) when the couplings and fields are known. In the inverse problem the reverse is done: the couplings and fields are computed using the (statistical) datum of the macroscopic quantities. This technique is known sometimes as Boltzmann machine learning, a special case of learning in statistical inference theory [1,2] when the probability measure is the Boltzmann-Gibbs one. In recent years, studies in deep learning for artificial intelligence have been approached in terms of inverse problem in statistical mechanics [3][4][5]. The techniques to study that case are of very different nature than those we treat in this work because the parameters to be identified are of very high dimension and the involved models concern the theory of disordered systems [6]. Although in this study we are only interested in computing three parameters, we believe that a robust understanding of the statistical mechanics low-dimensional inverse problem may shed some light in the general Boltzmann machine learning problem due to the presence of phase transitions for very large systems. A further reason of interest for the problem we deal with is that, in recent times, this method has attracted some attention due to it's ability to advance a useful novel approach for several applications like neural networks, protein structures, computer vision [7][8][9][10][11], and the socioeconomic sciences [12][13][14][15][16][17][18][19][20][21]. The system we consider here is made of Ising spins and, beside an homogeneous magnetic field and a constant two body interaction, it contains a constant three body term. One of the peculiarities of this model, which turns out to have a cubic Hamiltonian function, is that it lacks the * [email protected] † [email protected] ‡ [email protected] standard convexity property of its quadratic version and its direct and inverse problems are therefore outside the general methods of convex optimization problems. Taking into account the three-body term, we move from a generic graph (network) structure where we consider only dyadic or pairwise interactions into hypergraphs where faces are also considered [22][23][24]. This allows for the consideration of a large spectrum of applications that are closely related to real-world phenomena, such as team collaborations rather than collaborations between pairs (see [25]). According to [23,25] the presence of higherorder interactions, such as three or more body interactions, may have significant impact also on the dynamics of interacting networked systems and potentially lead to abrupt transitions between states. Abrupt transitions are a prevalent phenomenon in nature that can be found in everything from social networks to biology [25,26]. The model we consider is invariant under the permutation group but its extension to the case in which that symmetry is not present has been already considered in [27] with the same perspectives of the multi-populated quadratic models [15,28]. An intriguing feature of such model is that it shows a discontinuous first-order phase transition which is not present in the case of the standard quadratic mean-field model. To solve the inverse problem we first compute, exploiting the exact solution of the model [27,29], the analytical formulas for the system's macroscopic variables in the thermodynamic limit where they provide explicit expressions for the interaction couplings (cubic and quadratic) and the magnetic field. It is worth noticing that since the number of necessary relations to compute the free parameters is three we need to make observations up to the third moment of the probability distribution. To relate the analytical inversion with the (statistical) observations we use the maximum likelihood criteria and we advance a link between estimated and theoretical values. Finally, we test how well the model's free parameters are reconstructed using the inversion formulas and how their robustness is affected by both the system size and the number of independent samples simulated from the model's equilibrium configuration. The paper is organised as follows. The cubic mean-field model is introduced in Section II where it has been shown how to compute and test the robustness of the analytical inverse formulas using the maximum likelihood estimation procedure. Section III is devoted to the numerical testing of the robustness of the inversion formulas for unique stable solutions. In Section IV the case of metastable or multiple solutions for finite-size systems is discussed. The final section, Section V, provides a general conclusion and the model's future prospects. H N (σ) = − N i,j,k=1 K i,j,k σ i σ j σ k − N i,j=1 J i,j σ i σ j − N i=1 h i σ i . (1) Assuming mean-field interaction, we set K i,j,k = K 3N 2 , J i,j = J 2N and h i = h where K, J are the cubic and binary spin coupling and h is the external magnetic field. Hence, the Hamiltonian per particle is H N (σ) = −N K 3 m 3 N (σ) + J 2 m 2 N (σ) + hm N (σ) , (2) where m N (σ) = 1 N N i=1 σ i(3) is the magnetisation per particle of the configuration σ. The Boltzmann-Gibbs state on a configuration σ is given by P N,K,J,h (σ) = e −HN (σ) Z N ,(4) where Z N = σ∈ΩN e −HN (σ) is the partition function of the system. As a result, we obtain the pressure function per particle associated with the thermodynamic system as: p N = 1 N log Z N .(5) For a given observable f (σ) the Boltzmann-Gibbs expectation ω N (f (σ)) is defined as follows: ω N (f (σ)) = σ∈ΩN f (σ)e −HN (σ) Z N .(6) Furthermore, the pressure function (5) can be used to generate the moments of the system with respect to the Boltzmann-Gibbs measure. Hence, one obtains the following finite-size quantities: ∂p N ∂h = ω N (m N (σ)) (7) ∂ 2 p N ∂h 2 = χ N = N [ω N (m 2 N (σ)) − ω 2 N (m N (σ))](8) and ∂ 3 p N ∂h 3 = ψ N = N 2 [ω N (m 3 N ) − 3ω N (m N )ω N (m 2 N ) + 2ω 3 N (m N )](9) where ω N (m N (σ)), χ N and ψ N are the finite-size average magnetisation, susceptibility and third moment respectively. The considered model can be solved exactly [29] using the large deviations technique, which was proposed in [30]. The thermodynamic limit of (5) admits the following variational representation [29]: p(K, J, h) = lim N →∞ p N = sup m∈[−1,1] φ(m),(10) where φ(m) = U (m) − I(m) with U (m) = K 3 m 3 + J 2 m 2 + hm(11) is the energy contribution and I(m) = 1 − m 2 log 1 − m 2 + 1 + m 2 log 1 + m 2 (12) is the entropy contribution. The stationarity condition, that acts as a consistency equation, gives m = tanh(Km 2 + Jm + h),(13) and must be satisfied by the solutions of the variational principle (10). In order to solve the inverse problem analytically for a given configuration of spin particles, we first find the relation between the model parameters and the variational principle (10). Observe that, ∂p ∂h = m, i.e., m = tanh (Km 2 + Jm + h),(14)∂ 2 p ∂h 2 = χ = (1 − m 2 ) 1 − (1 − m 2 )(J + 2Km) and(15)∂ 3 p ∂h 3 = ψ = 2χ 2 (K(1 − 3m 2 ) − Jm)χ − m (1 − m 2 ) .(16) The peculiar feature of the cubic mean-field model is the presence of three distinct stable phases in the magnetic order parameter m. Unlike the usual quadratic model, here an unpolarised stable phase close to m = 0 appears beyond the usual two phases of positive and negative magnetization. From Figure 1 one can observe a triple point (K, J, h) = (0, 1, 0) where all the three phases meet [27]. Let us consider the model in its simplest form with zero quadratic coupling and magnetic field i.e. when J = h = 0 and only the cubic coupling in (2) is present. It is worth mentioning that when J = h = 0 and K is progressively increased from negative to positive, one encounters two transitions: from a negatively polarized phase to an unpolarized one and from an unpolarized phase to a positively polarized one (see Fig. 1; and Fig 1. of [27]). In Figure 2 we illustrate an example of critical behaviour for our model with the presence of phase transitions occurring at J = h = 0 when K is varied. The quantities, m, χ and ψ are the infinite volume limit average magnetisation, susceptibility and third moment corresponding to the finite-size quantities ω N , χ N and ψ N respectively in the thermodynamic limit. The system of equations (14), (15) and (16) has three unknowns K, J and h which can be solved. Having knowledge of m, χ and ψ one can compute the parameters (i.e. K, J and h) of the model through the following equations: At the critical point the susceptibility as seen in (b) and the third moment in (c) has a jump to 1 and a jump to around ±4 respectively. K = m (1 − m 2 ) 2 + ψ 2χ 3 ,(17)-4 -3 -2 -1 0 1 2 3 4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (a) Magnetization -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 1.2 (b) SusceptibilityJ = 1 1 − m 2 − 1 χ − 2Km(18) and the external magnetic field is then obtained from (13) as h = tanh −1 (m) − Km 2 − Jm.(19) Let us observe that, in the region of the parameter space where the consistency equation (13) has a unique stable solution the following holds: lim N →∞ ω N (m N (σ)) = m.(20) In analogy to the behaviour of the quadratic case [31], the Boltzmann-Gibbs measure (4) may be multimodal for some (K, J, h) in the parameter space for both the finitesize system and in the thermodynamic limit. In this case equation (20) fails to hold. We will discuss later how to handle such a case, following the work done in [31,32]. The procedure discussed so far deals with the analytical inverse problem. The remainder of this section will be devoted to the statistical procedure required to compute the estimators of K, J and h. We start by generating M independent configurations σ (1) , ..., σ (M) distributed according to (4) from the model's equilibrium configuration. Notice that the analytical inverse formulas of K, J and h in equations (17), (18) and (19) respectively, are valid on the infinite volume limit of the observables, i.e. m, χ and ψ. Hence, to compute the estimates of the model parameters K, J and h, the maximum likelihood estimation procedure will be adopted having knowledge of real data. This procedure ensures that the estimated model parameters maximize the probability of getting the given sample of spin configurations from the distribution. Furthermore, the analytical inverse procedure requires statistical approximation of the infinite volume limit quantities (i.e. m, χ and ψ) which are substituted by their finite-size forms ω N , χ N and ψ N . The maximum likelihood function for the measure (4) is defined as L(K, J, h) = P N,K,J,h {σ (1) , ..., σ (M) } = M l=1 P N,K,J,h {σ (l) } = M l=1 e −HN (σ (l) ) σ∈ΩN e −HN (σ) . This procedure will enable defining the finite-size magnetisation ω N (m N (σ)) in terms of the empirical average (i.e. m N ) for each of the M sampled spin configurations. Further, we have that ln L(K, J, h) = M l=1 (−H N (σ (l) )) − ln σ∈ΩN e −HN (σ) .(21) The derivatives with respect to the parameters K, J and h are given below as: ∂ ∂h ln L(K, J, h) = N M l=1 m N (σ (l) ) − ω(m N (σ)) ∂ ∂J ln L(K, J, h) = N 2 M l=1 m 2 N (σ (l) ) − ω(m 2 N (σ)) ∂ ∂K ln L(K, J, h) = N 3 M l=1 m 3 N (σ (l) ) − ω(m 3 N (σ)) and they vanish when ω N (m N (σ)) = 1 M M l=1 m N (σ (l) ) ω N (m 2 N (σ)) = 1 M M l=1 m 2 N (σ (l) ) ω N (m 3 N (σ)) = 1 M M l=1 m 3 N (σ (l) ).(22) The function L(K, J, h) is at its maximum when the first, second and third moments of the magnetization in equation (22) are obtained. It is worth noticing that m N (σ (l) ) = 1 N N i=1 σ (l) i for l = 1, . . . , M(23) are the total magnetizations of the M sample spin configurations. Now, the inverse problem can be solved when we make use of (17), (18), (19) and (22). The maximum likelihood procedure computes the estimators of the infinite volume quantities m, χ and ψ, from a sample data set through the following: m = 1 M M l=1 m N (σ (l) ),(24)χ = N 1 M M l=1 m 2 N (σ (l) ) − m 2(25) and ψ = N 2 1 M M l=1 m 3 N (σ (l) ) − 3 m 1 M M l=1 m 2 N (σ (l) ) + 2 m 3 .(26) We now define the estimators of the three parameters of the cubic mean-field model using the statistical estimators for the magnetization, susceptibility and third moment (24), (25) and (26) in the infinite volume limit relations among those quantities (17), (18) and (19) K = m (1 − m 2 ) 2 + ψ 2 χ 3 ,(27)J = 1 1 − m 2 − 1 χ − 2 K m,(28) and h = tanh −1 ( m) − K m 2 − J m.(29) At the critical point (K, J, h) = (0, 1, 0) where all the three phases meet the magnetization is zero and the infinite volume magnetic susceptibility χ and the third moment ψ defined by equations (15) and (16) respectively diverge. Hence, the inversion formulas (17), (18) and (19) does not hold as it will be illustrated at the end of the next section. We do not include the inversion formulas at the critical point in this work but the problem will be considered in future work. III. TEST FOR THE CASE OF UNIQUE SOLUTION In this section we are going to examine how the inversion equations perform for different and increasing choices of N and M , respectively the number of particles and sampled configurations. The specific case we consider is the inversion problem for those values of the triple (K, J, h) where there is a unique stable solution of (13). In this case, the Boltzmann-Gibbs distribution of the total magnetisation has a unique peak always centered around the analytic solution m: some examples are shown in Figure 3 for fixed N . The accuracy of the estimation increases as N and M increase. The parameters K, J and h are obtained from the computation of the finite-size quantities m N , χ N and ψ N using configurations extracted from the Boltzmann-Gibbs distribution of the data. Estimation of m N , χ N and ψ N for fixed triples of the parameters (K, J, h) and varying N ∈ [500, 10000] are shown in Figure 4. In the same figure, the thermodynamic limits of those quantities are also shown. From Figure 4 we can observe the monotonic behaviour of m N , χ N and ψ N as N increases. In Figure 5 we study the relationship between the absolute difference of the finite-size quantities and their corresponding thermodynamic values as a function of the system size N . We find evidence that the finite-size quantities m N , χ N and ψ N converge to their true values with a power law behaviour as N increases. The obtained results indicate that using N = 10000 one can estimate the infinite volume magnetisation, susceptibility and the third moment with vanishing error. We will proceed to use N = 10000 as the size for each of the M independent spin configurations σ (1) , ..., σ (M) . Further numerical tests will be performed to determine a suitable number of sample configurations M that can be used for reconstructing the model parameters using the inversion formulas. To obtain the standard deviations associated to the reconstruction of the estimators, we simulate from the model's equilibrium configuration 50 different instances In the sequel, we study the behaviour of the reconstructed parameter for fixed values of J and h and varying K (Figures 8 and 9) and also for fixed values of K and h and varying J (Figures 10 and 11). The simulations are performed using M = 20000, N = 10000 and error bars are standard deviations on 50 different M -samples of the same system. We find all the reconstructed parameter values in good agreement with the exact ones. We can observe that as the intensity of the cubic and quadratic coupling increases the error bars associated to the reconstructed parameters grow, as we can expect since in that region of the parameter space the system is more disordered due to the presence of multiple local stable states and the fluctuations are greater. Furthermore, Figure 12 show the reconstructed parameters as a function of N at the critical point (K = 0, J = 1, h = 0). It can be noticed that the reconstruction at the critical point for K and h agrees with their exact values with only a small percentage of error and that of J is underestimated. It worth observing that when K = h = 0 and J > 1 the consistency equation (13) has two stable solutions. In this case, for the finite-size system and in the thermodynamic limit, the Boltzmann-Gibbs distribution of the total magnetization presents two peaks each centered around one of the stable solutions. In such a case the inverse problem procedure discussed in Section II cannot be used for the reconstruction of the model parameters. We refer readers to [28] where this case has been studied using the spin flip approach due to symmetry of the solu- tion in both finite-size and infinite volume systems for the quadratic mean-field model. The clustering algorithm to be outlined in the next section provides a more general approach to handle the reconstruction of the model parameters when the phase space has multiple locally stable solution. IV. CLUSTERING ALGORITHM FOR METASTABLE STATE SOLUTIONS Here, we focus on cases where equation (13) has a metastable solution. This corresponds to the case where there are more than one locally stable solution of the consistency equation (13). For this model, equation (13) can have at most three solutions and φ has at most two local maxima for fixed (K, J, h). The existence of the metastable solution in the infinite volume limit is represented at finite N by the occurrence of an extra peak in the distribution. Therefore, while in the thermodynamic limit the Boltzmann-Gibbs distribution of the magnetisation is unimodal with the peak corresponding to the stable solution, in the finite size case also the peak corresponding to the metastable one is present and the distribution is bimodal. Hence, in this case, the inversion problem cannot be studied globally, as done in the previous section. Instead, the procedure has to be applied locally, that is to each subset of configurations clustered around the two local maxima. Given M spin configurations, σ (1) , ..., σ (M) , we perform the reconstruction by first partitioning the M configurations in clusters according to their local densities around each local maximum. More precisely, using the clustering algorithm discussed in [32][33][34][35][36] we divide the M configurations into different clusters using the mutual distances between their magnetizations of each configuration. Configurations form a cluster if the magnetization distances are less than a fixed threshold d c . The choice of the optimal threshold is obviously crucial: a too small threshold will produce too many clusters, while a too large one will give only one cluster. Given d c , for each configuration l the algorithm computes two quantities: the local density ρ l , defined as the number of magnetizations within the given distance d c to the magnetization of σ (l) , and the minimum distance δ l between the magnetization of configuration l and any other configuration with a higher density. The algorithm is based on the assumptions that the cluster centers are surrounded by points with a lower density, and that the centers are at a relatively large distance from each other. For each configuration, plotting the minimum distance δ as a function of the local density ρ provides a decision graph that gives the cluster centers: the cluster centers are the outliers in the graph. Finally, each remaining configuration is assigned to the same cluster of its nearest neighbor of higher density. In this study, we identify two clusters C k , k = 1, 2, using the optimal threshold d c = 0.001. Notice that it is not possible to observe three clusters in the inverse problem due to the analytical properties of the consistency equation (13). Then, for each cluster C k , k = 1, 2 we compute the estimates of the finite-size quantities, m, χ and ψ, and the corresponding K, J, h. More precisely, we can define the estimators of the finite-size quantities with reference to the clusters as follows: m C k = 1 M k l∈C k m N (σ (l) ),(30)χ C k = N 1 M k l∈C k m 2 N (σ (l) ) − m 2 C k(31) and ψ C k = N 2 1 M k l∈C k m 3 N (σ (l) ) − 3 m C k 1 M k l∈C k m 2 N (σ (l) ) + 2 m 3 C k ,(32) where M k is the size of the cluster C k , k = 1, 2 such that M 1 + M 2 = M . After obtaining the quantities above, we now compute the estimated values, K C k , J C k , h C k , using equations (27), (28) and (29) for each cluster and com-Observe that if a point (K, J, h) in the parameter space corresponds to a metastable solution (at finite volume) and it is sufficiently distant from the coexistence curve, we can expect a better reconstruction of the parameters by applying equations (27), (28) and (29) to the configurations in the largest cluster. However, if the point (K, J, h) is close to the coexistence curve, a better reconstruction can be expected using the density clustering algorithm, i.e. by using (33), (34) and (35). Figure 13 illustrates how the Boltzmann-Gibbs measure of the magnetization is changing with varying K, J and h in each column starting from the left respectively. A. Test for metastable state solutions The inverse problem is solved using the density clustering algorithm as discussed and identifying a suitable As is evident from Figure 14, the cluster centered around m 1 (i.e. C 1 ) has more configurations as compared to the other cluster centered around m 2 (i.e. C 2 ). We get the following reconstructed estimates for the parameter values by applying equations (27), (28) and (29) to the setups in both clusters (i.e. C 1 and C 2 ) according to formulas (33), (34) and ( Instead, we obtain the following reconstructed parameter values by applying equations (27), (28) and (29) just to the configurations in the more dense cluster C 1 : ( K, J, h) = (1.69 ± 0.23, 0.01 ± 0.06, 0.10 ± 0.004). Note that, the reconstructed parameters using only the configurations in the more dense cluster are in better agreement with the exact ones when compared to the reconstructed parameters on both clusters. This is an indication that the point (K, J, h) = (1.67, 0.01, 0.1) is sufficiently distant from the coexistence curve. Observe that if two clusters have the same density, we do not choose between them and the clustering algorithm provides an optimal reconstruction. Now, we perform reconstruction of the parameters using the cluster with largest size for fixed values of the model parameters and observe its performance for varying M in Figure 15. It can be observed that the reconstructed parameters are in good agreement to their corresponding exact values. As a last remark, note that, given a point (K, J, h) in a neighbourhood of the coexistence curve, one can observe a metastable state when the number of particles N is not large enough. In this case, the clustering algorithm is useful to reconstruct the parameters, but it has a high computational cost. This is easily overcome by using large number of particles, which cause the metastable state to vanish (see Figure 14) and the inversion formulas in equations (27), (28), (29) become efficient. V. CONCLUSION In this work we consider a mean-field statistical mechanics model with three-body interaction displaying a first order phase transition. We studied and solved the inverse problem and tested the statistical robustness of the inversion method. We numerically tested the inversion method for cases where the consistency equation (13) has a unique stable solution as well as more than one locally stable solution. For the case where the consistency equation (13) has multiple locally stable solution, we used the clustering algorithm to reconstruct the model parameters. Robustness was tested for different values of the number of particles N and samples M and reached the precision of a few percent for M = 2 × 10 4 . We plan to investigate in the future two extensions of the inverse problem: first to the critical point where some of the observables, such as χ and ψ, diverge and to the multipopulated version of the model that found applications to the description of human-AI ecosystems [27]. II. INVERSE PROBLEM FOR THE CUBIC MEAN-FIELD ISING MODEL Let us consider the Hamiltonian of an Ising model on N spin configurations, Ω N = {−1, +1} N , with cubic interaction and spin moments σ i = ±1, i = 1, . . . , N , defined as FIG. 1 . 1h = 0. Phase diagram of the stable solutions of(13) showing the coexistence curves. For J < 1, three distinct phases are observed: the negatively polarised phase (in blue), the zero or unpolarized phase (in gray), and the positively polarised phase (in red). As a result, in that region, a progressive increase in K from negative to positive values encounters two consecutive jumps. FIG. 2 . 2J = 0, h = 0. First three moments of the model as a function of K: In (a) the total magnetisation shows indication of phase transitions occurring at a critical point around ±2. K= -1.2 J= 0.1 h= - 0 K= 1.05 J= 0.8 h= 0. 01 FIG. 3 . 0013Boltzmann Gibbs distribution of the total magnetisation for N = 1000 and different set of triples (K, J, h). FIG. 4 . 4Finite-size average magnetization mN , susceptibility χN and third moment ψN as functions of N for three different set of triples (K, J, h). Blue crosses represent the values of mN (upper panels), χN (middle panels) and ψN (lower panels) for varying N .As N increases mN , χN and ψNapproach their true values in the thermodynamic limit given as the red horizontal lines for the chosen values of K, J and h. FIG. 5 . 5K = 1.8, J = 1.3, h = 0.2. Absolute error of the finite size quantities mN , χN and ΨN as functions of N together with the best power law fits. In the upper panel, |mN − m| is shown as a function of N together with the best fit aN b , where a = 0.28 ∈ (0.06, 0.50) and b = −1.37 ∈ (−1.49, −1.25) with a goodness of fit R 2 = 0.9829. The middle panel displays |χN − χ| as a function of N together with its corresponding best fit cN d , with c = 0.62 ∈ (0.14, 1.09), d = −1.37 ∈ (−1.49, −1.25) and R 2 = 0.9830 as goodness of fit. The lower panel represents |ψN − ψ| as a function of N together with its corresponding best fit gN f , with g = 1.47 ∈ (0.32, 2.62), f = −1.37 ∈ (−1.49, −1.25) and a goodness of fit R 2 = 0.9826. of the M −iid sample configurations, i.e. (σ (1) , . . . , σ (M) ), apply the maximum likelihood estimation procedure to each of them separately, solve the inverse problem using (27), (28) and (29) and then average the inferred values over the 50 different M -samples. The mean value of the estimators m, χ, ψ, and ( K, J, h) over the 50 different M -samples of spin configurations are denoted by m, χ, ψ, and ( K, J, h) respectively. The results are shown in Figures 6 and 7.Figures 6 and 7 illustrate that at M = 20000 we get smaller error bounds for the reconstruction as compared to lesser values of M . FIG. 6 .FIG. 7 . 67K = 0.5, J = 0.3, h = 0.1. Reconstructed average magnetization m, susceptibility χ and third moment ψ (blue crosses) as a function of M with standard deviation on 50 different M -sample and N = 10000. The continuous red line corresponds to m, χ and ψ in the thermodynamic limit. K = 0.5, J = 0.3, h = 0.1. K, J and h as a function of M for N = 10000. The blue crosses are the estimation of K, J and h with standard deviations on 50 different M -samples of configurations of the same system. The horizontal red line in each panel corresponds to the exact values of K, J and h. FIG. 8 . 8K as a function of K for N = 10000 and M = 20000. J = 0.3, h = 0.1 in left panel and J = 0.4, h = −0.3 in the right panel. The estimations of K are given as the blue crosses in both panels with standard deviations on 50 different M -samples of configurations of the same system. The red continuous line represents K = K. FIG. 9 .FIG. 10 .FIG. 11 . 91011J and h as a function of K for N = 10000 and M = 20000. J = 0.3, h = 0.1 in the left panels and J = 0.4, h = −0.3 in the right panels. The estimates of J and h are given as the blue crosses in all the panels with standard deviations on 50 different M -samples of configurations of the same system. The red continuous lines represent the exact values of J and h. J as a function of J for N = 10000 and M = 20000. K = 0.05, h = 0 in the left panel and K = 0.05, h = −0.02 in the right panel. The blue crosses are the reconstructed values of J in both panels with standard deviations on 50 different M -samples of configurations of the same system. The red continuous line represents the exact value J = J. K and h as a function of J for N = 10000 and M = 20000. K = 0.05, h = 0 in the left panels and K = 0.05, h = 0.02 in the right panels. The estimate of K and h are given as the blue crosses in all the panels with standard deviations on 50 different M -samples of configurations of the same system. The red continuous lines represent the exact values of K and h. FIG. 12 . 12K = 0, J = 1, h = 0. K, J and h as a function of N for M = 20000. The reconstructed estimates of K, J and h are given as the blue crosses on statistical error bars of 50 different M −samples. The red continuous line is the exact value of the parameters K, J and h in the respective panels. 1.67 J= 0.012 h= 0.103 FIG. 13. Boltzmann-Gibbs distribution of the total magnetisation with metastable states for fixed K, J, h at N = 1000. The peaks of the distribution are centered around the two solutions of the consistency equation. number of samples M for better reconstruction of the model parameters. The test is performed with M = 20000 and standard deviations are computed over 50 different M -samples from the same distribution. As an example, consider the reconstruction of the parameter values (K, J, h) = (1.67, 0.01, 0.1) for M = 20000 and N = 3000. The distribution of the magnetization at this point is given as the blue dashed curve in Figure 14, where the two peaks are centered around m 1 = 0.1311 and m 2 = 0.8973, the stable solution and the metastable solution of the consistency equation (13), respectively. FIG. 14 . 14K = 1.67, J = 0.01, h = 0.1. Boltzmann-Gibbs distribution of the total magnetization at fixed values of N . The peaks of the distribution are centered around the two solutions of the equation (13), with m1 = 0.1311 being the stable solution and m2 = 0.8973 the metastable solution. We can observe that the probability of the metastable solution vanishes to 0 as N goes to infinity (black continuous curve). The red dot-dashed line corresponds to the distribution for N = 1000, blue dashed line corresponds to the distribution for N = 3000 and the black continuous line for the distribution with N = 10000. ( K, J, h) = (1.76 ± 0.67, −0.11 ± 1.11, 0.15 ± 0.49). FIG. 15 . 15K = 1.67, J = 0.01 and h = 0.1. K, J and h as a function of M using the largest cluster and N = 3000. The reconstructed estimates, K, J and h, are blue crosses on statistical error bars on 50 different M -samples of configurations of the same system. The horizontal red lines in each panel correspond to the exact values of K, J and h. ACKNOWLEDGMENTSThe authors thank Claudio Giberti and Emanuele Mingione for useful discussions and G.O. appreciates Filippo Zimmaro for interesting discussions. Information theory and statistical mechanics. E T Jaynes, 10.1103/PhysRev.106.620Phys. Rev. 106620E. T. Jaynes, Information theory and statistical mechan- ics, Phys. Rev. 106, 620 (1957). D J C Mackay, Information Theory, Inference, and Learning Algorithms. Copyright Cambridge University PressD. J. C. MacKay, Information Theory, Inference, and Learning Algorithms (Copyright Cambridge University Press, 2003). Inverse statistical problems: from the inverse Ising problem to data science. H C Nguyen, R Zecchina, J Berg, 10.1080/00018732.2017.1341604Advances in physics. 66H. C. Nguyen, R. Zecchina, and J. Berg, Inverse statis- tical problems: from the inverse Ising problem to data science, Advances in physics 66, 197-261 (2017). From inverse problems to learning: a Statistical Mechanics approach. C Baldassi, F Gerace, L Saglietti, R Zecchina, 10.1088/1742-6596/955/1/012001Journal of physics. Conference series. 95512001C. Baldassi, F. Gerace, L. Saglietti, and R. Zecchina, From inverse problems to learning: a Statistical Mechanics approach, Journal of physics. Conference series 955, 012001 (2018). Higher-order interactions in statistical physics and machine learning: A model-independent solution to the inverse problem at equilibrium. S V Beentjes, A Khamseh, 10.1103/physreve.102.053314Physical Review E. 102S. V. Beentjes and A. Khamseh, Higher-order in- teractions in statistical physics and machine learn- ing: A model-independent solution to the inverse problem at equilibrium, Physical Review E 102, 10.1103/physreve.102.053314 (2020). Spin glass theory and beyond: An introduction to the replica method and its applications. M Mezard, G Parisi, M A Virasoro, World Scientific PublishingM. Mezard, G. Parisi, and M. A. Virasoro, Spin glass theory and beyond: An introduction to the replica method and its applications (World Scientific Publishing, 1987). Weak pairwise correlations imply strongly correlated network states in a neural population. E Schneidman, M Berry, R Segev, W Bialek, 10.1038/nature04701Nature. 4401007E. Schneidman, M. Berry, R. Segev, and W. Bialek, Weak pairwise correlations imply strongly correlated network states in a neural population, Nature 440, 1007 (2006). Constraint satisfaction problems and neural networks: A statistical physics perspective. M Mézard, T Mora, 10.1016/j.jphysparis.2009.05.013Journal of physiology. 103M. Mézard and T. Mora, Constraint satisfaction prob- lems and neural networks: A statistical physics perspec- tive, Journal of physiology, Paris 103, 107-113 (2009). High-resolution protein complexes from integrating genomic information with molecular simulation. A Schug, M Weigt, J N Onuchic, T Hwa, H Szurmant, 10.1073/pnas.0912100106PNAS. 10622124A. Schug, M. Weigt, J. N. Onuchic, T. Hwa, and H. Szurmant, High-resolution protein complexes from in- tegrating genomic information with molecular simula- tion, PNAS 106, 22124 (2009). Direct-coupling analysis of residue coevolution captures native contacts across many protein families. F Morcos, A Pagnani, B Lunt, A Bertolino, D S Marks, C Sander, R Zecchina, J N Onuchic, T Hwa, M Weigt, 10.1073/pnas.1111471108PNAS. 1081293F. Morcos, A. Pagnani, B. Lunt, A. Bertolino, D. S. Marks, C. Sander, R. Zecchina, J. N. Onuchic, T. Hwa, and M. Weigt, Direct-coupling analysis of residue coevo- lution captures native contacts across many protein fam- ilies, PNAS 108, E1293 (2011). Markov random field image models and their applications to computer vision. S Geman, C Graffigne, Proceedings of the International Congress of Mathematicians. the International Congress of MathematiciansS. Geman and C. Graffigne, Markov random field image models and their applications to computer vision, in Pro- ceedings of the International Congress of Mathematicians (1986) p. 1496-1517. Economic choices. D Mcfadden, 10.1257/aer.91.3.351American Economic Review. 91351D. McFadden, Economic choices, American Economic Review 91, 351 (2001). S N Durlauf, Proc. Natl. Acad. Sci. USA. Natl. Acad. Sci. USA9610582S. N. Durlauf, Proc. Natl. Acad. Sci. USA 96, 10582 (1999). Discrete Choice with Social Interactions. W A Brock, S N Durlauf, 10.1111/1467-937X.00168The Review of Economic Studies. 68235W. A. Brock and S. N. Durlauf, Dis- crete Choice with Social Interactions, The Review of Economic Studies 68, 235 (2001). Enhancing participation to health screening campaigns by group interactions. R Burioni, P Contucci, M Fedele, C Vernia, A Vezzani, Scientific Reports. 5R. Burioni, P. Contucci, M. Fedele, C. Vernia, and A. Vezzani, Enhancing participation to health screen- ing campaigns by group interactions, Scientific Reports 5 (2015). On a statistical mechanics approach to some problems of the social sciences. P Contucci, C Vernia, 10.3389/fphy.2020.585383Frontiers in Physics. 8P. Contucci and C. Vernia, On a statistical mechanics ap- proach to some problems of the social sciences, Frontiers in Physics 8, 10.3389/fphy.2020.585383 (2020). . A Barra, P Contucci, R Sandell, C Vernia, https:/arxiv.org/abs/https:/doi.org/10.1038/srep04174Scientific Reports. 4A. Barra, P. Contucci, R. Sandell, and C. Vernia, Scientific Reports 4, 4174 (2014), https://doi.org/10.1038/srep04174. . G Ignacio, A Barra, P Contucci, Math. Models Methods Appl. Sci. 191427G. Ignacio, A. Barra, and P. Contucci, Math. Models Methods Appl. Sci. 19, 1427 (2009). . R Burioni, P Contucci, M Fedele, C Vernia, A Vezzani, https:/arxiv.org/abs/https:/doi.org/10.1038/srep09904Sci. Rep. 5R. Burioni, P. Contucci, M. Fedele, C. Ver- nia, and A. Vezzani, Sci. Rep. 5, 1 (2019), https://doi.org/10.1038/srep09904. A statistical mechanics approach to the study of energy use behaviour. G Osabutey, A A Opoku, S Gyamfi, 10.1155/2020/7384053Journal of applied mathematics. 2020G. Osabutey, A. A. Opoku, and S. Gyamfi, A statistical mechanics approach to the study of energy use behaviour, Journal of applied mathematics 2020, 1-14 (2020). . A A Opoku, G Osabutey, C Kwofie, https:/arxiv.org/abs/https:/doi.org/10.1155/2019/3435626Journal of Probability and Statistics. 10A. A. Opoku, G. Osabutey, and C. Kwofie, Jour- nal of Probability and Statistics 2019, 10 (2019), https://doi.org/10.1155/2019/3435626. F Battiston, G Cencetti, I Iacopini, V Latora, M Lucas, A Patania, J.-G Young, G Petri, 10.1016/j.physrep.2020.05.004Networks beyond pairwise interactions: Structure and dynamics. 874F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lu- cas, A. Patania, J.-G. Young, and G. Petri, Networks beyond pairwise interactions: Structure and dynamics, Physics reports 874, 1-92 (2020). The physics of higher-order interactions in complex systems. F Battiston, E Amico, A Barrat, G Bianconi, G Ferraz De Arruda, B Franceschiello, I Iacopini, S Kéfi, V Latora, Y Moreno, M M Murray, T P Peixoto, F Vaccarino, G Petri, 10.1038/s41567-021-01371-4Nature physics. 17F. Battiston, E. Amico, A. Barrat, G. Bianconi, G. Ferraz de Arruda, B. Franceschiello, I. Iacopini, S. Kéfi, V. Latora, Y. Moreno, M. M. Murray, T. P. Peixoto, F. Vaccarino, and G. Petri, The physics of higher-order interactions in complex systems, Nature physics 17, 1093-1098 (2021). Dynamics on higher-order networks: a review. S Majhi, M Perc, D Ghosh, 10.1098/rsif.2022.0043Journal of the Royal Society, Interface. 1920220043S. Majhi, M. Perc, and D. Ghosh, Dy- namics on higher-order networks: a review, Journal of the Royal Society, Interface 19, 20220043 (2022). Simplicial closure and higherorder link prediction. A R Benson, R Abebe, M T Schaub, A Jadbabaie, J Kleinberg, 10.1073/pnas.1800683115Proc. Natl Acad. Sci. USA. 115A. R. Benson, R. Abebe, M. T. Schaub, A. Jadbabaie, and J. Kleinberg, Simpli- cial closure and higherorder link prediction, Proc. Natl Acad. Sci. USA 115, E11221-E11230 (2018). Evolutionary dynamics of higher-order interactions in social networks. U Alvarez-Rodriguez, F Battiston, G F De Arruda, Y Moreno, M Perc, V Latora, 10.1038/s41562-020-01024-1Nature human behaviour. 5U. Alvarez-Rodriguez, F. Battiston, G. F. de Arruda, Y. Moreno, M. Perc, and V. Latora, Evolutionary dy- namics of higher-order interactions in social networks, Nature human behaviour 5, 586-595 (2021). Human-AI ecosystem with abrupt changes as a function of the composition. P Contucci, J Kertész, G Osabutey, 10.1371/journal.pone.0267310PloS one. 17267310P. Contucci, J. Kertész, and G. Osabutey, Human-AI ecosystem with abrupt changes as a function of the com- position, PloS one 17, e0267310 (2022). Inverse problem robustness for multi-species mean-field spin models. M Fedele, C Vernia, P Contucci, 10.1088/1751-8113/46/6/065001J. of Phys. A: Math. and Theo. 4665001M. Fedele, C. Vernia, and P. Contucci, Inverse prob- lem robustness for multi-species mean-field spin models, J. of Phys. A: Math. and Theo. 46, 065001 (2013). PhD Thesis in preparation. G Osabutey, Alma Mater Studiorum -University. unpublishedG. Osabutey, PhD Thesis in preparation. Alma Mater Studiorum -University of Bologna, Italy. (unpublished). Entropy, large deviations and statistical mechanics. R S Ellis, SpringerBerlin Heidelberg New YorkR. S. Ellis, Entropy, large deviations and statistical me- chanics (Springer Berlin Heidelberg New York, 1985). . M Fedele, C Vernia, https:/link.aps.org/doi/10.1103/PhysRevE.96.042135Phys. Rev. E. 9642135M. Fedele and C. Vernia, Phys. Rev. E 96, 042135 (2017). . P Contucci, R Luzi, C Vernia, 10.1088/1751-8121/aa69efJ. Phys. A: Math. Theor. 50205002P. Contucci, R. Luzi, and C. Vernia, J. Phys. A: Math. Theor. 50, 205002 (2017). . N Ito, G A Kohring, Int. J. Modern Phys. C. 051N.Ito and G.A.Kohring, Int. J. Modern Phys. C 05, 1 (1994). . A Decelle, F Ricci-Tersenghi, 10.1103/PhysRevE.94.012112Phys. Rev. E. 9412112A. Decelle and F. Ricci-Tersenghi, Phys. Rev. E 94, 012112 (2016). . A Rodriguez, A Laio, https:/arxiv.org/abs/https:/www.science.org/doi/pdf/10.1126/science.1242072Science. 3441492A. Rodriguez and A. Laio, Science 344, 1492 (2014), https://www.science.org/doi/pdf/10.1126/science.1242072. . H C Nguyen, J Berg, 10.1103/PhysRevLett.109.050602Phys. Rev. Lett. 10950602H. C. Nguyen and J. Berg, Phys. Rev. Lett. 109, 050602 (2012).
[]
[ "Hypercontractivity Meets Random Convex Hulls: Analysis of Randomized Multivariate Cubatures", "Hypercontractivity Meets Random Convex Hulls: Analysis of Randomized Multivariate Cubatures" ]
[ "Satoshi Hayakawa [email protected] \nMathematical Institute\nUniversity of Oxford\n\n", "Harald Oberhauser [email protected] \nMathematical Institute\nUniversity of Oxford\n\n", "Terry Lyons [email protected] \nMathematical Institute\nUniversity of Oxford\n\n" ]
[ "Mathematical Institute\nUniversity of Oxford\n", "Mathematical Institute\nUniversity of Oxford\n", "Mathematical Institute\nUniversity of Oxford\n" ]
[]
Given a probability measure µ on a set X and a vector-valued function ϕ, a common problem is to construct a discrete probability measure on X such that the push-forward of these two probability measures under ϕ is the same. This construction is at the heart of numerical integration methods that run under various names such as quadrature, cubature, or recombination. A natural approach is to sample points from µ until their convex hull of their image under ϕ includes the mean of ϕ. Here we analyze the computational complexity of this approach when ϕ exhibits a graded structure by using so-called hypercontractivity. The resulting theorem not only covers the classical cubature case of multivariate polynomials, but also integration on pathspace, as well as kernel quadrature for product measures.
10.1098/rspa.2022.0725
[ "https://export.arxiv.org/pdf/2210.05787v1.pdf" ]
252,846,220
2210.05787
4db7920bc3b6cef216224bc4528129911ae67fd7
Hypercontractivity Meets Random Convex Hulls: Analysis of Randomized Multivariate Cubatures 11 Oct 2022 Satoshi Hayakawa [email protected] Mathematical Institute University of Oxford Harald Oberhauser [email protected] Mathematical Institute University of Oxford Terry Lyons [email protected] Mathematical Institute University of Oxford Hypercontractivity Meets Random Convex Hulls: Analysis of Randomized Multivariate Cubatures 11 Oct 2022arXiv:2210.05787v1 [math.PR] Given a probability measure µ on a set X and a vector-valued function ϕ, a common problem is to construct a discrete probability measure on X such that the push-forward of these two probability measures under ϕ is the same. This construction is at the heart of numerical integration methods that run under various names such as quadrature, cubature, or recombination. A natural approach is to sample points from µ until their convex hull of their image under ϕ includes the mean of ϕ. Here we analyze the computational complexity of this approach when ϕ exhibits a graded structure by using so-called hypercontractivity. The resulting theorem not only covers the classical cubature case of multivariate polynomials, but also integration on pathspace, as well as kernel quadrature for product measures. Introduction Let X be a random variable that takes values in a set X , and F ⊂ R X a linear, finite dimensional space of integrable functions from X to R. A cubature formula for (X, F ) is a finite set of points (x i ) ⊂ X and weights (w i ) ⊂ R such that E[f (X)] = n i=1 w i f (x i ) for all f ∈ F . (1) If the function class F is infinite-dimensional one can not hope for equality in (1) and instead aims to find an approximation that holds uniformly over F . We also denote µ = Law(X) and refer toμ = n i=1 w i δ xi as the cubature measure for (X, F ). The existence of such a cubature formula that further satisfies n ≤ 1 + dim P , w i ≥ 0 and n i=1 w i = 1 is guaranteed by what is often referred to as Tchakaloff's theorem although a more accurate nomenclature would involve Wald, Richter, Rogosinski, and Rosenbloom [62,57,49,50,51]; see [17] for a historical perspective. Arguably the most famous applications concerns the case when X is a subset of R d and F is the linear space of polynomials up to a certain degree, that is F is spanned by monomials up to a certain degree. However, more recent applications include the case when X is a space of paths and F is spanned by iterated Ito-Stratonovich integrals [39], or kernel quadrature [33,27] where X is a set that carries a positie definite kernel and F is a subset of the associated reproducing kernel Hilber space that is spanned by eigenfunctions of the integral operator induced by a kernel. Convex Hulls. If F is spanned by m functions ϕ 1 , . . . , ϕ m : X → R, then we can denote ϕ = (ϕ 1 , . . . , ϕ m ) : X → R m and see that (1) is equivalent to E[ϕ(X)] = n i=1 w i ϕ(x i ). If we restrict attention to non-negative weights that sum up to one (equivalently,μ is a probability measure) this is equivalent to that statement that E[ϕ(X)] ∈ conv{ϕ(x 1 ), . . . , ϕ(x n )}, (2) where we denote for an A ⊂ R m its convex hull as conv A = c 1 a 1 + · · · + c k a k k ≥ 1, a i ∈ A, c i ≥ 0, k i=1 c i = 1 . Random Convex Hulls. A natural and general approach to find points (x i ) ⊂ X for which (2) holds was recently proposed in [24]: draw N ≫ n independent random samples (X j ) N j=1 from µ and subsequently try to select a subset of n points (x i ). The success of this approach amounts the event that E[ϕ(X)] ∈ conv{ϕ(X 1 ), . . . , ϕ(X N )}, since then simple linear programming (LP) allows select the subset of x i 's resp. compute the remaining weights that determine a cubature formula. The following guarantees that for large enough N this event occurs with high probability Proposition 1 ( [24]). If X 1 , X 2 , . . . are independent copies of X, then the probability of the event (3) tends to 1 as N → ∞. Empirically, this approach turns out to work well already for "reasonable" magnitudes of N [24,27]. The aim of this article is to fill this gap and provide theoretical guarantees for the number of samples N for which this approach leads with high probability to a successful cubature construction; that is to provide a quantitative version of Proposition 1 that applies to common cases. Hypercontractivity. Our main tool is hypercontractivity. This allows to prove the existence of a constant C ′ m satisfying (mainly for p = 4) E[|f (X)| p ] ≤ C ′ m E |f (X)| 2 p/2 uniformly for a large class of functions f , and where X follows the product measure µ ⊗d . While hypercontractivity is classically studied for Gaussian, discrete, and uniform probability measures on hypercubes or hyperspheres [11,43,6,7]. We generalize it to function classes that have a certain graded structure. Contribution. Our main result is to provide an upper bound for the number of samples N such that an N -point i.i.d. sample of random vectors contains the expectation in its convex hull, i.e. the event (3) occurs, with a reasonable probability. Although the connection between the bound for N and the hypercontractivity of the given random vector/function class has implicitly been proven in a preceding study [26] in the form of Theorem 3, generic conditions for having a good hypercontractivity constant and why the magnitude of required N becomes reasonably small in practice have not been established or understood. In this paper, we address these questions by • extending the hypercontractivity for the Wiener chaos to what we call generalized random polynomials (Section 3) and • showing that this extension naturally applies to important examples in numerical analysis including classical cubature, cubature on Wiener space, and kernel quadrature (Section 4). We explain the intuition behind these points by introducing Theorem 1 and Example 1: Theorem 1 (informal) . Let µ be a probability measure on X . Suppose we have a "natural" function class F = d≥1 m≥0 F d,m , where F d,m denotes a set of functions from X d to R of "degree" up to m. Then, under some integrability assumptions, there exists for every m a constant C m = C m (µ, F ) > 0 such that the following holds: Let d and D be two positive integers and ϕ = (ϕ 1 , . . . , ϕ D ) : X d → R D with ϕ 1 , . . . , ϕ D ∈ F d,m . Then, for all integers N ≥ C m D, we have P(E[ϕ(X)] ∈ conv{ϕ(X 1 ), . . . , ϕ(X N )}) ≥ 1 2 , where X, X 1 , . . . , X N are i.i.d. samples from the product measure µ ⊗d on X d . Example 1. Although the "assumption" in the above statement is somewhat abstract, this applies to important examples as follows: • Classical Cubature [56]: µ is a probability measure with finite m moments and F d,m is the space of d-variate polynomials up to degree m . • Cubature on Wiener space [39]: µ is the Wiener measure and F d,m is spanned by up to m-times iterated Ito-Stratonovich integrals. • Kernel quadrature [33,27]: µ is a probability measure on set X that carries a positive definite kernel k and F d,m is spanned by the eigenfunctions (down to some eigenvalue) of the integral operator g → k ⊗d (·, x)g(x) dµ ⊗d (x), where k ⊗d is a tensor product kernel. Related work. If the measure µ has finite support, the problem (1) is also known as recombination. While in this case, the existence follows immediately from Caratheodory's theorem, the design of efficient algorithms to compute the cubature measure is more recent; we mention efficient deterministic algorithms [37,58,40] and randomized speedups [15]. For non-discrete measures, the majority of the cubature constructions are typically limited to algebraic approaches that cannot apply to general situations. Related to our convex hull approach but different, is a line of research aiming at constructing general cubature formulas with positive weights by using least-squares instead of the random convex hull approach [21,42]. As their theory is on the positivity of the resulting cubature formula given by solving a certain least squares problem, requires more (or efficiently selected) points than that needed for simply obtaining a positively weighted cubature. Hypercontractivity is the key technical tool for our estimates and its use seems to be novel in the context of cubature resp. random convex hull problems. Somewhat related to the special case of kernel quadrature, [41] proves a generalization error bound for kernel ridge regression with random features, however hypercontractivity is simply adopted as a technical assumption. Further, for low-degree polynomials of a sequence of random variables, Kim and Vu [34], Schudy and Sviridenko [53] give similar estimates on their higher order moments, but they mainly estimate the concentration of the moments, and do not generally analyze the curtosis-type values appearing in the hypercontractivity. Outline. In Section 2, we give briefly explain recent results on random convex hulls, and give some assertions that additionally follow from them. In Section 3, we introduce the Gaussian hypercontractivity and show its generalization that is suitable for multivariate cubatures. Section 4 gives some applications of Gaussian/generalized hypercontractibity to random convex hulls with product structure, including cubature on Wiener space and kernel quadrature. Finally, we conclude the paper in Section 5. All the omitted proofs are given in Appendix B. Random Convex Hulls Our main interest is the probability of the even (3) but it turns out to be more convenient to study a more general problem. Therefore we define Definition 1. Let X be a D-dimensional random vector and X 1 , X 2 , . . . be iid copies of X. For every integer N > 0 and θ ∈ R D define p N,X (θ) := P(θ ∈ conv{X 1 , . . . , X N }) and N X (θ) := inf{N | p N,X (θ) ≥ 1/2}. Both of these quantities are classically studied for symmetric X by Wendel [63], but more recently sharp inequalities for general X [60,26] as well and calculations on the Gaussian case [31] have been established. Using this notation, our main interests is the choice θ = E[ϕ(X)]. In the following two paragraphs we briefly discuss how bounds on N X (θ) can be derived with previous approaches; in Section 3 we then discuss the approach via hypercontractivity. Bounds via Tukey Depth. It turns out that a classical quantity from statistics, the so-called Tukey depth [59,52], is closely related to the two quantities. Definition 2. The Tukey depth of θ ∈ R D with respect to the distribution of X is defined as α X (θ) := inf c∈R D \{0} P c ⊤ (X − θ) ≤ 0 .(4) The relation between the above quantities is 26]). Let θ ∈ R D and X be an arbitrary D-dimensional random vector. Then, we Theorem 2 ([have 1 2α X (θ) ≤ N X (θ) ≤ 3D α X (θ) . The above can be used to provide a novel bound on N X (E[X]) for a general class of distributions called log-concave, Proposition 2. If X is a D-dimensional log-concave random vector, we have N X (E[X]) ≤ ⌈3eD⌉. Bounds via Moments. Theorem 2 gives a good intuition behind the random convex hulls, but computing the Tukey depth α X (θ) itself is in general a difficult task [16,64]. In [26] an alternative way to bound N X (θ) is provided by using the Berry-Esseen theorem [10,20,36]. Theorem 3 ([26]). Let X be an arbitrary D-dimensional random vector with E X 3 < ∞. If a constant K > 0 satisfies c ⊤ (X − E[X]) L 3 ≤ K c ⊤ (X − E[X]) L 2 for all c ∈ R D , then we have N X (E[X]) ≤ 17(1 + 9K 6 /4)D. This result still recovers a sharp bound N X (E[X]) = O(D) up to constant for a Gaussian, where we have detailed information about the marginals. The sort of inequality assumed in the statement is also called Khintchin's inequality (see e.g., [35,19]) and there are known values of B for a certain class of X such as a Rademacher vector. Indeed, we can easily show the following estimate under a clear independence structure: Proposition 3. Let X = (X 1 , . . . , X D ) ⊤ be a D-dimensional random vector whose coordinates are independent and identically distributed. If E[X 1 ] = 0 and X 1 L 4 ≤ K X 1 L 2 holds for a constant K > 0, then we have c ⊤ X L 4 ≤ K c ⊤ X L 2 for all c ∈ R D . Hypercontractivity The previous section provides bounds on N X but the assumptions-log-concavity or coordinate-wise independence-are too strong for many applications. We now develop an approach an appraoch via hypercontractivity; this results in bounds that apply under much less strict assumptions. Hypercontractivity: the Gaussian case. It is instructive to briefly review the classical results for Gaussian measures by following Janson [30] since we need several generalizations of this. Theorem 4 (Wiener Chaos Decomposition). Let H be a Gaussian Hilbert space 1 and let σ(H) be the σ-algebra generated by H. Then L 2 (Ω, σ(H), P) = ∞ n=0 H (n) , where H (n) := P n (H) ∩ P n−1 (H) ⊥ with P n (H) := {f (Y 1 , . . . , Y m ) | f is a polynomial of degree ≤ m, Y 1 , . . . , Y m ∈ H, m < ∞} with P −1 (H) := {0} and P n (H) denotes the completion in L 2 (Ω, F , P). Hence, for each X ∈ L 2 (Ω, σ(H), P), we have a unique decomposition X = ∞ n=0 X n such that X n ∈ H (n) . Theorem 5 (Hypercontractivity, [30], Theorem 5.8). For r ∈ [0, 1] denote T r : L 2 (Ω, σ(H), P) → L 2 (Ω, σ(H), P), X → ∞ n=0 r n X n . If p > 2 and 0 < r ≤ (p − 1) −1/2 , then we have T r (X) L p ≤ X L 2 . From this, we have the following moment bound on P n (H), which is also referred to as hypercontractivity, see for example [45]. Theorem 6. Let n ≥ 0 be an integer. For each p > 2, we have X L p ≤ (p − 1) n/2 X L 2 , X ∈ P n (H). Proof. Let X = n m=0 X m with X m ∈ H (m) . For 0 < r ≤ (p − 1) −1/2 , by Theorem 5, we have X 2 L p = T r n m=0 r −m X m 2 L p ≤ n m=0 r −m X m 2 L 2 = n m=0 r −2m X m 2 L 2 ≤ r −2n X 2 L 2 . We obtain the conclusion by letting r = (p − 1) −1/2 . We included the proof since we are going to generalize it in the next section. Hypercontractivity for Generalized Random Polynomials The phenomenon of hypercontractivity is not limited to the Gaussian setting. Indeed, the hypercontractivity of operators on the space of boolean functions (i.e., {−1, 1} n → R) associated with the uniform measure was established even before the Gaussian case [11,54]. Our focus is to obtain estimates analogous to Theorem 6 when a graded class of test function is given; we refer to such a class as generalized random polynomials. Generalized Random Polynomials. Definition 3. Under a probability space (Ω, G, P), a triplet G = (Y, Q, λ) is called GRP if it satisfies the following conditions: • Y is a random variable taking values in a topological space X . • Q = (Q m ) ∞ m=0 is a nondecreasing sequence of linear spaces of L 2 (P Y )-integrable functions X → R. Namely, if we let Q m (Y ) := {f (Y ) | f ∈ Q m }, then each Q m is a linear subspace of L 2 (P), with Q 0 ⊂ Q 1 ⊂ · · · ⊂ L 2 (P) . We additionally assume Q 0 is the set of constant functions. • λ = (λ m ) ∞ m=0 satisfies 1 = λ 0 > λ 1 ≥ λ 2 ≥ · · · ≥ 0. If G is a GRP, we also define deg G X := inf{1/λ m | m ≥ 0, X ∈ Q m (Y )}. Intuitively, each Q m is a generalization of degree-m polynomials and deg G indicates the "degree" of such functions (though Y plays a role in the latter). In the setting of actual polynomials like Wiener chaos, we can define λ m = b −m for a certain b > 1, and then we have deg X = log b deg G X for the usual degree of X as a random polynomial. Definition 4. Let G = (Y, Q, λ) be a GRP. We define H m (Y ) := Q m (Y ) ∩ Q m−1 (Y ) ⊥ in terms of L 2 (P) where Q −1 (Y ) := {0} and H ∞ := L 2 (Ω, σ(Y ), P) ∩ M m=0 Q m (Y ) ⊥ . We refer L 2 (Ω, σ(Y ), P) = ∞ m=0 H m (Y ) ⊕ H ∞ (Y ) as the orthogonal decomposition associated with G. Definition 5. Let G = (Y, Q, λ) be a GRP. The operator T (G) is defined as T (G) : L 2 (Ω, σ(Y ), P) → L 2 (Ω, σ(Y ), P), X → ∞ m=0 λ m X m , where (X m ) m∈N∪∞ with X m ∈ H m (Y ) is the orthogonal decomposition of X associated with the GRP G. We say that a GRP G = (Y, Q, λ) is (2, p; s)-hypercontractive if T (G) s X L p ≤ X L 2 , X ∈ L 2 (Ω, σ(Y ), P). Thus, T (G) s X = ∞ m=0 λ s m X m for s > 0 and if G is (2, p; s)-hypercontractive, it is (2, p; t)-hypercontractive for all t ≥ s as T (G) t−s is a contraction in L 2 . The formulation of G associated with "degree" concept given by λ then naturally extends to the multivariate case. Definition 6. We call a set of d GRPs, G (i) = (Y (i) , Q (i) , λ (i) ) for i = 1, . . . , d independent, if the Y (i) ' s are independent random variables taking values in X (i) 's. For d independent GRPs, their product is a GRP G = (Y, Q, λ) that is defined as follows • Y = (Y (1) , . . . , Y (d) ) ∈ X (1) × · · · × X (d) . • λ m is the (m + 1)-th largest value in the set d i=1 λ (i) mi λ (i) mi ∈ λ (i) , i = 1, . . . , d . • Q m = span f : (x 1 , . . . , x d ) → d i=1 f i (x i ) f i ∈ Q (i) mi , d i=1 λ (i) mi ≤ λ m . As Q m (Y ) ⊂ L 2 it follows from independence for each m that G = (Y, Q, λ) is indeed a GRP. We also denote it by G = G (1) ⊗ · · · ⊗ G (d) . Example 2. Consider the case when Q (i) m are degree-m polynomials of Y (i) and λ (i) m = t m for some t ∈ (0, 1) independent of i. This shows that the product GRP generalizes the multivariate random polynomials. Also, when Y (i) are i.i.d. and (Q (i) , λ (i) ) are the same for all i = 1, . . . , d, then we say G (i) are i.i.d. and we can particular write G ≃ (G (1) ) ⊗d . A straightforward generalization follows from the classical way of proving hypercontractivity. Nevertheless, it turns out to be very useful for treating multivariate hypercontractivity of our GRP setting. Theorem 7. Let r ∈ (0, 1] and p > 2. If d independent GRPs G (1) , . . . , G (d) are all (2, p; s)hypercontractive, then their product G = G (1) ⊗ · · · ⊗ G (d) is also (2, p; s)-hypercontracitve. Remark 1. We only use the (2, p; s)-hypercontractivity in this paper, but we can also deduce the same results for the general (q, p; s)-hypercontractivity with 1 ≤ q ≤ p < ∞ (for the operator norm of L q → L p ), analogous to e.g. Janson [30]. The following is a parallel result of Theorem 6 and the proof is almost identical. Proposition 4. Let s > 0 and p > 2. If G is a GRP that is (2, p; s)-hypercontractive, then we have X L p ≤ ( deg G X) s X L 2 for all X ∈ L 2 . Remark 2. Although we have treated general GRPs G = (Y, Q, λ) in these propositions, we are basically only interested in the moment inequality for X up to some "degree" fixed beforehand (in the case of Wiener chaos, it suffices to treat P n (H) for some finite n to obtain Theorem 6). Thus, our main interest is in "finite" GRPs, satisfying Q n = Q n+1 = · · · for some n, and their product in practice, which the might be better for readers to have in mind when reading the next proposition. We next show the following "converse" result for the relation of the hypercontractivity and moment estimate for a (truncated) GRP when p = 4. Proposition 5. Let G = (Y, Q, λ) be a GRP. Suppose there exists a s > 0 such that X m L 4 ≤ λ −s m X m L 2 , X m ∈ H m (Y ) holds for all m. If t > s satisfies m≥1 λ t−s m ≤ 1/ √ 3 and λ t 1 ≤ 1/2, then G is (2, 4; t)-hypercontractive. By using this, we can also prove the following as a non-quantitative result. Theorem 8. Let K > 0 and G be a GRP such that the space {X ∈ L 2 | deg G X ≤ K} is included in L 4 (Ω, F , P) and finite-dimensional. Then, there exists a constant C = C(G, K) such that for an arbitrary d, X L 4 ≤ C X L 2 holds if we have deg G ⊗d X ≤ K. Applications The generality of Proposition 5 and Theorem 8 allows to quantify the number of samples resp. probability of success of the random convex hull approach to the problem of cubature. Concretely, we give formal statements of Theorem 1 for (i) Classical Cubature, (ii) Cubature on Wiener Space, (iii) Kernel Quadrature. various cubature constructions. Classical Polynomial Cubatures When the GRP G are actual random polynomials, we recover the following result Proof. By introducing a truncated GRP given by a random variable X (1) , function spaces Q i of univariate polynomials up to degree i, and λ i = 2 −i for 0 ≤ i ≤ m, we can apply Theorem 8 to obtain the desired result. If we combine this with Theorem 3, we obtain the following result for polynomial cubatures: Corollary 2. Let m ≥ 1 be an integer and X (1) , X (2) , . . . be i.i.d. real-valued random variables with E |X (1) | 4m < ∞. Then, there exists a constant C m > 0 such that the following holds: Let d ≥ 1 be an integer and ϕ : R d → R D be a D-dimensional vector-valued function such that each coordinate is given by a polynomial up to degree m. If we let X Cubature on Wiener Space Cubature on Wiener space [39] is a weak approximation scheme for stochastic differential equations; at the hear of it is the construction of a finite measure on pathspace, such that the expectation of their first m-times iterated integrals matches those of Brownian motion. Some algebraic constructions are known for degree m = 3, 5 [39] as well as m = 7 [44]. The random convex hull approach applies in principle for any m, however, a caveate is that the discretization of paths becomes an issue in particular for high values of m; some experimental results are available in [25] for constructing them by using random samples of piecewise linear approximations of Brownian motion. In this section, we use hypercontractivity to estimate the number of samples needed in this approach to cubature via sampling. I α (x) := 0<t1<···<t k <1 dx α1 t1 · · · dx α k t k , I α (B) := 0<t1<···<t k <1 • dB α1 t1 · · · • dB α k t k , where the latter is given by the Stratonovich stochastic integral. Then, a degree m cubature formula on Wiener space for d-dimensional Brownian motion is a set of BV paths x 1 , . . . , x n : [0, 1] → R d+1 and convex weights w 1 , . . . , w n such that n i=1 w i I α (x i ) = E[I α (B)] for all multiindices α = (α 1 , . . . , α k ) ∈ ℓ≥1 {0, 1, . . . , d} ℓ with α := k + |{j | α j = 0}| ≤ m. Indeed, if we consider the Gaussian Hilbert space given by H := d i=1 1 0 f i (t) dB i t f 1 , . . . , f d ∈ L 2 ([0, 1]) , the iterated integral I α (B) with α ≤ m is in the m-th Wiener chaos P m (H) (see Section 3) as it can be expressed as a limit of polynomials of increments of B. We thus have the hypercontructivity given in Theorem 6 and the following assertion: Corollary 3. Let d, m ≥ 1 be integers and B be a d-dimensional Brownian motion. Then, for an arbitrary linear combination X = α ≤m c α I α (B) with c α ∈ R, we have X L 3 ≤ 2 m/2 X L 2 . As the bound is independent of the dimension d of the underlying Brownian motion, we have the following version of Theorem 1 by combining it with Theorem 3 as follows: The above allows to choose the number of candidate paths that need to be sampled. However, as mentioned above, one challenge that is specific to cubature on pathspace is that one cannot sample a true Brownian trajectory which leads to an additional discretization error. However, we conjecture that the number of random samples divided by D and the number of time partitions for piecewise linear approximation in constructing cubature on Wiener space can be independent of the underlying dimension d. Remark 3. One can also apply these estimates to fractional Brownian motion [48], though we also need to obtain the exact expectations of iterated integrals of fractional Brownian motion (we can find some results on the Ito-type iterated integrals without the time integral by B 0 t = t in the literature [5,Theorem 31]). Kernel Quadrature for Product Measures Let X be a topological space and k : X × X → R be a positive definite kernel with the reproducing kernel Hilbert space (RKHS) H k [9]. A kernel quadrature for a random variable X or equivalently a Borel probability measure µ (i.e., X ∼ µ) on X is a cubature formula for (H k , µ); that is, a set of points x i ∈ X and weights w i ∈ R such that µ n = w i δ xi minimizes worst-case error wce( µ n ; H k , µ) := sup f H k ≤1 E[f (X)] − n i=1 w i f (x i ) ,(5) which we might just denote by wce( µ n ), has been widely studied from the viewpoint of optimization [14,4,29] as well as sampling [3,8,27]. Tensor Product Kernels. When there are d pairs of space and kernel (X 1 , k 1 ), . . . , (X d , k d ), the tensor product kernel on the product space X 1 × · · · × X d is defined as (k 1 ⊗ · · · ⊗ k d )(x, y) := d i=1 k i (x i , y i ), x = (x 1 , . . . , x d ), y = (y 1 , . . . , y d ) ∈ X 1 × · · · × X d . This is indeed the reproducing kernel of the tensor product H k1 ⊗ · · · ⊗ H k d in terms of RKHS [9]. The most important example of this construction is when the underlying d kernels are the same, k ⊗d = k ⊗ · · · ⊗ k. Given a probability measure µ in the (conceptually univariate) space X , constructing a kernel quadrature for µ ⊗d with respect to k ⊗d is a natural multivariate extension of kernel quadrature that is widely studied in the literature [47,32,3,33], and corresponds to high-dimensional QMCs [18]. Mercer Expansions and Quadrature. The convergence rate of wce( µ n ) is typically described by using the Mercer expansion: k(x, y) = ∞ ℓ=1 σ ℓ e ℓ (x)e ℓ (y),(6) where (σ ℓ , e ℓ ) ∞ ℓ=1 are eigenpairs of the integral operator K : f → X k(·, y)f (y) dµ(y) in L 2 (µ) with σ 1 ≥ σ 2 ≥ · · · ≥ 0 and e ℓ L 2 (µ) = 1. Assumption A. The kernel k satisfies that the expansion (6) converges pointwise, ∞ ℓ=1 σ ℓ < ∞, and ( √ σ ℓ e ℓ ) ∞ ℓ=1 is an orthonormal basis of H k . Mild conditions already imply that Assumption A applies, e.g., supp µ = X , k is continuous, and x → k(x, x) is in L 1 (µ) is sufficent, see [55]. Under this assumption, an n-point kernel quadrature rule that exactly integrates the first n − 1 eigenfunctions satisfies the following theoretical guarantee: Proposition 6 ([27]). Under Assumption A, let µ n = (w i , x i ) n i=1 be a kernel quadrature with convex weights satisfying X e ℓ (x) dµ(x) = n i=1 w i e ℓ (x i ) for each ℓ = 1, . . . , n − 1. Then, by letting r n (x) := ∞ m=n σ m e m (x) 2 , we have wce( µ n ) 2 ≤ 4 sup x∈X r n (x). We have more favorable bounds on wce( µ n ) by assuming more, but the important fact here is that the event (3) for a vector-valued ϕ given by eigenfunctions e 1 , . . . , e n−1 enables us to construct an interesting numerical scheme. A similar approach, specialized to a Gaussian kernel over a Gaussian measure can be found in [33]. As the construction of such µ n for general k and µ relies on random sampling, we want to estimate N ϕ(X) (E[ϕ(X)]) for X ∼ µ and ϕ = (e 1 , . . . , e n−1 ). From RKHS to GRP. To make it compatible with the framework of GRPs introduced in the previous section, we further assume the following condition, which ensures that the kernel is in an appropriate scaling. Assumption A ′ . The kernel k satisfies Assumption A, σ 1 ≤ 1, and the strict inequality σ ℓ < 1 holds if e ℓ ∈ L 2 (µ) is not constant. Under Assumption A ′ , we can naturally define a GRP G = (Y, Q, λ) with Y following µ, Q m = span{1, e 1 , . . . , e m } and λ m = σ m for m ≥ 1. Note that it violates the condition λ 1 < 1 if σ 1 = 1 and e 1 is constant, but in that case we can simply decrement all the indices of (Q m , λ m ) by one. We call it the natural GRP for k and µ and denote it by G = G k,µ . Remark 4. The scaling given in Assumption A ′ is essential to the hypercontractivity under the framework of tensor product kernels when considering "eigenspace down to some eigenvalue." Indeed, if σ ℓ ≥ 1 for some nonconstant eigenfunction e ℓ , we have, for p > 2, e ⊗d ℓ L p (µ ⊗d ) e ⊗d ℓ L 2 (µ ⊗d ) = e ℓ L p (µ) e ℓ L 2 (µ) d which increases exponentially as d grows, whereas the corresponding eigenvalue is lower bounded by 1. So the hypercontractivity in our sense never gets satisfied if σ ℓ ≥ 1 for a nonconstant e ℓ . The following statement, written without GRPs, is what we can prove by using the hypercontractivity of GRPs. Proposition 7. Let k satisfy Assumption A ′ and Y 1 , Y 2 , . . . independently follow µ. For each δ > 0, define a set of random variables as S(δ) := span({1} ∪ {e ℓ1 (Y m1 ) · · · e ℓ k (Y m k ) | k ≥ 1, m 1 < · · · < m k , σ ℓ1 · · · σ ℓ k ≥ δ}). Then, if e ℓ (Y 1 ) L 4 < ∞ holds for all ℓ with σ ℓ ≥ δ, then there is a constant C = C(δ) > 0 such that X L 4 ≤ C X L 2 for all X ∈ S(δ). Proof. The finiteness of the dimension of eigenspace for Y 1 , i.e, the finiteness of ℓ satisfying σ ℓ ≥ δ follows from ∞ ℓ=1 σ ℓ < ∞ in Assumption A. Thus, Theorem 8 gives the conclusion. This assertion, of course, includes a hypercontractivity statement for an eigenspace of k ⊗d and µ ⊗d for a fixed d, but we can go further to a quantitative statement by imposing another assumption. Assumption B. The kernel k can be written as k = 1 + k 0 , where k 0 : X × X → R is a positive definite kernel satisfying X k 0 (x, y) dµ(y) = 0 for (µ-almost) all x ∈ X . Under Assumption A, this is simply equivalent to e 1 being constant. This assumption might seem artificial, but naturally arises in the following situations: (a) X is a compact group and µ is its Haar measure. k is a positive definite kernel given as k(x, y) = g(x −1 y), where g : X → R ≥0 and X g(x) dµ(x) = 1. (b) k 0 is a kernel called Stein kernel [46,2] with appropriate scaling. One theoretically sufficient condition for these assumptions can be described as follows: Proposition 8. Let X be compact metrizable and path-connected, supp µ = X , and k be continuous and nonnegative. If X k(x, y) dµ(y) = 1 holds for all x ∈ X , Assumption A ′ and B hold. From this proposition, for instance, an appropriately scaled exponential/Gaussian kernel over the n-sphere with the uniform measure satisfies Assumption A ′ and B. Under these two assumptions, the operator T (G k,µ ) in terms of GRPs corresponds to the integral operator K : f → X k(·, y)f (y) dµ(y), so the situation becomes even simpler. We can directly apply Proposition 5 by replacing λ's with σ's, but we also have the following sufficient conditions for the hypercontractivity without explicitly using the eigenvalue sequence. In the following, K 0 := σ 2 < 1 is the operator norm of K 0 : f → X k 0 (·, y)f (y) dµ(y) on L 2 (µ), and tr(K 0 ) := X k 0 (x, x) dµ(x). We may have the following quantitative condition for hypercontractivity. Proposition 9. Let k = 1 + k 0 satisfy Assumption A ′ and B. When K 0 > 0, if r, s ≥ 1 satisfy K 0 −(r+s) ≥ 2, K 0 −(r−1) ≥ √ 3 tr(K 0 ), K 0 −(s−1) ≥ k 0 L 4 (µ⊗µ) , then G k,µ is (2, 4; r + s)-hypercontractive. In particular, if we have sup x∈X |k 0 (x, x)| ≤ 1/ √ 3, then G k,µ is (2, 4; 2)-hypercontractive. Example 3 (Periodic Sobolev spaces over the torus.). Following Bach [3], we consider periodic kernels over [0,1]. Therefore let X = [0, 1], µ be the uniform distribution on X , and define k r,δ (x, y) = 1 + δ · (−1) r−1 (2π) 2r (2r)! B 2r (|x − y|)(7) for each positive integer s and δ ∈ (0, 1), where B 2r is the 2r-th Bernoulli polynomial [61]. δ = 1 is assumed in the original definition, but it violates Assumption A ′ (see also Remark 4). Albeit this slight modification, the kernel k r,δ gives an equivalent norm to the periodic Sobolev space in the literature. For δ ∈ (0, 1), k r,δ satisfies Assumption A ′ and B. The eigenvalues and eigenfunctions with respect to the uniform measure are known [3]; the eigenvalues are: 1 for the constant function, and δm −2r for c m (·) := √ 2 cos(2πm ·) and s m (·) := √ 2 sin(2πm ·) for m ≥ 1, 2, . . .. We now apply Proposition 5 with (for sake of concreteness) δ = 1/3. This gives c m L 4 (µ) = s m L 4 (µ) = (3/2) 1/4 . Thus, to satisfy the condition of Proposition 5, it suffices for s < t to satisfy 3 s ≥ (3/2) 1/4 , δ t−s ζ(2r(t − s)), 3 t ≥ 2, where ζ is Riemann's zeta function. Hence a simple numerical sufficient condition for this is s = 0.1 and t = 1.1 for r = 1, and t = log 3 2 ≤ 0.631 for r ≥ 2, which can be derived by letting 2r(t − s) ≥ 2. To sum up, in the case r ≥ 2, we only need O λ −0.631 D times of sampling for meeting (3) with probability over a half, if X ∼ µ ⊗d and each coordinate of ϕ : X d → R D is in the eigenspace of the eigenvalue λ. Concluding remarks We investigated the number of samples needed for the expectation vector to be contained in their convex hull from the viewpoint of product/graded structure. We showed that the fact that we empirically only need O(D) times of sampling for the D-dimensional random vector in practical examples can partially be explained by the hypercontractivity in the Gaussian case as well as the generalized situation including random polynomials and product kernels. There are also interesting questions for further research; for example, although in the asymptotic d → ∞ we established that the required number of sampling divided by D is independent of d, the constants are larger than what purely empirical estimates given in [24,27] (where 10D is sufficient in practice). Another direction, is the case of cubature of Wiener space, as one cannot actually sample from Brownian motion and discretization errors propage to higher order m; an promising research direction could be to study "approximate sampling" or consider unbiased simulations [28] for the iterated integrals. If each X ℓ,m can be written as a finite sum X ℓ,m = k X (1) k,ℓ,m X (2) k,ℓ,m with X (1) ) and X (2) k,ℓ,m ∈ H (2) m (Y (2) ), then by using Minkowski's integral inequality [23] and the (2, p; s)hypercontractivity of G (1) and G (2) , we have (1) k,ℓ,m ∈ H (1) ℓ (YT (G) s X L p = E Y (1)   E Y (2)   ℓ,m (λ (1) ℓ λ (2) m ) s X ℓ,m p     1/p = E Y (1)   E Y (2)   k,ℓ,m (λ (1) ℓ ) s X (1) k,ℓ,m (λ (2) m ) s X (2) k,ℓ,m p     1/p ≤ E Y (1)     E Y (2)    k,ℓ,m (λ (1) ℓ ) s X (1) k,ℓ,m X (2) k,ℓ,m 2    p/2     1/p (by G (2) ) ≤ E Y (2)   E Y (1)   k,ℓ,m (λ (1) ℓ ) s X (1) k,ℓ,m X (2) k,ℓ,m p   2/p    1/2 (by Minkowski) ≤ E Y (2)   E Y (1)    k,ℓ,m X (1) k,ℓ,m X (2) k,ℓ,m 2       1/2 = X L 2 . (by G (1) ) The general case follows from the limit argument. B.3 Proof of Proposition 4 Proof. Let G = (Y, Q, λ). Suppose deg G X < ∞ and let n be the minimum integer satisfying X ∈ Q n (Y ). Then, by decomposing X = n m=0 X m with X m ∈ H m (Y ), we obtain X L p = T (G) s n m=0 λ −s m X m L p ≤ n m=0 λ −s m X m L 2 ≤ λ −s m X L 2 , where we have used the (2, p; s)-hypercontractivity in the second inequality. B.4 Proof of Proposition 5 Proof. It suffices to consider X having the decomposition X = m X m with X m ∈ H m (Y ). Recall that we have assumed that Q 0 is the space of constant functions, so X 0 is a constant. First, we consider the case X 0 = 0. In this case, for t > s, we have T (G) t X 2 L 4 = m≥1 λ t m X m 2 L 4 ≤   m≥1 λ t−s m λ s m X m L 4   2 ≤   m≥1 λ t−s m X m L 2   2 ≤   m≥1 λ 2(t−s) m   X 2 L 2 . (Cauchy-Schwarz) Therefore, when m≥1 λ 2(t−s) m ≤ 1/ √ 3 we have T (G) t X L 4 ≤ 3 −1/4 X L 2(8) for all X satisfying X 0 = 0. In the case X 0 = 0, we can assume X 0 = 1 without loss of generality. Let W = X − 1 and Z = T (G) t W = T (G) t X − 1. Note that E[W ] = E[Z] = 0 holds by the orthogonality. We can explicitly expand the L 4 norm as follows: T (G) t X 4 L 4 = 1 + 6E Z 2 + 4E Z 3 + E Z 4 ≤ 1 + 8E Z 2 + 3E Z 4 . (AM-GM) We also have X 4 L 2 = E (1 + W ) 2 2 = (1 + E W 2 ) 2 = 1 + 2E W 2 + E W 2 2 . So it suffices to show 4E Z 2 ≤ E W 2 and 3E Z 4 ≤ E W 2 2 , but the latter immediately follows from (8). The former holds when λ t 1 ≤ 1/2: E Z 2 = m≥1 λ 2t m E X 2 m ≤ λ 2t 1 E W 2 . Therefore, we have completed the proof. B.5 Proof of Theorem 8 Proof. Let G = (Y, Q, λ) and X be the space in which Y takes values. By truncating Q and λ (i.e., ignoring Q m with 1/λ m > K), we can assume that Q(Y ) = {X ∈ L 2 | deg G X ≤ K}. Then, as dim Q < ∞, we can take a vector-valued measurable function ϕ = (ϕ 1 , . . . , ϕ N ) ⊤ : X → R N such that (ϕ i (Y )) N i=1 is an orthonormal basis of Q(Y ). Then, we have sup X∈Q(Y )\{0} X L 4 X L 2 = sup c∈R N \{0} c ⊤ ϕ(Y ) L 4 c ⊤ ϕ(Y ) L 2 = sup c∈R N , c =1 c ⊤ ϕ(Y ) L 4 < ∞, where the right-hand side is the supremum of a continuous functions over a compact domain, and so is indeed finite. Hence, we can apply Proposition 5, and there exists a constant s > 0 such that T (G) t X L 4 ≤ X L 2 , X ∈ Q(Y ), because λ 1 < 1 and (λ m ) m is of finite length now. So G = (Y, Q, λ) (with truncation by K) is actually (2, p; t)-hypercontractive and it extends to G ⊗d for any d by Theorem 7 (note that the truncation does not affect the random variables with deg G ⊗d X ≤ K). Then, we finally use Proposition 4 to obtain the desired result with C = K t . B.6 Proof of Proposition 8 Proof. Let f ∈ L 2 (µ) be an eigenfunction with eigenvalue λ ≥ 0 of the integral operator, i.e., it satisfies X k(x, y)f (y) dµ(y) = λf (x) (assume this equality holds for all x, not just µ-almost all). As Assumption A is met from the general theory [55], it suffices to show λ ≥ 1 if and only if f is constant. Note that f = 1 is an eigenfunction for λ = 1 by assumption. Assume λ ≥ 1. Since k is bounded from the assumption, for an (x n ) ∞ n=1 converging to x, we have f (x n ) = 1 λ X k(x n , y)f (y) dµ(y) → 1 λ X k(x, y)f (y) dµ(y) = f (x) by the dominated convergence theorem. Thus, f is continuous. Let F = max x∈X f (x). If x * ∈ f −1 ({F }), then 0 = F − f (x * ) = X k(x * , y) F − 1 λ f (y) dµ(y). Let s 0 be the minimum nonnegative number satisfying K 0 −s0 ≥ k 0 L 4 (µ⊗µ) . As K 0 ∈ (0, 1) from Assumption A ′ , s 0 is well-defined. Then, for s := 1 + s 0 and m ≥ 2, from (9), we have e m L 4 ≤ k 0 L 4 (µ⊗µ) σ m e m L 2 ≤ 1 σ m K 0 s0 e m L 2 ≤ σ −1−s0 m e m L 2 . Thus, the condition for s and t := r + s of Proposition 5 is satisfied, and so we have the desired conclusion. Corollary 1 . 1Let m be a positive integer and X (1) , X(2) , . . . be i.i.d. real-valued random variables with E |X (1) | 4m < ∞. Then, there exists a constant C m > 0 such thatf (X (1) , . . . , X (d) ) L 4 ≤ C m f (X (1) , . . . , X (d) ) L 2for any positive integer d and any polynomial f : R d → R with degree up to m. , . . . be independent copies of X (1:d) = (X (1) , . . . , X (d) ), we have P E ϕ(X (1:d) ) ∈ conv{ϕ(X integers N ≥ C m D. Random Convex Hulls of Iterated Integrals. For a bounded-variation (BV) path x = (x 0 , . . . , x d ) : [0, 1] → R d+1 and a d-dimensional standard Brownian motion B = (B 1 , . . . , B d ) with B 0 t := t, we define the iterated integrals as Corollary 4 . 4Let d, m ≥ 1 be integers and B, B 1 , B 2 , . . . be independent standard d-dimensional Brownian motions. Then, if ϕ(B) is a D-dimensional random vector such that each coordinate is given by a linear combination of (I α (B)) α ≤m , then we have P(E[ϕ(B)] ∈ conv{ϕ(B 1 ), . . . , ϕ(B N )}) ≥ 1 2 for all integers N ≥ 17(1 + 18 · 8 m−1 )D. A Gaussian Hilbert space is a closed linear subspace of L 2 (Ω, G, P) whose elements all follow Gaussian distributions. A Log-concave distributionsfor all x, y ∈ R d and t ∈ [0, 1]. A probability distribution with a log-concave density is also called log-concave, and this class includes the multivariate Gaussian/exponential/Wishart distributions, the uniform distribution over a convex domain, and many more univariate common distributions[1,12]. For the log-concave random vectors, the following result is known:Here, α X (E[X]) is the Tukey depth of E[X] with respect to the ditrtibution of X which is defined as(4). The case when X is uniform over a convex set is proven in Grünbaum[22], and Lovász and Vempala[38,Section 5]gives simpler proofs than the original result in Caplin and Nalebuff[13].B ProofsB.1 Proof of Proposition 3Proof. It suffices to consider the case X 1 L 4 < ∞. If we write c = (c 1 , . . . , c D ) ⊤ , then by using independence we haveas we clearly have K ≥ 1 (or X = 0 almost surely).B.2 Proof of Theorem 7Proof. We give the proof by generalizing the proof of Lemma 5.3 in Janson[30]. It suffices to prove the statement for d = 2, as the product of GRPs is associative. LetIf we denote the product by G = G (1) ⊗G(2). Then, for a random variable X = ℓ,m X ℓ,m with X ℓ,m ∈ H As k(x * , ·) is a probability density (recall k ≥ 0 from the assumption) with respect to µ and supp µ = X , we must have λ ≤ 1 and k(x * , y) = 0 for all y ∈ f −1 ({F }). Now, it suffices to prove f −1 ({F }) = X actually holds when λ = 1. Let K = max x,y∈X k(x, y). By taking an ε > 0 such thatTherefore, if f −1 ({F }) = X , f is disconnected (because X is path-connected), and it is contradiction. This completes the proof.B.7 Proof of Proposition 9We first prove the following lemma.Proof. By Minkowski's integral inequality, we haveFrom this lemma, we havefor each m ≥ 2.Proof of Proposition 9. It suffices to consider the case k 0 L 4 (µ⊗µ) < ∞. Note that λ ℓ−1 = σ ℓ for ℓ = 1, 2, . . . for the GRP G k,µ , so λ 1 = σ 2 = K 0 . Let r 0 be the minimum nonnegative number satisfying K 0 −r0 ≥ √ 3 tr(K 0 ). Then, for r := 1 + r 0 , we have Log-concave probability distributions: Theory and statistical testing. Duke University Dept of Economics Working Paper. M Y An, M. Y. An. Log-concave probability distributions: Theory and statistical testing. Duke Uni- versity Dept of Economics Working Paper, 1997. Stein's method meets statistics: A review of some recent developments. A Anastasiou, A Barp, F.-X Briol, B Ebner, R E Gaunt, F Ghaderinezhad, J Gorham, A Gretton, C Ley, Q Liu, L Mackey, C J Oates, G Reinert, Y Swan, arXiv:2105.03481arXiv preprintA. Anastasiou, A. Barp, F.-X. Briol, B. Ebner, R. E. Gaunt, F. Ghaderinezhad, J. Gorham, A. Gretton, C. Ley, Q. Liu, L. Mackey, C. J. Oates, G. Reinert, and Y. Swan. Stein's method meets statistics: A review of some recent developments. arXiv preprint arXiv:2105.03481, 2021. On the equivalence between kernel quadrature rules and random feature expansions. F Bach, The Journal of Machine Learning Research. 181F. Bach. On the equivalence between kernel quadrature rules and random feature expansions. The Journal of Machine Learning Research, 18(1):714-751, 2017. On the equivalence between herding and conditional gradient algorithms. F Bach, S Lacoste-Julien, G Obozinski, International Conference on Machine Learning. F. Bach, S. Lacoste-Julien, and G. Obozinski. On the equivalence between herding and conditional gradient algorithms. In International Conference on Machine Learning, pages 1355-1362, 2012. Operators associated with a stochastic differential equation driven by fractional Brownian motions. F Baudoin, L Coutin, Stochastic Processes and their Applications. 117F. Baudoin and L. Coutin. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Processes and their Applications, 117(5):550-574, 2007. Inequalities in Fourier analysis. W Beckner, Annals of Mathematics. 1021W. Beckner. Inequalities in Fourier analysis. Annals of Mathematics, 102(1):159-182, 1975. Sobolev inequalities, the Poisson semigroup, and analysis on the sphere s n. W Beckner, Proceedings of the National Academy of Sciences. the National Academy of Sciences89W. Beckner. Sobolev inequalities, the Poisson semigroup, and analysis on the sphere s n . Proceedings of the National Academy of Sciences, 89(11):4816-4819, 1992. Kernel quadrature with DPPs. A Belhadji, R Bardenet, P Chainais, Advances in Neural Information Processing Systems. 32A. Belhadji, R. Bardenet, and P. Chainais. Kernel quadrature with DPPs. In Advances in Neural Information Processing Systems, volume 32, pages 12907-12917, 2019. Reproducing kernel Hilbert spaces in probability and statistics. A Berlinet, C Thomas-Agnan, SpringerA. Berlinet and C. Thomas-Agnan. Reproducing kernel Hilbert spaces in probability and statistics. Springer, 2004. The accuracy of the Gaussian approximation to the sum of independent variates. A C Berry, Transactions of the American Mathematical Society. 491A. C. Berry. The accuracy of the Gaussian approximation to the sum of independent variates. Transactions of the American Mathematical Society, 49(1):122-136, 1941. Étude des coefficients de Fourier des fonctions de L p (G). A Bonami, 20Annales de l'institut FourierA. Bonami. Étude des coefficients de Fourier des fonctions de L p (G). Annales de l'institut Fourier, 20(2):335-402, 1970. S Boyd, S P Boyd, L Vandenberghe, Convex optimization. Cambridge University PressS. Boyd, S. P. Boyd, and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004. Aggregation and social choice: A mean voter theorem. Econometrica. A Caplin, B Nalebuff, Journal of the Econometric Society. A. Caplin and B. Nalebuff. Aggregation and social choice: A mean voter theorem. Econo- metrica: Journal of the Econometric Society, pages 1-23, 1991. Super-samples from kernel herding. Y Chen, M Welling, A Smola, Conference on Uncertainty in Artificial Intelligence. Y. Chen, M. Welling, and A. Smola. Super-samples from kernel herding. In Conference on Uncertainty in Artificial Intelligence, pages 109-116, 2010. A randomized algorithm to reduce the support of discrete measures. F Cosentino, H Oberhauser, A Abate, Advances in Neural Information Processing Systems. 33F. Cosentino, H. Oberhauser, and A. Abate. A randomized algorithm to reduce the support of discrete measures. In Advances in Neural Information Processing Systems, volume 33, pages 15100-15110, 2020. J A Cuesta-Albertos, A Nieto-Reyes, The random Tukey depth. Computational Statistics & Data Analysis. 52J. A. Cuesta-Albertos and A. Nieto-Reyes. The random Tukey depth. Computational Statis- tics & Data Analysis, 52(11):4979-4988, 2008. The multidimensional truncated moment problem: The moment cone. arXiv: Functional Analysis. P J Di Dio, K Schmudgen, P. J. di Dio and K. Schmudgen. The multidimensional truncated moment problem: The moment cone. arXiv: Functional Analysis, 2018. High-dimensional integration: The quasi-Monte Carlo way. J Dick, F Y Kuo, I H Sloan, Acta Numerica. 22J. Dick, F. Y. Kuo, and I. H. Sloan. High-dimensional integration: The quasi-Monte Carlo way. Acta Numerica, 22:133-288, 2013. Sharp comparison of moments and the log-concave moment problem. A Eskenazis, P Nayar, T Tkocz, Advances in Mathematics. 334A. Eskenazis, P. Nayar, and T. Tkocz. Sharp comparison of moments and the log-concave moment problem. Advances in Mathematics, 334:389-416, 2018. On the Liapunoff limit of error in the theory of probability. C.-G Esseen, Arkiv for Matematik, Astronomi och Fysik. C.-G. Esseen. On the Liapunoff limit of error in the theory of probability. Arkiv for Matematik, Astronomi och Fysik, A: 1-19, 1942. Stable high-order cubature formulas for experimental data. J Glaubitz, Journal of Computational Physics. 447110693J. Glaubitz. Stable high-order cubature formulas for experimental data. Journal of Compu- tational Physics, 447:110693, 2021. Partitions of mass-distributions and of convex bodies by hyperplanes. B Grünbaum, Pacific Journal of Mathematics. 104B. Grünbaum. Partitions of mass-distributions and of convex bodies by hyperplanes. Pacific Journal of Mathematics, 10(4):1257-1261, 1960. . G H Hardy, J E Littlewood, G Pólya, Inequalities, Cambridge University PressG. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities. Cambridge University Press, 1952. Monte Carlo cubature construction. S Hayakawa, Japan Journal of Industrial and Applied Mathematics. 38S. Hayakawa. Monte Carlo cubature construction. Japan Journal of Industrial and Applied Mathematics, 38:561-577, 2021. Monte Carlo construction of cubature on Wiener space. S Hayakawa, K Tanaka, Japan Journal of Industrial and Applied Mathematics. 392S. Hayakawa and K. Tanaka. Monte Carlo construction of cubature on Wiener space. Japan Journal of Industrial and Applied Mathematics, 39(2):543-571, 2022. Estimating the probability that a given vector is in the convex hull of a random sample. S Hayakawa, T Lyons, H Oberhauser, arXiv:2101.04250arXiv preprintS. Hayakawa, T. Lyons, and H. Oberhauser. Estimating the probability that a given vector is in the convex hull of a random sample. arXiv preprint arXiv:2101.04250, 2021. Positively weighted kernel quadrature via subsampling. S Hayakawa, H Oberhauser, T Lyons, 10.48550/arXiv.2107.09597Advances in Neural Information Processing Systems. 2022S. Hayakawa, H. Oberhauser, and T. Lyons. Positively weighted kernel quadrature via sub- sampling. In Advances in Neural Information Processing Systems, 2022. doi: 10.48550/arXiv. 2107.09597. Unbiased simulation of stochastic differential equations. P Henry-Labordere, X Tan, N Touzi, The Annals of Applied Probability. 276P. Henry-Labordere, X. Tan, and N. Touzi. Unbiased simulation of stochastic differential equations. The Annals of Applied Probability, 27(6):3305-3341, 2017. Optimally-weighted herding is Bayesian quadrature. F Huszár, D Duvenaud, Conference on Uncertainty in Artificial Intelligence. F. Huszár and D. Duvenaud. Optimally-weighted herding is Bayesian quadrature. In Confer- ence on Uncertainty in Artificial Intelligence, pages 377-386, 2012. Gaussian Hilbert spaces. S Janson, Cambridge University PressS. Janson. Gaussian Hilbert spaces. Cambridge University Press, 1997. Absorption probabilities for Gaussian polytopes and regular spherical simplices. Z Kabluchko, D Zaporozhets, Advances in Applied Probability. 522Z. Kabluchko and D. Zaporozhets. Absorption probabilities for Gaussian polytopes and regular spherical simplices. Advances in Applied Probability, 52(2):588--616, 2020. Convergence guarantees for kernelbased quadrature rules in misspecified settings. M Kanagawa, B K Sriperumbudur, K Fukumizu, Advances in Neural Information Processing Systems. 29M. Kanagawa, B. K. Sriperumbudur, and K. Fukumizu. Convergence guarantees for kernel- based quadrature rules in misspecified settings. Advances in Neural Information Processing Systems, 29:3296-3304, 2016. Gaussian kernel quadrature at scaled Gauss-Hermite nodes. T Karvonen, S Särkkä, BIT Numerical Mathematics. 594T. Karvonen and S. Särkkä. Gaussian kernel quadrature at scaled Gauss-Hermite nodes. BIT Numerical Mathematics, 59(4):877-902, 2019. Concentration of multivariate polynomials and its applications. J H Kim, V H Vu, Combinatorica. 203J. H. Kim and V. H. Vu. Concentration of multivariate polynomials and its applications. Combinatorica, 20(3):417-434, 2000. On the best constants in the Khintchine inequality for Steinhaus variables. H König, Israel Journal of Mathematics. 2031H. König. On the best constants in the Khintchine inequality for Steinhaus variables. Israel Journal of Mathematics, 203(1):23-57, 2014. An improvement of the Berry-Esseen inequality with applications to Poisson and mixed Poisson random sums. V Korolev, I Shevtsova, Scandinavian Actuarial Journal. 20122V. Korolev and I. Shevtsova. An improvement of the Berry-Esseen inequality with applications to Poisson and mixed Poisson random sums. Scandinavian Actuarial Journal, 2012(2):81-105, 2012. High order recombination and an application to cubature on Wiener space. C Litterer, T Lyons, The Annals of Applied Probability. 224C. Litterer and T. Lyons. High order recombination and an application to cubature on Wiener space. The Annals of Applied Probability, 22(4):1301-1327, 2012. The geometry of logconcave functions and sampling algorithms. L Lovász, S Vempala, Random Structures & Algorithms. 303L. Lovász and S. Vempala. The geometry of logconcave functions and sampling algorithms. Random Structures & Algorithms, 30(3):307-358, 2007. Cubature on Wiener space. T Lyons, N Victoir, Proceedings of the Royal Society of London Series A. 460T. Lyons and N. Victoir. Cubature on Wiener space. Proceedings of the Royal Society of London Series A, 460:169-198, 2004. Fast and accurate least-mean-squares solvers. A Maalouf, I Jubran, D Feldman, Advances in Neural Information Processing Systems. H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché Buc, E. Fox, and R. Garnett32A. Maalouf, I. Jubran, and D. Feldman. Fast and accurate least-mean-squares solvers. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32, pages 8305-8316, 2019. Generalization error of random feature and kernel methods: hypercontractivity and kernel matrix concentration. S Mei, T Misiakiewicz, A Montanari, Applied and Computational Harmonic Analysis. S. Mei, T. Misiakiewicz, and A. Montanari. Generalization error of random feature and kernel methods: hypercontractivity and kernel matrix concentration. Applied and Computational Harmonic Analysis, 2021. Stable high-order randomized cubature formulae in arbitrary dimension. G Migliorati, F Nobile, Journal of Approximation Theory. 105706G. Migliorati and F. Nobile. Stable high-order randomized cubature formulae in arbitrary dimension. Journal of Approximation Theory, page 105706, 2022. The free Markoff field. E Nelson, Journal of Functional Analysis. 122E. Nelson. The free Markoff field. Journal of Functional Analysis, 12(2):211-227, 1973. On implementation of high-order recombination and its application to weak approximations of stochastic differential equations. S Ninomiya, Y Shinozaki, Proceedings of the NFA 29th Annual Conference. the NFA 29th Annual ConferenceS. Ninomiya and Y. Shinozaki. On implementation of high-order recombination and its appli- cation to weak approximations of stochastic differential equations. In Proceedings of the NFA 29th Annual Conference, 2021. Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. The Annals of Probability. I Nourdin, G Peccati, G Reinert, 38I. Nourdin, G. Peccati, and G. Reinert. Invariance principles for homogeneous sums: univer- sality of Gaussian Wiener chaos. The Annals of Probability, 38(5):1947-1985, 2010. Control functionals for Monte Carlo integration. C Oates, M Girolami, N Chopin, Journal of the Royal Statistical Society. Series B: Statistical Methodology. 79C. Oates, M. Girolami, and N. Chopin. Control functionals for Monte Carlo integration. Journal of the Royal Statistical Society. Series B: Statistical Methodology, 79:695-718, 2017. Bayes-Hermite quadrature. A O&apos;hagan, Journal of Statistical Planning and Inference. 293A. O'Hagan. Bayes-Hermite quadrature. Journal of Statistical Planning and Inference, 29 (3):245-260, 1991. Some results on the signature and cubature of the fractional Brownian motion for H > 1 2. R Passeggeri, arXiv:1609.07352arXiv preprintR. Passeggeri. Some results on the signature and cubature of the fractional Brownian motion for H > 1 2 . arXiv preprint arXiv:1609.07352, 2016. Parameterfreie abschätzung und realisierung von erwartungswerten. H Richter, Blätter der DGVFM. 32H. Richter. Parameterfreie abschätzung und realisierung von erwartungswerten. Blätter der DGVFM, 3(2):147-162, 1957. Moments of non-negative mass. W W Rogosinski, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 245W. W. Rogosinski. Moments of non-negative mass. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 245(1240):1-27, 1958. Bulletin de la societe mathematique de France. P C Rosenbloom, 80Quelques classes de problèmes extrémaux. iiP. C. Rosenbloom. Quelques classes de problèmes extrémaux. ii. Bulletin de la societe math- ematique de France, 80:183-215, 1952. Rousseeuw and I. Ruts. The depth function of a population distribution. P J , Metrika. 493P. J. Rousseeuw and I. Ruts. The depth function of a population distribution. Metrika, 49 (3):213-244, 1999. Concentration and moment inequalities for polynomials of independent random variables. W Schudy, M Sviridenko, Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms. the twenty-third annual ACM-SIAM symposium on Discrete AlgorithmsSIAMW. Schudy and M. Sviridenko. Concentration and moment inequalities for polynomials of independent random variables. In Proceedings of the twenty-third annual ACM-SIAM sympo- sium on Discrete Algorithms, pages 437-446. SIAM, 2012. Hypercontractive semigroups and two dimensional selfcoupled Bose fields. B Simon, R Høegh-Krohn, Journal of Functional Analysis. 92B. Simon and R. Høegh-Krohn. Hypercontractive semigroups and two dimensional self- coupled Bose fields. Journal of Functional Analysis, 9(2):121-180, 1972. Mercer's theorem on general domains: On the interaction between measures, kernels, and RKHSs. I Steinwart, C Scovel, Constructive Approximation. 353I. Steinwart and C. Scovel. Mercer's theorem on general domains: On the interaction between measures, kernels, and RKHSs. Constructive Approximation, 35(3):363-417, 2012. Approximate calculation of multiple integrals. A H Stroud, Prentice-HallA. H. Stroud. Approximate calculation of multiple integrals. Prentice-Hall, 1971. Formules de cubature mécanique à coefficients non négatifs. V Tchakaloff, Bulletin des Sciences Mathématiques. 81V. Tchakaloff. Formules de cubature mécanique à coefficients non négatifs. Bulletin des Sciences Mathématiques, 81:123-134, 1957. Carathéodory cubature measures. M Tchernychova, University of OxfordPhD thesisM. Tchernychova. Carathéodory cubature measures. PhD thesis, University of Oxford, 2015. Mathematics and the picturing of data. J W Tukey, Proceedings of the International Congress of Mathematicians. the International Congress of MathematiciansVancouver2J. W. Tukey. Mathematics and the picturing of data. In Proceedings of the International Congress of Mathematicians, Vancouver, 1975, volume 2, pages 523-531, 1975. A continuous analogue of the upper bound theorem. U Wagner, E Welzl, Discrete & Computational Geometry. 262U. Wagner and E. Welzl. A continuous analogue of the upper bound theorem. Discrete & Computational Geometry, 26(2):205-219, 2001. Spline Models for Observational Data. G Wahba, Society for Industrial and Applied Mathematics. G. Wahba. Spline Models for Observational Data. Society for Industrial and Applied Mathe- matics, 1990. Limits of a distribution function determined by absolute moments and inequalities satisfied by absolute moments. A Wald, Transactions of the American Mathematical Society. 462A. Wald. Limits of a distribution function determined by absolute moments and inequalities satisfied by absolute moments. Transactions of the American Mathematical Society, 46(2): 280-306, 1939. A problem in geometric probability. J G Wendel, Mathematica Scandinavica. 111J. G. Wendel. A problem in geometric probability. Mathematica Scandinavica, 11(1):109-111, 1963. A new approach for the computation of halfspace depth in high dimensions. Y Zuo, Communications in Statistics-Simulation and Computation. 483Y. Zuo. A new approach for the computation of halfspace depth in high dimensions. Com- munications in Statistics-Simulation and Computation, 48(3):900-921, 2019.
[]
[ "Chiral Josephson effect in double layers: the role of particle-hole duality", "Chiral Josephson effect in double layers: the role of particle-hole duality" ]
[ "Klaus Ziegler \nInstitut für Physik\nUniversität Augsburg\nD-86135AugsburgGermany\n" ]
[ "Institut für Physik\nUniversität Augsburg\nD-86135AugsburgGermany" ]
[]
The Josephson effect of inter-layer s-wave pairing in a double layer of two chiral metals is considered. We employ the duality relation between electron-electron and electron-hole double layers to discuss the zero-energy eigenmodes at a domain wall and their coupling to the superfluid state. This is described in terms of the quasiparticle current and the supercurrent. It turns out that the degeneracy of the zero-energy eigenmodes is resolved by the coupling to the supercurrent. The duality relation between the electron-electron and electron-hole double layers leads to the same current distribution in both systems but to different zero-energy modes. arXiv:2301.12552v1 [cond-mat.supr-con]
10.1002/ctpp.202300014
[ "https://export.arxiv.org/pdf/2301.12552v1.pdf" ]
256,390,572
2301.12552
1627b9fe67510b82d940024e2ef1b96e40565264
Chiral Josephson effect in double layers: the role of particle-hole duality Klaus Ziegler Institut für Physik Universität Augsburg D-86135AugsburgGermany Chiral Josephson effect in double layers: the role of particle-hole duality (Dated: January 31, 2023) The Josephson effect of inter-layer s-wave pairing in a double layer of two chiral metals is considered. We employ the duality relation between electron-electron and electron-hole double layers to discuss the zero-energy eigenmodes at a domain wall and their coupling to the superfluid state. This is described in terms of the quasiparticle current and the supercurrent. It turns out that the degeneracy of the zero-energy eigenmodes is resolved by the coupling to the supercurrent. The duality relation between the electron-electron and electron-hole double layers leads to the same current distribution in both systems but to different zero-energy modes. arXiv:2301.12552v1 [cond-mat.supr-con] I. INTRODUCTION One of the most fascinating observations in condensed matter physics is the pairing effect, leading to phenomena such as superconductivity and superfluidity. Although the pairing effect is of quantum nature, its theoretical description in terms of a macroscopic order parameter field is given by a classical (mean-field) theory. Excitations in the form of quasiparticles, on the other hand, are represented by the Bogoliubov de Gennes (BdG) equation that describes a quantum wave function [1]. A related interesting phenomenon is the Josephson effect [2] that originates in a coupling between the macroscopic superfluid or superconducting order parameter with the quasiparticle modes [3][4][5][6][7][8]. It has been discussed for different systems, including electron-hole bilayers [9,10] and electron-hole double layers [11][12][13] and electron-hole double-bilayers [14,15]. This effect has the potential for the development of new technologies. For instance, it has been used to create and manipulate qubits in quantum computational devices [16][17][18]. An important aspect of the quasiparticles is their sensitivity to the underlying spatial structure in terms of geometry and topology, in particular, for zero-energy modes. The interplay of the Josephson effect with the topological properties of the quasiparticle modes was recently discussed for an electron-electron double layer (EEDL) [19] and for an electron-hole double layer (EHDL) [13] separately. The purpose of the present paper is study the connection of these two systems through the particle-hole duality and how this affects the zero-energy eigenmodes, the coupling of these modes to the superfluid and the distribution of currents. This analysis might be useful for future studies of more complex, multi-band systems with electron-electron and electron-hole pairing. II. MODEL: BOGOLIUBOV DE GENNES EQUATION/HAMILTONIAN The EEDL and the EHDL are dual to each other [20]. In the following we discuss these two cases separately and compare the resulting Josephson currents in Sect. IV. Both systems are treated within a BCS-like mean-field approach. This leads to an order parameter ∆ that characterizes the superconducting state of the EEDL and the superfluid state of the EHDL. Excitations in the form of quasiparticles are obtained form the corresponding BdG Hamiltonian, where the latter describes the quantum fluctuations about the mean-field approximation. In the following discussion we consider the inter-layer pairing but ignore the intra-layer pairing. This is a simplification which is plausible for a small distance of the layers and due to screening inside the layers but has been debated in the literature [21,22]. Moreover, inter-layer tunneling is suppressed by a dielectric between the layers. A. Chiral electron-electron double layer The EEDL comprises two electronic layers with a positively charged extra layer. The latter can either be an external gate (as visualized in Fig. 1a) or is provided by the positive charges inside the metallic layers. In both cases the entire system preserves charge neutrality. The electrons in the two layers repel each other due to the Coulomb interaction. The geometric constraint enable the electrons at fixed density Inter-layer tunneling is suppressed by a dielectric medium and inter-layer pairing requires a small distance of the layers to make the Coulomb interaction sufficiently strong. to form inter-layer Cooper pairs. This is formally supported by the duality transformation to the EHDL, in which the electron-hole pairs are subject to an attractive Coulomb interaction. In other words, the formation of inter-layer electron-hole pairs in the EHDL [11] is transformed into to inter-layer electronelectron pairs by the duality transformation [20]. Then the related quasiparticles are described by the BdG Hamiltonian of two layers with opposite chiralities reads [19] H EEDL = h 1 σ 1 + h 2 σ 2 ∆σ 2 ∆σ 2 h 1 σ 1 − h 2 σ 2 ,(1) where σ j are Pauli matrices, h j are tight-binding hopping matrices and ∆ is the pairing order parameter. For the subsequent discussion we assume a honeycomb lattice for the underlying spatial structure of the tight-binding model, such that the quasiparticle Hamiltonian describes graphene-like materials. Assuming translational invariance in y direction, the low-energy BdG Hamiltonian becomes with h 1 ∼ ihv F ∂ x , h 2 ∼hv F k y H EEDL ∼ ihv F ∂ x σ 1 +hv F k y σ 2 ∆(x)σ 2 ∆(x)σ 2 ihv F ∂ x σ 1 −hv F k y σ 2 ,(2) where v F is the Fermi velocity. Now we consider a domain wall in y direction at x = 0, as sketched in Fig. 2a: ∆(x) = sgn(x)|∆|. The resulting eigenvalue problem can be solved. At zero energy there is an exceptional point for the Hamiltonian (2), where the four-fold degeneracy coalesces to a two-dimensional eigenspace with two independent zero-energy modes (cf. App. A): Ψ 1 = 1 N    1 0 1 0    e −|∆||x|/hv F , Ψ 2 = 1 N    0 1 0 −1    e −|∆||x|/hv F(3) with the normalization N = 2v Fh /|∆|. Any superposition of the two zero-energy modes Φ = a 1 Ψ 1 + a 2 Ψ 2 with complex coefficients a j = |a j |e iϕj (and normalization |a 1 | 2 + |a 2 | 2 = 1) is also a zero-energy mode. Thus, the zero-energy eigenmodes are complex in general and only real for a special choice of the coefficients. These two modes are expressed separately for the top and for the bottom layer as Φ ↑ = a 1 a 2 e −|∆||x|/hv F N , Φ ↓ = a 1 −a 2 e −|∆||x|/hv F N ,(4) which will be used for the calculation of the Josephson currents in Sect. III. H EHDL = h 1 σ 1 + h 2 σ 2 ∆σ 3 ∆ * σ 3 h 1 σ 1 + h 2 σ 2 ,(5) where the chirality of the two layers is the same now and the pairing order parameter appears with a Pauli matrix σ 3 . This means that there is a coupling between the same metallic bands of the two layers with opposite sign though. The BdG Hamiltonian is dual to the BdG Hamiltonian of the EEDL in Eq. (1), and the duality transformation reads H EHDL (i∆) = V H EEDL (∆)V , V = σ 0 0 0 σ 1 ,(6) where the order parameter aquires a global imaginary unit. This implies for the eigenvalue equation H EEDL (∆)Ψ E = EΨ E H EHDL (i∆)V Ψ E = V H EEDL (∆)V V Ψ E = EV Ψ E ,(7) i.e., V Ψ E is eigenmode of H EHDL (∆) with eigenvalue E. Thus, the spectrum is invariant under the duality transformation, whereas the zero-energy eigenmodes are not. In particular, from Eq. (3) for the domain wall ∆(x) = isgn(x)|∆| an exceptional point at zero energy and a two-dimensional eigenspace of zero-energy modes V Ψ 1 = 1 N    1 0 0 1    e −|∆||x|/hv F , V Ψ 2 = 1 N    0 1 −1 0    e −|∆||x|/hv F .(8) From these modes we can construct again the zero-energy modes of the individual layers as V Φ = Φ ↑ Φ ↓ with Φ ↑ = a 1 a 2 e −|∆||x|/hv F N , Φ ↓ = −a 2 a 1 e −|∆||x|/hv F N .(9) III. JOSEPHSON CURRENTS With the help of the BdG equation we derive the continuity equation for the two layers separately as [3] ∂ t Φ σ · Φ σ + ∂ x I xσ = 0 (σ =↑, ↓),(10) where the total current I = j + j s is the sum of the quasiparticle current j and the supercurrent j s . For the y component we have ∂ y I yσ = 0 due to the uniform mode in the y direction. The quasiparticle current operator of a BdG Hamiltonian H BdG reads j x = ī h [H BdG , x]. The BdG equation of the EEDL yields for the continuity equation (10) in the top layer Fig. 2b) individually, we obtain a double torus with a cross-section visualized in b), where each torus has two domain walls. Then the currents wind along the two domain walls clockwise and counterclockwise around each torus, respectively. and in the bottom layer ∂ t Φ ↑ · Φ ↑ + ∂ x j x↑ = i ∆ h Ψ * ↓ σ 2 Ψ ↑ − i ∆ * h Ψ * ↑ σ 2 Ψ ↓ = 2sgn(x) |∆| 2 v Fh 2 Re(a 1 a * 2 )e −2|∆||x|/hv F(11)∂ t Φ ↓ · Φ ↓ + ∂ x j x↓ = i ∆ * h Ψ * ↑ σ 2 Ψ ↓ − i ∆ h Ψ * ↓ σ 2 Ψ ↑ = −2sgn(x) |∆| 2 v Fh 2 Re(a 1 a * 2 )e −2|∆||x|/hv F .(12) The expressions on the right-hand side of the equations are equal up to a minus sign. The quasiparticle currents j x↑ , j x↓ are directly calculated from the commutator j x = ī h [H EEDL , x], which gives j x↑ = −j x↓ = − |∆| h Re(a 1 a * 2 )e −2|∆||x|/hv F ,(13) such that we obtain from the continuity equations after integration along the x direction the x-components of the supercurrents as j s x↑,↓ (x) = ∓ |∆| h Re(a 1 a * 2 )(1 − e −2|∆||x|/hv F ).(14) The supercurrent component j s x↑,↓ (x) vanishes at the domain wall x = 0 and becomes ∓ |∆| h Re(a 1 a * 2 ) for |x| hv F /|∆|. Finally, from j y = ī h [H EEDL , y] we get the y-components of the quasiparticle currents j y↑ = j y↓ = −v F Φ ↓ · σ 2 Φ ↓ = − |∆| h Im(a 1 a * 2 )e −2|∆||x|/hv F .(15) The corresponding continuity equation of the EHDL reads for the top layer [13] ∂ t Φ ↑ · Φ ↑ + ∂ x j x↑ = i ∆ h Ψ * ↓ σ 3 Ψ ↑ − i ∆ * h Ψ * ↑ σ 3 Ψ ↓ = 2sgn(x) |∆| 2 v Fh 2 Re(a 1 a * 2 )e −2|∆||x|/hv F(16) and for the bottom layer ∂ t Φ ↓ · Φ ↓ + ∂ x j x↓ = i ∆ * h Ψ * ↑ σ 3 Ψ ↓ − i ∆ h Ψ * ↓ σ 3 Ψ ↑ = −2sgn(x) |∆| 2 v Fh 2 Re(a 1 a * 2 )e −2|∆||x|/hv F ,(17) since i(∆−∆ * ) = −2sgn(x)|∆|. Together with the commutators j x = ī h [H EHDL , x] and j y = ī h [H EHDL , y] we obtain for the current components of the EHDL the same expression as given in Eqs. (13) - (15). This agreement of the currents is a consequence of the duality relation between the EEDL and the EHDL. IV. DISCUSSION AND CONCLUSIONS The main result of our calculation is the relation between the supercurrent and the zero-energy eigenmodes, where the latter are characterized by the complex coefficients a 1 and a 2 in Eqs. (4) and (9), respectively. This relation reads for the supercurent away from the domain wall as j s x,↑↓ ∼ ∓ |∆| h Re(a 1 a * 2 ) = ∓ |∆||a 1 a 2 | h cos(ϕ 1 − ϕ 2 ) (|x| hv F /|∆|)(18) for the EEDL as well as for the EHDL, which depends on the phases of the coefficients a 1 and a 2 . Another interesting result is the angle α between the quasiparticle current and the domain wall in the top layer α = arctan(j y↑ /j x↑ ) = π 2 − ϕ 2 + ϕ 1 .(19) Thus, the relative phase of the eigenmode coefficients can be tuned by the angle α. The BdG Hamiltonians H EDDL and H EHDL lead to similar results. In particular, the Josephson currents are the same for both cases, whereas the zero-energy quasiparticle modes are different. The origin of this similarity is the duality relation (6), (7) of the two Hamiltonians and their eigenmodes. Moreover, both Hamiltonians get a sign change under the following transformation H EEDL → T H EEDL T = −H EEDL , T = σ 3 0 0 σ 3 ,(20) and H EHDL → T H EHDL T = −H EHDL , T = V T V = σ 3 0 0 −σ 3 ,(21) which reflects the chirality and the fact that their chiralities are not identical. All these results indicate that a domain wall or an edge affects the current distribution in the system. Thus, for the general case we must take into account all edges and domain walls, where the order parameter changes. On the other hand, we can avoid edges by choosing proper boundary conditions. In y direction we have already assumed periodic boundary conditions to create a uniform mode in this direction. Assuming two domain walls (cf. Fig. 2b)) and periodic boundary conditions in x direction for both layers individually, the resulting system is a double torus, which has no edges except for the domain walls, as visualized in Fig. 3. Then the coefficients a 1 , a 2 are fixed by the matching condition of the supercurrents in the regions between the domain walls. While these considerations give us an idea about the role of the Josephson effect in chiral double layers, a complete description requires a solution of the entire microscopic model through a self-consistent approach. Then the supercurrent is induced by an external current or external field, which is represented by a vector potential in the BdG Hamiltonian. This external field also affects the order parameter field ∆. In particular, the creation and measurement of currents in the EHDL was discussed in Ref. [23]. Appendix A: Coalescent eigenmodes For the eigenmodes of the BdG Hamiltonian H EEDL we make the ansatz Ψ(x) = ψe −bx , where ψ is a four-component spinor and b depends on x. This gives for k y = 0 withb = v Fh b the four-dimensional eigenvalue equation H EEDL ψ =    0 −ib 0 −i∆ −ib 0 i∆ 0 0 −i∆ 0 −ib i∆ 0 −ib 0    ψ = Eψ,(A1) which has the eigenvalue E − = − |∆| 2 −b 2 with the pair of eigenspinors ψ 1− =    1 0 b/∆ −i |∆| 2 −b 2 /∆    , ψ 2− =    0 1 i |∆| 2 −b 2 /∆ −b/∆    (A2) and the eigenvalue E + = |∆| 2 −b 2 with the pair of eigenspinors ψ 1+ =    1 0 b/∆ i |∆| 2 −b 2 /∆    , ψ 2+ =    0 1 −i |∆| 2 −b 2 /∆ −b/∆    .(A3) This indicates a two-fold degeneracy of the eigenvalues E ± , respectively. The limitb → ∆ yields E = 0 and the pairwise coalescent eigenspinors as ψ 1− → ψ 1+ →    1 0 1 0    , ψ 2− → ψ 2+ →    0 1 0 −1    . (A4) Thus, the eigenspace at E = 0 has only two dimensions, which represents an exceptional point [24]. This effect is known for solutions of the BdG Hamiltonian with point-and line defects [25]. FIG. 1 . 1An electron-electron double layer (a) and an electron-hole double layer (b) with inter-layer pairing due to Coulomb interaction, where the schematic gate in a) is positively charged and guarantees charge neutrality. FIG. 2 . 2Electronic double layer with domain walls, which is given by a sign jump of the pairing order parameter. The local currents (blue arrows) flow in the same (opposite) direction in the two layers parallel (perpendicular) to the domain wall.B. Chiral electron-hole double layerThe BdG Hamiltonian of the EHDL reads[13] FIG. 3 . 3After gluing both layers in Differential equations of quantum mechanics. I , Quart. Appl. Math. 80451I. Sigal, Differential equations of quantum mechanics, Quart. Appl. Math. 80, 451 (2022). Possible new effects in superconductive tunnelling. B Josephson, Physics Letters. 1251B. Josephson, Possible new effects in superconductive tunnelling, Physics Letters 1, 251 (1962). Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion. G E Blonder, M Tinkham, T M Klapwijk, Phys. Rev. B. 254515G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Transition from metallic to tunneling regimes in super- conducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion, Phys. Rev. B 25, 4515 (1982). A unified theory of clean josephson junctions. A Furusaki, M Tsukada, Physica B: Condensed Matter. 16519A. Furusaki and M. Tsukada, A unified theory of clean josephson junctions, Physica B: Condensed Matter 165-166, 967 (1990), lT-19. Josephson current through a superconducting quantum point contact shorter than the coherence length. C W J Beenakker, H Van Houten, Phys. Rev. Lett. 663056C. W. J. Beenakker and H. van Houten, Josephson current through a superconducting quantum point contact shorter than the coherence length, Phys. Rev. Lett. 66, 3056 (1991). Josephson current in s-wave-superconductor/sr2ruo4 junctions. Y Asano, Y Tanaka, M Sigrist, S Kashiwaya, Phys. Rev. B. 67184505Y. Asano, Y. Tanaka, M. Sigrist, and S. Kashiwaya, Josephson current in s-wave-superconductor/sr2ruo4 junctions, Phys. Rev. B 67, 184505 (2003). Solution of the bogoliubov-de gennes equations at zero temperature throughout the bcs-bec crossover: Josephson and related effects. A Spuntarelli, P Pieri, G Strinati, Physics Reports. 488111A. Spuntarelli, P. Pieri, and G. Strinati, Solution of the bogoliubov-de gennes equations at zero temperature throughout the bcs-bec crossover: Josephson and related effects, Physics Reports 488, 111 (2010). Andreev bound states and their signatures. J A Sauls, 1805.11069Philosophical Transactions of the Royal Society of London Series A. 376cond-mat.supr-conJ. A. Sauls, Andreev bound states and their signatures, Philosophical Transactions of the Royal Society of London Series A 376, 20180140 (2018), 1805.11069 [cond-mat.supr-con]. Dipolar superfluidity in electron-hole bilayer systems. A V Balatsky, Y N Joglekar, P B Littlewood, Phys. Rev. Lett. 93266801A. V. Balatsky, Y. N. Joglekar, and P. B. Littlewood, Dipolar superfluidity in electron-hole bilayer systems, Phys. Rev. Lett. 93, 266801 (2004). Josephson effect as a signature of electron-hole superfluidity in bilayers of van der waals heterostructures. F Pascucci, S Conti, D Neilson, J Tempere, A Perali, Phys. Rev. B. 106220503F. Pascucci, S. Conti, D. Neilson, J. Tempere, and A. Perali, Josephson effect as a signature of electron-hole superfluidity in bilayers of van der waals heterostructures, Phys. Rev. B 106, L220503 (2022). A new mechanism for superconductivity: pairing between spatially separated electrons and holes. Y E Lozovik, V I Yudson, Soviet Journal of Experimental and Theoretical Physics. 44389Y. E. Lozovik and V. I. Yudson, A new mechanism for superconductivity: pairing between spatially separated electrons and holes, Soviet Journal of Experimental and Theoretical Physics 44, 389 (1976). Thermal transport through an exciton-condensate josephson junction based on graphene double layers. C Zhang, G Jin, 10.1063/1.4831671Applied Physics Letters. 103202111C. Zhang and G. Jin, Thermal transport through an exciton-condensate josephson junction based on graphene double layers, Applied Physics Letters 103, 202111 (2013), https://doi.org/10.1063/1.4831671. K Ziegler, arXiv:2212.11161Josephson effect in excitonic chiral double layers (2022). K. Ziegler, Josephson effect in excitonic chiral double layers (2022), arXiv:2212.11161. High-temperature superfluidity in double-bilayer graphene. A Perali, D Neilson, A R Hamilton, Phys. Rev. Lett. 110146803A. Perali, D. Neilson, and A. R. Hamilton, High-temperature superfluidity in double-bilayer graphene, Phys. Rev. Lett. 110, 146803 (2013). Excitonic superfluid phase in double bilayer graphene. J I A Li, T Taniguchi, K Watanabe, J Hone, C R Dean, arXiv:1608.05846Nature Physics. 13751cond-mat.mes-hallJ. I. A. Li, T. Taniguchi, K. Watanabe, J. Hone, and C. R. Dean, Excitonic superfluid phase in double bilayer graphene, Nature Physics 13, 751 (2017), arXiv:1608.05846 [cond-mat.mes-hall]. Quantum-state engineering with josephson-junction devices. Y Makhlin, G Schön, A Shnirman, Rev. Mod. Phys. 73357Y. Makhlin, G. Schön, and A. Shnirman, Quantum-state engineering with josephson-junction devices, Rev. Mod. Phys. 73, 357 (2001). Quantum bits with josephson junctions. A F Kockum, F Nori, Fundamentals and Frontiers of the Josephson Effect. F. TafuriChamSpringer International PublishingA. F. Kockum and F. Nori, Quantum bits with josephson junctions, in Fundamentals and Frontiers of the Josephson Effect, edited by F. Tafuri (Springer International Publishing, Cham, 2019) pp. 703-741. A , arXiv:2211.10852Theory of superconducting qubits beyond the lumped element approximation (2022). A. Mizel, Theory of superconducting qubits beyond the lumped element approximation (2022), arXiv:2211.10852. Anomalous josephson effect of s-wave pairing states in chiral double layers. K Ziegler, A Sinner, Y E Lozovik, Phys. Rev. Lett. 128157001K. Ziegler, A. Sinner, and Y. E. Lozovik, Anomalous josephson effect of s-wave pairing states in chiral double layers, Phys. Rev. Lett. 128, 157001 (2022). Pairing transition in a double layer with interlayer Coulomb repulsion. A Sinner, Y E Lozovik, K Ziegler, arXiv:1912.13257Physical Review Research. 233085cond-mat.supr-conA. Sinner, Y. E. Lozovik, and K. Ziegler, Pairing transition in a double layer with interlayer Coulomb repulsion, Physical Review Research 2, 033085 (2020), arXiv:1912.13257 [cond-mat.supr-con]. Intralayer versus interlayer pairing in the copper oxide superconductors: The three-and four-layer problems. S H Liu, R A Klemm, Phys. Rev. B. 45415S. H. Liu and R. A. Klemm, Intralayer versus interlayer pairing in the copper oxide superconductors: The three-and four-layer problems, Phys. Rev. B 45, 415 (1992). Collective interlayer pairing and pair superfluidity in vertically stacked layers of dipolar excitons. M Zimmerman, R Rapaport, S Gazit, 10.1073/pnas.2205845119Proceedings of the National Academy of Sciences. 1192205845119M. Zimmerman, R. Rapaport, and S. Gazit, Collective interlayer pairing and pair superfluidity in vertically stacked layers of dipolar excitons, Proceedings of the National Academy of Sciences 119, e2205845119 (2022), https://www.pnas.org/doi/pdf/10.1073/pnas.2205845119. How to make a bilayer exciton condensate flow. J.-J Su, A H Macdonald, Nature Physics. 4799J.-J. Su and A. H. MacDonald, How to make a bilayer exciton condensate flow, Nature Physics 4, 799 (2008). Perturbation theory for linear operators. T Kato, Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics. BerlinSpringer2nd ed.T. Kato, Perturbation theory for linear operators; 2nd ed., Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics (Springer, Berlin, 1976). Exceptional points for chiral majorana fermions in arbitrary dimensions. I , Europhysics Letters). 11067005EPLI. Mandal, Exceptional points for chiral majorana fermions in arbitrary dimensions, EPL (Europhysics Let- ters) 110, 67005 (2015).
[]
[ "THE SYNCHROTRON LOW-ENERGY SPECTRUM ARISING FROM THE COOLING OF ELECTRONS IN GAMMA-RAY BURSTS", "THE SYNCHROTRON LOW-ENERGY SPECTRUM ARISING FROM THE COOLING OF ELECTRONS IN GAMMA-RAY BURSTS" ]
[ "A D Panaitescu \nIntelligence and Space Research\nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n", "W T Vestrand \nIntelligence and Space Research\nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n" ]
[ "Intelligence and Space Research\nLos Alamos National Laboratory\n87545Los AlamosNMUSA", "Intelligence and Space Research\nLos Alamos National Laboratory\n87545Los AlamosNMUSA" ]
[]
This work is a continuation of a previous effort (Panaitescu 2019) to study the cooling of relativistic electrons through radiation (synchrotron and self-Compton) emission and adiabatic losses, with application to the spectra and light-curves of the synchrotron Gamma-Ray Burst produced by such cooling electrons. Here, we derive the low-energy slope β LE of GRB pulse-integrated spectrum and quantify the implications of the measured distribution of β LE .If the magnetic field lives longer than it takes the cooling GRB electrons to radiate below 1-10 keV, then radiative cooling processes of power P (γ) ∼ γ n with n ≥ 2 (γ is the electron energy), i.e. synchrotron and inverse-Compton (iC) through Thomson scatterings, lead to a soft low-energy spectral slope β LE ≤ −1/2 of the GRB pulse-integrated spectrum F ǫ ∼ ǫ βLE below the peak-energy E γ , irrespective of the duration of electron injection t I . iC-cooling dominated by scatterings at the Thomson-Klein-Nishina transition of synchrotron photons below E γ has an index n = 2/3 → 1 and yield harder integrated spectra with β LE ∈ [0, 1/6], while adiabatic electron-cooling leads to a soft slope β LE = −3/4.Radiative processes that produce soft integrated spectra can accommodate the harder slopes measured by CGRO/BATSE and Fermi/GBM only if the magnetic field life-time t B is shorter than the time during which the typical GRB electron cools to radiate below 10 keV (i.e. less than several radiative cooling timescales t rad of that typical electron). In this case, there is a one-to-one correspondence between t B and β LE . To account for low-energy slopes β LE > −3/4, adiabatic electron-cooling requires a similar restriction on t B . In this case, the diversity of slopes arises mostly from how the electron-injection rate varies with time (temporal power-law injection rates yield power-law low-energy GRB spectra) and not from the magnetic field timescale.
10.3847/1538-4357/ac8b75
[ "https://export.arxiv.org/pdf/2209.10014v2.pdf" ]
252,408,854
2209.10014
8632ca55b7b58629ec91b3f44edec88388039822
THE SYNCHROTRON LOW-ENERGY SPECTRUM ARISING FROM THE COOLING OF ELECTRONS IN GAMMA-RAY BURSTS 6 Oct 2022 Draft version October 10, 2022 Draft version October 10, 2022 A D Panaitescu Intelligence and Space Research Los Alamos National Laboratory 87545Los AlamosNMUSA W T Vestrand Intelligence and Space Research Los Alamos National Laboratory 87545Los AlamosNMUSA THE SYNCHROTRON LOW-ENERGY SPECTRUM ARISING FROM THE COOLING OF ELECTRONS IN GAMMA-RAY BURSTS 6 Oct 2022 Draft version October 10, 2022 Draft version October 10, 2022arXiv:2209.10014v2 [astro-ph.HE] Preprint typeset using L A T E X style emulateapj v. 12/16/11 This work is a continuation of a previous effort (Panaitescu 2019) to study the cooling of relativistic electrons through radiation (synchrotron and self-Compton) emission and adiabatic losses, with application to the spectra and light-curves of the synchrotron Gamma-Ray Burst produced by such cooling electrons. Here, we derive the low-energy slope β LE of GRB pulse-integrated spectrum and quantify the implications of the measured distribution of β LE .If the magnetic field lives longer than it takes the cooling GRB electrons to radiate below 1-10 keV, then radiative cooling processes of power P (γ) ∼ γ n with n ≥ 2 (γ is the electron energy), i.e. synchrotron and inverse-Compton (iC) through Thomson scatterings, lead to a soft low-energy spectral slope β LE ≤ −1/2 of the GRB pulse-integrated spectrum F ǫ ∼ ǫ βLE below the peak-energy E γ , irrespective of the duration of electron injection t I . iC-cooling dominated by scatterings at the Thomson-Klein-Nishina transition of synchrotron photons below E γ has an index n = 2/3 → 1 and yield harder integrated spectra with β LE ∈ [0, 1/6], while adiabatic electron-cooling leads to a soft slope β LE = −3/4.Radiative processes that produce soft integrated spectra can accommodate the harder slopes measured by CGRO/BATSE and Fermi/GBM only if the magnetic field life-time t B is shorter than the time during which the typical GRB electron cools to radiate below 10 keV (i.e. less than several radiative cooling timescales t rad of that typical electron). In this case, there is a one-to-one correspondence between t B and β LE . To account for low-energy slopes β LE > −3/4, adiabatic electron-cooling requires a similar restriction on t B . In this case, the diversity of slopes arises mostly from how the electron-injection rate varies with time (temporal power-law injection rates yield power-law low-energy GRB spectra) and not from the magnetic field timescale. INTRODUCTION GRB-Pulse Temporal Properties GRB observations (e.g. Fenimore et al 1995, Norris et al 1996, Lee et al 2000 have established some essential/basic features of GRB pulses: i) they peak earlier at higher energies, ii) they are time-asymmetric, rising faster than they fall, with a rise-to-fall time ratio t r /t f in the range (0.1−0.9), iii) their temporal asymmetry is on average energyindependent, and iv) they last longer at lower energies, having a pulse duration-energy dependence δt γ ∼ ǫ −0.4 . Figures 5 and 6 of Panaitescu 2019 (P19) provide a limited assessment of the ability of adiabatic and synchrotron electron cooling to account for the above pulse features: i) peaks occurring earlier at higher energies is a trivial consequence for any electron cooling process, ii) both cooling processes yield pulses that are more timesymmetric at higher energies, in conflict with observations of most GRB pulses, iii) if the pulse duration δt γ dependence on energy arises from only electron cooling then, for a constant magnetic field, adiabatic cooling yields δt γ ∼ ǫ −0.4 (weaker than expected analytically) and synchrotron cooling leads to δt γ ∼ ǫ −0.5 (as expected), both being compatible with GRB observations. The geometrical curvature of the emitting surface leads to a spread in emission angles over the spherical surface of the GRB ejecta, increases all observer-frame timescales by ∼ 50%. Additionally, it delays the arrival-time of a photon emitted (toward the observer) from the fluid moving at a larger angle (relative to its radial direction of motion) due to a longer path to observer and reduces its energy (due to a lower relativistic boost). Therefore, the integration of emission over the angle of the fluid motion softens continuously the received emission by delaying the arrival of photons of lesser energy. Numerical calculations (P19) of GRB pulses show that the angular integration associated with the geometrical curvature of the emitting surface has the following effects on the pulse properties: i) contributes to pulses peaking earlier at higher energies (which is the continuous emission softening described above), ii) mitigates the wrong trend of pulses to be more timesymmetric at higher energy when synchrotron-cooling is dominant because, in that case, the synchrotron-cooling timescale t sy , being shorter than the adiabatic-cooling timescale t ad = 3 t ang (see §3.2.2), is also (likely) smaller than the angular time-spread t ang , thus the pulse rise and fall timescales t r and t f are set by the angular inte-gration, which does not induce an energy dependence of the ratio t r /t f , iii) is unable to compensate for pulses being more timesymmetric at higher energy when adiabatic cooling is dominant because the angular time-spread t ang is smaller than the adiabatic-cooling timescale t ad , thus the pulserise and fall timescales are not changed much by the angular integration, iv) leads to pulses lasting longer at lower energies (owing to the progressive softening of the received emission) and induces a pulse-duration energy-dependence δt γ (ǫ) ∼ ǫ −0.4 that is similar to that produced by each cooling process for a constant magnetic field. The integration of the received emission over the equal photon-arrival time is effective only if the emitting region extends an angle larger than Γ −1 , the inverse of the Lorentz factor at which that region moves toward the observer, and its effect is diminished if the emitting region is a bright-spot of angular extent well below Γ −1 . Therefore, the above evaluation of the pulse properties resulting when electron cooling is synchrotron-dominated applies only to GRB pulses that arise from bright-spots. However, given that the angular integration has little effect on the pulse properties when the electron cooling is adiabatic, the previous evaluation of those pulse properties is correct for both a bright-spot and an uniformlybright surface. Consequently, if the trend of numerically-calculated pulses to be more symmetric at higher energies is firmly established then its incompatibility with observations (for either electron cooling process) favors the hypothesis that GRB pulses arise from a uniformly-bright surface and that the electron cooling is synchrotron-dominated, i.e. disfavors a bright-spot origin for GRB pulses and an adiabatic-dominated electron cooling. However, the pulse timescales and properties depend on the evolution of the electron injection-rate R i and of the magnetic field B (the effect of monotonically-varying such quantities is illustrated by the pulse shapes and durations shown in figures 5 and 6 of P19), thus, a comprehensive numerical study of the pulse properties expected for various electron cooling processes might (not guaranteed) identify evolving injection-rates R i (t) and magnetic fields B(t) that accommodate all the basic GRB pulse features. This work shows the effect of a power-law evolving injection rate R i (t) ∼ t y on the GRB pulse-integrated spectrum, with emphasis on the diverse low-energy slopes that can be obtained from a decreasing R i in the case of adiabatic electron cooling. A decreasing magnetic field B(t) is important for reconciling with observations the pulse-duration dependence on energy resulting when the electron cooling is dominated by scatterings at the Thomson-Klein-Nishina transition of the synchrotron photons below the peak-energy E γ of the GRB spectrum. GRB Low-Energy Spectrum The GRB low-energy slope β LE (of the energy spectrum below its peak-energy E γ ) is measured by fitting the GRB count spectrum with various emipirical func-tions: i) a pure power-law (PL), ii) a power-law with an exponential cut-off (CPL), which is the Band function with a large high-energy spectral slope, iii) the Band function, which is a broken power-law with a fixed width for the transition between the asymptotic power-laws, iv) a smoothly broken power-law (SBPL), which has a free parameter for the width of the transition between the low-and high-energy power-laws. Power-Law GRB Low-Energy Spectrum Preece at al (2000) have analyzed 5500 pulse-integrated spectra at 25 keV -2 MeV of the 156 brightest (in peak flux or fluence) CGRO/BATSE GRBs, with 80% of bursts being fit with the Band and the SBPL functions, and have found a distribution for the low-energy slope of the pulse-peak spectra that is approximately a Gaussian For a larger sample of 8093 time-resolved spectra from 350 bright BATSE bursts fit with the CPL, Band, and SBPL, Kaneko et al (2006) found a distribution of the low-energy slope P (β LE ) (for their GOOD sample) similar to that of Preece et al (2000), with a weighted mean 1 β o = 0.00 and a variance σ = 0.14. A minority of 366 time-resolved spectra were fit with a PL and are significantly softer, with β (sof t) o = −0.74 ± 0.19. F ǫ (ǫ < E γ ) ∼ ǫ βLE P (β LE ) ∼ exp − (β LE − β o ) 2 2σ 2(1) The "parameter error" criterion used by Poolakkil et al (2021) for selecting the fitting function for Fermi/GBM peak-flux spectra at 10 keV-1 MeV leads to a bimodal distribution for the low-energy spectral slope β LE (of their GOOD sample): i) PL fits were used for the peakflux spectra of 2287 bursts, leading to a median slope β (sof t) o = −0.50 ± 0.18, ii) CPL, Band, and SBPL functions were used to fit the 1.0 s peak-flux spectra of 1897 bursts, leading to a median spectral slope β (hard) o = 0.31 ± 0.17. The analyses of Kaneko et al (2006) and Poolakkil et al (2021) are similar, as they used the same fitting functions and retained only those fits that led to lower parameter errors (the GOOD sample) and which had a higher statistical significance (the BEST sample), yet the two distributions of low-energy indices β LE are incompatible with each other, with the BATSE bursts being softer than the non-PL GBM bursts. The same is true for the sample of softer bursts that were fit with a PL. Poolakkil et al (2021) attribute the bimodality of the P (β LE ) distribution to the PL model being sufficient for the spectral fitting of the lower fluence GBM peak-flux spectra, probably because the break to a softer spectrum above the peak-energy E γ is lost for low S/N measurements at higher energies, which leads to a softer best-fit spectrum over the entire GBM window. Given that the bimodality of the P (β LE ) distribution for GBM bursts is "compromised" by the "insensitivity" of PL fitting to the true hardness of low-energy spectra for dimmer bursts, we will make further use of the P (β LE ) distribution for BATSE bursts, and we will forget (and forgive) the excitement caused by that the peaks of the GBM bimodal distribution at slopes 0.31 and -0.50 are very close to or exactly at the values expected for synchrotron emission from uncooled and cooled electrons, respectively. Broken Power-Law GRB Low-Energy Spectrum Before proceeding, we should note that strong evidence for electron cooling in the GRB low-energy spectra has been found by fitting the fluence-brightest GRBs spectra below the peak-energy E γ with a broken power-law instead of a single power-law : i) For 14 bright Swift GRBs with simultaneous observations by XRT (0.3-10 keV) and BAT (10-150 keV), Oganesyan et al (2017) have found that 2/3 of 86 instantaneous spectra are better fit with a double SBPL (three power-law segments) having a lower-energy break E b ∈ (2, 8) keV (and a peak-energy E γ ∈ (30, 500) keV, thus E b ≃ 0.03 E γ ) and spectral indices β (low) o = 0.33 ± 0.35 and β (high) o = −1.46 ± 0.20 below and above E b , respectively, with most of the remaining spectra fit adequately by a CPL with an average low-energy slope β o = −0.08 ± 0.23, ii) For ten Fermi (10 keV-3 MeV) long GRBs with the largest fluence, Ravasio et al (2019) have found that 70% of 75 instantaneous spectra are better fit by a double SBPL with a break-energy E b ∈ (20, 400) keV (and peak-energy E γ ∈ (300, 3000) keV, thus E b ≃ 0.1 E γ ) and spectral indices β Both of these works present evidence for a cooling-break of the synchrotron soectrum at energy E b < E γ corresponding to the ynchrotron emission from the lowestenergy cooled electrons, with the spectral indices below and above E b being very close to the expectations for the emission from synchrotron-cooling electrons: β (low) o = 1/3 and β (high) o = −1/2. The spectra simulated by Toffano et al (2021) have shown that i) such cooling breaks require SBPL fits if the burst is sufficiently fluence-bright (Φ = 3 × 10 −4 erg/cm 2 ), ii) for average or dim bursts (Φ ≤ 3 × 10 −5 erg/cm 2 ), the Band function provides a good fit because of the low S/N ratio, and iii) a Band fit yields intermediate low-energy spectral slopes, transiting from β LE = −1/2 (for the emission from the synchrotroncooling tail) to β LE = 1/3 (for the emission from uncooled electrons) when the cooling energy E b is increased from 0.01 E γ to 0.1 E γ , which would explain the diversity of slopes measured by BATSE and Fermi (Equation 1). However, this interpretation requires that the cooling energy of all bursts satisfies E b ∈ (0.01, 0.1) E γ because, otherwise, a low break-energy E b < 0.01 E γ would yield a peak of the P (β LE ) distribution at β LE = −1/2, while a high break-energy E b ∈ (0.1, 1) E γ would lead to a peak at β LE = 1/3, none of which is seen. What is done here In this work, we use the compatibility of the calculated pulse-integrated low-energy spectrum slope and observations (Equation 1) to set upper limits on the life-time t B of the magnetic field, when the electron-cooling stops (if it is radiative) and when the production of synchrotron emission ends, keeping in mind that values of the cooling energy E b within two decades below the peak-energy E γ could account for the diversity of GRB low-energy slopes. This incomplete complete electron cooling was first proposed by Oganesyan et al (2017) and Ravasio et al (2019) as the origin for the observed P (β LE ) distribution. The following work builds on that of P19, who have presented an analytical derivation of (and numerical results for) the low-energy slope of the instantaneous synchrotron spectrum for adiabatic, synchrotron, and inverse-Compton dominated electron cooling. Here, we present (for all three electron cooling processes) analytical derivations of the pulse light-curve at energies below the GRB's and of the low-energy slope of the pulseintegrated synchrotron spectrum. An important shortcoming of the basic synchrotron model for the GRB emission is that it cannot account for low-energy slopes harder than the β LE = 1/3 displayed by about i) 1/3 of CGRO/BATSE 25 keV-2 MeV time-resolved spectra (Preece et al 2000), ii) 1/10 of the 30 time-integrated 2-20 keV spectra of Xray Flashes and GRBs observed by BSAX/WFC (Kippen et al 2004) and BATSE, and iii) 1/4 of Fermi/GBM peak-flux 10 keV-1 MeV spectra of the BEST sample (Poolakkil et al 2021). Thus, if an yet-unidentified large systematic error σ(β LE ) ≃ 0.3 does not explain away the low-energy spectral slopes harder than β LE = 1/3, then the following formalism for studying the effects of electron cooling on the GRB synchrotron emission is relevant for a majority a GRBs but a deviation from that model (or another emission process) is needed for a substantial fraction of bursts. The shortest departures from that model harden the low-energy slope to β LE = 1 by relying on a very small electron pitch-angle α < γ −1 (with γ being the electron Lorentz factor), i.e. a pitch-angle less than the opening of the cone into which the cyclotron emission is relativistically beamed, as proposed by Lloyd & Petrosian (2000), or on a very small length-scale for the magnetic field, λ B < ρ L /γ (with ρ L being the electron gyration radius), so that electrons are deflected by angles less than γ −1 and produce a "jitter" radiation, as proposed by Medvedev (2000). A hard slope β LE = 1 is obtained if the GRB emission is the upscattering of self-absorbed lower-energy synchrotron photons (Panaitescu & Mészáros 2000), but the ǫF ǫ spectrum of the upscattered emission may be too broad compared to real GRB spectra. In addition to these models that employ synchrotron emission and explain measured low-energy slopes harder than β LE = 1/3, a photospheric black-body component (proposed by e.g. Mészáros & Rees 2000, used to account for most of spectrum of GRB 090902B by Ryde et al 2010, but being in general a sub-dominant component, e.g. Axelsson et al 2012 for GRB 110721A) can yield low-energy spectra as hard as β LE = 2, while a combination of synchrotron and thermal emission can lead to intermediate low-energy slopes β LE ∈ (1/3, 2) if the photospheric plus synchrotron GRB spectrum is fit with just the Band function. The issue of some measured lowenergy slopes being too hard for the synchrotron model may be also alleviated by the addition of a power-law component to the Band (strongest component) plus thermal (weakest component) decomposition (e.g. Guiriec et al 2015), although that has been proven for only a small number of bursts. Deficiency of Our Treatment A limitation of the following treatment of GRB pulses as synchrotron emission from a population of cooling relativistic electrons is that the effect of electron cooling on the pulse spectral evolution is calculated assuming that the typical energy γ i of the injected electrons is constant during the GRB pulse. Another default assumption (occasionally relaxed) is that the magnetic field B is also constant. These assumptions are needed for an easier calculation of electron cooling electrons but they imply that the peak-energy E γ of the ǫF ǫ instantaneous spectrum is constant and so will be the peak-energy of the integrated spectrum, if the low-energy slope is harder than β LE = −1. However, measurements of the pulse spectral evolution (e.g. Crider et al 1997, Ghirlanda, Celotti & Ghisellini 2003 show that the peak-energy E γ decreases monotonically throughout the pulse. Consequently, the following description of the spectral evolution due to electron cooling for a constant typical electron energy γ i and a constant magnetic field B is representative for real GRBs displaying a decreasing peakenergy E γ only if that decrease of the best-fit E γ value is the artifact of fitting the curvature below E γ of real instantaneous spectra with an empirical function of free or fixed smoothness for the transition between the two (low and high energy) power-laws. Magnetic Field Life-Time and Duration of Electron Injection The GRB low-energy slope and the GRB pulse duration (as well as the GRB-to-counterpart relative bright-ness and counterpart pulse duration) depend on the magnetic field life-time t B (real or apparent) and the duration over which relativistic electrons are injected into the region with magnetic field. For first-order Fermi acceleration at relativistic shocks, the duration t I of particle injection in the down-stream region is the sum of the shock life-time t sh (the time it takes the shock to cross the ejecta shell) and the duration it takes for a given particle to be accelerated, i.e. the time for it to diffuse (for a magnetic field perpendicular to the shock front) or to gyrate (for a magnetic field parallel to the shock surface) many times in the upstream and down-stream regions and undergo multiple shock-crossings. For magnetic fields generated by turbulence or twostream instability (Medvedev & Loeb 1999) at relativistic shocks, which decay in the down-stream region, the magnetic field intrinsic life-time t B would be the shock life-time t sh . However, if the particle injection is impulsive (shorter-lived) relative to the shock life and lasts t I < t B , then the apparent magnetic field life-time t B that a particle spends in the magnetic field region would be the time that it takes a particle to cross the downstream region where there is a magnetic field. The above suggest that the durations t B and t I may be correlated if particles are accelerated and if magnetic fields are produced at relativistic shocks. For generality (i.e. to include other mechanisms that produce magnetic fields and relativistic particles, such as magnetic reconnection -Zhang & Huirong 2011, Granot 2016), we consider that the two parameters t I and t B are independent. If electrons are re-accelerated (Kumar & McMahon 2008), the magnetic field life-time t B used here can be seen a surrogate for the re-acceleration timescale, as particles are allowed to cool only for that re-acceleration timescale, thus our assumption that synchrotron emission stops at t B does not affect the following results about the GRB low-energy spectral slope (or the brightness of the prompt counterpart). However, if the GRB pulse duration is set by the magnetic field life-time t B , electron reacceleration on a timescale t re−acc > t B could lead to GRB pulses longer than t B , thus a finite magnetic field life-time is not completely equivalent to the electron reacceleration. Table 1 lists the most often notations used here. THE ELECTRON-COOLING LAW For any cooling process, conservation of particles during their flow in energy can be written as ∂N ∂t + ∂ ∂γ N dγ dt = N i(2) with N (γ) = dN/dγ the particle distribution with energy and N i (γ i < γ) ∼ γ −p(3) the distribution of the injected electrons, set to zero below a typical/lowest electron energy γ i , and for SYnchrotron cooling, Q ∼ B 2 , with B the magnetic field, because the SY power is proportional to the energy density of the virtual photons that are upscattered to SY photons; for self-Compton cooling, Q ∼ B 2 τ , with τ the optical-thickness to electron scattering, because the inverse-Compton cooling power is proportional to the energy density of SY photons. − dγ dt = Q(t)γ n (4) Above γ i , Equation (2) has a broken power-law solution, the cooled-injected electron distribution having a break at the electron energy γ cr where the radiative cooling timescale equals the time elapsed since the beginning of electron injection (AD-cooling does not yield a "cooling-break" because the AD-cooling timescale is slightly larger than the system age). Going to higher energies, the exponent of the cooled-injected distribution decreases by unity at γ cr . The cooled-injected electron distribution is of importance for calculating the GRB spectral evolution and pulse shape (e.g. P19), but could also be relevant for the SY emission at lower energies, after electron injection stops and the injected electrons migrate toward lower energies, yielding a pulse decay, provided that n ≤ 1, because that injected distribution shrinks to quasi mono-chromatic for n > 1. For the SY spectrum and pulse shape (light-curve) at lower energies (X-ray and optical), we are interested in the cooled electron distribution below γ i (or cooling-tail) N (γ < γ i ) ∼ γ −m(5) Substitution of that power-law cooling-tail in the conservation Equation (2) leads to m = n: the exponent of the cooling-tail distribution with energy is equal to the exponent at which the electron energy appears in the cooling power for any n = 1, provided that a certain condition (dependent on the radiative cooling process) is satisfied (see P19). Adiabatic cooling, for which n = 1, does not yield m = n. A solution-continuity argument (based on the assumption that if the above result m = n is valid for n > 1 and n < 1, then it should also be valid for n = 1) seems reasonable but is wrong. SYNCHROTRON (SY) COOLING Synchrotron electron cooling is governed by − dγ dt = P sy (γ) m e c 2 = 1 6π σ e m e c γ 2 B 2(6) where σ e is the cross-section for electron scattering and B is the magnetic field strength. The photon SY characteristic energy at which an electron radiates is ǫ sy = 3 he 16 m e c Bγ 2(7) For a constant magnetic field B, integration of Equation (6) leads to the lowest electron energy γ m (t) = γ i 1+ t t sy,i −1 , t sy,i ≡ t sy (γ i ) = γ i m e c 2 P sy (γ i )(8) for an initial electron energy γ i , with t sy,i being the SYcooling timescale for the γ i electrons. Then, the SY photon energy ε m at which γ m electrons radiate and the transit-time t γǫ for a γ i electron radiating at the GRB peak-energy E γ ≃ 100 keV to cool to an energy for which its SY characteristic energy is ǫ sy = ǫ are ε m (t) = E γ 1 + t t sy,i −2 −→ t (sy) γǫ ≃ ǫ E γ −1/2 t sy,i(9) For later use, the SY-cooling law of equation (6) can be written − dγ dt sy = 1 t sy,i γ 2 γ i(10) and the SY-cooling timescale for an electron of energy γ radiating at SY energy ǫ is t sy (ǫ) = γ − dγ dt sy = γ i γ t sy,i = ǫ E γ −1/2 t sy,i = t (sy) γǫ (11) Thus, the SY-cooling timescale for an electron of energy γ is the transit-time from GRB to the SY characteristic energy ǫ(γ) at which that electron radiates. Cooled-Electrons Distribution (Cooling-Tail) At t < t sy,i , most electrons are at energies above γ i and have a distribution with energy that show the injected one (t < t sy,i ) N (γ i < γ) ∼ R i t γ i γ γ i −p(12) for a constant injection rate R i . At t sy,i < t < t I , if the magnetic field B is also constant, the cooled electron distribution of Equation (13) develops, and its normalization at γ i is constant because the number of electrons above γ i is that injected in the last cooling timescale t sy,i , N (γ > γ i ) = R i t sy,i , which is constant (t sy,i < t < t I , R i ∼ B 2 ) : N (γ m < γ < γ i ) ∼ R i t sy,i γ i γ γ i −2(13) with the lowest electron energy γ m m e c 2 given in Equation (8). The above condition for a power-law cooling-tail is satisfied if the magnetic field energy-density (∼ B 2 ) is a constant fraction of the internal energy of relativistic electrons (∼ n ′ e γ i ) because the comoving-frame density of those electrons should satisfy n ′ e ∼ R i . The growth of the above γ −2 cooling tail is confirmed by numerically tracking the SY cooling of electrons (Figure 1). At t > t I > t sy,i , the electron density at the peak of the cooled-electrons distribution is N (γ m ) = N (γ i ) dγ i dγ m = R i t sy,i γ i γ m γ i −2 ≃ R i (t + t sy,i ) 2 γ i t sy,i(14) with γ(t) given in Equation (8). Therefore, after electron injection stops, the peak of the cooled-electrons distribution slides on the same cooling curve γ −2 ( Figure 1). The width of the cooled-electrons distribution is ∆γ(t > t I ) ≃ R i t I N (γ m ) = t I t sy,i γ 2 m γ i → ∆γ γ m = t I t + t sy,i(15) Nearly the same result can be obtained easier by using the cooling law of Equation (8) to track the evolution of the cooling-tail bounds γ m − γ i at t > t I : ∆γ γ m (t > t I ) ≃ γ m (t − t I ) γ m (t) − 1 = t I t + t sy,i − t I(16) Thus, after the end of electron injection, the cooledelectrons distribution shrinks, becoming asymptotically mono-energetic at energy γ m . As shown in Figure 2 and in figure 2 of P19, if the power-law cooling-tail condition R i ∼ B 2 is not satisfied, then the cooled-electrons distribution becomes harder if R i increases or if B decreases faster than the powerlaw condition above. The former case leads to a GRB low-energy slope for the instantaneous spectrum that is harder than β LE = −1/2 but the latter does not because the decreasing peak-energy E γ brings at 10 keV the high-energy softer SY spectrum. Conversely, if R i decreases or if B increases, the distribution of cooled electrons becomes softer, yielding a spectral slope softer than β LE = −1/2 for the instantaneous spectrum. However, the hardening of the low-energy instantaneous spectrum for an increasing R i is a transient feature and disappears after several cooling timescales t sy,i because it depends on the differential/relative timederivative of the injection rate (dR i /dt)/R i ∼ y/t (for a power-law R i ∼ t y ), but lasts longer for faster evolving R i 's, as shown by how fast the spectral slopes β LE given in the legend of Figure 2 approach the asymptotic value β LE = −1/2. For that spectral hardening to become persistent, the logarithmic derivative of R i would have to be constant, which means an exponentially-increasing electron injection rate R i . Nevertheless, for an increasing rate R i , the hardening of the instantaneous spectrum lasts for a few/several cooling timescales t sy,i , thus such an R i yields a GRB low-energy slope for the integrated spectrum harder than β LE = −1/2 if the SY emission is integrated over a duration not much longer than 10 t sy,i . Instantaneous Spectrum and Pulse Light-Curve Pulse Rise The SY spectral peak flux f p ∼ BN e ∼ BR i min(t, t I ) at the photon energy ε p where the most numerous γ m electrons radiate is f p (t) ∼ F p (t sy,i ) t/t sy,i t < t I t I /t sy,i t I < t(17) with F p (t sy,i ) the flux at the peak-energy E γ of the GRB spectrum. For a constant on injection rate R i and constant magnetic field B, the flux F p increases linearly with time until t sy,i , then remains constant until the end of electron injection at t I (as indicated by the electron distributions of Figure 1). From Equation (9), the evolution of the spectral peak-energy ε p ∼ Bγ 2 m is approximately ε p (t) ≃ E γ t < t sy,i E γ (t sy,i /t) 2 t sy,i < t(18) The SY spectrum at a photon energy ǫ < E γ is (10). (This brute-force approach yields more accurate results than a sophisticated integration of the numerically unstable conservation Equation 2, and need not be computationally expensive). Magnetic field is B = 100 G, electrons are injected above energy γ i = 3×10 4 , with a p = 3 power-law distribution with energy. For a source Lorentz factor Γ = 100 and redshift z = 1, the peak-energy of the ǫF ǫ spectrum is E γ ≃ 100 keV, the observer-frame SY cooling-time is t (obs) sy,i = (z + 1)t sy,i /(2Γ) = 26 ms. for photons emitted by the fluid moving toward the observer (and a factor two larger for those emitted by a region moving at an angle Γ −1 relative to the direction toward the observer). The cooled electrons below γ i have a γ −2 distribution with energy, as shown analytically (Equation 13). The lowest electron energy γ m m e c 2 ∼ t −1 and their corresponding distribution N (γ m ) ∼ t 2 satisfy Equations (8) and (14). After electron injection stops (t > t I ), the width of the electron distribution (black lines) narrows as in Equation (15) and its peak slides on the same γ −2 line as during electron injection (t < t I ). The number of electrons producing SY emission at a given photon energy ǫ is constant after the epoch when the γ m electrons "migrate" to that ǫ and before the end of energy injection at t I ; thus the SY flux at energy ǫ will also be constant. Red lines show a variable electron injection rate R i that yields two GRB pulses. During their cooling energies, the gap between the two injections decreases and suggests that the GRB variability (with timescale t I /2 = 15 t sy,i ) will be lost in the optical counterpart produced when the GRB electrons reach optically-emitting energies (Panaitescu & Vestrand 2022). f ǫ ≃ f p    (ǫ/ε p ) 1/3 ǫ < ε p (t < t γǫ ) (ǫ/ε p ) −1/2 ε p < ǫ (t γǫ < t < t I ) 0 ε p < ǫ (t I + t γǫ < t)(19) with t γǫ the epoch when the spectral peak-energy ε p reaches the observing photon energy ǫ (Equation 9), and the last branch due to the exponential cut-off of the SY function. From the above three equations, it follows that, for an electron injection lasting shorter than the SY-cooling timescale (t I < t sy,i ), the pulse light-curve at ǫ < E γ and the instantaneous spectrum are -Synchrotron-dominated cooling of electrons injected at an increasing rate R i ∼ t y . Legend indicates the low-energy slope β LE (at 10 keV) and the peak-energy E γ (for B = 100 G, Γ = 100, z = 1) of the power-perdecade ǫF ǫ at three epochs (t = 3, 10, 30 t sy,i ) for the SY spectrum integrated up to those times. The faster that increase, the harder the cooled-electrons distribution below the typical electron energy γ i and the harder is the lowenergy SY spectrum. The low-energy spectrum softens progressively, with a hard slope β LE < ∼ 1/3 at t < 2 t sy,i , an average β LE ≃ 0 at t ≃ 5 t sy,i , and a soft slope β LE < ∼ −1/2 at t > ∼ 30 t sy,i , showing how the hardening of β LE vanishes at t ≫ t sy,i . Thus, the hardening of the low-energy spectrum for an increasing electron injection rate R i is a transient feature and spectra with harder slopes β LE > 0 require the cessation of SY emission before the asymptotic β LE = −1/2 is reached. The t I = 3t sy,i case shows that the softening of low-energy slope β LE is significantly faster if electron injection stops while the hardening produced by an increasing rate R i is still effective. f ǫ (t) F p (t I ) = ǫ E γ 1/3                t sy,i , as there is no significant cooling during that time, and if the magnetic field is constant. This indicates that a low-energy (25-100 keV) GRB pulse should display a very slow rise from the end of electron injection at t I and until the electron SY-cooling timescale t sy,i . Most GRB pulses are peaky (resembling a double, rising-andfalling exponential or Gaussian - Norris et al 1996), thus the lack of the above slow rise indicates that t I > ∼ t sy,i , unless the magnetic field evolution shapes the pulse rise. For an electron injection lasting longer than the SYcooling timescale (t I > t sy,i ) but shorter than the transittime (t I < t γǫ ) or, equivalently, for a sufficiently low observing energy ǫ <ǫ ≡ E γ (t sy,i /t I ) 2 , the pulse light-curve is Figure 1 by the overlapping cooling-tails. f ǫ (t) F p (t sy,i ) = ǫ E γ 1/3                indicated in That GRB pulses do not display the plateau expected for t I > t γǫ indicates that the electron injection timescale t I is not much larger than the transit-time t γǫ from the spectral peak-energy (of the pulse-integrated spectrum) E γ ≃ 100 − 200 keV to an observing energy ǫ = 25 − 100 keV : t I < ∼ t γǫ < ∼ (1 − 2) t sy,i . This conclusion rests on assuming a constant magnetic field and a constant electron injection rate. Putting together these two constraints on t I , it follows that the shape of GRB pulses requires that the electron injection timescale t I is comparable to the typical electron SY-cooling timescale t sy,i , a conclusion which is hard to explain. One might speculate that a correlation between t sy,i and t I could be induced if the injection timescale is proportional to the particle acceleration timescale, which for particles accelerated at shocks is proportional to the particle gyration timescale; then t I ∼ γ i /B. Adding that t sy,i ∼ 1/(γ i B 2 ) points to the magnetic field as the reason for a t I ∼ t sy,i correlation; however, the equality t sy,i ≃ t I would still be unexplained. Alternatively, the underlying assumption of a constant magnetic field (or varying on a timescale t B > max{t sy,i , t I }) is incorrect. If the magnetic field life-time t B < min{t I , t sy,i }, then the pulse shape is determined by the evolution of B, without any relation between the other timescales being implied by GRB observations. Pulse Fall After the transit-time t γǫ (for t I < t γǫ ) or after epoch t I + t γǫ (for t γǫ < t I ), all electrons radiate below the observing energy ǫ, the flux received from the region of angular extent Γ −1 moving toward the observer (the region of maximal relativistic boost Γ) is exponentially decreasing and the flux received becomes dominated by the emission from angles larger than Γ −1 . This "largerangle emission" (LAE) is progressively less enhanced relativistically and its decay can easily be calculated if the observer-frame pulse peak-time t p is shorter than the angular spread in the photon arrival-time t ang . In the case of a sufficiently short-lived emission, there is a one-toone correspondence between the angle of emission and the photon arrival-time, so that the LAE decay is (Ku- mar & Panaitescu 2000) f (LAE) ǫ (t >t p ) = f pk t t p −2+β(>ǫ) (f all)(23) where β(> ǫ) is the spectral slope at the higher (and higher) photon energy that gets less (and less) Doppler boosted to the observing energy ǫ, f pk = f ǫ (t γǫ ) =            F p (t I ) t I < t sy,i (< t γǫ ) F p (t sy,i ) t I t sy,i t sy,i < t I < t γǫ F p (t sy,i ) ǫ E γ −1/2 (t sy,i <)t γǫ < t I(24) is the pulse peak-flux of Equations (20) and (21), and t p = t p + t ang , t ang = R 2cΓ , t p ≃ t γǫ(25) are the comoving-frame pulse peak epoch, after being stretched linearly 3 by the spread in the photon arrival-time over the region of angular opening Γ −1 , the comoving-frame time-interval t ang corresponding to the observer-frame spread in the photon arrival-time t (obs) ang = Rθ 2 /2 = R/2cΓ 2 , and t p is the pulse peak-time, as shown by the pulse light-curves given in Equations (20) and (21). If t p ≫ t o (i.e. for any epoch well after the beginning of electron injection and of the SY emission), the integration over the spherical surface up to an angle θ = Γ −1 (beyond which the relativistic boost D = 2Γ/(1 + Γ 2 θ 2 ) decreases substantially) doubles the photon arrival-time t (obs) 0 corresponding to θ = 0 (i.e. from the fluid moving toward the observer). Thus, well after the initial adiabatic timescale, the angular integration increases t (obs) 0 by 50% on average and it can be shown that the integration over the spherical surface of the photon arrival-time weighed by the received flux yields a relative increase by 1/3. For GRB pulses, the peak epoch is t p = t sy,i and the peak flux is f (E γ , t sy,i ) = F p (t sy,i ) = F p (t I ), thus Equation (24) relates the low-energy pulse peak-flux f pk to the flux at the GRB pulse peak f (E γ , t p ), which is also the flux at the GRB peak-energy E γ . The conclusion that the pulse peak-time t p is comparable to the SY-cooling timescale t sy,i is based on the lack of slowly-rising and flat-top low-energy GRB pulses expected for a constant magnetic field and a constant electron injection rate. If the evolution of these quantities shapes the pulse lightcurve, then the pulse-peak epoch is t p = max{t I , t sy,i }, as shown by Equations (20) and (22). After noting that the comoving-frame angular timescale t ang is comparable to the AD-cooling timescale t ad = (3/2)t co = (3/2)R/(cΓ), with t co the comovingframe ejecta age, the condition that the electron cooling is SY-dominated (t sy,i < t ad ) is equivalent to the angular timescale setting the pulse duration (t sy,i < t ang ), as long as no other factors (duration of electron injection, magnetic field life-time) determine the pulse duration. Thus the pulse rise t r and fall t f timescales should always be comparable to t ang and GRB pulses should not be too time-asymmetric. Very asymmetric pulses, such as those with a measured ratio t r /t f < 0.2, require that the emitting surface extends much less than Γ −1 , i.e. the pulse emission arises from a bright-spot, and, as shown by numerically calculated pulses, a short electron injection timescale t I < ∼ t sy,i /10 or a magnetic field evolving on a timescale t B < t sy,i are responsible for the asymmetric pulse shape. For GRB pulses, the slope β in Equation (23) is that measured above the peak-energy E γ but, for lowerenergy (optical and X-ray) pulses, for which the pulse peaks at the transit-time t γǫ when a quasi-energetic cooled electron distribution "crosses" the observing energy, the above approximation of an infinitesimally short emission implies that, after t p = t γǫ , the pulse turnsoff exponentially because there would not be any cooled electrons to radiate above the observing energy ǫ and whose emission would be (less and less) relativistically boosted to energy ǫ. Relaxing the approximation of an infinitesimally short emission, the LAE received after the peak t p = t γǫ (if t I < t γǫ ) or after the plateau-end at t f lat = t γǫ + t I (if t γǫ < t I ) will be the integral over the ellipsoidal surface of equal arrival-time, with emission from the fluid moving at larger angles relative to the outflow originobserver direction radiating at earlier epochs, when the quasi-monoenergetic cooling-tail was radiating at a peakenergy ε m > ∼ ǫ (for t I < t γǫ ), hence β(ǫ) = 1/3, or when the high-energy end of the cooling tail was radiating at ε p > ∼ ǫ (for t γǫ < t I ), hence β(ǫ) = −1/2. Then, if the entire surface of the ejecta outflow is radiating at a uniform brightness, the LAE is that given in Equation (23) but with peak-time t p stretched by the angular time-spread t ang : (f all) f (LAE) ǫ (t > t p ) = F p (t sy,i )            ǫ E γ −5/6 t t sy,i −5/3    1 t I < t sy,i (< t γǫ ) t I t sy,i t sy,i < t I < t γǫ ǫ E γ −1/2 t t I −5/2 t γǫ < t I(26) Pulse Light-Curve Equations (20), (21), and (22) provide both the instantaneous spectrum and the pulse rise or light-curve at an energy below gamma-rays (a soft X-ray or optical pulse), for a constant electron injection rate R i and magnetic field B, and in the case of a bright-spot emission. The rise is followed by an exponential decay owing to the electron distribution having cooled to a quasimonoenergetic one and to the lack of the LAE. For a surface of uniform brightness, the same equations give the pulse rise light-curve if timescales are stretched by the angular time-spread t ang , and Equation (26) gives the pulse power-law decay from the LAE. The SY pulse light-curves for SY-dominated electron cooling are also given in Equations (A5)-(A7) for iCdominated electron cooling with an iC-power of exponent n > 1 (Appendix A1), if one sets n = 2 and replaces the iC-cooling timescale t ic with the SY-cooling timescale t sy,i . The above pulse light-curve equations show that the optical/X-ray pulse emission (instantaneous spectrum) displays a gradual softening, with the spectral slope 1/3 during the pulse rise evolving to -1/2,-5/6 after the pulse peak. The low-energy slope of GRB pulses softens from an initial β LE = 1/3 to β LE = −1/2 after (1 − 2)t sy,i , which may explain qualitatively the decrease of the count hardness-ratio measured for GRBs pulses (e.g. Bhat et al 1994, Band et al 1997. Pulse-Duration Dependence on Energy If the pulse duration is set by radiative cooling (Equation 11), then δt ǫ = t sy (ǫ) = ǫ E γ −1/2 t sy,i = t γǫ = t p(27) with the second to last equality following from (Equation 9) and the last from Equation (25). The equality of the pulse duration with the pulse-peak epoch stands naturally for any pulse whose rise or fall are not too fast or too slow, which is the case of the pulse light-curves given in Equations (20) and (21), and is an argument which applies to other cooling processes, not just SY. Thus, for SY-dominated electron cooling, the pulse duration should decrease with energy, with the expected dependence 4 δt ǫ ∼ ǫ −1/2 being close to that observed for GRB pulses δt γ ∼ ǫ −0.4 . However, as discussed above, when the electron cooling is SY-dominated (t sy,i < t ad ), the pulse duration may be set by the spread t ang = t ad /3 in the photon arrival-time caused by the spherical curvature of the emitting surface because t ang > t sy,i . Thus, an immediate consistency between the pulse duration dependence on energy δt ǫ given in Equation (27) and GRB observations is readily achieved only if the angular timescale is not dominant, e.g. if the emitting region is a small bright-spot of angular extent much less than the "visible" Γ −1 area moving toward the observer or if the pulse duration is determined by another timescale (duration of electron injection t I or magnetic field life-time t B ) longer than the angular time-spread t ang . Conversely, for a uniformly-bright spherically-curved emitting surface and for a radiative electron cooling, the pulse duration dependence on energy δt ǫ may be not set by the cooling timescale of that radiative process but by the continuous softening of the received emission induced by the differential relativistic boost (photons arriving later have less energy) of the emission from the region of angular opening Γ −1 moving toward the observer (corresponding to the pulse rise) and of the larger-angle emission from the fluid outside that Γ −1 region (corresponding to the pulse fall). Pulse-Integrated Synchrotron Spectrum By integrating the above instantaneous spectra over the entire pulse, i.e. past the peak epochs, one obtains the pulse-integrated spectrum. Due to its fast decay, the contribution of the larger-angle emission is a small fraction of the integral up to the pulse peak-epoch. For t γǫ < t I , the flat pulse-plateau flux is dominant and trivially sets the slope of the integrated spectrum F ǫ ∼ ǫ −1/2 . A more interesting situation occurs for t I < t γǫ , where F ǫ (t > t γǫ > t I ) = t 0 f ǫ (t ′ )dt ′ ≃ tγǫ tI f ǫ (t ′ )dt ′ ≃ F p (t sy,i ) t I t sy,i ǫ E γ 1/3 tγǫ tI t ′ t sy,i 2/3 dt ′ ≃ t γǫ f ǫ (t γǫ ) ≃ t I F p (t sy,i ) ǫ E γ −1/2 (28) with f ǫ (t γǫ ) = F p (t sy,i )(t I /t sy,i ) = f p being equal to the constant flux f p (t > t I ) at the peak ε p of the SY spectrum. (Integrating the instantaneous spectrum only until the pulse peak is a good approximation only for the emission from a bright-spot. If the emitting surface is of uniform brightness then, from Equation (26), one can show that the post-peak LAE fluence has the same spectrum ǫ −1/2 ). Thus, although the pulse instantaneous spectrum is hard during the pulse-rise, f ǫ (t < t γǫ ) ∼ ǫ 1/3 , a much softer integrated spectrum F ǫ (t > t γǫ ) ≃ [f ǫ (t γǫ ) = const] t γǫ ∼ t γǫ ∼ ǫ −1/2 is obtained because the transit- time t γǫ ∼ ǫ −1/2 over which the flux is integrated increases with a decreasing energy ǫ. Adding that the pulse duration δt ǫ should be comparable to the transit-time t γǫ , the above result suggests that the softness of the integrated spectrum can be seen as arising from the pulse duration dependence on the observing energy. Therefore, the pulse-integrated spectrum is F ǫ ∼ ǫ −1/2 irrespective of the ordering of electron injection time t I and electron transit-time t γǫ . This result was derived assuming a constant electron injection rate R i but it is valid even for a variable R i , as shown in Figure 2 for a powerlaw electron injection rate. A spectral slope β LE = −1/2 is about half-way on the soft side of the distribution of GRB low-energy slopes. The SY cooling-tail shown in Figure 1 shows the trivial fact that, for a long-lived electron injection, a GRB lowenergy spectral slope harder than β LE = −1/2 requires that electron cooling or, equivalently, the SY emission stops before the GRB-to-10-keV transit-time (Equation 11) t (sy) γ−10k = 3 E 1/2 γ,5 t sy,i(29) if the electron injection rate R i is constant, while Figure 2 suggests that the SY emission integrated up to ≃ 2t γ−10k should have a slope β LE > 0 for a rising R i (t). The same temporal upper limit on the electron cooling and SY emission is required by β LE > 0 when electron injection lasts shorter than the GRB-to-10-keV transit-time as, otherwise, the soft integrated spectrum of Equation (28) holds. That fact is also illustrated by the spectral slopes given in the legend of Figure 2 for the t I = 3t sy,i case, which shows a soft spectrum if it is integrated longer than t I . Therefore, a harder GRB low-energy slope β LE requires that the magnetic field fades on a shorter timescale and the low-energy slopes of the integrated spectra given in the legend of Figure 2 suggest that (SY) β LE =          1/3 t B < ∼ t sy,i ∈ (0, 1/3) t sy,i < ∼ t B < ∼ t (sy) γ−10k ∈ (−1/2, 0)t (sy) γ−10k < ∼ t B < ∼ 3 t (sy) γ−10k −1/2 3 t (sy) γ−10k < ∼ t B(30) This anti-correlation between the magnetic field lifetime t B and the hardness of the GRB low-energy slope applies to any cooling process because it arises from the softening (decrease of peak-energy) of the cooling-tail SY emission. The above conclusion that harder GRB low-energy spectral slopes β LE are the result of electrons not cooling below the lowest-energy channel (10-25 keV), offers a way to identify GRB pulses arising from bright-spots extending over much less than the visible region of the ejecta. In absence of a substantial electron cooling and of a significant spread in photon-arrival time (due to the small angular extent of a bright-spot), the GRB pulse duration would be more time-symmetric at higher energies and their duration should be less dependent on energy. INVERSE-COMPTON (IC) COOLING For a constant iC-cooling timescale of the GRB γ ielectrons, inverse-Compton (iC) cooling is governed by − dγ dt ic = P ic (γ) m e c 2 = 1 t ic,i γ n γ n−1 i , t ic,i ≡ t ic (γ i ) (31) with t ic,i the iC cooling timescale of the GRB γ i electrons. If the γ i electrons scatter their own photons in the Klein-Nishina regime (γ i E ′ γ > m e c 2 ), i.e. they cool mostly by scattering SY photons ǫ < E γ at the Thomson-Klein-Nishina (T-KN) transition, then their cooling begins with an index n = 2/3 and leads to a cooling tail N (γ < γ i ) ∼ γ −2/3 . When the lowest energy electrons γ m in the cooling tail begin scattering their own SY photons at the T-KN transition, their cooling exponent changes to n = 1 and a power-law segment of index m = 1 begins to grow (N (γ m < γ) ∼ γ −1 ), gradually replacing the pre-existing, higher-energy cooling-tail of index m = 2/3. When the γ i electrons begin to scatter the SY photons produced by the cooling γ m electrons, the entire cooling-tail has index m = 1 and is again a single power-law, albeit only until t ic,i (table 2 of P19). This m = 1 cooling-tail arising from iC-cooling dominated scatterings at the T-KN transition has been identified also by Nakar, Ando, Sari (2009) and Daigne, Bosnjak, Dubus (2011). If the γ i electrons scatter their SY photons in the Thomson regime (γ i E ′ γ < m e c 2 ), their iC-cooling has an index with n = min{(p + 1)/2, 2}, which changes progressively to n = min{(3p − 1)/4, 2} and n = min{p, 2} (table 1 of P19). The iC-cooled electron distribution (i.e. the solution to Equation 2 for N i = 0) is a power-law with the same exponent −n as that of the iC power in Equation (31), N (γ < γ i ) = a(t)γ −n , only if a(t) ≃ γ n−1 i R i t ic,i = const, i.e. if a(t) is time-independent. For SY-cooling, this condition becomes R i ∼ B 2 (Equation 13), which may have a good reason to be satisfied. For iC-dominated cooling, the same condition may be expressed as a relation between B, R i and γ i and has no obvious rationale. If the above condition for a power-law cooling-tail is not satisfied, then the cooling-tail should be curved, with the local slope n depending on the evolutions of the injection rate R i and magnetic field B, which could explain why the measured GRB low-energy spectral slopes β LE have a smooth distribution encompassing the values for β LE = −(n − 1)/2 listed above. Instantaneous and Integrated Spectra The SY instantaneous spectrum (= pulse light-curve) and integrated spectrum for iC-dominated electron cooling are derived in Appendix A, where a constant electron injection rate R i and magnetic field B were assumed. Then, the condition for the growth of a powerlaw cooled-electrons distribution, t ic,i ∼ R −1 i , is equivalent to a constant cooling timescale t ic,i for the typical GRB electron of energy γ i . Taken together, these three assumptions can easily be incompatible because the cooling timescale t ic,i depends on the injection rate R i and magnetic field B (this is not an issue for SY-dominated cooling because, in that case, t ic,i depends only on B). Given that the iC-cooling timescale is t ic,i ∼ t sy,i /Y ∼ (B 2 τ ) −1 with Y = P ic /P sy ∼ τ the Compton parameter and τ (t) ∼ t 0 R i (t ′ )dt ′ the electron optical-thickness to photon scattering, a constant t ic,i requires a decaying magnetic field B ∼ τ −1/2 that diverges at t = 0, when the electron injection begins and the optical-thickness is τ = 0. It is easy to recalculate the light-curves and spectra that account for an evolving magnetic field B(t), which requires to multiply all break energies and spectral peakflux densities by a factor B. However, the evolution of the magnetic field that ensures the power-law coolingtail condition R i t ic,i = const depends on the iC-cooling regime for the γ i electrons (the exponent n of the electron cooling law in Equation 31), thus a generalized treatment is not possible. Furthermore, specializing results to a particular B(t) limits the usefulness (if any !) of the results. Alternatively, one could assume a constant magnetic field, calculate the time-dependence of the cooling timescale t ic,i ∼ P −1 ic from the evolution of the iC-cooling power P ic ∼ τ , i.e. from the evolution of the scattering optical-thickness τ , and integrate the electron cooling law (Equation 31). However, the power-law cooling-tail condition t ic,i ∼ R −1 i will not be satisfied (unless a variable B is allowed, as discussed above for a constant t ic ) and the SY spectrum above the lowest break-energy will not be a power-law. Further use of that essential feature will lead to inaccurate results. In conclusion, there is no generalized/comprehensive and accurate way to calculate analytically iC-cooling SY spectra and light-curves. We return to all three constancy assumptions (for R i , B, t ic ), and recognize that the analytical results of Appendix A are only illustrative and of limited applicability. If the power-law cooling-tail condition is satisfied, then the cooling-tail and its SY emission spectrum are: N (γ < γ i ) ∼ γ −n , f ǫ (ǫ < E γ ) = F p ǫ E γ −(n−1)/2(32) the latter result holding for n > 1/3 (if n < 1/3, the SY emission from the cooled-electrons distribution is dominated by the highest energy γ i electrons and is f ǫ ∼ ǫ 1/3 , but such a hard cooling-tail is not expected to arise). Therefore, the SY instantaneous spectrum from the cooling-tail has a low-energy slope β LE = −(n − 1)/2. The smallest two values for the exponent n of the iCcooling law, are obtained if the γ i -electrons cool weakly through scatterings of sub-GRB peak-energy photons at the T-KN transition. For the smallest exponent n = 2/3, the resulting slope β LE = 1/6 is the hardest instantaneous SY spectrum arising from the cooling-tail and the only slope harder than the peak of the measured distribution P (β LE ). The next exponent n = 1 allows β LE = 0, which is at the peak of P (β LE ). All other exponents n > 1 occur when the γ i -electrons cool strongly by scattering photons in the Thomson regime, and yield slopes β LE < 0, on the softer half of the measured distribution P (β LE ). Equations (A3) and (A13) give the transit-time t (ic) γǫ ≃ t ic,i          ǫ E γ −(n−1)/2 n > 1 1 − ǫ E γ (1−n)/2 n < 1(33) For n > 1 (electron cooling dominated by iC scatterings in the Thomson regime), integration of the instantaneous spectrum over the pulse duration leads to an integrated spectrum of similar low-energy slope β LE = −(n − 1)/2, irrespective of the duration t I of electron injection relative to the gamma-to-X-ray transittime t γ−10k , therefore GRB low-energy slopes β LE > 0 require that electrons do not cool below 10 keV, i.e. a magnetic field life-time t B shorter than the GRB-to-10-keV transit-time t (ic) γ−10k ≃ 3 E 1/2 γ,5 t ic,i for n = 2. The dependence of the integrated spectrum slope β LE on the magnetic field lifetime is the same as for SY cooling (Equation 30) but with t ic,i instead of t sy,i . For n < 1 (electron cooling dominated by iC scatterings at the T-KN transition, with only n = 2/3 possible), Equations (A18) and (A19) show that the pulse instantaneous spectrum softens progressively but the spectral slope of the integrated spectrum is always that of the pulse rise, 1/3 if t B < t γǫ or 1/6 if t γǫ < t B . Equation (A21) shows that, even when the soft pulse-decay of spectral slope −(p − 1)/2 is at maximal brightness (t B > t p , thus the pulse emission is from the Γ −1 region moving toward the observer), the integrated spectrum still has the harder pre-peak slope 1/6. For a magnetic field life-time t B < t p , when the pulse decay is the faster decaying LAE (because emission from the fluid moving at angles larger than Γ −1 relative to the observer is less beamed relativistically), it is quite likely that the soft pulse-decay contribution to the pulse fluence is dominated by the pulse rise. Thus, the expected GRB low-energy spectral slope is (iC/T − KN : n < 1) : β LE =      1/3 t B <t (ic) γ−10k 1/6 t (ic) γ−10k <t B <t 2/3→1 0 t 2/3→1 <t I , t B t (ic) γ−10k ≃ 1 3 t ic,i(34) with t 2/3→1 given in Equation (A22) and with the middle branch second condition (t B < t 2/3→1 ) being effective only if t 2/3→1 < t I . That the cooling-tail for iC-dominated electron cooling cannot be a perfect power-law, and must have some curvature (see figure 3 of P19), implies that the actual lowenergy GRB spectral slope for iC-cooling through scatterings at the T-KN transition spans the range (0, 1/3). Pulse-Duration and Transit-Time Integration of the iC-cooling law of Equation (31) allows the calculation of the transit-time to a certain observing energy and of the pulse duration produced by the passage through the observing band of the SY characteristic energy of the electrons that produce the pulse peak. For an iC-cooling of exponent n > 1 (Appendix A1), the pulse peak is set by the passage of the minimal energy of the SY spectrum from the cooling-tail, while for n < 1 (Appendix A2), that epoch is set by the passage of the GRB γ i electrons after the end of electron injection at t I , provided that the electron-scattering (optical) thickness is approximately constant before t I (i.e. for a sufficiently fast decreasing electron injection rate) and that the magnetic field is also constant. For a constant cooling timescale t ic,i , i.e. in the case of a constant magnetic field B and a constant electron scattering thickness τ , the pulse duration resulting from the electron iC-cooling is δt ǫ = γ(ǫ) − dγ dt = γ i γ(ǫ) n−1 t ic,i = ǫ E γ (1−n)/2 t ic,i(35) after using Equation (31). For iC-cooling dominated by Thomson scatterings of ǫ ≃ E γ SY photons (n > 1), when the rate of electron cooling decreases faster with decreasing electron energy, pulses should last longer at lower energy: δt ǫ ∼ ǫ −(n−1)/2 , which is consistent with GRB observations if n = 2. Thus, the pulse duration δt ǫ is the same as the transit-time t γǫ (first branch of Equation 33). If the electron injection lasts shorter than the transit-time t I < t γǫ , Equation (A8) shows that the pulse peak-time t p is equal to the transit-time t γǫ : (n > 1) t p = t γǫ = δt ǫ ∼ ǫ (1−n)/2(36) For t I > t γǫ , the pulse peak is at either t γǫ or t γǫ + t I depending on the evolution of the injection rate and of the magnetic field. If iC-cooling is dominated by T-KN scatterings (n < 1) of ǫ ≪ E γ SY photons, then the rate of electron cooling decreases slower with decreasing electron energy, and pulses should last shorter at lower energy: δt ǫ ∼ ǫ (1−n)/2 = ǫ 1/6 (for the one and only n = 2/3), which is in contradiction with GRB observations 6 : δt ǫ ∼ ǫ −0.4 . The pulse peak-time (Equation A20) is set by the transit of the higher energy break ε p after the end of electron injection and the pulse duration δt ǫ is not equal to the transit-time t γǫ (second branch of Equation 33): (n < 1) t p = t γǫ + t I , δt ǫ = t ic,i − t γǫ(37) Further investigations to identify the conditions under which the iC-dominated electron cooling may explain the observed trend of GRB pulses to last longer at lower energy are presented in Appendix A3. The first conclusion is that an increasing scattering optical-thickness τ (t) affords some flexibility to the resulting energy-dependence of the pulse duration δt ǫ for iC-cooling with n > 1 but for n < 1 pulses should last longer at higher energy, in contradiction with observations. The second conclusion is that, for an iC-dominated cooling with n < 1, a decreasing magnetic field B should lead to a pulse duration dependence on energy that is compatible with observations. Somewhat surprising, the pulse duration dependence on energy is independent on how fast B(t) decreases, although that result may be an artifact of some approximations. The evolution of τ does not play any role, however how B(t) and τ (t) evolve sets the GRB low-energy slope β LE . The above conclusions are relevant for the SY emission from bright-spots of angular opening less than that of the "visible" region of angular extent Γ −1 , when all pulse properties could be determined by the electron iCcooling: i) For electron iC-cooling dominated by scatterings in the Thomson regime (n ≥ 2), same considerations apply for the pulse time-symmetry and pulse duration dependence on energy as for SY-dominated electron cooling (n = 2): the faster pulse-rise t 2/3 , t, t 5/3 implies a rise timescale t r that is set by the iC-cooling timescale t ic (Equations A5-A7), while the pulse-fall timescale t f is set by the pulse peak-time t p , which is the transit-time t γǫ ∼ ǫ (1−n)/2 , thus electron iC-cooling should lead to pulses with a rise-to-fall ratio t r /t f that increases with energy, i.e. to pulses which are more time-symmetric at higher energy if pulses rise faster than they fall (t r /t f < 1), which is in contradiction with observations, but the pulse duration dependence on energy (Equation 35) is consistent with measurements. ii) For iC-cooling dominated by scatterings at the T-KN transition (n = 2/3), the pulse-rise (1 − t/t ic ) −1 , t, t/(1 − t/t ic ) is faster than the pulse-fall (1 − t/t ic ) 3(p−1) (Equations A18 and A19), thus a rise-to-fall ratio t r /t f < 1 independent of energy is expected, which is in accord with observations, but pulses should last longer at higher energy (Equation 35), which is inconsistent with observations. Within the bright-spot emission scenario, the above incompatibilities may be solved by an evolving magnetic field; alternatively, those incompatibilities disappear if the GRB emission arises from a spherical surface of uniform brightness (in the lab-frame), in which case all pulse properties are determined by the spread in photon arrival-time and by the emission softening due to the spherical curvature of the emitting surface. ADIABATIC (AD) COOLING For a constant radial thickness of the already shocked GRB ejecta, the AD-cooling of relativistic electrons is γ m (t) = γ i 1 + t t o −2/3 −→ ε m (t) ≃ E γ t t o −4/3 (38) with ε m the SY characteristic energy ǫ sy (γ m ) (assuming a constant magnetic field), thus the AD-cooling law is − dγ dt ad = P ad (γ) m e c 2 = 2 3 γ t + t o(39) and the AD-cooling timescale is t ad = γ − dγ dt ad = 3 2 (t + t o )(40) for any electron energy. Equation (38) implies that, at the initial time t o (when electron injection begins), the electron transit-time from GRB emission to an observing energy ǫ is t (ad) γǫ = t o ǫ E γ −3/4(41) for a constant magnetic field. Unlike for SY and (most cases of) iC cooling, for AD cooling, where n = 1 (P ad ∼ γ), the conservation Equation (2) does not determine the γ-exponent of the power-law cooling-tail. Instead, that exponent can be determined from the continuity of the cooling-tail N (γ < γ i ) ∼ a(t)γ −m and the cooled injected distribution N (γ > γ i ) ∼ A(t)γ −p at the typical energy γ i of the injected electrons, where p is the exponent of the injected electron distribution: N i (γ > γ i ) ∼ R i γ −p . Substitution of the above two power-law electron distributions in the conservation Equation (2) and the use of the AD-cooling law of Equation (39) lead to da dt = − 2 3 (1 − m) a t + t o −→ a(t) ∼ (t + t o ) 2(1−m)/3 (42) dA dt + 2 3 (p − 1) A t + t o ∼ R i (t) (43) R i ∼ (t + t o ) −y −→ A(t) ∼ (t + t o ) 1−y(44) where a power-law injection rate R i was assumed, to allow for an easy solving of the differential equation for A(t). The two functions a(t) and A(t) are continuous at γ i only if they have the same time-dependence, which implies that m = 1 2 (3y − 1)(45) thus, the slope of the cooling-tail depends on the evolution of R i . The slope of the cooling-tail instantaneous SY spectrum, β = d ln f ǫ /d ln ǫ = min[1/3, −(m − 1)/2], is β LE = 1/3 (y < 5/9, m < 1/3) 3(1 − y)/4 (y > 5/9, m > 1/3)(46) For y > 5/9, the cooling-tail SY spectrum becomes softer for a faster-decreasing injection rate R i ; for y < 5/9, the cooling-tail is harder than N (γ < γ i ) ∼ γ −1/3 and its SY emission is overshined by that from the highestenergy γ i electrons in the cooling-tail, leading to a hard β LE = 1/3 spectrum. That is the case for a constant R i : y = 0 → m = −1/2. Equations (B6) and (B8) of Appendix B show that the instantaneous spectrum of AD-cooling electrons is harder during the pulse rise than during the pulse fall, with the pulse peak occurring at the time t γǫ (if y > 1) when the photon energy ε m ≡ ǫ sy (γ m ) crosses the observing energy ǫ or at the timet γǫ (if y < 1 -Equation B7) when the higher break-energy ε p of the last injected γ i -electrons crosses ǫ. For GRB spectra at the lowest observing energy (10 keV), these crossing epochs are t (ad) γ−10k ≃ 5.6E 3/4 γ,5 t o ,t (ad) γ−10k = t I t o t (ad) γ−10k > t (ad) γ−10k (47) Equation (B10) shows that the SY spectrum integrated over the entire pulse has a soft slope β LE = −3/4 (if the injected electron distribution has an index p > 5/2), being softer than that of the instantaneous spectrum (Equation 46) for a reason similar to that discussed above for the integrated spectrum from SY-cooling electrons. Consequently, for the integrated spectrum of ADcooling electrons to display a hard low-energy slope, the instantaneous spectrum must not be integrated past the crossing epochs t γ−10k andt γ−10k , i.e. the SY emission must stop and the magnetic field must disappear before the pulse-peak epochs t p given in Equation (B9): (R i ∼ t −y ) : β LE = 1 3 → t B < t (ad) γ−10k (y > 5/9) t (ad) γ−10k (y < 5/9)(48) The epoch t B when the magnetic field fades out is before the natural pulse-peak, thus t B becomes the pulse peak-epoch, after which the LAE emission describes the pulse decay, and the pulse duration δt ǫ has a weaker dependence on ǫ than given below. If the magnetic field lives longer than the crossing time t (ad) γ−10k , then a softer spectrum results after the crossing of the lower-end energy ε m of the cooling-tail SY spectrum (5/9 < y < 2) (49) and an even softer integrated spectrum is produced by the passage of the higher-end energy ε p of the cooling-tail SY spectrum β LE = 3 4 (1−y) ∈ − 3 4 , 1 3 → t (ad) γ−10k < t B <t (ad) γ−10kβ LE = −3/4 −→ t B > t (ad) γ−10k (y > 2) t (ad) γ−10k (p > 5/2)(50) with −p the exponent of the injected electron distribution with energy. For an injected distribution with p < 5/2, the integrated spectrum is dominated by the emission from the injected electrons, as they cool after the end of electron injection, with AD-cooling preserving the slope of their distribution with energy: N (γ p < γ) ∼ γ −p , thus β LE = −(p − 1)/2 ∈ (−1/2, −3/4) for p ∈ (2, 2.5) is harder than for the last case above. If the magnetic field lasts longer than the pulse-peak epoch t p given in Equation (B9), then the pulse duration corresponding to the cooling-law given in Equation (39) is δt ǫ = γ(ǫ) − dγ dt (t = t p ) ≃ 3 2 t p ≃ ǫ E γ −3/4 t I y < 1 t o y > 1(51) Thus, AD-dominated electron-cooling should yield pulses whose duration decreases with the observing energy ǫ, as is observed, but the resulting dependence δt ǫ ∼ ǫ −3/4 is stronger than measured. However, figure 5 of P19 shows that the numerically-calculated pulses display a duration dependence on energy that is weaker than in Equation (51) and consistent to that measured. That the comoving-frame angular time-spread t ang = R/(2cΓ) over the visible Γ −1 region of maximal relativistic boost (by a factor Γ) is always 3 times smaller than the current comoving-frame adiabatic timescale t ad = 1.5 t = 1.5 R/(cΓ), implies that, for AD-dominated electron cooling, all pulse properties are determined by the electron cooling and the above pulse duration dependence on energy is accurate for either a bright-spot emission or a uniform brightness surface. SYNCHROTRON AND ADIABATIC COOLING Equations (10) and (39) show that the SY and AD cooling powers are equal at the critical electron energy γ cr = 2t sy,i 3(t + t o ) γ i → γ<γ cr ,P ad >P sy (AD− cool) γ cr <γ,P ad <P sy (SY− cool) Below the critical electron energy γ cr , electrons cool adiabatically and the slope of the cooling-tail N (γ m < γ < γ cr ) is determined only by the history of the electron injection rate R i . Above γ cr , electrons cool radiatively and the slope of the cooling-tail N (γ cr < γ < γ i ) is set by the history of the electron injection rate R i and of the magnetic field B (which sets the radiative cooling power). At t = 0, the typical γ i electrons cool adiabatically if 3t o < 2t sy,i and radiatively if 2t sy,i < 3t o . Appendix C shows that the solution (Equation C5) to the AD+SY electron cooling implies that, if the γ i electrons are initially cooling adiabatically (γ i < γ cr (t = 0)), then their cooling remains adiabatic all times (γ m (t) < γ cr (t), with γ m (t = 0) = γ i ), while if the γ i electrons are initially cooling radiatively (γ cr (t = 0) < γ i ), then their cooling switches from radiative to adiabatic after a "critical" time t c (Equation C13) defined by γ cr (t c ) = γ m (t c ). Thus, in either case, the electrons cool adiabatically eventually, yet the exact electron cooling law (Equations C9-C11) is close to (1/3 of) that expected for SYdominated cooling: γ m (t) ∼ t −1 (Equation 8). It may be surprising that, if SY and AD electroncooling are considered separately, they lead to the opposite conclusion. The timescales for these two cooling processes, given in Equations (11) and (40), indicate that both cooling timescales increase linearly with time, but faster for AD-cooling (t ad ≃ 1.5t) than for SY (t sy (γ m ) ≃ t). Consequently, if the γ i electrons begin by cooling radiatively (t sy,i < t ad (t = 0) = 1.5 t o ), then t sy (γ m ) < t ad at any time, thus the electrons cool radiatively at all times. Conversely, if the γ i electrons cool adiabatically initially (1.5 t o < t sy,i ), then their cooling switches to SY-dominated at a (erroneous) critical time Thus, if the two electron cooling processes are treated as acting independently, the electron cooling becomes radiative at late times irrespective of which cooling process was dominant initially, in total contradiction with the expectations from the solution to the double-process cooling, which shows that electron cooling should always become adiabatic eventually. The reason for this discrepancy is the unwarranted (ab)use of the SY-cooling solution (Equation A1) in the calculation of the SYcooling timescale (Equation 11), which is correct only at early times and only if the electron-cooling begins in the SY-dominated regime, but is incorrect at later times, when the SY and AD cooling timescales t sy [γ(t)] and t ad ≃ t are comparable and when the exact electroncooling law (Equation C5) is inaccurately described by the SY-cooling of Equation (8). Despite this fundamental differences in the expectations for the single-and double-process cooling, the asymptotic SY solution at late times over-estimates the exact electron energy only by a factor up to 3. Thus, if one makes the mistake of using the SY-cooling solution whenever that process appears dominant, the resulting error is an over-estimation by up to an order of magnitude of the corresponding spectral break energies and by up to a factor 3 of the corresponding transit-times. The upper limits on the magnetic field life-time t B given in Equation (48) are valid if the cooling of the lowest-energy γ m electrons (for y > 5/9), or of the GRB γ i electrons after the end of electron injection (for y < 5/9), is described by the AD-cooling solution of Equation (38) until the corresponding transit-times given in Equation (47). If the electron cooling is ADdominated initially (t o < t sy,i ), then it remains so at any later time. However, the AD-cooling law of Equation (38) remains valid only until the switch-timet defined in Equation (C12), after which the electron cooling is de-scribed by the 1/3-SY solution, even though the electron cooling is AD-dominated. Thus, the results for the GRB low-energy slope β LE of §5 are applicable if the crossingtimes t (ad) γ−10k andt (ad) γ−10k are shorter than the switch-timẽ t, which lead to the same restriction: t o , t I < 0.2 t sy,i . Therefore, AD-cooling sets alone the GRB pulse lightcurve and integrated spectrum if the radiative (SY) cooling timescale is at least an order of magnitude larger than the initial ejecta age t o and the duration t I of electron injection. The evolution of the electron distribution undergoing AD and SY cooling, with the strength of AD cooling increasing from t o = 0.1t sy,i to t o = 0.01t sy,i , is shown in Figure 3 and supports the above conclusion. The low-energy slope β of the GRB instantaneous spectrum depends on the location of the SY characteristic energy ǫ sy (γ cr ) relative to the lowest-energy channel (10 keV) of GRB measurements or, equivalently, the location of the electron critical energy γ cr relative to the energy γ X ∼ γ i /3E 1/2 γ,5 of the electrons that radiate at 10 keV. From Equation (52), it follows that, if the γ ielectron cooling begins AD-dominated (t o ≪ t sy,i ), then their cooling remains AD-dominated (i.e. γ c > γ i ) until epoch t = (2/3)t sy,i and the cooling of γ X -electrons remains AD-dominated until epoch t ≃ 2 E 1/2 γ,5 t sy,i . For t > 2 t sy,i , SY-cooling sets the cooling-tail N (γ X < γ < γ i ) energy distribution below the GRB peak-energy E γ , leading to a softening of the SY emission to the expected asymptotic slope β LE = −1/2. The legend of Figure 3 shows that expected gradual softening of the instantaneous GRB low-energy slope. Thus, the condition for AD-cooling to set the lowenergy GRB spectral slope leads to an upper limit on the magnetic field life-time: t B < (2/3 − 2)t sy,i , to switch-off the soft SY emission at 10-100 keV produced by the soft cooling-tail above γ cr resulting through SY-dominated electron cooling. This condition on t B is satisfied on virtue of Equations 48 and 48) if electron cooling begins well in the AD-dominated regime t o ≪ t sy,i and if t I ≪ t sy,i . DISCUSSION GRB Low-Energy Slope β LE for SY Cooling Equation (30) and numerical calculations (Figure 2) show that a low-energy (10 keV) GRB spectral slope β LE < 1/3 of the pulse-integrated spectrum results from an incomplete/partial electron cooling due to the magnetic field life-time t B being comparable to the GRB-to-10-keV transit-time t (sy) γ−10k (Equation 29) that it takes the typical GRB electron (radiating initially at the GRB spectrum peak-energy E γ ∼ 100 keV) to cool to an energy for which the corresponding SY characteristic photon energy is 10 keV. More exactly, a slope β LE < ∼ 1/3 results for t B < 2 t sy,i , β LE ≃ 0 requires that t B ≃ (3−5) t sy,i , and β LE > ∼ −1/2 is obtained for 10 t sy,i < ∼ t B . For a constant electron injection-rate R i and magnetic field B, SY cooling over a duration longer than 3 t sy,i leads to a soft slope β LE = −1/2, irrespective of the du-ration t I over which electrons are injected: i) For t sy,i < t I , t B , the electron distribution develops a cooling-tail with energy distribution N (γ < γ i ) ∼ γ −2 at t > t sy,i , for which the SY instantaneous spectrum is f ǫ ∼ ǫ −1/2 and the integrated spectrum is the same. That GRB pulses do not have a flat plateau at their peak, starting at the transit-time t (sy) γ−10k and until the end of electron injection at t I , indicates that either R i or B are not constant, ii) For t I < t sy,i < t B , a power-law cooled electron distribution does not develop; instead that distribution shrinks to a mono-energetic one after t sy,i . Integration of the SY instantaneous spectrum f ǫ (ǫ < ε p ) ∼ ǫ 1/3 until after the SY characteristic energy ε p at which the cooled GRB electrons radiate decreases below 10 keV leads to an integrated spectrum with the same low-energy slope β LE = −1/2 as for a cooling-tail. This coincidence arises from that a cooling-law dγ/dt ∼ γ −n yields i) a cooling-tail N ∼ γ −n whose SY spectral slope is β = −(n − 1)/2 and ii) an electron cooling γ ∼ t −1/(n−1) for n > 1, a transit-time t γǫ ∼ ǫ −(n−1)/2 and an integrated spectrum F ǫ ≃ f ǫ (t γǫ )t γǫ ∼ t γǫ ∼ ǫ −(n−1)/2 . SY electron cooling can yield cooling-tails harder (softer) than N (γ < γ i ) ∼ γ −2 and corresponding SY spectra harder (softer) than β LE = −1/2 if electrons are injected at an increasing (decreasing) rate R i . Because the hardness of the 10-100 keV SY spectrum is set by the electrons injected during the last few cooling timescales (t (sy) γ−10k ∼ 3 t sy,i ), a variable electron injection rate R i can change the resulting cooling-tail only if the injection rate variability timescale is shorter than the transittime t (sy) γ−10k . This means that an electron injection rate R i that varies as a power-law in time, and which has a variability timescale equal to the current time, can alter the cooling-tail index only over a duration comparable to transit-time t (sy) γ−10k . Conversely, an electron injection rate R i that is a power-law in time does not change significantly over the second t (sy) γ−10k and leads to the standard slope β LE = −1/2. Consequently, a variable electron injection rate R i can change the above magnetic field life-times t B by a factor up to two. Harder (softer) cooling-tails can also be obtained if the magnetic field B decreases (increases), but a change in the low-energy slope β LE of the pulse-integrated spectrum is less feasible because a decreasing B leads to a decreasing SY spectrum peak-energy E γ which compensates the effect that a decreasing magnetic field has on the hardness of the cooling-tail, while an increasing B could lead to an increasing peak-energy E γ that is in contraction with observations. Thus, there is a direct mapping between the distribution of the GRB low-energy slope P (β LE ) and that of the magnetic field life-time P (t B ). The peak of the slope distribution P (β LE ) at β LE = 0 implies that the life-time distribution P (t B ) peaks at t B ≃ 3 t sy,i , which means that the generation of magnetic fields in GRB ejecta is tied to the cooling of the relativistic electrons. Figure 1 but for a stronger AD cooling of increasing strength: initial timescale (initial ejecta age) t o = 0.1t sy,i (blue) and t o = 0.01t sy,i (green). For a shorter initial time t o , the AD-cooling tail is more extended. Electron injection lasts until after the spectrum integration epochs (t I = t). Stars indicate the critical electron energy γ cr for which AD and SY losses are equal. For t o < ∼ 0.1t sy,i , the lowest energy γ m electrons cool adiabatically at all times and their energy decreases as in Equation (38) at t <t (Equation C12); at t >t, their cooling turns asymptotically to the (1/3)-SY cooling solution discussed in Appendix C, which is just that: 1/3 of the energy given in Equation (8) for synchrotron cooling. Electrons with energy γ < γ cr cool adiabatically, leading to a cooled-electrons distribution N (γ < γ cr ) ∼ γ 1/2 (for a constant electron injection rate R i ) and to a hard f ǫ ∼ ǫ 1/3 SY spectrum, while those with γ > γ cr cool radiatively, leading to N (γ cr < γ < γ i ) ∼ γ −2 (for a constant R i and magnetic field B). At t < 2 t sy,i , the γ X electrons radiating at 10 keV (mid X-rays) are below the critical energy (γ cr > γ X ≃ γ i /f ew) and their adiabatic-dominated cooling leads to a persistent GRB low-energy spectral slope β LE > 0, with softer slopes β LE resulting for a decreasing R i (t) (Equation 49). At t > 2 t sy,i , when γ cr < γ X , the synchrotron-dominated cooling of the mid-X-ray radiating electrons leads to a progressive softening of the instantaneous spectrum at 10 keV, asymptotically reaching the expected slope β LE = −1/2. Some of that softening is captured in the integrated spectrum slopes β LE listed in the legend. Indicated photon energies are for z = 1. The puzzling feature of the β LE − t B correlation is that the distribution of slopes β LE does not exhibit peaks at the extreme values β LE = 1/3 (corresponding to t B < t sy,i ) and β LE = −1/2 (corresponding to t B > 10 t sy,i ), which may be explained in part by the statistical uncertainty σ(β LE ) ≃ 0.1 of measuring the GRB low-energy slope β LE . 7.2. GRB Low-Energy Slope β LE for AD Cooling For AD cooling, the cooling-tail distribution is determined by the only factor at play, the electron injection rate, assumed here to be a power-law in time R i ∼ t −y , which provides all the flexibility needed, but using only one parameter. In contrast to SY-dominated electron cooling, where the dependence on the injection rate R i of the cooling-tail distribution is atransient feature, lasting for a few SYcooling timescales t sy,i , the power-law cooling-tail resulting for AD-cooling is a persistent feature because a substantial change in the rate R i is guaranteed to occur dur-ing an AD-cooling timescale, given that both timescales are the same (the current time). Similar to SY-cooling, for AD-dominated electron cooling, the passage of the peak-energy of the SY spectrum from the cooling-tail leads to a softer integrated spectrum with β LE = −3/4. Consequently, AD-cooling allows easier than SYcooling a range of spectral slopes for the instantaneous spectrum. That diversity is imprinted on the integrated spectrum if the cooling-tail contribution is dominant (which requires y < 2) and if the magnetic field has a life-time t B between the transit-times t γǫ (Equation 41) andt γǫ (Equation B7) corresponding to the low and high-energy ends ε m and ε p of the cooling-tail spectrum crossing the observing energy. Equation (49) for the GRB low-energy slope β LE (y) shows that the measured distribution of the GRB slope β LE ∈ (−3/4, 1/3) (which most of the range of GRB slopes) maps directly to the distribution of the exponent y ∈ (5/9, 2) of the electron injection rate R i ∼ t −y -P (y) = (3/4)P (β LE ) -for a magnetic field life-time Equation 47). For a t B outside the above range, the GRB low-energy slope can be a hard β LE = 1/3 (Equation 48) or a soft β LE = −3/4 (Equation 50), with even softer slopes β LE < −3/4 occurring if the integrated spectrum is dominated by the SY emission from GRB electrons of energy above γ i . t B ∈ (t (ad) γ−10k ,t (ad) γ−10k ) ( As for SY-dominated electron cooling, this conclusion comes with two puzzles: it implies a correlation between the magnetic field life-time t B and the cooling of the lowest and highest-energy electrons in the cooling-tail via the GRB-to-10-keV transit-times (Equation 47), and peaks in the P (β LE ) distribution at β LE = 1/3 and β LE = −3/2. GRB Low-Energy Slope β LE for iC Cooling If the typical GRB electrons of energy γ i cool through scatterings (of SY photons produced same electrons) in the Thomson regime (γ i E ′ γ < m e c 2 ), when the coolingpower exponent is n ≥ 2, the integrated spectrum shows the same features and dependence on the magnetic field life-time t B as for SY-dominated electron cooling (for which n = 2): i) crossing of the lowest-energy of the cooling-tail SY spectrum softens the integrated spectrum to the slope β LE = −(n − 1)/2, whether or not the electron injection lasts longer than the GRB-to-10-keV transit-time t (ic) γ−10k ≃ 3 t ic,i , i.e. whether the cooling-tail develops down to an energy for which the SY characteristic energy is below 10 keV or shrinks to a monoenergetic distribution before reaching the observing energy, ii) hard GRB low-energy spectra require an incomplete electron cooling due to a short-lived magnetic field, lasting about the transit-time t (ic) γ−10k (Equation A3), and there should be a one-to-one correspondence between the GRB low-energy slope β LE and the magnetic field lifetime t B , modulo a possible variation of the electron injection rate R i , whose effect lasts only for about t (ic) γ−10k , with t B ≃ t (ic) γ−10k accounting for the peak of the mea-sured P (β LE ) distribution at β LE = 0. The iC-cooling of GRB electrons through scatterings at the T-KN transition (γ i E ′ γ > m e c 2 ), when n = 2/3, has i) a similarity with the AD-dominated electron cooling (n = 1) in that an energy-wide cooling-tail persists after the end of electron injection, ii) a similarity with the SY-dominated electron cooling (n = 2) in that the crossing of either end of the coolingtail (at t (ic) γ−10k or at the pulse-peak epoch t p = t (ic) γ−10k +t I ) yields an integrated spectrum with the same slope β LE = 1/6 as for the SY emission from the cooling-tail, and iii) a unique feature in that, after the time t 2/3→1 of Equation (A22), the cooling-tail of exponent n = 2/3 is replaced by one with n = 1, provided that the electron injection lasts t I > t 2/3→1 , which leads to an instantaneous SY spectrum of slope β = 0 that yields an integrated spectrum of slope β LE = 0 (which is the peak of the measured low-energy slope distribution -Equation 1), if the magnetic field life-time satisfies t B > t 2/3→1 . IC-dominated electron cooling with n = 2/3, 1 cannot lead to integrated spectra with a low-energy slope β < 0 because the contribution to the integrated spectrum from the GRB electrons above γ i is smaller than that from the cooling-tail after the end of electron injection. Thus, one important feature of electron-cooling dominated by iC-scatterings at the T-KN transition (with n ≤ 1) is that, without the diversity in slopes β LE allowed by a variable electron injection rate, it can explain only the harder half of the measured distribution of GRB lowenergy slopes, with β LE ≥ 0 (Equation 34): the hardest slope β LE = 1/3 requires that t B < t (ic) γ−10k and the slope β LE = 0 at the peak of the P (β LE ) distribution requires that (t B , t I ) > t 2/3→1 . However, that electron cooling dominated by iCscatterings in the Thomson regime yields a GRB lowenergy slope β LE = −1/2 while iC-cooling dominated by scatterings occurring at the T-KN transition yields a persistent slope β LE = 0 suggest that diversity among bursts in the scattering regime that dominates the iC-cooling may yield intermediate slopes β LE . To that end, Daigne, Bosnjak, Dubus (2011) have illustrated how the transition from a soft low-energy spectrum to a harder one is obtained by i) replacing SY-cooling (Y (γ i ) < 1) or iCcooling in the Thomson regime (γ i E ′ γ < mc 2 , Y (γ i ) > 1) with iC-cooling at the T-KN transition (γ i E ′ γ > mc 2 , Y (γ i ) > 1) and by ii) increasing the Compton parameter, leading to β LE (Y < 1) = −1/2 to β LE (Y ≫ 1) = 0. CONCLUSIONS The aim of this work is to examine the implications of the low-energy slopes β LE measured for GRBs by CGRO/BATSE and Fermi/GBM within a simple model where relativistic electrons (of typical energy γ i m e c 2 ) in a magnetic field (B) produce SY emission in a relativistic source (of Lorentz factor Γ) and at some radius (R). Low-energy slope of instantaneous SY spectrum. That slope depends on the dominant electron-cooling process (Synchrotron, ADiabatic, iC-scatterings) and on how much electrons cool during the magnetic field life-time t B . For electron cooling dominated by radiative processes (SY, iC), the timescale t B sets how long electrons cool and radiate. For AD electron-cooling, the timescale t B determines only how long electrons radiate; they cool after t B but that is irrelevant if no emission is produced. In addition to the dominant electron-cooling process, the energy distribution of the cooling GRB electrons (the cooling-tail) that sets the GRB low-energy spectral slope β LE also depends on the history of the electron injection rate R i and of the magnetic field B. Furthermore, R i (t) and B(t) also determine the GRB pulse duration and shape. The initial assumption was that both quantities are constant until a certain time, t B and t I , respectively. This simplification does not change much the ability of radiative processes with a cooling-power P (γ) ∼ γ n of exponent n ≥ 2 to account for the GRB low-energy slope β LE , but a variable injection rate R i (t) is essential for allowing the AD-dominated electron cooling to account for more than two values for the slope β LE (1/3 and -3/4) and for iC-dominated cooling through scatterings at the T-KN transition of the synchrotron photons of energy below the GRB peak-energy E γ (n ≤ 1) to accommodate GRB low-energy slopes softer than β LE = 0. Hardest low-energy slope. If GRB electrons do not cool well below their initial energy γ i or do not radiate SY emission while they cool below γ i (either being due to a magnetic field life-time t B shorter than the initial electron-cooling timescale t rad ), the resulting slope β LE = 1/3 of the instantaneous spectrum is the hardest that SY emission (not self-absorbed, sic!) can produce, which is a trivial fact. Intermediate low-energy slope. Longer-lived magnetic fields yield softer slopes β LE for the integrated spectrum, with an anti-correlation between life-time t B and slope β LE (longer life-times lead to softer slopes) existing for t B ∈ (1, 10)t rad = (1/3, 3)t γ−10k , where t γ−10k ≃ 3 t rad is the transit-time for electrons to migrate from emitting SY radiation at E γ ≃ 100 keV (the GRB peak-energy) to 10 keV. Softest low-energy slope. For longer magnetic field lifetimes t B > 10 t rad , the slope β LE of the instantaneous spectrum settles at an asymptotic value that depends on the dominant electron-cooling process: for radiative cooling with a cooling power exponent n, the resulting slope is β LE = −(n − 1)/2 (for SY-cooling with n = 2, the slope β LE = −1/2 is a textbook result), for ADcooling, β LE = 0.75(1 − y) with y the exponent of the power-law electron injection rate R i ∼ t −y , provided that 5/9 < y < 2 (β LE = 1/3 for y < 5/9 and β LE = −3/4 for y > 2). Pulse-integrated spectrum. If electron injection lasts t I > t γ−10k , then the pulse-integrated spectrum has the same slope as the instantaneous spectrum for all radiative processes (another trivial fact), with a possible change from a cooling-tail with n = 2/3 to one with n = 1 for iC-cooling dominated by scatterings at the T-KN transition. For AD-cooling, the crossing of the lowest or highest-energy electrons in the cooling-tail below the observing energy leads to a soft slope β LE = −3/4. If the electron injection lasts t I < t rad then, for radiative processes with n ≥ 2, the cooling-tail width shrinks after the end of electron injection at t I and the passage of the quasi-monochromatic cooling-tail below the observing energy leads to a GRB pulse-integrated spectrum with the same low-energy spectral slope β LE = −(n − 1)/2 as for a long lived electron injection. For iC-scatterings at the T-KN transition (n ≤ 1) and ADcooling, the cooling-tail width increases or remains constant, respectively, in log(energy), and the previous results for the integrated spectrum for a longer-lived electron injection remain unchanged. Summarizing the above, a magnetic field life-time t B ∈ (1, 10) t rad = (1/3, 3) t γ−10k maps the GRB lowenergy slope β LE ∈ [−1/2, 1/3] if SY-cooling is dominant, β LE ∈ [−(n − 1)/2, 1/3] if iC-cooling in Thomson regime is dominant, β LE ∈ [0, 1/3] if iC-cooling at T-KN transition is dominant, and β LE ∈ [−3/4, 1/3] if ADcooling is dominant, with the softest values for the first two cooling processes applying to short-lived (t I < t rad ) electron injections, and all softest values applying to long-lived (t I > 10 t rad ) injections. The measured distribution P (β LE ) for the low-energy slopes of the pulse-integrated spectra does not have peaks at the above extreme values: the hard β LE = 1/3 and the soft β LE = −1/2, −3/4. That discrepancy is alleviated in part by the typically reported statistical uncertainty σ(β LE ) ≃ 0.1 in measuring the low-energy slope. Still, it is unlikely that spreading a multi-modal distribution of low-energy slopes with a kernel of dispersion 0.1 could lead to a smooth distribution P (β LE ) peaking at β LE = 0, particularly on its soft side with β LE < 0, displaying the largest gap being between the preferred values β LE = 0 and β LE = −1/2. Thus, absent some more substantial systematic errors in measuring the low-energy slope, the observed quasi-Gaussian P (β LE ) distribution requires that the distribution of magnetic field life-times P (t B ) among GRB pulses is restricted to mostly t B ∈ (1, 5) t rad and peaks at t B ≃ 3 t rad (which yields the peak of P (β LE ) at β LE ≃ 0), without a substantial fractions of pulses with t B < t rad or t B > 5 t rad . At this point, such a correlation between the magnetic field life-time t B and the cooling timescale of GRB electrons t rad is unwarranted and puzzling. The conclusion that intermediate GRB slopes β LE ∼ 0 require an incomplete electron cooling (meaning that electrons cool for a time t B that ranges from less than one cooling timescale t rad of the typical GRB electron to at most ten t rad ), is also suggested by the work of Kumar & McMahon (2008), who analyzed the 5-dimensional model parameter space for the hard β LE = 1/3 and soft β LE = −1/2 GRB low-energy slopes, but considering that the electron cooling stops after a re-acceleration timescale, which has the same effect on electron cooling as the disappearance of the magnetic field used here. At first sight, none of the possible mechanism for particle acceleration and magnetic field generation (at shocks, by instabilities, through magnetic reconnection) offers a reason for a correlation between that partial electron cooling on a timescale t B and the electron cooling timescale t rad . Inverse-Compton cooling comes in two flavors: 1) strong/fast cooling with an exponent n > 1, similar to SY-dominated cooling, where 1a) the electron energy decreases like a power-law in time, 1b) the cooled-electrons distribution shrinks to quasi mono-energetic after electron injection ends (the higher the exponent n, the faster the cooling-tail shrinks), and 1c) the passage of the peak-energy of the SY spectrum of the cooling-tail through the observing band softens the integrated spectrum, 2) weak/slow cooling with n < 1, similar to AD-dominated cooling (n = 1) in that the cooling-tail persists after the end of electron injection and the width (in log space) of that cooling-tail is practically constant, but different from an AD cooling-tail in that 2a) the electron cooling is slower than a power-law in time, and 2b) the passage of the coolingtail through the observing band does not lead to an integrated spectrum significantly softer than the instantaneous spectrum. A1. Strong iC-Cooling (n > 1) through Thomson Scatterings of High-Energy Photons This case is a generalization of SY-dominated cooling (n = 2) and is relevant for inverse-Compton cooling when the γ i -electrons scatter their own SY photons in the Thomson regime (n ≥ 2). Integrating the iC cooling law of Equation (31), one obtains γ m (t) = γ i 1 + t t ic,i −1/(n−1) , t ic,i ≡ γ i m e c 2 P ic (γ i ) ∼ γ 1−n i (A1) if the iC-cooling timescale t ic,i of the γ i electrons is time-independent. After the end of electron injection (t > t I > t ic,i ), Equation (A1) shows that the bounds of the cooling tail, γ m (t) for the electrons injected initially and γ M = γ m (t − t I ) for the electrons injected at t I , and its width evolutions are γ M γ m ≡ γ m (t − t I ) γ m (t) = 1 − t I t ic,i + t −1/(n−1) ≃ 1 + 1 n − 1 t I t if t ≫ t I (> t ic,i ) −→ ∆γ γ m = γ M − γ m γ m ∼ t I t ≪ 1 (A2) meaning that the cooling-tail becomes quasi-monoenergetic. The SY peak-energy for the lowest-energy electrons γ m and the transit-time from GRB peak-energy to an observing energy ǫ are ε m (t > t ic,i ) ≃ E γ t ic,i t 2/(n−1) , t (ic) γǫ ≃ t ic,i ǫ E γ −(n−1)/2 (A3) while the SY peak flux at ε m is f p (t) =            F p (t I )(t/t I ) t < t I < t ic,i f p (t I ) = F p (t I ) t I < t (t I < t ic,i ) F p (t ic,i )(t/t ic,i ) t < t ic,i < t I F p (t ic,i )(ε m /E γ ) −(n−1)/2 = F p (t ic,i )(t/t ic,i ) t ic,i < t < t I f p (t I ) = F p (t ic,i )(t I /t ic,i ) t ic,i < t I < t (A4) with F p (t I ) and F p (t ic,i ) being the GRB peak flux (the energy density at the spectral peak or at the pulse peak). The first and third branches show the linear increase of the numer of electrons radiating at the GRB peak-energy E γ (for a constant electron injection rate), the second and fifth branches arise from the constant number of electrons radiating at the peak-energy ε m after the end of electron injection, and the fourth branch arises from the linear increase of the number of electrons radiating at ε m , with the flux being independent of the exponent n of the cooling power, all cases assuming a constant magnetic field. Adding the SY spectrum of Equation (19), but with the slope β = −(n − 1)/2 above the peak-energy ε m , and the larger-angle emission emerging at t > t γǫ , t γǫ + t I (owing to the exponential cut-off of the synchrotron function and to the quasi-monoenergetic cooling-tail), the resulting instantaneous spectrum and pulse light-curve at an observing energy ǫ < E γ (below the GRB spectrum peak-energy) are (t I < t ic,i ) f ǫ (t) ≃ F p (t I ) ×                ǫ E γ 1/3    t/t I t < t I (rise) [1 + (t/t ic,i )] 2/(3n−3) t I < t < t ic,i (very slow rise) (t/t ic,i ) 2/(3n−3) t ic,i < t < t γǫ (slow rise) 1 t = t γǫ (peak) ǫ E γ −5(n−1)/6 t t ic,i −5/3 t γǫ < t (LAE − f all) (A5) (t ic,i < t I < t γǫ ) f ǫ (t) ≃ F p (t ic,i ) ×                ǫ E γ 1/3    t/t ic,i t < t ic,i (rise) (t/t ic,i ) (n−1/3)/(n−1) t ic,i < t < t I (f ast rise) (t I /t ic,i )(t/t ic,i ) 2/(3n−3) t I < t < t γǫ (slow rise) t I /t ic,i t = t γǫ (peak) ǫ E γ −5(n−1)/6 t I t ic,i t t ic,i −5/3 t γǫ < t (LAE − f all) (A6) (t ic,i < t γǫ < t I ) f ǫ (t) ≃ F p (t ic,i ) ×              ǫ E γ 1/3 t/t ic,i t < t ic,i (rise) (t/t ic,i ) (n−1/3)/(n−1) t ic,i < t < t γǫ (f ast rise) ǫ E γ −(n−1)/2    1 t γǫ < t < t I + t γǫ (top plateau) t t I −(n+3)/2 t I + t γǫ < t (LAE − f all)(A7) Thus, the pulse-peak flux and epoch are t p = t γǫ t I < t γǫ t γǫ − −(t γǫ + t I ) t γǫ < t I , f pk = f ǫ (t p ) =    F p (t I ) t I < t ic,i (< t γǫ ) F p (t ic,i )(t I /t ic,i ) t ic,i < t I < t γǫ F p (t ic,i )(ǫ/E γ ) −(n−1)/2 t γǫ < t I (A8) The LAE flux above was calculated by assuming that its asymptotic flux decay f (LAE) ǫ ∼ t −2+β is continuous at the pulse peak of Equation (A8) f (LAE) ǫ (t > t p ) = f pk × (t/t p ) −2+1/3 t I < t γǫ (t/t p ) −2−(n−1)/2 t γǫ < t I(A9) From Equations (A5) and (A6), for a sufficiently short electron injection (t I < t γǫ ) or a sufficiently low observing energy ǫ, the SY spectrum integrated until after the pulse peak-time t γǫ is F ǫ (t > t γǫ > t I ) ≃ t p f pk ≃ t γǫ ǫ E γ 1/3 t γǫ t ic,i 2/(3n−3) = 1 × F p (t I ) t I < t ic,i (< t γǫ ) F p (t ic,i )(t I /t ic,i ) t ic,i < t I < t γǫ = ǫ E γ −(n−1)/2 × t ic,i F p (t I ) t I < t ic,i (< t γǫ ) t I F p (t ic,i ) t ic,i < t I < t γǫ(A10) after using Equation (A3), with the pre pulse-peak and post pulse-peak (LAE) fluxes having comparable contributions to the pulse fluence. Thus, for t I < t γǫ , the addition of instantaneous f ǫ ∼ ǫ 1/3 hard spectra until the transit-time t γǫ , when the typical energy ε m of the SY emission from the quasi-monoenergetic cooling-tail crosses the observing energy ǫ, leads to a much softer integrated spectrum F ǫ (t < t γǫ ) ∼ ǫ −(n−1)/2 . Furthermore, it can be shown that the addition of instantaneous f ǫ ∼ ǫ −5(n−1)/6 LAE soft spectra after the transit-time t γǫ leads to a harder integrated spectrum with the same spectral slope −(n − 1)/2. Thus, the passage of the peak-energy ε m of the cooling-tail SY emission softens the contribution of the pre pulsepeak emission to the integrated spectrum and hardens the contribution of the post pulse-peak emission, bringing them to the same integrated spectrum F ǫ (t > t p ) ∼ ǫ −(n−1)/2 , which is the slope of the cooling-tail SY instantaneous spectrum while electrons were injected at t < t I . This coincidence, which is also obtained for a weaker cooling process of exponent n < 1 (next section), arises from the correlation between the evolution of the peak-energy ε m (Equation A12) and the SY spectral slope β(> ε m ), both of which are set by the exponent n of the electron-cooling law. In the case of a sufficiently long electron injection (t I > t γǫ ) or a sufficiently high observing energy ǫ, Equation (A7) gives for the SY spectrum integrated until after the pulse plateau F ǫ (t > t I > t γǫ ) ≃ t p f pk = t I F p (t ic,i ) ǫ E γ −(n−1)/2 (A11) as for the above t I < t γǫ case. The integrated spectrum has the same slope as the instaneous spectrum because the former is dominated by the emission at the pulse plateau (t γǫ , t γǫ + t I ), whose duration t I is independent of the observing energy ǫ, so that the plateau flux f pk (third branch of Equation A8) imprints its energy dependence on the integrated spectrum. The initial assumption of a constant cooling timescale t ic,i for the typical GRB electron of energy γ i allows all the results for SY-dominated electron cooling shown in §3 to be recovered by setting the cooling power exponent n = 2. A2. Weak iC-Cooling (n < 1) through Scatterings of Low-Energy Photons at the Thomson-Klein-Nishina Transition The only case with n < 1 is that of the electron-cooling dominated by iC-scatterings if the γ i electrons scatter their SY photons in the KN regime, and at times before the iC-cooling timescale t ic,i of the γ i electrons. In this case, the iC-cooling power P ic (γ) ∼ γ n has an exponent n ≃ 2/3. From the iC-cooling law of Equation (31), the lowest electron energy γ m , its SY photon energy ε m , and the transittime t γǫ from emission at gamma-ray energy E γ to the observing energy ǫ are: γ m (t < t ic,i ) = γ i 1 − t t ic,i 3 , t ic,i ≡ 3γ i m e c 2 P ic (γ i ) (A12) ε m (t < t ic,i ) = E γ 1 − t t ic,i instantaneous spectrum and pulse light-curve at observing energy ǫ (t I < t γǫ ) f ǫ (t) = F p (t I )                              ǫ E γ 1/3 1 − t t ic,i −1        t t I t < t I (f ast rise) 1 − t − t I t ic,i −1 t I < t < t γǫ (rise) ǫ E γ 1/6 1 − t − t I t ic,i −1 t γǫ < t < t γǫ + t I (slow rise) 1 t γǫ + t I (peak) ǫ E γ −(p−1)/2 1 − t − t I t ic,i 3(p−1) t I + t γǫ < t (f all) (A18) (t γǫ < t I < t ic,i ) f ǫ (t) = F p (t I )                          ǫ E γ 1/3 t t I 1 − t t ic,i −1 t < t γǫ (f ast rise) ǫ E γ 1/6    t/t I t γǫ < t < t I (rise) 1 − t − t I t ic,i −1 t I < t < t I + t γǫ (slow rise) 1 t = t γǫ + t I (peak) ǫ E γ −(p−1)/2 1 − t − t I t ic,i 3(p−1) t I + t γǫ < t (f all)(A19) with the flux at t > t γǫ + t I as given on the last line of Equation (A18). This shows that pulse peak-epoch and peak-flux are t p = t γǫ + t I , f pk = f ǫ (t p ) = F p (t I )(A20) where F p (t I ) is the GRB pulse peak-flux (or the GRB peak spectral energy). The peak epoch t p corresponds to the passage of the high-energy end ε p of the cooling-tail, after the end of electron injection. Equations (A18) and (A19) show that the SY spectrum integrated until the transit-time t γǫ (when the lowest energy ε m of the power-law SY spectrum from the cooling-tail crosses the observing energy ǫ) is F ǫ ∼ ǫ 1/3 and that, after t γǫ , it is F ǫ ∼ ǫ 1/6 , with the ε p crossing at the peak-time t p = t γǫ + t I yielding a contribution with the same spectral slope 1/6: F ǫ (t > t p ) − F ǫ (t = t p ) ∼ ǫ −(p−1)/2 (1 − t γǫ /t ic,i ) 3p−2 ∼ ǫ 1/6 , after using Equation (A13). The same two equations show that the slow pre-peak rise and post-peak fall after max(t γǫ , t I ) are always dominant over the preceding fluence, the integrated flux being F ǫ (t ≫ t p ) > ∼ F p (t I )t ic,i ǫ E γ 1/6        1 6 ln E γ ǫ + 1 3p − 2 (t γǫ < t I < t ic,i ,ǫ < ǫ) ln 1 + t I t ic,i E γ ǫ 1/6 + 1 3p − 2 (t I < t γǫ < t ic,i , ǫ <ǫ)ǫ ≡ E γ 1 − t I t ic,i 6 (A21) with the last term representing the contribution from the pulse fall, which can be dominant over the pre-peak contribution depending on the observing energy ǫ and on the index p of the injected electron distribution. The above derivations pertain to the case when the lowest-energy γ m electrons in the cooling-tail cool mostly by scattering SY photons of energy m e c 2 /γ m < ε m at the T-KN limit. Those photons have a f ǫ ∼ ǫ 1/3 distribution with energy, leading to P ic (γ) ∼ γ 2/3 and to a cooling-tail distribution N (γ m < γ < γ i ) ∼ γ −2/3 , with γ m given in Equation (A12) for n = 2/3. The iC-cooling power of the γ m electrons switches exponent from n = 2/3 to n = 1 when the γ m electrons scatter their own SY photons of energy e sy (γ m ) at the T-KN transition (P19), i.e. when E ′ γ (γ m /γ i ) 2 = m e c 2 /γ m , with E ′ γ the GRB spectral peak-energy in the comoving frame. After that epoch, a softer distribution N (γ m < γ) ∼ γ −1 grows above the low-energy end of the cooling-tail, up to an electron energy that increases in time, i.e. the harder distribution N (γ < γ i ) ∼ γ −2/3 of the cooling-tail below the high-energy end γ i shrinks progressively. When the γ i electrons scatter the lowest energy ε m SY photons at the T-KN transition, i.e. when ε m = m e c 2 /γ i , the entire cooling-tail becomes N (γ m < γ < γ i ) ∼ γ −1 . Adding that, for n = 1, the cooling of the γ m electrons is a exponential in time (with timescale t ic,i ) that continues after the modified power-law cooling given in Equation (A12), it can be shown that the n = 2/3 initial iC cooling-tail is completely replaced by a softer n = 1 cooling-tail at epoch t 2/3→1 = t ic,i 1 − m e c 2 γ i E ′ γ 1/9 + 1 6 ln γ i E ′ γ m e c 2(A22) For the n = 1 cooling-tail to develop up to the GRB typical electron energy γ i , electron injection must last longer than the iC switch-time t 2/3→1 : t I > t 2/3→1 (first condition). For the n = 1 cooling-tail SY emission to dominate the integrated spectrum, the iC switch-time t 2/3→1 must occur before the pulse-peak epoch t p = t I + t γǫ , with the transit-time t γǫ for iC-cooling with n = 2/3: t I > t 2/3→1 − t γǫ (second condition). The second condition is satisfied if the first one is fullfilled, thus, the n = 1 cooling-tail yields an integrated spectrum of slope β LE = 0 only if t I > t 2/3→1 . To assess the robustness of the above result regarding GRBs with a hard low-energy slope β LE > 0 arising from iC-dominated electron cooling with n < 1, we consider next the case when the scattering optical-thickness τ is not constant. For an electron injection rate R i ∼ t y , the above case of a constant τ (leading to a constant iC-cooling timescale t ic (γ i )) corresponds to y < −1. For y > −1, when τ ∼ t y+1 increases, the cooling timescale t ic,i ∼ τ −1 decreases and the iC-cooling law of Equation (31) becomes A3. Pulse Duration and Transit-Time − dγ dt = 1 t ic,i (t I ) t t I y+1 γ n γ n−1 i (y ≥ −1) (A23) with the cooling timescale t ic,i (t I ) at the end of electron injection containing all relevant and unspecified quantities: magnetic field B and optical-thickness τ (t I ). The above equation can be integrated to derive γ(t) and, by using ǫ/E γ = [γ(t γǫ )/γ i ] 2 as definition for the transittime t γǫ , one obtains t (ic) γǫ = t I y + 2 n − 1 ǫ E γ (1−n)/2 − 1 t ic,i t I 1/(y+2) (y ≥ −1)(A24) The pulse duration can be calculated as in Equation (35): δt ǫ = ǫ E γ (1−n)/2 t I t γǫ y+1 t ic,i (t I ) (y ≥ −1)(A25) For y = −1, Equations (A24) and (A25) give timescales for a constant scattering optical-thickness τ . For an increasing τ (t), the transit-time t γǫ has a weaker dependence on the observing energy ǫ than for a constant τ (Equation 33) because the exponent 1/(y + 2) < 1, while the pulse duration δt ǫ picks-up an energy-dependent factor (t I /t γǫ ) y+1 . For iC-cooling dominated by scatterings in the Thomson regime (n > 1), Equations (A24) and (A25) lead to δt ǫ = n − 1 y + 2 t γǫ ∼ ǫ E γ −(n−1)/(2y+4) (y ≥ −1)(A26) thus, the trend of pulses to last shorter at higher energies still stands even when the scattering optical-thickness τ increases. The measured energy dependence of the pulse duration, δt γ ∼ ǫ −0.4 , has an exponent between the values −0.25 and −0.50 expected for iC-dominated electron cooling in the Thomson regime (n = 2), for a constant electron injection rate R i (y = 0) or a constant optical-thickness τ (y = −1), respectively. For iC-cooling dominated by scatterings at the T-KN transition (n < 1), if the electron injection rate R i decreases sufficienly fast (faster than 1/t), then the cooling-tail will be curved downward, with most of the flux being produced by the lowest energy γ m electrons of the tail. Thus, the pulse peak-time will be the transit-time that it takes the γ m electrons to cool to a SY-emitting energy equal to observing energy ǫ, and the pulse duration will be set by their cooling rate when their SY characteristic energy ε m drops to ǫ. The exact evolution of the injection rate R i is not relevant for the electron iC-cooling because the scattering optical-thickness τ is practically constant. Equations (A24) and (A25) still apply, but with y = −1, for which they reduce to Equations (33) and (35), thus, in the limit ǫ ≪ E γ , the transit-time t γǫ is very weakly dependent on the observing energy ǫ, and the pulse duration δt ǫ ∼ ǫ (1−n)/2 increases with ǫ, remaining at odds with GRB observations. For ǫ < ∼ E γ (just below to the GRB peak-energy), the term t γǫ in the denominator of Equation (A25) makes δt ǫ increase with ǫ even stronger than ǫ (1−n/2 , thus the incompatibility with observations becomes more severe. If R i does not decrease faster than 1/t, then the cooling-tail will be close to a power-law or will be curved upward, with most of the flux arising from the highest energy γ p electrons of the tail. Thus, the pulse peak-epoch will be the transit-time for the γ p electrons to "reach" the observing energy ǫ after the end of electron injection at t I , and the pulse duration δt ǫ will be set by the cooling rate of the γ p electrons when their SY characteristic ε p reaches ǫ, at t (ic) γǫ > t I . Because the scattering optical-thickness is constant after the end of electron injection, the iC-cooling is independent of the history of electron injection rate R i (t < t I ), and so are the transit-time t γǫ and pulse duration δt ǫ arising from the cooling of the γ p electrons after t I . Consequently, the pulse duration is still as given in Equation (35), and the incompatibility with observations remains unchanged. A3.2 Decreasing Magnetic Field B(t) An evolving magnetic field B(t) affects the transit-time t γǫ and the pulse duration δt ǫ in two ways: first, it determines the energy density of the seed SY photons to be upscattered and iC-cool the electrons, thus it determines the electron cooling and, second, it determines the evolution of the ends of the SY spectrum from the cooling-tail, whose passage through the observing band defines t γǫ and δt ǫ . The case of electron iC-cooling occurring mostly through scatterings in the Thomson regime (n > 1) will not be considered further because, as shown above, it can account for the observed trend of GRB pulse duration to decrease with observing energy, provided that electrons cool to below the observing energy, and we focus on iC-cooling dominated by scatterings of the sub-GRB SY photons at the T-KN transition (n < 1), a case that fails to accommodate that observational feature for a constant magnetic field. For iC-cooling with n < 1, the history of the electron injection rate R i (t) is irrelevant for the cooling of the electrons that determine the transit-time and the pulse duration, thus these two quantities depend only on the evolution of the magnetic field, which we will assume to be a power-law B(t) = B i (t/t I ) x , normalized at the end of electron injection. It can be shown that the iC-cooling power satisfies P ic (γ) ∼ τ (Bγ) n , either for n < 1 (equation 45 of P19) or for n > 1 (equation 41 of P19), thus, for the above B(t), the iC-cooling law is − dγ dt = 1 t ic,i (t I ) t t I nx γ n γ n−1 i (A27) which can be integrated (from t = 0 to t = t γǫ if τ is constant and from t = t I to t = t I + t γǫ if τ increases) and, after using ǫ E γ = γ(t γǫ ) γ i 2 B(t γǫ ) B i = γ(t γǫ ) γ i 2 t γǫ t I x (A28) will lead to an algebraic equation for the transit-time t γǫ , which can be solved in asymptotic regimes: (n < 1, B ∼ t x ) t (ic) γǫ ≃ t I ×          ǫ E γ 1/xǫ ≪ ǫ < E γ (x < 0) t ic,i t I 1/(nx+1) ǫ ≪ǫ (− 1 n < x < 0)ǫ ≃ E γ t ic,i t I x/(nx+1)(A29) The second branch of Equation (A29) shows that, for a sufficiently low observing energy ǫ, the transit-time is independent of ǫ. In the limit x → 0 (constant B), one hasǫ(x = 0) = E γ , thus the first branch of Equation (A29) disappears and the second branch reduces to the transit-time given in Equation (33) in the limit ǫ ≪ E γ , The condition x < 0 arises from requiring that the transit-time decreases with observing energy ǫ; in the opposite case (x > 0), an increasing magnetic field would compensate the electron cooling and lead to break energies (at either end of the cooling tail's SY spectrum) that increase, and there is no transit of the break energies to an observing energy ǫ < E γ . The working condition x > −1/n for the second branch leads to simple temporal power-law dependence for the cooling-equation solution γ(t); for x < −1/n, the corresponding equation for t γǫ becomes even more complicated; however, the result given in the first branch stands for x < −1/n as well. Once the transit-time t γǫ is known, the pulse duration can be calculated by using the cooling-law of Equation (A27): δt ǫ = ǫ E γ (1−n)/2 t I t γǫ (n+1)x/2 t ic,i (t I )(A30) leading to δt ǫ ≃ t ic,i (t I ) ×          ǫ E γ −n=−2/3ǫ < ǫ < E γ (x < 0) t I t ic,i( For a sufficiently low observing energy (second branch), the pulse duration still increases with energy, however, for energies just below the GRB spectral peak E γ , a decreasing magnetic field "opens" the first branch above, for which δt ǫ ∼ ǫ −n ∼ ǫ −2/3 is consistent with (or stronger than) the observed trend of pulse duration to decrease with energy. and (R i ∼ t −y , 5/9 < y) f ǫ (t) = F p (t I )                              ǫ E γ 1/3 t I t o y−1 t t o 4/9 t < t γǫ (slow rise) t I t o y−1 t = t γǫ (peak if y > 1) ǫ E γ 3(1−y)/4 t t I 1−y t γǫ < t <t γǫ (y < 1|slow rise; y > 1|f all) 1 t =t γǫ (peak if y < 1) ǫ E γ −(p−1)/2 t t I −2(p−1)/3t γǫ < t (f all)(B8) For any y, f ǫ (t γǫ ) = F p (t I ) follows from Equation (B4) becauset γǫ > t I (Equation B7), i.e. the high-energy break ε p decreases below the observing energy ǫ only after electron injection stops. For y < 5/9, the cooling-tail emission is dimmer than the ǫ 1/3 low-energy SY flux produced by the γ p electrons and the entire spectrum is as if the cooling-tail did not exist, thus the epoch t γǫ when ε m crosses the observing band is irrelevant. For y > 5/9, the light-curve depends on the evolution of the lowest-energy characteristics (ε m and f m ), which are unchanged across t I , thus there is no light-curve break at the epoch t I when electron injection stops and the epoch t I is irrelevant. (B8) show that the pulse rise is harder (β r ≥ 0) than its decay (β f ≤ 0), and that the pulse peak-epoch and peak-flux are Equations (B6) and t p = t (ad) γǫ y < 1 t (ad) γǫ y > 1 = ǫ E γ −3/4 t I y < 1 t o y > 1 , f pk = F p (t I ) × 1 y < 1 (t I /t o ) y−1 y > 1 (B9) The pulse-integrated spectrum is dominated by the emission prior to when the flux decay becomes faster than t −1 . That happens at t γǫ if y > 2 (leading to F ǫ ∼ ǫ −3/4 , see below), att γǫ if y < 2 (leading to F ǫ ∼ ǫ −3/4 , see below) and if the injected electron distribution has an exponent p > 5/2, but continues through the pulse decay at t >t γǫ if p < 5/2 (leading to F ǫ ∼ ǫ −(p−1)/2 ). After some calculations, the integrated spectrum is found to be The above slope β LE = −3/4 of the integrated spectrum arises from the passage of the higher spectral break ε p , because the fluence from the pulse fall is most often dominant, yielding F ǫ ≃t γǫ f ǫ (t γǫ ) ∼t γǫ (ǫ 1/3t 4/9 γǫ ) ∼ ǫ −3/4 . However, even if when the pulse fluence is dominated by the crossing of the lower spectral break ε m (i.e. for y > 3.4), the slope of the integrated spectrum would be the same β LE = −3/4 because the transit-times t γǫ (Equation 41) andt γǫ (Equation B7) have the same dependence on the observing energy ǫ. As for SY cooling, the softness of the pulse-integrated spectrum is a consequence of the cooling-time increase with observing energy: t (ad) γǫ ∼ ǫ −3/4 . The last branch above simply states that, if the injected electron distribution is sufficiently hard (p < 5/2), then the pulse fluence is mostly from the cooled-injected distribution, and not from the cooling-tail. In this case, the integrated spectrum will have the slope β LE = −(p − 1)/2 of the cooled-injected distribution, which has the slope of the injected power-law energy distribution because AD cooling shifts distributions to lower energies while preserving their slopes. (R i ∼ t −y ) F ǫ (t >t γǫ ) = F p (t I )t I ×          ǫ E γ C. SYNCHROTRON (SY) AND ADIABATIC (AD) COOLING Equations (10) and (39) xg x−1 dg dt + g 2x γ o t sy (γ o ) + 2 3 g x t + t o = 0 (C2) asymptotic solutions can be derived in three regimes: i) t ≪ t o → X − 1 ≃ t/(3t o ) ≪ 1 leading to the SY-solution, ii) t ≫ t o → X ≃ (t/t o ) 1/3 ≫ 1 leading to the AD and the 1/3-SY solutions, and iii) X − 1 ≪ Z/2, leading to the AD-solution. Depending on the relative strength of the AD and SY losses at t = 0, quantified by Z, the solution given in Equation (C6) for AD+SY cooling has the following asymptotic regimes: (Z < 2 4/3 − 2 ≃ 0.52) : γ(t) ≃ γ sy t ≪ t o (early SY solution) 1 3 γ sy t o ≪ t (late 1/3 − SY solution) (C9) (0.52 < Z < 1) : γ(t) ≃      γ sy t ≪ t o (early SY solution) γ ad t o < t <t ∈ (1, 19 8 )t o (transient AD solution) 1 3 γ syt ≪ t (late 1/3 − SY solution)(C10) (1 < Z) : γ(t) ≃ γ ad t ≪t (early AD solution) 1 3 γ sy ( 19 8 t o <)t ≪ t (late 1/3 − SY solution) (C11) wheret ≡ t o Z 2 + 1 3 − 1 (C12) is the AD-SY solution switch-time, when X − 1 = Z/2. The above electron cooling through the three asymptotic regimes is indicated in Figure C4 by horizontal arrows (increasing time toward right). Which asymptotic solutions are encountered depends on the parameter Z. The AD and SY solutions are separated by the Z = 2(X − 1) line corresponding to t =t. The above asymptotic solutions have some interesting features: i) Electron cooling is asymptotically described at early times by the SYnchrotron solution γ sy only if Z < 1, which means t sy (γ o ) < t ad (t = 0), i.e. only if the electron cooling is initially SY-dominated (obviously). Furthermore, the SY-cooling solution is accurate only at times t ≪ t o , when X − 1 ≃ t/3t o , for which the right-hand side term of Equation (C5) shows an AD-cooling term smaller than the SY-cooling term, owing to that Z < 1. ii) Electron cooling is described by the ADiabatic solution γ ad only for t ≪t, i.e. only for 2(X − 1) ≪ Z, when the right-hand side of Equation (C6) is unity. This condition is sufficient for an asymptotic AD-solution at early times if Z > 1, i.e. if the electron cooling is initially AD-dominated (obviously), but is not sufficient if Z < 1, i.e. if the electron cooling is initially SY-dominated. In the latter case, competition between the adiabatic term X 2 and the mixed term 1 + 2(X − 1)/Z appearing in Equation (C6) allows the AD-solution to set in at t ≃ t o . iii) As can be seen in Figure C4, irrespective of which cooling process is dominant initially, electron cooling is asymptotically described at late times t ≫ max(t o ,t) by the 1/3-SYnchrotron solution. Condition t ≫ t o implies X ≫ 1, which implies that X 2 [1 + 2(X − 1)/Z] ≃ X 2 (1 + 2X/Z), and condition t ≫t implies X ≫ Z/2, which leads to X 2 (1 + 2X/Z) ≃ 2X 3 /Z ≃ 2t/Zt o = 3t/t sy (γ o ), thus γ ≃ (1/3)γ o t sy (γ o )/t ≃ γ sy /3. In other words, for sufficiently late times, the product of the AD-cooling term X 2 and the modified SY-cooling term 1 + 2(X − 1)/Z ≃ 2X/Z is proportional to the "pure" SY-cooling term, leading to a SY-cooling solution despite that, at late times, the electron cooling is guaranteed to be AD-dominated, as shown below. From Equations (11) and (C6), the SY-cooling timescale of the SY+AD-cooling electron is t sy (γ) = t sy (γ o )γ o /γ = 1.5Zt o γ o /γ = 3t + t sy (γ o )(t/t o ) 2/3 at t ≫ t o , while the AD-cooling timescale (Equation 40) t ad = 1.5(t + t o ) is independent of the electron energy. That t sy > 3t and t ad > ∼ 1.5t guarantees that, after some time, t sy > t ad and the electron cooling will be eventually AD-dominated even if it started in a SY-dominated regime (t sy (γ o ) < t ad (t o )). The condition t sy (γ) = t ad implies X = 2 − Z, which defines a critical time and a critical electron energy (Z < 1) : t cr = t o [(2 − Z) 3 − 1] ∈ (0, 7)t o , γ(t cr ) ≡ γ cr = Z X 3 γ o = 2t sy (γ o ) 3(t + t o ) γ o < γ o , t sy (γ) t ad = γ cr γ(C13) For Z < 1, the electron cooling is SY-dominated until t cr ; at t cr , the electron energy is γ cr and the powers of the two cooling processes are equal; after t cr , AD-cooling is dominant. For Z > 1, when the electron cooling is AD-dominated at t = 0, it can be shown using Equation (C6) that t sy (γ) > t ad at any time, thus the electron cools adiabatically at all times. Depending on which cooling process is dominant initially (SY-cooling if Z < 1, AD-cooling for Z > 1), the dominance at later times is established as following: Z < 1 : [SY − cool : t sy (γ(t)) < t ad (t)] and [γ(t) = SY − sol] f or [γ > γ cr , t < t cr , X < 2 − Z] [AD − cool : t sy (γ) > t ad ] and [γ(t) = 1 3 SY − sol] f or [γ < γ cr , t > t cr , X > 2 − Z] peaking at β o = 0.0 and with a dispersion σ ≃ 0.40 (halfwidth at half-maximum of 0.45). 0.20, while the remaining 30% of spectra are well-fit by a SBPL with an average low-energy slope β o = −0.02 ± 0.19. 1. 3 . 3Limitations of the Standard Synchrotron Model 1.3.1. The Low-Energy Spectral Slope Fig. 1 . 1-Evolution of the electron distribution for synchrotron-dominated electron cooling and for a constant magnetic field B, obtained by tracking the cooling of infinitesimal electron injections on a (time x energy) grid, using Equation Fig. 2.-Synchrotron-dominated cooling of electrons injected at an increasing rate R i ∼ t y . Legend indicates the low-energy slope β LE (at 10 keV) and the peak-energy E γ (for B = 100 G, Γ = 100, z = 1) of the power-perdecade ǫF ǫ at three epochs (t = 3, 10, 30 t sy,i ) for the SY spectrum integrated up to those times. The faster that increase, the harder the cooled-electrons distribution below the typical electron energy γ i and the harder is the lowenergy SY spectrum. The low-energy spectrum softens progressively, with a hard slope β LE < ∼ 1/3 at t < 2 t sy,i , an average β LE ≃ 0 at t ≃ 5 t sy,i , and a soft slope β LE < ∼ −1/2 at t > ∼ 30 t sy,i , showing how the hardening of β LE vanishes at t ≫ t sy,i . Thus, the hardening of the low-energy spectrum for an increasing electron injection rate R i is a transient feature and spectra with harder slopes β LE > 0 require the cessation of SY emission before the asymptotic β LE = −1/2 is reached. The t I = 3t sy,i case shows that the softening of low-energy slope β LE is significantly faster if electron injection stops while the hardening produced by an increasing rate R i is still effective. Fig. 3 . 3-Adiabatic and Synchrotron cooling, for the same parameters as in APPENDIX A. SPECTRA AND LIGHT-CURVES OF SYNCHROTRON EMISSION FROM INVERSE-COMPTON (IC) COOLING ELECTRONS A3. 1 1Constant B and Increasing τ (t) (Decreasing tic(γi)) n+1)x/(2nx+2) ǫ E γ (1−n)/2=1/6 ǫ <ǫ (−3/2 = −1/n < x < 0) lead to(SY + AD) − dγ dt = − dγ dt sy − dγ dt ad = γ 2 γ o t sy (γ o ) + 2 3 γ t + t o(C1)for an electron of initial energy γ o . In above equation, t sy (γ o ) = 7.7×10 8 /(γ o B 2 ) (Equation 8) is the SY-cooling timescale for the γ o electrons. With the substitution γ = g x , the electron-cooling equation becomes TABLE 1 1Glossary of more frequently used notations is the electron cooling law for the corresponding cooling process. For ADiabatic cooling, Q is a constant;Electron energy γ i typical energy of injected electrons γm lowest energy of cooled electrons γp energy of electrons radiating at εp γcr critical electron energy, where tsy (γcr) = t ad Spectral quantities β LE GRB low-energy slope (below Eγ) βoγ optical-to-gamma effective spectral slope Eγ peak energy of GRB νFν spectrum Fp flux at Eγ ǫ observing energy ǫsy SY characteristic energy fǫ spectral flux density Fǫ pulse-integrated spectral flux density εp peak energy of the Fν SY spectrum fp SY flux at εp εm SY energy for the γm electrons fm SY flux at εm Electron timescales t ad AD cooling timescale t rad radiative cooling timescale of γ i electrons tsy SY cooling timescale t ic iC cooling timescale t sy,i SY cooling timescale for γ i electrons t ic,i iC cooling timescale for γ i electrons tγǫ transit time from GRB Eγ energy to ǫ t γ−10k transit time from GRB to mid X-rays (10 keV) tcr epoch when tsy(γm) = t ad Other timescales t B magnetic field life-time t I electron injection duration tp pulse peak epoch tang angular spread in photon arrival-time δtγ duration of GRB pulse δtǫ pulse duration at energy ǫ This is the variance-weighted average of the three median slopes found for the above three fitting functions. However, the individual distributions do not display any visible skewness, thus the median slope should be very close to the variance-weighted average slope, for each of the three sets. t t I t < t I (rise) 1+ t t sy,i 2/3 t I < t < t sy,i (very slow rise) t t sy,i 2/3 t sy,i ≪ t < t γǫ (slow rise)(20)where F p (t I ) = F p (t sy,i ) because the GRB peak flux (or flux density at E γ ) does not change much from the end of electron injection at t I to one SY-cooling timescale t t sy,i t < t sy,i (rise) t t sy,i 5/3 t sy,i < t < t I (f ast rise) t I t sy,i t t sy,i 2/3 t I < t < t γǫ (slow rise)(21)Lastly, for an electron injection lasting longer than the transit-time (t I > t γǫ ) or for an observing energy ǫ > ǫ, the first two rising branches of Equation (21) remain unchanged (with t γǫ instead of t I ) and the third rising branch is replaced by a plateauf ǫ (t γǫ < t < t I + t γǫ ) = F p (t sy,i ) ǫ E γ −1/2 (plateau)(22)for t γǫ < t I . The constancy of the SY flux at t > t γǫ is This linearity can be easily proven by calculating the delay in arrival-time between a photon emitted at time t by the fluid moving directly toward the observer and a photon emitted at time t + δt by the fluid moving at angle Γ −1 relative to the direction toward the observer. However, a different recipe for adding timescales will result if times are weighed by the intensity of the emission produced at that time and at a certain angular location. This dependence δtǫ derived for energies below the GRB peakenergy Eγ , which is assumed constant for the duration of the entire electron injection, applies also for GRB channels above Eγ, where the pulse peak epoch tp is the duration over which electrons accumulate without significant cooling, i.e. tp is the SY-cooling timescale for electrons radiating at ǫ > Eγ . The integration of emission over the equal arrival-time surface may induce a decreasing pulse duration with observing energy, and could reverse the above expected trend, thus this limitation of iC cooling applies mostly to the emission from a bright-spot t cr defined by t sy,i [γ m (t cr )] = t ad (t cr ) (which leads to t cr = 2t sy,i − 3 t o ), after which t sy [γ m (t)] < t ad and the electron cooling should become SY-dominated. t γǫ ≡ t I ǫ E γ −3/4 = t I t o t γǫ (B7) assuming a constant magnetic field B and a constant iC-cooling timescale t ic,i ≡ t ic (γ i ).In constrast with the n > 1 case, for n < 1, the observing energy ǫ is "reached" before the cooling timescale t ic,i of the γ i electrons, and the spectrum f ǫ ∼ ǫ 1/6 of the SY emission from the cooled electron distribution rises (instead of falling), peaking at ε p = E γ , the GRB energy peak. Thus, the flux at ε m iswhere the evolution of the GRB peak flux F p (t < t I ) stands for a constant electron injection rate R i and a constant magnetic field B.The γ i -electrons injected at t I cool followingusing Equation (A12), thus the width of the cooling-tail increases slowly at t I < t ≪ t ic,i , implying that the cooling-tail is stretched and becomes harder. That hardening being slow, we will ignore it and assume that the cooling-tail remains a power-law of exponent −n. The SY spectrum peaks at the characteristic SY energy of the γ p -electrons, ε p = ǫ sy (γ p ), and the peak flux at that energy is approximately constantbecause most electrons (of constant number) radiate at ε p .From the SY spectrum corresponding to the broken power-law electron distributionand the above evolutions of break energies ε m , ε p and of the coresponding fluxes f m and f p , one can calculate the During electron injection (at t < t I ) at the power-law rate R i ∼ (t + t o ) −y , the cooling-tail extends from the lowest energy γ m given in Equation(38)to the minimal electron injection energy γ i , thus, the cooling-tail SY emission extends from photon energies ε m (t) (Equation 38) to ε p (t < t I ) = E γ and has the slope β given in Equation(46). For y > 1, m = (1 − 3y)/2 < −1 and most cooled electrons are at γ m ; for y < 1, m > −1 and most cooled electrons are at γ i ; irrespective of y.Thus, for a constant magnetic field B, the flux densities f ǫ ∼ Bγ(ǫ)N (γ) at the ends of the cooling-tail's characteristic synchrotron energies ε m and ε p = E γ arewith F p (t I ) the flux density at the GRB peak-energy E γ when electron injection ends. For y < 1, i.e. for low-energy slopes β LE (t < t I ) > 0, the epoch t I also marks the GRB pulse peak, thus F p (t I ) is also the GRB pulse peak-flux (or pulse-peak flux) for GRBs with a harder low-energy slope during the pulse rise.After electron injection ends (at t > t I ), the lowest-energy electrons γ m continue to cool adiabatically according to Equation(38), thus ε m evolves is as in Equation(38), while the γ i electrons begin cooling adiabatically at t I following the same law but with time measured since t I and for the current system age t I + t o :(B3) thus the width of the cooling-tail is constant. From the electron AD-cooling law γ(t > t I ) ∼ t −2/3 , it can be shown that the cooling-tail slope −m remains unchanged at t > t I . From γp γm dγN (γ) = const at t > t I , it can be shown that the flux densities at the at the ends ε m and ε p of cooling-tail remain constant after t I . Thus, the flux density at the lowest energy of the cooling-tail is same as in Equation(B2), while the flux density at the higher-energy break ε p isafter using Equation (B2).Adding the SY spectrum from a broken power-law electron distribution consisting of a cooling-tail N (γ m < γ < γ p ) ∼ γ (1−3y)/2 and a cooled-injected electron distribution N (γ p < γ) ∼ γ −p isthe instantaneous spectrum (pulse light-curve) at lower energies can be calculated:with t γǫ the epoch when the lowest energy electrons ε m radiate at the observing energy ǫ (Equation 41), andt γǫ the epoch when the higher spectral break at ε p crosses the observing energy ǫ:which is a first-order linear differential equation of the formonly if x = −1:where the constant can be determined from the initial condition γ(t = 0) = γ o .Thus, the solution to the SY and AD-cooling is :which can be written aswhere γ ad is the solution (Equation 38) to the AD cooling law. Thus, the solution to the SY and AD cooling is a modified adiabatic-cooling solution, with denominator containing information about both AD and SY cooling. The reason for this structure for the cooling solution is that the linear term of the first-order LDE for electron cooling (Equation C4) contains only AD-cooling. In the limit t ≪ t o , the denominator loses dependence on AD cooling and the solution becomeswhich shows SY cooling (Equation 8) and AD cooling (Equation 38) operating independently.The inverse-Compton cooling law of Equation (31) can be added to the AD and SY cooling terms of Equation (C1), but an approximative calculation of the inverse-Compton power is possible only until the overall cooling timescale t rad of the typical γ i electron (i.e. before the cooling-tail develops) because, in the opposite case t > t rad , the Compton parameter Z depends on the minimal electron energy γ m of the cooling-tail, whose evolution is not known in advance (unless iC-cooling is weaker than SY and AD-cooling and Equation C5 can be used as an approximation of γ m ). For t < t rad , the iC power depends on the electron energy as P ic ∼ γ 2/3 for an electron that scatter the SY photons E γ produced by the typical γ i -electron in the Klein-Nishina regime (γE ′ γ > mc 2 ); then iC-cooling introduces a tγ 2/3 term in Equation (C1), with time t entering through the proportionality of the iC power on the electron scattering optical-thickness τ .In this case, the substitution γ = 1/g that is required for the SY-cooling term of Equation (C1) to yield a first-order linear differential equation (LDE) brings an iC-cooling term proportional to tg 4/3 , which cannot be combined with the AD-cooling term g/(t + t o ) to lead to the term a(t)g of the LDE (C1). However, for electrons that scatter E γ photons in the Thomson regime (γE ′ γ < mc 2 ), the iC power has the same P ic ∼ γ 2 dependence as the SY power and the substitution γ = 1/g used above leads to a cooling law of the form given in Equation (C3) but with a slightly more complex term b(t) = const + kt.Then, the first integral given in Equation (C3) can be calculated analytically and the electron cooling subject to all three processes iswhere t ic (γ o ) is the iC-cooling timescale at the epoch t I when the electron injection ends and the scattering opticalthickness τ is maximal.Again, the solution (Equation C8) to the full electron-cooling law (as inEquation C1but with an extra term for iC-cooling in the Thomson regime: −(dγ/dt) iC = γ 2 /[γ o t ic (γ o )]) is a modified AD-cooling solution, with the decrease of the electron energy γ expedited by a SY and an iC-cooling term, each expressed as their strength (1/t sy or 1/t ic ) relative to that of AD-cooling (1/t o ) at the beginning of electron injection.Because Equation (C8) has a limited applicability, as it describes electron cooling only before the cooling-tail develops significantly, we will not investigate it any further. Instead, we return to Equation (C6) for AD and SY-cooling, whose . M Axelsson, L Baldini, G Barbiellini, ApJ. 75731Axelsson M., Baldini L., Barbiellini G. et al, 2012, ApJ 757, L31 . D Band, ApJ. 486928Band D., 1997, ApJ 486, 928 . P Bhat, G Fishman, C Meegan, ApJ. 426604Bhat P., Fishman G., Meegan C. et al, 1994, ApJ 426, 604 . A Crider, E Liang, I Smith, ApJ. 47939Crider A., Liang E., Smith I. et al, 1997, ApJ 479, L39 . F Daigne, Z Bosnjak, G Dubus, 526110Daigne F., Bosnjak Z., Dubus G., 2011, AA 526, 110 . E Fenimore, J Zand, J Norris, J Bonnell, R Nemiroff, ApJ. 448101Fenimore E., Zand J., Norris J., Bonnell J., Nemiroff R., 1995, ApJ 448, L101 . G Ghirlanda, A Celotti, G Ghisellini, A&A. 406879Ghirlanda G., Celotti A., Ghisellini G., 2003, A&A 406, 879 . J Granot, ApJ. 81620Granot J., 2016, ApJ 816, L20 . S Guiriec, C Kouveliotou, F Daigne, ApJ. 807148Guiriec S., Kouveliotou C., Daigne F. et al, 2015, ApJ 807, 148 . Y Kaneko, R Preece, M Briggs, ApJS. 166298Kaneko Y., Preece R., Briggs M. et al, 2006, ApJS 166, 298 . R Kippen, J Zand, P Woods, AIP Conf. Proc. 727119Kippen R., in't Zand J., Woods P. et al, 2004, AIP Conf. Proc. 727, 119 . P Kumar, A Panaitescu, ApJ. 54151Kumar P., Panaitescu A., 2000, ApJ 541, L51 . P Kumar, E Mcmahon, MNRAS. 38433Kumar P., McMahon E., 2008, MNRAS 384, 33 . A Lee, E Bloom, V Petrosian, ApJS. 1311Lee A., Bloom E., Petrosian V., 2000, ApJS 131, 1 . R Lloyd, V Petrosian, ApJ. 543722Lloyd R., Petrosian V., 2000, ApJ 543, 722 . M Medvedev, A Loeb, ApJ. 526706Medvedev M., Loeb A., 1999, ApJ 526, 706 . M Medvedev, ApJ. 540704Medvedev M., 2000, ApJ 540, 704 . P Mészáros, M Rees, ApJ. 530292Mészáros P., Rees M., 2000, ApJ 530, 292 . E Nakar, S Ando, R Sari, ApJ. 703675Nakar E., Ando S., Sari R., 2009, ApJ 703, 675 . J Norris, R Nemiroff, J Bonnell, ApJ. 459393Norris J., Nemiroff R., Bonnell J. et al, 1996, ApJ 459, 393 . G Oganesyan, L Nava, G Ghirlanda, A Celotti, ApJ. 846137Oganesyan G., Nava L., Ghirlanda G., Celotti A., 2017, ApJ 846, 137 . A Panaitescu, P Mészáros, ApJ. 54417Panaitescu A., Mészáros P., 2000, ApJ 544, L17 . A Panaitescu, ApJ. 88619Panaitescu A., 2019, ApJ 886, 106 (P19) . A Panaitescu, W T Vestrand, arXiv.org/abs/2209.11847ApJ. Panaitescu A., Vestrand W.T., 2022, ApJ, accepted (arXiv.org/abs/2209.11847) . S Poolakkil, R Preece, C Fletcher, ApJ. 91360Poolakkil S., Preece R., Fletcher C. et al, 2021, ApJ 913, 60 . R Preece, S Briggs, R Mallozzi, ApJS. 12619Preece R., Briggs S., Mallozzi R. et al, 2000, ApJS 126, 19 . M Ravasio, G Ghirlanda, L Nava, G Ghisellini, A&A. 62560Ravasio M., Ghirlanda G., Nava L., Ghisellini G., 2019, A&A 625, A60 . F Ryde, M Axelsson, B B Zhang, ApJ. 709172Ryde F., Axelsson M., Zhang B.B. et al, 2010, ApJ 709, L172 . M Toffano, G Ghirlanda, L Nava, A&A. 652123Toffano M., Ghirlanda G., Nava L. et al, 2021, A&A 652, A123 . B Zhang, Y Huirong, ApJ. 72690Zhang B., Huirong Y., 2011, ApJ 726, 90 . B Spectra, Light-Curves, Of, Emission, Adiabatically-Cooling, Electrons, B. SPECTRA AND LIGHT-CURVES OF SYNCHROTRON EMISSION FROM ADIABATICALLY-COOLING ELECTRONS The horizontal (green) arrows represent the cases identified in Equations (C9)-(C11). The AD-solution (γ ad of Equation C6) applies mostly to Z > 1 (AD-cooling dominant initially over SY-losses) and lasts untilt given by Equation (C12) and shown by the solid (purple) line. The SY-solution (γ sy of Equation C7) occurs only for Z < 1 (i.e. if SY losses are dominant over AD cooling initially) and lasts until t o . For either cooling process being dominant initially (i.e. for any Z), the electron cooling eventually turns to the 1/3-SY solution, at a time max{t o ,t}, and that transition occurs even when the electron cooling is AD-dominated after that time. Dotted (black) line shows the epoch when AD and SY cooling powers are equal. depending on the initial ratio Z of SY to AD cooling power (γ o = γ(t = 0) is the initial electron energy). (Constant magnetic field is assumed for the former). at t = t cr (Equation C13), but that epoch is irrelevant for the electron-cooling, which is (almost) always the 1/3-SY solution across t crFig. 4.-Passage of electron cooling γ(t) (for increasing X(t)) through three possible asymptotic solutions (AD, SY, and 1/3-SY), depending on the initial ratio Z of SY to AD cooling power (γ o = γ(t = 0) is the initial electron energy). (Constant magnetic field is assumed for the former). The horizontal (green) arrows represent the cases identified in Equations (C9)-(C11). The AD-solution (γ ad of Equation C6) applies mostly to Z > 1 (AD-cooling dominant initially over SY-losses) and lasts untilt given by Equation (C12) and shown by the solid (purple) line. The SY-solution (γ sy of Equation C7) occurs only for Z < 1 (i.e. if SY losses are dominant over AD cooling initially) and lasts until t o . For either cooling process being dominant initially (i.e. for any Z), the electron cooling eventually turns to the 1/3-SY solution, at a time max{t o ,t}, and that transition occurs even when the electron cooling is AD-dominated after that time. Dotted (black) line shows the epoch when AD and SY cooling powers are equal, at t = t cr (Equation C13), but that epoch is irrelevant for the electron-cooling, which is (almost) always the 1/3-SY solution across t cr . AD − cool : t sy (γ) > t ad ] and [γ(t) = AD − sol] f or. t <t, X − 1 < Z/2] [AD − cool : t sy (γ) > t ad ] and [γ(t) = 1Z > 1 : [AD − cool : t sy (γ) > t ad ] and [γ(t) = AD − sol] f or [t <t, X − 1 < Z/2] [AD − cool : t sy (γ) > t ad ] and [γ(t) = 1 . Sy − Sol] F Or, t < t, X − 1 > Z/2SY − sol] f or [t < t, X − 1 > Z/2] Comparing these expectations with the expanded solution (Equations C9-C11) for electron cooling, we note that the condition for the SY-cooling solution to be asymptotically displayed at early times, t ≪ t o , is more restrictive than the condition for SY cooling to be dominant: X < 2 − Z. Similarly, the conditions for the AD-cooling solution to be asymptotically manifested at early times: t ≪t (or X ≪ Z/2 + 1) and Z > 1. are more restrictive than the condition for AD-cooling to be dominant: X > 2 − ZComparing these expectations with the expanded solution (Equations C9-C11) for electron cooling, we note that the condition for the SY-cooling solution to be asymptotically displayed at early times, t ≪ t o , is more restrictive than the condition for SY cooling to be dominant: X < 2 − Z. Similarly, the conditions for the AD-cooling solution to be asymptotically manifested at early times: t ≪t (or X ≪ Z/2 + 1) and Z > 1, are more restrictive than the condition for AD-cooling to be dominant: X > 2 − Z. -(C11) shows that a change in the evolution of the electron energy is not tied to the competition between the two cooling processes, which defines the critical electron energy γ cr (Equation C13) that is crossed by the cooling electron at the critical time t cr , but by the interplay between the SY and AD terms in the solution (Equation C6) to the two-process cooling equation. Finally, at late times: t ≫ max(t, t cr ), when AD-cooling is dominant (X > 2 − Z for t > t cr. Furthermore, the expanded solution in Equations (C9). the solution to electron cooling is 1/3 of the SY-cooling solutionFurthermore, the expanded solution in Equations (C9)-(C11) shows that a change in the evolution of the electron energy is not tied to the competition between the two cooling processes, which defines the critical electron energy γ cr (Equation C13) that is crossed by the cooling electron at the critical time t cr , but by the interplay between the SY and AD terms in the solution (Equation C6) to the two-process cooling equation. Finally, at late times: t ≫ max(t, t cr ), when AD-cooling is dominant (X > 2 − Z for t > t cr ), the solution to electron cooling is 1/3 of the SY-cooling solution
[]
[ "LOWER BOUNDS ON THE ERROR PROBABILITY FOR INVARIANT CAUSAL PREDICTION", "LOWER BOUNDS ON THE ERROR PROBABILITY FOR INVARIANT CAUSAL PREDICTION" ]
[ "Ieee International ", "On ", "Learning ", "Agu Signal Processing " ]
[]
[]
It is common practice to collect observations of feature and response pairs from different environments. A natural question is how to identify features that have consistent prediction power across environments. The invariant causal prediction framework proposes to approach this problem through invariance, assuming a linear model that is invariant under different environments. In this work, we make an attempt to shed light on this framework by connecting it to the Gaussian multiple access channel problem. Specifically, we incorporate optimal code constructions and decoding methods to provide lower bounds on the error probability. We illustrate our findings by various simulation settings.Index Terms-Invariance, Gaussian multiple access channel, error exponent, lower bound.1To simplify notation, we skip settings when m is polynomial in n (or anything slower than exponential), which also belong to the zero-rate setup.
10.1109/mlsp55214.2022.9943384
[ "https://arxiv.org/pdf/2206.14362v2.pdf" ]
250,113,414
2206.14362
880ec7f3339e48144449be0ec2d03a6ea779ef03
LOWER BOUNDS ON THE ERROR PROBABILITY FOR INVARIANT CAUSAL PREDICTION 22-25, 2022 Ieee International On Learning Agu Signal Processing LOWER BOUNDS ON THE ERROR PROBABILITY FOR INVARIANT CAUSAL PREDICTION XI'AN, CHINA22-25, 2022Index Terms-InvarianceGaussian multiple access channelerror exponentlower bound It is common practice to collect observations of feature and response pairs from different environments. A natural question is how to identify features that have consistent prediction power across environments. The invariant causal prediction framework proposes to approach this problem through invariance, assuming a linear model that is invariant under different environments. In this work, we make an attempt to shed light on this framework by connecting it to the Gaussian multiple access channel problem. Specifically, we incorporate optimal code constructions and decoding methods to provide lower bounds on the error probability. We illustrate our findings by various simulation settings.Index Terms-Invariance, Gaussian multiple access channel, error exponent, lower bound.1To simplify notation, we skip settings when m is polynomial in n (or anything slower than exponential), which also belong to the zero-rate setup. INTRODUCTION Invariant causal prediction (ICP) [1] is a recently proposed framework on linear models for selecting features that are stable across different environments. It is motivated by the idea of invariance, which is often referred to as modularity, autonomy [2,3,4,5,6], or stability [7,5]. The main assumption for invariant prediction is that there exists a linear model invariant across environments, with an unknown noise distribution and arbitrary dependence among predictors. Roughly speaking, it assumes for all environments e ∈ E there exists Y e = X e γ + Z e , where Z e is distributed according to an unknown F (that does not depend on e) and Z e is independent of a set of true predictors X e S * indexed by set S * ⊆ {1, ..., m} (with |S * | = k). With a total of n observations from different environments, the goal of ICP is to approximate S * in a computationally efficient manner. There is a rich line of works on support recovery from an information-theoretic perspective [8,9] (and we only list the most relevant ones due to space limit). The ICP framework is different from the traditional support recovery settings in that (a) the set of predictors may not be unique (see Discussion in [1]), (b) the number of potential predictors m may not grow with number of observations n, and (c) it allows for arbitrary dependencies among features (while in support recovery settings, the measurement matrix is usually assumed to be i.i.d. [8,9]; partly because the i.i.d. codebook is optimal for the Gaussian multiple access channel when m is exponential in n). In this work, we leverage information-theoretic techniques for Gaussian point-to-point and multiple access channels to shed light on the character of ICP. This connection enables us to use power constraints to differentiate environments and apply optimal codeword constructions and decoding methods to obtain lower bounds on the error probability. To make this connection possible, we consider the simplest setting that guarantees a unique support such that it becomes reasonable to discuss the lower bound on the error probability of support recovery; at the same time we try to keep the dependency between variables as general as possible. Specifically, we assume that both Gaussian noise and k are known, and study cases when the channel gains are known or need to be estimated. We focus mainly on the zero-rate setting, where the number of potential predictors m does not grow with the number of samples n 1 . In the zero-rate setting, the optimal code is known to be the simplex code (detailed in Section 3.1), which we use to understand the impact of different environments on ICP. The analysis for the positive-rate case (when m is exponential in n) is possible by adopting Fano's inequality [10] and is discussed briefly in Section 4. BACKGROUND AND PROBLEM SETTING Methods for Invariant Causal Prediction Consider a setting in which we have different experimental conditions and let E denote the index set of all possible environmental conditions. In each e ∈ E, let the feature vector X e = (X e 1 , . . . , X e m ) ∈ R m×1 and the response Y e ∈ R form a joint distribution (X e , Y e ). For a set S ⊆ {1, . . . , m}, X e S denotes a random vector containing all variables X e i , i ∈ S. In this work, we focus on the simplest setting where there are only two environments, i.e., |E| = 2. The main assumption for ICP is that there exist a vector of coefficients γ = (γ 1 , . . . , γ m ) with support S * := {i : γ i = 0} that, for both environments, satisfies: X e has an arbitrary distribution and Y e = X e γ + Z e , Z e ∼ F and Z e ⊥ ⊥ X e S * ,(1) where the (zero mean and finite variance) noise term Z e follows the same distribution F across both environments 2 . Variables in the vector X e S * are referred to as causal predictors, and the number of causal predictors (i.e., the cardinality of the support S * ) is |S * | = k. Upon receiving n observations of (X e , Y e ) and in each environment, the goal of ICP is to recover the support S * . Two ICP methods are proposed in [1], known both here and in the original work as Method I and Method II. The idea behind these and their variants (e.g. [11]) is to iterate over all subsets of variables X S , S ∈ {1, . . . , m} and test each subset for invariance. It is shown in [1] that, with high probability, the resulting intersection of invariant sets will be a subset of S * . Generally, only the test for invariance differentiates ICP algorithms. Method I fits linear models in each environment and uses a test on regression coefficients to assert invariance. Method II fits a linear model and, for each environment, tests the mean and variance of the residuals to determine invariance. Error Exponent and Gaussian MAC Error exponent for the Gaussian channel. We briefly review some classical results on error exponent from Shannon [12] for the point-to-point Gaussian channel. Consider a Gaussian point-to-point channel in which the sender has access to a codebook C = {c 1 , c 2 , . . . , c m } for m messages, where c j ∈ R n and m is the number of codewords in C. To transmit information, the sender first chooses a codeword and then sends the i-th element of the chosen codeword at transmission time i as the input symbol X i . We assume the peak energy constraint n i=1 x i (l) 2 ≤ nP for all messages 1 ≤ l ≤ m; this constraint allows us to model different environments in a natural manner. The receiver obtains Y i = hX i + Z i , where h is the channel gain, and the Z i are each assumed to be i.i.d. N (0, σ 2 z ). After n transmissions, the receiver needs to determine which codeword in codebook C was sent. Definition 1 (Error exponent). We define error exponent as the rate of decay for the error probability of the optimal sequence of (m, n) codes. i.e., E m := lim sup n→∞ − 1 n ln P * e (m, n),(2) where P * e (m, n) denotes the best error probability over all (m, n) codes. 2 We skip the intercept term in the model for simplicity of presentation. As shown by Shannon in [12,Equation (82)], when h = 1, a lower bound on P * e (m, n) for communicating using a codebook of m codewords over a point-to-point channel is P * e (m, n) ≥ 1 2 Φ − m 4(m − 2) · nP 2 ,(3) where Φ(x) denotes the cumulative distribution function (CDF) of the standard Gaussian distribution. Accordingly, the error exponent is upper bounded by m 4(m−1) P (which follows from [12,Equation (81)] as (82) therein is a slightly loose bound). In fact, it is well-known that E m = m 4(m−1) P is optimal for zero-rate settings (which include the case of interest in this paper, i.e., when m is fixed and does not grow with n), since it can be achieved using a regular simplex code on the sphere of radius √ nP along with minimum distance decoding. The zero-rate error exponent and simplex code play a fundamental role in communication problems such as the Gaussian channel with noisy feedback [13]. To the best of our knowledge, the optimal error exponent is unknown for the Gaussian multiple access channel under the zero-rate setting. Gaussian MAC and support recovery. One variant of the Gaussian multiple access channel (MAC) can be formulated similarly, where all the k senders share the same codebook C = {c 1 , c 2 , . . . , c m }. The receiver obtains Y i = h 1 X 1,i + h 2 X 2,i + · · · + h k X k,i + Z i ,(4) where h l is the channel gain for sender l, and X l,i is the input symbol from sender l at time i. From now on, we will simply refer to this common codebook setting as the Gaussian MAC. It is well-known that this Gaussian MAC setting and the support recovery problem in linear models are equivalent (apart from unknown channel gains h l 's needing to be estimated properly [8,9]). We list the key similarities between these two and ICP as follows (see also details from [8]): (1) The k senders relate to the elements in S * or, similarly, the non-zero coefficients in γ; (2) The nonzero entries in γ can be seen as channel gains; (3) The goal of codeword recovery for a Gaussian MAC is to determine the index of the sent codeword in the codebook; similarly, the support recovery problem must determine the support S * of the coefficients γ. Connecting ICP with the Gaussian MAC Even though the three similarities mentioned above are shared between the Gaussian MAC and ICP, the two problems, unlike the support recovery and codeword recovery in a Gaussian MAC, are not equivalent. This is because (1) the notion of environment and invariance in ICP; (2) ICP is very general in its assumptions. e.g, the distributions X e i and Z e are arbitrary. Careful assumptions must be made and resulting differences carefully thought-out and accounted for in order to shed light on ICP via the Gaussian MAC. We outline several differences mainly due to the generality of ICP. 1. Codewords in a Gaussian MAC are often assumed to be independently distributed. In an ICP setting, however, dependencies between predictor variables are allowed. 2. In a Gaussian MAC, channel gains are known (e.g. from pilot sequences or feedback). In the case of ICP, the non-zero coefficients in γ are actually unknown and need to be estimated. 3. In ICP, the number of invariant causal predictors, k, is not known. This is in contrast to a Gaussian MAC where the number of senders is always known. 4. In a Gaussian MAC, the noise distribution is known to be i.i.d. N (0, σ 2 z ). In ICP, the distribution of the noise can be arbitrary. Because of the generality of ICP, S * might not be unique (see Discussion in [1]), which is why most ICP methods iterate over all subsets then intersect accepted sets. This is a sharp contrast with both Gaussian MAC and ICP. Therefore, as a first attempt to connect Gaussian MAC and ICP, we need to restrict ourselves to some tractable settings where S * is unique. We study the following class of ICP problems: (1) ICP noise is known and distributed according to N (0, σ 2 z ); (2) the non-zero coefficients in γ are known (this can be relaxed in the next section), and (3) k is known. These constraints guarantee a unique S * , enabling us to lower bound the error probability of recovering S * by leveraging information-theoretic techniques. We will mainly focus on a natural setting for ICP, the zero-rate case, when m is fixed. Then, we present analysis for the positive-rate case, when m grows exponentially with n. It is noteworthy that even these two settings are highly non-trivial and we report only partial theoretical solutions with heuristic algorithms. LOWER BOUNDS: ZERO-RATE CASE In an ICP setting, it is not natural to assume the number of predictors need grow with the sample size. The more relevant setting is the zero-rate case, where m is fixed and does not grow with n. Thus, this is the primary setting explored in this work. Additionally, from an algorithmic perspective, it would quickly become computationally infeasible to run ICP if m grows exponentially as most ICP methods would iterate over an exponentially increasing number of subsets. In order to leverage the lower bound for the Gaussian point-to-point channel in (3), we start with the setting where the channel gain is known and k = 1 (the unknown channel gain setting is covered in Section 3.2). Suppose that (1) holds with two environments (n/2 samples per environment), Z e ∼ N (0, 1), and a channel gain of one. We are now ready to present a lower bound on the error probability of recovering S * in ICP. 1 2 Φ − m 4(m − 2) · n(P + d/2) 2 .(5) Proof. First, it is sufficient to lower bound P * e (m, n) for the Gaussian point-to-point channel to obtain a lower bound in the ICP problem. This is because by definition, P * e (m, n) corresponds to optimal (over all possible (m, n) codes) error probability, which implies the best environments for support recovery in ICP. Furthermore, P * e (m, n) is computed with known noise distribution and known channel gain, which are not given in ICP. Now, without loss of generality, assume n n is even. With additional power constraints on each codeword such that, for any given codeword, n/2 i=1 x 2 i ≤ nP/2 and n i=n/2+1 x 2 i ≤ n(P + d)/2. Thus, the peak energy constraint is n i=1 x 2 i ≤ n(P + d/2), and the result then follows directly from (3). Using the bound in (5), we compare the performance of existing methods (Methods I and II) to the lower bound on the probability of error for ICP. The setup for this comparison is as follows. Each predictor in environment one follows a uniform distribution such that X e1 i ∼ U [0, √ P ]. Pre- dictors in environment two are distributed such that X e2 i ∼ U [0, √ P + d]. Sampling from distributions such that these ensures the constraints in Proposition 1 are met. Unless otherwise noted, the probability of error is estimated by averaging the results over 1000 instances for all simulated experiments. Results from the comparison can be seen in Figure 1. Given the arguments in the proof of Proposition 1, it is not surprising to see that Method I and Method II behave suboptimally. In the following section, we discuss the optimal codes and optimal procedures for recovering S * such that the lower bound in (5) is almost achieved (as [12,Equation (82)] is a slightly loose bound). Optimal Codes and Differences in Environment In this section, we discuss codes used to achieve the optimal upper and lower bounds for ICP and how these codes relate to the distance between environments. In Proposition 1, we chose to model the differences in environment as codewords having different power constraints. This, however, is not the only interpretation for an environment in a MAC setting. For example, as we show in Section 4, environment can also be viewed as a shift in the mean of each codeword. In this section, we take a general view as to the definition of difference between environments, and do so in terms of optimal and worst-case codes with the goal of understanding several general principles relating to environment. As mentioned, the bound in Proposition 1 can be achieved using a regular simplex code on the sphere of radius √ nP . In such a code, each codeword in X satisfies the total power constraint n i=1 X 2 i = nP and is the same Euclidean distance from all other codewords. For example, when m = 3, X 1 = √ nP · [0, 1, 0, · · · , 0 ] , X 2 = √ nP · [−1/2, − 3/2, 0, · · · , 0 ] , X 3 = √ nP · [1/2, − 3/2, 0, · · · , 0 ] . We refer to a codebook constructed in this way as X sim . Similarly, the worst-case code that can be constructed, assuming it satisfies the same total power constraint, is one such that all codewords are equal. That is, X i = [ √ P , √ P , · · · , √ P ] . We refer to a codebook constructed in this manner as X unif . Since Gaussian noise is symmetric, the optimal decoding scheme, referred to as the minimum distance decoding (MDD), is such that the codeword geometrically closest to the received signal is the decoded codeword [12]. Recall, that minimum distance decoding is compatible with these designed codes in the setting in question, but may not be optimal in general. We now discuss the environments of codebooks X sim and X unif . Since X sim and X unif represent best and worst-case codes, their environments must also represent the best and worst environments, respectively. Recall the first environment in the simplex codebook, relating to its first n/2 rows, contains at most m(m − 1) non-zero elements (assuming n > 2(m−1)). Note any simplex code, such as the example above for m = 3, can have have m(m − 1) non-zero elements, simply by rotating the simplex. The second environment in the simplex codebook, relating to its second n/2 rows, contains only zero elements. The portions of each codeword belonging to environment one lie on a sphere of radius √ nP while the portions of each codeword for environment two all lie at the origin. Thus, distances between the portions of the codewords belonging to environment one and that of environment two can not get any larger without first increasing the power constraint, i.e., environments in the optimal code X sim are as different as possible. In the case of X unif , since all elements of each codeword are equal, the worst-case code, X unif , has identical environments. Thus, we see that optimal codes correspond to environments that are different while worst-case codes correspond to environments that are the same. To further examine how distance between environments effects the probability of error, we form a codebook X inter whose difference in environments is larger than that of X unif but smaller than that of X sim using X inter = aX unif + (1 − a)X sim ,(6) where a ranges from 0 to 1. The value of a can be seen as a measure of how different the two environments are, where a = 1 being the least different and a = 0, the most. An example of the transition between X sim and X unif can be seen in Figure 2. When a = 0, we have the optimal case where X sim is paired with minimum distance decoding. When a = 1, we have the optimal decoding scheme but the worst code, X unif . In this case, the minimum distance decoding reverts to simply selecting, at random, any one of the codewords. Thus, for any n, the probability of error converges to m−1 m . Consequently, we come to see that no increase in sample size will improve the accuracy of ICP when data from environment one and environment two are equivalent, which is consistent with [1]. Aside from the point where two environments are identical, we find the error probability consistently drops as the sample size n increases, suggesting that, given a large enough n, one can achieve a probability of error equal to zero. In other words, given large enough n, one can always tell the minute differences in environment such that ICP can recover S * . Unknown Channel Gains Oftentimes in a Gaussian MAC setting, the channel gains are assumed to be known; while in ICP, the non-zero coefficients in γ need to be estimated. Thus, we examine two methods for ICP in which the distribution of the noise is known but the coefficients in γ are not. The first is a natural extension of minimum distance decoding where gains are estimated via ordinary least squares (OLS), while the second is a simple extension of a support recovery approach in [8]. For case in which k = 1, a natural extension to minimum distance decoding would be to estimate the one non-zero coefficient in γ using OLS for each possible S * . Then, as in minimum distance decoding, the S * that produces the estimate closest to the received signal Y e is the acceptedŜ * . The consequence of unknown coefficients in γ is seen by comparing this procedure using the optimal code X sim to the lower bound in (3) (see Figure 3). This procedure can be extended to the cases when k > 1 by examining the distance between the received signal Y e and its estimateŶ e S = i∈Sγ i X e i for all subsets S ⊆ {1, . . . , m} such that |S| = k. The estimateŜ * is the subset S belonging to theŶ e S closest to Y e . We refer to this procedure, which is outlined in Algorithm 1, as OLS+MDD. If we assume k = 1, and that the gains are known (i.e., OLS is unnecessary), OLS+MDD is reduced to minimum distance decoding. While likely not optimum for k ≥ 1, as minimum distance decoding is for k = 1, OLS+MDD may provide a good estimate of S * under unknown gains. Similarly, the optimal code for k = 1, X sim , is likely not optimal for k ≥ 2. However, we report results using X sim so as to compare results with the optimal k = 1 case. Algorithm 1 OLS+MDD Input: Received signal Y e , codebook X e , and number of causal predictors k Output:Ŝ * for every subset of variables X e S in X e such that |S| = k do Estimate non-zero coefficients in γ assuming S * = S using OLS estimates PredictŶ e S = i∈Sγ i X e i Compute R S = ||Y e −Ŷ e S || 2 end for Output: The S associated with the smallest R S Results for simulations done using Algorithm 1 and X sim can be seen in Figure 4. As might be expected for k ≥ 2, the probability of error is greater than that of k = 1. One factor contributing to this increase is that the number of subsets grows with k. For example, when k = 1 and m = 10, there are only 10 possibilities for S * . When k = 5, one must select S * from 252 possible choices. Similarly, as k grows, accurately estimating gains is challenging as it becomes more likely to find some combination of variables in X S that explain Y (including ones that are not causal predictors of Y ). As a second heuristic, for the k = 1 case, we include a slight extension to a support recovery method proposed in [8,Equations (19) and (20)] where codeword variance is simply replaced with sample variance. The procedure is as follows. For a codeword j, consider an estimateγ of γ in whicĥ γ = σ2 j −1 · 1 n ||Y || 2 − σ 2 z ,(7) whereμ j := (1/n) n i=1 X i,j is the sample mean andσ 2 j := (1/(n − 1)) n i=1 (X i,j −μ j ) 2 is the sample variance. Then, declare thatŜ * = {j} if it is the unique index such that 1 n ||Y − (−1) qγ XŜ * || 2 ≤ σ 2 z + 2σ2 j(8) for all j where q is either 1 or 2 and is fixed such that it is greater than 0. If for every j ∈ {1, . . . , k} there is none that meets the above criteria, or if there are multiple, one is picked arbitrarily. Note that an extension to this method exists in [8] that allows for k ≥ 2. However, due to the performance of the OLS+MDD approach, we choose not to include any comparisons to this method. When comparing the performance of this support recovery method, and the OLS+MDD method, we find OLS+MDD greatly outperforms the support recovery method when tested on X inter for 0 ≤ a ≤ 1 (see Figure 5). In fact, as a grows, OLS+MDD behaves as MDD in the optimal case where the gains are assumed to be known. LOWER BOUNDS: POSITIVE-RATE CASE As discussed previously, the bound in (5) refers to the zero rate case where m does not grow with n. However, for cases in which it is permissible to allow m to be large, and thus the rate to be positive, there exists several straightforward extensions derived from results in [8]. This allows us to analyze a lower bound on the error probability for k ≥ 1. In particular, we examine two cases of primary interest, when differences in environment constitute shifts in mean and variance for X e being both random and fixed. For simplicity of notation, we assume the noise variance to be 1. Now by considering a deterministic X e , the next result follows similarly and we omit the proof due to space limit. Corollary 2. In an ICP setting, suppose the set of predictors X e is a deterministic matrix in R n×m where each column X e j for j ∈ {1, . . . m} obeys the energy constraint i x 2 i,j ≤ 1. Assume that, after being generated, the second n/2 rows of X e are shifted by µ d . i.e., for i > n/2 + 1, X e i,j = x i,j + µ d . Then, for any T ⊆ {1, 2, . . . k}, the error probability P e of recovering S * can be lower bounded by, m =1 x i, ). REFERENCES Proposition 1 . 1Suppose each predictor X e i for i ∈ {1, . . . m} is a deterministic vector in R n that obeys the power constraint x 2 i ≤ nP/2 for environment 1 and x 2 i ≤ n(P +d)/2 for Fig. 1 : 1Robustness of linear ICP methods with respect to the lower bound in Proposition 1. m = 3, P = 0.1, and d = 1.environment 2. The error probability of correctly recovering S * can be lower bounded as follows, Fig. 2 : 2Error prob. using X inter with m = 3 and P = 0.1. Fig. 3 : 3Error probability using X sim compared to the lower bound in(3)where m = 3, k = 1, and P = 0.1. Fig. 4 : 4Error probability using X sim with m = 3 and P = 0.1. Fig. 5 : 5Error probability using X inter with m = 3, k = 1, P = 0.1, and n = 100. MDD refers to the optimal minimum distance decoding approach where the gains are known. Corollary 1 . 1Consider an ICP setting with noise Z ∼ N (0, 1). Suppose each predictor X e i is independently distributed and has mean 0 and variance 1 for environment one and mean µ x and variance 1 + σ 2 d for environment two. For any T ⊆ {1, 2, . . . k}, the error probability P e of recovering S * can be lower bounded by,P e ≥ |T | log m − n 4 log(a + 1)((1 + σ 2 d )a + 1) − c n /b, where a = j∈T γ 2 j , b = log m k , and c n = log k! + 1 + n log m |T | / |T |−1 q=0 (m − (k − |T |) − q) .Proof. It follows immediately from [8, Theorem 2] using Fano's inequality and we only outline the key differences from [8, Equation (41)] |T | log m ≤P e log k! + log(2πe) + 1 + n n , where n = 1 n log m |T | / |T |−1 q=0 (m − (k − |T |) − q) . P e ≥ |T | log m − n 4 log a + τ (m) + 1 − n 4 log η(µ d ) a + τ (m) + 1 − c n b,where (a, b, c n ) are defined the same as before in Causal inference by using invariant prediction: identification and confidence intervals. Jonas Peters, Peter Bühlmann, Nicolai Meinshausen, Journal of the Royal Statistical Society. Series B (Statistical Methodology). Jonas Peters, Peter Bühlmann, and Nicolai Mein- shausen. Causal inference by using invariant prediction: identification and confidence intervals. Journal of the Royal Statistical Society. Series B (Statistical Methodol- ogy), pages 947-1012, 2016. The probability approach in econometrics. Trygve Haavelmo, Econometrica: Journal of the Econometric Society. 115Trygve Haavelmo. The probability approach in econo- metrics. Econometrica: Journal of the Econometric So- ciety, pages iii-115, 1944. . John Aldrich. Autonomy. Oxford Economic Papers. 411John Aldrich. Autonomy. Oxford Economic Papers, 41(1):15-34, 1989. The logic of causal inference: Econometrics and the conditional analysis of causation. D Kevin, Hoover, Economics & Philosophy. 62Kevin D Hoover. The logic of causal inference: Econo- metrics and the conditional analysis of causation. Eco- nomics & Philosophy, 6(2):207-234, 1990. Judea Pearl. Models, reasoning and inference. CambridgeUniversityPress19Cambridge, UKJudea Pearl. Models, reasoning and inference. Cam- bridge, UK: CambridgeUniversityPress, 19, 2000. Jonas Peters. Bernhard Schölkopf, Dominik Janzing, arXiv:1206.6471Eleni Sgouritsa, Kun Zhang, and Joris Mooij. On causal and anticausal learning. arXiv preprintBernhard Schölkopf, Dominik Janzing, Jonas Pe- ters, Eleni Sgouritsa, Kun Zhang, and Joris Mooij. On causal and anticausal learning. arXiv preprint arXiv:1206.6471, 2012. Identifying the consequences of dynamic treatment strategies: A decision-theoretic overview. Philip Dawid, Vanessa Didelez, Statistics Surveys. 4A Philip Dawid and Vanessa Didelez. Identifying the consequences of dynamic treatment strategies: A decision-theoretic overview. Statistics Surveys, 4:184- 231, 2010. Limits on support recovery of sparse signals via multipleaccess communication techniques. Yuzhe Jin, Young-Han Kim, D Bhaskar, Rao, IEEE Transactions on Information Theory. 5712Yuzhe Jin, Young-Han Kim, and Bhaskar D Rao. Lim- its on support recovery of sparse signals via multiple- access communication techniques. IEEE Transactions on Information Theory, 57(12):7877-7892, 2011. Limits on support recovery with probabilistic models: An informationtheoretic framework. Jonathan Scarlett, Volkan Cevher, IEEE Transactions on Information Theory. 631Jonathan Scarlett and Volkan Cevher. Limits on support recovery with probabilistic models: An information- theoretic framework. IEEE Transactions on Information Theory, 63(1):593-620, 2016. Elements of information theory. M Thomas, Cover, John Wiley & SonsThomas M Cover. Elements of information theory. John Wiley & Sons, 1999. Invariant causal prediction for nonlinear models. Christina Heinze-Deml, Jonas Peters, Nicolai Meinshausen, Journal of Causal Inference. 62Christina Heinze-Deml, Jonas Peters, and Nicolai Mein- shausen. Invariant causal prediction for nonlinear mod- els. Journal of Causal Inference, 6(2), 2018. Probability of error for optimal codes in a gaussian channel. Claude E Shannon, Bell System Technical Journal. 383Claude E Shannon. Probability of error for optimal codes in a gaussian channel. Bell System Technical Jour- nal, 38(3):611-656, 1959. Gaussian channel with noisy feedback and peak energy constraint. Yu Xiang, Young-Han Kim, IEEE transactions on information theory. 598Yu Xiang and Young-Han Kim. Gaussian channel with noisy feedback and peak energy constraint. IEEE trans- actions on information theory, 59(8):4746-4756, 2013.
[]
[ "Probing the physics and history of cosmic reionization with the Sunyaev-Zel'dovich Effect", "Probing the physics and history of cosmic reionization with the Sunyaev-Zel'dovich Effect" ]
[ "S Colafrancesco [email protected] \nINAF -Osservatorio Astronomico di Roma\nvia Frascati 33I-00040MonteporzioItaly\n\nSchool of Physics\nUniversity of the Witwatersrand\nPrivate Bag 3, 2050-JohannesburgSouth Africa\n", "P Marchegiani \nINAF -Osservatorio Astronomico di Roma\nvia Frascati 33I-00040MonteporzioItaly\n\nSchool of Physics\nUniversity of the Witwatersrand\nPrivate Bag 3, 2050-JohannesburgSouth Africa\n", "M S Emritte \nSchool of Physics\nUniversity of the Witwatersrand\nPrivate Bag 3, 2050-JohannesburgSouth Africa\n" ]
[ "INAF -Osservatorio Astronomico di Roma\nvia Frascati 33I-00040MonteporzioItaly", "School of Physics\nUniversity of the Witwatersrand\nPrivate Bag 3, 2050-JohannesburgSouth Africa", "INAF -Osservatorio Astronomico di Roma\nvia Frascati 33I-00040MonteporzioItaly", "School of Physics\nUniversity of the Witwatersrand\nPrivate Bag 3, 2050-JohannesburgSouth Africa", "School of Physics\nUniversity of the Witwatersrand\nPrivate Bag 3, 2050-JohannesburgSouth Africa" ]
[]
Context. The evolution of the universe during the Dark Ages (DA) and the Epoch of Reonization (EoR) marks an important transition in the history of the universe but it is not yet fully understood. Aims. We study here an alternative technique to probe the DA and EoR that makes use of the Comptonization of the CMB spectrum modified by physical effects occurring during this epoch related to the emergence of the 21-cm radiation background. Inverse Compton scattering of 21-cm photon background by thermal and non-thermal electrons residing in the atmospheres of cosmic structures like galaxy clusters, radiogalaxy lobes and galaxy halos, produces a specific form of Sunyaev-Zel'dovich effect (SZE) that we refer to as SZE-21cm. Methods. We derive the SZE-21cm in a general relativistic approach which is required to describe the correct spectral features of this astrophysical effect. We calculate the spectral features of the thermal and non-thermal SZE-21cm in galaxy clusters and in radiogalaxy lobes, and their dependence on the history of physical mechanisms occurring during the DA and EoR. We study how the spectral shape of the SZE-21cm can be used to establish the global features in the mean 21-cm spectrum generated during and prior to the EoR, and how it depends on the properties of the (thermal and non-thermal) plasma in cosmic structures. Results. We find that the thermal and non-thermal SZE-21cm have peculiar spectral shapes that allow to investigate the physics and history of the EoR and DA. Its spectrum depends on the gas temperature (for the thermal SZE-21cm) and on the electrons minimum momentum (for the non-thermal SZE-21cm). The global SZE-21cm signal can be detected (in ∼ 1000 hrs) by SKA1low in the frequency range ν ∼ > 75 − 90 MHz, for clusters in the temperature range 5 to 20 keV, and the difference between the SZE-21cm and the standard SZE can be detected by SKA1 or SKA2 at frequencies depending on the background model and the cluster temperature.Conclusions. We have shown that the detection of the SZE-21cm can provide unique information on the DA and EoR, and on the cosmic structures that produce the scattering; the frequencies at which the SZE-21cm shows its main spectral features will indicate the epoch at which the physical processes related to the cosmological 21-cm signal occurred and shed light on the cosmic history during the DA and EoR by using local, well-known cosmic structures like galaxy clusters and radio galaxies.
10.1051/0004-6361/201424904
[ "https://arxiv.org/pdf/1607.07723v1.pdf" ]
55,742,347
1607.07723
279ffd8e24914defe2fb31a3d71a218fc002c5f9
Probing the physics and history of cosmic reionization with the Sunyaev-Zel'dovich Effect 26 Jul 2016 March 26, 2018 S Colafrancesco [email protected] INAF -Osservatorio Astronomico di Roma via Frascati 33I-00040MonteporzioItaly School of Physics University of the Witwatersrand Private Bag 3, 2050-JohannesburgSouth Africa P Marchegiani INAF -Osservatorio Astronomico di Roma via Frascati 33I-00040MonteporzioItaly School of Physics University of the Witwatersrand Private Bag 3, 2050-JohannesburgSouth Africa M S Emritte School of Physics University of the Witwatersrand Private Bag 3, 2050-JohannesburgSouth Africa Probing the physics and history of cosmic reionization with the Sunyaev-Zel'dovich Effect 26 Jul 2016 March 26, 2018Received / AcceptedAstronomy & Astrophysics manuscript no. sz21cm˙accepted c ESO 2018Cosmology: cosmic microwave background; Galaxies: clusters: theory Context. The evolution of the universe during the Dark Ages (DA) and the Epoch of Reonization (EoR) marks an important transition in the history of the universe but it is not yet fully understood. Aims. We study here an alternative technique to probe the DA and EoR that makes use of the Comptonization of the CMB spectrum modified by physical effects occurring during this epoch related to the emergence of the 21-cm radiation background. Inverse Compton scattering of 21-cm photon background by thermal and non-thermal electrons residing in the atmospheres of cosmic structures like galaxy clusters, radiogalaxy lobes and galaxy halos, produces a specific form of Sunyaev-Zel'dovich effect (SZE) that we refer to as SZE-21cm. Methods. We derive the SZE-21cm in a general relativistic approach which is required to describe the correct spectral features of this astrophysical effect. We calculate the spectral features of the thermal and non-thermal SZE-21cm in galaxy clusters and in radiogalaxy lobes, and their dependence on the history of physical mechanisms occurring during the DA and EoR. We study how the spectral shape of the SZE-21cm can be used to establish the global features in the mean 21-cm spectrum generated during and prior to the EoR, and how it depends on the properties of the (thermal and non-thermal) plasma in cosmic structures. Results. We find that the thermal and non-thermal SZE-21cm have peculiar spectral shapes that allow to investigate the physics and history of the EoR and DA. Its spectrum depends on the gas temperature (for the thermal SZE-21cm) and on the electrons minimum momentum (for the non-thermal SZE-21cm). The global SZE-21cm signal can be detected (in ∼ 1000 hrs) by SKA1low in the frequency range ν ∼ > 75 − 90 MHz, for clusters in the temperature range 5 to 20 keV, and the difference between the SZE-21cm and the standard SZE can be detected by SKA1 or SKA2 at frequencies depending on the background model and the cluster temperature.Conclusions. We have shown that the detection of the SZE-21cm can provide unique information on the DA and EoR, and on the cosmic structures that produce the scattering; the frequencies at which the SZE-21cm shows its main spectral features will indicate the epoch at which the physical processes related to the cosmological 21-cm signal occurred and shed light on the cosmic history during the DA and EoR by using local, well-known cosmic structures like galaxy clusters and radio galaxies. Introduction Departures of the Cosmic Microwave Background (CMB) frequency spectrum from a pure blackbody encode information about the thermal history of the early universe before the epoch of recombination when it emerged from the last scattering surface. The evolution of the universe after this epoch proceeds through the period of the Dark Ages (DA) that ends ∼ 400 million years later, when the first galaxies formed and start emitting ionizing radiation. Send offprint requests to: S. Colafrancesco The transition period at the end of the DA marks the Epoch of Reionization (EoR). During this epoch, radiation from the very first luminous sources (e.g., early stars, galaxies, and quasars) succeeded in ionizing the neutral hydrogen gas that had filled the universe since the recombination event (see, e.g., Barkana and Loeb 2001, Bromm and Larson 2004, Ciardi and Ferrara 2005, Choudhury and Ferrara 2006, Furlanetto et al. 2006, Morales and Wyithe 2010. The current constraints suggest that the EoR roughly occurs within the redshift range of z ≈ 6 − 20. This cosmic period is not yet completely understood and various astrophysical probes have been suggested to shed light on this epoch for early structure formation (see Zaroubi 2013 for a review). Information from the DA period is not explicitly contained in the CMB because baryonic matter and radiation have already decoupled, and the bulk of baryonic matter in the universe during this period is in the form of neutral hydrogen gas in the inter galactic medium (IGM). Rather than target observations at the first galaxies and quasars that are the rare, early products of gravitational collapse, it is then necessary to detect directly the presence of the ubiquitous hydrogen gas. One of the methods of achieving this detection is to search for signatures of the (highly redshifted) 21-cm hyperfine transition line of neutral hydrogen (see, e.g., Loeb & Zaldarriaga 2004;Cooray 2004;Bharadwaj & Ali 2004;Carilli et al. 2004;Furlanetto & Briggs 2004;Furlanetto et al. 2006;Pritchard & Loeb 2010Liu et al. 2013). The 21-cm signal from the DA would appear as a faint, diffuse background detectable at frequencies below 200 MHz (for redshifts z > 6). Thus, measuring the brightness temperature of the redshifted 21-cm background could yield information about both the global and local properties of the IGM. Determining the average brightness temperature over a large solid angle as a function of redshift would eliminate any dependence on local density perturbations and constrain the history of the neutral fraction of hydrogen in the IGM. It has been noted that there are several problems related to the observation of the 21-cm background. Firstly, this signal is faint, of the order of tens of mK relative to the CMB (see, e.g., Furlanetto et al. 2006), and until now only upper limits have been obtained (see, e.g., Paciga et al. 2013, Dillon et al. 2014, Parsons et al. 2014. The second problem is related to the presence of galactic and extragalactic foregrounds whose amplitude can be also about four order of magnitude larger than this signal (see, e.g., de Oliveira-Costa et al. 2008). These problems make difficult to study this signal with the present-day and new generation of radio interferometers, since they are not sensitive to the mean signal, but only to its inhomogeneity, and thus require a very precise calibration and knowledge of foregrounds to remove their contribution (see, e.g., discussion in Furlanetto et al. 2006). Various methods have been proposed to overcome these problems. One possibility is to study the 21-cm fluctuations to measure the mean background through their redshift-space anisotropies (Barkana & Loeb 2005a); this method can be used with the next generation instruments like the Square Kilometer Array (SKA) (see, e.g., McQuinn et al. 2006). A second method is to measure the contrast between the 21-cm signal and the bubbles of ionized plasma present during the EoR, and use their contrast to measure the mean amount of neutral gas (see, e.g., Furlanetto et al. 2006 and references therein). An alternative method that we want to discuss extensively in this paper is to use the SZE-21cm, i.e. the spectral distortion of the CMB spectrum modified by physical effects occurring during the epoch related to the emergence of the 21-cm radiation background, induced by in-verse Compton scattering off the intervening electrons in the atmospheres of various cosmic structures, like galaxy clusters, radiogalaxy lobes and galactic halos. A preliminary attempt to calculate the SZE-21cm has been presented by Cooray (2006). This calculation turns out to be not adequate for a correct description of the SZE-21cm for two reasons: i) the photon background model used for the modification to the CMB caused by mechanisms working during the DA and EoR is unphysical, because it contains a number of artificial discontinuities, under-resolves the main features of interest at ν ∼ 70 MHz and contains an unphysical reionization history that produces substantial 21-cm signal down to redshifts z < 2 (i.e., at frequencies > 300 MHz); ii) it is performed in the non-relativistic approximation of the Compton scattering process of CMB photons in the hot intra-cluster medium of galaxy clusters thus neglecting any effect induced by the relativistic corrections to this scattering, by multiple scattering effects and by the scattering of additional non-thermal electrons in clusters, as explicitly reported by Cooray (2006). Such problems in the Cooray (2006) calculations lead to an incorrect description of the SZE-21cm that has important consequences in using this cosmological probe. In fact, to take full advantage of the SZE-21cm study, it is necessary to use a full relativistic formalism, its generalization to any order of magnitude in the plasma optical depth τ and the possibility to include also the combination of various electron populations residing in cosmic structures (see, e.g., Colafrancesco et al. 2003). It is also necessary to use a wider and more physically motivated set of models for the 21-cm background, including also other physical processes that can change this background, such as the effect of Dark Matter heating. Finally, it is worth considering the effect of changing the redshifts at which the different physical processes took place. In this paper we perform such a more complete study following the previous lines of investigation. First, to describe the CMB spectrum modified by the 21cm cosmological background, we use the results of the 21cmFAST code (Mesinger, Furlanetto & Cen 2011) that include realistic physical effects and also additional mechanisms, such as the heating induced by Dark Matter annihilation (e.g., Valdes et al. 2013;Evoli et al. 2014). Secondly, we perform the calculations in the full relativistic formalism for the derivation of the SZE (see, e.g., Colafrancesco et al. 2003 for details), that is suitable to calculate the SZE-21cm in detail, and to derive the precise information about its spectral properties over a wide frequency range and in a wide set of cosmic structures. This general treatment allows, therefore, to increase both the number and the redshift distribution of objects that can be studied with this method, including galaxy clusters with high temperatures (which are the best targets for maximizing the SZE-21cm signal and are more subject to relativistic effects), with radio halos, cool-cores and other complex morphologies, as well as other extragalac-tic sources with non-thermal electron distributions such as radio galaxies lobes. The plan of the paper is the following: in Sect 2 we present the general, full relativistic derivation of the SZE-21cm and the models for the frequency distribution of the global 21-cm background we use in the paper. These are new crucial elements of the derivation of the SZE-21cm that have never been provided up to date. In Sect. 3 we discuss the results of our calculations for various scenarios of the radiation background emerging from the DA and EoR, considering various astrophysically motivated scenarios. We also discuss here, for the first time, the derivation and the possibility to observe both the thermal and the non-thermal SZE-21cm. We discuss our results in the light of the future radio interferometric experiments like the SKA in Sect.4, and we summarize our conclusions in Sect.5. Throughout the paper, we use a flat, vacuumdominated cosmological model with Ω m = 0.315, Ω Λ = 0.685 and H 0 = 67.3 km s −1 Mpc −1 . Derivation of the SZE-21cm General derivation of the SZE for a modified CMB spectrum The spectral distortion due to the SZE of the CMB is given in the general form by I(x) = +∞ −∞ I 0 (xe −s )P (s)ds(1) (see Colafrancesco et al. 2003 for a general derivation of the SZE), where x = hν/(kT 0 ) is the normalized photon frequency, T 0 is the CMB temperature, P (s) is the photon redistribution function (yielding the probability of a logarithmic shift s = ln(ν ′ /ν) in the photon frequency due to the inverse Compton scattering process) that depends on the electron spectrum producing the CMB Comptonization, and I 0 (x) is the specific intensity of the incident CMB radiation field. The redistribution function P (s), that contains the relativistic corrections required to describe correctly the Compton scattering produced by high temperature or relativistic electrons, is given by the sum of the probability functions to have n scatterings, P n (s), weighted by the corresponding Poissonian probability: P (s) = +∞ n=0 e −τ τ n n! P n (s),(2) where the optical depth is given by the integral along the line of sight ℓ of the electron density τ = σ T n e dℓ ,(3) where n e is the plasma electron density. Each function P n (s) is given by the convolution product of n single scattering probability functions P 1 (s): P n (s) = P 1 (s) ⊗ . . . ⊗ P 1 (s) n times ,(4) where P 1 (s) = ∞ 0 f e (p)P s (s, p)dp,(5) and where f e (p) is the electron momentum distribution function (normalized as to have ∞ 0 f e (p)dp = 1), and P s (s, p) is the function that gives the probability to have a frequency shift s by an electron with adimensional momentum p = βγ, and is given by the physics of the inverse Compton scattering process (see, e.g., Enßlin & Kaiser 2000, Colafrancesco et al. 2003. The function P (s) that we use in our approach can be calculated at the desired approximation order in the plasma optical depth τ or via a general relativistic method by using Fourier transform properties (see Colafrancesco et al. 2003 for details), at variance with the case discussed in Cooray (2006) that is only a non-relativistic approximation for values τ ≪ 1. Once the Comptonized spectrum given by eq.(1) is calculated, the general form of the SZE is given by the difference: ∆I(x) = I(x) − I 0 (x).(6) For the incoming radiation spectrum I 0 (x) it is possible, in our general derivation, to use any radiation field. In the original derivation of the SZE the incoming spectrum is given by the standard CMB spectrum I 0,st (x) = 2 (kT 0 ) 3 (hc) 2 x 3 e x − 1 ,(7) that, inserted in eq.(1) and using eq.(6), allows to obtain the standard SZE ∆I st (x). Our general derivation allows to use the CMB spectrum modified by other physical effects, such as the possible effect of the photon decay (Colafrancesco & Marchegiani 2014), the effect of non-planckian deviation of the CMB due to the effect of the plasma frequency in an ionized medium (Colafrancesco, Emritte & Marchegiani 2015), or -as we study in this paper -by the modifications of the CMB provided by mechanisms yielding the 21-cm radiation field. For the case of the CMB spectrum modified by the effects during the DA and EoR, the expression of the CMB, written as a function of the frequency ν, is given by I 0,mod (ν) = I 0,st (ν) + δI(ν),(8) where the modification to the CMB spectrum, δI(ν), can be expressed in terms of brigthness temperature change relative to the CMB, defined as: δT (ν) = c 2 2kν 2 δI(ν).(9) In the next Sect. 2.2 we discuss how to obtain the function δI(ν). Using eqs. (1) and (6), the SZE-21cm reads: ∆I mod (ν) = I mod (ν) − I 0,mod (ν) .(10) In the following, we will express the SZE using the brightness temperature change relative to the CMB: ∆T (ν) = c 2 2kν 2 ∆I(ν),(11) that is valid for both the standard, ∆T st (ν) and the SZE-21cm, ∆T mod (ν). The CMB spectrum modified during the DA and EoR The CMB radiation spectrum is modified during the DA and EoR by various physical mechanisms: subsequent to recombination, the temperature of neutral gas is coupled to that of the CMB, and no changes in the CMB spectrum can be observed. At redshifts below 200 the gas cools adiabatically, its temperature drops below that of the CMB, and neutral hydrogen resonantly absorbs CMB photons through the spin-flip transition (Field 1959, Scott and Rees 1990, Loeb and Zaldarriaga 2004. Heating effects of the neutral gas may also occur at high redshifts. As the first Dark Matter (DM) clumps form in the early Universe, the DM WIMP annihilation can in fact produce a substantial heating of the surrounding IGM (Valdes et al. 2013). At much lower redshifts, gas temperature is also expected to heat up again the IGM as luminous sources turn on and their UV and soft X-ray photons re-ionize and heat the gas (Chen and Miralda-Escude 2004). An additional spectral signature is also expected from the Ly-α radiation field produced by first sources (Barkana and Loeb 2005b) that is coupled to the CMB spectrum through the Wouthuysen-Field effect (Wouthuysen 1952, Field 1959, producing a suppression of the radiation field (see Furlanetto et al. 2006 for details). As a result of these physical mechanisms, the CMB spectrum is modified depending on the redshift at which these mechanisms take place. The spectral shape of the brightness temperature change relative to the CMB (see eq. 9) is shown in Fig.1, where the background radiation models are calculated with numerical simulations performed using the 21cmFAST code (Evoli, private communication) for different assumptions on the physical processes occurring during the EoR, and without and with DM annihilation effects. The first one (solid line) is a fiducial model without Dark Matter, with standard assumptions on the properties of heating by cosmic structures (see Valdes et al. 2013 andEvoli et al. 2014 for details), without considering the effect of gas collisions which can be observed at frequencies ν < 30 MHz, and therefore can not be detected with a ground-based telescope like SKA. This fiducial model takes into account the effects of the Ly-α radiation field at z ∼ 30 − 20, and the effects of UV ionization and X-ray photon heating at z ∼ 20 − 6. We use this modified CMB radiation field scheme as a benchmark case for the sake of a general discussion of the SZE-21cm. A second model without Dark Matter that we consider here assumes extreme values for the heating by cosmic structures and, as a result, the deep brightness decrease caused by the coupling of the spin temperature of the IGM with the Ly-α photons is damped, while the emission at higher frequencies is amplified. We finally consider two models with the fiducial parameters and with the heating effects produced by Dark Matter annihilation (Valdes et al. 2013): in these models, the strongest effect is produced by small mass Dark Matter halos, so we consider a model with minimum halo mass M min = 10 −3 M ⊙ , and one with M min = 10 −6 M ⊙ , which is more effective in damping the Ly-α coupling effect. The Dark Matter model used here is a WIMP with mass of 10 GeV and annihilation channel e + /e − with cross-section < σV >= 10 −26 cm 3 /s. The SZE-21cm spectrum in the benchmark background radiation model The modified CMB spectrum (eqs. 8 and 9, where δT is shown in Fig.1) is then scattered by electrons (of both thermal and non-thermal nature) residing in the atmospheres of various cosmic structures, like galaxy clusters, radiogalaxy lobes and galactic halos, and the SZE-21cm is produced. In Figure 2 we show an example of the SZE-21cm, ∆T mod , calculated in a galaxy cluster with a temperature of kT = 7 keV, and using the benchmark model for the modified CMB spectrum shown in Fig.1. Figure 2 shows that in some frequency bands the SZE-21cm is stronger than the standard one, whereas in other bands it is weaker. This behaviour is mainly related to the curvature of the input spectrum δT (see Fig.1): in the frequency range where the input spectrum has a negative curvature (for ν ∼ < 55 MHz and 90 ∼ < ν ∼ < 140 MHz for our fiducial model), the SZE-21cm is smaller than the standard one (i.e. ∆T mod − ∆T st < 0), while at frequencies where the curvature is positive (55 ∼ < ν ∼ < 90 MHz and ν ∼ > 140 MHz) we have ∆T mod − ∆T st > 0. This is due to the fact that the inverse Compton scattering produces a shift in the frequency of photons and, as a consequence, the amplitude of the SZE at a certain frequency depends on the distribution of the photons around that frequency (see, e.g., the shape of the function P 1 (s) defined in eq. 5 in Colafrancesco et al. 2003). As a result, at the frequency where the curvature of the input spectrum is negative, a smaller number of photons are present around that frequency with respect to the case of the standard CMB spectrum (where the spectral curvature, in brightness temperature units, is zero), and the resulting SZE-21cm is smaller than the standard one; on the other hand, where the curvature is positive a larger number of photons is present and the SZE-21cm is higher than the standard one. We also find that the minimum point in the input radiation spectrum (ν ∼ 70 MHz) corresponds to a maximum point in the SZE-21cm; this is due to the fact that a minimum point in the input spectrum means a smaller number of photons with respect to the standard CMB: as a consequence, when subtracting the input spectrum to calculate the SZE-21cm (see eq.10), the resulting emission is stronger than for the standard SZE. The opposite behaviour is observed at the frequencies where the input radiation spectrum has its maximum points (ν ∼ 45 and 120 MHz), that are close to the minimum points of the SZE-21cm; in this case, the correspondence is less precise with respect to the previous case because the maximum points in the input spectrum are less sharped than the minimum one, and the convolution of photons with those at surrounding frequencies produces a slight shift in the frequency of the minimum points in the SZE-21cm. In the following we discuss more detailed and new results obtained for the specific case of the SZE-21cm produced by i) thermal electron populations, that provide the dominant contribution to the SZE observed in galaxy clusters, and by ii) non-thermal electrons populations, that are present in clusters that show non-thermal activity (i.e. radio halos or relics) and in the extended lobes of radiogalaxies. This can be done by using the corresponding functions f e (p) in eq.(5), i.e. a maxwellian distribution for a thermal population and a power-law distribution for a nonthermal population. A specific analysis on the relevance of relativistic effects in the SZE-21cm is also presented. The SZE-21cm: detailed spectral analysis In the following we discuss first our results obtained for the benchmark modified background radiation scenario (solid line in Fig. 1), and then for the set of other modified radiation background models shown in Fig.1. We start our discussion, for the sake of clarity, by showing the spectral shape of the standard SZE, ∆T st , for the unmodified CMB spectrum. Figure 3 shows the standard SZE, not modified by the 21-cm line radiation field, in units of brigthness temperature relative to the CMB for the case of a galaxy cluster with thermal plasma at temperature kT = 5 keV and with optical depth τ = 5×10 −3 , and for the case of a non-thermal plasma with a single power-law spectrum N (p) ∼ p −s for p ≥ p 1 , with s = 3.5, p 1 = 10 and τ = 1 × 10 −4 . We notice that the standard SZE is a constant line in units of CMB brightness temperature in the Rayleigh-Jeans (RJ) regime (hν ≪ kT CMB ) for both the case of a thermal SZE and for the case of a non-thermal, relativistic plasma typical of the radiogalaxy lobes (we assume here a steep spectrum S ν ∝ ν −αR with α R = (s − 1)/2 = 1.25) The thermal SZE-21cm, ∆T mod , is shown in Fig. 4 for the case of thermal plasma in galaxy clusters for four different electron temperatures of 5, 10, 15 and 20 keV. We find that the spectral shape of the thermal SZE-21cm changes for different electron temperatures, consistently with the effects of relativistic corrections that are fully considered in our approach, while its amplitude increases with the cluster temperature, which is consistent with the notion that the SZE amplitude increases with increasing the cluster Compton parameter y = σT mec 2 dℓP e , that reads y ∝ kT · τ for the case of a thermal intracluster medium (Colafrancesco et al. 2003). We verified the level of the error done when using a non-relativistic approach to calculate the SZE-21cm w.r.t. our full relativistic approach. In Fig.5 we show the percentage difference between the results of the two calculations for clusters with electron temperatures of 20, 15 and 7 keV calculated with the relativistic and the nonrelativistic approaches (as in the case of Cooray 2006). We find that the percentage difference is different from 0 (i.e., the case in which the non-relativistic calculation gives the same result than the relativistic one) at almost all frequencies. As discussed in details in the Appendix, we also note that the percentage difference has local maxima (in absolute value) in correspondence of the points where the second derivative of the input spectrum has its maxima and minima, i.e. at ν ∼ 50, 60, 77 and 95 MHz (see lower panel in Fig.A.1). This is related, as discussed for the shape of the SZE-21cm, to the fact that the SZE is produced by a convolution of the input spectrum photon distribution with photons at surrounding frequencies. The non-relativistic calculation considers a shape of the function P (s) which is narrower than the one in the relativistically correct calculation (see, e.g., Birkinshaw 1999, Colafrancesco et al. 2003. Therefore, when the curvature (positive or negative) of the input radiation spectrum is maximum, the error done by convolving the input spectrum with a function P (s) narrower than the correct one is larger, because it implies to lose the contribution from the photons with farther frequencies. As a consequence, the more the input spectrum is different from a straight line, the larger is the error done by using the non-relativistic calculation. In the Appendix we expand these considerations by discussing also the other three input models considered for the input radiation spectrum we use in this paper. For the case of a cluster with a temperature of 20 keV, the percentage difference reaches at its local maxima/minima values of the order of ≈ 65%, ≈ 60%, ≈ 100% and ≈ 50% at frequencies ν ≈ 50, 60, 77, 95 MHz, respectively, which introduce therefore substantial modifications in the value of the SZE-21cm calculated in the non-relativistic approach. For the other temperatures, the percentage error is smaller, but still of the order of at least 30% at the previous frequencies, and at ∼ 77 MHz the percentage error is ∼ 100% independently on the cluster temperature. For this reason we conclude that in order to perform a correct study of the SZE-21cm it is mandatory to use the full relativistic formalism as described in our paper. In Figure 6 we show the difference between the value of ∆T for the thermal SZE-21cm and the standard thermal SZE on the unmodified CMB (note that this last SZE is a constant value for all frequencies in the considered range, as shown in Fig.3) in order to highlight the spectral difference between the two effects and between the thermal effects calculated for different electron temperatures. We notice that the main differences appear around 50 MHz and in the range ≈ 60 − 80 MHz (reflecting the Lyα spin coupling effect), and in the range 100−150 MHz (reflecting the UV ionization effect during the EoR). To investigate the non-thermal SZE-21cm effect produced by non-thermal (or relativistic) electrons residing, e.g., in the radiogalaxy lobes or in cluster radio halos/relics, we consider an electron population with a single power-law spectrum with index s = 3.5 and various values of the minimum electron momentum p 1 . Figure 7 shows the non-thermal SZE-21cm for values p 1 = 0.1, 1, 5 and 10. The non-thermal SZE-21cm has an amplitude that increases (in modulus) with increasing values of p 1 , for a constant value of τ . We show in Fig. 8 the difference between the non-thermal SZE-21cm and the standard non-thermal SZE where the CMB spectrum is not modified. The largest differences of the non-thermal SZE-21cm with respect to the standard one take place at frequencies similar to the thermal case, and the differences with the thermal case are more important for high values of p 1 , i.e. when the scattering electrons are more energetic. We also check how the shape of the SZE-21cm depends on the frequency of the modifications to the overall radiation field, that depends on the assumed redshift range in which the various mechanisms operating during the DA and EoR act to modify the original CMB spectrum. To this purpose, for an illustrative description of the possible redshift-dependence of the overall modified background model, we show the frequency shape of the resulting SZE-21cm when the redshift of the input modified radiation field is varied. To this aim, we use a typical galaxy clus- ter with a thermal electron plasma at a temperature of 7 keV and optical depth τ = 5 × 10 −3 , and we compare the total SZE-21cm as previously discussed with the one in which the background spectrum is shifted globally in frequency by a factor 3 (see Fig. 9). With this illustrative example, we are considering the possibility that the redshifts at which the various phenomena (e.g., collisions, Ly-α interactions, UV ionization) can be different from the ones assumed in the benchmark model. Thus, from the frequency at which the different effects in the SZE-21 cm are observed, it is possible to derive the redshift at which these effects took place, and in principle determine the full cosmic history of the DA and EoR. By using the other models of the modified radiation background described in Sect. 2.2, we obtained the results shown in Figure 10, where the thermal SZE-21cm spectrum for clusters with 5 and 20 keV is plotted, and in Figure 11, where instead the non-thermal SZE-21cm with s = 3.5 and p 1 = 0.1 and 10 is plotted. As we can see, while the spectral shape of the non-thermal SZE-21 cm is very similar to the thermal one for p 1 = 0.1, for high values of p 1 the main difference is the damping of the features produced by the Ly-α spin coupling effect at ∼ 60 and 100 MHz. The effect of considering a higher heating rate, both from usual astrophysical sources and from DM, is to increase the temperature of the IGM, to which the spin temperature is linked by the Ly-α coupling, and as a result the peak in the SZE-21cm in the 60-80 MHz frequency range is damped, with different spectral shapes depending on the Dark Matter properties. These results therefore show that the SZE-21cm can be also considered as a tool to probe both the amount of DM in the universe and the minimal mass of DM halos collapsed at early epochs. The DM abundance can be probed using the amplitude and the spectral shape of the SZE-21cm in two best frequency ranges: around ∼ 50 MHz and at ≈ 60 − 90 MHz, where the sensitivity to the DM density is higher. The sensitivity to M min for the DM halos is best achievable at ν ≈ 50 − 70 MHz where the effect of M min increases the amplitude of the SZE-21cm and shifts its maximum in frequency. Discussion In the full relativistic description of the SZE-21cm we found that the following properties are important for the correct use of this technique: i) The scattering properties of high-energy electrons need a full relativistic treatment: avoiding this will generate percentage differences up to about 100 % at the relevant frequencies where this effect can be observed. This is ensured in our approach through a self-consistent computation of the SZE-21cm. ii) We find that the amplitude of the SZE-21cm and its variations w.r.t. the standard SZE (using the non-modified CMB spectrum) are larger for clusters with high temperature (see Fig.6) and for non-thermal electron plasmas with high values of the minimum momentum p 1 (see Fig.8), i.e. when the high-energy electrons are more important. iii) Studying the detailed spectrum of the SZE-21cm allows to derive precise information on the epochs at which the CMB has been modified and on the physical mechanism that produced such modifications during the DA and EoR (see Fig.9). iv) The thermal and non-thermal SZE-21cm have peculiar spectral shapes (see Figs. 6,[8][9][10][11]. Thus, it is possible, in principle, to derive information also on the existence and the properties of the electron population in cosmic structures also from very low-ν observations of the SZE. We note that this property is complementary with the results of previous studies, in accordance with which the properties of non-thermal electrons can be derived from the study of the SZE at high frequencies (see, e.g., Colafrancesco, Marchegiani & Buonanno 2011 for the case of the Bullet Cluster). Differential analysis technique and foreground contamination Observations of the SZE-21cm can be carried out with radio interferometers since the modification associated with low-redshift scattering can be established from differential observations towards and away from galaxy clusters and other cosmic structures containing diffuse thermal and non-thermal plasmas. Unlike an experiment to directly establish the cosmic 21-cm frequency spectrum at low radio frequencies involving a total intensity measurement of the sky, the differential observations with a radio interferometer are less affected by issues such as the exact calibration of the observed intensity using an external source, and the confusion from galactic foregrounds that are uniform over angular scales larger than a typical cluster, such as the Galactic synchrotron background at low radio frequencies. Also, since the SZE does not depend on redshift, it is more suitable to study sources located at large distances, allowing to reduce the importance of the cluster radio emissions (both diffuse and point-like sources) with respect to the SZE, and allowing to detect a larger number of sources, thus increasing the possibility to obtain more precise results by studying this effect in many sources at cosmological scales. The resulting modification to the 21-cm spectrum due to the thermal SZE-21cm is expected at the level of a few tenths mK brightness temperature relative to the CMB. Therefore, such a small modification challenges an easy detection, but for upcoming radio interferometers (like the SKA), the specific spectral signatures would allow to produce a relatively clean detection. In addition, multiobject SZE-21cm observations could be facilitated by the fact that the instantaneous field-of-view of upcoming interferometers is expected to be more than 100 square degrees and one expects to detect simultaneously hundreds, or more, massive clusters in such wide fields. Therefore, the SZE-21cm effect can be effectively used to establish the global features in the mean 21-cm spectrum generated during and prior to the EoR. We note that it is also possible to produce cluster population studies with the SZE-21cm (e.g., cluster counts and redshift distribution) and use them as cosmological probes. These goals make desirable to build a technique allowing to study a large number of objects (including galaxy clusters in merging and relaxed states, radio halo and cooling flow clusters, radio galaxy lobes), and to study objects at high redshift. Even if the differential measurements of the SZE-21cm avoid contamination from foreground/background emissions on scales larger than the cluster/radiogalaxy size, another possible source of contamination is the synchrotron radio emission within galaxy clusters and radio galaxies lobes. This contamination should decrease for objects at large distances, because the synchrotron emission varies with the luminosity distance as D −2 L , whereas the SZE does not vary with the distance of the source. For nearby objects, the synchrotron emission at low frequencies can be much stronger than the SZE. In Fig. 12 we show a com-parison between two cases of the SZE-21 cm (for thermal plasma with temperature of 5 and 20 keV and optical depth τ = 5 × 10 −3 ), a spectrum similar to that of the Coma radio halo (approximated as a perfect power law), and a spectrum of a Coma-like cluster located at z = 1. We note the at all frequencies we are interested, the synchrotron emission is much larger than the SZE for a nearby cluster like Coma; so, it is necessary to study the cluster radio halo spectrum to separate the two contributions. At higher-z, however, the radio halo flux decreases rapidly while the SZE-21cm remains unchanged thus providing a lower level of contamination and an easier subtraction procedure. Another possible source of contamination is given by the point radio sources in galaxy clusters; in this case, the goal to separate this contribution from the SZE-21 cm signal is easier, since it is possible to use both the spectral information we have at other frequencies and the spatial information, in order to remove the contribution from point sources. Fig. 12. The SZE-21cm (in units of Brightness Temperature relative to the CMB and in absolute value) for a thermal plasma with temperature kT = 20 keV (solid line) and 5 keV (dashed line), and with τ = 5 × 10 −3 , compared with a spectrum similar to that of Coma radio halo (long-dashed line), and with the same spectrum for a Coma-like cluster located at z = 1. Detectability with SKA We discuss now the detectability of the SZE-21 cm with the SKA1-low instrument. We extracted the performance of SKA1-low from the SKA1 System Baseline Design document (see Dewdney et al. 2012). First of all, we calculate the loss of signal at small angular radii produced by the finite extension of the interferometer. For this purpose, we calculate the SZE flux from an isothermal cluster with a gas density profile given by a β-profile: n e (r) = n e,0 1 + r r c 2 − 3 2 β (12) (Cavaliere & Fusco-Femiano 1976). For such a cluster, the optical depth at a projected distance θ from the center of the cluster is given by the expression: (Colafrancesco et al. 2003), where θ c = r c /D A and D A is the angular diameter distance of the cluster. We assume τ 0 = 5 × 10 −3 , β = 0.75, θ c = 300 arcsec and calculate the flux up to an angular size θ max = 10θ c . The reference spatial resolution of SKA1-low at 110 MHz, corresponding to a minimum baseline of 50 km, is θ min ∼ 11 arcsec. Since at first order in τ the SZE-21cm is proportional to the product of the SZE spectral function and of the cluster optical depth (see, e.g., Colafrancesco et al. 2003), we can estimate that the lack of sensitivity for angular scales θ < θ min is given by the ratio between the optical depth integrated in this small θ range and the total one, and it implies a signal loss of the order of τ (θ) = τ 0 1 + θ θ c 2 1 2 − 3 2 β(13)θmin 0 2πθτ (θ)dθ θmax 0 2πθτ (θ)dθ ∼ 1.1 × 10 −4 .(14) To have an idea about the intensity of the signal we should expect, we plot in Fig. 13 the surface brightness profiles of the standard SZE at the frequency of 110 MHz for the optical depth profiles in the eq.(13), with the same parameters values described above, and for the temperatures of 20, 15, 10, and 5 keV. Therefore, in the inner part (e.g., within a radius of ∼ 20 arcmin) of a galaxy cluster with high temperature we can estimate a SZE signal of the order of ∼ 10 µJy and, as a consequence, the loss of signal due to the finite baseline configuration of the SKA1 is of the order of ∼ nJy, and therefore does not affect our results. To study the detectability of the SZE-21 cm signal, we compare the flux calculated for the modified CMB spectrum, ∆I mod , and the one calculated for the non-modified CMB spectrum, ∆I st , with the sensitivities of SKA-50%, SKA1, and SKA2 for 100 kHz bandwith, 1000 hrs of integration, 2 polarizations, no taper, no weight. We show the result in Figures 14-17 for the different radiation background models we use in this paper. For our benchmark model, the SZE-21cm can be detected with SKA1-low with 1000 hrs integration time at frequencies ν ∼ > 75 MHz for clusters with very high temperature (kT = 20 keV) and at ν ∼ > 90 MHz for low temperature clusters (kT = 5 keV). With SKA-50% the SZE-21cm can be detected at higher frequencies (85 and 100 MHz for hot and cold clusters respectively), and with SKA2 it can be detected at small frequencies (50 and 80 MHz), giving the possibility to study the EoR and the Dark Ages until very high redshift (z ∼ 30). The possibility to discriminate between the SZE-21 cm and the standard SZE signals is more challenging: the difference between the two signals is always at most of the order of few µJy (see lower panel in Figure 14), so it requires to measure the signal with high precision, and at frequencies where the differences are larger. Good frequency channels for this purpose can be found at ∼75 MHz (where the SZE-21cm is lower than the standard SZE because of the Ly-α coupling effect), and at 100-110 MHz (where the SZE-21cm is stronger because of the UV ionization effect). Because of the better sensitivity of SKA1-low at its high frequency band, the best frequency range where we can obtain information on the SZE-21 cm is ν ∼ > 100 MHz. However, also in this frequency range the difference between the two signals is of the order of µJy, so very deep observations, and very accurate data analysis procedures are required for this purpose, together with the fact that it is necessary to use clusters with high values of electron temperature and optical depth. We further show that with SKA2 the difference between the modified and the standard SZE can be detected at frequency ν ∼ > 60 MHz in galaxy clusters with temperature kT ∼ > 15 keV and at ∼ > 65 MHz in clusters with temperature kT ∼ > 10 keV for an integration time of 1000 hrs. For the other models we use, detecting the difference between the modified and the standard SZE is more challenging. In general, it is not possible to detect this difference with SKA1; only in the case of the model with Dark Matter with M min = 10 −3 M ⊙ it would be possible detect this difference by increasing the integration time by a factor ∼ 3 for the hottest clusters at a frequency around 110 MHz. With SKA2, the detection is possible at frequencies 85-120 MHz (only for cluster temperatures kT > 10 keV) Both panels are using for the modified CMB the fiducial model without Dark Matter (solid line in Fig.1). Both panels are for thermal plasma with temperatures kT = 20 (green), 15 (black), 10 (red) and 5 (cyan) keV, and calculated for τ0 = 5 × 10 −3 , θc = 300 arcesc, β = 0.75, θmax = 10θc, compared with the SKA-50%, SKA1-low, and SKA 2 sensitivities for 100 kHz bandwith, 1000 hrs of integration, 2 polarizations, no taper, no weight (thick lines). A promising strategy can be designed to study the SZE at higher frequencies (with experiments like, e.g., SPT, ACT, Millimetron) in order to derive precise information on the parameters of the ICM, and then use these constraints to obtain a better estimate of the properties of the SZE-21 cm with SKA1-low and SKA2. Conclusions The goal of obtaining information on the physical processes occurred during the DA and EoR by measuring the SZE-21 cm with SKA is challenging, but possible if pursued with good theoretical and observational strategies. Observations have to be carried out towards high temperature and high optical depth clusters to maximize both the overall signal and the difference between the standard and the modified SZE. The best frequency ranges of observation of the SZE-21cm are between ∼ 90 and 120 MHz, where the difference between the standard and the modified SZE is maximum. In our benchmark model, the sensitivity of SKA1-low is good enough to detect this difference with 1000 hours of integration, whereas for the other background models the difference between the standard and the modified SZE can be detected only with SKA2 for the same integration time in frequencies bands that depend on the background model and the temperature of the cluster. Together with very deep observations, a very accurate theoretical analysis is required, where the full formalism to calculate the SZE and detailed models for describing the effect of the cosmological 21-cm background on the CMB spectrum have to be used. In addition, we find that a very important strategy will be the detailed study of the SZE at higher frequencies in order to estimate the gas parameters to be used as prior constraints for the study of the SZE-21 cm at low frequencies. Observations in the frequency bands of SKA1-mid are also very important to disentangle the SZE from the cluster synchrotron emission. In this respect, the use of highredshift clusters can alleviate the problem, since the radio emission decreases as D −2 L , whereas the SZE is not depending on the cluster distance. The detection of the non-thermal SZE-21 cm appears to be more challenging, since the signal is much fainter with respect to the thermal one, especially regarding the difference between the standard and the modified SZE, that can be also a factor of ∼ 10 2 smaller with respect to the thermal case. However, the different spectral features can allow, in principle, a detection of this signal and hence an estimate of non-thermal cluster properties independently of measurements in other spectral bands. We note here that it is possible to strategize the search of this signal in objects where the non-thermal components are dominant, such as in the case of radio galaxies lobes. In this case, objects with more energetic electrons (i.e. with harder radio spectra), large optical depth (for which a good indication could be a strong radio luminosity) and high redshift are preferable. The independence of the SZE from the redshift can allow the study of the SZE-21cm in a larger number of objects spread over a wider redshift range, therefore producing statistical studies aimed at maximizing the detectable signal, and detect the properties of the 21-cm background and of the early DM halos over a large set of spatial directions, allowing in such a way a better understanding of the full cosmic history of the physical processes occurring in the Dark Ages and the Epoch of Reionization. For the standard SZE the input radiation is a Planck black-body spectrum which at low frequency has a constant brightness temperature, and the resulting SZE ∆T st is a constant as well (see Fig.3). It is important to note that the Planck spectrum is a smooth function, and we noticed that because of this smoothness the difference between the use of a relativistic approach and a nonrelativistic approach in computing the SZE is smaller for low electron temperatures and at low frequencies (see, e.g., Colafrancesco et al. 2003). However, when computing the SZE-21 cm, the shape of the input radiation spectrum plays an important role for the determination of the error done in the calculation of the SZE-21cm using a nonrelativistic approach. To discuss this issue, we show the spectra of the SZE-21cm calculated with the relativistic and the nonrelativistic approach for the four input models we are using in this paper, and for a reference electron temperature of 7 keV (see ). The SZE-21cm is also compared with the standard SZE calculated with the relativistic and the non-relativistic approach. We show that the use of the non-relativistic approach introduces an overall numerical error into the standard SZE, and that this error is amplified in a frequency-dependent way for the SZE-21cm. To better study the frequency dependence of this error, we also show the percentage error done in these cases, and we compare these results with the properties of the input spectrum. As discussed in Sect.3, we expect that the most important factor in determining the error done with the non-relativistic approach is the curvature of the input radiation spectrum: if the input spectrum has a large curvature this implies that using a function P (s) that is narrower than the correct relativistic one (like in the nonrelativistic approach) gives a result that is more different from the correct one w.r.t. the case where the input radiation spectrum is smooth, like in the case of the standard CMB. To check this conclusion, we compare the percentage error for the four models with the second derivative of the input radiation spectrum. As expected, we observe that the percentage difference between the relativistic and non-relativistic spectrum has maximum points lying at frequencies where there is a peak in the second derivative of the input radiation spectrum, corresponding to point of maximum curvature. For the first model, we observe that there are two peaks in the frequency range 60-80 MHz in the case of the nonrelativistic SZE-21cm. The existence of these peaks depends on the fact that the input radiation spectrum has two peaks in its second derivative, and the effect of using the non-relativistic Kernel introduces numerical artifacts due to the fact we are convolving the input radiation spectrum with a very narrow Kernel (see Birkinshaw et al. 1999). With the correct relativistic Kernel, the input spectrum is convolved with a wider function and the two peaks are then smoothed in only one peak. Therefore, the use of a non-relativistic approach gives origin not only to a numerical error in the value of the computed SZE, but also in its spectral shape and this error increases for increasing electron temperatures. In the other models we consider in our paper, the second derivative of the input radiation spectrum has only one peak at frequencies ν ∼ 60 − 70 MHz, and as a result also the non-relativistic SZE-21cm has only one peak in this spectral range. It can be seen that there are peaks/troughs in the percentage difference at frequencies whereby peaks/troughs are in the second derivative of the input spectrum (e.g. at ν ∼ 153 MHz for the second model). This shows that the smoothness of the input radiation spectrum is an important aspect which produces differences in computing the SZE spectrum using a relativistic approach or a non-relativistic approach. To conclude, we have shown in this Appendix that there is a substantial numerical error when computing the SZE using a non-relativistic approach, in particular when the input radiation spectrum is not a smooth function, as in the case of the modified CMB giving rise to the 21-cm background. This means that when using SZE of cosmic structures to study the cosmological 21-cm, it is imperative to use a full relativistic computation in order to obtain the correct SZE amplitude and its spectral shape. Upper panel: thermal SZE-21cm for kT = 7 keV and τ = 5 × 10 −3 calculated with the relativistic approach (solid line) and the non-relativistic approach (dashed line), compared with the standard SZE calculated with the relativistic approach (long-dashed line) and the non-relativistic approach (dotted line). Middel panel: percentage difference between the relativistic result and the non-relativistic one. Lower panel: second derivative of the input spectrum. Fig. 1 . 1Modified CMB spectrum emerging from the DA and EoR, in units of brightness temperature relative to the CMB (Evoli, private communication): a fiducial model without Dark Matter (solid line; this is our benchmark model), an extreme model without Dark Matter (dashed line), a fiducial model with Dark Matter with Mmin = 10 −3 M⊙ (dot-dashed line), where Mmin is the mass of the smallest DM subhalo, and a fiducial model with Dark Matter with Mmin = 10 −6 M⊙ (three dots-dashed line). Fig. 2 . 2The SZE-21cm (in units of brightness temperature relative to the CMB) for a thermal plasma at temperature kT = 7 keV and with τ = 5 × 10 −3 (solid line). With the dashed line the standard SZE ∆Tst for the same parameters is plotted for comparison. Fig. 3 . 3The standard SZE (in units of brightness temperature relative to the CMB) for a thermal plasma (kT = 5 keV and τ = 5×10 −3 ; solid line) and for a non-thermal plasma (s = 3.5, p1 = 10 and τ = 1×10 −4 ; dashed line). The SZE is shown in the radio frequency range where the 21-cm radiation background are visible. Fig. 4 . 4The SZE-21cm (in units of brightness temperature relative to the CMB) for thermal plasma at temperature kT = 5, 10, 15 and 20 keV, shown by the solid, dashed, dot-dashed and dash-three dots lines, respectively. A constant value τ = 5 × 10 −3 has been used in the calculations. Fig. 5 . 5Percentage difference between the relativistic result and the non-relativistic one for the SZE-21cm for galaxy clusters with temperatures of 20 keV (solid line), 15 keV (dashed line) and 7 keV (dot-dashed line). Fig. 6 . 6Difference between the SZE-21cm and the standard SZE (in units of brightness temperature relative to the CMB) for thermal plasma at temperature kT = 5, 10, 15 and 20 keV, shown by the solid, dashed, do-dashed and dash-three dots lines, respectively, as inFig.4. A constant value τ = 5 × 10 −3 has been used in the calculations. Fig. 7 . 7The SZE-21cm (in units of brightness temperature relative to the CMB) for non-thermal electrons with a power-law spectrum with s = 3.5 and p1 = 0.1, 1, 5 and 10, shown by the solid, dashed, dot-dashed and dash-three dots lines, respectively. A constant value τ = 1 × 10 −4 has been used in the calculations. Fig. 8 . 8Difference between the SZE-21cm and the standard SZE (in units of brightness temperature relative to the CMB) for non-thermal electrons with a power-law spectrum with s = 3.5 and p1 = 0.1, 1, 5 and 10, shown by the solid, dashed, dot-dashed and dash-three dots lines, respectively. A constant value τ = 1 × 10 −4 has been used in the calculations. Fig. 9 . 9For an illustrative description of the possible redshiftdependence of the overall modified background model, we show the SZE-21cm (in units of brightness temperature relative to the CMB) for a thermal plasma at temperature kT = 7 keV and with τ = 5×10 −3 for the modified CMB spectrum with the values of z taken from the original model (solid line), and for a modified CMB spectrum in which all components are globally shifted in frequency by a factor 3 (dashed line). Fig. 10 . 10The SZE-21cm (in units of brightness temperature relative to the CMB) for a thermal plasma at temperature kT = 5 keV (upper panel) and 20 keV (lower panel) and with τ = 5 × 10 −3 for a modified CMB spectrum with a fiducial model without Dark Matter (solid line), an extreme model without Dark Matter (dashed line), a fiducial model with Dark Matter with Mmin = 10 −3 M⊙ (dot-dashed line), and a fiducial model with Dark Matter with Mmin = 10 −6 M⊙ (three dots-dashed line). Fig. 11 . 11The SZE-21cm (in units of brightness temperature relative to the CMB) for a non-thermal plasma with s = 3.5 and p1 = 0.1 (upper panel) and 10 (lower panel) and with τ = 1 × 10 −4 for a modified CMB spectrum with a fiducial model without Dark Matter (solid line), an extreme model without Dark Matter (dashed line), a fiducial model with Dark Matter with Mmin = 10 −3 M⊙ (dot-dashed line), and a fiducial model with Dark Matter with Mmin = 10 −6 M⊙ (three dots-dashed line). Fig. 13 . 13Surface brightness profile of the standard SZE in absolute value for thermal plasma with temperatures kT = 20 (solid line), 15 (dashed), 10 (dot-dashed) and 5 (three dotsdashed) keV, and calculated for τ0 = 5×10 −3 , θc = 300 arcesc, β = 0.75, θmax = 10θc. Fig. 14 . 14Upper panel: the spectra of the fluxes of the SZE-21cm ∆I mod (in units of µJy and in absolute value with the solid lines) and the SZE for a non-modified CMB ∆Ist (dashed lines). Lower panel: the absolute value of the difference between the SZE-21cm and the standard SZE for a non-modified CMB. and ∼ > 145 MHz for the case of extreme heating without Dark Matter, at 65-75 MHz (only for kT ∼ 20 keV) and 95-145 MHz in the case of the model with Dark Matter with M min = 10 −3 M ⊙ , and at 95-135 MHz and ∼ > 150 MHz (for kT ∼ > 10 keV) in the case of the model with Dark Matter with M min = 10 −6 M ⊙ . Fig. 15 . 15Like Fig.14 but for an extreme model without Dark Matter for the modified CMB (dashed line in Fig.1). Fig. 16 . 16Like Fig.14 but for a fiducial model with Dark Matter with Mmin = 10 −3 M⊙ for the modified CMB (dot-dashed line in Fig.1). Fig. 17 . 17Like Fig.14 but for a fiducial model with Dark Matter with Mmin = 10 −6 M⊙ for the modified CMB (three dotsdashed line in Fig.1). Fig. A. 1 . 1Results for the first model (solid line) ofFig.1. Fig. A. 2 . 2Like Fig.A.1 for the second model (dashed line) ofFig.1. Fig. A. 3 . 3Like Fig.A.1 for the third model (dot-dashed line) of Fig.1. Fig. A. 4 . 4Like Fig.A.1 for the fourth model (three dotsdashed line) of Fig.1. SZE-21cm is calculated by using a non-relativistic approach, as a function of the properties of the input spectrum, using the four models shown inFig.1. Acknowledgements. S.C. acknowledges support by the South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation and by the Square Kilometre Array (SKA). P.M. and M.S.E. acknowledge support from the DST/NRF SKA post-graduate bursary initiative. We thank C. Evoli for providing the numerical files of the models inFig.1, and A.Ferrara, M. Birkinshaw and A. Tailor for useful discussions. We thank the Referee for several useful comments and suggestions that allowed us to improve the presentation of our results.Appendix A: The relation between the error done by using the non-relativistic calculation and the properties of the input radiation field.In this Appendix, we estimate the error done when the . R Barkana, A Loeb, Phys. Rept. 349125Barkana, R. & Loeb, A., 2001, Phys. Rept., 349, 125 . R Barkana, A Loeb, ApJ. 62465Barkana, R. & Loeb, A., 2005a, ApJ, 624, L65 . R Barkana, A Loeb, ApJ. 6261Barkana, R. & Loeb, A., 2005b, ApJ, 626, 1 . S Bharadwaj, S S Ali, MNRAS. 352142Bharadwaj, S., & Ali, S.S., 2004, MNRAS, 352, 142 . M Birkinshaw, Phys. Rep. 31097Birkinshaw, M. 1999, Phys. Rep., 310, 97 . V Bromm, R B Larson, ARA&A. 4279Bromm, V., & Larson, R.B., 2004, ARA&A, 42, 79 . C L Carilli, NewAR. 481029Carilli, C.L., et al., 2004, NewAR, 48, 1029 . A Cavaliere, R Fusco-Femiano, A&A. 49137Cavaliere, A. & Fusco-Femiano, R, 1976, A&A, 49, 137 . X L Chen, J Miralda-Escude, ApJ. 6021Chen, X.L., & Miralda-Escude, J., 2004, ApJ, 602, 1 . T R Choudhury, A Ferrara, arXiv:astro-ph/0603149Choudhury, T.R., & Ferrara, A., 2006, arXiv:astro-ph/0603149 . B Ciardi, A Ferrara, SSRv. 116625Ciardi, B., & Ferrara, A., 2005, SSRv, 116, 625 . S Colafrancesco, Acta Pol. 53560Colafrancesco, S. 2013, Acta Pol., 53, 560 . S Colafrancesco, P Marchegiani, A&A. 5622Colafrancesco, S. & Marchegiani, P., 2014, A&A, 562, L2 . S Colafrancesco, P Marchegiani, E Palladino, A&A. 397Colafrancesco, S., Marchegiani, P. and Palladino, E., 2003, A&A, 397, 27-52 . S Colafrancesco, P Marchegiani, R Buonanno, A&A. 5271Colafrancesco, S., Marchegiani, P. & Buonanno, R., 2011, A&A, 527, L1 . S Colafrancesco, M S Emritte, P Marchegiani, JCAP. 6Colafrancesco, S., Emritte, M.S., & Marchegiani, P., 2015, JCAP, 05, 006 . A Cooray, Phys. Rev. D. 7063509Cooray, A., 2004, Phys. Rev. D, 70, 063509 . A Cooray, Phys. Rev. D. 73103001Cooray, A., 2006, Phys. Rev. D, 73, 103001 . A De Oliveira-Costa, MNRAS. 388247de Oliveira-Costa, A. et al., 2008, MNRAS, 388, 247 . P Dewdney, W Turner, R Millenaar, R Mccool, J Lazio, T Cornwell, SKA baseline design documentDewdney, P., Turner, W., Millenaar, R., McCool, R., Lazio, J. & Cornwell, T., 2012, SKA baseline design document, http://www.skatelescope.org/wp-content/uploads/2012/07/SKA-TEL . J S Dillon, Phys. Rev. D. 8923002Dillon, J.S. et al., 2014, Phys. Rev. D, 89, 023002 . T A Enßlin, C R Kaiser, A&A. 360417Enßlin, T.A. & Kaiser, C.R., 2000, A&A, 360, 417 . C Evoli, A Mesinger, A Ferrara, JCAP. 1124Evoli, C., Mesinger, A., & Ferrara, A., 2014, JCAP, 11, 024 . G B Field, ApJ. 129536Field, G.B., 1959, ApJ, 129, 536 . S R Furlanetto, F H Briggs, 481039NewARFurlanetto, S.R., & Briggs, F.H., 2004, NewAR, 48, 1039 . S R Furlanetto, S P Oh, F H Briggs, Phys. Rep. 433181Furlanetto, S.R., Oh, S.P., & Briggs, F.H., 2006, Phys. Rep., 433, 181 . A Liu, Phys. Rev. D. 8743002Liu, A., et al., 2013, Phys. Rev. D, 87, 043002 . A Loeb, R Barkana, ARA&A. 3919Loeb, A., & Barkana, R., 2001, ARA&A, 39, 19 . A Loeb, M Zaldarriaga, Phys. Rev. Lett. 92211301Loeb, A., & Zaldarriaga, M., 2004, Phys. Rev. Lett. 92, 211301 . M Mcquinn, ApJ. 653815McQuinn, M., et al., 2006, ApJ, 653, 815 . A Mesinger, S Furlanetto, R Cen, MNRAS. 411955Mesinger, A., Furlanetto, S., & Cen, R., 2011, MNRAS, 411, 955 . M F Morales, J S B Wyithe, ARA&A. 48127Morales, M.F., & Wyithe, J.S.B., 2010, ARA&A, 48, 127 . G Paciga, MNRAS. 433639Paciga, G. et al., 2013, MNRAS, 433, 639 . A R Parsons, ApJ. 788106Parsons, A.R. et al. 2014, ApJ, 788, 106 . J R Pritchard, A Loeb, Phys. Rev. D. 8223006Pritchard, J.R., & Loeb, A. 2010, Phys. Rev. D, 82, 023006 . J R Pritchard, A Loeb, Rep. Progr. Phys. 7586901Pritchard, J.R., & Loeb, A. 2012, Rep. Progr. Phys., 75, 086901 . P A Shaver, R A Windhorst, P Madau, A G De Bruyn, A&A. 345380Shaver, P.A., Windhorst, R.A., Madau, P. & de Bruyn, A.G., 1999, A&A, 345, 380 . D Scott, M Rees, MNRAS. 990510Scott, D. & Rees, M.J. 990, MNRAS, 247, 510 . M Valdes, MNRAS. 4291705Valdes, M., et al., 2013, MNRAS, 429, 1705 . S A Wouthuysen, AJ. 5731Wouthuysen, S.A., 1952, AJ, 57, 31 The First Galaxies. S Zaroubi, Astrophysics and Space Science Library. 39645Zaroubi, S. 2013, The First Galaxies, Astrophysics and Space Science Library, 396, 45
[]
[ "Quantum computing online workshops and hackathon for Spanish speakers: A case study", "Quantum computing online workshops and hackathon for Spanish speakers: A case study" ]
[ "Alberto Maldonado-Romo \nCentro de Investigación en Computación\nDepartment of Computer Science\nInstituto Politécnico Nacional Mexico City\nMexico\n", "Lia Yeh \nUniversity of Oxford Oxford\nUK\n" ]
[ "Centro de Investigación en Computación\nDepartment of Computer Science\nInstituto Politécnico Nacional Mexico City\nMexico", "University of Oxford Oxford\nUK" ]
[]
We discuss the challenges and findings of organizing an online event in Spanish, consisting of a series of introductory workshops leading up to a quantum hackathon for Latin America. 220 Spanish speakers were registered, 66% of whom self-identified as being at an introductory level of quantum computing. We gain a better picture of the impact of quantum computing in Latin America, and the importance of generating educational resources in Spanish about quantum computing. Additionally, we report results on surveying the participants by country; educational status; self-reported levels of quantum computing, linear algebra, and Python competency; and their areas of interest within quantum.This event was organized by Quantum Universal Education with the Centro de Investigación en Computación del Instituto Politécnico Nacional (CIC-IPN) as the host institution, in collaboration with a number of organizations and companies: IBM Quantum, Xanadu, Multiverse Computing, Quantum Universal Education, Quantum Hispano, QMexico, Haq.ai, Dive in Learning. This was part of a larger event, the Qiskit Fall Fest 2021, as one of several hackathons organized around the world in a similar span of time. In each Qiskit Fall Fest hackathon, participants were challenged to form teams of up to 5, to develop in 5 days a project using the IBM Qiskit framework.
10.1109/qce53715.2022.00096
[ "https://export.arxiv.org/pdf/2302.12119v1.pdf" ]
253,803,726
2302.12119
2d35b63f62a48e91dc4a3c69fc114cfcd2ed4ee9
Quantum computing online workshops and hackathon for Spanish speakers: A case study Alberto Maldonado-Romo Centro de Investigación en Computación Department of Computer Science Instituto Politécnico Nacional Mexico City Mexico Lia Yeh University of Oxford Oxford UK Quantum computing online workshops and hackathon for Spanish speakers: A case study Index Terms-quantum educationquantum computingwork- shophackathonSpanishLatin America We discuss the challenges and findings of organizing an online event in Spanish, consisting of a series of introductory workshops leading up to a quantum hackathon for Latin America. 220 Spanish speakers were registered, 66% of whom self-identified as being at an introductory level of quantum computing. We gain a better picture of the impact of quantum computing in Latin America, and the importance of generating educational resources in Spanish about quantum computing. Additionally, we report results on surveying the participants by country; educational status; self-reported levels of quantum computing, linear algebra, and Python competency; and their areas of interest within quantum.This event was organized by Quantum Universal Education with the Centro de Investigación en Computación del Instituto Politécnico Nacional (CIC-IPN) as the host institution, in collaboration with a number of organizations and companies: IBM Quantum, Xanadu, Multiverse Computing, Quantum Universal Education, Quantum Hispano, QMexico, Haq.ai, Dive in Learning. This was part of a larger event, the Qiskit Fall Fest 2021, as one of several hackathons organized around the world in a similar span of time. In each Qiskit Fall Fest hackathon, participants were challenged to form teams of up to 5, to develop in 5 days a project using the IBM Qiskit framework. I. INTRODUCTION Accompanying the rise in applications of quantum technology, there has been growing interest in quantum education and workforce development initiatives [1]. These beginning and/or grassroots efforts have the potential to be very impactful, particularly in this formative period, where traditional educational offerings have yet to catch up to train the skills demanded for in these new industries [2]. As with any emerging technical subject, most students and professionals across all ages and stages of learning who would like to begin to learn it, will not find a course offering at their institution to gain exposure to the subject. This makes it important that there be informal educational opportunities open to those who would like to learn, and welcoming to those new to the topic, especially as one must first be exposed to the topic in order to then become interested in learning it. Much progress has been made on addressing these needs in recent years, from online extracurricular courses for middle and high school students [3] to games and software tools designed to teach quantum science concepts [4]. This report serves to introduce the approach of hackathons as a gateway for quantum computing learning and outreach. Hackathons are events where participants form teams of one to up to around five participants, who compete to create a functioning project from start to finish, over the course of the event, which can range from one day to a few weeks in duration. Hackathons are a widely adopted approach to facilitating learning and collaboration in a short but focused time period, with hundreds of hackathons taking place each year. Definitions of hackathons, or "hacking marathons," may vary greatly depending on whether the intended purpose(s) is innovation, collaboration, competition, business solutions, software prototyping, education, outreach, research, or fun [4], [5]. Despite the growing number of quantum hackathons worldwide, there is limited documentation and evaluation of their organization and effectiveness. A particular subcategory of quantum hackathons has been studied for educational purposes: quantum games hackathons, and their more relaxed and less competitive counterpart, quantum game jams. Ref. [4] presented a comprehensive overview of quantum gamesgames with quantum elements in their game mechanics -for education and outreach, with a section describing an online Quantum Games Hackathon. Ref. [6] evaluated each quantum game created in the course of five quantum game jams on criteria such as playability and educational value. II. A SPANISH-LANGUAGE VIRTUAL HACKATHON FOR LATIN AMERICA As an emerging area worldwide, quantum computing has seen recent growth in the educational materials accessible in a variety of forms, including but not limited to: formal courses, textbooks, self-paced online courses, video series, programming tutorials, workshops, games, and comics. However, all these learning resources and opportunities are predominantly restricted to the English language. For one to become interested in quantum and feel a sense of belonging, quantum education needs to recognize, and strive to attune to, each and every diversity of identity and background. In order to ©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. drive equitable and inclusive education in quantum science and technology, the challenges of language, geographical, and socioeconomic barriers must be addressed. In Mexico, the Cuarantena Cuantica (literal translation: Quarantine Quantum) quantum computing seminar [7] was held virtually in January 2021 by the Sociedad Científica Juvenil (SCJ), i.e. the Juvenile Scientific Society of Mexico. It was followed by the Qiskit Summer Jam 2021 Mexico quantum hackathon [8] held in August 2021 as a collaboration between SCJ and QMexico, a QCousins branch of the QWorld non-profit organization. These and the Quantum Latino 2021 event [9] were the direct antecedents of the workshop series and hackathon described in this report, as quantum computing events respectively centered on Mexico and Latin America. With the importance of supporting more accessible pathways to exposure to quantum concepts in a collaborative and welcoming environment in mind, this event was organized with beginner friendliness as the key priority. The goal was to encourage participation at all levels, especially for those for whom this was their first quantum event or even exposure to technical quantum concepts and quantum programming. For this reason, we put together a series of workshops in Spanish and English in the week preceding the hackathon. By design, the first workshop, offered once in Spanish and once in English by the Quantum Universal Education not-forprofit organization, was Introduction to Quantum Computing; it began by explaining what a qubit is logically and what it can look like physically, with the quantum circuits introduced alongside small illustrations of colorful cats, accompanied by a live code demonstration in the quantum programming language Qiskit. As we conclude in the section on our participant survey findings, this decision of focusing on introductory workshops was crucial to the objective of increasing awareness of quantum computing in Latin America. With these goals in mind, we proceed to describe the organizational structure, dissemination of event sign-up information, and other considerations for the planning of the event, in hopes of informing future Spanish language and/or hackathon quantum educational events. A. Overview of the event organization The hackathon was part of IBM Quantum Education's Qiskit Fall Fest 2021, which coordinated the initiative to encourage and support 18 hackathons organized in roughly the same time frame around the world. IBM Quantum Education provided guidance in planning a hackathon in the form of courses and a hackathon guide [10]. These courses supported the planning of this hackathon to design it online, and to make it open to all Spanish speakers. Topics covered included the dissemination of information, the tools to broadcast, and platforms to manage and link people together whether that be in-person or online. Some tools were Hype Innovation Management Software, a web platform used to manage, for each hackathon: team formation, project ideation, and project submission; and Discord, a community and messaging social media platform used by each hackathon for schedule, announcements, networking, mentoring, memes, and other communication. Considering the locations of countries in Latin America, to create a schedule that the most people across these time zones can realistically participate in, we used the UTC-6 time zone. The event, while being run online, was hosted thanks to the Centro de Investigación en Computación del Instituto Politécnico Nacional (CIC-IPN). For reasons due to the pandemic situation, the in-person conditions were not suitable. With regards to the CIC-IPN organization, the event was part of the CORE International Congress 2022, an international congress focused on computer science, organized by students from the Centro de Investigación en Computación (CIC) in Mexico City [11]. In conjunction with the Instituto Politécnico Nacional (IPN), this facilitated extending the scope of participation to different institutions in the city, country, and different countries. To reach Spanish speakers online, information on the event and how to register were disseminated through a number of online channels, as a collaboration between various organizations and communities each with hundreds or thousands of people involved. Communities of special mention for the unique roles each played in supporting this event include Quantum Universal Education, Quantum Hispano, the QMexico chapter of the QWorld non-profit, and the Qiskit Advocates program of IBM Quantum Community. The social networks and messaging platforms utilized included Facebook, Twitter, LinkedIn, Discord, Whatsapp, Slack and others, with a majority of registrants having heard of the event through Facebook (see Figure 1). The announcement campaign was conducted in both English and Spanish, using phrases such as introduction to quantum computing; programming quantum algorithms; quantum computing hackathon; and create and learn your first quantum computing project. For there to be an expectation as to the prerequisite knowledge recommended, the suggested skills to have an introductory grasp of were: how to program in Python, to know linear algebra, and to know probability; to know about quantum mechanics or quantum computation was optional, given the purpose of the workshops. The event was scheduled such that the workshops took place across six consecutive days. The last day of workshops focused on industry, where presenters from Multiverse Computing and Xanadu offered workshops by Spanish speakers where they indicated the aspects and requirements of a job in a quantum computing company, and different tips for starting out in this area. The week of workshops was followed by the hackathon, which took place across five consecutive days. Over the course of the hackathon, people new to quantum would have ample time to get to know each other and get started in the creation of their first quantum computing project with virtual sessions, and not feel rushed to ask general questions and concerns about quantum computing or about their projects. B. During the Event The event consisted of two parts in order to involve people who had no knowledge of quantum computing, but had some general knowledge of Python programming, linear algebra, and optionally general topics of quantum physics. For the event, a server was created on the Discord social community platform, in which only registered participants had access to the join link. After entering the Discord server, participants could select the option to attend the workshops, where they were then automatically provided access to the channels where the links, resources, and question and answer sessions regarding each workshop were posted. Likewise, participants could select the option to attend the hackathon, to access channels where they can introduce themselves, describe their areas of interest, form teams, and discuss possible projects and ideas. For each team to come up with their project proposal, they are given a channel for their team to work, visible to just their team members and to the hackathon volunteers and mentors to respond to any questions and concerns. For all participants, whether they opted to take part in the workshops, hackathon, or both (See Figure 2 for these choices at the time of registration), there were general channels for: announcements about quantum computing news, sharing how to get started with tutorials and applications, suggesting project ideas, and memeing. These were used as spaces to motivate and support people to get comfortable, challenge themselves, learn from and with each other, and let their imagination take off. They were encouraged to ask or inquire more about topics in quantum, and to take a look at the linked open educational resources accessible to them geared towards those new to quantum concepts, for example from IBM Quantum Education and Quantum Universal Education. C. Series of Workshops The series of workshops was designed to introduce at a beginner level different applications of quantum computing, such as optimization, chemistry, algorithms, machine learning, and video games. These are areas focused on in previous Qiskit hackathons, as areas for which there are a variety of applications and more accessible learning materials. In addition to technical workshops, a panel was added to shed light on the landscape of quantum computing in Latin America, and opportunities for work and perspectives shared by Spanishspeaking people who are in the industry. The purpose of these workshops is to introduce the state of the art in quantum computing to Spanish speaking audiences, for which the original materials have to be adapted to Spanish and adapted to an introductory level. For this we considered the criterion of inviting people who meet one of three requirements: 1) be a Qiskit advocate with a particular interest in an area of quantum computing; 2) be someone who works in the quantum computing industry being people from for instance IBM Quantum, Xanadu and Multiverse Computing; or 3) have completed an internship or mentorship in one of the quantum computing communities such as QWorld. Each of the selected workshop speakers facilitated a workshop in their area of expertise. For accessibility of the learning in general, the workshops were coordinated to be introductory and limit the need for prerequisite knowledge of quantum where possible. Moreover, the more introductory topics were scheduled earlier in the week, and the programming examples were encouraged to use the Qiskit framework to ensure continuity of understanding between workshops. The series of workshops and their descriptions can be seen in Table 1. D. Hackathon In organizing a hackathon, a number of logistical components are involved. In this section, we describe the aspects of team formation and project ideation; mentors; and project evaluation and prizes. Additionally, we provide brief descriptions of and links to the winning projects. Companies and governments are investing in its research and development since its use represents the answer to tasks that involve excessively complex calculations in artificial intelligence, chemistry, cryptography, among others. Actors such as IBM and Microsoft are looking to train developers who can program their quantum computers. To this end, they make their platforms freely available to us. The challenge now is to know the resources, materials, and events that one can develop all their skills. This presentation will address the opportunity to have a vision of this large and complex world. Estado actual y perspectiva de los juegos cuanticos Language: Spanish Workshop presenter: Anamaría García Hernández Description: The field of quantum games is developing rapidly. In this project we collect information about quantum games at present and classify them into different categories. The results obtained will be discussed, as well as future projections. Creacion de oracles para algoritmos Language: Spanish Workshop presenter: Emilio Pelaez Description: In this workshop, we will explore the construction of oracles for different algorithms. Giving examples on concrete algorithms such as Grover's search algorithm and the 3-SAT problem, and defining the concept of an oracle formally and how we can translate it into a circuit efficiently. Calculando observables físicos con VQE Language: Spanish Workshop presenter: Siddhartha Morales In this workshop, we will explore how to use variational algorithms to solve some interesting physical problems, such as the ground state of a molecule and of an atomic nucleus. We will see how to use the quantum variational algorithm, as well as how to create our own ansatz, test different optimizers and how to send the work to a real quantum computer. Algoritmo BB84 Language: Spanish Workshop presenter: Luis Martínez Description: In this workshop, we will see a brief introduction to the area of quantum cryptography. With special emphasis on the BB84 key exchange algorithm. We will also review some primitives that are important in this field. For this workshop it is not necessary to know classical cryptography as the basic concepts will be reviewed in the workshop. Knapsack Problem Language: Spanish Workshop presenter: Claudia Zendejas Morales Having quantitative information to make decisions leads to more and better profits. Studying the knapsack problem (KP) allows us to find solutions to combinatorial optimization problems by modelling a situation analogous to filling a knapsack. Its applications range from transportation and logistics problems to financial investments. In this workshop, we will see how to solve the knapsack problem with quantum computing. Criptografía murió RSA? Language: Spanish Workshop presenter: Daniel Sierra-Sosa Description: One of the areas of interest in Quantum Computing is information security. It involves the fundamental principles and techniques of quantum computing, notions of information theory, algorithms (Grover's search and Shor's factorization), and applications of Quantum Computing such as quantum encryption and key distribution. This workshop will aim to explore and discuss real-world scenarios related to information security in times of quantum technologies, participants will understand the opportunities and challenges in this area and will have a hands-on experience on the IBM Q Experience platform. Quantum Game Development at an introductory level Language: English Workshop presenter: Wen-Sen Lu Description: It is our privilege to explore the cutting-edge quantum computational space during the NISQ era with QISKit. Looking back into the history, especially in the 1970's, arcade game developers already started the machine-level programming and prepared themselves as the future coders even if the hardware was still limited. In the meanwhile, game-driven breakthrough for the classical hardware, such as the first 3D acceleration chip Super FX in Nintendo super-NES home console, also demonstrated the possibilities where new hardware could be inspired by the game developers. In this talk, I will start with my personal experience to quickly walk us through the process of quantum game development: looking for ideas from existing games, selecting development tools, and putting together two example codes in PICO-8 (Lua) to quickly demonstrate the classical and quantum counterpart of the game dev, respectively. Haq.ai Language: English Workshop presenter: Adam Fattal Description: haq.ai is a platform oriented for everyone interested in the field of quantum computing that wants to sharpen their quantum programming skills. Through a wide collection of problems in many topics and with different difficulties, users can develop their abilities. They can learn to efficiently decompose quantum circuits through methods presented in literature, harness the power of numerical computation in the field of quantum information, explore fun problems that require using popular algorithms and protocols, and much more. We aim to make the journey through quantum computation more interactive, while we don't offer a whole educational component, we offer a great supplement to a conventional quantum computing education. Introducción al Aprendizaje de Máquina Cuántico Language: Spanish Workshop presenter: Alberto Maldonado Romo Description: In this workshop we will see a small introduction of how to pass classical information to qubits to be able to treat them and get to general models such as neural networks in their quantum version. Situación de la computación cuántica en América Látina Language: Spanish Workshop presenters: Jazmin Esteva, Bruno Ramírez, Dr. Javier Orduz Description: Quantum computing has had a great impact on the world and many companies have focused on this area, but what is happening in Latin America? Introducción al QAOA Language: Spanish Workshop presenter: Victor Onofre Description: QAOA is one of the hybrid quantum-classical algorithms that have been proposed to take full advantage of current quantum resources. In this workshop, we will explain its application to the MAX-CUT problem. Quantum Enhanced Monte Carlo Simulations Language: Spanish Workshop presenter: Cristina Sanz Fernández Description: Case study of how quantum computing can be useful to us today. Monte Carlo simulations are a widely used tool both in research (physics, chemistry, etc.) and in practical applications in our day-today life (finance, meteorology, telecommunications, etc.). In this talk, I specifically explain how quantum leads to a quadratic improvement of Monte Carlo calculations. Carreras en la computación cuántica Xanadu Language: Spanish Workshop presenter: Catalina Albornoz Description: In this talk, we will discuss the different career opportunities in the field of quantum computing, and what skills can lead you to that dream job. specific channels for participants to search for teammates, with teams limited to 2 to 5 members. 48 hours before the hackathon, during team formation, mentors were involved in proposing and bouncing off ideas with the participants. This process included understanding, for each participant, what workshop they liked the most, what geographical location they are from, what skills they have and would like to learn, and their past project experiences. Suggested project ideas and feedback on proposed ideas were made to each participating team, especially in conversation with those who had doubts as to where to start with proposing their project. • Mentors: The mentors were selected from the Qiskit advocate program, most of them being the same people who gave workshops, or people who could speak or write in Spanish to provide one-on-one support to the teams. • Project evaluation and prizes: The hackathon judges evaluated the projects, consisting of the project files and a short recorded talk explaining the motivation and demoing the project, according to the following four criteria, each of equal weight. -Technical Challenge: Did the team challenge themselves and try to learn and implement something new to them or to the area? -Impact: Does the project have a high potential for impact? E.g. industrial application, educational value, theoretical interest, etc. -Creativity: Does the project go beyond the scope of a typical hackathon project? Unlike the technical challenge criterion, the creativity criterion incentivizes imagination in forms beyond technical (eg. artistic, originality, user friendliness, etc.). -Presentation: Is the project functioning and thorough? Is the presentation of the project thoughtfully explained and easy to understand? There are some observed similarities between the above four hackathon judging criteria, and those of the quantum games hackathon by the Quantum AI Foundation of 1) Correctness, 2) Playability, 3) Originality, and 4) Quality and complete-ness [4]. To improve the learning experience, each team was asked which workshops they enjoyed, and what topics they were interested in, so that to the extent possible, they could be assigned a mentor(s) suiting those specializations. E. Winning projects A total of 10 projects were ideated by the 29 hackathon participants, of which 8 projects were submitted as code along with recorded presentation for judging [12]. Prizes were awarded to the top three projects, with an additional prize for the Best Education Hack awarded by Quantum Universal Education. The winning projects along with brief description are in Table 2, and links to each project are in the bibliography. Second Place Description: The way to find the optimal resource allocation for users for which throughput is calculated based in resource blocks (chunks of frequency) and modulation (a sort of channel indicator). Variational Quantum Circuits in a Protein Network Graph [15] Third Place Description: The main idea of the proposal is to map biochemical interactions inside a 3D protein structure into a graph network. Quantum classifier for medical data [16] Best Education Hack Description: In this project, we propose an introduction to quantum machine learning using a variational quantum algorithm for classification, applying it to two medical datasets. Additionally, we report a follow-up to the 14 winning participants six months after the conclusion of the hackathon. First, we remark that of the 14, 9 expressed preference for future quantum computing events to be hybrid, in comparison to 2 for in-person and 3 for virtual. In the response to the question How much time had passed between your first exposure to quantum science, computing, or engineering, and participating in this event (in October 2021)? 6 of 14 responded 0-6 months. Despite the high percentage of beginners, in response to the question On a scale of 1 (not interested) to 5 (very interested), what is your interest in a career in quantum? 2 of the 14 winning participants indicated a 3, 4 indicated a 4, and 8 indicated a 5. Most notably, 9 out of the 14 hackathon winners responded Yes to the question Since participating in the Qiskit Fall Fest CIC-IPN Hackathon have you participated in any quantum-related event? This shows that the hackathon winners felt empowered to continue to actively learn quantum science following the event, which they did through participating in quantum-related school projects, challenges, summer schools, and other quantum hackathons. III. SURVEY RESPONSES OF PARTICIPANTS To guide inclusive and beginner-friendly hackathon organization, four questions were kept in mind [17]: • Who is eligible to apply? • Who is it marketed to? • Who actually attends? • Who is it prepared to support? For participants for whom it is their first exposure to what they can do with quantum computing, it is understandable to feel not yet ready or comfortable to engage in a quantum hackathon. Regardless of whether they chose to participate in the hackathon (which anecdotally was a very positive experience for not just those experienced in quantum, but those new to quantum as well), the organizers considered the engagement and sincerity of participation in the workshops to be just as, if not more, important than the hackathon component. A. Demographic information The participation student status data indicates that the event, open to students and non-students of all levels of education and experience in quantum computing, attracted participation from all levels. This shows that there are no noticeable gaps in the demographic reached, compared to that targeted. We observe that amongst high school participants, the selfidentified level of quantum computing was comparable to that of other participants with other levels of education and occupational status. This raises the possibility of an alternate approach to quantum hackathons: Instead of targeting a certain level of education (e.g. high school) amongst which exposure to quantum computing may vary greatly, another option is to target beginners in quantum computing regardless of level of education, instead categorized by self-identified level of experience in quantum computing (see Figure 3). This approach is informed by the student-run hackathon 〈Womxn/Hacks〉, which in 2019 hosted ≈ 200 female-identifying and nongender-binary undergraduate and graduate students, ≈ 2/3 of whom self-identified as beginners in programming, and nearing 3/4 of whom were pursuing neither computer science nor computer engineering degrees -≈ 1/3 of the total participants were arts and humanities students [17]. The primary advantage of this approach is that individuals who have had their first exposure to or gained confidence to learn the topic at a later stage in life, who are disproportionately likely to be underrepresented in that topic, can have the opportunity to learn. For an approach like 〈Womxn/Hacks〉's which awarded prizes for both beginner and advanced levels, more advanced students in an earlier stage of education who may have exhausted the learning opportunities in quantum available to high schoolers have the option to seek the challenge to learn more by forming a team with others more experienced in other aspects and competing for advanced category prizes. Fig. 3: The survey data is first plotted by the number of participants for each self-reported level of quantum computing knowledge, ranging from 1 (little to none) to 5 (expert). Within each of these five levels, the data was then plotted by number of participants for each educational level (if they were a student) or occupation (for non-students). This shows that 66% of the participants identified as beginners in quantum computing -1 or 2 out of 5. The country participation data indicates that the country with the most participation was Mexico. This is as the organizers anticipated, with a number of possible explanations that this could be attributed to. First, the advertisement for the event was distributed and shared by a number of Mexican organizations and reaching audiences of Mexican identity. Second, while this was a fully online event, there is the status of the host university and members of the event organizers being Mexican, and hence increased technical and administrative support for those time zones. Although the event time zones had workshops and final project presentations scheduled at more central times of day with respect to time zones around the world, the event being virtually hosted in Mexico may have resulted in people in time zones further from Mexico being less inclined to participate, due to anticipated inconvenience or impracticality of time zone difference. With that said, the organizers were pleasantly surprised by the number of countries represented in the participants. This not only indicates that educational opportunities in quantum are appealing to students of all levels and from many geographical locations, but that channels to reach out to them all exist and should be utilized more. Combining this with the information that a majority of participants heard about the event through Facebook, this indirectly alludes to the fact that online Spanish-speaking communities for learning and interest in quantum encompass a greater diversity than previously realized (See Figure 4). Regarding the question guiding hackathon organization, "Who is it prepared to support?": To reach more persons in other countries, a recommendation for future online and international Spanish-language hackathons is to assemble an organizing team across more countries. However, the amount of overhead in supporting more time zones can make organization logistically more complex, and so a balance is needed. B. Self-reported amount of experience in quantum, linear algebra, python Comparison of the data on self-reported level of quantum, linear algebra, and Python competence establishes that participants of this event are generally more versed in linear algebra and Python, whilst being new to quantum computing. More participants identified as level 1 out of 5, the lowest level, in quantum computing, than any other level. While programming skills are expected of persons interested in a hackathon, the data on linear algebra being very similar to that is interesting. This supports the perception of both linear algebra and programming as prerequisite skills to learning quantum, as participants felt competent in these two skills whilst being beginners in quantum (see Figure 5). It is worth noting that the participants are mostly concentrated at an introductory level in quantum computing, and most of them consider that they have intermediate to advanced knowledge about Python and linear algebra. This is in line with the communication about the event, where it was emphasized that beginners to quantum were very welcome, and it was recommended to have some familiarity with Python and linear algebra for the workshops. Across all participants, the subject area the most participants were interested in was by far quantum algorithms, following by quantum machine learning and quantum cryptography. This finding is interesting because it shows correspondence with a survey of 57 companies in the quantum industry, which identified quantum algorithm development as the skill most relevant across job roles in quantum computing [18]. C. Feedback We collected feedback from 16 participants a week after the hackathon. First, we note several comments regarding the data collection and storage to bear in mind when we run this workshop series and hackathon again. At the time, the objective of soliciting feedback was to gauge from a small sample of participants the general perception of the event, in addition to identifying any areas for improvement. In retrospect, it would have better informed future similar initiatives to collect feedback promptly at two points: at the conclusion of the week of workshops, and at the conclusion of the hackathon. By surveying participants a week after the hackathon, we were less likely to hear from participants of only the workshops part of the event. Another complications we had was inexperience with regards to data privacy laws to storing participant identifying information and thus opting to be on the safe side of not collecting sensitive information. For this reason, we did not have sufficient data to draw conclusions about changes before and after the hackathon, for instance in participants' self-reported quantum computing level. In the future, we would also like to know participants' field(s) of study for students undergraduate level and above, and aspiring topic(s) for high school students. We report three observations pertaining to the participant feedback data. The first is the favorite workshop(s) of each surveyed participant, presented in Table III where each row lists the favorite workshop(s) of a particular participant, sorted from lowest (top of the table) to highest (bottom) by selfreported quantum computing level, and within the same level sorted by education level. As expected, the Introduction to quantum computing workshop was frequently favorited in the top half of the table but not in the bottom half. The most popular topics within the workshops were QML, QAOA, and Quantum Enhanced Monte Carlo simultation. The second observation is the highly positive perception amongst participants about the event organization. 10 of the 16 participants answered the optional short reply question to solicit feedback. In spite of the question being about feedback, there were only two constructive criticisms received: that the Quantum Machine Learning workshop was too technical, and the online meeting links could be sent more in advance. All the responses were in Spanish. Their machine translation to English are provided in Table IV. Like Table III, Table IV is sorted from lowest (top of the table) to highest (bottom) by self-reported quantum computing level, and within the same level sorted by education level. We hope that by sharing their expressed wishes to see more workshops and events in Spanish, it can be seen that there is avid interest in learning quantum computing in Mexico and Latin America. Everything was excellent, I really noticed the effort of the organizers. I wish there were more events like this one. I was interested in the Quantum Machine Learning workshop, but it was very technical and I got lost. They were days of great learning in the quantum computing workshops, great organization of the event, I am very happy that I decided to propose my own initiative in the Hackathon, I want to live the experience and what better than in my language (Spanish) is really a plus that motivates me even more to continue learning. Thank you very much! Very good project, it is a breakthrough for quantum computing in the country. Nothing more. Thanks to the organizers, I appreciate the order that they had with everything and their effort is noticeable! I was delighted that these workshops were given in Spanish, it facilitates the dissemination of quantum computing in Latin America. You were very good, the only recommendation might be that the link was a little earlier because in some meetings it was 10 minutes earlier. I think this is an excellent initiative. Keep it up :D IV. CONCLUSION Science communication and outreach was done after the event, not only by the Qiskit Blog [19], but also outside the field of quantum reaching non-technical and Spanish speaking audiences. In the aftermath of the event, the main event organizer was interviewed in Spanish by a national television channel in Mexico to talk about the event, and the CORE International Congress published a magazine article in Spanish about the event [20]. Finally, the host institute IPN conducted an interview in Spanish with the local participants who won third place [21]. In addition to the benefits of raising literacy in quantum technologies, and encouraging students to interact with industry and researchers, these events motivate participants to gain confidence in further pursuits learning quantum. Furthermore, such events can connect participants with more opportunities. For example, at the hackathon, one of the first place winners was encouraged to apply to the Qiskit advocates program by IBM Quantum, which he has since become a part of after passing the Qiskit developer certification exam. We conclude by stating that in the participant feedback survey, for the question, "Would you like to see more events like this in Spanish?", 100% of the responses were "Yes". There is much interest in learning quantum computing and gaining relevant skills from Latin America, and lessons were learned in how to develop opportunities to catalyze this. Fig. 1 : 1Percentage of registered attendees by education level. Fig. 2 : 2Percentage of registered attendees interested only in the hackathon (4.07%), only in the workshops (40.7%), and in both the workshops and the hackathon (55.2%). Fig. 4 : 4Percentage of participants by country. TABLE I : ISeries of WorkshopsIntroduction to quantum computing Language: English / Spanish Workshop presenter: Lia Yeh / Alberto Maldonado Romo Description: There is a growing trend towards quantum computing. TABLE II : IIWinning Projects Threerra: A Qiskit module for three-level systems [13] First Place Description: Created a module to allow users to both create unitary operations acting onto three-level systems (qutrits) using Qiskit Pulse, and to execute them on real hardware available through the IBM Quantum platform. Quantum Radio Resource Scheduling for 4G and 5G simulators [14] TABLE IV : IVParticipant optional feedback to the organizers I wish there were more workshops and the resources were more open and available. Very interesting the topics presented. • Team formation and choosing projects: The workshop session included activities such as a six-phase introduction, discussion of topics of interest, and the creation of (a) Participants' knowledge of quantum computing compared to their interest in different areas of quantum computing. ACKNOWLEDGMENTThis event would not have been possible without the active involvement and support of many people. We would like to thank our fellow hackathon organizers, Jose Navarro and David Pérez, along with CIC-IPN for their collaboration as the host organization. We would also like to thank all of the workshop presenters inTable IIand hackathon mentors. We thank IBM Quantum, Multiverse Computing, Xanadu, QMexico, QuantumHispano, Haq.ai, and Quantum Universal Education for sponsoring this event. We thank IBM Quantum Education for providing resources and guidance in organizing the hackathon as part of Qiskit Fall Fest 2021, especially Brian Ingmanson, Anamaría García Hernández, Katie Pizzolato, and Josie Kies. Finally, we thank the workshop series and hackathon participants for their feedback, and for their enthusiasm which made the event a joy to organize.(b) Participants' knowledge of Python compared to their interest in different areas of quantum computing.(c) Participants' knowledge of Linear Algebra compared to their interest in different areas of quantum computing. Quantum Information Science and Technology Workforce Development National Strategic Plan. Subcommittee on Quantum Information Science. Committee On Science of the National Science & Technology Council"Quantum Information Science and Technology Workforce Development National Strategic Plan," Subcommittee on Quantum Information Sci- ence, Committee On Science of the National Science & Technology Council, February 2022. https://www.quantum.gov/wp-content/uploads /2022/02/QIST-Natl-Workforce-Plan.pdf Preparing for ther quantum revolution: What is the role of higher education?. M R J Fox, B M Zwickl, H J Lewandowski, Physical Review Physics Education Research. 16M. R. J. Fox, B. M. Zwickl, and H. J. Lewandowski, "Preparing for ther quantum revolution: What is the role of higher education?", Physical Review Physics Education Research, vol. 16, October 2020. Qubit by Qubit 2021 Impact Report. The Coding School"Qubit by Qubit 2021 Impact Report," The Coding School, December 2021. Quantum games and interactive tools for quantum technologies outreach and education. Z C Seskir, P Migdał, C Weidner, A Anupam, N Case, N Davis, C Decaroli, İ Ercan, C Foti, P Gora, K Jankiewicz, B R La Cour, J Y Malo, S Maniscalco, A Naeemi, L Nita, N Parvin, F Scafirimuto, J F Sherson, E Surer, J R Wootton, L Yeh, O Zabello, M Chiofalo, 10.1117/1.OE.61.8.081809SPIE Opt. Eng. 618Z. C. Seskir, P. Migdał, C. Weidner, A. Anupam, N. Case, N. Davis, C. Decaroli,İ. Ercan, C. Foti, P. Gora, K. Jankiewicz, B. R. La Cour, J. Y. Malo, S. Maniscalco, A. Naeemi, L. Nita, N. Parvin, F. Scafirimuto, J. F. Sherson, E. Surer, J. R. Wootton, L. Yeh, O. Zabello, and M. Chiofalo, "Quantum games and interactive tools for quantum technologies outreach and education," SPIE Opt. Eng., vol. 61, no. 8, pp. 1-38, July 2022. https://doi.org/10.1117/1.OE.61.8.081809 Code camps and hackathons in education -literature review and lessons learned. J Porras, A Knutas, J Ikonen, A Happonen, J Khakurel, A Herala, 10.24251/hicss.2019.93352nd Hawaii International Conference on System Sciences. J. Porras, A. Knutas, J. Ikonen, A. Happonen, J. Khakurel, and A. Herala, "Code camps and hackathons in education -literature review and lessons learned," 52nd Hawaii International Conference on System Sciences, January 2019. http://dx.doi.org/10.24251/hicss.2019.933 Quantum game jammaking games with quantum physicists. A Kultima, L Piispanen, M Junnila, 10.1145/3464327.3464349Academic Mindtrek 2021. Association for Computing MachineryA. Kultima, L. Piispanen, and M. Junnila, "Quantum game jam - making games with quantum physicists," in Academic Mindtrek 2021, Association for Computing Machinery, pp. 134-144. https://doi.org/10 .1145/3464327.3464349 Semana de la computación cuántica y aplicaciones de la física cuántica, Sociedad Científica Juvenil. Cuarantena Cuantica, Cuarantena Cuantica: Semana de la computación cuántica y aplicaciones de la física cuántica, Sociedad Científica Juvenil, January 2021. https: //www.facebook.com/SCJ.MX/photos/3555268284521727 Quantum Latino 2021. Quantum Latino 2021. https://quantum-latino.com/quantum-latino-20 21/ Qiskit Hackathon Guide. IBM Quantum Community. "Qiskit Hackathon Guide," IBM Quantum Community, December 2021. https://raw.githubusercontent.com/BrianIngmanson/Qiskit-Hackathon- Guide/main/Qiskit%20University%20Hackathon%20Guide.pdf CORE International Congress 2022, Centro de Investigación en Computación del Instituto Politécnico Nacional (CIC-IPN). CORE International Congress 2022, Centro de Investigación en Com- putación del Instituto Politécnico Nacional (CIC-IPN), September 2021. https://www.core.cic.ipn.mx/ Recording of Workshop at Major League Hackcon VIII. E Martinez, L Yeh, L Zeng, N Y Lara, Developing a Beginner-Friendly HackathonE. Martinez, L. Yeh, L. Zeng, and N. Y. Lara, "Developing a Beginner- Friendly Hackathon," Recording of Workshop at Major League Hackcon VIII, September 2020. https://www.youtube.com/watch?v=EuBR6 jDk a8&t=196s Assessing the Needs of the Quantum Industry. C Hughes, D Finke, D.-A German, C Merzbacher, P M Vora, H J Lewandowski, arXiv preprintC. Hughes, D. Finke, D.-A. German, C. Merzbacher, P. M. Vora, and H. J. Lewandowski, "Assessing the Needs of the Quantum Industry," arXiv preprint, August 2021. https://arxiv.org/abs/2109.03601
[]
[ "Carousel Personalization in Music Streaming Apps with Contextual Bandits", "Carousel Personalization in Music Streaming Apps with Contextual Bandits" ]
[ "Walid Bendada ", "Guillaume Salha ", "Théo Bontempelli ", "France Deezer " ]
[]
[]
Media services providers, such as music streaming platforms, frequently leverage swipeable carousels to recommend personalized content to their users. However, selecting the most relevant items (albums, artists, playlists...) to display in these carousels is a challenging task, as items are numerous and as users have different preferences. In this paper, we model carousel personalization as a contextual multi-armed bandit problem with multiple plays, cascade-based updates and delayed batch feedback. We empirically show the effectiveness of our framework at capturing characteristics of real-world carousels by addressing a large-scale playlist recommendation task on a global music streaming mobile app. Along with this paper, we publicly release industrial data from our experiments, as well as an open-source environment to simulate comparable carousel personalization learning problems.
10.1145/3383313.3412217
[ "https://arxiv.org/pdf/2009.06546v2.pdf" ]
221,655,834
2009.06546
b623569fc4dcf45bfdb2c8ded237504fa4bc0868
Carousel Personalization in Music Streaming Apps with Contextual Bandits Walid Bendada Guillaume Salha Théo Bontempelli France Deezer Carousel Personalization in Music Streaming Apps with Contextual Bandits 10.1145/3383313.3412217ACM Reference Format: Walid Bendada, Guillaume Salha, and Théo Bontempelli. 2020. Carousel Personalization in Music Streaming Apps with Contextual Bandits. In Fourteenth ACM Conference on Recommender Systems (RecSys '20), September 22-26, 2020, Virtual Event, Brazil. ACM, New York, NY, USA, 9 pages. https://CCS Concepts: • Information systems → Recommender systemsPersonalizationContent ranking• Computing methodologies → Learning paradigmsOnline learning settingsSequential decision making Additional Key Words and Phrases: Multi-Armed Bandits with Multiple Plays, Contextual Bandits, Cascade Models, Expected Regret, Carousel Personalization, Playlist Recommendation, Music Streaming Platforms, A/B Testing Media services providers, such as music streaming platforms, frequently leverage swipeable carousels to recommend personalized content to their users. However, selecting the most relevant items (albums, artists, playlists...) to display in these carousels is a challenging task, as items are numerous and as users have different preferences. In this paper, we model carousel personalization as a contextual multi-armed bandit problem with multiple plays, cascade-based updates and delayed batch feedback. We empirically show the effectiveness of our framework at capturing characteristics of real-world carousels by addressing a large-scale playlist recommendation task on a global music streaming mobile app. Along with this paper, we publicly release industrial data from our experiments, as well as an open-source environment to simulate comparable carousel personalization learning problems. INTRODUCTION Recommending relevant and personalized content to users is crucial for media services providers, such as news [28], video [8] or music streaming [37] platforms. Indeed, effective recommender systems improve the users' experience and engagement on the platform, by helping them navigate through massive amounts of content, enjoy their favorite videos or songs, and discover new ones that they might like [4,37,46]. As a consequence, significant efforts were initiated to transpose promising research on these aspects to industrial-level applications [8,11,17,27,32,45]. In particular, many global mobile apps and websites, notably from the music streaming industry, currently leverage swipeable carousels to display recommended content on their homepages. These carousels, also referred to as sliders or shelves [32], consist in ranked lists of items or cards (albums, artists, playlists...). A few cards are initially displayed to the users, who can click on them or swipe on the screen to see some of the additional cards from the carousel. Selecting and ranking the most relevant cards to display is a challenging task [12,15,31,32], as the catalog size is usually significantly larger than the number of available slots in a carousel, and as users have different preferences. While being close to slate recommendation [16,18,20,40] and to learning to rank settings [30,34,36], carousel personalization also requires dealing with user feedback to adaptively improve the recommended content via online learning strategies [2,6,13], and integrating that some cards from the carousel might not be seen by users due to the swipeable structure. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. In this paper, we model carousel personalization as a multi-armed bandit with multiple plays [2] learning problem. Within our proposed framework, we account for important characteristics of real-world swipeable carousels, notably by considering that media services providers have access to contextual information on user preferences, that they might not know which cards from a carousel are actually seen by users, and that feedback data from carousels might not be available in real time. Focusing on music streaming applications, we show the effectiveness of our approach by addressing a large-scale carousel-based playlist recommendation task on the global mobile app Deezer 1 . With this paper, we also release industrial data from our experiments, as well as an open-source environment to simulate comparable carousel personalization learning problems. This paper is organized as follows. In Section 2, we introduce and formalize our multi-armed bandit framework for carousel personalization. We detail our data, our playlist recommendation task and our experimental setting in Section 3. We present and discuss our results in Section 4, and we conclude in Section 5. A CONTEXTUAL MULTI-ARMED BANDIT FRAMEWORK FOR CAROUSEL PERSONALIZATION In this section, after reviewing key notions on multi-armed bandits with multiple plays, we introduce our framework. Background on Multi-Armed Bandits with Multiple Plays Multi-armed bandits are among the most famous instances of sequential decision making problems [23,38,39]. Multiarmed bandits with multiple plays [2,22] involve K entities called arms. At each round t = 1, 2, ...,T , a forecaster has to select a set S t ⊂ {1, ..., K } of L < K arms (while L = 1 in the single play version of the problem [38]). The forecaster then receives some rewards from the selected arms, that we assume to be binary. The reward associated to an arm i ∈ S t is a sample drawn from a Bernoulli(p i ) distribution, with p i ∈ [0, 1] being an unknown parameter. Bernoulli distributions of arms 1, ..., K are assumed independent, which we later discuss. The objective of the forecaster is to maximize the sum of rewards received from the selected arms over time. It requires identifying the optimal set δ * (L) ⊂ {1, ..., K } of the L arms associated to the top-L highest Bernoulli parameters, i.e. the L highest expected rewards, as fast as possible. In such problems, the forecaster faces an exploration-exploitation dilemma. As the environment does not reveal the rewards of the unselected arms, the forecaster needs to try all arms over time to identify the best ones (exploration). However, selecting underperforming arms also leads to lower expected rewards, which encourages the forecaster to repeatedly select the assumed best ones (exploitation). Over the past years, several strategies have been proposed and studied, aiming at providing efficient trade-offs between these two opposite objectives when sequentially selecting sets S t . Notable examples include the Upper Confidence Bound (UCB) [3,6,26,44] and Thompson Sampling [5,22,42] algorithms (see Section 3). The expected cumulative regret Reg(T ) = T t =1 i ∈δ * (L) p i − i ∈S t p i , which represents the expected total loss endured by the forecaster by selecting non-optimal sets of arms at rounds 1 to T , is a common measure to compare the performances of strategies addressing this top-L best arms identification problem [2,6,22,38,39,44]. Multi-Armed Bandits with Multiple Plays for Carousel Personalization Throughout this paper, the K arms will correspond to a list of K cards/items, such as a catalog of albums or playlists in a music streaming app. They can be recommended to N users through a swipeable carousel containing L ≪ K slots. As users have various preferences, different cards can be displayed to different users. The L recommended cards from the carousel of each user, i.e. the L selected arms for each user, are updated at regular intervals or rounds, whose frequency depends on the technical constraints of the platform. We aim at optimizing display-to-stream rates, i.e. at identifying the L cards for which each user is the most likely to click and then to stream the underlying content, at least once during the round. When a card i is displayed to a user u, such streaming activity, i.e. a reward of 1, occurs during the round with an unknown probability p ui ∈ [0, 1]. Here, we assume that the number of cards, the number of users, and the display-to-stream probabilities p ui are fixed ; we later discuss these assumptions. A naive way to tackle this problem would consist in simultaneously running N standard bandit algorithms, aiming at individually identifying the top-L cards with highest p ui probabilities for each user u. This approach is actually unsuitable and would require a too long training time to reach convergence. Indeed, the number of display-to-stream parameters to estimate would be K × N , which is very large in practice as platforms often have millions of active users. In Section 2.3, we describe two strategies to address this problem by leveraging contextual information on user preferences. Leveraging Contextual Information on User Preferences 2.3.1 Semi-Personalization via User Clustering. First, let us assume that we have access to a clustering of users, constructed from users' past behaviours on the platform. Each user belongs to one of the Q groups C 1 , C 2 , ..., C Q with Q ≪ N . For instance, on a music streaming app, users from a same group would have homogeneous musical tastes. We propose to assume that users from a same group have identical expected display-to-stream probabilities for each card: ∀c ∈ {C 1 , ..., C Q }, ∀u ∈ c, ∀i ∈ {1, ..., K }, p ui = p ci .(1) Then, we simultaneously run Q bandit algorithms, one for each cluster, to identify the top-L best cards to recommend to each group. This strategy reduces the number of parameters to estimate to K × Q, which is significantly fewer than K × N in practice. Moreover, thanks to such users gathering, platforms receive more feedback on each displayed card w.r.t. the previous naive setting. This ensures a faster and more robust identification of optimal sets. However, the empirical performance of this strategy also strongly depends on the quality of the underlying user clustering. Contextual Multi-Armed Bandits. Instead of relying on clusters, let us now assume that we directly have access to a D-dimensional attribute vector x u ∈ R D for each user u. These vectors aim at summarizing user preferences on the platform, e.g. their musical tastes (in terms of genres, moods, countries...) for a music streaming app. We assume that the expected display-to-stream probabilities of a user u are functions of his/her attribute vector: ∀i ∈ {1, ..., K }, p ui = σ (x T u θ i ),(2) where θ 1 , ..., θ K are D-dimensional weight vectors to learn for each of the K arms, and where σ (·) is the sigmoid function: σ (x) = 1/(1 + e −x ) . This corresponds to the contextual bandit setting [1,7,28], a popular learning paradigm for online recommender systems [12,28,29,32,35,41,43,47,48]. Strategies to learn weight vectors are detailed e.g. in [5,32]. As D ≪ N in practice, such strategy also significantly reduces the number of parameters, to K × D. By design, users with similar preferences will have close expected display-to-stream probabilities. Moreover, all N users can end up with different optimal carousels, contrary to the aforementioned semi-personalized clustering approach. Capturing Characteristics of Real-World Carousels: Cascade-Based Updates, Delayed Feedback In our framework, we also aim at capturing other important characteristics of real-world swipeable carousels. In particular, while standard bandit algorithms usually consider that the forecaster receives rewards (0 or 1) from each of the L selected arms at each round, in our setting some selected cards might actually not be seen by users. As illustrated in Figure 1, only a few cards, say L init < L, are initially displayed on a user's screen. The user needs to swipe right to see additional cards. As we later verify, ignoring this important aspect, and thus returning a reward of 0 for all unclicked cards at each round whatever their rank in the carousel, would lead to underestimating display-to-stream probabilities. In this paper, we assume that we do not exactly know how many cards were seen by each user. Such assumption is consistent with Deezer's actual usage data and is realistic. Indeed, on many real-world mobile apps carousels, users usually do not click on any button to discover additional cards, but instead need to continuously swipe left and right on the screen. As a consequence, the card display information is ambiguous, and is technically hard to track with accuracy. Here, to address this problem, we consider and later evaluate a cascade-based arm update model. We draw inspiration from the cascade model [9], a popular approach to represent user behaviours when facing ranked lists of recommended items in an interface, with numerous applications and extensions [21,24,25,49]. At each round, we consider that: • An active user who did not stream any card during the round only saw the L init first ones. • An active user who streamed the i th card, with i ∈ {1, ..., L}, saw all cards from ranks 1 to max(L init , i). For instance, let L init = 3 and L = 12. The reward vectors obtained from users who a) did not stream during the round, b) only streamed the 2 nd card, and c) streamed the 2 nd and 6 th cards, are as follows, with X denoting no reward: a : [0, 0, 0, X , X , X , X , X , X , X , X , X ] b : [0, 1, 0, X , X , X , X , X , X , X , X , X ] c : [0, 1, 0, 0, 0, 1, X , X , X , X , X , X ] Last, to be consistent with real-world constraints, we assume that rewards are not processed on the fly but by batch, at the end of each round e.g. every day. We study the impact of such delayed batch feedback in our upcoming experiments. Related Work Bandits are very popular models for online recommendation [28, 29, 33-36, 41, 43, 47]. In particular, [12] and [32] also recently studied carousel personalization in mobile apps. [32] introduced a contextual bandit close to our Section 2.3.2. However, their approach focuses more on explainability, they do not model cascade-based displays and do not integrate semi-personalized strategies. [12] also considered contextual bandits inspired from [32] for playlist recommendation in carousels, but did not provide details on their models. They instead aimed at predicting the online ranking of these models from various offline evaluations. Last, other different sets of ordered items have been studied [16, 18-20, 40, 49]. EXPERIMENTAL SETTING In the following, we empirically evaluate and discuss the effectiveness of our carousel personalization framework. Playlist Recommendation on a Global Music Streaming App We study a large-scale carousel-based playlist recommendation task on the global mobile app Deezer. We consider K = 862 playlists, that were created by professional curators from Deezer with the purpose of complying with a specific music genre, cultural area or mood, and that are among the most popular ones on the service. Playlists' cover images constitute the cards that can be recommended to users on the app homepage in a carousel, updated on a daily basis, with L = 12 available slots and L init = 3 cards initially displayed. Figure 1 provides an illustration of the carousel. To determine which method would best succeed in making users click and stream the displayed playlists, extensive experiments were conducted in two steps. First, offline experiments simulating users' responses to carousel-based recommendations were run, on a simulation environment and on data that we both publicly release 2 with this paper (see Section 3.2). We believe that such industrial data and code release will benefit the research community and future works. Then, an online large-scale A/B test was run on the Deezer app to validate the findings of offline experiments. A Simulation Environment and Dataset for Offline Evaluation of Carousel-Based Recommendation For offline experiments, we designed a simulated environment in Python based on 974 960 fully anonymized Deezer users. We release a dataset in which each user u is described by a feature vector x u of dimension D = 97, computed internally by factorizing the interaction matrix between users and songs as described in [14] and then adding a bias term. A k-means clustering with Q = 100 clusters was also performed to assign each user to a single cluster. In addition, for each user-playlist pair, we release a "ground-truth" display-to-stream probability p ui = σ (x T u θ i ) where, as in [5], the D-dimensional vectors θ i were estimated by fitting a logistic regression on a click data history from January 2020. Simulations proceed as follows. At each round, a random subset of users (20 000, in the following) is presented to several sequential algorithms a.k.a. policies to be evaluated. These policies must then recommend an ordered set of L = 12 playlists to each user. Streams, i.e. positive binary rewards, are generated according to the aforementioned display-to-stream probabilities and to a configurable cascading browsing model capturing that users explore the carousel from left to right and might not see all recommended playlists. At the end of each round, all policies update their model based on the set of users and on binary rewards received from displayed playlists. Expected cumulative regrets of policies [2,22,39] w.r.t. the optimal top-L playlists sets according to p ui probabilities are computed. Algorithms In our experiments, we evaluate semi-personalized versions of several popular sequential decision making algorithms/policies, using the provided Q = 100 clusters, and compare their performances against fully-personalized methods. As detailed in Section 2.3.1, users within a given cluster share parameters for all semi-personalized policies; they are the ones whose names end with -seg in the following list. We consider the following methods: • random: a simple baseline that randomly recommends L playlists to each user. • ϵ-greedy-seg: recommends playlists randomly with probability ϵ, otherwise recommends the top-L with highest mean observed rewards. Two versions, ϵ-greedy-seg-explore (ϵ=0.1) and ϵ-greedy-seg-exploit (ϵ=0.01) are evaluated. • etc-seg: an explore then commit strategy, similar to random until all arms have been played n times, then recommends the top-L playlists. Two versions, etc-seg-explore (n = 100) and etc-seg-exploit (n = 20) are evaluated. • kl-ucb-seg: the Upper Confidence Bound (UCB) strategy [3,6,26], that tackles the exploration-exploitation trade-off by computing confidence intervals for the estimation of each arm probability, then selecting the L arms with highest upper confidence bounds. Here, we use KL-UCB bounds [10], tailored for Bernoulli rewards. • ts-seg: the Thompson Sampling strategy [5,42], in which estimated display-to-stream probabilities are samples drawn from Beta distributions [42], whose parameters are updated at each round in a Bayesian fashion, such that variance tends towards zero and expectation converges to empirical mean as more rewards are observed. Two versions, ts-seg-naive (prior distributions are Beta(1, 1), i.e. Uniform(0, 1)) and ts-seg-pessimistic (priors are Beta(1, 99)) are evaluated. As the UCB algorithm [6], Thompson Sampling is backed by strong theoretical guarantees [22] on speeds of expected cumulative regrets in the multi-armed bandit with multiple plays setting. • ts-lin: an extension of Thompson Sampling [5] to the linear contextual framework from Section 2.3.2. We follow the method of [5] to learn θ i vectors for each arm i from Gaussian prior distributions. Two versions, ts-lin-naive (0 means for all dimensions of the prior) and ts-lin-pessimistic (-5 mean for the bias dimension prior) are evaluated. By default, policies always abide by the cascade model introduced in Section 2.4, meaning they do not update the parameters relative to recommended playlists that the cascade model labels as unseen. For comparison, we also implemented versions of these policies that do not abide by this behaviour. In the following, they are labelled no-cascade. cumulative regret random etc-seg-explore etc-seg-exploit kl-ucb-seg ts-lin-naive ts-lin-pessimistic -greedy-seg-explore -greedy-seg-exploit ts-seg-naive ts-seg-pessimistic Figure 2 provides cumulative regrets over 100 rounds for the different policies, recommending playlists via our offline environment. Both etc-seg-explore and etc-seg-exploit behave as badly as random in the exploration phase, then, shortly after starting to exploit, they both reach competitive performances as illustrated by the brutal flattening of their cumulative regret curves, with etc-seg-exploit transitioning 50 rounds earlier. EXPERIMENTAL RESULTS Offline Evaluation Semi-Personalization vs Personalization. The later strategy also outperforms kl-ucb-seg, which shape suggests slow learning throughout the whole experiment. Moreover, both ts-lin-pessimistic and ts-lin-naive appear to stabilize to non-flat linear cumulative regret curves after only a few rounds. Pessimistic policies are overall more effective than their naive counterparts, which is due to their lower prior display-to-stream probabilities, that are more realistic. Overall, several semi-personalized policies eventually outclassed fully-personalized alternatives, with ts-seg-pessimistic already outperforming them all at the end of the first 25 rounds. This method manages to effectively exploit information and to quickly rank playlists, which is an interesting result, as fully-personalized contextual models were actually the only ones able to learn the exact display-to-stream probabilities (see generative process in Section 3.2), and as both frameworks have comparable numbers of parameters (K × Q vs K × D). While fully-personalized methods have been the focus of previous works on carousel recommendation [12,32], our experiments emphasize the empirical benefit of semi-personalization via user clustering that, assuming good underlying clusters, might appear as a suitable alternative for such large-scale real-world applications. Impact of Delayed Batch Feedback. In our experiments, to be consistent with real-world constraints, rewards are not processed on the fly but by batch, at the end of each round. We observe that, for semi-personalization, such setting tends to favor stochastic policies, such as the ts-seg or ϵ-greedy-seg ones, w.r.t. deterministic ones such as kl-ucb-seg. Indeed, as kl-ucb-seg selects arms in a deterministic fashion, it always proposes the same playlists to all users of a same cluster until the round is over. On the contrary, stochastic policies propose different playlists sets within a same cluster, ensuring a wider exploration during the round, which might explain why kl-ucb-seg underperforms in our experiments. ignored the cascade display model, and thus returned a 0 reward for all unstreamed playlists at each round, whatever their rank in the carousel. Only two policy pairs are displayed in Figure 3 for brevity. For both of them, the no-cascade variant is outperformed by policies integrating our proposed cascade-based update model from Section 2.4. This result validates the relevance of capturing such phenomenon for our carousel-based personalization problem. Online Experiments An industrial-scale A/B test has been run in February 2020, to verify whether results from the simulations would hold on the actual Deezer mobile app. The 12 recommended playlists from each user's carousel were updated on a daily basis on the app. Due to industrial constraints, only a subset of policies, from (naive) Thompson Sampling, were tested in production. Also, for confidentiality reasons, we do not report the exact number of users involved in each cohort, nor the precise display-to-stream rates. Instead, results are expressed in Figure 4 in terms of relative display-to-stream rates gains w.r.t. random-top-100, an internal baseline that randomly recommends 12 playlists from a subset of 100, pre-selected for each cluster from internal heuristics. Results confirm the superiority of the proposed multi-armed bandit framework for personalization, notably the semi-personalized strategy, and the empirical benefit of integrating a cascade model for arms updates, although users might actually have more complex behaviours on the platform. CONCLUSION AND DISCUSSION In this paper, we modeled carousel personalization as a contextual multi-armed bandit problem with multiple plays. By addressing a challenging playlist recommendation task, we highlighted the benefits of our framework, notably the integration of the cascade model and of semi-personalization via user clustering. Along with this paper, we publicly release a large-scale dataset of user preferences for curated playlists on Deezer, and an open-source environment to recreate comparable learning problems. We believe that such release will benefit future research on carousel personalization. In particular, we assumed that the number of users and cards was fixed throughout the rounds, which is a limit, that could initiate future studies on the integration of new users or new recommendable content in swipeable carousels. Moreover, our work, as most previous efforts, also assumes that arms/cards distributions are fixed and independent, which might be unrealistic. A playlist's relative interest might depend on its neighbors in the carousel, and individually selecting the top-L playlists does not always lead to the best set of L playlists, e.g. in terms of musical diversity. Future works in this direction would definitely lead towards the improvement of carousel personalization. Fig. 1 . 1Swipeable carousels on the app. Fig. 2 . 2Offline evaluation of top-12 playlist recommendation: expected cumulative regrets of policies over 100 simulated rounds. The empirical gain of ts-seg-pessimistic w.r.t. others is statistically significant at the 1% level (p-value <0.01). vs No-Cascade. All policies from Figure 2 abide by the cascade model introduced in Section 2.4. In Figure 3 , 3we report results from follow-up experiments, aiming at measuring the empirical benefit of taking into account this cascading behaviour of users when browsing a sequence of playlists. We compared policies to alternatives that greedy-seg-explore-no-cascade -greedy-seg-explore-cascade ts-seg-pessimistic-no-cascade ts-seg-pessimistic-cascadeFig. 3. Offline evaluation: comparison of cascade vs no-cascade, over 100 simulated rounds. Differences at final round are statistically significant at the 1% level (p-value <0.01).Fig. 4. Online A/B test evaluation: relative display-to-stream gains w.r.t. random-top-100 baseline (see Section 4.2). Differences are statistically significant at the 1% level (p-value <0.01).0.90 0.95 1.00 1.05 1.10 1.15 1.20 relative display-to-stream rate ts-seg-cascade ts-seg-no-cascade ts-lin-cascade random-top-100 https://www.deezer.com Data and code are available at: https://github.com/deezer/carousel_bandits Taming the monster: A fast and simple algorithm for contextual bandits. Alekh Agarwal, Daniel Hsu, Satyen Kale, John Langford, Lihong Li, Robert Schapire, Proceedings of the 31st International Conference on Machine Learning. the 31st International Conference on Machine LearningAlekh Agarwal, Daniel Hsu, Satyen Kale, John Langford, Lihong Li, and Robert Schapire. 2014. Taming the monster: A fast and simple algorithm for contextual bandits. In Proceedings of the 31st International Conference on Machine Learning. 1638-1646. Asymptotically efficient allocation rules for the multiarmed bandit problem with multiple plays-Part I: IID rewards. Venkatachalam Anantharam, Pravin Varaiya, Jean Walrand, IEEE Trans. Automat. Control. 32Venkatachalam Anantharam, Pravin Varaiya, and Jean Walrand. 1987. Asymptotically efficient allocation rules for the multiarmed bandit problem with multiple plays-Part I: IID rewards. IEEE Trans. Automat. Control 32, 11 (1987), 968-976. Finite-time analysis of the multiarmed bandit problem. Peter Auer, Nicolo Cesa-Bianchi, Paul Fischer, Machine Learning. 473Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. 2002. Finite-time analysis of the multiarmed bandit problem. Machine Learning 47, 2-3 (2002), 235-256. Recommender systems survey. Knowledge-Based Systems. Jesús Bobadilla, Fernando Ortega, Antonio Hernando, Abraham Gutiérrez, 46Jesús Bobadilla, Fernando Ortega, Antonio Hernando, and Abraham Gutiérrez. 2013. Recommender systems survey. Knowledge-Based Systems 46 (2013), 109-132. An empirical evaluation of Thompson sampling. Olivier Chapelle, Lihong Li, Advances in Neural Information Processing Systems. Olivier Chapelle and Lihong Li. 2011. An empirical evaluation of Thompson sampling. In Advances in Neural Information Processing Systems. 2249-2257. Combinatorial multi-armed bandit and its extension to probabilistically triggered arms. Wei Chen, Yajun Wang, Yang Yuan, Qinshi Wang, Journal of Machine Learning Research. 17Wei Chen, Yajun Wang, Yang Yuan, and Qinshi Wang. 2016. Combinatorial multi-armed bandit and its extension to probabilistically triggered arms. Journal of Machine Learning Research 17, 1 (2016), 1746-1778. Contextual bandits with linear payoff functions. Wei Chu, Lihong Li, Lev Reyzin, Robert Schapire, Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics. the Fourteenth International Conference on Artificial Intelligence and StatisticsWei Chu, Lihong Li, Lev Reyzin, and Robert Schapire. 2011. Contextual bandits with linear payoff functions. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics. 208-214. Deep neural networks for Youtube recommendations. Paul Covington, Jay Adams, Emre Sargin, Proceedings of the 10th ACM Conference on Recommender Systems. the 10th ACM Conference on Recommender SystemsPaul Covington, Jay Adams, and Emre Sargin. 2016. Deep neural networks for Youtube recommendations. In Proceedings of the 10th ACM Conference on Recommender Systems. 191-198. An experimental comparison of click position-bias models. Nick Craswell, Onno Zoeter, Michael Taylor, Bill Ramsey, Proceedings of the 2008 International Conference on Web Search and Data Mining. the 2008 International Conference on Web Search and Data MiningNick Craswell, Onno Zoeter, Michael Taylor, and Bill Ramsey. 2008. An experimental comparison of click position-bias models. In Proceedings of the 2008 International Conference on Web Search and Data Mining. 87-94. The KL-UCB algorithm for bounded stochastic bandits and beyond. Aurélien Garivier, Olivier Cappé, Proceedings of the 24th Annual Conference on Learning Theory. the 24th Annual Conference on Learning TheoryAurélien Garivier and Olivier Cappé. 2011. The KL-UCB algorithm for bounded stochastic bandits and beyond. In Proceedings of the 24th Annual Conference on Learning Theory. 359-376. The Netflix recommender system: Algorithms, business value, and innovation. A Carlos, Neil Gomez-Uribe, Hunt, ACM Transactions on Management Information Systems (TMIS). 6Carlos A Gomez-Uribe and Neil Hunt. 2015. The Netflix recommender system: Algorithms, business value, and innovation. ACM Transactions on Management Information Systems (TMIS) 6, 4 (2015), 1-19. Offline evaluation to make decisions about playlist recommendation algorithms. Alois Gruson, Praveen Chandar, Christophe Charbuillet, James Mcinerney, Samantha Hansen, Damien Tardieu, Ben Carterette, Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining. the Twelfth ACM International Conference on Web Search and Data MiningAlois Gruson, Praveen Chandar, Christophe Charbuillet, James McInerney, Samantha Hansen, Damien Tardieu, and Ben Carterette. 2019. Offline evaluation to make decisions about playlist recommendation algorithms. In Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining. 420-428. C H Steven, Doyen Hoi, Jing Sahoo, Peilin Lu, Zhao, arXiv:1802.02871Online learning: A comprehensive survey. arXiv preprintSteven CH Hoi, Doyen Sahoo, Jing Lu, and Peilin Zhao. 2018. Online learning: A comprehensive survey. arXiv preprint arXiv:1802.02871 (2018). Collaborative filtering for implicit feedback datasets. Yifan Hu, Yehuda Koren, Chris Volinsky, Proceedings of the Eighth IEEE International Conference on Data Mining. the Eighth IEEE International Conference on Data MiningIeeeYifan Hu, Yehuda Koren, and Chris Volinsky. 2008. Collaborative filtering for implicit feedback datasets. In Proceedings of the Eighth IEEE International Conference on Data Mining. Ieee, 263-272. Contextual fact ranking and its applications in table synthesis and compression. Silu Huang, Jialu Liu, Flip Korn, Xuezhi Wang, You Wu, Dale Markowitz, Cong Yu, Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data MiningSilu Huang, Jialu Liu, Flip Korn, Xuezhi Wang, You Wu, Dale Markowitz, and Cong Yu. 2019. Contextual fact ranking and its applications in table synthesis and compression. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. 285-293. SLATEQ: A tractable decomposition for reinforcement learning with recommendation sets. Eugene Ie, Vihan Jain, Jing Wang, Sanmit Narvekar, Ritesh Agarwal, Rui Wu, Heng-Tze Cheng, Tushar Chandra, Craig Boutilier, Proceedings of the 28th International Joint Conference on Artificial Intelligence. the 28th International Joint Conference on Artificial IntelligenceAAAI PressEugene Ie, Vihan Jain, Jing Wang, Sanmit Narvekar, Ritesh Agarwal, Rui Wu, Heng-Tze Cheng, Tushar Chandra, and Craig Boutilier. 2019. SLATEQ: A tractable decomposition for reinforcement learning with recommendation sets. In Proceedings of the 28th International Joint Conference on Artificial Intelligence. AAAI Press, 2592-2599. Music personalization at Spotify. Kurt Jacobson, Vidhya Murali, Edward Newett, Brian Whitman, Romain Yon, Proceedings of the 10th ACM Conference on Recommender Systems. the 10th ACM Conference on Recommender SystemsKurt Jacobson, Vidhya Murali, Edward Newett, Brian Whitman, and Romain Yon. 2016. Music personalization at Spotify. In Proceedings of the 10th ACM Conference on Recommender Systems. 373-373. Optimizing slate recommendations via slate-CVAE. Ray Jiang, Sven Gowal, Timothy A Mann, Danilo J Rezende, arXiv:1803.01682arXiv preprintRay Jiang, Sven Gowal, Timothy A Mann, and Danilo J Rezende. 2018. Optimizing slate recommendations via slate-CVAE. arXiv preprint arXiv:1803.01682 (2018). Beyond greedy ranking: Slate optimization via list-CVAE. Ray Jiang, Sven Gowal, Timothy A Mann, Danilo J Rezende, Proceedings of the Eighth International Conference on Learning Representations. the Eighth International Conference on Learning RepresentationsRay Jiang, Sven Gowal, Timothy A Mann, and Danilo J Rezende. 2019. Beyond greedy ranking: Slate optimization via list-CVAE. Proceedings of the Eighth International Conference on Learning Representations (2019). Non-stochastic bandit slate problems. Satyen Kale, Lev Reyzin, Robert E Schapire, Advances in Neural Information Processing Systems. Satyen Kale, Lev Reyzin, and Robert E Schapire. 2010. Non-stochastic bandit slate problems. In Advances in Neural Information Processing Systems. 1054-1062. DCM bandits: Learning to rank with multiple clicks. Sumeet Katariya, Branislav Kveton, Csaba Szepesvari, Zheng Wen, Proceedings of the 33rd International Conference on Machine Learning. the 33rd International Conference on Machine LearningSumeet Katariya, Branislav Kveton, Csaba Szepesvari, and Zheng Wen. 2016. DCM bandits: Learning to rank with multiple clicks. In Proceedings of the 33rd International Conference on Machine Learning. 1215-1224. Optimal regret analysis of Thompson sampling in stochastic multi-armed bandit problem with multiple plays. Junpei Komiyama, Junya Honda, Hiroshi Nakagawa, Proceedings of the 32nd International Conference on Machine Learning. the 32nd International Conference on Machine LearningJunpei Komiyama, Junya Honda, and Hiroshi Nakagawa. 2015. Optimal regret analysis of Thompson sampling in stochastic multi-armed bandit problem with multiple plays. Proceedings of the 32nd International Conference on Machine Learning (2015). Algorithms for multi-armed bandit problems. Volodymyr Kuleshov, Doina Precup, arXiv:1402.6028arXiv preprintVolodymyr Kuleshov and Doina Precup. 2014. Algorithms for multi-armed bandit problems. arXiv preprint arXiv:1402.6028 (2014). Cascading bandits: Learning to rank in the cascade model. Csaba Branislav Kveton, Zheng Szepesvari, Azin Wen, Ashkan, Proceedings of the 32nd International Conference on Machine Learning. the 32nd International Conference on Machine LearningBranislav Kveton, Csaba Szepesvari, Zheng Wen, and Azin Ashkan. 2015. Cascading bandits: Learning to rank in the cascade model. In Proceedings of the 32nd International Conference on Machine Learning. 767-776. Multiple-play bandits in the position-based model. Paul Lagrée, Claire Vernade, Olivier Cappe, Advances in Neural Information Processing Systems. Paul Lagrée, Claire Vernade, and Olivier Cappe. 2016. Multiple-play bandits in the position-based model. In Advances in Neural Information Processing Systems. 1597-1605. Asymptotically efficient adaptive allocation rules. Advances in Applied Mathematics. Tze Leung Lai and Herbert Robbins6Tze Leung Lai and Herbert Robbins. 1985. Asymptotically efficient adaptive allocation rules. Advances in Applied Mathematics 6, 1 (1985), 4-22. Large-scale real-time product recommendation at Criteo. Romain Lerallut, Diane Gasselin, Nicolas Le Roux, Proceedings of the 9th ACM Conference on Recommender Systems. the 9th ACM Conference on Recommender SystemsRomain Lerallut, Diane Gasselin, and Nicolas Le Roux. 2015. Large-scale real-time product recommendation at Criteo. In Proceedings of the 9th ACM Conference on Recommender Systems. 232-232. A contextual-bandit approach to personalized news article recommendation. Lihong Li, Wei Chu, John Langford, Robert E Schapire, Proceedings of the 19th International Conference on World Wide Web. the 19th International Conference on World Wide WebLihong Li, Wei Chu, John Langford, and Robert E Schapire. 2010. A contextual-bandit approach to personalized news article recommendation. In Proceedings of the 19th International Conference on World Wide Web. 661-670. Unbiased offline evaluation of contextual-bandit-based news article recommendation algorithms. Lihong Li, Wei Chu, John Langford, Xuanhui Wang, Proceedings of the fourth ACM International Conference on Web Search and Data Mining. the fourth ACM International Conference on Web Search and Data MiningLihong Li, Wei Chu, John Langford, and Xuanhui Wang. 2011. Unbiased offline evaluation of contextual-bandit-based news article recommendation algorithms. In Proceedings of the fourth ACM International Conference on Web Search and Data Mining. 297-306. Learning to rank for information retrieval. Tie-Yan Liu, Foundations and Trends in Information Retrieval. 3Tie-Yan Liu et al. 2009. Learning to rank for information retrieval. Foundations and Trends in Information Retrieval 3, 3 (2009), 225-331. An introduction to entity recommendation and understanding. Hao Ma, Yan Ke, Proceedings of the 24th International Conference on World Wide Web. the 24th International Conference on World Wide WebHao Ma and Yan Ke. 2015. An introduction to entity recommendation and understanding. In Proceedings of the 24th International Conference on World Wide Web. 1521-1522. Explore, exploit, and explain: personalizing explainable recommendations with bandits. James Mcinerney, Benjamin Lacker, Samantha Hansen, Karl Higley, Hugues Bouchard, Proceedings of the 12th ACM Conference on Recommender Systems. the 12th ACM Conference on Recommender SystemsAlois Gruson, and Rishabh MehrotraJames McInerney, Benjamin Lacker, Samantha Hansen, Karl Higley, Hugues Bouchard, Alois Gruson, and Rishabh Mehrotra. 2018. Explore, exploit, and explain: personalizing explainable recommendations with bandits. In Proceedings of the 12th ACM Conference on Recommender Systems. 31-39. Dynamic clustering of contextual multi-armed bandits. T Trong, Nguyen, W Hady, Lauw, Proceedings of the 23rd ACM International Conference on Conference on Information and Knowledge Management. the 23rd ACM International Conference on Conference on Information and Knowledge ManagementTrong T Nguyen and Hady W Lauw. 2014. Dynamic clustering of contextual multi-armed bandits. In Proceedings of the 23rd ACM International Conference on Conference on Information and Knowledge Management. 1959-1962. Online learning to rank for sequential music recommendation. L Bruno, Alberto Pereira, Gustavo Ueda, Penha, L T Rodrygo, Nivio Santos, Ziviani, Proceedings of the 13th ACM Conference on Recommender Systems. the 13th ACM Conference on Recommender SystemsBruno L Pereira, Alberto Ueda, Gustavo Penha, Rodrygo LT Santos, and Nivio Ziviani. 2019. Online learning to rank for sequential music recommendation. In Proceedings of the 13th ACM Conference on Recommender Systems. 237-245. Contextual combinatorial bandit and its application on diversified online recommendation. Lijing Qin, Shouyuan Chen, Xiaoyan Zhu, Proceedings of the 2014 SIAM International Conference on Data Mining. SIAM. the 2014 SIAM International Conference on Data Mining. SIAMLijing Qin, Shouyuan Chen, and Xiaoyan Zhu. 2014. Contextual combinatorial bandit and its application on diversified online recommendation. In Proceedings of the 2014 SIAM International Conference on Data Mining. SIAM, 461-469. Learning diverse rankings with multi-armed bandits. Filip Radlinski, Robert Kleinberg, Thorsten Joachims, Proceedings of the 25th International Conference on Machine Learning. the 25th International Conference on Machine LearningFilip Radlinski, Robert Kleinberg, and Thorsten Joachims. 2008. Learning diverse rankings with multi-armed bandits. In Proceedings of the 25th International Conference on Machine Learning. 784-791. Current challenges and visions in music recommender systems research. Markus Schedl, Hamed Zamani, Ching-Wei Chen, Yashar Deldjoo, Mehdi Elahi, International Journal of Multimedia Information Retrieval. 7Markus Schedl, Hamed Zamani, Ching-Wei Chen, Yashar Deldjoo, and Mehdi Elahi. 2018. Current challenges and visions in music recommender systems research. International Journal of Multimedia Information Retrieval 7, 2 (2018), 95-116. Introduction to multi-armed bandits. Aleksandrs Slivkins, Foundations and Trends in Machine Learning. 122Aleksandrs Slivkins et al. 2019. Introduction to multi-armed bandits. Foundations and Trends in Machine Learning 12, 1-2 (2019), 1-286. Reinforcement learning: An introduction. S Richard, Andrew G Sutton, Barto, Richard S Sutton and Andrew G Barto. 2011. Reinforcement learning: An introduction. (2011). Off-policy evaluation for slate recommendation. Adith Swaminathan, Akshay Krishnamurthy, Alekh Agarwal, Miro Dudik, John Langford, Damien Jose, Imed Zitouni, Advances in Neural Information Processing Systems. Adith Swaminathan, Akshay Krishnamurthy, Alekh Agarwal, Miro Dudik, John Langford, Damien Jose, and Imed Zitouni. 2017. Off-policy evaluation for slate recommendation. In Advances in Neural Information Processing Systems. 3632-3642. Ensemble contextual bandits for personalized recommendation. Liang Tang, Yexi Jiang, Lei Li, Tao Li, Proceedings of the 8th ACM Conference on Recommender Systems. the 8th ACM Conference on Recommender SystemsLiang Tang, Yexi Jiang, Lei Li, and Tao Li. 2014. Ensemble contextual bandits for personalized recommendation. In Proceedings of the 8th ACM Conference on Recommender Systems. 73-80. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. William R Thompson, Biometrika. 254William R Thompson. 1933. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. Biometrika 25, 3/4 (1933), 285-294. Efficient ordered combinatorial semi-bandits for whole-page recommendation. Yingfei Wang, Hua Ouyang, Chu Wang, Jianhui Chen, Tsvetan Asamov, Yi Chang, Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence. the Thirty-First AAAI Conference on Artificial IntelligenceYingfei Wang, Hua Ouyang, Chu Wang, Jianhui Chen, Tsvetan Asamov, and Yi Chang. 2017. Efficient ordered combinatorial semi-bandits for whole-page recommendation. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence. Regional Multi-Armed Bandits. Zhiyang Wang, Ruida Zhou, Cong Shen, Proceedings of the 21st International Conference on Artificial Intelligence and Statistics. the 21st International Conference on Artificial Intelligence and StatisticsZhiyang Wang, Ruida Zhou, and Cong Shen. 2018. Regional Multi-Armed Bandits. In Proceedings of the 21st International Conference on Artificial Intelligence and Statistics. 510-518. Graph convolutional neural networks for web-scale recommender systems. Rex Ying, Ruining He, Kaifeng Chen, Pong Eksombatchai, L William, Jure Hamilton, Leskovec, Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data MiningRex Ying, Ruining He, Kaifeng Chen, Pong Eksombatchai, William L Hamilton, and Jure Leskovec. 2018. Graph convolutional neural networks for web-scale recommender systems. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. 974-983. Deep learning based recommender system: A survey and new perspectives. Shuai Zhang, Lina Yao, Aixin Sun, Yi Tay, ACM Computing Surveys (CSUR). 52Shuai Zhang, Lina Yao, Aixin Sun, and Yi Tay. 2019. Deep learning based recommender system: A survey and new perspectives. ACM Computing Surveys (CSUR) 52, 1 (2019), 1-38. Latent contextual bandits and their application to personalized recommendations for new users. Li Zhou, Emma Brunskill, Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence. the Twenty-Fifth International Joint Conference on Artificial IntelligenceLi Zhou and Emma Brunskill. 2016. Latent contextual bandits and their application to personalized recommendations for new users. In Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence. 3646-3653. Online learning to rank in stochastic click models. Masrour Zoghi, Tomas Tunys, Mohammad Ghavamzadeh, Branislav Kveton, Csaba Szepesvari, Zheng Wen, Proceedings of the 34th International Conference on Machine Learning. JMLR. org. the 34th International Conference on Machine Learning. JMLR. orgMasrour Zoghi, Tomas Tunys, Mohammad Ghavamzadeh, Branislav Kveton, Csaba Szepesvari, and Zheng Wen. 2017. Online learning to rank in stochastic click models. In Proceedings of the 34th International Conference on Machine Learning. JMLR. org, 4199-4208. Cascading bandits for large-scale recommendation problems. Shi Zong, Hao Ni, Kenny Sung, Nan Rosemary Ke, Zheng Wen, Branislav Kveton, Proceedings of the Conference on Uncertainty in Artificial Intelligence. the Conference on Uncertainty in Artificial IntelligenceShi Zong, Hao Ni, Kenny Sung, Nan Rosemary Ke, Zheng Wen, and Branislav Kveton. 2016. Cascading bandits for large-scale recommendation problems. In Proceedings of the Conference on Uncertainty in Artificial Intelligence. 835-844.
[ "https://github.com/deezer/carousel_bandits" ]
[ "Salient Instance Segmentation via Subitizing and Clustering", "Salient Instance Segmentation via Subitizing and Clustering" ]
[ "Jialun Pei ", "He Tang ", "Chao Liu ", "Chuanbo Chen " ]
[]
[]
The goal of salient region detection is to identify the regions of an image that attract the most attention. Many methods have achieved state-of-the-art performance levels on this task. Recently, salient instance segmentation has become an even more challenging task than traditional salient region detection; however, few of the existing methods have concentrated on this underexplored problem. Unlike the existing methods, which usually employ object proposals to roughly count and locate object instances, our method applies salient objects subitizing to predict an accurate number of instances for salient instance segmentation. In this paper, we propose a multitask densely connected neural network (MDNN) to segment salient instances in an image. In contrast to existing approaches, our framework is proposal-free and category-independent. The MDNN contains two parallel branches: the first is a densely connected subitizing network (DSN) used for subitizing prediction; the second is a densely connected fully convolutional network (DFCN) used for salient region detection. The MDNN simultaneously outputs saliency maps and salient object subitizing. Then, an adaptive deep feature-based spectral clustering operation segments the salient regions into instances based on the subitizing and saliency maps. The experimental results on both salient region detection and salient instance segmentation datasets demonstrate the satisfactory performance of our framework. Notably, its AP r @0.7 reaches 60.14% in the salient instance dataset, surpasses the state-of-the-art methods by about 5%.
10.1016/j.neucom.2020.04.022
[ "https://arxiv.org/pdf/1909.13240v1.pdf" ]
203,593,326
1909.13240
573218b76b6873e4e4cd9cd160eff2bec0d598bb
Salient Instance Segmentation via Subitizing and Clustering Jialun Pei He Tang Chao Liu Chuanbo Chen Salient Instance Segmentation via Subitizing and Clustering 1Index Terms-Saliency detectioninstance segmentationsubitizingmultitask networks The goal of salient region detection is to identify the regions of an image that attract the most attention. Many methods have achieved state-of-the-art performance levels on this task. Recently, salient instance segmentation has become an even more challenging task than traditional salient region detection; however, few of the existing methods have concentrated on this underexplored problem. Unlike the existing methods, which usually employ object proposals to roughly count and locate object instances, our method applies salient objects subitizing to predict an accurate number of instances for salient instance segmentation. In this paper, we propose a multitask densely connected neural network (MDNN) to segment salient instances in an image. In contrast to existing approaches, our framework is proposal-free and category-independent. The MDNN contains two parallel branches: the first is a densely connected subitizing network (DSN) used for subitizing prediction; the second is a densely connected fully convolutional network (DFCN) used for salient region detection. The MDNN simultaneously outputs saliency maps and salient object subitizing. Then, an adaptive deep feature-based spectral clustering operation segments the salient regions into instances based on the subitizing and saliency maps. The experimental results on both salient region detection and salient instance segmentation datasets demonstrate the satisfactory performance of our framework. Notably, its AP r @0.7 reaches 60.14% in the salient instance dataset, surpasses the state-of-the-art methods by about 5%. I. INTRODUCTION T HE seminal work by Itti et al. illustrates that the most arresting objects are usually visually salient [1]. Salient object detection models aim at highlighting the most attractive regions of an image, simulating the visual attention process in human visual systems [2]. In contrast to image segmentation and target recognition, saliency maps are determined by only the most salient objects [3]. Salient object detection has recently attracted considerable interest in computer vision tasks such as scene understanding [4], image editing [5], image retrieval [6], video summarization [7] and robotic perception [8]. Over the past two decades, saliency detection has played an increasingly important role in computer vision problems. The accuracy of salient object detection has improved rapidly due to the renaissance of convolutional neural network (CNN) models [9], [10], which have shown superior performance over traditional solutions [11]- [13]. Due to the multilevel and multiscale features extracted by CNNs, the most salient objects can be captured with high precision [14]. However, the previous works like [15]- [17] focused only on the salient regions and overlooked the individual instances of salient regions. Currently, a new challenging task has gradually attracted widespread attention: instance-level salient object segmentation (or salient instance segmentation for short), which was first proposed by et al. [18]. Compared to salient object detection, salient instance segmentation attempts to discriminate individual object instances in the salient regions of an image or scene. The resulting instance-level saliency maps foster a more detailed analysis by labeling each instance with an accurate pixelwise segmentation map. Compared to general instance segmentation, salient instance segmentation predicts salient object instances without categorizing the objects and has more specific and in-depth application areas than does salient object detection, including target recognition, driver assistance and image captioning [19]. Salient instance segmentation is challenging because the salient regions exhibit boundaries that are often obscured or overlapped with other instances that possess similar features. Moreover, the high occlusion probability and the diverse shape deformations have a negative effect on the ability to distinguish intersecting instances [20]. In general, the existing methods locate individual instances in salient regions with the aid of proposals. Li et al. combined salient object proposals and contours to generate the final instances [18]. S4Net also uses RPN (Region Proposal Network) to generate a number of proposals, and then uses SID (salient instance discriminator) to segment the salient instances [21]. However, the proposalbased approach has some important drawbacks. First, region proposal schemes typically generate hundreds of candidate bounding boxes, and filtering these numerous proposals to obtain an optimized subset is inefficient [22], [23]. Second, the proposed bounding boxes do not produce boundaries for strongly occluded objects. Consequently, the post-processing algorithm has considerably difficulty detecting highly accurate object boundaries. In this paper, we propose a proposal-free method to effectively solve the salient instance segmentation task. The pipeline of the salient instance segmentation process used by the proposed method is illustrated in Fig. 1. Different from existing methods based on object proposals, the proposalfree method applies salient object subitizing process rather than proposals to avoid the aforementioned disadvantages of proposals [24]. In addition, some proposal methods used for object detection are category-dependent, which makes them inappropriate for use in salient instance segmentation because the goal is to distinguish individual instances rather than perform category-level segmentation. We focus on category-arXiv:1909.13240v1 [cs.CV] 29 Sep 2019 independent salient instance segmentation and propose a multitask neural network that applies two parallel branches a densely connected subitizing network (DSN) to predict the number of instances and a densely connected fully convolutional network (DFCN) for salient region detection. Then, an adaptive deep feature-based spectral clustering operation combines the subitizing and saliency maps to generate promising instance-level salient object segmentation results [25]. The proposed salient instance segmentation task is divided into three subtasks: saliency detection, subitizing prediction, and clustering of the salient instances. First, we use a DSN to perform salient object subitizing. We were motivated to use a subitizing model to segment salient instances by Zhang et al. [24]. The fully connected layer of DenseNet is applied to predict the number of instances, and the accuracy rate of this approach is better than that of GoogleNet, which was applied in [24]. Second, we extend DenseNet [26] to our DFCN by adding upsampling layers. It is worth noting that a fully convolutional network (FCN) is regarded as a CNN extension that recovers the full resolution of the input images, while CNNs are appropriate for image classification prediction tasks [27]. To reduce the resolution loss and retain the multiscale features from the downsampling path, we build skip connections between the downsampling and upsampling paths. In addition, a fully connected Conditional Random Field (CRF) works as a pixelwise refined model to purify the salient region maps [28]. Our network outputs full-resolution salient maps with high accuracy and outperforms other state-of-the-art methods. We provide salient region detection result comparisons in Section IV. Third, the subitizing operation provides support for the initial number of clusters k of the spectral clustering algorithm. Therefore, spectral clustering can be used to merge the salient region maps and perform subitizing to output the final results. To improve the spectral clustering effectiveness, the deep features of DFCN are extracted as input to the spectral clustering algorithm. In addition, we apply the simple linear iterative clustering (SLIC) algorithm to the clustering process, which boosts the instance edge segmentation effect during the kmeans spectral clustering step [29]. At the clustering step in spectral clustering, we adopt a k-means algorithm that improves the selection of initial clustering centers. The proposed MDNN achieves excellent performance and surpasses all the existing methods when applied to the only available salient instance segmentation dataset (dataset1K) in [18]. Moreover, our saliency model outperforms other state-of-the-art models on public salient object detection datasets. In summary, our contributions are as follows: • We propose a multitask densely connected neural network (MDNN) to address the challenging salient instance segmentation task. The proposed method is proposal-free and category-independent, which improve the robustness of the algorithm. • A subitizing process is adopted to predict the number of instances using a DSN instead of region proposal methods. Moreover, we detect salient objects by upgrading DenseNet to a DFCN that produces accurate, fullresolution saliency maps. • We apply an adaptive spectral clustering operation to produce the final salient instances. We improve the adjacency matrix W of spectral clustering by adding multiscale deep features, which improves the resulting performance. In addition, we develop an adaptive quantile strategy instead of using random selection that allows us to flexibly and automatically select the initial clustering centers. The remainder of this paper is organized as follows. Section II introduces the related works. Section III describes the theoretical basis and the architecture of the proposed method. Section IV presents the experimental comparisons and discussions. Finally, Section V concludes the paper. II. RELATED WORK Thanks to the deep convolutional neural networks that have been developed for salient object detection, the current salient object segmentation method is good enough to function as an intermediate process for salient instance segmentation [30]. However, due to the category-independent nature of salient object segmentation, salient instance segmentation cannot be regarded as simply a subbranch of instance segmentation. To understand the task of salient instance segmentation, in essence, we can consider salient object detection, instancelevel semantic segmentation and salient instance segmentation as three different types of image segmentation problems. A. Salient object detection Salient object detection is usually divided into two phases: the first phase detects the most salient regions in the image, and the second phase accurately segments salient objects from the salient regions [31]. Liu et al. [32] and Achanta et al. [33] presented salient region detection as an outgrowth of the binary object segmentation problem. Traditional salient object detection depends primarily on traditional machine learning methods such as bottom-up and top-down methods based on multilevel features [34] [37]. With the development of deep CNNs, salient object detection has gradually become dominated by deep neural network model methods. Kruthiventi et al. used a fully convolutional neural network to simultaneously perform eyefixation and salient object detection [38]. Li et al. introduced semantic features into a multitask convolutional network to assist in salient object detection [39]. These latest end-toend networks improve both accuracy and efficiency compared to the earlier traditional methods. Using newer neural network models, the salient object detection problem can be determined using a fully convolutional neural network similar to semantic segmentation. The proposed multitask densely connected neural network (MDNN) sets the output layer of the DFCN to two channels, which respectively classify the salient regions and background in the image. B. Instance-Aware semantic segmentation The earlier instance-aware semantic segmentation is defined as a multitask operation consisting of object detection and semantic segmentation. The limitation of this task is that object detection estimates the bounding boxes only by using segmentation methods, and semantic segmentation simply predicts the category of each pixel. To overcome this constraint, Hariharan et al. presented simultaneous detection and segmentation (SDS) to detect all instances of a category in an image [40]. Subsequently, Dai et al. used a multitask cascade network to detect the proposals and then extracted contours to classify the objects [41]. In [42], the authors proposed a model named instance-sensitive fully convolutional networks that integrate feature proposals in different locations into a complete mask, which is different from traditional FCNs. In particular, a Mask R-CNN framework was presented to promote the Faster R-CNN architecture by adding a branch to predict an object mask [43], [44]. Despite the relatively fast development of semantic instance segmentation, these frameworks rely on defined categories. In contrast, for categoryindependent salient instance segmentation, it is unnecessary to predefine classes before segmentation. C. Salient instance segmentation Salient object detection should must be able to solve two problems: they must be able to determine the number of objects and detect the location of each object, both of which are important aspects of salient instance segmentation. Zhang et al. utilized a CNN to generate salient object proposals and predicted the proposal locations with a subset optimization framework based on the maximum a posteriori principle [45]. This approach promoted the salient objects from region-level to instance-level; thus, each instance could correspond to a bounding box for the first time. Li et al. were the first to address the instance-level salient object segmentation task [18]. The developed method used the multiscale refinement network (MSRNet) to produce the saliency maps and contour maps and then used subset optimization to refine the number of proposals. A fully connected CRF performed postprocessing to generate the final results. In addition, they compiled a new dataset for salient instance segmentation. However, their method depended heavily on the CRF process for edge detection, and the quality of the optimization proposals was affected by the subset optimization.Recently, Fan et al. proposed an end-to-end neural network to solve the problem of salient instance segmentation, called S4Net [21]. The method first used a single-shot object detector to detect the approximate position of the instances, and proposed to use a ternary masking instead of the traditional binary masking, which is similar to the center-around difference theory in the saliency detection [46]. All the above-mentioned works depend on region proposals to locate and predict the number of instances. However, this approach is both inefficient and complicated due to the number of proposals generated and the need to execute post- Fig. 3 is the schema of the SE block. X and X represent the original feature and the recalibration of feature respectively. r is the reduction ratio, which value is set to 16 in this paper. processing steps. To avoid this negative effect, we propose a proposal-free multitask NN framework to generate the salient instance segmentation results. Performing subitizing instead of the proposal method can boost the robustness of the final results. III. PROPOSAL-FREE SALIENT INSTANCE SEGMENTATION A. Overall framework Fig. 2 shows the overall architecture of the proposed proposal-free salient instance segmentation method. The framework consists of three main processes. (1) a densely connected subitizing network (DSN) that predicts the number of instances; (2) a densely connected fully convolutional network (DFCN) that performs salient region detection; and (3) adaptive spectral clustering using deep features to perform salient instance segmentation. Given an input image, we first use the DSN to predict the number of salient instances. The proposed DFCN is tailored to perform salient region detection. By extending the upsampling layers in the network and adding skip connections between the downsampling and upsampling paths the DFCN achieves a better pixelwise salient detection capability. A fully connected CRF [47] is adopted as a postprocessor to refine the salient region maps produced by the DFCN. Then, salient instances are segmented from the subitizing and salient regions based on adaptive spectral clustering. B. Densely connected subitizing network for instance-number prediction The main idea of the proposal-free method is to use salient object subitizing instead of an object region proposal method. Although object proposal methods locate the objects and estimate the number of instances by optimizing the number of bounding boxes, subitizing can directly predict the number of salient objects from the salient regions already located by the salient region detection task [48]. In other words, optimized object proposals may include nonsalient objects, while in contrast, subitizing directly predicts the salient number of objects. However, background distraction is usually the main challenge, which causes detection errors [49]. Since Zhang et al. [24] proposed salient object subitizing, the operation has been neither widely advanced nor widely applied thus far. In light of the concept of instance-level salient object segmentation, it is appropriate to use subitizing to accurately enumerate the number of instances. As mentioned earlier, the purpose of the DSN in the proposed MDNN is to predict the number of instances in an input image. Different from the way AlexNet and GoogleNet in SOS predicted the number of salient objects [50], we construct the DSN based on DenseNet. Compared with AlexNet and GoogleNet, DenseNet requires fewer parameters and preserves richer features based on how it performs concatenation [26]. More importantly, we embed the SE (Squeeze-and-Excitation) block for improving the representational power of the densely subitizing network by explicitly modelling the interdependencies between the channels of its convolutional features. The detailed DSN structure for subitizing is illustrated in Fig. 3. In our work, the subitizing task is viewed as a classification problem. First, we resize the input image to 224 × 224 to adapt the image to the downsampling process. The size of the initial convolution kernel is 7 × 7 with a stride of 2. Followed by a 3 × 3 max pooling layer, the first SE block is connected, which is illustrated in the bottom of Fig. 3. It is designed to improve the representational capacity of a network by enabling it to perform dynamic channelwise feature recalibration. We consider the input features X ∈ R H * W * C as X = [x 1 , x 2 , ..., x c ] where X i ∈ R H * W is the i-th channel of X and C is the total channel number. Firstly, the global average pooling is worked on each x i to obtain a channel-wise feature vector v ∈ R C . Then, two fully connected (FC) layers are used to capture channelwise dependencies. Like [51], we encode the channel-wise feature vector by forming a bottleneck with two FC layers around the non-linearity to limit model complexity and aid generalization. After adjusting the channel weight by the sigmoid operation, the output features are adapted to the inputspecific descriptor v. the Squeeze-and-Excitation process is described as: X = (ϕ(f c 2 (η(f c 1 (v, W 1 )), W 2 ))) · X,(1) where η refers to the ReLU function, ϕ refers to sigmoid operation, f c refers to FC layers, W 1 ∈ R c r ×c and W 2 ∈ R c× c r . The parameter W 1 is used to reduce the dimension of the feature and W 2 is worked to recover the dimension of the feature for adapting the next layer. The reduction ratio r in W 1 and W 2 is an important hyperparameter. According to [51], r is set to 16 for all experiments in this paper. We added a total of 7 SE blocks in the structure of DSN. Followed by the first SE block, the feature X is transmitted to the dense block which embrace 6 layers. Each layer in the dense block contains a 1×1 convolutional layer followed by a 2 × 2 average pooling layer. The main advantage of this DSN lies in how it performs concatenation manner; it connects each layer to every other layer in a feed-forward fashion [26]. Given that a dense block contains L layers, we define a nonlinear transformation H l (·) as a successive composite function. The function embraces three operations: batch normalization (BN) [52], a rectified linear unit (ReLU) [53] and a 3×3 convolution [26]. We assume that x l is the output of the l-th layer, which is defined as: x l = H l ([x 0 , x 1 , x 2 , ..., x l−1 ]),(2) where [x 0 , x 1 , x 2 , ..., x l−1 ] represents the series of connected feature maps. This type of connectivity pattern alleviates feature loss and encourages feature reuse. As shown in Fig. 3, the proposed network has four dense blocks, each of which embeds 6, 12, 48 and 32 layers, respectively. The feature map sizes produced in these four dense blocks are 56 × 56, 28 × 28, 14 × 14 and 7 × 7, respectively. In addition, we insert three transitional layers between two contiguous dense blocks composed of 1 × 1 convolution followed by 2 × 2 average pooling. At the end of the network, the classification layer consists of a 7 × 7 average pool, a 1000-D fully connected layer and a 4-D fully connected layer. Normally, the human visual system is able to consider only up to 4 salient instances simultaneously without thinking [49]. Using the dataset from [18], we do not know how many salient instance images to identify, so we set the last FC layer to 4-D to predict 1, 2, 3 and 4+ salient instances in the image data. C. Densely connected fully convolutional network for salient region detection The goal of the salient region detection task is to label the salient and interesting regions from the image [32]. Because the final mission is to segment the salient instances, it is important for the salient detection procedure to produce precise salient regions. We proposed the densely connected fully convolutional network (DFCN) to detect salient regions; this network was inspired by DenseNet and FCNs [27]. The DFCN is described in Fig. 4. The framework pipeline looks similar to a U-shape after adding the deconvolution process based on DenseNet [54]. The proposed DFCN takes advantage of the concatenation process in the dense blocks to build an FCN-like architecture for generating saliency maps of arbitrary sizes. To further preserve the downsampling layer features, we build bridges between the upsampling and downsampling layers called "skip connections" to ameliorate the resolution loss and boost the completeness of the feature at the upsampling step. Given an input image, the size of the first convolution is 3 × 3; and 5 dense blocks are concatenated with the same number of transition layers. The transition layers are structured the same as those in DenseNet, but the number of layers in the dense blocks are different, with 4, 5, 7 10 and 12 layers, respectively. The bottom of the network embeds a 15-layer dense block followed by an upsampling layer. An upsampling operation consisting of a 3 × 3 transposed convolution with a stride of 2. The upsampling path is used not only to recover the input spatial resolutions but also to upsample the feature maps. The output of the upsampling layer is combined with the features from the skip connections to form the input of the next dense block. In the upsampling path, the dense block corresponds to the downsampling path, and the upsampling layer compensates for the pooling operation in the transition layers. The input to the last dense block consists of the summed information from all the previous dense blocks. Following a 1 × 1 convolution, a softmax layer is used to provide the salient region maps. Because the FCN-based salient region segmentation result is coarse and does not delineate the object borders, we use the fully connected CRFs to refine the segmentation prediction results [47]. Each pixel in the salient object map is further finely classified into labels of salient region or background by employing the energy function of CRF: E(S) = − i logP (s i ) + i,j ϕ p (s i , s j ),(3) where S represents a salient object map for all pixels, and P (s i ) is the probability of pixel x i belonging to the label s i , which indicates the saliency likelihood of pixel x i . The pairwise cost ϕ p (s i , s j ) for two labels s i and s j is defined as: ϕ p (s i , s j ) = ω 1 exp − |p i − p j | 2 2θ 2 α − |I i − I j | 2 2θ 2 β + w 2 exp − |p i − p j | 2θ 2 γ 2(4) where ω 1 and ω 2 indicate the relative weights corresponding to the two parts of Eq. (4). The first part represents the appearance kernel, and the second part quantifies the smoothness kernel. The hyperparameters θ α , θ β and θ γ are the standard deviation values that control the Gaussian kernels; p i and p j are the position vectors; and I i , I j are the respective RGB vectors of the pixels x i and x j . In this paper, we set the values of ω 1 , ω 2 , θ α , θ β and θ γ to 30, 30, 61, 13 and 1, respectively. The model causes the fully connected CRF to enforce the structural consistency of the segmentation output and refines the areas to generate smoother contours [55]. As shown in the example in Fig. 4, compared to the prediction map, the handle of the handbag is both finer and more accurate after the CRF process. It is essential to detect the salient regions precisely to improve the next instance segmentation. D. Salient instance clustering When a known number of instances exist, it is natural to consider using a clustering algorithm to perform salient instance segmentation. Spectral clustering based on the graph model can produce more accurate and effective results than other, simpler clustering algorithms [56]. The number of instances predicted by the DSN indicates a reasonable cluster number for the k value used in spectral clustering. In this phase, we utilize the features extracted from the DFCN during the spectral clustering instead of the low-level features. In addition, the pixels in the input image are replaced by the superpixels generated by the SLIC algorithm and used during the clustering process [29]. In general, k-means clustering is the last procedure in spectral clustering. Considering the drawbacks of using the local minimum to select the initial clustering centers, we also use fractile points to determine the initial clustering centers instead of simply making random selections. First, we overlay the original image with the salient region map and use that as the input image to isolate the salient regions and mitigate background influence. Second, we found that the instances are segmented integrally and simply when the number of superpixels is set to between 200 and 300 [27]; consequently, all the pixels in the input image are classified into superpixels with a size of 250. After the preprocessing step, we construct a single graph G = (V, E), where V consists of the nodes in the input image and E is a set of undirected edges [57]. The affinity matrix is defined by ω ij = e − c i − c j σ 2 1 + λ · ||d i − d j || i, j ∈ V,(5) where λ is a parameter to control the spatial distance, which is set to 3 in our implementation. d i − d j is the Euclidean distance between pixel i and j. The parameter σ 2 controls how rapidly the affinity ω ij declines with the distance between c i and c j , and we set σ 2 to 10 [26]. c i and c j denote the means of the superpixels corresponding to two nodes in the feature map produced by DFCN. We extract 6 candidate deep features in DFCN and resize those feature maps so that their size corresponds with the input image; the locations of 6 feature maps are shown in Fig. 4. We discussed the effectiveness of different feature maps working on the spectral clustering in Section IV-B. Given image X, where all the superpixels are represented by (v 1 , v 2 , , v n ), the degree matrix is D = diag{d 11 , , d nn }, d ii = j ω ij . In conjunction with D, we find the smallest k eigenvalues using D −1/2 LD −1/2 and then generate the corresponding eigenvectors U . Finally, k-means clustering is used to cluster the eigenvectors U and obtain the salient instance segmentation results. The traditional k-means clustering algorithm randomly selects k pixel points from an image to be the initial cluster centers. However, randomly selecting initial cluster centers can cause the clustering process to become stuck in local minima, which prevents the salient instances from being completely segmented. Therefore, we use the fractile concept to find the initial cluster centers and avoid the negative influence of random selection on the final results. Following statistical methods, we arrange all the values from small to large and divide them into four equal parts. The values at the three split points are the quartiles. Thus, we divide all the pixels of the eigenvectors U into k equal parts from small to large and adopt the center point of each part as the initial cluster center. The method provides some guidance when selecting the initial cluster centers. First, we arrange all the pixels of an image in ascending order based on their values into the characteristic matrix U . Second, we let Q be a fractile of this vector, where Q = Q 1 , Q 2 , , Q i . The value of Q i is calculated as follows: Q i = 50 k + (i − 1) 100 k · U, i = 1, 2, . . . , k,(6) where k is the number of clusters, and U is the vector of the characteristic matrix arranged by ascending values. When the value of k is 4, we will obtain four fractiles: Q 1 ,Q 2 ,Q 3 ,and Q 4 . These Q values are what we adopt for the cluster centers. It is both convenient to assign the initial pixels through this initial guidance method and it also enhances the stability of salient instance segmentation. The major procedures by adaptive spectral clustering method are summarized in Algorithm 1. E. Implementation Details In this section, we present more details regarding the training phase of MDNN. We employed a weighted crossentropy function as the loss function to train the network, which is actuated as follows: L (y, y) = − 1 N N i=1 c c=1 y c i log y c i ,(7) whereŷ c i denotes the probability of pixel i belonging to class c (c = 4 in the DSN and c = 2 in the DFCN), and y c i indicates the ground truth label for pixel i. Because we use separate training sets for the two subtask networks, the parameters of these two networks cannot be shared. Both networks are trained using stochastic gradient descent Algorithm 1 Salient Instance Clustering Input: An image, the corresponding salient object map and a subitizing k 1: Refine the image to filter out the background depending on the saliency map. 2: Segment the image produced in step 1 into superpixels and build a graph G. 3: The feature map extracted by the DFCN is embed to compute the degree matrix D and affinity matrix W using Eq. (5). 4: Calculate the normalized Laplace matrix and add the number of instances k to obtain the corresponding eigenvectors U . 5: Use the improved k-means clustering to classify the eigenvectors U and resize the label map by the pixel list in the SLIC algorithm to obtain the salient instance results. Output: The salient instance segmentation map, in which each instance is represented by one label. (SGD) [58]. During the training phase, the weight decay is empirically set to 5×10 −4 , and the momentum is 0.95 without dampening. For the DSN, the initial learning rate is set to 0.001, which is then divided by a factor of 0.1 every 7 epochs. The weight decay is applied to the weights in the convolution and fully connected layers. In addition, we use a mini-batch size of 8 for 100 epochs. In DFCN, the learning rate is set to 10 −7 , and the weight decay is applied only to the convolution weights. Different from the DSN, the mini-batch size is set to 6 for 100 epochs. The entire procedure is repeated iteratively for training. IV. EXPERIMENTS In this section, we thoroughly explain the details of the experimental process and evaluate the results of various procedures on different performance metrics, including subitizing, salient region detection and salient instance segmentation tasks. To demonstrate the validity of our approach, we execute our MDNN method on several public datasets and compare it equitably with the state-of-the-art methods equitably. In addition, we analyze the different strategies used in our approach and adopt the optimal solutions in the subsequent experiments. Finally, we discuss a set of experimental results using qualitative and quantitative criteria. A. Experimental settings As described in Section III-C, our proposed MDNN is implemented in the PyTorch framework on 2 NVIDIA GeForce GTX 1080Ti GPUs with 22 GB of memory. Because none of the existing image datasets contain both subitizing and salient object maps, the subnetworks cannot all be trained together; thus, their parameters cannot be optimized simultaneously. In the experiment, we use 500 images for training, 200 for validation and 300 for testing from the index of dataset1k. Before using the dataset1k, we labeled the number of instances for the subitizing task to reflect the ground-truth. Because the sole available salient instance segmentation dataset1k contains only 500 images for training, we extended the training set by adding the datasets SOS [24] and MSRA-B [59] for the subitizing and salient region detection subnetworks, respectively. For the DSN, the SOS dataset, which includes 13,707 images proposed by Zhang et al. [24], was added for training. For fair comparison, we generated synthetic images to train the CNN model before fine-tuning on the real data, which is same as [24]. Besides, we resized all the images and groundtruth maps to 256×256 and randomly cropped the region into a 224×224 square image. During validation, the images were resized to 224 × 224 regardless of their original aspect ratios. For the DFCN, we combined the MSRA-B (5000 images) into the training sets [59]. Different from the previous network, we resized the images to 340 × 340 and randomly cropped the regions into 300 × 300 square images. The MDNN is finetuned by flipping the training sets horizontally at a probability of 0.5, and we did not perform any random augmentations during validation. Training the two subtask networks takes approximately 7 hours for the DSN and approximately 10 hours for the DFCN. During the test phase, it requires approximately 0.8 seconds per image to produce a salient region map and determine the number of instances and another 0.5 seconds to segment the instances using the proposed spectral clustering on a 340×340 test image. The overall time cost is 1.3 seconds per image, which is considerably less than MSRNet [18], which requires more than 20 seconds per image. B. Results and comparisons Ablation Study: To investigate the effectiveness of different structures in our method, we conduct the ablation study. First, to verify the most appropriate basic network structures, we trained four candidate models based on DenseNet [26], named DenseNet-121, DenseNet-161, DenseNet-169 and DenseNet-201. These models have different numbers of layers in the dense blocks [26]. All the compared models are trained with the same settings as our DSN. Their average precision (AP) results are listed in Table. I, which shows that our DSN achieves the best performance in terms of the AP on the test images. DenseNet-121 performs the worst because it has fewer layer in its dense blocks. The other two models exhibit similar performances behind our network. DenseNet-201 achieved 77.5% of the AP scores which is better than other version of DenseNet, consequently we choose DenseNet-201 as the backbone of DSN. The good performance of DSN is also benefited by inserting the SE block. In order to further improve the performance, we experiment two ways to embed the SE blocks in DSN: insert them within the dense block or append them to head and tail of each dense block. The results of different connection modes using SOS dataset are displayed in Table. II. We can see that the connection mode of inter-SE is generally better than the intra-SE mode no matter how many instances in a scene. Finally, we choose the inter-SE mode that make the SE block is appended to head and tail of each dense block in DSN. In addition, to evaluate the performance of different feature maps in the spectral clustering operation, we also test six feature maps extracted from DFCN to generate the final salient instance segmentation. The marked number of feature maps corresponds to Fig. 4, feature map 1, 2 and 3 belongs to the front of DFCN and feature map 4, 5 and 6 comes from the latter layers of DFCN. As we know that the first few layers of features extracted by the DFCN includes relatively more low-level features, while latter maps contain more high-level features. In this experiment, we added these feature maps in the spectral clustering to obtain the performance metrics AP r @0.5 and AP r @0.7 (Eq. (10)) reported in Table. III. Table. III shows that the feature map 6 achieves the beat results. It also demonstrates that the latter feature maps have higher AP r scores than the former features. The reason is the input feature to the dense block in DFCN consists of the summed information from all the previous dense blocks, so the latter map contains more abundant feature including lowlevel and high-level features. Evaluation of subitizing: To evaluate the number of instances, we first input 300 test images from the index of dataset1k to the DSN. To demonstrate the accuracy of the predicted subitizing, Fig. 5 shows a confusion matrix for the distribution of the results by the DSN. Different from IV: Comparison of maximum F-measure (larger is better), MAE scores (smaller is better), S-measure (larger is better) and E-measure (larger is better). The best three scores in each row are shown in red, blue, and green, respectively. SOS [20], the DSN excludes the number 0 because dataset1K have no image without a salient object. The matrix presents the percentage of results compared to the ground truth. The recall value is stable and is as expected when the number of instances is below 4. The accuracy rate for category 1 was the highest (98%). The recall values for categories 2 and 3 are stable and exhibit a gradual downward trend. When the images contain 4 or more objects, the accuracy decreases to (51%). It is interesting to note that most incorrect results are located near correct results. This finding demonstrates that we can continue to segment the corresponding number of instances while losing few instances. It is worth noting that the prediction ratio for category 1 in the 4+ images is (10%), which can be interpreted as one object of the image being substantially larger than the other 3 objects, causing the algorithm to fail to count them correctly. To further demonstrate the efficiency of our algorithm, we tested the DSN on the SOS dataset (with 13,707 images) compared with the CNN-Syn-FT model provided by Zhang et al. [24].To ensure a fair comparison, we used the two-stage fine-tuning scheme with the real and synthetic image data to train DSN. For pre-training using the synthetic images, we generated 34,000 for each number in 1−4 as same as [24]. In addition, DSN training and testing parameters were conducted in line with those of the CNN-Syn-FT. Table. V shows the average precision scores. We can see that DSN is comprehensive superior to CNN-Syn-FT and the mean AP scores of our network is 5 percentage points better than CNN-Syn-FT. Especially, its accuracy scores on category 3 is higher than CNN-Syn-FT about 11 percent. Evaluation of salient region detection: The salient region detection task has a direct impact on the salient instance segmentation results. To validate the performance of DFCN, we conducted experiments on four publicly available benchmark datasets annotated with pixelwise ground-truth labeling: DUT-OMRON [57], ECSSD [59], MSRA-B [32], SOD [60] and SED2 [60]. For these experiments, we adopted four of the most representative evaluation metrics to evaluate the salient region maps. The first metric is the F-measure, which evaluates the quality of salient region maps after binarizing them using a specific threshold. The F-measure is computed as follows: F β = (1 + β 2 )P recisionRecall β 2 P recision + Recall ,(8) and β 2 was set to 0.3 [61]. In this experiment, we used the maximum F-measure (maxF) instead of the average Fmeasure (aveF) because of the adaptive threshold. The second metric is mean absolute error (MAE), which reflects the mean pixelwise difference between the saliency map and the ground truth [62] and is calculated as M AE = 1 W × H W x=1 H y=1 S(x, y) − G(x, y) ,(9) where S and G denote the saliency map and the ground truth, respectively. W and H are the image size parameters. This metric of assessing the salient regions compared with the ground truth is intended to reflect the salient instance detection quality. A smaller MAE value denotes better quality prediction results. The two other metrics are structure measure (S-measure) [63] and Enhanced measure (E-measure) [64], which have added to serve as the supplement of the F-measure and MAE. The higher that S-measure and E-measure are, the better results are the methods. For comparison with other salient region detection methods, we performed a horizontal evaluation with 12 classic or stateof-the-art methods, including GC [65], GMR [58], DRFI [66], FT [59], BMS [37], MDF [10], MTDNN [39], DCL [28], MSRNet [18], DSS [67], PAGR [68] and [69]. Among these methods, the first five are implemented based on traditional machine learning, while the rest of approaches operate using deep learning models. We conducted the experiments by executing the publicly available source code provided by the original authors. Table. IV shows the results from our DFCN compared with those of the other methods on the metrics w.r.t maxF, MAE, S-measure and E-measure on five benchmark datasets. Clearly, DFCN performs stably and favorably when compared to the state-of-the-art methods in most cases. More specifically, the proposed network performs best with regard to the MAE value, which is highly important for the salient instance detection task. On a macro level, the deep-learning approaches achieve better performances than do the traditional machine learning methods. In addition, the last column in Table. IV reports the results after CRF [47], which comprehensively improves the earlier results measurements whether considering any one of the metrics; however, the magnitudes of the increases are not large. Evaluation of salient instance segmentation: Because salient instance segmentation is a new and challenging task, few performance measures exist to perform a quantitative verification. Therefore, to evaluate the effectiveness of our approach for salient instance segmentation, we refer to the metrics for semantic segmentation provided by [40]. Different from the mean average precision (MAP) for semantic segmentation, salient object instances are independent of category; consequently, we adopt the AP r metric instead of the M AP r metric at intersection-over-union (IoU) scores of 0.5 to 0.9. Taking the threshold of IoU at 0.5 as an example, we first need to calculate the per image precision of each instance from the dataset. Fixing the precision value at IoU > 0.5, the sum of these precision values divided by the number of all ground-truth instances obtains the AP r at 0.5 IoU scores. This operation is followed by the following calculation: AP r @0.5 = 1 N K i=1 precision(i) IoU (i) ≥ 0.5,(10) where K is the number of instances in which the value of IoU is greater than 0.5, and N denotes the total number of ground-truth instances in the dataset. Due to datasets limitations, we adopted only the dataset including 1,000 images provided by [18]. We used the published results on the dataset1K test set directly; all the other settings were the same as those used in [18] to ensure a fair comparison. We report the results of the AP r of the metrics with IoU scores of 0.5 to 0.9 predicted by the state-of-theart methods in Table. VI. The proposed method performs better than MSRNet [18] and S4Net [21] when segmenting the salient instances, especially when the AP r metric is greater than the IoU score of 0.6. due to the related code of [18] is not available, we cannot get its results of AP r metric. The detailed AP r scores for different numbers of instances per image by S4Net [21] and our method are listed in Table. VII, it reveals that the performances showed a downward trend as the number of instances in the images increased. Comparing with S4Net, our results outperform against them generally, while the AP r @0.5 score for the number of 4+ instances is smaller than [21]. We can clearly see that using the Non-maximum Suppression (NMS) method to estimate the number of proposals is not stable enough compared with DSN. The Table. VII also illustrates that the accuracy rate of the subitizing prediction has a positive influence on the salient instance segmentation. Overall, the performance of MDNN is as expected and it outperforms the state-of-the-art methods. We also qualitatively evaluated our method on the salient instance segmentation dataset. Fig. 6 visualizes the predicted salient instance segmentation results using our approach. Due to the lack of published code for the only related work (MSRNet [18]), we were not able to visually compare the results with those of any existing method. Instead, we chose representative segmentation results for analysis in Fig. 6. To be consistent with the ground truth, the segmented instances are labeled with different colors corresponding to the ground truth. The proposed approach achieves satisfactory performance regardless of how complicated the original images are. The first three rows show instances that have similar internal features and are close together in one image. The fourth row shows two overlapping instances with complex color and texture features; however, the proposed method still segments them effectively. The middle rows show some images with messy backgrounds, and the instances are confused by the background. For example, in the fifth row, the woman is wearing a dark-colored coat and partially blends in with the tall tree behind her. In addition, the color of the sheep in the three last row image blends with the rocks and trees in the background. Other segmentation results reveal other challenging cases, including occlusions, diverse phenotypes and different views. The salient instances are segmented distinctly and consistently with the ground truth annotations. Visual Comparison: We also qualitatively evaluated our method and S4Net [21] on the salient instance segmentation dataset. Fig. 6 visualizes the predicted salient instance segmentation results and we chose representative segmentation results for analysis. To be consistent with the ground truth, the segmented instances are labeled with different colors corresponding to the ground truth. The proposed approach achieves satisfactory performance regardless of how complicated the original images are. Compared to S4Net [21], our method gets the better segmentation results which are Fig. 6: Examples of salient instance segmentation results by the proposed method and S4Net [21]. In each image, the different colors indicate different instances. closer to the ground truth, especially when the number of instances is below 3. To be specific, the first three rows show the performance of different methods when the number of instances is 1. We can see it very intuitively that the proposed model can predict the number of instances accurately while S4Net generated unsatisfactory results because of estimating the number of instances incorrectly [21]. The followed four rows show instances that have similar internal features and are close together in one image. The eighth and ninth rows show two overlapping instances with complex color and texture features; however, the proposed method still segments them effectively. The followed rows show some images with messy backgrounds, and the instances are confused by the background. For example, in the fourth from the bottom row, the woman is wearing a dark-colored coat and partially blends in with the tall tree behind her. In addition, the color of the birds in the three last row image blends with the rocks and trees in the background. Other segmentation results reveal other challenging cases, including occlusions, diverse phenotypes and different views. The salient instances of our method are segmented distinctly and consistently with the ground truth annotations. In contrast, the effectiveness of S4Net [21] is slightly inferior to the proposed method. it reports that our novel method is highly suitable for solving the salient instance segmentation task. However, the lack of datasets id an urgent problem that limits the generalizability of our approach. Some segmentation failure examples are shown in Fig. 7. These examples illustrate that some results were segmented unsatisfactorily because of an incorrect number of instances. Inaccurate salient regions can also lead to errors for the final instances, such as the results in the third and fifth rows. In addition, the image in the third row has diverse instance types, and the entire scene is chaotic, which can cause inexact results. When the instances are too small, the adaptive deep feature-based spectral clustering neglects them, which leads to segmentation failure. We plan to address these failure examples to handle tough instances in future work. V. CONCLUSION AND FUTURE WORKS Salient instance segmentation is a new and challenging task that is an extension of salient region detection. Thus far, limited research has been conducted for salient instance segmentation since the first work by Li et al. [18]. In this paper, we introduced a proposal-free method that performs salient instance segmentation. Rather than using region proposal methods, the proposal-free network, named MDNN, incorporates the DSN and the DFCN. The MDNN can directly predict the salient regions and the number of instances in a category-independent manner. Adaptive spectral clustering is applied to the feature maps extracted from the DFCN to generate the final salient instances. The experimental results show that the proposed method achieves significant improvements for salient instance segmentation. More concretely, the subtask networks that perform saliency region detection and the subitizing perform well against most previous state-of-theart methods. In the future, we plan to expand the instancelevel salient object datasets and produce more appropriate evaluation metrics for measuring the quality of the final salient instance results. Fig. 1 : 1The pipeline of the proposed method illustrates the salient instance segmentation. Fig. 2 : 2The proposed overall MDNN framework for salient instance segmentation. The upper dashed box is the subitizing network for predicting number of instances. The bottom dotted box contains the DFCN and the fully connected CRF for producing saliency maps. The skip connections are used to reuse hierarchical features in the upsampling path. The feature from the last dense block layers are extracted for spectral clustering. Fig. 3 : 3Details of the DSN network in MDNN, the bottom of Fig. 4 : 4The architecture of the U-shaped DFCN. The numbers in the dense block boxes indicate the number of layers contained in each dense block. The number 1 through 6 indicate the indexes of the candidate feature maps of the proposed adaptive spectral clustering, which discussed in the ablation study of Section IV. Fig. 5 : 5Confusion matrix of the proposed DSN. Each cell includes the accuracy percentage (recall). Each row corresponds to the ground-truth number, while each column shows the predicted results. The values on the diagonal line represent the correct result percentages. Fig. 7 : 7Illustration of failure examples generated by our method. to build saliency models. For instance, Cheng et al. extracted salient regions by computing the global contrast in the image [35]; Perazzi et al. considered salient object detection as a filtering problem [36]; Zhang et al. achieved high performance results based on a Boolean map approach TABLE I : IComparison of the DSN and different versions of DenseNet model. The second row represents the average precision (%) of the predicted number of instances.DSN DenseNet-121 DenseNet-161 DenseNet-169 DenseNet-201 79.4 68.7 72.5 70.3 77.5 TABLE II : IIThe AP score performance of DSN with different connection types of SE block. Inter-SE means SE block is appended to head and tail of each dense block and Intra-SE means SE block is connected inside the dense block. The highest scores in each row are labeled in bold.Method 0 1 2 3 4+ mean intra-SE 98.1 96.7 80.2 72.7 77.5 85 inter-SE 98.3 97.1 82.4 75.3 76.4 85.9 TABLE III : IIIQuantitative comparisons with the results obtained by varying the feature maps in the spectral clustering algorithm. The bold values indicate the best performance.Metrics Feature1 Feature2 Feature3 Feature4 Feature5 Feature6 AP r @0.5(%) 50.89 53.18 57.46 62.74 69.24 73.46 AP r @0.7(%) 38.2 40.66 43.83 48.96 56.18 60.14 TABLE TABLE V : VAverage Precision comparison (%) of the subitiz- ing task. Category 0 was added for this comparison, and the separated and mean of the AP scores are reported. The highest scores in each row are labeled in bold. Method 0 1 2 3 4+ mean CNN-Syn-FT [24] 93.5 93.8 77.4 64.3 73 80.4 DSN 98.3 97.1 82.4 75.3 76.4 85.9 TABLE VI : VIQuantitative comparisons with existing methods by the AP r metric at IoU threshold of 0.5 to 0.9. The bold values indicate the best performances.Method AP r 0.5(%) AP r 0.6(%) AP r 0.7(%) AP r 0.8(%) AP r 0.9(%) MSRNet 65.32 - 52.18 - - S4Net 74.16 66.21 55.34 36.95 12.74 Ours 73.46 67.3 60.14 46.25 23.51 TABLE VII : VIIComparing the AP r metrics for different num- bers of instances per image found by the existing methods on dataset1K. The bold values indicate the best performances. Method Metrics 1 2 3 4+ S4Net AP r 0.5(%) 81.98 74.70 63.71 49.25 Ours AP r 0.5(%) 83.44 75.20 64.23 43.22 S4Net AP r 0.7(%) 65.64 54.24 47.57 31.54 Ours AP r 0.7(%) 70.09 58.08 58.53 32.50 ACKNOWLEDGMENTSThis research was supported by the National Natural Science Foundation of China Grant 61902139. A model of saliency-based visual attention for rapid scene analysis. L Itti, C Koch, E Niebur, IEEE Transactions on Pattern Analysis & Machine Intelligence. 11L. Itti, C. Koch, and E. Niebur, "A model of saliency-based visual attention for rapid scene analysis," IEEE Transactions on Pattern Analysis & Machine Intelligence, no. 11, pp. 1254-1259, 1998. A universal framework for salient object detection. J Lei, B Wang, Y Fang, W Lin, P Le Callet, N Ling, C Hou, IEEE Transactions on Multimedia. 189J. Lei, B. Wang, Y. Fang, W. Lin, P. Le Callet, N. Ling, and C. Hou, "A universal framework for salient object detection," IEEE Transactions on Multimedia, vol. 18, no. 9, pp. 1783-1795, 2016. Salient object detection: A survey. A Borji, M.-M Cheng, Q Hou, H Jiang, J Li, arXiv:1411.5878arXiv preprintA. Borji, M.-M. Cheng, Q. Hou, H. Jiang, and J. Li, "Salient object detection: A survey," arXiv preprint arXiv:1411.5878, 2014. Repfinder: finding approximately repeated scene elements for image editing. M.-M Cheng, F.-L Zhang, N J Mitra, X Huang, S.-M Hu, ACM Transactions on Graphics (TOG). 29483ACMM.-M. Cheng, F.-L. Zhang, N. J. Mitra, X. Huang, and S.-M. Hu, "Repfinder: finding approximately repeated scene elements for image editing," in ACM Transactions on Graphics (TOG), vol. 29, no. 4. ACM, 2010, p. 83. Hierarchical semantic labeling for taskrelevant rgb-d perception. C Wu, I Lenz, A Saxena, Robotics: Science and systems. C. Wu, I. Lenz, and A. Saxena, "Hierarchical semantic labeling for task- relevant rgb-d perception." in Robotics: Science and systems, 2014. Database saliency for fast image retrieval. Y Gao, M Shi, D Tao, C Xu, IEEE Transactions on Multimedia. 173Y. Gao, M. Shi, D. Tao, and C. Xu, "Database saliency for fast image retrieval," IEEE Transactions on Multimedia, vol. 17, no. 3, pp. 359- 369, 2015. A generic framework of user attention model and its application in video summarization. X Hua, L Lu, H Zhang, H District, IEEE Transaction on multimedia. 75X.-s. Hua, L. Lu, H.-j. Zhang, and H. District, "A generic framework of user attention model and its application in video summarization," IEEE Transaction on multimedia, vol. 7, no. 5, pp. 907-919, 2005. Calibration-free gaze sensing using saliency maps. Y Sugano, Y Matsushita, Y Sato, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEEY. Sugano, Y. Matsushita, and Y. Sato, "Calibration-free gaze sensing using saliency maps," in 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE, 2010, pp. 2667-2674. Saliency detection by multi-context deep learning. R Zhao, W Ouyang, H Li, X Wang, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionR. Zhao, W. Ouyang, H. Li, and X. Wang, "Saliency detection by multi-context deep learning," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2015, pp. 1265-1274. Visual saliency based on multiscale deep features. G Li, Y Yu, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionG. Li and Y. Yu, "Visual saliency based on multiscale deep features," in Proceedings of the IEEE conference on computer vision and pattern recognition, 2015, pp. 5455-5463. Iterative feedback control-based salient object segmentation. S Huo, Y Zhou, J Lei, N Ling, C Hou, IEEE Transactions on Multimedia. 206S. Huo, Y. Zhou, J. Lei, N. Ling, and C. Hou, "Iterative feedback control-based salient object segmentation," IEEE Transactions on Mul- timedia, vol. 20, no. 6, pp. 1350-1364, 2017. Salient object segmentation via effective integration of saliency and objectness. L Ye, Z Liu, L Li, L Shen, C Bai, Y Wang, IEEE Transactions on Multimedia. 198L. Ye, Z. Liu, L. Li, L. Shen, C. Bai, and Y. Wang, "Salient object segmentation via effective integration of saliency and objectness," IEEE Transactions on Multimedia, vol. 19, no. 8, pp. 1742-1756, 2017. Unsupervised salient object detection via inferring from imperfect saliency models. R Quan, J Han, D Zhang, F Nie, X Qian, X Li, IEEE Transactions on Multimedia. 205R. Quan, J. Han, D. Zhang, F. Nie, X. Qian, and X. Li, "Unsupervised salient object detection via inferring from imperfect saliency models," IEEE Transactions on Multimedia, vol. 20, no. 5, pp. 1101-1112, 2017. Deep salient object detection with dense connections and distraction diagnosis. H Xiao, J Feng, Y Wei, M Zhang, S Yan, IEEE Transactions on Multimedia. 2012H. Xiao, J. Feng, Y. Wei, M. Zhang, and S. Yan, "Deep salient object detection with dense connections and distraction diagnosis," IEEE Transactions on Multimedia, vol. 20, no. 12, pp. 3239-3251, 2018. Amulet: Aggregating multi-level convolutional features for salient object detection. P Zhang, D Wang, H Lu, H Wang, X Ruan, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionP. Zhang, D. Wang, H. Lu, H. Wang, and X. Ruan, "Amulet: Aggre- gating multi-level convolutional features for salient object detection," in Proceedings of the IEEE International Conference on Computer Vision, 2017, pp. 202-211. Dhsnet: Deep hierarchical saliency network for salient object detection. N Liu, J Han, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionN. Liu and J. Han, "Dhsnet: Deep hierarchical saliency network for salient object detection," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 678-686. A bi-directional message passing model for salient object detection. L Zhang, J Dai, H Lu, Y He, G Wang, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionL. Zhang, J. Dai, H. Lu, Y. He, and G. Wang, "A bi-directional message passing model for salient object detection," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 1741-1750. Instance-level salient object segmentation. G Li, Y Xie, L Lin, Y Yu, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionG. Li, Y. Xie, L. Lin, and Y. Yu, "Instance-level salient object segmen- tation," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2017, pp. 2386-2395. Deep visual-semantic alignments for generating image descriptions. A Karpathy, L Fei-Fei, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionA. Karpathy and L. Fei-Fei, "Deep visual-semantic alignments for generating image descriptions," in Proceedings of the IEEE conference on computer vision and pattern recognition, 2015, pp. 3128-3137. Proposal-free network for instance-level object segmentation. X Liang, L Lin, Y Wei, X Shen, J Yang, S Yan, IEEE transactions on pattern analysis and machine intelligence. 40X. Liang, L. Lin, Y. Wei, X. Shen, J. Yang, and S. Yan, "Proposal-free network for instance-level object segmentation," IEEE transactions on pattern analysis and machine intelligence, vol. 40, no. 12, pp. 2978- 2991, 2018. S4net: Single stage salient-instance segmentation. R Fan, M.-M Cheng, Q Hou, T.-J Mu, J Wang, S.-M Hu, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionR. Fan, M.-M. Cheng, Q. Hou, T.-J. Mu, J. Wang, and S.-M. Hu, "S4net: Single stage salient-instance segmentation," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2019, pp. 6103-6112. Bing: Binarized normed gradients for objectness estimation at 300fps. M.-M Cheng, Z Zhang, W.-Y Lin, P Torr, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionM.-M. Cheng, Z. Zhang, W.-Y. Lin, and P. Torr, "Bing: Binarized normed gradients for objectness estimation at 300fps," in Proceedings of the IEEE conference on computer vision and pattern recognition, 2014, pp. 3286-3293. Selective search for object recognition. J R Uijlings, K E Van De Sande, T Gevers, A W Smeulders, International journal of computer vision. 1042J. R. Uijlings, K. E. Van De Sande, T. Gevers, and A. W. Smeulders, "Selective search for object recognition," International journal of com- puter vision, vol. 104, no. 2, pp. 154-171, 2013. Salient object subitizing. J Zhang, S Ma, M Sameki, S Sclaroff, M Betke, Z Lin, X Shen, B Price, R Mech, INTERNATIONAL JOURNAL OF COMPUTER VISION. J. Zhang, S. Ma, M. Sameki, S. Sclaroff, M. Betke, Z. Lin, X. Shen, B. Price, and R. Mech, "Salient object subitizing," INTERNATIONAL JOURNAL OF COMPUTER VISION, 2017. On spectral clustering: Analysis and an algorithm. A Y Ng, M I Jordan, Y Weiss, Advances in neural information processing systems. A. Y. Ng, M. I. Jordan, and Y. Weiss, "On spectral clustering: Analysis and an algorithm," in Advances in neural information processing systems, 2002, pp. 849-856. Densely connected convolutional networks. G Huang, Z Liu, L Van Der Maaten, K Q Weinberger, Proceedings of the IEEE confer. the IEEE conferG. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger, "Densely connected convolutional networks," in Proceedings of the IEEE confer- ence on computer vision and pattern recognition, 2017, pp. 4700-4708. Fully convolutional networks for semantic segmentation. J Long, E Shelhamer, T Darrell, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionJ. Long, E. Shelhamer, and T. Darrell, "Fully convolutional networks for semantic segmentation," in Proceedings of the IEEE conference on computer vision and pattern recognition, 2015, pp. 3431-3440. Deep contrast learning for salient object detection. G Li, Y Yu, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionG. Li and Y. Yu, "Deep contrast learning for salient object detection," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 478-487. Slic superpixels compared to state-of-the-art superpixel methods. R Achanta, A Shaji, K Smith, A Lucchi, P Fua, S Süsstrunk, IEEE transactions on pattern analysis and machine intelligence. 34R. Achanta, A. Shaji, K. Smith, A. Lucchi, P. Fua, and S. Süsstrunk, "Slic superpixels compared to state-of-the-art superpixel methods," IEEE transactions on pattern analysis and machine intelligence, vol. 34, no. 11, pp. 2274-2282, 2012. Salicon: Reducing the semantic gap in saliency prediction by adapting deep neural networks. X Huang, C Shen, X Boix, Q Zhao, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionX. Huang, C. Shen, X. Boix, and Q. Zhao, "Salicon: Reducing the semantic gap in saliency prediction by adapting deep neural networks," in Proceedings of the IEEE International Conference on Computer Vision, 2015, pp. 262-270. State-of-the-art in visual attention modeling. A Borji, L Itti, IEEE transactions on pattern analysis and machine intelligence. 35A. Borji and L. Itti, "State-of-the-art in visual attention modeling," IEEE transactions on pattern analysis and machine intelligence, vol. 35, no. 1, pp. 185-207, 2013. Learning to detect a salient object. T Liu, Z Yuan, J Sun, J Wang, N Zheng, X Tang, H.-Y Shum, IEEE Transactions on Pattern analysis and machine intelligence. 332T. Liu, Z. Yuan, J. Sun, J. Wang, N. Zheng, X. Tang, and H.-Y. Shum, "Learning to detect a salient object," IEEE Transactions on Pattern analysis and machine intelligence, vol. 33, no. 2, pp. 353-367, 2011. Frequencytuned salient region detection. R Achanta, S Hemami, F Estrada, S Süsstrunk, IEEE International Conference on Computer Vision and Pattern Recognition. CONFR. Achanta, S. Hemami, F. Estrada, and S. Süsstrunk, "Frequency- tuned salient region detection," in IEEE International Conference on Computer Vision and Pattern Recognition (CVPR 2009), no. CONF, 2009, pp. 1597-1604. Context-aware saliency detection. S Goferman, L Zelnik-Manor, A Tal, IEEE transactions on pattern analysis and machine intelligence. 34S. Goferman, L. Zelnik-Manor, and A. Tal, "Context-aware saliency detection," IEEE transactions on pattern analysis and machine intelli- gence, vol. 34, no. 10, pp. 1915-1926, 2012. Global contrast based salient region detection. M.-M Cheng, N J Mitra, X Huang, P H Torr, S.-M Hu, IEEE Transactions on Pattern Analysis and Machine Intelligence. 373M.-M. Cheng, N. J. Mitra, X. Huang, P. H. Torr, and S.-M. Hu, "Global contrast based salient region detection," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 37, no. 3, pp. 569-582, 2015. Saliency filters: Contrast based filtering for salient region detection. F Perazzi, P Krähenbühl, Y Pritch, A Hornung, 2012 IEEE conference on computer vision and pattern recognition. IEEEF. Perazzi, P. Krähenbühl, Y. Pritch, and A. Hornung, "Saliency filters: Contrast based filtering for salient region detection," in 2012 IEEE conference on computer vision and pattern recognition. IEEE, 2012, pp. 733-740. Saliency detection: A boolean map approach. J Zhang, S Sclaroff, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionJ. Zhang and S. Sclaroff, "Saliency detection: A boolean map approach," in Proceedings of the IEEE international conference on computer vision, 2013, pp. 153-160. Saliency unified: A deep architecture for simultaneous eye fixation prediction and salient object segmentation. S S Kruthiventi, V Gudisa, J H Dholakiya, R Venkatesh, Babu, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionS. S. Kruthiventi, V. Gudisa, J. H. Dholakiya, and R. Venkatesh Babu, "Saliency unified: A deep architecture for simultaneous eye fixation prediction and salient object segmentation," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 5781-5790. Deepsaliency: Multi-task deep neural network model for salient object detection. X Li, L Zhao, L Wei, M.-H Yang, F Wu, Y Zhuang, H Ling, J Wang, IEEE Transactions on Image Processing. 258X. Li, L. Zhao, L. Wei, M.-H. Yang, F. Wu, Y. Zhuang, H. Ling, and J. Wang, "Deepsaliency: Multi-task deep neural network model for salient object detection," IEEE Transactions on Image Processing, vol. 25, no. 8, pp. 3919-3930, 2016. Simultaneous detection and segmentation. B Hariharan, P Arbeláez, R Girshick, J Malik, European Conference on Computer Vision. SpringerB. Hariharan, P. Arbeláez, R. Girshick, and J. Malik, "Simultaneous detection and segmentation," in European Conference on Computer Vision. Springer, 2014, pp. 297-312. Instance-aware semantic segmentation via multi-task network cascades. J Dai, K He, J Sun, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionJ. Dai, K. He, and J. Sun, "Instance-aware semantic segmentation via multi-task network cascades," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 3150-3158. Instance-sensitive fully convolutional networks. J Dai, K He, Y Li, S Ren, J Sun, European Conference on Computer Vision. SpringerJ. Dai, K. He, Y. Li, S. Ren, and J. Sun, "Instance-sensitive fully convolutional networks," in European Conference on Computer Vision. Springer, 2016, pp. 534-549. Mask r-cnn. K He, G Gkioxari, P Dollár, R Girshick, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionK. He, G. Gkioxari, P. Dollár, and R. Girshick, "Mask r-cnn," in Proceedings of the IEEE international conference on computer vision, 2017, pp. 2961-2969. Fast r-cnn. R Girshick, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionR. Girshick, "Fast r-cnn," in Proceedings of the IEEE international conference on computer vision, 2015, pp. 1440-1448. Unconstrained salient object detection via proposal subset optimization. J Zhang, S Sclaroff, Z Lin, X Shen, B Price, R Mech, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionJ. Zhang, S. Sclaroff, Z. Lin, X. Shen, B. Price, and R. Mech, "Un- constrained salient object detection via proposal subset optimization," in Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp. 5733-5742. A feature-integration theory of attention. A M Treisman, G Gelade, Cognitive psychology. 121A. M. Treisman and G. Gelade, "A feature-integration theory of attention," Cognitive psychology, vol. 12, no. 1, pp. 97-136, 1980. Efficient inference in fully connected crfs with gaussian edge potentials. P Krähenbühl, V Koltun, Advances in neural information processing systems. P. Krähenbühl and V. Koltun, "Efficient inference in fully connected crfs with gaussian edge potentials," in Advances in neural information processing systems, 2011, pp. 109-117. What is an object?. B Alexe, T Deselaers, V Ferrari, in 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEEB. Alexe, T. Deselaers, and V. Ferrari, "What is an object?" in 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE, 2010, pp. 73-80. Delving into salient object subitizing and detection. S He, J Jiao, X Zhang, G Han, R W Lau, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionS. He, J. Jiao, X. Zhang, G. Han, and R. W. Lau, "Delving into salient object subitizing and detection," in Proceedings of the IEEE International Conference on Computer Vision, 2017, pp. 1059-1067. Imagenet classification with deep convolutional neural networks. A Krizhevsky, I Sutskever, G E Hinton, Advances in neural information processing systems. A. Krizhevsky, I. Sutskever, and G. E. Hinton, "Imagenet classification with deep convolutional neural networks," in Advances in neural information processing systems, 2012, pp. 1097-1105. Squeeze-and-excitation networks. J Hu, L Shen, G Sun, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionJ. Hu, L. Shen, and G. Sun, "Squeeze-and-excitation networks," in Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp. 7132-7141. Batch normalization: Accelerating deep network training by reducing internal covariate shift. S Ioffe, C Szegedy, arXiv:1502.03167arXiv preprintS. Ioffe and C. Szegedy, "Batch normalization: Accelerating deep network training by reducing internal covariate shift," arXiv preprint arXiv:1502.03167, 2015. Deep sparse rectifier neural networks. X Glorot, A Bordes, Y Bengio, Proceedings of the fourteenth international conference on artificial intelligence and statistics. the fourteenth international conference on artificial intelligence and statisticsX. Glorot, A. Bordes, and Y. Bengio, "Deep sparse rectifier neural networks," in Proceedings of the fourteenth international conference on artificial intelligence and statistics, 2011, pp. 315-323. U-net: Convolutional networks for biomedical image segmentation. O Ronneberger, P Fischer, T Brox, International Conference on Medical image computing and computer-assisted intervention. SpringerO. Ronneberger, P. Fischer, and T. Brox, "U-net: Convolutional net- works for biomedical image segmentation," in International Confer- ence on Medical image computing and computer-assisted intervention. Springer, 2015, pp. 234-241. Semantic image segmentation with deep convolutional nets and fully connected crfs. L.-C Chen, G Papandreou, I Kokkinos, K Murphy, A L Yuille, arXiv:1412.7062arXiv preprintL.-C. Chen, G. Papandreou, I. Kokkinos, K. Murphy, and A. L. Yuille, "Semantic image segmentation with deep convolutional nets and fully connected crfs," arXiv preprint arXiv:1412.7062, 2014. Self-tuning spectral clustering. L Zelnik-Manor, P Perona, Advances in neural information processing systems. L. Zelnik-Manor and P. Perona, "Self-tuning spectral clustering," in Advances in neural information processing systems, 2005, pp. 1601- 1608. Saliency detection via graph-based manifold ranking. C Yang, L Zhang, H Lu, X Ruan, M.-H Yang, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionC. Yang, L. Zhang, H. Lu, X. Ruan, and M.-H. Yang, "Saliency detection via graph-based manifold ranking," in Proceedings of the IEEE conference on computer vision and pattern recognition, 2013, pp. 3166-3173. Backpropagation applied to handwritten zip code recognition. Y Lecun, B Boser, J S Denker, D Henderson, R E Howard, W Hubbard, L D , Neural computation. 14Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel, "Backpropagation applied to handwritten zip code recognition," Neural computation, vol. 1, no. 4, pp. 541-551, 1989. Hierarchical saliency detection. Q Yan, L Xu, J Shi, J Jia, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionQ. Yan, L. Xu, J. Shi, and J. Jia, "Hierarchical saliency detection," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2013, pp. 1155-1162. Design and perceptual validation of performance measures for salient object segmentation. V Movahedi, J H Elder, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition-Workshops. IEEEV. Movahedi and J. H. Elder, "Design and perceptual validation of performance measures for salient object segmentation," in 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition-Workshops. IEEE, 2010, pp. 49-56. Saliency detection via multiscale global cues. X Lin, Z.-J Wang, L Ma, X Wu, IEEE Transactions on Multimedia. X. Lin, Z.-J. Wang, L. Ma, and X. Wu, "Saliency detection via multi- scale global cues," IEEE Transactions on Multimedia, 2018. Salient object detection: A benchmark. A Borji, M.-M Cheng, H Jiang, J Li, IEEE transactions on image processing. 2412A. Borji, M.-M. Cheng, H. Jiang, and J. Li, "Salient object detection: A benchmark," IEEE transactions on image processing, vol. 24, no. 12, pp. 5706-5722, 2015. Structure-measure: A New Way to Evaluate Foreground Maps. D.-P Fan, M.-M Cheng, Y Liu, T Li, A Borji, IEEE International Conference on Computer Vision (ICCV). IEEED.-P. Fan, M.-M. Cheng, Y. Liu, T. Li, and A. Borji, "Structure-measure: A New Way to Evaluate Foreground Maps," in IEEE International Conference on Computer Vision (ICCV). IEEE, 2017, pp. 4548-4557, http://dpfan.net/smeasure/. Enhanced-alignment Measure for Binary Foreground Map Evaluation. D.-P Fan, C Gong, Y Cao, B Ren, M.-M Cheng, A Borji, International Joint Conference on Artificial Intelligence (IJCAI). D.-P. Fan, C. Gong, Y. Cao, B. Ren, M.-M. Cheng, and A. Borji, "Enhanced-alignment Measure for Binary Foreground Map Evaluation," in International Joint Conference on Artificial Intelligence (IJCAI), 2018, pp. 698-704, http://dpfan.net/e-measure/. Efficient salient region detection with soft image abstraction. M.-M Cheng, J Warrell, W.-Y Lin, S Zheng, V Vineet, N Crook, Proceedings of the IEEE International Conference on Computer vision. the IEEE International Conference on Computer visionM.-M. Cheng, J. Warrell, W.-Y. Lin, S. Zheng, V. Vineet, and N. Crook, "Efficient salient region detection with soft image abstraction," in Proceedings of the IEEE International Conference on Computer vision, 2013, pp. 1529-1536. Salient object detection: A discriminative regional feature integration approach. H Jiang, J Wang, Z Yuan, Y Wu, N Zheng, S Li, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionH. Jiang, J. Wang, Z. Yuan, Y. Wu, N. Zheng, and S. Li, "Salient object detection: A discriminative regional feature integration approach," in Proceedings of the IEEE conference on computer vision and pattern recognition, 2013, pp. 2083-2090. Deeply supervised salient object detection with short connections. Q Hou, M.-M Cheng, X Hu, A Borji, Z Tu, P H Torr, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionQ. Hou, M.-M. Cheng, X. Hu, A. Borji, Z. Tu, and P. H. Torr, "Deeply supervised salient object detection with short connections," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2017, pp. 3203-3212. Progressive attention guided recurrent network for salient object detection. X Zhang, T Wang, J Qi, H Lu, G Wang, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionX. Zhang, T. Wang, J. Qi, H. Lu, and G. Wang, "Progressive attention guided recurrent network for salient object detection," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 714-722. R3net: Recurrent residual refinement network for saliency detection. Z Deng, X Hu, L Zhu, X Xu, J Qin, G Han, P.-A Heng, Proceedings of the 27th International Joint Conference on Artificial Intelligence. the 27th International Joint Conference on Artificial IntelligenceAAAI PressZ. Deng, X. Hu, L. Zhu, X. Xu, J. Qin, G. Han, and P.-A. Heng, "R3net: Recurrent residual refinement network for saliency detection," in Proceedings of the 27th International Joint Conference on Artificial Intelligence. AAAI Press, 2018, pp. 684-690.
[]
[ "Auditing Source Diversity Bias in Video Search Results Using Virtual Agents", "Auditing Source Diversity Bias in Video Search Results Using Virtual Agents" ]
[ "Aleksandra Urman ", "Mykola Makhortykh ", "Roberto Ulloa ", "\nUniversity of Bern University of Zurich\nSwitzerland\n", "\nUniversity of Bern\nSwitzerland\n" ]
[ "University of Bern University of Zurich\nSwitzerland", "University of Bern\nSwitzerland" ]
[ "Companion Proceedings of the Web Conference 2021 (WWW '21 Companion)" ]
We audit the presence of domain-level source diversity bias in video search results. Using a virtual agent-based approach, we compare outputs of four Western and one non-Western search engines for English and Russian queries. Our findings highlight that source diversity varies substantially depending on the language with English queries returning more diverse outputs. We also find disproportionately high presence of a single platform, YouTube, in top search outputs for all Western search engines except Google. At the same time, we observe that Youtube's major competitors such as Vimeo or Dailymotion do not appear in the sampled Google's video search results. This finding suggests that Google might be downgrading the results from the main competitors of Googleowned Youtube and highlights the necessity for further studies focusing on the presence of own-content bias in Google's search results.KEYWORDS source diversity bias, algorithmic auditing, web search ACM Reference Format:
10.1145/3442442.3452306
[ "https://arxiv.org/pdf/2106.02715v1.pdf" ]
235,324,811
2106.02715
deeb2184750b7dd71b4a83b18b1c312a7b76d4af
Auditing Source Diversity Bias in Video Search Results Using Virtual Agents ACMCopyright ACM2021. April 19-23, 2021 Aleksandra Urman Mykola Makhortykh Roberto Ulloa University of Bern University of Zurich Switzerland University of Bern Switzerland Auditing Source Diversity Bias in Video Search Results Using Virtual Agents Companion Proceedings of the Web Conference 2021 (WWW '21 Companion) Ljubljana, Slovenia; New York, NY, USAACM52021. April 19-23, 202110.1145/3442442.3452306GESIS -Leibniz Institute for the Social Sciences Germany We audit the presence of domain-level source diversity bias in video search results. Using a virtual agent-based approach, we compare outputs of four Western and one non-Western search engines for English and Russian queries. Our findings highlight that source diversity varies substantially depending on the language with English queries returning more diverse outputs. We also find disproportionately high presence of a single platform, YouTube, in top search outputs for all Western search engines except Google. At the same time, we observe that Youtube's major competitors such as Vimeo or Dailymotion do not appear in the sampled Google's video search results. This finding suggests that Google might be downgrading the results from the main competitors of Googleowned Youtube and highlights the necessity for further studies focusing on the presence of own-content bias in Google's search results.KEYWORDS source diversity bias, algorithmic auditing, web search ACM Reference Format: INTRODUCTION The public tends to put high trust in the information retrieved via search engines [1]. However, search engine outputs are prone to different forms of bias [11,21] which can result in distorted representation of subjects which users are searching for [12,25]. One way of uncovering bias in web search outputs is to engage in algorithmic auditing [24]. Though a few studies have audited bias in text and image search (e.g., [21,22,26]), to the best of our knowledge no audits of video search results were conducted. However, video search is important in the societal context since people increasingly consume news via online videos [20] and treat video hosting platforms as a preferred environment for news finding [30]. The fact that video information can have powerful influence on users and even affect their behaviors [10,17] prompts the need for auditing whether video search is subjected to bias. To address this gap, we investigate the presence of source diversity bias -that is a systematic prevalence of specific information sources [11] -in video search outputs. By consistently prioritizing the same set of sources independently of search queries, search algorithms can diminish the quality of outputs by making them less representative [25] and negatively affecting the user experience [6]. Source diversity bias is also related to the phenomenon of search concentration, namely the tendency of search engines to prioritize few well-established domains over other sources [19] that, according to media, often results in search companies promoting their own services (e.g., YouTube in the case of Google [33]). By diminishing the diversity in the composition of source domains, companies can consolidate global media monopolies [19] through gaining unfair advantage over their competitors. To examine whether video search outputs are subjected to source diversity bias, we audit search results coming from five search engines -four Western (Bing, DuckDuckGo, Google, and Yahoo) and one non-Western (Yandex) -in response to English and Russian queries. Including Yandex along with queries in Russiana language dominating the main markets of Yandex -allowed us to test whether (some of) our observations can be attributed to structural differences between Western and non-Western markets (e.g., almost monopolistic status of Google in the former and its competition with Yandex in post-Soviet countries). We use virtual agent-based auditing approach to prevent search outputs from being affected by search personalization and search randomization. Then, using a selection of metrics, we assess the level of source domain diversity in search outputs and investigate whether there is evidence of certain engines prioritizing specific information sources. Specifically, we examine whether search engines tend to promote platforms associated with their parent companies (e.g., Alphabet for Google or Microsoft for Bing) or downgrade the competitors as was claimed by earlier research [19]. RELATED WORK The problem of auditing systematic skewness of web search outputs is increasingly recognized in the field of information retrieval (IR) [11,12,16,25]. Existing studies primarily look at it from one of the two perspectives: user bias and retrieval bias. User bias concerns skews in user perceptions of search outputs [15]. Retrieval bias relates to a skewed selection of search results [12,16]. One form of retrieval bias, which the current paper focuses on, is source diversity bias. Originally discussed in the context of search engines' tendency to prioritize web pages with the highest number arXiv:2106.02715v1 [cs.IR] 4 Jun 2021 of visitors [16], source diversity bias is currently investigated in the context of prioritization of certain categories of sources in response to particular types of queries (e.g., [23]). A disproportionate visibility of specific types of web resources can diminish overall quality of search results [6] and provide unfair advantage to companies and individuals that own specific search engines [16] -e.g., through own-content bias, -or direct most of the traffic to a handful of well-established sources, a phenomenon also known as search concentration [19]. To date, source diversity bias has been primarily investigated in the context of text search results [7,31,32] with the focus exclusively on one engine -Google. At the same time, few comparative studies that were conducted highlight substantial cross-engine differences in search diversity bias levels [18,23,34]. In the case of video search results, there is no systematic comparative assessment of source diversity bias. In 2020, the Wall Street Journal [33] and, subsequently, Mozilla [3] have found that Youtube appears among the top-3 featured "carousel" results in text search 94% of the time [3]. These findings highlight search concentration [19] around Youtube in Google's "carousel" results. However, it is unclear, first, whether the distribution of domains is similar in dedicated video search results. And second, how Google's results compare to those of its competitors -a comparison that is necessary to establish whether Youtube's dominance in video search results from the way Google's algorithm works exhibiting own-content bias. We aim to address these gaps with the present study. METHODS In this study, we have opted for a combination of methods for bias detection in web search outlined by Edelman [8]: 1) comparative analysis of the results provided by multiple search engines across a variety of search queries in two languages; 2) identification of skewness of results towards specific domains and whether the skewness, if observed, can be explained by market structure incentives. We have used video search results from the 4 biggest Western search engines by market share -Google, Yahoo, Bing, DuckDuckGo, -and one major non-Western search engine -Yandex [2]. Since Yandex has the largest presence in post-Soviet states where large shares of populations speak Russian as a first language, we utilized queries in both, English and Russian, to conduct the search and estimate whether there are differences in the observations. There were 62 queries in total -31 English queries with translations into Russian, -that concerned contemporary events (i.e., the US presidential elections and coronavirus), conspiracy theories (i.e., Flat Earth), and historical events (i.e., Holocaust) 1 . Below we outline the details on the data collection and analysis. Data collection To collect the data, we utilized a set of virtual agents -that is software simulating user browsing behavior and recording its outputs. The benefits of this approach, which extends algorithmic auditing methodology introduced by Haim et al. [13], is that it allows controlling for personalization [14] and randomization [23] factors. For the current study, we built a network of 84 CentOS virtual machines based in the Frankfurt region of Amazon Elastic Compute Cloud (EC2). On each machine, we deployed 2 virtual agents (one in Chrome browser and one in Mozilla Firefox browser), thus providing us with 188 agents overall. Each agent was made of two browser extensions: a tracker and a bot. The tracker collected the HTML and the metadata of all pages visited in the browser and immediately sent it to a storage server. The bot emulated a sequence of browsing actions that consisted of (1) visiting a video search engine page, (2) entering one of the 62 queries, and (3) scrolling down the search result page to load at least 50 images. Before searching for a new query, the browsers were cleaned to prevent the search history affecting the search outputs, and there was a 7-minute break between searches to mitigate potential effects of previous searches on the results. The study was conducted on February 26, 2020. We equally distributed the agents between five search engines: Google, Bing, Yahoo, Yandex, and DuckDuckGo (DDG) 2 . Because of technical issues (e.g., bot detection mechanisms), some agents did not manage to complete their routine. The overall number of agents per engine which completed the full simulation routine and returned the search results differed by query -sometimes the search engine would detect automation and temporarily ban the agent. This was particularly often the case with Yandex, where for some queries all 34 deployed agents successfully finished the routine while for others (a minority of queries) Yandex only returned the results for 10 agents and banned the rest. The mean number of agents who completed the full routine by engine across all queries is the following: Bing (29), DDG (34), Google (33), Yahoo (31), and Yandex (17). After the data was collected, we have extracted top-10 individual video links obtained by each agent for each search query and proceeded with the analysis using this data. Our decision to rely on the top-10 results only is motivated by the fact that users tend to pay the most attention to the first few results -i.e., those on the first results page [27]. A comparison of search results by browser has demonstrated that there are no major between-browser differences -a finding in contrast with those observed for text search results [23] -thus for the analysis we have proceeded aggregating the results for both browsers. Data analysis 3.2.1 Source diversity. To assess whether there is evidence suggesting that diversity bias -that is, lack of source diversity, -is present in the sampled results on domain level, we have calculated how many distinct source domains are, on average, present in the results for each query. To account for the potential randomization due to so-called "Google Dance" [5] in the results, for each query we aggregated the calculation over the results obtained from each individual autonomous agent. Afterwards, we have calculated mean numbers of distinct sources separately for English and Russian queries to establish whether there are differences in the observations depending on the language of the search. We have also qualitatively examined the results per query to find out whether there are distinct patterns with regard to query categories. Search concentration. To establish whether there is evidence of search concentration in the collected video results, we have calculated, first, the share of times different domains appear as the top result for each search query, and second, the proportion of times different domains appear among the top-10 search results per query at all. We suggest that the consistent appearance of a specific domain or few specific domains at the very top of search results and, in general, among top-10 results more frequently than the others would indicate search concentration. In addition, we have scrutinized the results with regard to owncontent bias exhibited by Google according to the media [33]. We aimed to establish whether Google's results lend evidence of own-content bias either through the promotion of Youtube in the results or the demotion of the results provided by its main competitors. One way to assess that is to compare Google's results with those obtained through other search engines [8] and see whether there are major differences in the observed frequencies of appearance of different domains between them. Thus, we compared the proportion of times each domain -Youtube and its competitors such as Vimeo, Dailymotion and Rutube, -appears in the results provided by different engines. As Fig.1 shows, there are differences in the level of source diversity exhibited by the examined search engines. Google has consistently presented more diverse video results in terms of source domains than its competitors. Yandex has taken a second place in terms of domain diversity. This domain diversity hierarchy is similar for both English and Russian queries, albeit for the Russian queries the observed domain diversity is a bit lower on all five search engines than for the English queries. FINDINGS 4.1 Source diversity Qualitative analysis of the domain diversity by query has demonstrated that there are no consistent patterns with regard to the proportion of distinct sources by query category in our dataset. Hence, we suggest that domain diversity is affected more by the algorithms used by each search engine examined and, probably, the data they are trained on -the latter might explain the observed differences between Russian and English queries, -rather than by the specific topics of the search queries. Search concentration As shown in Fig.2, Youtube has been featured as the top result most frequently in all cases but one -namely, the results for English queries on Yandex where Vimeo surfaced as the first result most often. Remarkably, on Google itself Youtube appeared as the top result less frequently than on other platforms, a finding in contrast with those made by The Wall Street Journal [33] and Mozilla [3] in the context of featured video "carousel" on the first page of Google's text results. DuckDuckGo, Yahoo, Yandex are the three engines exhibiting sizeable differences in the prominence of Youtube as the top result between English and Russian queries with Youtube being featured as the top result more frequently in response to English queries. We suggest that the findings reported in Fig.2 lend evidence to search concentration bias in video search results on the examined search engines, with the effect in our sample being stronger for English than for Russian queries. The findings reported in Fig.3 suggest that search engines feature different arrays of domains -with the exception of Youtube -in search results. Google tends to retrieve results from legacy media in both Russian and English more frequently than other search engines, a finding in line with the previous research [7,31,32]. Other search engines also include some legacy media, though to a lower extent than Google, as well as social media (e.g., Ok.ru, Facebook.com), and several video portals that are Youtube's competitors -Dailymotion, Vimeo and Rutube. None of these potential Youtube competitors, however, appear in the top-10 results on Google in our dataset at all despite their presence on other search engines. Vimeo, Dailymotion and Rutube all appear at least once among the top-10 results on all search engines except Google. This finding suggests that Google might downgrade Youtube's direct competitors, however an analysis based on a broader spectrum of queries is necessary to estimate the scope and persistence of this result. CONCLUSIONS AND FUTURE WORK Top 10 outputs of video search for most of search engines except Google show limited source diversity. By relying on average on 2-3 unique sources to retrieve top results for English queries, search engines create a situation in which users' information choices are shaped by a few content providers. This raises concerns about search engines facilitating consolidation of power on the information markets. The only exception among the five search engines examined is Google, where the degree of source diversity is almost twice as high. This effect can be attributed to Google putting substantial effort into diversifying search results in response to earlier criticisms. With 6 unique domains per 10 top results, Google follows its declared principle of having no more than 2 results coming from the same domain in the top results [4]. The finding also suggests that low diversity on other search engines is likely attributed to the absence of diversification mechanisms which Google implements. The importance of integrating diversification measurements is highlighted by high degree of source diversity bias. The top results for all search engines (except Yandex in English and Google) are dominated by one platform, namely YouTube. Its systematic prevalence reinforces the platform's almost monopolistic status. It is problematic considering that the platform is already used as a major news source among certain shares of the population [20], despite earlier audits demonstrating that its algorithms might lead to user radicalization [29] and, affected by users' viewing history, aggressively promote pseudoscientific content to users who have watched pseudoscientific videos before [28]. Search concentration around Youtube can only help cement its monopoly and the associated effects. Google fairs better than other search engines in terms of domainlevel source diversity and, at a first glance, does not exhibit owncontent bias since Youtube is less prominent in its results than on other search engines. However, in our sample, Google is the only search engine that did not provide any results from Youtube's major competitors. It is thus plausible that Google indeed, as media reports suggested [33], might be downgrading the results coming from the major competitors of Youtube thus exhibiting own-content bias manifested not in promoting Youtube but in lowering the prominence of its competitors in search results. However, it could also be that the obtained results with regards to Google and the absence of Youtube's competitors in its outputs are specific to the topics addressed by our sample of queries and is absent in other contexts. Hence, to make a robust conclusion about the presence or absence of own-content bias in Google's video search results, further studies encompassing broader sets of queries are necessary. We suggest that our observations highlight the need for such studies. Further, the present analysis is based on a snapshot experiment on a limited selection of queries. We believe that our findings warrant subsequent longitudinal audits of video search results to assess the persistence of our observations and, potentially, the changes that occur overtime. Such audits are crucial to inform the decisions of policy-makers and regulators. This is especially timely and pressing given the recent anti-trust cases against Google [9] and calls for putting tech giants under scrutiny, among other, in the context of their market power. Figure 1 : 1Mean number of distinct domains in top-10 video search results (Y-axis) per query, grouped by query languages (legend) and search engines (X-axis) Figure 2 : 2Domain most frequently appearing as the top result by query group and search engine; % of time it appears as the top result. In Fig.3 we list the domains most frequently featured among top-10 search results on each of the examined search engines for English and Russian queries. As with the top-1 result, the domain most frequently featured in top-10 is Youtube on all engines but Yandex in response to English queries where the most frequent domain is Vimeo. The share of other domains in search results is comparatively marginal -less than 10% -in almost all cases with the exception of Ok.ru for Russian queries on Yandex. Youtube thus emerges as the most prominent domain in search results. Apart from it, there is no other domain that is among the top 5 domains most frequently appearing among the first 10 results on all search engines. Figure 3 : 3Domains most frequently appearing among top-10 results by query group and search engine and % of time a domain appears among top-10 results. We used the following queries: coronavirus, bernie sanders, joe biden, pete buttigieg, elizabeth warren, michael bloomberg, donald trump, us elections, syria conflict, ukraine conflict, yemen conflict, holocaust, holodomor, armenian genocide, second world war, first world war, artificial intelligence, big data, virtual reality, vaccination, vaccination benefits, vaccination dangers, george soros, illuminati, new world order, Flat Earth, UFO, Aliens, misinformation, disinformation, fake news. All queries were entered into search engines in lower case. For the searches in Russian, we used the exact translations of the English queries listed below into Russian verified by two native Russian speakers. For all engines, the ".com" version of the image search engine was used (e.g., google.com). 2020 Edelman Trust Barometer. Search Engine Market Share Worldwide. n.d.. n.d.[n.d.]. 2020 Edelman Trust Barometer. https://www.edelman.com/trustbarometer [2] [n.d.]. Search Engine Market Share Worldwide. https://gs.statcounter.com/search- engine-market-share YouTube Dominates Google Video in 2020. YouTube Dominates Google Video in 2020. https://moz.com/blog/youtube- dominates-google-video-results-in-2020 Google search update aims to show more diverse results from different domain names. 2019. Google search update aims to show more diverse results from different domain names. https://searchengineland.com/google-search-update-aims-to- show-more-diverse-results-from-different-domain-names-317934 The Search: How Google and Its Rivals Rewrote the Rules of Business and Transformed Our Culture. John Battelle, ID: FR9PAAAAMAAJPortfolio. Google-Books. John Battelle. 2005. The Search: How Google and Its Rivals Rewrote the Rules of Business and Transformed Our Culture. Portfolio. Google-Books-ID: FR9PAAAAMAAJ. How algorithmic popularity bias hinders or promotes quality. Giovanni Luca Ciampaglia, Azadeh Nematzadeh, Filippo Menczer, Alessandro Flammini, 10.1038/s41598-018-34203-2Scientific Reports. 815951Nature Publishing GroupGiovanni Luca Ciampaglia, Azadeh Nematzadeh, Filippo Menczer, and Alessandro Flammini. 2018. How algorithmic popularity bias hinders or promotes quality. Scientific Reports 8, 1 (Oct. 2018), 15951. https://doi.org/10.1038/s41598-018- 34203-2 Number: 1 Publisher: Nature Publishing Group. . Nicholas Diakopoulos, Daniel Trielli, Jennifer Stark, Sean Mussenden, n.d.Nicholas Diakopoulos, Daniel Trielli, Jennifer Stark, and Sean Mussenden. [n.d.]. I Vote For-How Search Informs Our Choice of Candidate. 22I Vote For-How Search Informs Our Choice of Candidate. ([n. d.]), 22. Benjamin Edelman, BIAS IN SEARCH RESULTS?: DIAGNOSIS AND RESPONSE. 17Benjamin Edelman. [n.d.]. BIAS IN SEARCH RESULTS?: DIAGNOSIS AND RESPONSE. 7 ([n. d.]), 17. . Gilad Edelman, n.d.Gilad Edelman. [n.d.]. Google's Antitrust Cases: A Guide for the Perplexed. Wired. Google's Antitrust Cases: A Guide for the Perplexed. Wired ([n. d.]). https://www.wired.com/story/google-antitrust-lawsuits-explainer/ Viral videos and virtue: Moral elevation inductions shift affect and interpersonal goals in daily life. Thane M Erickson, Adam P Mcguire, Gina M Scarsella, Tara A Crouch, Jamie A Lewis, Ashley P Eisenlohr, Tasha J Muresan, 10.1080/17439760.2017.1365163The Journal of Positive Psychology. 13Thane M. Erickson, Adam P. McGuire, Gina M. Scarsella, Tara A. Crouch, Jamie A. Lewis, Ashley P. Eisenlohr, and Tasha J. Muresan. 2018. Viral videos and virtue: Moral elevation inductions shift affect and interpersonal goals in daily life. The Journal of Positive Psychology 13, 6 (Nov. 2018), 643-654. https://doi.org/10.1080/17439760.2017.1365163 Publisher: Routledge _eprint: https://doi.org/10.1080/17439760.2017.1365163. Search Engine Bias and the Demise of Search Engine Utopianism. E Goldman, 10.1007/978-3-540-75829-7_8Web Search: Multidisciplinary Perspectives. Amanda Spink and Michael ZimmerBerlin, HeidelbergSpringerE. Goldman. 2008. Search Engine Bias and the Demise of Search Engine Utopianism. In Web Search: Multidisciplinary Perspectives, Amanda Spink and Michael Zimmer (Eds.). Springer, Berlin, Heidelberg, 121-133. https://doi.org/ 10.1007/978-3-540-75829-7_8 James Grimmelmann, Some Skepticism About Search Neutrality. Faculty Scholarship. James Grimmelmann. 2010. Some Skepticism About Search Neutrality. Faculty Scholarship (Jan. 2010). https://digitalcommons.law.umaryland.edu/fac_pubs/ 1417 Abyss or Shelter? On the Relevance of Web Search Engines' Search Results When People Google for Suicide. Mario Haim, Florian Arendt, Sebastian Scherr, 10.1080/10410236.2015.1113484Health Communication. 32Mario Haim, Florian Arendt, and Sebastian Scherr. 2017. Abyss or Shelter? On the Relevance of Web Search Engines' Search Results When People Google for Suicide. Health Communication 32, 2 (Feb. 2017), 253-258. https://doi.org/10. 1080/10410236.2015.1113484 Measuring personalization of web search. Aniko Hannak, Piotr Sapiezynski, Arash Molavi Kakhki, Balachander Krishnamurthy, David Lazer, Alan Mislove, Christo Wilson, 10.1145/2488388.2488435Proceedings of the 22nd international conference on World Wide Web -WWW '13. the 22nd international conference on World Wide Web -WWW '13Rio de Janeiro, BrazilACM PressAniko Hannak, Piotr Sapiezynski, Arash Molavi Kakhki, Balachander Krish- namurthy, David Lazer, Alan Mislove, and Christo Wilson. 2013. Measuring personalization of web search. In Proceedings of the 22nd international conference on World Wide Web -WWW '13. ACM Press, Rio de Janeiro, Brazil, 527-538. https://doi.org/10.1145/2488388.2488435 Domain bias in web search. Samuel Ieong, Nina Mishra, Eldar Sadikov, Li Zhang, 10.1145/2124295.2124345Proceedings of the fifth ACM international conference on Web search and data mining (WSDM '12). the fifth ACM international conference on Web search and data mining (WSDM '12)New York, NY, USAAssociation for Computing MachinerySamuel Ieong, Nina Mishra, Eldar Sadikov, and Li Zhang. 2012. Domain bias in web search. In Proceedings of the fifth ACM international conference on Web search and data mining (WSDM '12). Association for Computing Machinery, New York, NY, USA, 413-422. https://doi.org/10.1145/2124295.2124345 Shaping the Web: Why the Politics of Search Engines Matters. Lukas Introna, Helen Nissenbaum, 10.1080/01972240050133634The Information Society. 163Lukas Introna and Helen Nissenbaum. 2000. Shaping the Web: Why the Politics of Search Engines Matters. The Information Society 16, 3 (July 2000), 169- 185. https://doi.org/10.1080/01972240050133634 Publisher: Routledge _eprint: https://doi.org/10.1080/01972240050133634. Michiko Izawa, WHAT MAKES VIRAL VIDEOS VIRAL?: ROLES OF EMOTION, IMPRESSION, UTILITY, AND SOCIAL TIES IN ONLINE SHARING BEHAVIOR. 56Michiko Izawa. [n.d.]. WHAT MAKES VIRAL VIDEOS VIRAL?: ROLES OF EMOTION, IMPRESSION, UTILITY, AND SOCIAL TIES IN ONLINE SHARING BEHAVIOR. ([n. d.]), 56. The business and politics of search engines: A comparative study of Baidu and Google's search results of Internet events in China. Min Jiang, 10.1177/1461444813481196New Media & Society. 162SAGE PublicationsMin Jiang. 2014. The business and politics of search engines: A comparative study of Baidu and Google's search results of Internet events in China. New Media & Society 16, 2 (March 2014), 212-233. https://doi.org/10.1177/1461444813481196 Publisher: SAGE Publications. Search Concentration, Bias, and Parochialism: A Comparative Study of Google, Baidu, and Jike's Search Results From China. Min Jiang, 10.1111/jcom.12126Journal of Communication. 64Min Jiang. 2014. Search Concentration, Bias, and Parochialism: A Comparative Study of Google, Baidu, and Jike's Search Results From China. Journal of Communication 64, 6 (2014), 1088-1110. https://doi.org/10.1111/jcom.12126 _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/jcom.12126. Online News Video Consumption. Antonis Kalogeropoulos, 10.1080/21670811.2017.1320197Digital Journalism. 6Routledge _eprintAntonis Kalogeropoulos. 2018. Online News Video Consumption. Digital Jour- nalism 6, 5 (May 2018), 651-665. https://doi.org/10.1080/21670811.2017.1320197 Publisher: Routledge _eprint: https://doi.org/10.1080/21670811.2017.1320197. Unequal Representation and Gender Stereotypes in Image Search Results for Occupations. Matthew Kay, Cynthia Matuszek, Sean A Munson, 10.1145/2702123.2702520Proceedings of the 33rd Annual ACM Conference on Human Factors in Computing Systems (CHI '15). the 33rd Annual ACM Conference on Human Factors in Computing Systems (CHI '15)New York, NY, USAAssociation for Computing MachineryMatthew Kay, Cynthia Matuszek, and Sean A. Munson. 2015. Unequal Repre- sentation and Gender Stereotypes in Image Search Results for Occupations. In Proceedings of the 33rd Annual ACM Conference on Human Factors in Computing Systems (CHI '15). Association for Computing Machinery, New York, NY, USA, 3819-3828. https://doi.org/10.1145/2702123.2702520 Search bias quantification: investigating political bias in social media and web search. Juhi Kulshrestha, Motahhare Eslami, Johnnatan Messias, Muhammad Bilal Zafar, Saptarshi Ghosh, Krishna P Gummadi, Karrie Karahalios, 10.1007/s10791-018-9341-2Information Retrieval Journal. 221Juhi Kulshrestha, Motahhare Eslami, Johnnatan Messias, Muhammad Bilal Zafar, Saptarshi Ghosh, Krishna P. Gummadi, and Karrie Karahalios. 2019. Search bias quantification: investigating political bias in social media and web search. Information Retrieval Journal 22, 1 (April 2019), 188-227. https: //doi.org/10.1007/s10791-018-9341-2 How search engines disseminate information about COVID-19 and why they should do better. Mykola Makhortykh, Aleksandra Urman, Roberto Ulloa, Mykola Makhortykh, Aleksandra Urman, and Roberto Ulloa. 2020. How search engines disseminate information about COVID-19 and why they should do better. 10.37016/mr-2020-017COVID-19 and Misinformation. 1Harvard Kennedy School Misinformation Review 1, COVID-19 and Misinformation (May 2020). https://doi.org/10.37016/mr-2020-017 Automation, Algorithms, and Politics| Auditing for Transparency in Content Personalization Systems. Brent Mittelstadt, International Journal of Communication. 1012Brent Mittelstadt. 2016. Automation, Algorithms, and Politics| Auditing for Transparency in Content Personalization Systems. International Journal of Communication 10, 0 (Oct. 2016), 12. https://ijoc.org/index.php/ijoc/article/view/ 6267 Number: 0. Assessing bias in search engines. Abbe Mowshowitz, Akira Kawaguchi, 10.1016/S0306-4573(01)00020-6Information Processing & Management. 38Abbe Mowshowitz and Akira Kawaguchi. 2002. Assessing bias in search engines. Information Processing & Management 38, 1 (Jan. 2002), 141-156. https://doi.org/ 10.1016/S0306-4573(01)00020-6 Competent Men and Warm Women: Gender Stereotypes and Backlash in Image Search Results. Jahna Otterbacher, Jo Bates, Paul Clough, 10.1145/3025453.3025727Proceedings of the 2017 CHI Conference on Human Factors in Computing Systems (CHI '17). the 2017 CHI Conference on Human Factors in Computing Systems (CHI '17)New York, NY, USAAssociation for Computing MachineryJahna Otterbacher, Jo Bates, and Paul Clough. 2017. Competent Men and Warm Women: Gender Stereotypes and Backlash in Image Search Results. In Proceedings of the 2017 CHI Conference on Human Factors in Computing Systems (CHI '17). Association for Computing Machinery, New York, NY, USA, 6620-6631. https://doi.org/10.1145/3025453.3025727 Bing Pan, Helene Hembrooke, Thorsten Joachims, Lori Lorigo, Geri Gay, Laura Granka, 10.1111/j.1083-6101.2007.00351.xGoogle We Trust: Users' Decisions on Rank, Position, and Relevance. 12Bing Pan, Helene Hembrooke, Thorsten Joachims, Lori Lorigo, Geri Gay, and Laura Granka. 2007. In Google We Trust: Users' Decisions on Rank, Position, and Relevance. Journal of Computer-Mediated Communication 12, 3 (2007), 801-823. https://doi.org/10.1111/j.1083-6101.2007.00351.x _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1083-6101.2007.00351.x. It is just a flu. Kostantinos Papadamou, Savvas Zannettou, Jeremy Blackburn, Emiliano De Cristofaro, Gianluca Stringhini, Michael Sirivianos, arXiv:2010.11638arXiv: 2010.11638Assessing the Effect of Watch History on YouTube's Pseudoscientific Video Recommendations. Kostantinos Papadamou, Savvas Zannettou, Jeremy Blackburn, Emiliano De Cristofaro, Gianluca Stringhini, and Michael Sirivianos. 2020. "It is just a flu": Assessing the Effect of Watch History on YouTube's Pseudoscientific Video Rec- ommendations. arXiv:2010.11638 [cs] (Nov. 2020). http://arxiv.org/abs/2010.11638 arXiv: 2010.11638. Auditing radicalization pathways on YouTube. Raphael Manoel Horta Ribeiro, Robert Ottoni, West, A F Virgílio, Wagner Almeida, Meira, 10.1145/3351095.3372879Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency (FAT* '20). the 2020 Conference on Fairness, Accountability, and Transparency (FAT* '20)New York, NY, USAAssociation for Computing MachineryManoel Horta Ribeiro, Raphael Ottoni, Robert West, Virgílio A. F. Almeida, and Wagner Meira. 2020. Auditing radicalization pathways on YouTube. In Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency (FAT* '20). Association for Computing Machinery, New York, NY, USA, 131-141. https://doi.org/10.1145/3351095.3372879 Many Americans Get News on YouTube, Where News Organizations and Independent Producers Thrive Side by Side. Galen Stocking, Michael Patrick Van Kessel, Katerina Eva Barthel, Maya Matsa, Khuzam, Galen Stocking, Patrick van Kessel, Michael Barthel, Katerina Eva Matsa, and Maya Khuzam. 2020. Many Americans Get News on YouTube, Where News Organizations and Independent Producers Thrive Side by Side. https: //www.journalism.org/2020/09/28/many-americans-get-news-on-youtube- where-news-organizations-and-independent-producers-thrive-side-by-side/ Search as News Curator: The Role of Google in Shaping Attention to News Information. Daniel Trielli, Nicholas Diakopoulos, 10.1145/3290605.3300683Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems (CHI '19). the 2019 CHI Conference on Human Factors in Computing Systems (CHI '19)Glasgow, Scotland UkAssociation for Computing MachineryDaniel Trielli and Nicholas Diakopoulos. 2019. Search as News Curator: The Role of Google in Shaping Attention to News Information. In Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems (CHI '19). Association for Computing Machinery, Glasgow, Scotland Uk, 1-15. https: //doi.org/10.1145/3290605.3300683 Googling Politics: Parties, Sources, and Issue Ownerships on Google in the 2017 German Federal Election Campaign. Julian Unkel, Mario Haim, 10.1177/0894439319881634Social Science Computer Review. 0894439319881634Julian Unkel and Mario Haim. 2019. Googling Politics: Parties, Sources, and Issue Ownerships on Google in the 2017 German Federal Election Campaign. Social Science Computer Review (Dec. 2019), 0894439319881634. https://doi.org/10.1177/ 0894439319881634 Searching for Video? Google Pushes YouTube Over Rivals. Kirsten Sam Schechner West, John Grind, Wall Street Journal. Sam Schechner West, Kirsten Grind and John. 2020. Searching for Video? Google Pushes YouTube Over Rivals. Wall Street Journal (July 2020). https://www.wsj. com/articles/google-steers-users-to-youtube-over-rivals-11594745232 Querying the Internet as a mnemonic practice: how search engines mediate four types of past events in Russia. Media. Andrei Zavadski, Florian Toepfl, 10.1177/0163443718764565Culture & Society. 41SAGE Publications LtdAndrei Zavadski and Florian Toepfl. 2019. Querying the Internet as a mnemonic practice: how search engines mediate four types of past events in Russia. Media, Culture & Society 41, 1 (Jan. 2019), 21-37. https://doi.org/10.1177/ 0163443718764565 Publisher: SAGE Publications Ltd.
[]
[ "RoMQA: A Benchmark for Robust, Multi-evidence, Multi-answer Question Answering", "RoMQA: A Benchmark for Robust, Multi-evidence, Multi-answer Question Answering" ]
[ "Victor Zhong \nUniversity of Washington † Meta AI\n\n", "Weijia Shi \nUniversity of Washington † Meta AI\n\n", "Wen-Tau Yih \nUniversity of Washington † Meta AI\n\n", "Luke Zettlemoyer \nUniversity of Washington † Meta AI\n\n" ]
[ "University of Washington † Meta AI\n", "University of Washington † Meta AI\n", "University of Washington † Meta AI\n", "University of Washington † Meta AI\n" ]
[]
We introduce RoMQA, the first benchmark for robust, multi-evidence, multi-answer question answering (QA). RoMQA contains clusters of questions that are derived from related constraints mined from the Wikidata knowledge graph. RoMQA evaluates robustness of QA models to varying constraints by measuring worst-case performance within each question cluster. Compared to prior QA datasets, RoMQA has more human-written questions that require reasoning over more evidence text and have, on average, many more correct answers. In addition, human annotators rate RoMQA questions as more natural or likely to be asked by people. We evaluate state-of-theart large language models in zero-shot, few-shot, and fine-tuning settings, and find that RoMQA is challenging: zero-shot and few-shot models perform similarly to naive baselines, while supervised retrieval methods perform well below gold evidence upper bounds. Moreover, existing models are not robust to variations in question constraints, but can be made more robust by tuning on clusters of related questions. Our results show that RoMQA is a challenging benchmark for large language models, and provides a quantifiable test to build more robust QA methods.
10.48550/arxiv.2210.14353
[ "https://export.arxiv.org/pdf/2210.14353v2.pdf" ]
253,116,788
2210.14353
b09a0e0398023683da479afc31df31440abb8f3e
RoMQA: A Benchmark for Robust, Multi-evidence, Multi-answer Question Answering Victor Zhong University of Washington † Meta AI Weijia Shi University of Washington † Meta AI Wen-Tau Yih University of Washington † Meta AI Luke Zettlemoyer University of Washington † Meta AI RoMQA: A Benchmark for Robust, Multi-evidence, Multi-answer Question Answering We introduce RoMQA, the first benchmark for robust, multi-evidence, multi-answer question answering (QA). RoMQA contains clusters of questions that are derived from related constraints mined from the Wikidata knowledge graph. RoMQA evaluates robustness of QA models to varying constraints by measuring worst-case performance within each question cluster. Compared to prior QA datasets, RoMQA has more human-written questions that require reasoning over more evidence text and have, on average, many more correct answers. In addition, human annotators rate RoMQA questions as more natural or likely to be asked by people. We evaluate state-of-theart large language models in zero-shot, few-shot, and fine-tuning settings, and find that RoMQA is challenging: zero-shot and few-shot models perform similarly to naive baselines, while supervised retrieval methods perform well below gold evidence upper bounds. Moreover, existing models are not robust to variations in question constraints, but can be made more robust by tuning on clusters of related questions. Our results show that RoMQA is a challenging benchmark for large language models, and provides a quantifiable test to build more robust QA methods. Introduction A high quality compositional question answering (QA) model should be robust to small variations in the underlying meaning of input questions. Consider the question "which pianists born in Paris play Western classical music?" To show robust understanding, a QA model should not only be able to correctly answer this direct question, but also a wide range of related queries that different in only a few constraints (e.g. who was a pianist born in Paris?, who was a Western classical pianist, not born in Paris?). Prior compositional QA datasets do not evaluate the robustness of QA models to variations in question constraints. We introduce RoMQA, a benchmark for Robust, Multi-evidence, multi-answer QA, which * Corresponding author [email protected] explicitly evaluates for robustness to small question perturbations. Figure 1 shows examples from RoMQA. RoMQA differs from previous work in a number of ways. Evaluates robustness to constraint variations. RoMQA contains clusters of related questions that are used to measure robustness to varying implicit question constraints. For each cluster, we compute a robustness score that is the the minimum score over the questions it contains. In order to perform well on RoMQA robustness evaluation, a model must be able to understand many different combinations of the implicit constraints that define the cluster, such as what it means to be a pianist, to be born in Paris, and to play Western classical music. To our knowledge, RoMQA is the first QA benchmark that evaluates this type of robustness. More complex questions. Human questions often have many answers and cannot be answered from a single text. When compared to existing datasets, RoMQA questions have more answers (mean 108.6, median 11), cover more diverse topics, and require more pieces of evidence text (mean 41.6, median 24). RoMQA also contains entity-linked, relation-extracted text that provide provenance for the constraints, showing the questions are answerable with multi-evidence reasoning from the text corpus. More natural human written questions. Compared to prior multi-answer compositional QA datasets, RoMQA provides an order of magnitude more human-written questions. Human evaluations show that these questions are more natural, as gauged by how likely a person is to ask the question. Qualitatively, RoMQA questions are less likely to contain overly precise constraints, unusual attribute comparisons, or overly large numbers of referential hops. We evaluate state-of-the-art large language Finally, no test model is robust to variations in question constraints. The best performing retrieval method obtains a worse-case related question test score of 37.9 F1 in the closed setting -a 25.9 F1 absolute drop compared to evaluating questions independently. Training on clusters of related questions, such RoMQA clusters, improves model robustness over training on unrelated questions. However the robustness gap remains large -closing this gap will likely require significant advances in natural language understanding. We open-source RoMQA at github. com/facebookresearch/romqa. RoMQA We describe RoMQA construction and how it differs from prior compositional QA datasets. Dataset construction RoMQA construction has three goals. First, we want a diverse selection of question topics. Sec-ond, these questions should require reasoning over multiple pieces of evidence. Third, we need to understand what implicit constraints the questions contain in order to evaluate robustness to varying constraints. At a high level, RoMQA construction involves 1) sampling constraints from knowledge base (KB) triples, 2) clustering related constraints, 3) sampling implicit constraints that form logical queries, and 4) annotating language questions. Sampling constraints from a KB. We create RoMQA questions from Wikidata (Vrandečić and Krötzsch, 2014) that are answerable given entity-linked and relation-extracted text (Elsahar et al., 2018). Wikidata consists of subjectproposition-object triples such as Gilbert_Amy occupation pianist. We convert these triples into entity-relation constraints. For instance, the previous example is decomposed into constraints Gilbert_Amy occupation obj and pianist occupation subj. Clustering related constraints. A cluster of related constraints share at least two answer entities. For example, occupation pianist subj and place_of_birth Paris subj are in the same cluster because they share the same answers Gilbert_Amy and Claude_Helffer (Parisborn pianists). As Wikidata has a skewed proposition distribution, we resample cluster constraints with probability inversely proportional to their proposition frequency in the KB (Appendix A). This down-samples over-represented propositions such as country. We keep clusters with ≥3 constraints to be able to generate many related questions from each cluster. We discard clusters of potentially spuriously related constraints with a single shared answer. 10k clusters are randomly chosen for training and 15k clusters for evaluation. RoMQA A film composed by S. Thaman and produced by Ganesh Babu. Who did not play for the Carolina Panthers but was a linebacker and was on the Chicago Bears? Which members of the Royal Society received the Order of Australia, but were not employed by the University of Oxford? Sub-orbital spaceflight that launched from Cape Canaveral Air Force Station Launch Complex 5. Launched by Mercury-Redstone Launch Vehicle Who is an athlete who participated in diving, and was born in Stockholm? HotpotQA Are Random House Tower and 888 7th Avenue both used for real estate? Which American singer and songwriter has a mezzo-soprano vocal range, Tim Armstrong or Tori Amos? WFMT FM radio transmits from the second tallest building in the United States, which is located where? Who was the recipient of a prize also given to a player for Chinese club Tianjin Quanjian? Which of Tara Strong major voice role in animated series is an American animated television series based on the DC Comics fictional superhero team, the "Teen Titans"? ComplexWebQuestions What university has more than 15,835 undergraduates and is the university Derek Fisher attended? Who influenced Whitman's poetry who was the public speaker who spoke about the American Civil War? What is the main train station called in the governmental jurisdiction where the government includes the position Mayor of San Francisco? Which country that borders Russia has the smallest ISO? What country that's a neighbor of Russia is a governmental jurisdiction where Erik Asanbayev holds a governmental office? QAMParI Where did a Roman Catholic archbishop of San Francisco attend school? At what institution did a Bishop of Derby receive their education? For which movie did Mani Ratnam work on the script and serve as producer? What Type VII C/41 and Type VII ships was in both the vessel classes of German? Philip Kaufman was responsible for both writing and directing of which movie? Sampling constraints to form logical queries. We generate up to 5 logical queries using each cluster. For each logical query, we copy the cluster and remove constraints with probability 0.1 and negate with 0.1. We negate sparingly because initial trials showed that a large number of negative constraints resulted in unnatural questions. We further remove redundant constraints (e.g. American presidents born in the US), and uniformly subsample up to 4 constraints. This constitutes a logical query with multiple conjunctions and subtractions. For instance, the cluster {occupation pianist subj, born_in Paris subj} can form a logical query occupation pianist subj AND born_in Paris subj. We discard overly general queries with ≥5000 answers. Creating natural language questions. Mechanical Turk crowd-workers annotate logical queries marked with Wikidata titles, descriptions, and aliases into questions. Appendix B Figure 11 shows the interface. Two more annotators verify each annotation to confirm that it matches the logical query. We keep only annotations with 100% agreement, resulting in 11% being discarded. After verification, we additionally discard clusters with ≤2 questions. Dataset analyses and comparison We Dataset size and question complexity Table 2 shows that only RoMQA evaluates robustness to input variations. Moreover, only RoMQA and QAMParI are human annotated with multiple answers and gold evidence. However, QAMParI provides 2,000 human-written questions for evaluation while RoMQA provides 11k for training and 17k for evaluation. Figure 2 shows the distribution of answer, evidence, and question sizes. First, RoMQA questions requires finding many answers. On average, RoMQA questions have 108 answers -at least 10x larger than others. Second, RoMQA requires reasoning over a much more evidence documents. On average, entities in the RoMQA answer set combine for a total of 52 evidence sentences. Third, RoMQA questions are longer with more words. Figure 3 shows that in a random sample of 500 questions, RoMQA refers to more unique noun phrases apart from HotpotQA. Naturalness human evaluation Prior work sometimes sacrifice question naturalness in pursuit of complexity. Table 1 Proportion of votes for "someone would ask this question" Figure 4: The distribution of questions naturalness ratings by 3 annotators on 1,000 randomly sampled questions from the development set of each dataset. Each annotator rates four questions shuffled in random order, one from each dataset. he annotator is asked whether they would ask the question themselves, and if they think someone else would ask the question. clude unusual constraints such as IDs (e.g. . . . has the smallest ISO? 1 ), overly precise constraints (e.g. . . . is an American animated television series based on the DC Comics fictional superhero team, the "Teen Titans") and an excessive number of referential expressions (e.g. . . . in the governmental jurisdiction where the government includes the position Mayor of San Francisco). We compare the naturalness of 1,000 randomly sampled human written questions from each dataset. Each annotator is shown a randomly sampled question from each dataset, shuffled in random order. The annotator is asked "how likely would you ask this question?" and "how likely do you think another person would ask this question?" Each question is annotated by 3 crowdworkers. The breakdown of ratings across questions is shown for each dataset in Figure 4. On average, annotators consider RoMQA to be significantly more natural than HotpotQA and ComplexWebQuestions. Table 3: Input format given the question "Who was a pianist born in Paris". For closed setting, the candidate "Gilbert Amy" is used as an example. Evidence includes retrieved sentences for the retrieval model or gold evidence for the upperbound model. Experiments How do existing QA systems perform on RoMQA? Are they robust to variations in question constraints? We answer these questions by experimenting with state-of-the-art models in zero-shot, few-shot, and supervised learning settings. Evaluation Open vs closed settings Given a question in RoMQA, a QA system should produce a set of answer entities. In the closed setting, the system is given 100 candidate entities and must identify which ones answer the question. Negative candidates are potentially difficult for a model because they can match any constraint in the question. In the open setting, candidates are not given. Evaluation metrics RoMQA questions have many answers. For the closed setting, we evaluate predictions using F 1 and accuracy. F 1 measures set overlap between predicted and gold answers, while accuracy measures whether they match exactly. For the open setting, we evaluate preci-sion@K (P 10 ) for two reasons. First, precision gives partial credit when it is too hard to enumerate the full set. Second, the user may ask a question with many answers (e.g. hundreds or thousands) with the intent of only seeing some examples (e.g. list British footballers). For each score, we additionally have a robustness variant. Let Q = {q 1 , q 2 . . . q n } denote a cluster of n related questions. The question q i has the corresponding predicted answer set p i and gold answer set g i . A robustness score is the worst-case score across the cluster. For instance, the robust F 1 is F 1 R (Q) = min i (F 1 (p i , g i )). We compute similar robustness scores for accuracy and precision@K. Models We evaluate three classes of models. The input format for each class is shown in Table 3. Zero-shot. We consider a naive closed setting baseline that predicts all candidates as answers. We also include state-of-the-art prompting models tk-instruct-3B (Wang et al., 2022) and opt-instruct-175B (Zhang et al., 2022). In the closed setting, they generate yes/no given the question and a candidate. In the open setting, they generate answers given the question. In the closed setting, the context includes an equal number of randomly sampled positive and negative examples. We compare the scores for the candidate answering vs. not answering the current question. These scores are calibrated using channel calibration (Min et al., 2022). In the open setting, the model context includes ≤10 subsampled answers for each example. Supervised learning. We tune BART-large with and without retrieved evidence (Lewis et al., 2020) and show standard deviation across 5 random seeds. For the closed setting which considers a candidate entity, we use a two-stage hybrid retrieval because dense retrievers under-perform sparse retrievers on rare, precise entities (Sciavolino et al., 2021). We first retrieve documents with BM25 using entity title as the query. We then use DPR (Karpukhin et al., 2020) to retrieve document sentences whose cosine similarity with the question exceed a threshold (0.65) tuned on the training set. Finally, we fine-tune to classify whether each candidate belongs to the answer set. In the open setting, we do not use classification models because it must decide over all possible (2.9M) entities per question and is computationally prohibitive. 2 Instead, we directly retrieve evidence with DPR and fine-tune the model to generate the answer set as a 1024-token sequence. Upper bound with gold evidence. We provide a performance upper bound by training supervised models with gold evidence -sentences that provide provenance to an implicit question constraint. For instance, consider the example in Figure 1. Claude Hellfer is an answer to the question "Who was a pianist born in Paris?", whereas David Fray is not. In this case, the gold evidence includes "Claude Helffer. . . was a French pianist", "Helffer was born in Paris", and "David Fray. . . is a French classical pianist". Because the gold evidence only contains sentences that provide provenance to an implicit constraint, it does not contain the sentence "David Fray was born in Tarbes, near the Pyrenees." In other words, given gold evidence, the QA model does not need to filter out candidates (e.g. David Fray) because of evidence that conflict with implicit constraints (e.g. born in Tarbes instead of Paris). Instead, it only needs to verify that the evidence references all implicit constraints. Consequently, the gold evidence setting is overly optimistic in that part of the reasoning is completed by a perfect retriever. While no such retriever currently exists, this setting nevertheless provides an upper bound estimate for RoMQA. Table 4 and Table 5 show performance on RoMQA closed and open settings. RoMQA is challenging for state-of-the-art large-scale models. Moreover, these models are not robust to variations in question constraints. The best models significantly trail the gold evidence upper bound, showing there is significant room future work. Results Zero-shot and few-shot models perform similarly to naive predict-all baseline. In the closed setting, each system is given a set of 100 candidate entities and must identify which entity belong to the answer set. We find that state-of-the-art pretrained instruction prompting models perform on par with the naive baseline of simply predicting that every candidate belongs to the answer set. This occurs both with instructing prompting and in-context learning models, and suggests that they can not effectively reason about the types of compositional questions found in RoMQA. Both closed and open settings remain challenging with supervised training. When given 11k annotated examples, large retrieval models perform better than zero-shot and few-shot LMs. However, supervised performance also trails the gold evidence upper bound. This suggests that there is significant room for modeling improvements that retrieve and compose evidence in order to answer compositional questions. What types of questions do the best-performing supervised systems struggle with? Figure 5 plots Pearson correlation with F 1 in the closed setting, and shows that systems generally struggles with more precise questions. 3 First, when the question has many answers, the model has an easier time producing some correct answers). Second, the model performs better on more general propositions that co-occur with many different unique entities. Third, the model struggles with questions with more implicit constraints. Methods are not robust to question constraint variations. All methods drop in performance when considering the worst-case performance among clusters of related questions. This suggests that large LM-based QA systems are not robust to variations in question constraints. Figure 6 shows what types of questions result in robustness drops. Compared to other questions in the same cluster, a question is easier if it contains more answers, and harder if it specifies more implicit constraints. Training on clusters of related questions (e.g. RoMQA clusters) is one way to improve model robustness. Given clusters of questions with related implicit constraints, in the first setting we train on unrelated questions -one question from each cluster for a total of K training examples. In the second setting, we train on related questions -K consecutive examples from entire clusters. Table 6 shows that while the diversity from training on unrelated questions marginally improves overall performance, training on clusters of related questions results in more robust models. Nevertheless, the robustness drops remain significant. Considering that variations in RoMQA questions are reasonable questions humans would write, as opposed to artificially created adversarial questions, our findings suggests that there is a practical need for developing more robust QA systems. precision recall f1 Figure 6: Correlation with robustness drop (F1 -cluster mean F1) on the closed setting. The axes denote deviation from the cluster means. Among a cluster of related questions, a more precise question with more constraints or fewer answers tend to be harder for the model than related questions with more answers or less constraints. Building context for open setting is very challenging. While the closed setting RoMQA challenges current state-of-the-art models, the open setting remains an even greater challenge. One core difficulty associated with open setting RoMQA is that it is difficult to compute the evidence set required the answer the question. Consider Figure 1's question "Who was a Western classical music pianist, not born in Paris". The obvious way a human would answer this question is substracting the set of people born in Paris from the set of Western classical music pianists. However, both of these sets are very large. Our results show that an end-to-end large language model struggles in reasoning over such large sets. Related Work Question answering datasets Existing QA datasets focus on answering from a single passage (Rajpurkar et al., 2016;Joshi et al., 2017;Kwiatkowski et al., 2019;Sciavolino et al., 2021) to answering over multiple pieces of evidence (Yang et al., 2018;Welbl et al., 2018;Thorne et al., 2018). Recent datasets further emphasize answering questions that have multiple answers (Min et al., 2020;Amouyal et al., 2022). RoMQA combines the latter two settings in that it requires answering questions over multiple pieces of evidence to provide multiple answers. Compared to prior datasets, RoMQA questions require robust reasoning over more pieces of evidence to provide more answers. Robustness evaluation NLP systems have previous been show to lack robustness. They are susceptible to character based attacks that comprise of both nonsensical inputs (Jia and Liang, 2017), ran- , and semantically equivalent inputs adversarially selected to disrupt system behaviour (Ribeiro et al., 2018;Zhao et al., 2018;Iyyer et al., 2018). In contrast, the questions in RoMQA are not adversarial -they are written with reasonable information-seeking intent. Zero-shot/few-shot learning Recent work has also shown that large pretrained LMs can perform zero-shot and few-shot inference (Brown et al., 2020;Wang et al., 2022;Zhang et al., 2022). For the former, the LM performs inference given a prompt or an instruction. For the latter, the LM is additionally given a sample of training examples as demonstration. We use both settings as baselines for RoMQA, and find that there is significant room for improvement in large-scale pretraining to capture compositional reasoning over multiple pieces of evidence text. Conclusion In order to build effective NLP models, we must move towards evaluations that test model robustness to variations in the input. We presented RoMQA, the first benchmark for robust, multi-evidence, multi-answer QA. RoMQA evaluates robustness of models to varying question constraints by testing for worst-case performance among clusters of related questions. Compared to prior QA datasets, RoMQA has more natural human-written questions that require reasoning over more evidence text to more answers. RoMQA is challenging for state-of-the-art large LMs in zero-shot, few-shot, and supervised settings, and provides a quantifiable test to build more robust QA methods. We want questions that cover diverse topics, however Wikidata has a very skewed proposition distribution, with a long tail of rare propositions. Hence, we down-sample frequent propositions. Let P prop (x) denote the percentage of triples that contain the proposition x. We define the average proposition probability as P prop = 1 |X | x P prop (x). Given a constraint with proposition x, we remove it with probability r = 1 − min 1, Pprop Pprop(x) 1 2 . In particular, those with below average frequency are not removed, and those with above average frequency are removed with increasing likelihood. After removing constraints based on propositions, we randomly sample up to 10 constraints using a distribution over their inverse proposition probabilities 1 Pprop . Figure 7 shows the distribution over cluster sizes after resampling. Figure 10 shows that resampling results in a more diverse set of questions by emphasizing rarer propositions in the knowledge graph. B Annotation RoMQA questions are annotated by crowdworkers on Amazon Mechanical Turk from the US, Canada, UK, Australia, and New Zealand. We require that annotators have ≥95% approval rating and have done a minimum of 5000 HITs. Questions are submitted for annotation in batches of 500. For each batch, a sample of 10 questions from each worker is inspected by the authors. If ≥2 of annotations in the sample are incorrect, then response from the worker in that batch are marked for re-annotation. The final set of annotations are additionally verified by 2 more crowd-workers to confirm that they correspond to constraints. We keep only examples with 100% agreement, which corresponds to 89% of the annotated data. C Dataset Statistics Cluster sizes. Figure 7 shows the distribution of cluster sizes in RoMQA. During the sampling procedure, we remove small clusters of ≤3 questions and avoid large clusters of ≥7 questions. Implicit constraint distribution. Figure 9 shows the distribution of positive and negative implicit constraints in RoMQA questions. Most questions have 2 positive constraints, and 0-1 negative constraints. We limit questions to 7 constraints during sampling. In practice, nonsensical questions with too many constraints are filtered out during verification. Figure 8 shows the distribution of implicit constraints vs. the size of the answer set. More precise questions with more implicit constraints typically have fewer answers. In general, RoMQA questions may have more than 1000 answers, though the vast majority contain less than 1000 answers. The outlier questions with more than 1000 answers are not shown in the figure. D Experiment setup For zero-shot and few-shot models, we use API services provided by the original authors. For supervised models, we fine-tune BART-large models with learning rate 1e − 6 on V100 GPUs. For classification in the closed setting, we train with batch size 100 and evaluate with batch size 1000. For generation in the open setting, we train and evaluate with batch size 2 and decode with beam size 3. Figure 11: Mechanical Turk annotation interface for question writing. The annotator is shown a collection of positive and negative constraints in random order. Each constraints consists of an entity, a proposition, and a direction. Entities and propositions are expanded with descriptions and aliases from Wikidata. A subset of answer entities is listed to disambiguate answer types. Figure 2 : 2compare RoMQA to prior compositional QA datasets: HotpotQA (Yang et al., 2018), Com-plexWebQuestions (CWQ; Talmor and Berant, 2018), and QAMParI (Amouyal et al., 2022). Dataset comparison over question, evidence, and answer size distributions. Figure 3 : 3Question diversity as measured by the number of unique noun-phrases in 500 randomly sampled questions from the development set of each dataset. The batches are randomly sampled 4 times to compute standard deviation. illustrates artifacts in randomly sampled questions from HotpotQA, ComplexWebQuestions, and QAMParI. They in- Few-shot in-context learning. We evaluate tk-instruct-3B (Wang et al., 2022), opt-instruct-175B (Zhang et al., 2022), and GPT3 (text-davinci-002; Brown et al. (2020)) with as many in-context examples as model context allows (4, 8, and 8 respectively). Input format is similar to that of the zero-shot setting, with the addition of in-context examples. Figure 5 : 5Correlation with model performance (F1) on the closed setting. Imprecise questions with many answers are easier to answer (higher F1). Questions based on general propositions that co-occur with many different entities are easier to answer. Questions with more constraints are more difficult to answer. Figure 7 : 7Number of constraints per cluster. Figure 8 :Figure 9 : 89Answer set size vs constraint count. Positive vs negative constraint count. Figure 10 : 10Most common propositions, before and after subsampling. Subsampling downsamples overly represented propositions in the knowledge graph and results in a more diverse set of propositions and question topics. W Table 1 : 1Randomly sampled examples from RoMQA and other compositional QA datasets. Human evaluations show that people are more likely to ask RoMQA questions than those from other compositional QA datasets. Qualitatively, RoMQA questions exhibit fewer artifacts such as overly precise constraints (e.g. 15,835 undergraduates), overly numerous references (e.g. is an American animated. . . based on. . . the "Teen Titans"), and unusual attribute comparisons (e.g. smallest ISO).Dataset Train Dev Test Human written Multi answer Gold evidence Robustness evaluation RoMQA (Ours) 11k 7k 11k Yes Yes Yes Yes HotpotQA 90k 7k 7k Yes No Yes No CWQ 28k 3k 3k Yes Yes No No QAMParI 64k 1k 1k Eval only Yes Yes No Table 2 : 2Dataset size and question complexity. Model class Setting Input format Zero-shot Closed Gilbert Amy [SEP] Who was a pianist born in Paris? Open Who was a pianist born in Paris? Few-shot Closed Katie Bell [SEP] Who is an athlete who participated in diving, and was born in Stockholm? [SEP] False [newline] . . . Gilbert Amy [SEP] Who was a pianist born in Paris? [SEP] True Open Who is an athlete who participated in diving, and was born in Stockholm? [SEP] Johan Jansson . . . [newline] . . . Who was a pianist born in Paris? [SEP] Supervised Closed Gilbert Amy [SEP] Who was a pianist born in Paris? Open Who was a pianist born in Paris? Sup+evidence Closed Gilbert Amy [SEP] Who was a pianist born in Paris? [SEP] Gilbert Amy (born 29 August 1936) is a French composer and conductor . . . Open Who was a pianist born in Paris? [SEP] Gilbert Amy (born 29 August 1936) is a French composer and conductor . . . Table 4 : 4Model performance on closed setting RoMQA. Metrics are set F1, set accuracy, and their robustness counterparts (i.e. worst case measure over cluster of related questions). Each model is given 100 candidate entities and must decide which entity belongs to the answer set. The retrieval model additionally observes sentences retrieved via BM25 followed by DPR. Zeroshot and few-shot are binary-classifiers calibrated with channel calibration. Supervised models fine-tune BART-large on the training data to classify the answer set on a per-entity basis. Table 6 : 6Training supervised retrieval models on related vs unrelated questions. For unrelated questions training, one question is taken from each cluster for a total of 2695 ques- tions. For related questions training, entire clusters of related questions are included until there are 2695 questions. While training on a more diverse set of unrelated questions results in marginally higher overall performance, training on related questions results in more robust models. dom sentence/word triggers (Jia and Liang, 2017; Wallace et al., 2019) ISO codes are 2-3 character-long codes that represent names of countries and their subdivisions. For reference, inference over the entire test set with 100 candidate entities per example using the classification model requires 10 hours on a Volta 32GB GPU. Pearson correlation is a measure of linear correlation. A Pearson coefficient of 1 or -1 implies positive or negative linear correlation, while 0 implies no linear dependency.
[]
[ "MEASURES, ANNULI AND DIMENSIONS", "MEASURES, ANNULI AND DIMENSIONS" ]
[ "Zoltán Buczolich ", "Stéphane Seuret " ]
[]
[]
Given a Radon probability measure µ supported in R d , we are interested in those points x around which the measure is concentrated infinitely many times on thin annuli centered at x. Depending on the lower and upper dimension of µ, the metric used in the space and the thinness of the annuli, we obtain results and examples when such points are of µ-measure 0 or of µ-measure 1.The measure concentration we study is related to "bad points" for the Poincaré recurrence theorem and to the first return times to shrinking balls under iteration generated by a weakly Markov dynamical system.The study of thin annuli and spherical averages is also important in many dimension-related problems, including Kakeya-type problems and Falconer's distance set conjecture.
10.1007/s00209-023-03230-9
[ "https://export.arxiv.org/pdf/2111.09379v3.pdf" ]
244,345,624
2111.09379
ea5bf981a85bd34d432d32e55c93d683ee832ab6
MEASURES, ANNULI AND DIMENSIONS Zoltán Buczolich Stéphane Seuret MEASURES, ANNULI AND DIMENSIONS Given a Radon probability measure µ supported in R d , we are interested in those points x around which the measure is concentrated infinitely many times on thin annuli centered at x. Depending on the lower and upper dimension of µ, the metric used in the space and the thinness of the annuli, we obtain results and examples when such points are of µ-measure 0 or of µ-measure 1.The measure concentration we study is related to "bad points" for the Poincaré recurrence theorem and to the first return times to shrinking balls under iteration generated by a weakly Markov dynamical system.The study of thin annuli and spherical averages is also important in many dimension-related problems, including Kakeya-type problems and Falconer's distance set conjecture. Introduction and main results In the following, µ is a Radon probability measure supported in R d and dim H denotes the Hausdorff dimension. We denote by B(x, r) the closed ball {y ∈ R d : x − y ≤ r}, which obviously depends on the norm · chosen on R d . We consider norms which are equivalent with the most common Euclidean one, ||.|| 2 . Definition 1.1. For every x ∈ R d , 0 < r < 1 and δ ≥ 1, define the annulus (1.1) A(x, r, δ) = B(x, r) \ B(x, r − r δ ). We say that P µ (x, r, δ, η) holds when µ A(x, r, δ) ≥ η · µ B(x, r) . Finally, we set E µ (δ, η) = {x ∈ R d : P µ (x, r n , δ, η) holds for a sequence (r n ) n≥1 → 0}. Intuitively, around points belonging to E µ (δ, η), the measure µ concentrates a substantial part of its local mass on a very thin annulus (since r δ < < r). The larger δ, the thinner the annulus: E µ (δ , η) ⊂ E µ (δ, η) when δ ≥ δ. Our goal is to investigate the size of the sets E µ (δ, η). In this paper, we only consider diffuse measures, i.e. without any Dirac mass: µ(B(x, 0)) = 0, for every x ∈ R d . In this case, µ(E µ (1, η)) = 1 for every η ∈ [0, 1]. The question we investigate hereafter concerns the size of E µ (δ, η) for δ > 1, and it appears that the answer depends on the measure µ, the thinness δ and the norm used to define the annuli, in a subtle manner. The sets E µ (δ, η) appear in various places. For instance, in [2], it is proved that if µ is the Sinai-Ruelle-Bowen measure associated with a non-uniformly hyperbolic dynamical system (X, T, µ), then the elements of E µ (δ, η) are "bad points" for the Poincaré recurrence theorem, in the sense that given r > 0, when x ∈ E µ (δ, η), the iterates T j x of x come back inside B(x, r) not as often as expected. More recently, Pawelec, Urbański, and Zdunik [17] investigated the first return times to shrinking balls under iteration generated by a weakly Markov dynamical systems, and had to deal with what they call the Thin Annuli Property. This property has several versions in [17], and is very similar to belonging to the complementary set of our sets E µ (δ, η), except that the exponent δ in P µ (x, r, δ, η) depends on r, and may tend to infinity when r tends to 0. In weaker versions of the Thin Annuli Property there are also restrictions on the range of radii. The conditions we impose to the elements of E µ (δ, η) are stronger, that is, they imply that the so-called Full Thin Annuli Property of [17] holds. In the same paper, the authors prove (Theorem C) that every finite Borel measure µ in a Euclidean space R d , satisfies the Thick Thin Annuli Property (this means that for arbitrary measures the range of radii for which the Thin Annuli Property holds is more limited). Our theorems below state that in many situations (for instance, for all measures µ with large lower dimensions), Theorem C can be improved. Let us also mention that Theorem D of [17] shows that certain measures coming from conformal geometrically irreducible Iterated Function Systems satisfy the Full Thin Annuli Property. We will come back to this later in the introduction. Similar questions appear also when studying orbit distribution of various groups acting on R 2 (Theorem 3.2 of [18]). See also [8,19] for other occurrences of such questions. Connections with other works are also made later in the introduction. We start by proving that, regardless of the norm in R d , measures with large lower dimension do not charge annuli at small scales if the exponent δ defining the annuli is sufficiently large, where "sufficiently large" depends on the lower and upper dimensions of µ, whose definitions are recalled now. ∃ r x > 0, ∀ 0 < r < r x , µ(B(x, r)) ≤ r α } and dim(µ) = inf{β ≥ 0 : for µ-a.e x, ∃ r x > 0, ∀ 0 < r < r x , µ(B(x, r)) ≥ r β }. Our first result is the following. Theorem 1.3. Let µ be a probability measure on R d such that dim(µ) > d − 1. For every δ > dim(µ)−(d−1) dim(µ)−(d−1) and η ∈ (0, 1], one has µ E µ (δ, η) = 0. Hence, for mono-dimensional measures µ satisfying dim(µ) = dim(µ) > d − 1, µ E µ (δ, η) = 0 for every δ > 1. Observe that Theorem 1.3 holds true regardless of the underlying metric used to define A(x, r, δ). Also, in dimension d = 1, Theorem 1.3 is simpler and rewrites as follows: µ E µ (δ, η) = 0 for every δ > dim(µ) dim(µ) and η ∈ (0, 1], Next theorem shows that Theorem 1.3 is optimal if the ||.|| ∞ metric is used. Theorem 1.4. Suppose that the metric generated by the norm ||.|| ∞ = max{|x i | : i = 1, ..., d} is used to define the annuli A(x, r, δ) in (1.1). For every d − 1 < d < d < d and every η ∈ (0, 1), there exists a probability measure µ on R d such that dim(µ) = d, dim(µ) = d and (1.2) µ E µ d − (d − 1) d − (d − 1) , η = 1. Remark 1.5. In Theorem 1.4 the case d = d is trivial since as noticed above, µ(E µ (1, η)) = 1 is always true for any non-atomic measure. Still in the ||.|| ∞ case, we further investigate what happens for measures of lower dimension less than d − 1. The quite surprising result is that for such measures µ, the worse scenario may always happen, in the sense that it is possible that µ charges only points around which the mass is infinitely often concentrated on small annuli. Theorem 1.6. Suppose that d ≥ 2 and that the metric ||.|| ∞ is used. For every d ≤ d − 1, d ≤ d ≤ d, every η ∈ (0, 1) and every δ > 1, there exists a probability measure µ on R d such that dim(µ) = d, dim(µ) = d and (1.3) µ E µ (δ, η) = 1. Although we do not explicitly state it, an adaptation of their proofs show that Theorems 1.4 and 1.6 remain true when the frontiers of the annuli A(x, r, δ) in the given metric are finite unions of convex parts of hyperplanes, for instance in the case ||.|| = ||.|| 1 = d i=1 |x i |. While the proof of Theorem 1.3 deals with all measures satisfying its assumptions, the proofs of Theorems 1.4 and 1.6 are constructive (they are both based on the same arguments): we explicitly build measures such that (1.2) or (1.3) are true. Coming back to Theorem 1.3, it is striking that when the Euclidean metric is used, the uniform bound for δ can be improved, in the sense that µ(E µ (δ, η)) = 0 even for δ smaller than d−(d−1) d−(d−1) . Next theorem illustrates this fact when d = 2, even when d < d − 1 = 1. Suppose that µ is a Radon probability measure such that dim µ = d and dim µ = d. Then µ(E µ (30, η)) = 0 for any η ∈ (0, 1). In the above Theorem 1.7, taking d = 1.01 and d = 1.99, one sees that d − (d − 1) d − (d − 1) = 1.99 − 1 1.01 − 1 = 99. By Theorem 1.4, one might expect the existence of a probability measure µ for which µ(E µ (99, η)) = 1. Since E µ (30, η) ⊃ E µ (99, η) the result of Theorem 1.7 goes well beyond the bound in (1.2) and shows that in the Euclidean metric, annuli are sufficiently "independent/decorrelated" so that Theorem 1.3 can be sharpened significantly -observe that Theorem 1.7 holds for all measures satisfying its assumptions. The heuristic intuition explaining the difference between Theorems 1.6 and 1.7 is that when an annulus with a cubic shape centered at a point x is translated by a very small distance, a large part of the translated annulus is still contained in a cubic annulus centered at x with comparable sidelength. But this does not hold true anymore for annuli with spherical shape. More generally, it is standard that dealing with the Euclidean norm is often more complicated than with polyhedral norms in many dimensional problems (we come back to this below). Our key tool to prove Theorem 1.7 is Lemma 5.1, which is an estimate of the size of intersecting annuli. This type of estimates were considered by many authors see, for example [1], [13] or especially Lemma 3.1 of [21]. The order we obtain in Lemma 5.1 is slightly better than the ones available in the literature, and optimal as we remark in Section 5. Also, it is striking that Theorem 1.7 deals also with lower and upper dimensions for µ that are less than 1 = d − 1, emphasizing the difference between the ||.|| ∞ metric (and Theorem 1.4) and the Euclidean metric. The values 0.89 and 30 we obtain are not optimal, and obtaining exact bounds in Theorem 1.7 for the Euclidean metric in dimension d seems to be a challenging and interesting open problem. Question 1.8. Suppose that the Euclidean metric is used in R d . For every 0 < d ≤ d ≤ d, find the best 1 ≤ δ = δ(d, d) such that for every probability measure µ supported inside [0, 1] d , for every δ > δ, for every η ∈ (0, 1), µ E µ (δ , η) = 0. Given our previous results, it is natural to conjecture the following: Conjecture 1.9. When d > d − 1, the optimal δ(d, d) is such that δ(d, d) < d−(d−1) d−(d−1) . Application to dynamical systems. Suppose that (T, X, µ, ) is a metric measure preserving dynamical system, that is (X, ) is a metric space and T : X → X is a Borel measurable map preserving a Borel probability measure µ on X. Given a ball B(x, r) and y ∈ X, τ B(x,r) (y) := min{n ≥ 1 : T n (y) ∈ B(x, r)}, is the first entry time of y to B(x, r). When y ∈ B(x, r), it is called the first return time of y to B(x, r). The entry and return times τ B(x,r) (y) are studied in [17], for Weakly Markov systems (T, X, µ, ) (we refer to [17] for precise definitions). From Theorems 1.3 and 1.7, it follows that for certain measures the Full Thin Annuli Property from [17] is satisfied. This way, based on Theorem B of [17] one can state the following theorem. Theorem 1.10. Let (T, X, µ, ) be a Weakly Markov system, with X ⊂ R d . If one of the following two conditions is satisfied: (i) dim(µ) > d − 1 and the metric used is any of the equivalent metrics used in R d ; (ii) d = 2, the metric is the Euclidean and dim(µ) ≥ 0.89; then the distributions of the normalized first entry time and first return time converge to the exponential one law, that is (1.4) lim r→0 sup t>0 µ y ∈ X : τ B(x,r) (y) > t µ(B(x, r)) − e −t = 0 and (1.5) lim r→0 sup t>0 1 µ(B(x, r)) µ y ∈ B(x, r) : τ B(x,r) (y) > t µ(B(x, r)) − e −t = 0. In [17], the conclusions are true for every Weakly Markov system (T, X, µ, ), with the restriction that the limits in (1.4) and (1.5) are taken only on a subsequence of radii (more precisely, the limit is lim r→0, r∈Rx , where R x is a β-thick class of radii -for the exact details see [17]). Our main improvement is to show that the limit holds true for every measure provided that one of the two conditions is satisfied. Let us mention that the problems concerning intersecting annuli in Section 5 are reminiscent to questions arising when studying for instance Falconer's distance set conjecture (for which recently many striking results were recently obtained [4,5,9,7,15]) and the (circular) Kakeya problem [10,11]. Distribution of measures on annuli plays an important role in these problems as well, mainly through the study of cubic or spherical averages, and it is a standard issue that the choice of the norm influences the results. In Section 6 we discuss this in more detail, in particular, based on standard arguments [4,15,20] in Fourier and potential theory, the following proposition is proved. Proposition 1.11. Let t > 1/2, and let µ be a finite t-regular measure on R 2 , i.e. a Radon measure satisfying (1.6) c t r t ≤ µ(B(x, r)) ≤ C t r t , ∀x ∈ spt(µ) and 0 < r < diam(spt µ). Assume that µ has compact support. Then, for δ = 4, for every η > 0 (1.7) lim r→0 µ({x ∈ R 2 : P µ (x, r, 4, η) holds)}) = 0. Unfortunately, this convergence in measure, or similar arguments, do not help improving our bounds (in fact, as far as we checked they do not even yield Theorems 1.3 to 1.7). But it is quite interesting that similar issues arise in both problems. The paper is organized as follows. In Section 2, Theorem 1.3 is proved. The proof is natural and quite short, based on the Radon measure version of Lebesgue's density theorem, Corollary 2.1. In Section 3, we explicitly build a measure µ such that µ(E µ (δ, η)) = 1, for any choice of δ and η. The construction is based on two subdivision schemes A and B that allow to spread the mass of a cube on its boundaries in a controlled manner. In Section 4, we show that the construction of Section 3 can be adapted to prove Theorem 1.6. In Section 5, the Euclidean case (and Theorem 1.7) is studied. Finally, in Section 6 we prove Proposition 1.11 and explain why such arguments, though interesting, are for the moment not strong enough to reach Theorems 1.3 to 1.7. Proof of Theorem 1.3 Before starting the proof we recall with slight change of notation part (1) of 2.14 Corollary from [14]. Corollary 2.1. Suppose that µ is a Radon measure on R n and E ⊂ R n is µ measurable. Then the limit lim r 0 µ(E ∩ B(x, r)) µ(B(x, r)) exists and equals 1 for µ-almost all x ∈ E and equals 0 for µ-almost all x ∈ R n \ E. Proof of Theorem 1.3. Fix δ > (d − (d − 1))/(d − (d − 1)) and ε > 0 so small that (2.1) (d − ε + (d − 1))δ > d + ε + (d − 1). Observe that there exists a constant C d > 0, depending on d and the chosen norm only, such that for any ball B(x, r), the associated annulus A(x, r, δ) can be covered by at most C d r (d−1)(1−δ) smaller balls B of radius r δ . Also, choose 0 < η < 1, and consider E µ (δ, η). Proceeding towards a contradiction, suppose that µ(E µ (δ, η)) > 0. Consider for every r, ε > 0 the set (2.2) D ε,r = {x ∈ R d : ∀ 0 < s ≤ r, s d+ε ≤ µ(B(x, s)) ≤ s d−ε }. By definition, for every ε > 0, the set E µ (δ, η) ∩ p≥1 D ε,1/p has full µ-measure in E µ (δ, η). This holds especially for ε fixed in (2.1). We put E = E µ (δ, η) ∩ D ε,1/p with a choice of a sufficiently large p such that µ(E) ≥ µ(E µ (δ, η))/2 > 0. Finally, we choose 0 < γ < 1/p so small that for every 0 ≤ r ≤ γ, (2.3) r (d−ε−(d−1))δ ≤ η2 ε−d 2C d r d+ε−(d−1) . For every x ∈ E, there exists r x > 0, such that (2.4) r x < γ/2, r d+ε x ≤ µ(B(x, r x )) ≤ r d−ε x and P µ (x, r x , δ, η) holds. By using Corollary 2.1 we can also assume that for µ-almost all x ∈ E, we have chosen r x so small that µ(E ∩ B(x, r x )) > (1 − η/10)µ(B(x, r x )) and hence (2.5) µ(B(x, r x ) \ E) < (η/10)µ(B(x, r x )). Let us write B x = B(x, r x ) and A x = A(x, r x , δ) for every x ∈ E. Such a ball satisfies by (2.4) (2.6) r d+ε x ≤ µ(B x ) ≤ r d−ε x together with P µ (x, r x , δ, η). Since µ(E) > 0, it is possible to select an x ∈ E for which (2.5) holds. Since P µ (x, r x , δ, η) is satisfied, we have (2.7) η · µ(B x ) ≤ µ(A x ). In addition, by definition of C d , A x is covered by at most C d (r x ) (d−1)(1−δ) many balls B of radius r δ x . For each of these balls B, either µ(E ∩ B) = 0, or µ(E ∩ B) > 0 and in this case E ∩ B ⊂ B(y, 2r δ x ) for some y ∈ E. Observe that by δ ≥ 1 and (2.4), we have 2r δ x < 2r x < γ < 1/p. By (2.2) µ(E ∩ B) ≤ µ(B(y, 2r δ x )) ≤ (2r δ x ) d−ε . Hence, summing over the (at most C d (r x ) (d−1)(1−δ) ) balls that cover A x , we get by (2.3) and (2.6) that µ(A x ∩ E) ≤ C d (r x ) (d−1)(1−δ) (2r δ x ) d−ε ≤ η 2 r d+ε x ≤ η 2 µ(B x ). Since by (2.5) and (2.7) µ(A x ∩ E) ≥ µ(A x ) − µ(B x \ E) ≥ 0.9 · η · µ(B x ) we obtain a contradiction with the previous equation. 3. Proof of Theorem 1.4 In this section we construct a Cantor-like measure µ which satisfies the assumptions of Theorem 1.4. The main idea is that the construction steps leading to µ are similar to the ones of a standard Cantor set and measure, except that for some exceptional steps where we impose that some annuli carry the essential weight of the mass. 3.1. Preliminaries. Fix 0 < η < 1, d − 1 < d < d < d, and put (3.1) δ = d − (d − 1) d − (d − 1) > 1, where the equality is equivalent to (3.2) (d − 1)(1 − δ) + δd = d. We call D n , n = 0, 1, ... the family of half-open dyadic cubes of side length 2 −n , that is cubes Q = d i=1 [k i · 2 −n , (k i + 1)2 −n ), k i ∈ Z, i = 1, ..., d. Observe that Q contains exactly one of its vertices, namely the one with coordinates (k 1 · 2 −n , ..., k d · 2 −n ). We call this vertex the smallest vertex of Q and denote it by v min,Q . The sum of the coordinates of v min,Q is denoted by s(v min,Q ), that is (3.3) s(v min,Q ) = k 1 · 2 −n + ... + k d · 2 −n . Definition 3.1. For every cube Q ∈ D m , for every n > m denote by ∂ n Q ⊂ D n the set of n-boundary cubes of Q, that is those Q ∈ D n which are included in Q, but at least one of their neighbors is not included in Q. We denote by d n (Q) the number of D n cubes in ∂ n Q. Out of the 2d faces of Q, we call ∂ n,1 Q the face consisting of those cubes Q ∈ ∂ n Q for which the first coordinate of its smallest vertex is k 1 · 2 −n : ∂ n,1 Q will be called the smallest face of Q. We also put ∂ m,1 Q = {Q}. We denote by d n,1 (Q) the number of D n cubes in ∂ n,1 Q. It is clear that (3.4) d n,1 (Q) = 2 (d−1)(n−m) . In the rest of this section, we construct a sequence of mass distributions (µ m ) m≥1 which converges to a measure µ that will satisfy the assumptions of Theorem 1.4. We put µ 0 ([0, 1) d ) = 1, and for Q = [0, 1) d , Q ∈ D 0 , we impose that µ 0 (Q) = 0. At the mth step, the mass distribution µ m will be defined by fixing the µ mweight of every cube Q ∈ D m , and this µ m -mass will be uniformly distributed inside every such Q. Then µ m+1 will be a refinement of µ m in the sense that (3.5) for every Q ∈ D m , µ m+1 (Q) = µ m (Q). Due to Kolmogorov's extension theorem (see for example [16], [20] or [12]) this ensures the weak convergence of (µ m ) to a measure µ defined on [0, 1] d . Set η * = √ η > η. Fix a constant c d > 0 so large that c d ≥ 2 10d+1 , η * > c −1 d (3.6) 1 − η > 1 − η * ≥ c −1 d , and η * 2 d+1+δ ≤ c d . (3.7) Denote by D + m the subset of D m containing those cubes Q ∈ D m for which the measure µ m (Q) > 0. The sequence of measures (µ m ) m≥1 will satisfy that for some C > 1, for every m ≥ 1, for all Q ∈ D + m , C −1 2 −md ≤ µ m (Q) ≤ C2 −md . We are going to alternate between two subdivision schemes. The subdivision scheme of type A is meant to distribute quite uniformly the mass of a cube into some of its subcubes, while the subdivision scheme of type B will concentrate the mass of Q into a very thin "layer" close to the boundary near the smallest face of Q and around its center. Subdivision scheme of type A. Assuming that µ m is defined on D m , this scheme A is applied to one individual cube Q ∈ D m to define a measure µ m+1 on the subcubes Q ∈ D m+1 included in Q. This subdivision scheme A distinguishes three cases: (A1) If µ m (Q) = 0, then for any Q ⊂ Q with Q ∈ D m+1 , we put µ m+1 (Q ) = 0. (A2) If 2 −d µ m (Q) ≥ 2 −(m+1)d then for any Q ⊂ Q with Q ∈ D m+1 , we set µ m+1 (Q ) = 2 −d µ m (Q). (A3) If 2 −d µ m (Q) < 2 −(m+1)d , then we concentrate all the mass on the subcube Q ∈ D m+1 included in Q whose smallest vertex is the same as that of Q. In other words, µ m+1 (Q ) = µ m (Q) and v min,Q = v min,Q . For all the other cubes Q ⊂ Q, Q ∈ D m+1 , we put µ m+1 (Q ) = 0. It is clear that with this process µ m+1 (Q) = µ m (Q), so µ m+1 is indeed a refinement of µ m on Q. Remark that (A2) tends to spread the mass of Q uniformly on its subcubes (hence to make the local dimension increase since d < d) while (A3) tends to concentrate the mass (hence to make the local dimension decrease from generation m to generation m + 1). Lemma 3.2. Assume that µ m satisfies (3.8) c −2 d 2 −md ≤ µ m (Q) ≤ c 2 d 2 −md , with Q ∈ D m , and apply subdivision scheme A to define µ m+1 on the sub- cubes Q ⊂ Q, Q ∈ D m+1 . Then, for every Q ∈ D m+1 with Q ⊂ Q, such that µ m+1 (Q ) = 0, (3.8) holds with the measure µ m+1 and generation m + 1, i.e. (3.9) c −2 d 2 −(m+1)d ≤ µ m+1 (Q ) ≤ c 2 d 2 −(m+1)d . Proof. Assume that we are in situation (A2). Hence, initially we had The construction ensures that for Q ⊂ Q, Q ∈ D m+1 , 2 −(m+1)d ≤ µ m+1 (Q ) = 2 −d µ m (Q) ≤ 2 −d c 2 d 2 −md ≤ c 2 d 2 −(m+1)d which implies (3.9) (the last inequality holds since d < d). Assume that we are now in situation (A3), which implies that c −2 d 2 −md ≤ µ m (Q) < 2 d 2 −(m+1)d < 2 d 2 −(m+1)d . Thus µ m+1 (Q ) = µ m (Q) for one selected Q ⊂ Q, Q ∈ D m+1 , and by (3.6) c −2 d 2 −(m+1)d ≤ c −2 d 2 −md ≤ µ m+1 (Q ) < 2 d 2 −(m+1)d < c 2 d 2 −(m+1)d . We prove now that if we apply scheme A a sufficiently large number of times, then we obtain cubes Q ∈ D n which all satisfy (3.10) either 2 −nd ≤ µ n (Q) < c d 2 −nd , or µ n (Q) = 0. Lemma 3.3. Assume that µ m satisfies (3.8) with Q ∈ D m , and apply subdi- vision scheme A to Q to define µ m+1 on the subcubes Q ⊂ Q, Q ∈ D m+1 , then apply subdivision scheme A to all Q ⊂ Q, Q ∈ D m+1 , to define µ m+2 on all subcubes Q ⊂ Q, Q ∈ D m+2 , etc... There exists an integer φ(Q) > m such that for every n ≥ φ(Q) and every cube Q ∈ D n with Q ⊂ Q, (3.10) holds for Q and µ n . Proof. We separate two cases depending on whether at step m we need to apply (A2) or (A3). The second case can be reduced to the first. Indeed, taking into consideration (3.8), suppose that we have c −2 d 2 −md ≤ µ m (Q) < 2 d 2 −(m+1)d and we start with subdivision (A3). Then for Q ⊂ Q, Q ∈ D m+1 , either µ m+1 (Q ) = 0, or c −2 d 2 −(m+1)d ≤ µ m+1 (Q ) = µ m (Q) < 2 d 2 −(m+1)d < c 2 d 2 −md . At the next step, we either have to apply division step (A3), or we can apply division step (A2). It is also clear that after finitely many steps we get to a situation when the first time step (A2) must be applied. In that case, at level n ≥ m, we have exactly one Q ⊂ Q, Q ∈ D n such that (3.11) 2 d 2 −(n+1)d ≤ µ n (Q ) < c 2 d 2 − nd and for all other descendants Q ⊂ Q, Q ∈ D n , µ n (Q ) = 0. Then we can start an argument which is the same as if we started with a subdivision (A2) from the very beginning. Observe that d < d and (3.11) imply that 2 −nd < 2 d 2 −(n+1)d ≤ µ n (Q ). For ease of notation we suppose that at step m we can already start with a subdivision step (A2), that is (3.11) holds with m instead of n. Now we apply (A2) to Q and µ m , and iteratively to all subcubes of Q of generation n > m, as long as µ n (Q ) ≥ 2 d 2 −(n+1)d for Q ∈ D n . Observe that for a cube Q ∈ D n with Q ⊂ Q, as long as 2 −d µ n (Q ) ≥ 2 −(n+1)d , that is µ n (Q ) ≥ 2 d 2 −(n+1)d > 2 d 2 −(n+1)d = 2 −nd , the mass of every subcube Q ⊂ Q of next generation is such that µ n+1 (Q ) = 2 −d µ n (Q ). Hence (3.12) log µ n+1 (Q ) log 2 −(n+1) = −d log 2 + log µ n (Q ) −(n + 1) log 2 > log µ n (Q ) log 2 −n . This means that the local dimension increases from generation n to generation n + 1. The construction ensures that all the subcubes of Q at a given generation n > m have the same µ n -mass. Further, by (3.12), the sequence log µn(Q ) log 2 −n (for Q ⊂ Q, Q ∈ D n ) is strictly increasing. Assuming that this process (A2) is iterated a number of times very large when compared to m, we would have log µn(Q ) log 2 −n ∼ log(2 −dn ) log 2 −n = d so µ n (Q ) ∼ 2 −dn < < 2 d 2 −(n+1)d , since d < d. Hence, after a finite number of iterations, we necessarily have µ n (Q ) < c d 2 −nd . Call φ(Q) ≥ m the first integer such that for all Q ⊂ Q, Q ∈ D φ(Q) , 2 −φ(Q)d ≤ µ φ(Q) (Q ) ≤ c d 2 −φ(Q)d . Remark 3.4. In case we started with subdivision steps (A3) before getting to a subcube in which we could apply (A2), then we define φ(Q) starting from this subcube. Recall that for ease of notation at the beginning of this part of the argument,we supposed that we start with subdivision steps (A2) at the mth step. Recall that at generation φ(Q), all the cubes Q ⊂ Q, Q ∈ D φ(Q) , have the same µ φ(Q) -mass. This also shows that (3.10) holds at generation φ(Q). Assume that (3.10) holds at generation n ≥ φ(Q) for Q ∈ D n , Q ⊂ Q. Then • if 2 −d µ n (Q ) ≥ 2 −(n+1)d , then we apply (A2) and for every Q ∈ D n+1 , Q ⊂ Q , 2 −(n+1)d ≤ µ n+1 (Q ) ≤ 2 −d c d 2 −nd ≤ c d 2 −(n+1)d . • if 2 −d µ n (Q ) ≤ 2 −(n+1)d , then we apply (A3) and for every Q ∈ D n+1 , either µ n+1 (Q ) = 0 or 2 −(n+1)d ≤ 2 −nd ≤ µ n+1 (Q ) ≤ 2 d 2 −(n+1)d ≤ c d 2 −(n+1)d . In all cases, (3.10) holds at generation n + 1. Hence the result. Subdivision scheme of type B. Let m ∈ N. Consider some Q ∈ D m , and assume that, (3.13) 2 −md ≤ µ m (Q) < c d 2 d 2 −md holds for µ m and Q at generation m. The purpose of the second subdivision scheme is to concentrate the mass µ m (Q) on two subparts of Q, first in a thin region close to the (inner part of the) boundary of Q (very close to its smallest ψ(m)-face), and second around its center. More precisely, we will assign η * of the initial mass µ m (Q) to part of an annulus very thin close to the border of Q (on Figure 1 this is the thin blue shaded rectangular region), and 1 − η * in a small cube located around the center of Q (on Figure 1, this is the small blue shaded central square). The remaining subcubes of Q will receive zero µ-mass. Distributing (part of ) the mass on the smallest face: Choose the smallest integer ψ(m) such that (3.14) 2 −(m+1)δ−1 ≤ 2 −ψ(m) < 2 −(m+1)δ . Since δ > 1, we have ψ(m) > m + 1. Consider ∂ ψ(m),1 Q the set of ψ(m)-boundary cubes on the smallest face of Q. Recalling (3.4), we have (3.15) d ψ(m),1 (Q) = 2 (d−1)(ψ(m)−m) . For all Q ∈ ∂ ψ(m),1 Q, Q ∈ D ψ(m) , we put (3.16) µ ψ(m) (Q ) = η * · 1 d ψ(m),1 (Q) µ m (Q). Combining (3.2), (3.13), (3.14), (3.15) and (3.16), we see that for all Q ∈ ∂ ψ(m),1 Q, µ ψ(m) (Q ) ≥ η * 2 −(d−1)(ψ(m)−m) 2 −md = η * 2 −(d−1)ψ(m) 2 −m(d−(d−1)) = η * 2 −(d−1)ψ(m) 2 −mδ(d−(d−1)) ≥ η * 2 −(d−1)ψ(m) 2 −ψ(m)(d−(d−1)) > c −1 d 2 −ψ(m)d , (3.17) where (3.6) has been used for the last lower bound. By (3.14) , (m+1)δ +1 ≥ ψ(m), hence − (d − 1)ψ(m) − mδ(d − (d − 1)) ≤ −ψ(m)d + (d − (d − 1))ψ(m) − mδ(d − (d − 1)) ≤ −ψ(m)d + (d − (d − 1))((m + 1)δ + 1) − mδ(d − (d − 1)) (3.18) = −ψ(m)d + (d − (d − 1))(δ + 1). 2 −m−1 c Q c Q 2 −ψ ′ (m) fraction 1 − η * of the fraction η * of the x r x 2 −ψ(m) initial mass of the cube initial mass of the cube µ ψ(m) (Q ) ≤ η * 2 −(d−1)(ψ(m)−m) c d 2 d 2 −md = η * 2 d c d 2 −(d−1)ψ(m) 2 −m(d−(d−1)) η * 2 d c d 2 −(d−1)ψ(m) 2 −mδ(d−(d−1)) ≤ η * 2 d c d 2 −ψ(m)d · 2 (1+δ)(d−(d−1)) ≤ c 2 d 2 −ψ(m)d . (3.19) Finally, for all Q ∈ ∂ ψ(m),1 Q we have (3.20) c −2 d 2 −ψ(m)d ≤ µ ψ(m) (Q ) ≤ c 2 d 2 −ψ(m)d . In particular, these cubes satisfy (3.8). Intuitively, starting with a cube Q such that µ(Q) ∼ |Q| d , we end up with many small cubes Q ⊂ Q, all located on the border of Q, and such that µ(Q ) ∼ |Q | d . Distributing part of the mass close to the center of Q: Let ψ (m) ≥ m be the unique integer satisfying (3.21) 2 −md−1 ≤ 2 −ψ (m)d ≤ 2 −md . Then it is easy to see that when m becomes large, m < ψ (m) < ψ(m) (intuitively, ψ (m) ∼ md/d while ψ(m) ∼ mδ and by (3.2), δ > d/d). We assume that m is so large that (1 − 2 −(ψ (m)−m) − 2 −(ψ(m)−m) ) d−1 > (1 − 2 −(ψ (m)−m)+1 ) d−1 > η * . (3.22) We denote by c Q the center of Q and by Q the (unique) dyadic cube of generation ψ (m) that contains c Q . Then, since we deal with dyadic cubes, c Q is the smallest vertex of Q, that is c Q = v min, Q . We put (3.23) µ ψ (m) ( Q) = (1 − η * )µ m (Q). By using (3.6), (3.7), (3.13), (3.21) and (3.22), we obtain µ ψ (m) ( Q) ≥ (1 − η * )2 −md ≥ (1 − η * )2 −ψ (m)d ≥ c −2 d 2 −ψ (m)d (3.24) and µ ψ (m) ( Q) < 2 d c d 2 −md < 2 d+1 c d 2 −ψ (m)d ≤ c 2 d 2 −ψ (m)d . We deduce that Q and ψ (m) satisfy equation (3.20), i.e. c −2 d 2 −ψ (m)d ≤ µ ψ (m) ( Q) ≤ c 2 d 2 −ψ (m)d . (3.25) Hence one can apply subdivision scheme A to it, iteratively, for all integers n such that ψ (m) < n ≤ ψ(m). At the end of the process, by Lemma 3.2, we get a collection of cubes Q ∈ D ψ(m) and a measure µ ψ(m) such that they all satisfy either µ ψ(m) (Q ) = 0, or c −2 d 2 −ψ(m)d ≤ µ ψ(m) (Q ) ≤ c 2 d 2 −ψ(m)d . (3.26) Definition 3.5. If Q ∈ D ψ(m) is such that Q ⊂ Q ⊂ Q (where Q is the cube of D ψ (m) containing c Q ), then Q is called a B-central cube at scale ψ(m) associated to Q ∈ D m . By construction, (3.27) µ Q ⊂Q: Q is a B-central cube at scale ψ(m) Q = (1 − η * )µ(Q). Lemma 3.6. If, for some large integer m, Q ∈ D ψ(m) is B-central, then for any x ∈ Q , there exists r x such that P µ ψ(m) (x, r x , δ, η) holds and 2 −m−1 ≤ r x < 2 −m−1 · 1.125. Remark 3.7. At inequalities (3.28) and (3.29) in the next proof, we will use that µ(B(x, 2 −m )) = µ(Q), which means essentially that µ ψ(m) (and µ) charges only the cube Q and not its neighbors at generation m. This will be a consequence of our construction in Section 3.4. Remark 3.8. Lemma 3.6 is stated for the measure µ ψ(m) , but it also holds for the measure µ obtained at the end of the construction. This simply follows from (3.5). Proof. For simplicity, we write µ for µ ψ(m) . As explained above, this abuse of notation is justified by (3.5). Let Q be a B-central cube, and x ∈ Q . We shall prove that, for some Figure 1 where x is marked with a dot in the B-central shaded blue cube, an arrow with dashed line is pointing at x, the label x is written at the bottom end of this arrow, the distance r x is marked by a solid left-right arrow, and the boundary of A(x, r x , δ) is shown with dotted lines. r x > 0, µ(A(x, r x , δ)) ≥ η · µ(B(x, r x )). Write x = (x 1 , ..., x d ) and Q = d i=1 [k i · 2 −m , (k i + 1) · 2 −m ). See Set r x := x 1 − k 1 · 2 −m . Since x − c Q ∞ ≤ 2 −ψ (m) and c Q = ((k 1 + 1 2 )2 −m , ..., (k d + 1 2 )2 −m ), we see that 2 −m−1 ≤ r x ≤ 2 −m−1 + 2 −ψ (m) . By construction, B(x, r x ) contains the largest part of ∂ ψ(m),1 Q. Indeed, if Q is a cube of ∂ ψ(m),1 Q containing one y ∈ ∂ ψ(m),1 Q with x − y ∞ > r x , then c Q − y ∞ ≥ x − y ∞ − c Q − x ∞ ≥ r x − c Q − x ∞ ≥ 2 −m−1 − 2 −ψ (m) . Denote by c Q the projection of c Q onto the smallest face of Q, see Figure 1. Since the supremum norm is used, the cubes of ∂ ψ(m),1 Q that may not intersect B(x, r x ) are thus located outside of B( c Q , 2 −m−1 − 2 −ψ (m) ). Observe that the intersection of B( c Q , 2 −m−1 − 2 −ψ (m) ) with the smallest face of Q is a (d−1)-dimensional cube of side length 2 −m −2 1−ψ (m) , and that its projection onto the smallest face of Q is of (d − 1)-dimensional volume (2 −m − 2 1−ψ (m) ) d−1 . Observe also that r δ x ≥ 2 −δ(m+1) > 2 −ψ(m) so all the above cubes Q belonging simultaneously to B(x, r x ) and ∂ ψ(m),1 Q also are included in A(x, r x , δ). These cubes are forming a single layer, and their projection onto the smallest face of Q is of (d − 1)-dimensional volume 2 −(d−1)ψ(m) . Recalling that d ψ(m),1 (Q) = 2 (d−1)(ψ(m)−m) , we obtain that A(x, r x , δ) contains more than (2 −m − 2 1−ψ (m) ) d−1 2 −(d−1)ψ(m) ≥ d ψ(m),1 (Q)(1 − 2 −(ψ (m)−m)+1 ) d−1 ≥ η * d ψ(m),1 (Q) many cubes from ∂ ψ(m),1 Q, where (3.22) is also used. Hence, recalling (3.16), (3.28) µ(A(x, r x , δ)) > η * d ψ(m),1 (Q) · η * 1 d ψ(m),1 (Q) µ m (Q) = (η * ) 2 µ m (Q) = ηµ(Q). Finally, by Remark 3.7, our construction ensures that µ(B(x, r x )) ≤ µ(Q). Thus (3.29) µ(A(x, r x , δ)) ≥ ηµ(B(x, r x )), i.e. P µ (x, r x , δ, η) holds. Finally, the fact that 2 −m−1 ≤ r x < 2 −m−1 · 1.125 follows from 2 −m−1 ≤ r x ≤ 2 −m−1 + 2 −ψ (m) and 2 −ψ (m)+m tends to zero when m → +∞. 3.3.3. Giving a zero-mass to the other cubes, and defining the measure µ ψ(m) on Q. For the remaining cubes at generation ψ(m), i.e. those Q ⊂ Q, Q ∈ D ψ(m) , Q ∈ ∂ ψ(m),1 (Q) and Q ⊂ Q, we put µ ψ(m) (Q ) = 0. The measure µ ψ(m) (Q ) is now defined for all Q ⊂ Q, Q ∈ D ψ(m) . Observe that by (3.16), the properties of the subdivision scheme A and (3.23), we have Q ⊂Q, Q ∈D ψ(m) µ ψ(m) (Q ) = µ m (Q), i.e. we indeed distributed the mass of Q onto the cubes Q ⊂ Q, Q ∈ D ψ(m) . 3.3.4. Defining inside Q the measure µ n for m < n < ψ(m). Forgetting for a while the details of the construction, and just focusing on the result, starting from Q ∈ D m with a given µ m -weight satisfying (3.10), we end up with a measure µ ψ(m) well defined on all cubes Q ⊂ Q, Q ∈ D ψ(m) . Since we jumped from level m to ψ(m), we can easily define µ n for m < n < ψ(m), but only inside Q. Indeed, for such an integer n, the measure µ n is simply defined by using µ ψ(m) as "reference" measure: for Q ∈ D n , Q ⊂ Q, we set (3.30) µ n (Q ) = Q ⊂Q , Q ∈D ψ(m) µ ψ(m) (Q ). It is easily checked that this definition is consistent with the definition of µ ψ(m) that we gave in (3.23) for instance. Next lemma shows that all the "intermediary" measures µ n so defined share the same scaling properties as µ m and µ ψ(m) . Consider first Q ∈ ∂ n,1 Q, i.e. a cube located on the border of Q. Choose an arbitrary Q ∈ ∂ ψ(m),1 Q, Q ⊂ Q. By (3.20) and d − 1 < d, one obtains µ n (Q ) = 2 (ψ(m)−n)(d−1) µ n (Q ) ≤ 2 (ψ(m)−n)(d−1) c 2 d 2 −ψ(m)d < c 2 d 2 (ψ(m)−n)d · 2 −ψ(m)d < c 2 d 2 −nd .µ n (Q ) = 2 (ψ(m)−n)(d−1) η * 2 (m−ψ(m))(d−1) µ m (Q) ≥ η * 2 (m−n)(d−1) · 2 −md > η * 2 (m−n)d · 2 −md = η * 2 −nd > c −2 d 2 −nd , hence (3.8) holds for Q . For the B-central cubes, suppose that m < n < ψ (m) and Q ∈ D n , Q ⊃ Q, where Q was defined after (3.22). Then µ n (Q ) = µ ψ(m) ( Q) = µ ψ (m) ( Q) = (1 − η * )µ m (Q) and by (3.25) (3.31) µ n (Q ) = µ ψ (m) ( Q) ≤ c 2 d 2 −ψ (m)d < c 2 d 2 −nd . On the other hand by (3.7) and (3.13) (3.32) µ n (Q ) = (1 − η * )µ m (Q) > (1 − η * )2 −md > (1 − η * )2 −nd > c −2 d 2 −nd , hence (3.8) holds for Q . Finally, when ψ (m) ≤ n < ψ(m) and Q ∈ D n and Q ⊂ Q, the fact that (3.8) holds for Q follows from the application of subdivision scheme A to the cube Q which satisfies (3.8). Part (ii) follows from (3.20) and (3.25). Observe that the construction ensures that, as claimed at the beginning of this section, Q ∈∂ ψ(m),1 Q µ ψ(m) (Q ) = η * · µ m (Q). Construction of the measure of Theorem 1.4. Recall that µ 0 is the Lebesgue measure on the cube [0, 1] d . By definition µ 0 satisfies (3.8). Step 1: We apply Subdivision Scheme A, m 1 times to the cube [0, 1] d , where m 1 ≥ φ([0, 1] d ) is such that Lemma 3.3 and (3.22) simultaneously hold for m = m 1 . We obtain a measure µ m 1 defined on cubes of D m 1 , such that for every Q ∈ D m 1 , either µ m 1 (Q) = 0 or the first half of (3.10) holds true with µ m 1 . Now, for each cube Q ∈ D m 1 , we select the cube Q of generation m 1 = m 1 + 1 located at its smallest vertex and we set µ m 1 (Q) = µ m 1 (Q ), so that µ m 1 satisfies (3.13). This last step ensures that the cubes at generation m 1 supporting µ m 1 are isolated, and Lemma 3.6 (together with Remark 3.7) applies. Now we are able to iterate the construction: Step 2k: We apply Subdivision scheme B to all cubes Q ∈ D m 2k−1 . Call m 2k = ψ(m 2k−1 ). We obtain a measure µ m 2k defined on D m 2k such that the properties of Lemma 3.3 hold for all dyadic cubes Q ∈ D n , m 2k−1 ≤ n ≤ m 2k . Step 2k + 1: We apply Subdivision scheme A to all cubes of generation m 2k . By Lemma 3.3, for each cube Q ∈ D m 2k , there exists an integer φ(Q) such that for n ≥ φ(Q), for every cube Q ∈ D n (3.10) holds for Q and µ n . Setting m 2k+1 = max{φ(Q) : Q ∈ D m 2k }, we are left with a measure µ m 2k+1 such that for all cubes Q ∈ D m 2k+1 (3.10) holds for Q and µ m 2k+1 . Setting m 2k+1 = m 2k+1 +1, for each cube Q ∈ D m 2k+1 , we select the cube Q of generation m 2k+1 located at its smallest vertex and we set µ m 2k+1 (Q) = µ m 2k+1 (Q ), so that µ m 2k+1 satisfies (3.13) and the cubes at generation m 2k+1 supporting µ 2k+1 are isolated. We are now ready to construct the set and the measure satisfying the conditions of Theorem 1.4. Call Q n (x) the unique dyadic cube Q ∈ D n that contains x. Q. For every x ∈ C, for every n, (3.33) c −2 d 2 −nd ≤ µ n (Q n (x)) ≤ c 2 d 2 −nd , and there exist two strictly increasing sequences of integers (j n (x)) n≥1 and (j n (x)) n≥1 satisfying B(x, r))) log r . (3.34) 2 −j n (x)d ≤ µ j n (x) (Q) < c d 2 −j n (x)d and (3.35) c −2 d 2 −j n (x)d ≤ µ j n (x) (Q) < c 2 d 2 −j n (x)d From the previous proposition, we easily deduce the following property. Corollary 3.11. For every x ∈ C, d µ (x) = d and d µ (x) = d. In particular, the measure µ satisfies d = dim(µ) < dim(µ) = d. Proposition 3.12. For µ-almost every x, there exist infinitely many integers n such that Q 2n (x) is a B-central cube at Step m 2n . The construction ensures that µ-almost all points are regularly located in a B-central cube, in the sense of Definition 3.5. Proof. For n ≥ 1, call A n = x ∈ C : Q m 2n (x) is a B-central cube at generation m 2n associated with a cube Q ∈ D m 2n−1 . By construction, and recalling (3.27), we get µ(A n ) = 1 − η * . Also, it is clear from the uniformity of the construction that the sequence (A n ) n≥1 is independent when seen as events with respect to the probability measure µ: for every finite set of integers (n 1 , n 2 , ..., n p ), µ(A n 1 ∩ A n 2 ∩ · · · ∩ A np ) = (1 − η * ) p . Applying the (second) Borel-Cantelli lemma (see for example [6], Section 7.3), we obtain that µ-almost every point belongs to an infinite number of sets A n . Hence the result. Remark 3.13. Observe that the same proof gives that µ-almost every point belongs to an infinite number of sets A c n . Corollary 3.14. The measure µ satisfies µ(E µ (δ, η)) = 1. This follows from Proposition 3.12 and Lemma 3.6. Proof of Theorem 1.6 We deal with the case where dim µ ≤ d − 1. In this situation, as stated in Theorem 1.6, there is no more restriction on δ. However we can argue almost like in the proof of Theorem 1.4. Fix an arbitrary δ > 1. As before, we are going to use various subdivision schemes to build a measure fulfilling our properties. The Subdivision scheme of type A in Section 3.2 is left unchanged. The Subdivision scheme of type B in Section 3.3 requires some adjustments (in particular, one cannot use (3.1) any more). The problem comes from the fact that when dim µ is less than d − 1, when trying to spread the mass of a given cube Q ∈ D m such that µ m (Q) ∼ 2 −md to the smaller cubes Q located on its smallest face ∂ ψ(m),1 Q, it is not possible to impose that µ m (Q ) ∼ 2 −md for all Q ∈ ∂ ψ(m),1 Q, since d ψ(m),1 (Q)2 −ψ(m)d ∼ 2 −(d−1)(ψ(m)−m) 2 −ψ(m)d > > 2 −md . In other words, the mass of the initial cube Q is not large enough to give the sufficient weight to each Q . So we introduce a Subdivision scheme of type C to solve this issue. We discuss the modifications needed to adapt Subsection 3.3 to this situation d < d − 1, and give the main ideas to proceed -some proofs are omitted, since they are exactly similar to those of Section 3. 4.1. Subdivision scheme of type C. As explained above, a new subdivision scheme is introduced, by essentially modifying a little bit Subdivision scheme of type B. Assume that Q ∈ D m satisfies (3.13). The integer ψ(m) is defined by (3.14), as in the previous Section. We know that (3.15) holds. As argued above, one can see that using (3.13) In this case we also define ∂ ψ(m),1,0 Q = ∅, and d ψ(m),1,0 (Q) = 0. For all m < n ≤ ψ(m) and Q ∈ ∂ n,1,+ Q := ∂ n,1 Q, using d ≤ d − 1 ≤ d and (3.13), one gets (4.1) 1 d ψ(m),1 (Q) µ m (Q) < c d 2 d 2 −md · 2 −(d−1)(ψ(m)−m) ≤ c d 2 d 2 −md · 2 −d(ψ(m)−m) = c d 2 d 2 −ψ(m)d .µ n (Q ) = η * 1 d ψ(m),1,+ (Q) · µ m (Q) · 2 (d−1)(ψ(m)−n) (4.3) < η * 2 −(d−1)(ψ(m)−m) · c d 2 d 2 −md · 2 (d−1)(ψ(m)−n) = η * c d 2 d 2 m(d−1) · 2 −md · 2 −(d−1)n ≤ η * c d 2 d · 2 −nd < c 2 d 2 −nd . Similarly, µ n (Q ) > η * 2 −(d−1)(ψ(m)−m) · 2 −md · 2 (d−1)(ψ(m)−n) (4.4) = η * 2 −(d−1)(n−m) 2 −md ≥ η * 2 −d(n−m) 2 −md ≥ η * 2 −nd > c −2 d 2 −nd . This means that (3.8) will remain true for these cubes. To resume, the scheme is in this case the same as before. • Case d ≤ d < d − 1: An extra care is needed. Recall Definition 3.1. Consider first the cubes Q ∈ ∂ m+1,1 Q. For the other cubes Q ⊂ Q, Q ∈ D m+1 , set µ m+1 (Q ) = 0. (i) If η * 2 −(d−1) µ m (Q) ≥ 2 −(m+1)d , then for any Q ∈ ∂ m+1,1 Q, define µ m+1 (Q ) = η * 2 −(d−1) µ m (Q ). For Q ⊂ Q, Q ∈ D m+1 , but Q ∈ ∂ m+1,1 Q, set µ m+1 (Q ) = 0. (ii) If η * 2 −(d−1) · µ m (Q) < 2 −(m+1)d , Next we iterate the process. Suppose that m + 1 < n ≤ ψ(m), and that µ n−1 (Q ) is defined for all Q ∈ ∂ n−1,1 Q. (C1) If µ n−1 (Q ) = 0 then for all subcubes Q ∈ D n of Q , put µ n (Q ) = 0. (C2) If 2 −(d−1) µ n−1 (Q ) ≥ 2 −nd then for any Q ⊂ Q with Q ∈ ∂ n,1 Q, we set µ n (Q ) = 2 −(d−1) µ n−1 (Q ). For Q ⊂ Q , Q ∈ D n but Q ∈ ∂ n,1 Q, we set µ n (Q ) = 0. (C3) If 2 −(d−1) · µ n−1 (Q ) < 2 −nd , then, as above, we select the cube (4.5) Q ∈ Q , Q ∈ ∂ n−1 Q with maximal s(v min, Q ) among those cubes satisfying (4.5). Next, we set µ n ( Q) = µ n−1 (Q ). For the other cubes Q ⊂ Q , Q ∈ D n , we impose µ n (Q ) = 0. The motivation for the choice of Q is that Q is located (if we look at our two-dimensional Figure 1) in the upper left "Northwest" direction from c Q . We select Q in the "upper" corner of Q on the boundary of Q in order to get as many as possible of the charged cubes into A(x, r x , δ) in Lemma 3.6. Recall that on Figure 1 the central cube Q with smallest vertex c Q is also located "above" (in the direction "Northeast") from this vertex and x is located in Q. We repeat the above steps for n = m + 1, ..., ψ(m) and denote by ∂ n,1,+ Q those cubes in ∂ n,1 Q for which µ n (Q ) > 0. The adjustment at step n = m+1 implies that we distribute a mass of η * · µ m (Q) on the ∂ ψ(m),1,+ Q cubes and in this case (4.2) holds as well. By (3.13) initially verified by Q and the first step of our induction, for n = m and n = m + 1, one has (4.6) η * · 2 −nd < µ n (Q) < c d 2 d · 2 −nd . Suppose that Q ∈ ∂ n,1,+ Q satisfies (4.6). Consider Q ∈ ∂ n+1,1,+ Q such that Q ⊂ Q . If Step (C2) was used to define µ n+1 (Q ), then on the one hand, µ n (Q ) ≥ 2 (d−1) 2 −(n+1)d , and hence (4.7) µ n+1 (Q ) ≥ 2 −(n+1)d . On the other hand, since d ≤ d − 1, one sees that (4.8) µ n+1 (Q ) = 2 −(d−1) µ n (Q ) < c d 2 d · 2 −(d−1) 2 −nd ≤ c d 2 d 2 −(n+1)d . If (C3) was used, then for the only Q ∈ ∂ n+1,1,+ Q satisfying Q ⊂ Q , one has (4.9) µ n+1 ( Q) = µ n (Q ) < 2 (d−1) · 2 −(n+1)d < c d 2 d · 2 −(n+1)d . On the other hand, from (4.6) it also follows that (4.10) η * 2 −(n+1)d < η * 2 −nd < µ n (Q ) = µ n+1 ( Q). Thus, by induction, (4.6) holds true. Hence (3.8) holds for any Q ∈ ∂ n,1,+ Q for any n = m + 1, ..., ψ(m). distribute µ ψ(m) (Q ) onto some subcubes of Q and to define µ n on Q for ψ(m) < n ≤ Ψ(m) = ψ (m). By an immediate application of Lemma 3.2, since µ ψ(m) (Q ) satisfied (3.8), the same inequality (3.8) (with m replaced by n) remains true for all n ∈ (ψ(m), Ψ(m)] for any Q ⊂ Q , Q ∈ D + n . In particular, the following analog of Lemma 3.6 holds. Lemma 4.2. If, for some large integer m, Q ∈ D Ψ(m) is C-central, then for any x ∈ Q , there exists r x such that P µ Ψ(m) (x, r x , δ, η) holds and 2 −m−1 ≤ r x < 2 −m−1 · 1.125. Proof. We discuss only the changes in the argument of the original proof of Lemma 3.6 when ψ(m) < Ψ(m) = ψ (m). In this new situation, A(x, r x , δ) contains more than (1 − 2 −(ψ (m)−m)+1 ) d−1 d Ψ(m),1 (Q) > η * d Ψ(m),1 (Q) many cubes from ∂ Ψ(m),1 Q. Now even if we had to use Step (C3), our choice of the cube with maximum s(v min, Q ) after (4.5) implies that (1 − 2 −(ψ (m)−m)+1 ) d−1 d Ψ(m),1,+ (Q) > η * d Ψ(m),1,+ (Q) holds in this case as well. This finally yields µ(A(x, r x , δ)) > η * · η * µ m (Q) > η · µ m (Q). Proof. The proof is similar to that of Lemma 3.9, up to some minor modifications that are left to the reader. Construction of the measure of Theorem 1.4. The measure µ is built exactly as in Section 3.4. Proposition 3.12 can be proved in this case as well. Proposition 4.4. For µ-almost every x, there exist infinitely many integers n such that Q 2n (x) is a C-central cube at Step m 2n . The conclusions and the arguments are similar to those developed in Section 3.4, we only sketch the proof to get Theorem 1.4 As a consequence of Proposition 4.4, µ-a.e. x belongs to a C-central cube infinitely often, and (3.25) holds infinitely often. For such an x, there exists an increasing sequence of integers (j n (x)) n≥1 such that the following version of (3.35) holds (4.11) c −2 d 2 −j n (x)d ≤ µ j n (x) (Q) < c 2 d 2 −j n (x)d . By Lemma 3.2 and the construction, there exists another increasing sequence of integers (j n (x)) n≥1 satisfying (3.34). Then part (i) of Lemma 4.3 yields the dimension estimate (3.33), which concludes the proof. 5. Lemma 5.1 and the proof of Theorem 1.7 5.1. Intersection of thin Euclidean annuli. The idea of the proof of Theorem 1.7 is based on the observation that if balls (in the 2-dimensional case, disks) of comparable radii ∼ 2 −n are centered not too close (in the next lemma, the distance between their centres is at least 2 −5n ), then the annuli corresponding to these balls are intersecting each other in a set of small diameter, see Figure 2. This follows from the strictly convex shape of the corresponding balls. It is illustrated by Lemma 5.1 below, which prevents that measures with different upper and lower dimensions charge thin annuli. Lemma 5.1. There exists an integer N corr such that if n ≥ N corr then for every 2 −n−1 ≤ r 1 , r 2 ≤ 2 −n and r 2 , 30) consists of at most two connex sets, each of them is of diameter, D * corr,n less than 24 · 2 −13.5n . This type of estimations appear at several places. For example from Lemma 3.1 of Wolff's survey [21] one gets an estimate of D * corr,n of the form C W 2 −12.5n with a constant C W not depending on n. The order of this estimate is slightly smaller than ours. (5.1) 2 −5n ≤ ||z 1 − z 2 || 2 ≤ 2 −n /30, A(z 1 , r 1 , 30) ∩ A(z 2 , We point out that the order 2 −13.5n is optimal. Indeed, taking R = A((0, 2 −n ), 2 −n , 30) ∩ A((0, 2 −n − 2 −5n ), 2 −n − 2 −5n , 30), then one can verify that the diameter of R can be estimated from below by C · 2 −13.5n with a constant not depending on n. For this, considering a triangle with sides a = 2 −n − 2 −5n , b = 2 −5n and c = 2 −n − 2 −30n , it z 1 = (0, y 1 ) is easily seen that half of the diameter of R is larger than the altitude m b of the triangle perpendicular to the side b. Using Heron's formula the area of the triangle is A = s(s − a)(s − b)(s − c) with s = (a + b + c)/2 and m b = 2A/b. Plugging in the above constants, one obtains the announced estimate. z 2 = (0, y 2 ) z 1 = (0, y 1 ) z 2 = (0, y 2 ) A B C D 16 · 2 −13.5n Since the notation of Lemma 3.1 of [21] is different from ours, we detail a bit the way one can obtain an estimate for D * corr,n by using that lemma. Let d W = |z 1 − z 2 | + |r 1 − r 2 | and ∆ W = ||z 1 − z 2 | − |r 1 − r 2 ||. We can suppose that r 1 ≥ r 2 , and the assumptions of Lemma 5.1 imply that r 2 ≥ r 1 /2. The argument of Lemma 3.1 of [21] gives an estimate D * corr,n ≤ C W r −30n 1 (r −30n 1 + ∆ W )(r −30n 1 + d W ) with C W independent of n. The assumptions of Lemma 5.1 imply that 0 ≤ ∆ W and 2 −5n ≤ d W , and these inequalities cannot be improved. Hence the above estimate implies D * corr,n ≤ C * W r −30n 1 r 15n 1 r 2.5n 1 = C W r −12.5n 1 ≤ C W 2 −12.5n . A similar order estimate can be obtained from Marstrand's paper [13]. In [1] a similar type of question is studied but it is less straightforward which order one can obtain for D * corr,n . We now turn to the main result and prove Theorem 1.7. Finally, Lemma 5.1 is proved in Section 5.3. 5.2. Proof of Theorem 1.7. Without limiting generality we can suppose that the Borel probability measure µ is supported on [0, 1] 2 . Proceeding towards a contradiction, suppose that µ(E µ (30, η)) > 0. For ease of notation, let E = E µ (30, η). Since E is fixed in the rest of the proof, µ(E) will be regarded as a positive constant. Since dim µ = d ∈ [0.89, 2] and dim µ = d ∈ [0.89, d], for µ-a.e. x ∈ E there exist ρ x > 0 such that (5.2) for any 0 < r ≤ ρ x , r d+0.01 ≤ µ(B(x, r)) ≤ r d−0.01 ≤ r 0.88 . For those xs for which ρ x , as defined above, does not exist, we set ρ x = 0. We can also suppose that for each x we select ρ x as half of the supremum of those ρs for which (5.2) holds with ρ instead of ρ x . This way it is not too difficult to see that the mapping ρ : x ∈ [0, 1] 2 → ρ x ∈ R + is Borel µ-measurable. The first step consists of finding a ball containing points of E with a very precisely controlled behavior, see Lemma 5.3 below. To prove it, let us start with a simple technical lemma. Lemma 5.2. Suppose E ⊂ R 2 is a µ-measurable set, and let ρ : R 2 → R, x → ρ x be a µ-measurable function such that µ({x ∈ E : ρ x > 0}) = µ( E). Then for any 0 < γ < 1, for µ-a.e. x ∈ E, there exists R x > 0 such that for any 0 < r < R x , , r)). (5.3) µ {x ∈ B(x, r) ∩ E : ρ x ≥ R x } > γ · µ(B(x Proof. Consider E n = {x ∈ E : ρ x > 1 n }, for n ≥ 1. Then µ( E\∪ n∈N E n ) = 0 by assumption. Fix now n ∈ N. By Corollary 2.1 µ( E n ∩ B(x, r)) µ(B(x, r)) → 1 for µ-a.e. x ∈ E n . For those x ∈ E n for which the above limit holds true, it is thus enough to choose 0 < R x < 1 n such that µ( E n ∩ B(x, r)) µ(B(x, r)) > γ holds for any 0 < r < R x . Using Lemma 5.2 with γ = 1 − η 2 applied to E, for µ-a.e. x ∈ E there exists an R x > 0 such that for any 0 < r < R x , (5.4) µ({x ∈ B(x, r) ∩ E : ρ x ≥ R x }) > 1 − η 2 µ(B(x, r)). Consider a large natural number N 0 > 10, whose precise value will be chosen later. By using the definition of E = E µ (30, η), for µ-a.e. x ∈ E one can select 0 < r x ≤ min{2 −N 0 , R x /10, ρ x /10} such that (5.5) µ(A(x, r x , 30)) ≥ η · µ(B(x, r x )). Recalling that all r x s are less than 2 −N 0 , and that µ-a.e. r x is strictly positive, there exists at least one integer n 0 ≥ N 0 such that (5.6) µ({x ∈ E : 2 −n 0 −1 ≤ r x < 2 −n 0 }) ≥ 1 10n 2 0 µ(E). Consider now the covering of {x ∈ E : 2 −n 0 −1 ≤ r x < 2 −n 0 } by the balls {B(x, r x ) : x ∈ E, 2 −n 0 −1 ≤ r x < 2 −n 0 }. By the measure theoretical version of Besicovitch's covering theorem (see [3], Theorem 20.1 for instance), there exists a constant C 2 > 0, depending only on the dimension 2, such that one can extract a finite family of disjoint balls of radius 4 · 2 −n 0 , denoted by B = {B i : i = 1, .., M }, such that, calling E = E ∩ ∪ B i ∈B B i , one has µ(B i ) > 0 for every i, and (5.7) µ( E n 0 ) ≥ C 2 n 2 0 µ(E), where E n 0 = {x ∈ E : 2 −n 0 −1 ≤ r x < 2 −n 0 }. Lemma 5.3. There exists a constant C > 0 depending only on the dimension and a ball B * = B(x * , 2 −n 0 /100) such that (5.8) µ B x * , 2 −n 0 /100 ∩ E n 0 ≥ C n 2 0 2 −2n 0 µ(E). Proof. Since E ⊂ [0, 1] 2 , the balls of B are disjoint and are of radius less than 2 −10 , their cardinality M is less than 2 2n 0 , and there exists one ball B i such that (5.9) µ(B i ∩ E n 0 ) ≥ 1 2 2n 0 +2 µ( E n 0 ) ≥ C 2 2 2n 0 n 2 0 µ(E). Write B i = B(x i , r x i ). Since r x i /100 < 2 −n 0 /100 < 2 −n 0 −1 ≤ r x i , there exists a constant C 3 > 0 which depends only on the dimension such that for some x * ∈ B i ∩ E, the ball with center x * and radius 2 −n 0 /100 supports a proportion C 3 > 0 of the µ-mass of the initial ball, i.e. (5.10) µ(B(x * , 2 −n 0 /100)) ≥ µ(B(x * , 2 −n 0 /100) ∩ E n 0 ) ≥ C 3 µ(B i ∩ E n 0 ). The last statement simply follows from the fact that B i can be covered by finitely many balls of radius 2 −n 0 /100, this finite number of balls being bounded above independently of x i and r x i . This and (5.9) imply the result. Further, as a second step, we seek for a lower estimate of the number M * n 0 of disjoint balls B(y j , 2 −5n 0 ), j = 1, ..., M * n 0 such that y j ∈ B(x * , 2 −n 0 /100)∩ E n 0 . The rest of the proof consists of showing that annuli centered at y j , j = 1, ..., M * n 0 , will charge B(x * , 4 · 2 −n 0 ) with too much measure, yielding a contradiction. Lemma 5.1 will play a key role here. Lemma 5.4. When N 0 is sufficiently large, for every n 0 ≥ N 0 , call M * n 0 the maximal number of disjoint balls B(y j , 2 −5n 0 ), j = 1, ..., M * n 0 such that y j ∈ B(x * , 2 −n 0 /100) ∩ E n 0 . Then M * n 0 ≥ 2 2.1n 0 . Proof. First, observe that, when y j ∈ B(x * , 2 −n 0 /100) ∩ E n 0 , by (5.2) (5.11) µ(B(y j , 2 −5n 0 )) ≤ 2 −0.88·5n 0 = 2 −4.4n 0 . Next, there exist two positive constants C 3 and C 4 depending only on the dimension such that B(x * , 2 −n 0 /100) ∩ E n 0 is covered by C 3 families F 1 , ..., F C 3 containing pairwise disjoint balls of the form B(y j , 2 −5n 0 ). At least one of these families, say F 1 , satisfies that B(y j ,2 −5n 0 )∈F 1 µ(B(y j , 2 −5n 0 )) ≥ 1 C 3 µ(B(x * , 2 −n 0 /100) ∩ E n 0 ). By (5.8), B(y j ,2 −5n 0 )∈F 1 µ(B(y j , 2 −5n 0 )) ≥ C C 3 n 2 0 2 −2n 0 µ(E) . Then, from (5.11) one deduces that (5.12) M * n 0 ≥ C C 3 n 2 0 2 −2n 0 µ(E) 1 2 −4.4n 0 ≥ 2 2.1n 0 , when N 0 is chosen sufficiently large. Next, as a third step, we study the annuli A(y i , r y i , 30) associated with the points y i , i = 1, ..., M * n 0 . Observe that these points y i satisfy the assumptions of Lemma 5.1 and in particular equation (5.1) with n = n 0 , as soon as N 0 ≥ N corr . Lemma 5.5. For every x ∈ B(x * , 2 −n 0 /100) ∩ E n 0 , set (5.13) A(x, r x , 30) = x ∈ A(x, r x , 30) ∩ E : ρ x > 10r x . Then, for some constant C > 0 that depends only on the dimension one has (5.14) µ A(x, r x , 30) ≥ η 2 − η · C n 2 0 µ(E) · 2 −2n 0 . Proof. Since r x ≤ R x /10, from (5.4) and (5.5) we infer that for µ-a.e. x ∈ E, one has µ A(x, r x , 30) = µ({x ∈ A(x, r x , 30) ∩ E : ρ x > 10r x }) ≥ η 2 µ(B(x, r x )). The last inequality holds for every r x such that (5.4) holds true. Recalling that for x ∈ E n 0 , 2 −n 0 −1 ≤ r x < 2 −n 0 , one deduces that for any x ∈ B(x * , 2 −n 0 /100) ∩ E n 0 , B(x * , 2 −n 0 /100) ⊂ B(x, r x /4). Hence, by (5.8), (5.15) µ(B(x, r x /4)) ≥ µ(B(x * , 2 −n 0 /100) ∩ E n 0 ) ≥ C n 2 0 µ(E) · 2 −2n 0 . It is also clear that B(x, r x /4) ∩ A(x, r x , 30) = ∅ for such xs. So, the fact that µ( A(x, r x , 30)) ≥ η 2 µ(B(x, r x )) ≥ η 2 µ( A(x, r x , 30)) + µ(B(x, r x /4)) implies by using (5.15) that µ( A(x, r x , 30)) ≥ η 2 − η µ(B(x, r x /4)) ≥ η 2 − η · C n 2 0 µ(E) · 2 −2n 0 , and the result follows. We are now ready to combine the previous arguments to prove Theorem 1.7. As in Lemma 5.4, select points (y j ), j = 1, ..., M * n 0 , such that the balls B(y j , 2 −5n 0 ) ⊂ B(x * , 4·2 −n 0 ) are pairwise disjoint and y j ∈ B(x * , 2 −n 0 /100)∩ E n 0 . By construction, the annuli A(y i , r y i , 30) and the sets A(y i , r y i , 30) satisfy the assumptions of Lemmas 5.1 and 5.5. Also, for any x ∈ A(y i , r y i , 30) ∩ A(y j , r y j , 30), one has ρ x > 10 · 2 −n 0 −1 . Hence, for any r < 10 · 2 −n 0 −1 , by (5.2) one necessarily also has µ(B(x , r)) ≤ r 0.88 . Then, as stated by Lemma 5.1, the diameter of each of the (at most) two connex parts of A(y i , r y i , 30) ∩ A(y j , r y j , 30) is smaller than 2 −13n 0 < 24 · 2 −13.5n 0 . These connex parts are included in an annulus A(y i , r y i , 30), so it is a very thin region (the width of the annulus is less than 2 −30n 0 ). Hence, the intersection of E with the union of the two connex parts can be covered by at most M * * balls of the form B(z , 2 −30n 0 ), where z ∈ A(y i , r y i , 30) ∩ A(y j , r y j , 30), (5.16) (5.17) and C * * is a positive constant depending only on the dimension. Also, by (5.13) and (5.16), one sees that z > 2 −n 0 −1 · 10 > 2 −30n 0 for all z s. Hence, (5.2) yields (5.18) µ(B(z , 2 −30n 0 )) ≤ 2 −30n 0 ·0.88 = 2 −26.4n 0 . M * * ≤ C * * 2 −13n 0 2 −30n 0 = C * * 2 17n 0 , This together with (5.17) imply that µ A(y i , r y i , 30) ∩ A(y j , r y j , 30) ≤ M * * · 2 −26.4n 0 < C * * 2 17n 0 · 2 −26.4n 0 = C * * 2 −9.4n 0 . (5.19) In addition, (5.14) gives (5.20) µ A(y i , r y i , 30) · µ A(y j , r y j , 30) ≥ η 2 − η 2 · C 2 n 4 0 µ 2 (E) · 2 −4n 0 . Hence by (5.19) for large n 0 , (5.21) µ( A(y i , r y i , 30) ∩ A(y j , r y j , 30)) < 2 −5.3n 0 µ( A(y i , r y i , 30)) · µ( A(y j , r y j , 30)). Finally, all sets A(y i , r y i , 30) are included in B(x * , 4 · 2 −n 0 ), and their cardinality by Lemma 5.4 is greater than 2 2.1n 0 . So, one has µ(B(x * , 4 · 2 −n 0 )) ≥ 2 2.1n 0 i=1 µ( A(y i , r y i , 30)) − 2 2.1n 0 i,j=1: i =j µ( A(y i , r y i , 30) ∩ A(y i , r y i , 30)) ≥ 2 2.1n 0 i=1 µ( A(y i , r y i , 30)) 1 − 2 2.1n 0 j=1 2 −5.3n 0 µ( A(y j , r y j , 30)) ≥ 2 2.1n 0 i=1 µ( A(y i , r y i , 30))(1 − 2 2.1n 0 · 2 −5.3n 0 ), where at the last step we simply used that µ( A(x j , r x j , 30)) ≤ 1. Then, (5.14) yields that when n 0 ≥ N 0 is sufficiently large, µ(B(x * , 4 · 2 −n 0 )) ≥ 2 2.1n 0 i=1 µ( A(y i , r y i , 30))(1 − 2 2.1n 0 · 2 −5.3n 0 ) ≥ 1 2 2 2.1n 0 i=1 µ( A(y i , r y i , 30)) ≥ 2 2.1n 0 η 2 − η · C n 2 0 µ(E)2 −2n 0 , which is greater than 1 as soon as N 0 (hence n 0 ) is chosen sufficiently large. This contradicts the fact that µ(R 2 ) = 1, and completes the proof. Remark 5.6. It would be natural to check if a version of Theorem 1.7 holds in which the constant 0.89 can be pushed down to a value closer to zero, maybe at the price that δ = 30 is replaced by a larger number. We point out that the estimates (5.17) and (5.19) show that in our arguments the order of estimate of D * corr,n in Lemma 5.1 is crucial. In (5.19) in the end of the inequality, we need a (sufficiently large) negative power of 2. If 0.89 is replaced by 0.56 = 17/30, using (5.17) and (5.18) one can see that in the crucial estimate (5.19), the power of 2 will become non-negative. Since the order of the estimate in Lemma 5.1 is best possible, then one cannot improve significantly (5.17) and hence the other estimates depending on it (by tighter estimates the exponent 30 − 13 = 17 in (5.17) can be replaced by 30 − 13.5 + ε = 16.5 + ε). 5.3. Proof of Lemma 5.1. Without limiting generality we can suppose r 1 ≥ r 2 and can choose a coordinate system in which z 1 = (0, y 1 ), z 2 = (0, y 2 ) and y 1 = r 1 . See Figure 2 for an illustration (the figures are of course distorted, since 2 −30n is much smaller than 2 −n , so on a correct figure one of them cannot be shown, due to pixel size limitations). In the proof, when constants are said to depend on the dimension 2 only, they do not depend on other parameters -similar constants exist in higher dimensions as well. With this notation, (5.1) means that (5.22) 2 −5n ≤ |y 1 − y 2 | ≤ 2 −n /30. Of course the last inequality holds for sufficiently large ns. (5.23) If A(z 1 , r 1 , 30) ∩ A(z 2 , r 2 , 30) is included in the strip [−8 · 2 −13.5n , 8 · 2 −13.5n ] × R, then its diameter is less than 24 · 2 −13.5n . Assume now that A(z 1 , r 1 , 30) ∩ A(z 2 , r 2 , 30) is not included in the strip [−8 · 2 −13.5n , 8 · 2 −13.5n ] × R. We consider one of the two connex parts of A(z 1 , r 1 , 30) ∩ A(z 2 , r 2 , 30), the one located in the right half-plane. We denote its closure by C R . See Figures 2 and 3. The other part is symmetric and a similar estimate is valid for it. Assume that r 1 is fixed and r ∈ [r 2 − r 30 2 , r 2 ]. Denote by M 1 (r) = (x 1 (r), y 1 (r)) the intersection of the circles of radii r 1 − r 30 1 and r, centered respectively at z 1 and z 2 . We put A = M 1 (r 2 ) = (x A , y A ), and B = M 1 (r 2 − r 30 2 ) = (x B , y B ). Observe that the points M 1 (r), r ∈ [r 2 − r 30 2 , r 2 ] are located on the arc with endpoints A and B. On Figure 2 the point A is the point of the closed region C R with the largest abscissa. Then x A = x 1 (r 2 ) > 8 · 2 −13.5n . However, as the left half of Figure 3 illustrates, for r 1 ≈ r 2 it may happen that not A = M 1 (r 2 ) = (x A , y A ) is the point of C R with the largest abscissa. On the left half of Figure 3 this point is C. In the sequel we suppose that x A = x 1 (r 2 ) > 8 · 2 −13.5n . The other cases can be treated analogously, the main point is that, on the boundary of C R , there is at least one point with abscissa larger than 8 · 2 −13.5n . The abscissa x 1 (r) satisfies the implicit equation F 1 (r, x 1 (r)) := y 2 − y 1 + r 2 − (x 1 (r)) 2 − (r 1 − r 30 1 ) 2 − (x 1 (r)) 2 = 0. Observe that by our assumption, the intersection point lies in the first quadrant x 1 (r) > 0. By implicit differentiation and after simplification, (x 1 ) (r) = − ∂ 1 F 1 (r, x 1 (r)) ∂ 2 F 1 (r, x 1 (r)) = r (r 1 − r 30 1 ) 2 − (x 1 (r)) 2 x 1 (r)(y 1 − y 2 ) . From the last equation, one deduces by (5.22), r ∈ [r 2 − r 30 2 , r 2 ], 2 −n−1 ≤ r 1 , r 2 ≤ 2 −n that By integration, |(x 1 (r)) 2 −(x 1 (r 2 )) 2 | ≤ 2·2 3n |r−r 2 | ≤ 2 3n+1 2 −30n ≤ 2 −27n+1 , and hence |(x 1 ) (r)x 1 (r)| ≤ r (r 1 − r 30 1 ) 2 − (x 1 (r)) 2 (y 1 − y 2 ) ≤ 2 −2n 2 −5n ≤ 2 3n .|x 1 (r) − x 1 (r 2 )| ≤ 2 −27n+1 |x 1 (r) + x 1 (r 2 )| . Finally using that x 1 (r 2 ) = x A ≥ 8 · 2 −13.5n the previous equation gives that for any r ∈ [r 2 − r 30 2 , r 2 ] (5.24) |x 1 (r) − x 1 (r 2 )| ≤ 2 −27n+1 2 13.5n−3 = 2 −13.5n−2 , so |x A − x B | < 2 −13.5n−2 and x B > 8 · 2 −13.5n − 2 −13.5n−2 = 7.75 · 2 −13.5n−1 . Assume now that r 2 is fixed and r ∈ [r 1 − r 30 1 , r 1 ]. Denote by M 2 (r) = (x 2 (r), y 2 (r)) the intersection of the circles of radii r and r 2 − r 30 2 , centered respectively at z 1 and z 2 . We put D = M 2 (r 1 ) = (x D , y D ). Observe that the points M 2 (r), r ∈ [r 1 − r 30 1 , r 1 ] are located on the arc with endpoints B and D. Using the implicit equation F 2 (r, x 2 (r)) := y 2 − y 1 + (r 2 − r 30 2 ) 2 − (x 2 (r)) 2 − r 2 − (x 2 (r)) 2 = 0, the same considerations show that for any r ∈ [r 1 − r 30 1 , r 1 ] when n ≥ N corr is sufficiently large (5.25) |x 2 (r) − x 2 (r 2 )| ≤ 2 −27n+1 2 13.5n /7.75 = 2 −13.5n+1 /7.75, so |x B − x D | ≤ 2 −13.5n+1 /7.75, and x D > 7.75 · 2 −13.5n − 2 −13.5n+1 /7.75 > 7 · 2 −13.5n . One can also consider the curve M 3 (r) = (x 3 (r), y 3 (r)) connecting the points D and C on the boundary of C R and the curve M 4 (r) = (x 4 (r), y 4 (r)) connecting the points C and A on the boundary of C R . Estimates analogous to equations (5.24) and (5.25) are valid for these curves as well with constants 7.75 and 7.5 decreased to 7 and 6. Since C R is compact, we can choose points P = (x P , y P ) and Q = (x Q , y Q ) on the boundary of C R such that the distance between P and Q equals the diameter of C R . Since these points can be connected by no more than three of the above mentioned arc segments we deduce that (5.26) |x P − x Q | ≤ 2 −13.5n . The analogous estimate (5.27) |y P − y Q | ≤ 2 −13.5n is also true. In fact in this case the calculations are even simpler, and most of the details are left to the reader. We mention here only that, for example, for the function y 1 (r) one has a much simpler implicit equation (5.28) F 3 (r, y 1 (r)) := r 2 1 − r 2 − (y 1 (r) − y 1 ) 2 + (y 1 (r) − y 2 ) 2 = 0 and by implicit differentiation (5.29) (y 1 ) (r) = r y 1 − y 2 . From this and (5.23) one concludes that the diameter of C R is less than 24 · 2 −13.5n . This concludes the proof, since the symmetric part (i.e. when x A < 0) is treated similarly. Methods related to Falconer's distance set problem The study of thin annuli and spherical averages is an important issue in many dimension-related problems, including Kakeya-type problems and Falconer's distance set conjecture. Recall that the distance set D(E) of the set E ⊂ R d is defined by D(E) := {||x − y|| 2 : x, y ∈ E}. Falconer's distance set problem is about finding bounds of Hausdorff measure and dimension of D(E) in terms of those of E. Examples of Falconer show that if s ≤ d/2 then there are sets E ⊂ R d such that dim H E = s and D(E) is of zero (one-dimensional) Lebesgue measure. It is conjectured that E has positive Lebesgue measure as soon as dim H E > d/2. In one of the most recent results in the plane (d = 2) [7], it is proved that if E is compact and dim H E > 5/4 then D(E) has positive Lebesgue measure. For further details about Falconer's distance set problem we also refer to [4] and Chapters 4, 15 and 16 of [15]. Using standard arguments from [4,15], which is a different approach from the one developed earlier in this paper, we can prove Proposition 1.11. Proof. Let t > 1/2, 0 < η < 1 and let µ be a finite t-regular measure on R 2 with compact support satisfying (1.6). Since we work in R 2 the local dimension of µ cannot exceed 2, so 2 ≥ t > 1/2. Since µ is t-regular, for every s < t, the s-energy of µ defined by I s (µ) = (R 2 ) 2 ||x − y|| −s 2 dµ(x)dµ(y) is finite. Recall also that Following Falconer's argument (Theorem 2.2 of [4]) (see also [15,Lemma 12.13]), one gets by (6.1), (6.2) and (6.4) that, keeping in mind that 1/2 < s ≤ 5/4 < 3/2 (R 2 ) 2 µ(A(x, r, δ))dµ(x) = for some constant C > 0 that depends on µ and might change from line to line. Consequently, by Chebyshev's inequality, and the lower bound in (1.6), we have µ({x : µ(A(x, r, δ)) ≥ η · µ(B(x, r))}) ≤ µ({x : µ(A(x, r, δ) ≥ η · cr t }) (6.5) ≤ C µ r 1/2+δ(s−1/2) I s (µ)/(ηr t ) = C µ η −1 I s (µ)r t−1/2 , (6.6) where at the last equality we used (6.3) and C µ is a suitable constant not depending on r. Hence the right-hand side tends to zero as r 0. This shows (1.7) with an additional decay rate faster than r t−1/2 , and thus completes the proof. However the above convergence in measure of Proposition 1.11 is not fast enough to hope to recover Theorems 1.3 to 1.7, at least for the moment. Let us justify this claim. Consider a measure µ supported on [0, 1] 2 (to ease the argument) such that the assumptions of Proposition 1.11 hold. We would like to apply (1.7) to deduce some estimate for the measure of E µ (δ, η). For this, consider equi-distributed points (r k,m ) m=0,...,2 3k in the interval [r k,0 = 2 −k−1 , r k,2 3k = 2 −k ]. The distance between two consecutive r k,m and r k,m+1 is 2 −4k−1 . If x is such that P µ (x, r, 4, η) holds true for r ∈ [2 −k−1 , 2 −k ], then P µ (x, r k,m , 4, η/2) holds for some m. So, the set of points {x : P µ (x, r, 4, η) holds true for some r ∈ [2 −k−1 , 2 −k ]} has µ-measure less than 2 3k k=0 µ({x : P µ (x, r k,m , 4, η/2) holds true} ≤ 2 3k k=0 Cη −1 2 −k(t−1/2) ≤ C2 k(7/2−t) by (6.6). Unfortunately, keeping in mind that t ≤ 2 we have k 2 k(7/2−t) = +∞ and the Borel-Cantelli lemma cannot be applied (by far!) to prove Theorem 1.3 or Theorem 1.7. Trying to optimize the choice of s or δ (instead of 4) does not help either, using similar arguments. Definition 1 . 2 . 12Let µ be a Radon probability measure on R d . The lower and upper dimensions of µ are defined as dim(µ) = sup{α ≥ 0 : for µ-a.e x, QFigure 1 . 1Subdivision scheme B for a cube Q ∈ D m Using (3.1), (3.7), (3.13), (3.14) and (3.18), one gets Lemma 3. 9 . 9Assume that µ m satisfies (3.13) for some Q ∈ D m , and apply the subdivision scheme B to define µ m+1 , ..., µ ψ(m) on the subcubes of Q of generation m + 1, ..., ψ(m).Then: (i) for every n ∈ {m, ..., ψ(m)}, for every Q ∈ D n such that Q ⊂ Q and µ n (Q ) = 0, (3.8) holds for Q with the measure µ n . (ii) for every cube Q ∈ D ψ(m) such that Q ⊂ Q and µ ψ(m) (Q ) = 0, there exists n ∈ {m, ..., ψ(m)} and a (unique) cube Q n ∈ D n such that Q ⊂ Q n ⊂ Q and (3.20) holds for µ n and Q n .Proof. (i) The cases where n = m and n = ψ(m) follow from (3.13),(3.17),(3.19), and (3.26). Suppose m < n < ψ(m). Two cases are separated depending on whether we deal with the border or the B-central cubes. For the estimate from below (3.6), (3.13), (3.15), (3.16) and (3.30) yield Proposition 3 . 10 . 310The sequence (µ n ) n≥1 converges to a measure µ which is supported by a Cantor-like set C defined by C = n≥1 Q∈Dn:µn(Q) =0 that the upper and lower local dimensions of the measure µ are defined by d µ (x) = lim sup r 0 log(µ(B(x, r))) log r and d µ (x) = lim inf r 0 log(µ( First, the way the mass is distributed on the border of Q (Subsection 3.3.2) is modified as follows. 4.1.1. Distributing (part of ) the mass on the smallest face: Two cases are separated. • Case d < d − 1 ≤ d: We set d ψ(m),1,+ (Q) = d ψ(m),1 (Q) = 2 (d−1)(ψ(m)−m)and for any Q ∈ ∂ ψ(m),1,+ Q := ∂ ψ(m),1 Q, put consider the only cube Q with "maximal" smallest vertex v min, Q among those cubes satisfying Q ∈ Q, Q ∈ ∂ m+1,1 Q. "Maximal" means with largest possible coordinates -this makes sense since the borders of the cube are parallel to the axes. If it is easier to understand this way, for such a vertex, the sum of its coordinates s(v min, Q ) (defined in (3.3)) is maximal. Then put µ m+1 ( Q) = η * µ m (Q). 4. 1 . 3 . 13Giving a zero-mass to the other cubes, and defining the measure µ Ψ(m) on Q. The measure µ Ψ(m) is extended inside Q as µ Ψ(m) in subsection 3.3.3.4.1.4.Defining inside Q the measure µ n for m < n < Ψ(m).The measures µ n are also defined as in Subsection 3.3.4. Lemma 4. 3 . 3Assume that µ m satisfies (3.13) for some Q ∈ D m , and apply the subdivision scheme C to define µ m+1 , ..., µ Ψ(m) on the subcubes of Q of generation m + 1, ..., Ψ(m).Then:(i) for every n ∈ {m, ..., Ψ(m)}, for every Q ∈ D n such that Q ⊂ Q and µ n (Q ) = 0, (3.8) holds for Q with the measure µ n . (ii) for every cube Q ∈ D Ψ(m) such that Q ⊂ Q and µ Ψ(m) (Q ) = 0, there exists n ∈ {m, ..., Ψ(m)} and a (unique) cube Q n ∈ D n such that Q ⊂ Q n ⊂ Q and (3.20) holds for µ n and Q n . Figure 2 . 2Positions of intersecting annuli. Figure 3 . 3Special position of annuli and calculating the diameter of their intersection. on t imply 1/2 < s ≤ 5/4. Set χ(x) = 1 1 [r−r δ ,r] (|x|). By Lemma 2.1 of [4],(6.4) |χ(ξ)| ≤ Cr 1/2 |ξ| −1/2 min(r δ , |ξ| −1 ). ≤ Cr 1/2+δ(s−1/2) I s (µ), Theorem 1.7. Suppose that d = 2 and that the Euclidean metric ||.|| 2 is used.Let d ∈ [0.89, 2] and d ∈ [0.89, d]. d 2 −(m+1)d ≤ µ m (Q) ≤ c 2 d 2 −md . AcknowledgmentsThe authors thank Benoît Saussol for asking the question treated in this paper, Jean-René Chazottes for interesting discussions and relevant references, Marius Urbański for informing us about[17]and an anonymous reader for turning our attention to the results and methods developed for Falconer's distance problem. Indeed, recall that intuitively, ψ (m) ∼ md/d and ψ(m) ∼ mδ, but now δ > 1 can be such that 1 < δ < d/d. Hence let us introduce Ψ(m) = max{ψ (m), ψ(m)}. As before. call Q the cube of D ψ (m) containing the center c Q of Qm). Indeed, recall that intuitively, ψ (m) ∼ md/d and ψ(m) ∼ mδ, but now δ > 1 can be such that 1 < δ < d/d. Hence let us introduce Ψ(m) = max{ψ (m), ψ(m)}. As before, call Q the cube of D ψ (m) containing the center c Q of Q. (m) is such that Q ⊂ Q ⊂ Q (where Q is the cube of D ψ (m) containing c Q ), Q is called a C-central cube at scale Ψ(m) associated to Q ∈ D m. Definition 4.1. If Q ∈ D ΨDefinition 4.1. If Q ∈ D Ψ(m) is such that Q ⊂ Q ⊂ Q (where Q is the cube of D ψ (m) containing c Q ), Q is called a C-central cube at scale Ψ(m) associated to Q ∈ D m . then we proceed analogously to Section 3.3.2, i.e. we put as in (3.23) µ ψ (m) ( Q) = (1 − η * )µ m (Q), and apply subdivision scheme A to Q and its subcubes until generation ψ(m). If Ψ(m) = ψ(m) ≥ ψ (m). If Ψ(m) = ψ(m) ≥ ψ (m), then we proceed analogously to Section 3.3.2, i.e. we put as in (3.23) µ ψ (m) ( Q) = (1 − η * )µ m (Q), and apply subdivision scheme A to Q and its subcubes until generation ψ(m). If Ψ(m) = ψ (m) and ψ (m) > ψ(m) then we also set as in (3.23) ≤ ψ(m). If Ψ(m) = ψ (m) and ψ (m) > ψ(m) then we also set as in (3.23) ≤ ψ(m). Hence, we apply Subdivision scheme A to the cubes Q ∈ ∂ ψ(m),1,+ Q to References. Hence, we apply Subdivision scheme A to the cubes Q ∈ ∂ ψ(m),1,+ Q to References Averages in the plane over convex curves and maximal operators. J Bourgain, J. Analyse Math. 47J. Bourgain. Averages in the plane over convex curves and maximal operators. J. Analyse Math., 47:69-85, 1986. Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems. J R Chazottes, P Collet, Erg. Th. Dyn. Syst. 331J. R. Chazottes and P. Collet. Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems. Erg. Th. Dyn. Syst., 33(1):49-80, Feb. 2013. Real Analysis. E Di Benedetto, BirkhauserE. di Benedetto. Real Analysis. Birkhauser, 2016. On the Hausdorff dimensions of distance sets. K J Falconer, Mathematika. 322K. J. Falconer. On the Hausdorff dimensions of distance sets. Mathematika, 32(2):206- 212 (1986), 1985. Dimensions of intersections and distance sets for polyhedral norms. K J Falconer, Real Anal. Exchange. 3025K. J. Falconer. Dimensions of intersections and distance sets for polyhedral norms. Real Anal. Exchange, 30(2):719-726, 2004/05. Probability and random processes. G R Grimmett, D R Stirzaker, Oxford University PressNew Yorkthird editionG. R. Grimmett and D. R. Stirzaker. Probability and random processes. Oxford Uni- versity Press, New York, third edition, 2001. On Falconer's distance set problem in the plane. L Guth, A Iosevich, Y Ou, H Wang, Invent. Math. 2193L. Guth, A. Iosevich, Y. Ou, and H. Wang. On Falconer's distance set problem in the plane. Invent. Math., 219(3):779-830, 2020. Limiting distribution and error terms for the number of visits to balls in nonuniformly hyperbolic dynamical systems. N Haydn, K Wasilewska, Discrete Contin. Dyn. Syst. 365N. Haydn and K. Wasilewska. Limiting distribution and error terms for the number of visits to balls in nonuniformly hyperbolic dynamical systems. Discrete Contin. Dyn. Syst., 36(5):2585-2611, 2010. . A Iosevich, I , A. Iosevich and I.. Laba. k-distance sets, Falconer conjecture, and discrete analogs. Integers. 52Laba. k-distance sets, Falconer conjecture, and discrete analogs. Integers, 5(2), 2005. Small union with large set of centers. T Keleti, arXiv:1701.02762ArXiv PreprintT. Keleti. Small union with large set of centers. ArXiv Preprint arXiv:1701.02762. Squares and their centers. T Keleti, D Nagy, P Shmerkin, J. Anal. Math. 1432T. Keleti, D. Nagy, and P. Shmerkin. Squares and their centers. J. Anal. Math, 143(2):643-699, 2018. A survey of the mathematical theory. J Lamperti, Applied Mathematical Sciences. 23Springer-VerlagStochastic processesJ. Lamperti. Stochastic processes. Springer-Verlag, New York-Heidelberg, 1977. A survey of the mathematical theory, Applied Mathematical Sciences, Vol. 23. Packing circles in the plane. J M Marstrand, Proc. London Math. Soc. 553J. M. Marstrand. Packing circles in the plane. Proc. London Math. Soc. (3), 55(1):37- 58, 1987. Geometry of Sets and Measures in Euclidean Spaces. P Mattila, Cambridge University PressP. Mattila. Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, 1995. Fourier analysis and Hausdorff dimension. P Mattila, of Cambridge Studies in Advanced Mathematics. CambridgeCambridge University Press150P. Mattila. Fourier analysis and Hausdorff dimension, volume 150 of Cambridge Stud- ies in Advanced Mathematics. Cambridge University Press, Cambridge, 2015. Stochastic differential equations. B , Springer-VerlagBerlinUniversitextsixth edition, 2003. An introduction with applicationsB. Øksendal. Stochastic differential equations. Universitext. Springer-Verlag, Berlin, sixth edition, 2003. An introduction with applications. Thin annuli property and exponential distribution of return times for weakly markov systems. L Pawelec, M Urbański, A Zdunik, Fund. Math. 2513L. Pawelec, M. Urbański, and A. Zdunik. Thin annuli property and exponential dis- tribution of return times for weakly markov systems. Fund. Math., 251(3):269-316, 2020. On quasi-invariant transverse measures for the horospherical foliation of a negatively curved manifold. B Schapira, Erg. Th. Dyn. Syst. 241B. Schapira. On quasi-invariant transverse measures for the horospherical foliation of a negatively curved manifold. Erg. Th. Dyn. Syst., 24(1):227 -255, 2004. Distribution of orbits in R 2 of a finitely generated group of SL2(R). B Schapira, B. Schapira. Distribution of orbits in R 2 of a finitely generated group of SL2(R). . Amer. J. Math. 1366Amer. J. Math, 136(6):1497-1542, 2014. An introduction to measure theory. Terence Tao, Graduate Studies in Mathematics. 126American Mathematical SocietyTerence Tao. An introduction to measure theory, volume 126 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011. Recent work connected with the Kakeya problem. T Wolff, Id: 0000-0001-5481-8797address: [email protected] URL. Princeton, NJ; Providence, RI; Budapest, Hungary EmailDepartment of Analysis, ELTE Eötvös Loránd, UniversityProspects in mathematicsT. Wolff. Recent work connected with the Kakeya problem. In Prospects in math- ematics (Princeton, NJ, 1996), pages 129-162. Amer. Math. Soc., Providence, RI, 1999. Department of Analysis, ELTE Eötvös Loránd, University, Pázmány Péter Sétány 1/c, 1117 Budapest, Hungary Email address: [email protected] URL: http://buczo.web.elte.hu, ORCID Id: 0000-0001-5481-8797 . Stéphane Seuret, address: [email protected] (UMR. 8040Université Paris-EstStéphane Seuret, Université Paris-Est, LAMA (UMR 8040), UPEMLV, UPEC, CNRS, F-94010, Créteil, France Email address: [email protected]
[]
[ "Computing the strong metric dimension for co-maximal ideal graphs of commutative rings *", "Computing the strong metric dimension for co-maximal ideal graphs of commutative rings *" ]
[ "R Shariyari \nDepartment of Mathematics, Science and Research Branch\nIslamic Azad University\nTehranIran\n", "R Nikandish \nDepartment of Mathematics\nJundi-Shapur University of Technology\nDezfulIran\n\nJundi-Shapur Research Institute\nJundi-Shapur University of Technology\nDezfulIran\n", "A Tehranian \nDepartment of Mathematics, Science and Research Branch\nIslamic Azad University\nTehranIran\n", "H Rasouli [email protected] \nDepartment of Mathematics, Science and Research Branch\nIslamic Azad University\nTehranIran\n" ]
[ "Department of Mathematics, Science and Research Branch\nIslamic Azad University\nTehranIran", "Department of Mathematics\nJundi-Shapur University of Technology\nDezfulIran", "Jundi-Shapur Research Institute\nJundi-Shapur University of Technology\nDezfulIran", "Department of Mathematics, Science and Research Branch\nIslamic Azad University\nTehranIran", "Department of Mathematics, Science and Research Branch\nIslamic Azad University\nTehranIran" ]
[]
Let R be a commutative ring with identity. The co-maximal ideal graph of R, denoted by Γ(R), is a simple graph whose vertices are proper ideals of R which are not contained in the Jacobson radical of R and two distinct vertices I, J are adjacent if and only if I + J = R. In this paper, we use Gallai , s Theorem and the concept of strong resolving graph to compute the strong metric dimension for co-maximal ideal graphs of commutative rings. Explicit formulae for the strong metric dimension, depending on whether the ring is reduced or not, are established. *
10.1142/s0219498824500488
[ "https://export.arxiv.org/pdf/2208.08095v1.pdf" ]
251,622,428
2208.08095
248ea067330a4bb2b9d0885c85ab2e90ee1789c2
Computing the strong metric dimension for co-maximal ideal graphs of commutative rings * Aug 2022 R Shariyari Department of Mathematics, Science and Research Branch Islamic Azad University TehranIran R Nikandish Department of Mathematics Jundi-Shapur University of Technology DezfulIran Jundi-Shapur Research Institute Jundi-Shapur University of Technology DezfulIran A Tehranian Department of Mathematics, Science and Research Branch Islamic Azad University TehranIran H Rasouli [email protected] Department of Mathematics, Science and Research Branch Islamic Azad University TehranIran Computing the strong metric dimension for co-maximal ideal graphs of commutative rings * 17Aug 2022 Let R be a commutative ring with identity. The co-maximal ideal graph of R, denoted by Γ(R), is a simple graph whose vertices are proper ideals of R which are not contained in the Jacobson radical of R and two distinct vertices I, J are adjacent if and only if I + J = R. In this paper, we use Gallai , s Theorem and the concept of strong resolving graph to compute the strong metric dimension for co-maximal ideal graphs of commutative rings. Explicit formulae for the strong metric dimension, depending on whether the ring is reduced or not, are established. * Introduction The concept of metric dimension which enables an observer to uniquely recognize the current position of a moving enemy in a network was initiated by Harary and Melter [11]. This parameter has found various applications in the other fields of sciences (see for instance [7,15]). From then on many graph theorists have been attracted by computing the metric dimension of graphs (see for example [6,12,14,16]). In 2004, Sebö and Tannier ( [26]) introduced a more restricted parameter than metric dimension called strong metric dimension. Computing the strong metric dimension of graphs has appeared in some publications (see [17,18,24] for more information). Both of the strong background and wide range of applications motivate some of algebraic graph theorists to study metric and strong metric dimensions of graphs associated with algebraic structures, see [5,8,9,10,13,19,20,23,25]. In this paper, we deal with the problem of finding the strong metric dimension for co-maximal graph associated with a commutative ring. Throughout this paper, all rings R are commutative with identity. The sets of all maximal ideals, ideals with non-zero annihilators, Jacobson radical and nilpotent elements of R are denoted by Max(R), A(R), J(R), N il(R), respectively. If T is a subset of a ring R, then the symbol T * denotes T \ {0}. Moreover, Ann R (T ) = {r ∈ R| rT = 0}. The ring R is called reduced if N il(R) = {0}. By G = (V, E), we mean a simple and undirected graph G with the vertex set V = V (G) and edge set E = E(G). A connected graph is a graph in which there exists at least one path between any two vertices. Distance between two distinct vertices x, y, denoted by d(x, y), is the length of the shortest path between x and y and diam(G) = max{d(x, y) |x, y ∈ V } is called the diameter of G. In the graph G, let V 0 ⊆ V and E 0 ⊆ E. The induced subgraph by V 0 , denoted by G[V 0 ], is a subgraph of G whose vertex set and edge set are V 0 and E 0 = {{u, v} ∈ E | u, v ∈ V 0 }, respectively. Let x ∈ V . Then the open and closed neighborhood of x are denote by N (x) and N [x], respectively. A complete graph is a graph in which each pair of vertices are adjacent. We use K n to denote a complete graph of order n. A set S of vertices of a graph G is a vertex cover of G if every edge of G has one end in S. The vertex cover number of G, denoted by α(G), is the smallest cardinality of a vertex cover of G. The independence number of a graph G, denoted by β(G), is the largest cardinality of an independent set. For a graph G, S ⊆ V (G) is called a clique if the induced subgraph on S is complete. The number of vertices in the largest clique of a graph G is called the clique number of G and denoted by ω(G). For a connected graph G, let S = {v 1 , v 2 , . . . , v k } be an ordered subset of V (G) and v ∈ V (G) \ S. The metric representation of v with respect to S is the k-vector D(v|S) = (d(v, v 1 ), d(v, v 2 ), . . . , d(v, v k )). For S ⊆ V (G), if for every u, v ∈ V (G) \ S, D(u|S) = D(v|S) implies that u = v, then S is called a resolving set for G. The metric basis for G is a resolving set S of minimum cardinality and the number of elements in S is called the metric dimension of G and denoted by dim M (G). A vertex w of a connected graph G strongly resolves two vertices u, v of G if there exists a shortest path from u to w containing v or a shortest path from v to w containing u. A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G and denoted by sdim M (G). For all undefined notions from graph theory, we refer the reader to [28]. Let R be a ring. The co-maximal ideal graph of R, denoted by Γ(R), is a graph whose vertices are proper ideals of R which are not contained in the Jacobson radical of R and two distinct vertices I, J are adjacent if and only if I + J = R. The concept of co-maximal ideal graph of a commutative ring was first introduced and studied in [30]. Since then co-maximal ideal graphs of commutative rings have been studied by several authors, for instance see [2,29,31]. It is worth mentioning that co-maximal graphs for lattices in [1], for groups in [3], for matrix algebras in [21], for two generated groups in [22] and for noncommutative rings in [27] were investigated. This paper is devoted to study the strong metric dimension of a co-maximal ideal graph and it is organized as follows. In Section 2, we completely determine sdim M (Γ(R)) in terms of the number of maximal ideals of R, in case R is reduced. In Section 3, we focus on the strong metric dimension of Γ(R), when R is a non-reduced ring. Reduced rings case In this section, we present strong metric dimension formula for a co-maximal ideal graph, when R is reduced. We begin with a series of lemmas. Lemma 2.1 Let G be a connected graph and diam(G) = d < ∞. Then the following statements hold. (1) dim M (G) is finite if and only if G is finite. (2) If W ⊂ V (G) is a strong resolving set of G and u, v ∈ V (G) such that N (u) = N (v) or N [u] = N [v], then u ∈ W or v ∈ W . (3) If W ⊂ V (G) is a strong resolving set of G and u, v ∈ V (G) such that d(u, v) = diam(G), then u ∈ W or v ∈ W . Proof. (1) To prove the non-trivial direction, assume that dim M (G) is finite and for some non-negative integer n, let W = {I 1 , I 2 , . . . , I n } be a metric basis for G. Since diam(G) = d < ∞, for every x ∈ V (G) \ W , there are only (d + 1) n choices for D(x|W ). Thus |V (G)| ≤ (d + 1) n + n and hence G is finite. (2) and (3) are obvious. Let G be a graph. It is easily seen that every strong resolving set is also a resolving set, which leads to dim M (G) ≤ sdim M (G). Hence, we have the following immediate corollary. The following well-known result, due to Gallai, which states a relationship between the independence number and the vertex cover number of a graph G has a key role in this paper. Lemma 2.2 (Gallai , s Theorem) For any graph G of order n, α(G) + β(G) = n. A vertex u of G is maximally distant from v (in G) if for every w ∈ N (u), d(v, w) ≤ d(u, v). If u is maximally distant from v and v is maximally distant from u, then we say that u and v are mutually maximally distant. The boundary of G is defined as ∂(G) = {u ∈ V (G)| there exists v ∈ V (G) such that u, v are mutually maximally distant}. We use the notion of strong resolving graph introduced by Oellermann and Peters-Fransen in [24]. The strong resolving graph of G is a graph G SR with vertex set V (G SR ) = ∂(G) where two vertices u, v are adjacent in G SR if and only if u and v are mutually maximally distant. It was shown in [24] that the problem of finding the strong metric dimension of a graph G can be transformed into the problem of computing the vertex cover number of G SR . (1) Since (K n ) SR = K n , sdim M (K n ) = n − 1. (2) Let R = Z 2 × Z 2 × Z 2 . Suppose that X = {(Z 2 , Z 2 , 0), (0, Z 2 , Z 2 ), (Z 2 , 0, Z 2 )} and Y = {(Z 2 , 0, 0), (0, Z 2 , 0), (0, 0, Z 2 )}. It is not hard to see that for any u ∈ X, there is no v ∈ V (Γ(R)) such that u and v are mutually maximally distant, whereas each pair of vertices in Y are mutually maximally distant. This follows that ∂(Γ(R)) = {(Z 2 , 0, 0), (0, Z 2 , 0), (0, 0, Z 2 )} and Γ(R) SR = K 3 . Since α(Γ(R) SR ) = 2, Lemma 2.3 follows that sdim M (Γ(R)) = 2. On the other hand, W = {(Z 2 , 0, 0), (0, Z 2 , 0)} is the minimum strong resolving set, i.e., sdim M (Γ(R)) = 2. t (Z 2 , 0, 0) t (0, Z 2 , Z 2 ) t (Z 2 , Z 2 , 0) t(Z 2 , 0, Z 2 ) ❅ ❅ ❅ t (0, 0, Z 2 ) ✟ ✟ ✟ t (0, Z 2 , 0) ❍ ❍ ❍ t (Z 2 , 0, 0) t t t ❅ ❅ ❅ t (0, 0, Z 2 ) ✜ ✜ ✜ ✜ ✜ ✜ ✜ t (0, Z 2 , 0) ❭ ❭ ❭ ❭ ❭ ❭ ❭ Γ(R) Γ(R) SR It was proved in [30,Theorem 2.4] that diam(Γ(R)) ≤ 3. Hence we omit the elementary proof of the next lemma. Lemma 2.4 Let R ∼ = F 1 × · · · × F n , where F i is a field for every 1 ≤ i ≤ n and let I, J be two vertices of Γ(R). Then the following statements hold. In order to present our results we need to introduce some more terminologies. Let R be a ring and Γ(R) be the co-maximal ideal graph of R. We define Γ(R) * * as the graph with vertex set V (Γ(R) * * ) = V (Γ(R)) such that two vertices I, J are adjacent in Γ(R) * * if and only if I + J = R, I J and J I. Also, we define Γ(R) * as follows: Let Γ(R) * = Γ(R), if Γ(R) is complete, and otherwise Γ(R) * is obtained from Γ(R) * * by removing all its isolated vertices. Let R ∼ = R 1 × · · · × R n , where R i is a ring for every 1 ≤ i ≤ n. Let I = (I 1 , I 2 , . . . , I n ) be an ideal of R. By N ZC(I), we mean the number of nonzero components of I. Lemma 2.5 Let R be a reduced ring and sdim M (Γ(R)) < ∞. Then the following statements hold. (1) If |Max(R)| = 2, then Γ(R) * * = Γ(R) * = Γ(R) = Γ(R) SR = K 2 . (2) If |Max(R)| ≥ 3, then I is an isolated vertex in Γ(R) * * if and only if I ∈ Max(R). (3) If |Max(R)| = n ≥ 3, then Γ(R) * * = H + K 1 + K 1 + · · · + K 1 (n times), where H is a connected graph. (4) Γ(R) * = Γ(R) SR . Proof. (1) Since sdim M (Γ(R)) is finite, R has finitely many ideals, by Corollary 2.1 and so [4,Theorem 8.7] implies that R ∼ = F 1 × · · · × F n , where F i is a field for every 1 ≤ i ≤ n = |Max(R)|. If n = 2, since Γ(R) = K 2 , we have Γ(R) * * = Γ(R) * = Γ(R) = Γ(R) SR = K 2 . (2) First we show that I is an isolated vertex in Γ(R) * * , for every I ∈ Max(R). Assume that I is adjacent to J in Γ(R) * * . Then I J and J I. Hence I +J = R, a contradiction. This implies that I, J are mutually maximally distant and hence I is adjacent to J in Γ(R) SR . Therefore, the induced subgraph on S is a clique in Γ(R) SR . Now, we show that if I ∈ Max(R) ∪ S, then I is adjacent to some vertices of S in Γ(R) SR . But this is obvious, because it is not hard to see that d(I, J) = 3 = daim(Γ(R)), for some J ∈ S. Therefore, ∂(Γ(R)) = V (Γ(R)) \ Max(R) and so the claim is proved. To complete the proof, it is enough to show that the adjacency between vertices in Γ(R) * is in a one to one correspondence between vertices in Γ(R) SR and vice versa. Assume that I, J ∈ V (Γ(R) * ) and I is adjacent to J. Thus I +J = R, I J and J I. Lemma 2.6 Suppose that R ∼ = F 1 × · · · × F n , where F i is a field for every 1 ≤ i ≤ n and S is the largest independent set of Γ(R) SR . Then the followings hold. (1) ∂(Γ(R)) = V (Γ(R)) \ Max(R). (2) N ZC(I) = n − 2, for some I ∈ S. (3) β(Γ(R) SR ) = n − 2, if n ≥ 3. Proof. (1) By Lemma 2.5, it is obvious. (2) Assume that S = {I 1 , I 2 , . . . , I t } is the largest independent set of Γ(R) SR . Clearly, 1 ≤ N ZC(I) ≤ n − 2, for every I ∈ V (Γ(R) SR ). With no loss of generality, assume that N ZC(I t ) ≥ N ZC(I i ) for every 1 ≤ i ≤ t and I i ∈ S. We claim that N ZC(I t ) = n − 2. Assume to the contrary, N ZC(I t ) ≤ n − 3. Since I t is not adjacent to I i , for every 1 ≤ i ≤ t and I i ∈ S, we deduce that I i ⊆ I t or I t ⊆ I i or I t + I i = R. If I t ⊆ I i , for some I i ∈ S, then N ZC(I i ) ≥ N ZC(I t ), a contradiction. This implies that I i ⊆ I t or I t + I i = R, for every I i ∈ S and 1 ≤ i ≤ t. Assume that S 1 = {I ∈ S | I ⊆ I t } and S 2 = {I ∈ S | I + I t = R}. Since N ZC(I t ) ≤ n − 3, by replacing one of the zero components of I t by F i , we get J ∈ V (Γ(R) SR ). Hence I ⊆ J, for every I ∈ S 1 and I + J = R, for every I ∈ S 2 . This implies that S ∪ {J} is a independent set of Γ(R) SR , a contradiction. Therefore, N ZC(I t ) = n − 2 which completes the proof. (3) Assume that A 1 = {I ∈ V (Γ(R) SR ) | N ZC(I) = 1}, A 2 = {I ∈ V (Γ(R) SR ) | N ZC(I) = 2}, A 3 = {I ∈ V (Γ(R) SR ) | N ZC(I) = 3}, . . .≤ i ≤ [ n − 2 2 ] − 1, we conclude that Γ(R) SR [A i ] is complete, by Fact 2. Fact 4. For every [ n − 2 2 ] ≤ i ≤ n − 2, assume that S i ⊆ A i is the largest set of A i such that for every I, J ∈ S i , I + J = R. Then |S i | = [ n n − i ]. Continue the proof in the following steps: Step 1. Put I 1 = (F 1 , 0, . . . , 0), I 2 = (F 1 , F 2 , 0, . . . , 0), . . . , I i = (F 1 , F 2 , . . . , F i , . . . , 0), where i = [ n − 2 2 ] − 1 and W 1 = {I 1 , I 2 , . . . , I i }. Then W 1 is an independent set of Γ(R) SR [∪A i ]. Since by Fact 3 Γ(R) SR [A i ] is complete and |W 1 ∩ A i | = 1, we deduce that W 1 is the largest independent set of Γ(R) SR [∪A i ], for i = 1 up to i = [ n − 2 2 ] − 1 (if n is odd up to [ n − 2 2 ]). This implies that β(Γ(R) SR [∪A i ]) = |W 1 | = [ n − 2 2 ] − 1. Step 2. Step 3. Continue the procedure in Step 2 up to t = n − 2 − [ n − 2 2 ] and get W t = {(F 1 , 0, . . . , 0), (F 1 , F 2 , 0, . . . , 0), . . . , (F 1 , F 2 , . . . , F n−2 , 0, 0)}. Therefore, β(Γ(R) SR [∪ n−2 1 A i ]) = β(Γ(R) SR ) = |W t | = n − 2. We are now in a position to state the main result of this section. Theorem 2.1 Suppose that R is a reduced ring. If sdim M (Γ(R)) is finite, then (1) If |Max(R)| = 2, then sdim M (Γ(R)) = 1. (2) If |Max(R)| = n ≥ 3, then sdim M (Γ(R)) = 2 n − 2n. Proof. Since sdim M (Γ(R)) is finite, R has finitely many ideals, by Corollary 2.1, and so [4,Theorem 8.7] implies that R ∼ = F 1 × · · · × F n , where F i is a field for every 1 ≤ i ≤ n = |Max(R)|. If n = 2, then Γ(R) = K 2 and so sdim M (Γ(R)) = 1. If n ≥ 3, then by Lemmas 2.2 and 2.3, sdim M (Γ(R)) = α(Γ(R) SR ) = V (Γ(R) SR ) − β(Γ(R) SR ). On the other hand, ∂(Γ(R)) = 2 n −2−n and β(Γ(R) SR ) = n−2, by Lemma 2.6. Therefore, sdim M (Γ(R)) = 2 n − 2 − n − (n − 2) = 2 n − 2n. Non-reduced rings case In this section, we study the strong metric dimension of Γ(R), when R is non-reduced. As we have seen in Corollary 2.1, sdim M (Γ(R)) is finite if and only if Γ(R) is finite. Hence in this section, we focus on rings with finitely many ideals. Obviously, such rings are Artinian and it follows from [4,Theorem 8.7] that there exists a positive integer n such that R ∼ = R 1 × · · · × R n , where R i is an Artinian local ring for every 1 ≤ i ≤ n. If every R i is non-reduced, then it is not a field and so |A(R i ) * | ≥ 1 (It is not hard to see that if R is Artinian and non-reduced, then every proper ideal has a non-zero annihilator). Now, let R be a such ring. Suppose that I = (I 1 , . . . , I n ) and J = (J 1 , . . . , J n ) are two vertices of Γ(R). Define the relation ∼ on A(R) * as follows: I ∼ J, whenever for each 1 ≤ i ≤ n, The proof of the following lemma is obvious. Lemma 3.2 Suppose that R ∼ = R 1 × · · · × R n , where R i is an Artinian local ring and |A(R i ) * | ≥ 1, for every 1 ≤ i ≤ n. Then the following statements hold. "I i ⊆ N il(R i ) ifLemma 3.1 Suppose that R ∼ = R 1 × · · · × R n ,(1) Γ(R) ′ = H ′ + K |A(R 1 )| + K |A(R 2 )| + · · · + K |A(Rn)| , where H ′ = ∅ if n = 2 and H ′ is connected if n ≥ 3. (2) Γ(R) ′ = Γ(R) SR . Next, it is proved that Γ(R) ′ \ A is a connected graph. For this, let Proof. Let A 1 = {I ∈ V (Γ(R)) | [I] = [(N il(R 1 ), R 2 , . . . , R n )]}, A 2 = {I ∈ V (Γ(R)) | [I] = [(R 1 , N il(R 2 ), R 3 , . . . , R n )]}, . . .S = {(R 1 , 0, . . . , 0), (0, R 2 , 0, . . . , 0), . . . , (0, . . . , 0, R n )}. Using a proof technique similar to Lemma 2.5 implies that the induced subgraph on S is a clique in Γ(R) ′ and every I ∈ V (Γ(R) ′ ) \ (S ∪ A) is adjacent to some vertices of S in Γ(R) ′ . Therefore, Γ(R) ′ = H ′ + K |A(R 1 )| + K |A(R 2 )| + · · · + K |A(Rn)| . By an easy verification, if n = 2, then H ′ = ∅. 3 Suppose that R ∼ = R 1 × · · · × R n , where R i is an Artinian local ring and |A(R i ) * | ≥ 1, for every 1 ≤ i ≤ n. Then the following statements hold. (1) ∂(Γ(R)) = V (Γ(R)). (2) β(Γ(R) SR ) = 2n − 2. Proof. (1) is obvious. (2) By the proof of Lemma 3.2, Γ(R) SR = H ′ + K |A(R 1 )| + K |A(R 2 )| + · · · + K |A(Rn)| . It is known that β(K |A(R 1 )| + K |A(R 2 )| + · · · + K |A(Rn)| ) = n. Let Now, we are ready to state the following result. A = {(I 1 , . . . , I n ) ∈ V (H ′ ) | I i ∈ {0, R 1 , . . . , R n } for every 1 ≤ i ≤ n}. Theorem 3.1 Suppose that R ∼ = R 1 × · · · × R n , where R i is an Artinian local ring and |A(R i ) * | ≥ 1, for every 1 ≤ i ≤ n. Then sdim M (Γ(R)) = |V (Γ(R))| − 2n + 2. Proof. The proof follows from Lemmas 2.2, 2.3 and 3.3. The following example is related to Theorem 3.1. We close this paper with the following result which completely characterizes sdim M (Γ(R)), when R is non-reduced. Corollary 3.1 Let R ∼ = R 1 × · · · × R n × F 1 × · · · × F m , be a ring, n ≥ 1, m ≥ 1 where R i is an Artinian local ring such that for every 1 ≤ i ≤ n, |A(R i ) * | ≥ 1 and each F i is a field. Then sdim M (Γ(R)) = |V (Γ(R))| − 2n − 2m + 2. Proof. The proof here is a refinement of the arguments in proofs of Theorem 3.1 and Lemma 2.5. By similar proofs, one may get Γ(R) SR = H ′′ + K |A(R 1 )| + K |A(R 2 )| + · · · + K |A(Rn)| , β(H ′′ ) = n + m − 2 and β(K |A(R 1 )| + K |A(R 2 )| + · · · + K |A(Rn)| ) = n. Thus β(Γ(R) SR ) = 2n + m − 2. On the other hand, since |V (Γ(R) SR )| = |V (Γ(R))| − m, we deduce that sdim M (Γ(R)) = |V (Γ(R))| − m − (2n + m − 2) = |V (Γ(R))| − 2n − 2m + 2. Corollary 2. 1 1Let R be a ring. Then sdim M (Γ(R)) is finite if and only if Γ(R) is finite. Lemma 2.3 ([24, Theorem 2.1]) For any connected graph G, sdim M (G) = α(G SR ). The next example illustrates the validity of Lemma 2.3. ( 1 ) 1d(I, J) Γ(R) = 2 if and only if I ∩ J = 0 and I + J = R. ( 2 ) 2d(I, J) Γ(R) = 3 if and only if I ∩ J = 0 and I + J = R. ( 3 ) 3If I ⊆ J or J ⊆ I, or I + J = R, then I and J are not mutually maximally distant. Let A = V (Γ(R) * * ) \ Max(R). To complete the proof, we show that Γ(R) * * [A] is a connected graph. To see this, let S = {(F 1 , 0, . . . , 0), (0, F 2 , 0, . . . , 0), . . . , (0, . . . , 0, F n )}. If I, J ∈ S, then I + J = R, I J and J I. Thus I is adjacent to J in Γ(R) * * [A]. Therefore, the induced subgraph on S is a clique in Γ(R) * * [A]. Hence suppose that I ∈ V (Γ(R) * * ) \ S ∪ Max(R). Then I is adjacent to some vertices of S in Γ(R) * * [A], because at least two components of I are zero. ( 3 ) 3By(2), it is obvious.(4) First we claim that V (Γ(R) * ) = V (Γ(R) SR ). By parts(2)and (3), V (Γ(R) * ) = V (Γ(R)) \ Max(R). We prove that V (Γ(R) SR ) = V (Γ(R)) \ Max(R). Let I ∈ Max(R). It is shown that there is no J ∈ V (Γ(R)) such that I and J are mutually maximally distant. If not, I J and J I, by Lemma 2.4. Since I ∈ Max(R), I + J = R in Γ(R). Hence d(I, J) = 1. If J = Ann(I), then d(I, Ann(I)) = 1, but Ann(I) ⊆ J and so d(J, Ann(I)) = 1, a contradiction. If J = Ann(I), for some Ann(I) ⊆ K, then d(I, Ann(I)) = 1, but Ann(I) ⊆ K and so d(K, Ann(I)) = 1, a contradiction. Hence I is an isolated vertex in Γ(R) SR and so V (Γ(R) SR ) ∩ Max(R) = ∅. Next, assume that S = {I ∈ V (Γ(R)) | N ZC(I) = 1} and I, J ∈ S. Then d(I, J) Γ(R) = 3 = diam(Γ(R)). Since I +J = R, d(I, J) Γ(R) = 1. Indeed, d(I, J) Γ(R) ∈ {2, 3}. If d(I, J) Γ(R) = 3, then d(I, J) = daim(Γ(R)). So I, J are mutually maximally distant. Hence I is adjacent to J in Γ(R) SR . Therefore, suppose that d(I, J) Γ(R) = 2 and K ∈ N Γ(R) (I). Since I +K = R and J I, K ∩J = 0. By Lemma 2.4, d(K, J) Γ(R) ≤ 2. Thus d(K, J) Γ(R) ≤ d(I, J) Γ(R) . Similarly, d(L, I) Γ(R) ≤ d(I, J) Γ(R) = 2, for every L ∈ N (J). So I, J are mutually maximally distant and hence I is adjacent to J in Γ(R) SR . Finally,l let I, J ∈ V (Γ(R) SR ) and I is adjacent to J. Thus I, J are mutually maximally distant. By Lemma 2.4, I + J = R, I J and J I and so I is adjacent to J in Γ(R) * . A n− 2 = 2{I ∈ V (Γ(R) SR ) | N ZC(I) = n − 2}. Consider the following facts: Fact 1. For every 1 ≤ i ≤ n − 2 and I, J ∈ A i , since N ZC(I) = N ZC(J), we conclude that I J and J I. Fact 2 . 2If I, J ∈ A i for every 1 ≤ i ≤ n − 2 and I is not adjacent to J, then I + J = R, by Fact 1. Fact 3 . 3Since there is no I, J ∈ A i such that I + J = R, for every 1 Consider A i , where i = [ n − 2 2 ] and n is even. Then we have |S i | = 2. Without loss of generality, assume that S i = {I = (F 1 , F 2 , . . . , F i , . . . , 0), J = (0, . . . , 0, F i+1 , . . . , F n )}. This implies that J is adjacent to some vertices of W 1 , but W 2 = W 1 ∪ {I} is the largest independent set of Γ(R) SR [∪A i ] for i = 1 up to i = [ n − 2 2 ]. Hence β(Γ(R) SR [∪A i ]) = |W 2 | = [ and only if J i ⊆ N il(R i )". Clearly, ∼ is an equivalence relation on A(R) * . The equivalence class of I is denoted by [I]. Suppose that I = (I 1 , . . . , I n ) is and ideal of R. Then by I ′ = (I ′ 1 , . . . , I ′ n ), we mean a new ideal obtained from I whose all nonzero nilpotent components are replaced by 0. We define Γ(R) ′ as the graph with vertex set V (Γ(R) ′ ) = V (Γ(R)) such that two vertices I, J are adjacent in Γ(R) ′ if and only if [I] = [J] or I ′ + J ′ = R, I ′ J ′ and J ′ I ′ . where R i is an Artinian local ring for every 1 ≤ i ≤ n. If I and J are vertices of Γ(R) with [I] = [J], then N [I] = N [J]. A n = {I ∈ V (Γ(R)) | [I] = [(R 1 , . . . , R n−1 , N il(R n ))]} and A = ∪A i . Obviously, if I, J ∈ A i , then [I] = [J] and so I is adjacent to J in Γ(R) ′ . This implies that Γ(R) ′ [A i ] is a complete graph, for every 1 ≤ i ≤ n. Suppose that I ∈ A i and J ∈ A i . We show that I is not adjacent to J in Γ(R) ′ . If not, since [I] = [J], we must have I ′ + J ′ = R, I ′ J ′ and J ′ I ′ , a contradiction, as I ′ + J ′ = R. ( 2 ) 2Let I ∈ V (Γ(R) ′ ). Since |[I]| ≥ 2, I ∼ J and I = J. Since N (I) = N (J), we deduce that I and J are mutually maximally distant and hence I ∈ V (Γ(R) SR ). Thus V (Γ(R) ′ ) = V (Γ(R) SR ). We show that Γ(R) ′ = Γ(R) SR . Let I, J ∈ V (Γ(R) ′ ) and I is adjacent to J. If [I] = [J], then I, J are mutually maximally distant and so I is adjacent to J in Γ(R) SR . If [I] = [J], then I ′ + J ′ = R, I ′ J ′ and J ′ I ′ . By a similar proof to that of Lemma 2.5, I ′ , J ′ are mutually maximally distant. Since [I] = [I ′ ] and [J] = [J ′ ], I is adjacent to J in Γ(R) SR . Finally, suppose that I, J ∈ V (Γ(R) SR ) and I is adjacent to J. If [I] = [J], then N (I) = N (J) and so I, J are mutually maximally distant. If [I] = [J], then I + J = R, I J and J I. Thus I ′ + J ′ = R, I ′ J ′ and J ′ I ′ . Hence I is adjacent to J in V (Γ(R) ′ ). Then H ′ [A] ∼ = H, where H = Γ(R) * * − n(K 1 ) (as in part (3) of Lemma 2.5). Thus |A ∩ [I]| = 1, for every equivalence class [I]. On the other hand, since H ′ [[I]] is complete for every I ∈ H ′ , β(H ′ ) = β(Γ(R) SR [A]). Since Γ(R) SR [A] ∼ = H and β(H) = n − 2, we deduce that β(H ′ ) = β(Γ(R) SR [A]) = n − 2. Therefore, β(Γ(R) SR ) = β(H ′ ) + n = 2n − 2. Let R = Z 4 × Z 4 × Z 8 . Then[(Z 4 , Z 4 , (0))] = {(Z 4 , Z 4 , (0)), (Z 4 , Z 4 , (2)), (Z 4 , Z 4 , (4))},[(Z 4 , (0), Z 8 )] = {(Z 4 , (0), Z 8 ), (Z 4 , (2), Z 8 )}, [((0), Z 4 , Z 8 )] = {((0), Z 4 , Z 8 ), ((2), Z 4 , Z 8 )}, [((0), (0), Z 8 )] = {((0), (0), Z 8 ), ((2), (0), Z 8 ), ((0), (2), Z 8 ), ((2), (2), Z 8 )}, [(Z 4 , (0), (0))] = {(Z 4 ,(0), (0)), (Z 4 , (0), (2)), (Z 4 , (0), (4)), (Z 4 , (2), (0)), (Z 4 , (2), (2)), (Z 4 , (2), (4))}, [((0), Z 4 , (0))] = {((0), Z 4 , (0)), ((0), Z 4 , (2)), ((0), Z 4 , (4)), ((2), Z 4 , (0)), ((2), Z 4 , (2)), ((2), Z 4 , (4))} and A = {(Z 4 , Z 4 , (0)), (Z 4 , (0), Z 8 ), ((0), Z 4 , Z 8 ), ((0), (0), Z 8 ), (Z 4 , (0), (0)), ((0), Z 4 , (0))}. Let J, K ∈ [I]. Then N [J] = N [K] and so by Lemma 2.1, J ∈ W or K ∈ W , where W is a strong resolving set of Γ(R). Thus one may assume that V (Γ(R)) \ A ⊆ W . Let V 1 = 1{((0), Z 4 , (0)), (Z 4 , (0), (0)), ((0), (0), Z 8 )} andV 2 = {(Z 4 , (0), Z 8 ), ((0), Z 4 , Z 8 ), (Z 4 , Z 4 , (0))}. Since d(I, J) = daim(Γ(R)), for every I, J ∈ V 1 , we conclude that I ∈ W or J ∈ W . Thus one may let V (Γ(R)) \ V 3 ⊆ W , where V 3 = V 2 ∪ {((0), (0), Z 8 )}.Since for every I, J ∈ V 3 we have N [I] = N [J], I, J are strongly resolved by some vertices of W = A(R) * \ (V 3 ). This means that sdim M (Γ) = |W | = 23 − 4 = 19. On the other hand, by Theorem 3.1 it is easily seen that sdim M (Γ(R)) = 23 − 6 + 2 = 19. Acknowledgements. Reza Nikandish in this work has been financially supported by the research of Jundi-Shapur Research Institute. The grant number was 01-100-1-1400. The comaximal graph of a lattice. M Afkhami, K Khashyarmanesh, Bull Malays. Sci. Soc. 372M. Afkhami, K. Khashyarmanesh, The comaximal graph of a lattice, Bull Malays. Sci. Soc. 37 (2) (2014) 261-269. A note on co-maximal ideal graph of commutative rings. S Akbari, B Miraftab, R Nikandish, Ars. Combin. 134S. Akbari, B. Miraftab, R. Nikandish, A note on co-maximal ideal graph of commutative rings, Ars. Combin. 134 (2017) 261-265. Co-maximal graphs of subgrous of groups. S Akbari, B Miraftab, R Nikandish, Canad. Math. Bull. 601S. Akbari, B. Miraftab, R. Nikandish, Co-maximal graphs of subgrous of groups, Canad. Math. Bull. 60 (1) (2017) 12-25. Introduction to Commutative Algebra. M F Atiyah, I G Macdonald, Addison-Wesley Publishing CompanyM. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Pub- lishing Company (1969). Base size, metric dimension and other invariants of groups and graphs. R F Bailey, P J Cameron, Bull. London Math. Soc. 43R. F. Bailey, P. J. Cameron, Base size, metric dimension and other invariants of groups and graphs, Bull. London Math. Soc. 43 (2011) 209-242. On k-dimensional graphs and their bases. P S Buczkowski, G Chartrand, C Poisson, P Zhang, Period. Math. Hung. 46P. S. Buczkowski, G. Chartrand, C. Poisson, P. Zhang, On k-dimensional graphs and their bases, Period. Math. Hung. 46 (2003) 9-15. Resolvability in graphs and the metric dimension of a graph. G Chartrand, L Eroh, M A Johnson, O R Oellermann, Disc. Appl. Math. 105G. Chartrand, L. Eroh, M. A. Johnson, O. R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Disc. Appl. Math. 105 (2000) 99-113. The metric dimension of the total graph of a finite commutative ring. D Dolžan, Canad. Math. Bull. 59D. Dolžan, The metric dimension of the total graph of a finite commutative ring, Canad. Math. Bull. 59 (2016) 748-759. The metric dimension of the annihilating-ideal graph of a finite commutative ring. D Dolžan, Bull. Aust. Math. Soc. 103D. Dolžan, The metric dimension of the annihilating-ideal graph of a finite commutative ring, Bull. Aust. Math. Soc. 103 (2021) 362-368. On the strong metric dimension of annihilator graphs of commutative rings. Sh, R Ebrahimi, A Nikandish, H Tehranian, Rasouli, Bull. Malays. Math. Sci. Soc. 44Sh. Ebrahimi, R. Nikandish, A. Tehranian, H. Rasouli, On the strong metric dimension of annihilator graphs of commutative rings, Bull. Malays. Math. Sci. Soc. 44 (2021) 2507-2517. On the metric domension of a graph. F Harary, R A Melter, Ars Combin. 2F. Harary, R. A. Melter, On the metric domension of a graph. Ars Combin. 2 (1976) 191-195. On the metric dimension of circulant graphs. M Imran, A Q Baig, S A U H Bokhary, I Javaid, Appl. Math. Lett. 25M. Imran, A. Q. Baig, S. A. U. H. Bokhary, and I. Javaid, On the metric dimension of circulant graphs, Appl. Math. Lett. 25 (2012) 320-325. On strong metric dimension of zero-divisor graphs of rings. M Bhat, S Pirzada, Korean J. Math. 27M. Imran Bhat, S. Pirzada, On strong metric dimension of zero-divisor graphs of rings, Korean J. Math. 27 (2019) 563-580. On the metric dimension of Cartesian powers of a graph. Z Jiang, N Polyanskii, J. Comb. theory Ser. A. 165Z. Jiang, N. Polyanskii, On the metric dimension of Cartesian powers of a graph, J. Comb. theory Ser. A. 165 (2019) 1-14. Localization in graphs. S Khuller, B Raghavachari, A Rosenfeld, CS-TR- 3326University of Maryland at College ParkTechnical reportS. Khuller, B. Raghavachari, A. Rosenfeld, Localization in graphs, Technical report CS-TR- 3326, University of Maryland at College Park, (1994). Some binary products and integer linear programming for k-metric dimension of graphs. S Klavžar, F Rahbarnia, M Tavakolid, Appl. Math. Comput. 40915126420S. Klavžar, F. Rahbarnia, M. Tavakolid, Some binary products and integer linear programming for k-metric dimension of graphs, Appl. Math. Comput. 409 (15) (2021) 126420. On the strong metric dimension of corona product graphs and join graphs. D Kuziak, I G Yero, J A Rodrguez-Velzquez, Discrete Appl. Math. D. Kuziak, I. G. Yero, J. A. Rodrguez-Velzquez, On the strong metric dimension of corona product graphs and join graphs, Discrete Appl. Math. (2013) 161 1022-1027. On the strong metric dimension of the strong products of graphs. D Kuziak, I G Yero, J A Rodrguez-Velquez, Open Math. D. Kuziak, I. G. Yero, J. A. Rodrguez-Velquez, On the strong metric dimension of the strong products of graphs, Open Math. (2015) 13 64-74. The strong metric dimension of the power graph of a finite group. X Ma, M Feng, K Wang, Discrete Appl. Math. 239X. Ma, M. Feng, K. Wang, The strong metric dimension of the power graph of a finite group, Discrete Appl. Math. 239 (2018) 159-164. Strong metric dimensions for power graphs of finite groups. X Ma, L Zhai, 10.1080/00927872.2021.1924764Comm. Algebra. X. Ma, L. Zhai, Strong metric dimensions for power graphs of finite groups, Comm. Algebra (2021) DOI: 10.1080/00927872.2021.1924764. Co-maximal ideal graphs of matrix algebras. B Miraftab, R Nikandish, Bol. Soc. Mat. Mex. 24B. Miraftab, R. Nikandish, Co-maximal ideal graphs of matrix algebras, Bol. Soc. Mat. Mex. 24 (2018) 1-10. Co-maximal graphs of two generate groups. B Miraftab, R Nikandish, J. Algebra. Appl. 18413 pagesB. Miraftab, R. Nikandish, Co-maximal graphs of two generate groups, J. Algebra. Appl. 18 (4) (2019) (13 pages). Metric and strong metric dimension in cozerodivisor graphs. R Nikandish, M J Nikmehr, M Bakhtyiari, Mediterr. J. Math. 18112R. Nikandish, M. J. Nikmehr, M. Bakhtyiari, Metric and strong metric dimension in cozero- divisor graphs, Mediterr. J. Math. 18 112 (2021). The strong metric dimension of graphs and digraphs. O R Oellermann, J Peters-Fransen, Discrete Appl. Math. 155O. R. Oellermann, J. Peters-Fransen, The strong metric dimension of graphs and digraphs, Discrete Appl. Math. 155 (2007) 356-364. On the metric domension of a zero-divisor graph. S Pirzada, R Raja, Comm. Algebra. 454S. Pirzada, R. Raja , On the metric domension of a zero-divisor graph, Comm. Algebra 45 (4) (2017) 1399-1408. On metric generators of graphs. A Sebö, E Tannier, Math. Oper. Res. 29A. Sebö, E. Tannier, On metric generators of graphs, Math. Oper. Res. 29 (2004) 383-393. Some properties of comaximal right ideal graph of a ring. S Shen, W Liu, L Feng, Appl. Math. Comput. 33315S. Shen, W. Liu, L. Feng, Some properties of comaximal right ideal graph of a ring, Appl. Math. Comput. 333 (15) (2018) 225-230. Introduction to Graph Theory. D B West, Prentice HallUpper Saddle River2nd ed.D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall, Upper Saddle River (2001). Graph properties of co-maximal ideal graphs of commutative rings. T S Wu, M Ye, Q Liu, J Guo, J. Algebra. Appl. 14313 pagesT. S. Wu, M. Ye, Q, Liu, J. Guo, Graph properties of co-maximal ideal graphs of commutative rings, J. Algebra. Appl. 14 (3) (2015) (13 pages). Co-maximal ideal graphs of commutative rings. M Ye, T S Wu, J. Algebra. Appl. 16614 pagesM. Ye, T. S. Wu, Co-maximal ideal graphs of commutative rings, J. Algebra. Appl. 16(6) (2012) (14 pages). Implements of graphs blow-up in co-maximal ideal graphs. M Ye, T S Wu, Q Liu, H Yu, Comm. Algebra. 42M. Ye, T. S. Wu, Q. Liu, H. Yu, Implements of graphs blow-up in co-maximal ideal graphs, Comm. Algebra 42 (2014) 2476-2483.
[]
[ "A complete radio study of SNR G15.4+0.1 from new GMRT observations", "A complete radio study of SNR G15.4+0.1 from new GMRT observations" ]
[ "L Supan [email protected] \nInstituto de Astronomía y Física del Espacio (IAFE\nCONICET-UBA)\nCC 67, Suc. 281428Buenos AiresArgentina\n", "G Castelletti \nInstituto de Astronomía y Física del Espacio (IAFE\nCONICET-UBA)\nCC 67, Suc. 281428Buenos AiresArgentina\n", "B C Joshi \nNational Centre for Radio Astrophysics (NCRA-TIFR)\nPost Bag No 3, Ganeshkhind, Pune -411007India\n", "M P Surnis \nNational Centre for Radio Astrophysics (NCRA-TIFR)\nPost Bag No 3, Ganeshkhind, Pune -411007India\n", "D Supanitsky \nInstituto de Astronomía y Física del Espacio (IAFE\nCONICET-UBA)\nCC 67, Suc. 281428Buenos AiresArgentina\n" ]
[ "Instituto de Astronomía y Física del Espacio (IAFE\nCONICET-UBA)\nCC 67, Suc. 281428Buenos AiresArgentina", "Instituto de Astronomía y Física del Espacio (IAFE\nCONICET-UBA)\nCC 67, Suc. 281428Buenos AiresArgentina", "National Centre for Radio Astrophysics (NCRA-TIFR)\nPost Bag No 3, Ganeshkhind, Pune -411007India", "National Centre for Radio Astrophysics (NCRA-TIFR)\nPost Bag No 3, Ganeshkhind, Pune -411007India", "Instituto de Astronomía y Física del Espacio (IAFE\nCONICET-UBA)\nCC 67, Suc. 281428Buenos AiresArgentina" ]
[]
Aims. The supernova remnant (SNR) G15.4+0.1 is considered to be the possible counterpart of the γ-ray source HESS J1818−154. With the goal of getting a complete view of this remnant and understanding the nature of the γ-ray flux, we conducted a detailed radio study that includes the search for pulsations and a model of the broadband emission for the SNR G15.4+0.1/HESS J1818−154 system. Methods. Low-frequency imaging at 624 MHz and pulsar observations at 624 and 1404 MHz towards G15.4+0.1 were carried out with the Giant Metrewave Radio Telescope (GMRT). We correlated the new radio data with observations of the source at X-ray and infrared wavelengths from XMM-Newton and Herschel observatories, respectively. To characterize the neutral hydrogen (HI) medium towards G15.4+0.1, we used data from the Southern Galactic Plane Survey. We modelled the spectral energy distribution (SED) using both hadronic and leptonic scenarios. Results. From the combination of the new GMRT observations with existing data, we derived a continuum spectral index α=−0.62 ± 0.03 for the whole remnant. The local synchrotron spectra of G15.4+0.1, calculated from the combination of the GMRT data with 330 MHz observations from the Very Large Array, tends to be flatter in the central part of the remnant, accompanying the region where the blast wave is impinging molecular gas. No spectral index trace was found indicating the radio counterpart to the pulsar wind nebula proposed from X-ray observations. In addition, the search for radio pulsations yielded negative results. Emission at farinfrared wavelengths is observed in the region where the SNR shock is interacting with dense molecular clumps. We also identified HI features forming a shell that wraps most of the outer border of G15.4+0.1. Characteristic parameters were estimated for the shocked HI gas. We found that either a purely hadronic or leptonic model is compatible with the broadband emission known so far.
10.1051/0004-6361/201425284
[ "https://arxiv.org/pdf/1502.01224v1.pdf" ]
55,514,324
1502.01224
5c9396868376152d61666464f9b52f3321180e4b
A complete radio study of SNR G15.4+0.1 from new GMRT observations 4 Feb 2015 February 5, 2015 L Supan [email protected] Instituto de Astronomía y Física del Espacio (IAFE CONICET-UBA) CC 67, Suc. 281428Buenos AiresArgentina G Castelletti Instituto de Astronomía y Física del Espacio (IAFE CONICET-UBA) CC 67, Suc. 281428Buenos AiresArgentina B C Joshi National Centre for Radio Astrophysics (NCRA-TIFR) Post Bag No 3, Ganeshkhind, Pune -411007India M P Surnis National Centre for Radio Astrophysics (NCRA-TIFR) Post Bag No 3, Ganeshkhind, Pune -411007India D Supanitsky Instituto de Astronomía y Física del Espacio (IAFE CONICET-UBA) CC 67, Suc. 281428Buenos AiresArgentina A complete radio study of SNR G15.4+0.1 from new GMRT observations 4 Feb 2015 February 5, 2015Received <date>; Accepted <date>Astronomy & Astrophysics manuscript no. castelletti-astroph c ESO 2015ISM: individual objects: SNR G154+01, HESS J1818−154-ISM: supernova remnants-Gamma rays: ISM-Radio con- tinuum: ISM Aims. The supernova remnant (SNR) G15.4+0.1 is considered to be the possible counterpart of the γ-ray source HESS J1818−154. With the goal of getting a complete view of this remnant and understanding the nature of the γ-ray flux, we conducted a detailed radio study that includes the search for pulsations and a model of the broadband emission for the SNR G15.4+0.1/HESS J1818−154 system. Methods. Low-frequency imaging at 624 MHz and pulsar observations at 624 and 1404 MHz towards G15.4+0.1 were carried out with the Giant Metrewave Radio Telescope (GMRT). We correlated the new radio data with observations of the source at X-ray and infrared wavelengths from XMM-Newton and Herschel observatories, respectively. To characterize the neutral hydrogen (HI) medium towards G15.4+0.1, we used data from the Southern Galactic Plane Survey. We modelled the spectral energy distribution (SED) using both hadronic and leptonic scenarios. Results. From the combination of the new GMRT observations with existing data, we derived a continuum spectral index α=−0.62 ± 0.03 for the whole remnant. The local synchrotron spectra of G15.4+0.1, calculated from the combination of the GMRT data with 330 MHz observations from the Very Large Array, tends to be flatter in the central part of the remnant, accompanying the region where the blast wave is impinging molecular gas. No spectral index trace was found indicating the radio counterpart to the pulsar wind nebula proposed from X-ray observations. In addition, the search for radio pulsations yielded negative results. Emission at farinfrared wavelengths is observed in the region where the SNR shock is interacting with dense molecular clumps. We also identified HI features forming a shell that wraps most of the outer border of G15.4+0.1. Characteristic parameters were estimated for the shocked HI gas. We found that either a purely hadronic or leptonic model is compatible with the broadband emission known so far. Introduction The radio source G15.4+0.1 was first identified as a supernova remnant (SNR) during the 90 cm multiconfiguration Very Large Array (VLA) survey of the Galactic plane (Brogan et al. 2006). This remnant is found to positionally coincide with extended emission (∼8.5 arcmin in size) detected in the TeV γ-ray band with the High Energy Stereoscopic System (H.E.S.S.) (Hofverberg et al. 2011). The TeV source, catalogued as HESS J1818−154, is one of the faintest HESS sources detected to date. On the basis of the morphological correspondence between the brightest hotspot of HESS J1818−154 and the inner part of G15.4+0.1, it has been proposed that a pulsar wind nebula (PWN) powered by a still undetected rotating neutron star could be responsible for the TeV γ rays (Hofverberg et al. 2011). Recently, Castelletti et al. (2013) (C13) found morphological and kinematical evidence for shocked CO gas in a dense molecular cloud located in the foreground of the SNR G15.4+0.1. Because of the location of the cloud (partially overlapping the HESS source) and the signatures of interaction between the molecular gas and the remnant, the authors argued that the gamma radiation might be arising from hadronic interactions produced by the passage of the supernova shock front through Send offprint requests to: G. Castelletti the molecular cloud. In that paper, based on 21 cm HI absorption measurements along the line of sight to G15.4+0.1, those authors also redetermined the distance to this remnant placing it at 4.8 ± 1.0 kpc, instead of the 10 ± 3 kpc estimated by Hofverberg et al. (2011) using the uncertain Σ−∆ relation for supernova remnants. Later on, Abramowski et al. (2014) used the XMM-Newton telescope to investigate the X-ray counterpart to the γ-ray source HESS J1818−154. The new X-ray image of the remnant shows extended emission in the interior of the radio SNR shell, in spatial coincidence with the discovered HESS source and the southern part of the molecular cloud. Although the X-ray emission is fitted satisfactorily by both the thermal and non-thermal models, these authors interpreted the X-rays as emission from a PWN. In addition, the X-ray observations lead to the detection of five point-like sources in the region of the γ-ray emission. From their X-ray properties, only two were thought to be galactic sources that could be candidates for the central compact object of the SNR. The analysis of periodicity in the signal from these sources, however, has not shown evidence of X-ray pulsations. In this paper, based on new data obtained with the Giant Metrewave Radio Telescope (GMRT), we provide the first comprehensive analysis of the morphological and spectral radio properties of SNR G15.4+0.1, which includes the results of the search A&A proofs: manuscript no. castelletti-astroph for radio pulsations in the region of the source HESS J1818−154. In addition, we report on the study of the spatial correspondences between the radio, infrared, and X-ray emission bands, and the analysis of the neutral hydrogen distribution in the environment of the remnant. On the basis of the flux measurements from radio to γ ray, we discuss scenarios in which the TeV emission from this source originates from either hadrons interacting with dense interstellar material or leptonic emission. Observations and imaging Radio continuum and neutral hydrogen observations The SNR G15.4+0.1 was observed using the GMRT at 624 MHz on June 17 and 18, 2012 (Project code 22-015). The project also includes GMRT observations of the remnant at 1420 MHz. The image at this frequency was presented for the first time in C13, and so it is not shown again here. The data were collected in a total bandwidth of 33.33 MHz split into 512 spectral channels. The flux density calibrator used was 3C 286, for which a flux density of 14.92 Jy was set using the Perley-Butler 2010 VLA values 1 . Regular observations of the source 1822−096 were used for phase and bandpass calibration. Reduction and imaging were carried out with the NRAO Astronomical Image Processing System (AIPS) package. The data from each day were fully reduced separately and then combined in a single data set. The reduction of the data consisted of the removal of radio frequency interference (RFI) by hand rather than with the automatic flagging routines, and bandpass and gain calibrations. After initial calibration, the data were averaged in terms of frequency by collapsing the bandwidth to a number of 100 spectral channels. For the concatenated 624 MHz data set, we employed widefield imaging based on a pseudo-three-dimensional multi-facet algorithm. In addition, we used a multi-scale CLEAN algorithm in AIPS, with four different scale sizes. Several self-calibration and imaging iterations were made to obtain the final image. The resulting GMRT image at 624 MHz has an angular resolution of 7 ′′ .05 × 4 ′′ .55, PA=44 • .42, and an rms noise level of 0.15 mJy beam −1 . We note the significantly high sensitivity (a factor of 10 better than the published image at 330 MHz) of this image at 624 MHz, being reported for the first time. On the other hand, the interstellar medium in the direction of G15.4+0.1 was investigated using data from the Southern Galactic Plane Survey in the 21 cm emission line produced by HI (SGPS, McClure-Griffiths et al. 2005). The SGPS images, combining data from the Australia Telescope Compact Array (ATCA) and the Parkes 64 m single-dish telescope, are valuable in tracing large scale HI structures with an angular resolution of 3 ′ .3 × 2 ′ .1, PA=135 • , and a velocity channel spacing of 0.82 km s −1 . Radio time-series observations Time-series observations towards the reported centroid of the HESS source, l = 15 • .41, b = 0 • .17 (Hofverberg et al. 2011), were carried out with the GMRT at 1404 MHz on May 11 and 12, 2012, and at 624 MHz on June 17 and 18, 2012 (Project code 22-015), simultaneously with the imaging observations. At each frequency, the GMRT was simultaneously used in two different modes to obtain two different time-series. In one mode, the output of 26 antennas were combined in an incoherent array (IA) with an effective beam of 43 ′ and 24 ′ at 624 and 1404 MHz, respectively. The five X-ray point sources, identified in the XMM-Newton observations (Abramowski et al. 2014), all lie between 2 ′ -5 ′ from the centre of this beam and were covered in the search for pulsation. A higher sensitivity phased array (PA) of 15 antennas, with an effective beam of 100 ′′ and 40 ′′ at 624 and 1404 MHz, respectively, and pointed at the centroid of the HESS position, was used in the other mode. Both time-series were sampled every 61 µs with 512 channels across a 33.33 MHz bandpass at each frequency of observation. To accomplish the periodic requirement of phase calibration for both imaging and phased array observations, the time-series were acquired in 13 non-contiguous integrations of about 20 minute duration each at 624 and 1404 MHz, respectively. In order to assess the sensitivity of the search for radio pulsations, as well as the dispersion measures (DM) and the flux density estimates, we also carried out time-series observations in direction of three known pulsars: B1642−03, B1937+21, and J1901−0906. X-rays observations With the aim of performing a multiwavelength analysis of the emission from G15.4+0.1, we reprocessed data extracted from the on-line archive of the XMM-Newton observations 2 . The observations in the direction of the remnant were made on October 10, 2012, during revolution 2351 (Obs-ID: 0691390101). The data were reduced using the XMM-Newton Science Analysis System (SAS) software package Version 13.5.0. We applied standard processing to this data set to obtain clean event files. In order to mitigate the impact of high background flare activity, an appropriate screening was applied, extracting a light curve for photons above 10 keV, for the entire field of view. We selected events with FLAG = 0 and PATTERN 12 for both EPIC MOS1 and MOS2 cameras (Turner et al. 2001). The resulting total effective exposure time of observation with the MOS1 and MOS2 cameras was 30 ks. In order to reveal the diffuse emission, point-like sources within the field of view were extracted by using the dmfilth tool of the CIAO (v. 4.6) reduction package 3 with POISSON statistics to fill the excluded regions. Finally, a single combined image in the energy range 1-8 keV from both cameras was obtained, which was then smoothed with a Gaussian kernel to a resolution of ∼1 ′ .4. Results 3.1. Total intensity image of G15.4+0.1 at 624 MHz Figure 1 shows the new image of SNR G15.4+0.1 obtained at 624 MHz using the GMRT. It constitutes the first ever highresolution and sensitivity image of this source at low radio frequencies. The shortest baseline available in the GMRT observations at 624 MHz is ∼197λ, which implies that spatial scales of the G15.4+0.1 emission up to ∼17.4 arcmin are well sampled in the resulting image. The radio emission from this remnant has the appearance of an irregular shell somewhat elongated in the Galactic north-east to south-west 4 direction (i.e. from upper left to lower right of Fig. 1) with an average diameter of 14.5 arcmin (i.e. ∼20 pc at the distance of 4.8 kpc derived in C13). The same overall total intensity morphology is also observed in the images of this remnant at 330 and 1420 MHz reported by Brogan et al. (2006) and in C13 based on VLA and GMRT observations, respectively. At 624 MHz, the emission is brighter and patchy along the Galactic northwestern rim and from the northeastern to the central part of the radio shell, overlapping the region where shocked CO gas has been identified (C13); see also Fig. 4 in current work. The bright spot at l = 15 • .52, b = 0 • .19 corresponds to the 624 MHz radio counterpart of the bipolar bubble HII region G015.520+0.188 (Anderson et al. 2011), for which Supan et al. (2014) have determined a distance of 3.0 ± 0.3 kpc based on the new GMRT observations and HI emission line data. The radio emission becomes fainter and uniform towards the southern part of G15.4+0.1. Unlike the north edge, no sharp boundary is apparent in the south limb of the remnant. A gap in the radio emission is also evident along the western edge of the remnant. To accurately determine the flux density integrated over the whole 624 MHz emission from G15.4+0.1, we subtracted the contribution of the aforementioned HII region and the bright knot located at l∼15 • .48, b∼0 • .20, which is likely to be an extragalactic source, previously identified as NVSS J181806−152321 (NED database) 5 . The integrated flux density for G15.4+0.1 based on the new GMRT observations at 624 MHz is 8.0 ± 1.1 Jy using the Perley & Butler (2013) flux density scale. The error in our estimate reflects the uncertainty both in the determination of background emission and in the selection of the area over which we integrate the remnant flux. Pulsar search The time-series data for both the incoherent and coherent array for each frequency were analyzed using SIGPROC 6 pulsar data analysis software using a 16 core 32-node High Performance Computing cluster (HPC) at the National Centre for Radio Astrophysics (NCRA, India). First, the channels affected by RFI were flagged from the data. The usable bandwidth of the observations was somewhat less than 33.33 MHz owing to the presence of narrowband RFI channels, which were removed, and was different for incoherent and phased array observations. These were then dedispersed to 1792 trial dispersion measures 7 (DMs) covering a range of 0-815 pc cm −3 at 624 MHz. The DMs were spaced more coarsely at 1404 MHz, owing to a small amount of dispersion smear across each channel. Data were dedispersed to 360 trial DMs at this frequency covering 0-1200 pc cm −3 . A harmonic search was performed on the dedispersed timeseries for each DM. Because of the non-contiguous nature of the time-series, an incoherent addition of spectra from individual time-series segments was carried out before the harmonic search. Known RFI periodicities, such as power-line frequencies at 50 and 100 Hz, were excised from the resultant spectra before the harmonic search. The resultant candidate periodicities were carefully examined using a visual representation of the search results as shown in Fig. 2. The search typically produced 60 to 70 candidates at each frequency for the phased array observations, while the number of candidates for the incoherent array were larger. No pulsations above a threshold signal-to-noise ratio (S/N) of 8 were detected in either phased array or incoherent array observations at both frequencies. The sensitivity of the pulsation search was assessed using observations of the three known pulsars, PSRs B1642−03, B1937+21, and J1901−0906, as mentioned in Section 2.2. A procedure similar to that outlined in Cañellas et al. (2012) was used for this purpose. The sensitivity of pulsation search depends on the duty cycle (ratio of pulse width to pulsar period) of the pulsar. As SNR G15.4+0.1 lies in the direction of the Galactic centre, any associated pulsar is likely to be scatter-broadened, particularly at 624 MHz, which affects the duty cycle of the pulsar. For example, using NE2001 models (Cordes & Lazio 2002) with a distance of 4.8 kpc to the SNR (C13), we get an estimate of temporal broadening of about 13 ms at 624 MHz, whereas this value is about 300 µs at 1404 MHz. Hence, we have plotted our sensitivity curves as a function of duty cycle in Fig. 3 to provide a more meaningful comparison in the presence of temporal broadening. The individual curves correspond to the incoherent array and the phased array observations at 624 and 1404 MHz, respectively. These curves provide 8 standard deviation upper limits, for pulsed emission with 10% duty cycle, of 250 and 300 µJy in the phased array observations at 624 and 1404 MHz, respectively, from any putative pulsar associated with the SNR G15.4+0.1/HESS J1818−154 system. Upper limits of 700 and 500 µJy in the incoherent array observations at 624 and 1404 MHz, respectively, are also implied for the five identified X-ray point sources within the extent of the SNR. We note that the incoherent array beam covers the entire remnant. A multiwavelength view of SNR G15.4+0.1 We have explored the spatial co-existence of synchrotron radio features and sites of emission at infrared and X-ray wave-A&A proofs: manuscript no. castelletti-astroph lengths. In order to compare radio and far-infrared (FIR) morphologies, we made use of archival data at 70 and 250 µm acquired using the SPIRE instrument on Herschel Space Observatory 8 (Griffin et al. 2010). Far-infrared images are a very useful tool with which to delineate shock heated dust, particularly in regions where the SNR shock front is hitting dense interstellar medium. The spatial correspondence between the new radio image at 624 MHz and the FIR images is shown in Fig. 4a; yellow contours are included to help locate the molecular cloud discovered by C13. To perform the radio/IR comparison, the 70 µm data were convolved to the same angular resolution as the 250 µm map (18 ′′ ). From Fig. 4a, the correlation between the CO molecular cloud and the spatial distribution of the emis-sion at IR wavelengths is clearly evident. Indeed, the elongated structures that appear in the FIR images is completely bounded within the dense CO structure detected at velocities between 46 and 50.3 km s −1 (C13). Moreover, significant enhancements in the dust emission at the two wavebands are located inside the two molecular clumps (referred to in C13 as clumps A and B). Such coupling between the dust emission in the FIR and the CO line emission provides additional support to the presence of dense molecular gas associated with the SNR. We thus interpret the infrared emission as marking the SNR-molecular cloud interface in G15.4+0.1. Evidence of a similar situation has also been observed in the northern shock front of Cas A, in the region where the remnant encounters a density enhancement in the interstellar medium (ISM) (Hines et al. 2004). To obtain a more complete description of the infrared distribution in G15.4+0.1, we also used data at 3.6, 4.5, 5.8, and 8 µm taken with the IRAC camera aboard the Spitzer Space Telescope along with MIPSGAL images at 24 µm. Aside from the striking correspondence observed at FIR wavebands with the CO cloud, no discernible emission was detected in the near-or mid-infrared wavebands, which correlates with either the radio surface brightness or the proposed PWN in X-rays (the comparisons are not shown here). This result contrasts with that observed in several PWNe in the Galaxy, for which there is a spatial agreement between synchrotron radio, X-rays, and near-infrared emission distribution (see e.g. the multiwavelength view of the PWNe associated with SNRs 3C 58, Slane et al. (2008); Crab Sandberg & Sollerman (2009); G21.5−0.9, Zajczyk et al. (2012); and G292.0+1.8, Zharikov et al. (2013)). The comparison of the emission in radio and X-ray bands was first presented by Abramowski et al. (2014) using the image of the remnant taken from the 330 MHz VLA survey of the Galactic plane (GPS, Brogan et al. 2005a) and XMM-Newton observations. Here, we redo this analysis in the light of the new GMRT data by combining the synchrotron emission at 624 MHz from the source, which shows a highly structured interior compared to that observed at the lower radio frequency, with XMM-Newton reprocessed data obtained from X-ray energies between 1.0 and 8.0 keV. The radio/X-ray morphological correlation is shown in Fig. 4b. Contours for the molecular gas interacting with the SNR (C13) together with the region where γ-ray emission was observed are also presented in the radio/X-ray comparison. Fig. 4b reveals little correspondence between both emitting plasmas. The X-rays associated with G15.4+0.1 do not show evidence of a limb-brightened X-ray shell, but most of the Xray emission appears broadly contained within the radio rims of G15.4+0.1. Moreover, the comparison shows X-ray emission extending slightly beyond the projected radio boundaries of the remnant. In addition, weak and diffuse X-ray emission is present filling the spatial gap observed in radio all along the western limb of the remnant. Figure 4b also demonstrates a significant lack of correspondence between the X-rays and the molecular material, delineated by yellow contours. This result may not be surprising if obscuration due to the molecular cloud detected between the observer and the remnant occurs. On the contrary, the morphology in X-rays overlaps part of the γ-ray emission area delineated by the white circle in Fig. 4b. Such a X-ray/γ ray correlation led Abramowski et al. (2014) to the conclusion that the X-ray emission is the counterpart to the TeV source originating in a pulsar wind nebula of a yet undiscovered pulsar. This interpretation, however, may not have a robust observational support since the authors remark that both thermal and non-thermal models are equally satisfactory to fit the spectrum of the diffuse X-rays in G15.4+0.1, which made somewhat uncertain the true origin of the detected X-ray emission. Finally, we found no radio counterparts corresponding to the five X-rays point sources detected using XMM-Newton observations at the location of the emission in γ rays (Abramowski et al. 2014). Radio spectral analysis in SNR G15.4+0.1 The integrated spectrum To determine the global spectral index (S ∝ ν α ) of G15.4+0.1, we have used the new flux density estimate at 624 MHz together with that obtained from GMRT data at 1420 MHz (C13) and measurements at 330, 1400, and 2700 MHz as determined from observations carried out with the VLA and Effelsberg telescopes (Brogan et al. 2006). In Table 1 we summarized the flux densities integrated over the whole extension of this remnant at the mentioned frequencies. The listed values constitute all the measurements published up to present for the source. A correction factor was applied to set all the values to the Perley & Butler (2013) absolute flux density scale. In particular, the correction to the flux density estimates at 330 and 624 MHz was made using the low frequency extension proposed by those authors. Figure 5 shows the resulting radio continuum spectrum of the remnant; the determinations from GMRT observations are plotted as open pentagons symbols. A weighted fit to all the available flux densities yields a spectral index α=−0.62 ± 0.03, which is compatible with the value derived previously by Brogan et al. (2006). This spectral behaviour is consistent with that measured in other shell type SNRs (Kothes et al. 2006). Spatial variations in the radio spectral index We have investigated the spectral changes as a function of the position within the remnant by using the image at 624 MHz and the one at 330 MHz (see panel (f) in Fig. 6) kindly provided by C. Brogan (Brogan et al. 2006). The analysis was done by constructing tomographic maps between them (Katz-Stone & Rudnick 1997). MHz, respectively, and α t represents a trial spectral index value. If the assumed spectral index coincides with the "true" value, one should expect the S t image to be equal to zero. On the contrary, features in the residual map that appear positive (negative) will be associated with local spectral indices that are steeper (flatter) than the assumed α t value. The technique has the advantage of delineating spectral features that can overlap along the line of sight while avoiding any zero level dependence. In Fig. 6, we reproduce the gallery of tomographic maps between 330 and 624 MHz. To construct them, the radio images were clipped at the 3σ level of their respective noise levels. In order to obtain a reliable determination of the spectral properties, the range of the spatial scales measured at 624 MHz was matched in the uv-coverage to that of the 330 MHz data. Additionally, the images were aligned and interpolated to identical projections to avoid positional offsets. The α t value shown in the maps varies from +0.1 to −0.7. For the sake of illustration the 330 MHz total intensity image of G15.4+0.1 is shown in the bottom right frame in Fig. 6. Spatial variations in the spectral index are recognizable across the synchrotron emission from G15.4+0.1. The central part of the remnant shows components of spectral index flatter (α varying between ∼+0.1 and −0.5) than the values in the ridge. We note that the very flat spectral component observed as a dark area in the southeastern border of the remnant (l∼15 • .40, b∼0 • .15), within which the spectral index appears to have posi- tive values (see panel corresponding to α t =+0.1 in Fig. 6), corresponds to a bordering clipped area in the map and should not be considered a real spectral component owing to the decrease in the signal-to-noise ratio observed in this region in the VLA 330 MHz image in comparison with the image at 624 MHz. In general, the central flattening traced in the 330/624 MHz comparison matches the zone where the interaction with the discovered CO molecular cloud is taking place (see Fig. 4). Variations in the spectrum are generally expected in regions where the SN shock is impacting high density interstellar material as a consequence of the density increase and the deceleration of the shock in the interacting regions. For instance, spectral flattening were found between 74 and 330 MHz for the two well-known SNRs/molecular clouds systems marking the ionized interface of molecular clouds interacting with the shock front in 3C 391 (Brogan et al. 2005b) and IC 443 (Castelletti et al. 2011). The tomographies also show a conspicuous bullet-like feature standing out as a bright spectral component (i.e. positive in the S t maps) located at the northeast part of the shell (l∼15 • .48, b∼0 • .25). It is characterized by a spectral index even steeper than −0.7 because it appears as a positive residual in all tomographic images. We found no catalogued extragalactic objects at the position of this feature. Along the westernmost edge (left side) of the SNR, there is an indication of a positive correlation between the local spectral index and the total radio intensity features in the sense that the bright ridge tends to be associated with steeper indices. A similar situation was found in young SNRs (see e.g. the analysis of SNRs G39.2−0.3 and G41.1−0.3 in Anderson & Rudnick 1993). In those cases, it was suggested that the radio brightness and spectra are being regulated by different mechanisms. In addition to this behaviour, most of the structures in the tomographies traced at α t =−0.5 and α t =−0.7 disappear against the background indicating that the spectral behaviour can be characterized by a mean spectral index ∼−0.6, which is consistent with the integrated spectral index of this SNR derived by fitting the total flux densities from 330 to 2700 MHz (see Sect. 4.1). Overall, the observed radio continuum spectral distribution across G15.4+0.1 is compatible with what is expected in the framework of diffusive shock acceleration models. Neutral gas distribution around G15.4+0.1 HI brightness morphology and physical parameters We explored the interstellar medium around G15.4+0.1 in the 21 cm line to distinguish some morphological correlations between the distribution of the neutral gas and the remnant that might account for the SNR shock effects on the surrounding medium. Figure 7 includes velocity channels maps from the SGPS data cube covering an interval from 41.6 to ∼108 km s −1 . All the velocities are measured with respect to the local standard of rest (LSR). Each frame was obtained by integrating the data cube every 8.2 km s −1 . An appropriate background level equal to the mean value of each integrated image was subtracted from every map. A contour line delineating the outer boundary of the radio continuum emission at 624 MHz from G15.4+0.1 was overlaid to the HI distribution for ease of comparison. The distribution of the neutral gas in the velocity range from 41.6 to 50 km s −1 only shows bright clouds spread over the field without any correspondence with the radio continuum shell of G15.4+0.1. A similar behaviour is found between 0 and 41.6 km s −1 and therefore, the images are not shown. In the images corresponding to velocities between 50 and 91.1 km s −1 , we observed the best morphological signature of neutral gas related to the SNR. Indeed, at ∼54 km s −1 , the HI distribution appears to be distorted at the position of the synchrotron emission from G15.4+0.1. Moreover, it is in the 58.1-91.1 km s −1 range where we see a cavity in the HI emission that closely matches G15.4+0.1. In particular, at velocities from 58.1 to 74.6 km s −1 , the HI distribution has an almost complete shell-like morphology along the eastern and western peripheries (left and right sides) of the remnant, which looks broken toward the northeastern border. The presence of this shell is compatible with the central systemic velocity v sys = 60 km s −1 derived in C13 for the SNR. The expanding shock wave of the SNR on the HI surroundings might be responsible for the creation of the observed HI structure. An alternative origin for the HI shell may be a pre-existing stellar wind bubble formed for instance by a group of OB stars, one of which could have been the SN precursor of G15.4+0.1. In this particular scenario, the brightest synchrotron emission observed L. Supan et al.: A complete radio study of SNR G15.4+0.1 from new GMRT observations Fig. 6. Tomographic images for SNR G15.4+0.1 constructed from data at 330 and 624 MHz. The data at both frequencies were matched to the same uv-coverage. The radio spectral index α t is indicated at the top right corner of the panels. The grey colour bar in the middle column represents the scale of the residuals components, which is the same for all tomographies. In the greyscale representation, spectral components for which α is steeper than the assumed α t are shown as light regions (positive residuals), while flatter components are seen as dark features (negative residuals). A contour delineating the radio continuum emission at 330 MHz is included for reference. The total intensity image of G15.4+0.1 at 330 MHz is plotted in the bottom right frame (Brogan et al. 2006). towards the northwest face of the remnant could be where the shock is interacting with the inner wall of the bubble. Furthermore, the fact that 13 CO emission was detected on the shell's central face of G15.4+0.1 at approximately 50 km s −1 (C13) may indicate that the molecular cloud is outside the front wall of atomic hydrogen; perhaps the inner wall is also the place where the synchrotron emission originates. Examples of bubbles created by the action of the precursor's stellar winds, inside of which evolves a SNR shock wave, are those discovered surrounding the remnants Kes 79 (Giacani et al. 2009), 3C 434.1 (Foster et al. 2004), G106.3+2.7 (Kothes et al. 2001). In any case, whether the HI shell is formed by neutral gas accelerated by the SNR shock or by stellar winds from the SN progenitor, the X-ray emission detected inside the radio shell of the SNR reinforces the idea that the center has been evacuated. It is also worth noting that at higher velocities there is no evidence of the naturally expected cap-like features associated with an HI shell. The non-detection of possible caps does not imply that these structures do not exist. They could be undetected because of confusion caused by unrelated foreground and background inhomogeneities in the ISM. The caps, produced by accelerated HI gas, have been observed either in emission as central concentrations projected within the remnant (Velázquez et al. 2002) or in HI self-absorption in an expanding stellar wind bubble wall (Foster et al. 2004). At velocities higher than ∼91.1 km s −1 , the 21 cm line emission does not correlate with the SNR, and neutral gas is preferentially oriented parallel to the Galactic plane. In what follows, we derive a number of parameters characterizing the HI shell. For this structure, centered at l∼15 • .42, b∼0 • .16, we assigned a radius of ∼9 ′ .8 or ∼13.7 pc at a distance of 4.8 kpc. If we assume that the HI emission is optically thin, the integration for velocities between 50 and 91 km s −1 , for which we observed the most evident signatures of associa-tion between the HI gas and the SNR, yields an average column density N H ≃ 4 × 10 20 cm −2 . The total mass of atomic gas that forms the shell is about 1.9 × 10 3 M ⊙ . This value is in fact an upper limit due to the impossibility of disentangling related and unrelated HI gas. If the HI shell was formed by ISM swept up by the SNR shock and assuming that before the SN event the total neutral mass was uniformly distributed inside the volume of the HI shell, the derived mean density of hydrogen nuclei in the ambient environment is η 0 ∼7 cm −3 , which is larger than the typical interstellar hydrogen densities of ∼1 cm −3 (averaged over the cold, warm, and hot gas-phase constituents of the ISM, McKee & Ostriker 1977). Our result corresponds to a reasonable kinetic energy supplied by the SN explosion into the the surrounding medium of about 7.6 × 10 48 erg. This value was estimated by adopting an expansion velocity for the HI shell of ∼20 km s −1 , based on the systemic velocity of the neutral gas derived in C13 (v sys ∼60 km s −1 ) and the HI-continuum correspondences seen between ∼50 and 91 km s −1 . Taking into account errors in the selection of the boundaries for integration, confusion from background or foreground unrelated HI gas, as well as uncertainties in the determination of the distance, we derived the parameters for the HI shell with a mean relative error of 45%. As a by-product, on the basis of the new radio continuum image at 624 MHz and the results derived from the HI data, we calculated physical parameters of the remnant G15.4+0.1. By combining the ambient density η 0 ∼7 cm −3 , the radius of G15.4+0.1 R SNR ∼7 ′ .2 (∼10.1 pc at the distance of 4.8 kpc) with the velocity of the shock, v sh , we can roughly calculate the energy released by the SN explosion to the ISM from the expression given in the model discussed by Chevalier (1974), E SN = 5.3 × 10 43 η 1.12 0 v 1.40 sh R 3.12 SNR erg.(1) A&A proofs: manuscript no. castelletti-astroph To estimate the shock velocity, we used v sh = 66.5 ( R SNR 21.9 ) −2.23 km s −1 and obtained v sh = 370 km s −1 for a shock radius set to R SNR ∼10.1 pc. We understand that this value represents a reliable bound for the velocity of the SN shock before impacting the dense molecular cloud located in front of G15.4+0.1; the velocity range of 46-50.3 km s −1 for the cloud corresponds to a distance of ∼4.2 kpc (C13). To test the robustness of our interpretation, we also estimated the shock velocity using the physical parameters derived for the 13 CO cloud interacting with SNR G15.4+0.1, as calculated with the expression v sh = v cl (η cl /η int ) 1/2 , where η cl and v cl are the density and the expansion velocity of cloud, respectively, and η int is the intercloud density assumed to be 1 cm −3 (Dubner et al. 1999). In C13, it was demonstrated that the CO cloud interacting with G15.4+0.1 has a density of ∼1.5 ×10 3 cm −3 and it is expanding at ∼10 km s −1 . In these circumstances, we derived a shock velocity of v sh ≃ 380 km s −1 , which agrees within the uncertainties of the method with that obtained using HI data. Finally, for an average shock velocity of 375 km s −1 and using the ambient density along with the radius of the shell derived from our study, we obtain an initial energy of ∼2.6 ×10 51 erg, which agrees with the standard value. To account for the age of G15.4+0.1, we employed the dynamical evolution model of Chevalier (1974) and obtained t ≃ ( R S NR 21.9 ) 3.23 10 5 ≃ 8.2 ×10 3 yr, where we used a radius for the SNR of R S NR ∼10.1 pc at a distance of about 4.8 kpc. This result differs from that calculated by Abramowski et al. (2014) (2.5×10 3 yr) based on a Sedov-Taylor and the assumption of an ambient density around G15.4+0.1 of 1 cm −3 . Modelling the broadband spectrum of G15.4+0.1 The analysis in Abramowski et al. (2014) to determine the origin of the γ-ray emission in the region of G15.4+0.1 excludes a hadronic contribution. In modelling the spectral energy distribution (SED), these authors used radio data at 330 and 1400 MHz from the GPS (Brogan et al. 2005a) and the Multi-Array Galactic Plane Imaging Survey (MAGPIS, Helfand et al. 2006), respectively, whose fluxes were treated as upper limits, X-ray data from XMM-Newton, and γ-ray observations obtained with H.E.S.S. In this section, we first demonstrate that, with the available data, a hadronic model in which the TeV γ-ray flux is produced through the interaction of accelerated protons with ambient protons, mainly from the molecular cloud located foreground G15.4+0.1, is still compatible with the broadband spectrum of the G15.4+0.1/HESS J1818−154 system. In addition, in the light of the new GMRT data, we revisited the leptonic model presented in Abramowski et al. (2014) to fit the multiwavelength spectral data. The energy spectrum of the accelerated particles (electrons and protons) in the emission region is described by a power law with an exponential cut-off, dn e,p dE = K e,p E −Γ e,p exp −E/E cut e,p ,(2) where the subscripts e and p refer to the particle species (electrons or protons), and E, K, Γ, and E cut are the particle energy, a normalization constant, the spectral index, and the cut-off energy, respectively. We refer to Aharonian et al. (2010) and references therein for calculation details of the synchrotron emissivity produced by electrons. Regarding the inverse Compton (IC) and the non-thermal Bremsstrahlung emission, we followed the method given in Jones (1968) for the former, while the method presented in Baring et al. (1999) was used to model the electronelectron Bremsstrahlung interaction and in Koch & Motz (1959) and Sturner et al. (1997) for the electron-ion Bremsstrahlung process. On the other hand, we determine the γ-ray spectrum due to the neutral pion decays by using the ppfrag program (Kachelrie s & Ostapchenko 2012). In the calculations, the low and high energy parts of the spectrum were separately considered. For energies of the γ rays smaller than 50 GeV, we used the parametrization presented in Kamae et al. (2007), while for larger values of energy we based our calculations in the interaction model QGSJET-II-04 9 (Ostapchenko 2011). The XMM-Newton observations show faint X-ray emission, about 9 arcmin in size, which partially coincides with the size of the HESS J1818−154 source (see Fig. 4). The origin of the X-rays was considered non-thermal and based on their spatial correspondence with the γ rays, an association between them was suggested (Abramowski et al. 2014). We model the spectrum of G15.4+0.1 from radio to the TeV γ rays by considering a non-thermal nature of the X-ray emission. This implies that the whole X-ray flux and part of the radio flux is produced by synchrotron radiation of electrons in the turbulent magnetic field in the region interior to the SNR shell, the PWN proposed in Abramowski et al. (2014). We refer to this zone as Region I. The remaining part of the radio flux is also generated by synchrotron radiation, but corresponding to a different population of electrons that are placed in the shell and referred to hereafter as Region II. Figure 8 shows the broadband spectrum for the SNR G15.4+0.1/HESS J1818−154 system. The radio data for Regions I and II include the observations presented in Brogan et al. (2006) and the new GMRT data at 624 and 1420 MHz. The spectrum also includes the best fit to the X-ray data using a power law taken from Abramowski et al. (2014), and upper limit γ-ray fluxes obtained by Fermi-LAT (Acero et al. 2013) and H.E.S.S (Abramowski et al. 2014). Although upper limits on the fluxes at GeV energies are represented, they were not used in fitting the spectrum as they are too far from HESS data. The best fit to the radio data corresponding to the interior SNR region is shown by a dashed curve labelled I. The intensity of the magnetic field and the cut-off energy are fixed during the fitting procedure in order to be consistent with the power-law fit to the X-ray data. The normalization constant K and the spectral index Γ in Eq. (2) are taken as free fit parameters. The intensity of the magnetic field used is B I = 25 µG and the cutoff energy is log(E I cut,e /eV) = 12.83. These values are chosen in such a way that the inverse Compton component has a subdominant role at very high energies. It is worth mentioning that scenarios with larger values of the magnetic field intensity and smaller cut-off energies of the electron component, such that (E I cut,e ) 2 × B I = 1.1536 × 10 21 eV 2 G, also describe properly the X-ray data, making the inverse Compton component even smaller. The best-fit model for Region I yields a spectral index Γ I e = 2.35 ± 0.14. We also considered a contribution to the leptonic mechanism from accelerated electrons in the SNR shell. The radio flux, corresponding to the shell region, is obtained by fitting the total radio flux, by using the best fit to the radio data corresponding to the internal region. In this case, we fixed the cut-off energy and the intensity of the magnetic field to log(E II cut,e /eV) = 11.6 and B II = 62.5 µG, respectively. This choice of the parameters prevents the synchrotron flux that originated in the shell region from contributing to the X-ray part of the SED. We note that this combination of the two parameters is not unique. Smaller values of the cut-off energy, which are consistent with the data, produce smaller values of the high energy flux originated by the inverse Compton process. The spectral index obtained from the fit is Γ II e = 2.16 ± 0.08. It is evident that in our model the inverse Compton process alone cannot explain the emission at TeV energies from Region I or II. Figure 8 also includes the spectrum corresponding to the Bremsstrahlung radiation. In this case, we assumed that only Region II has enough density to produce a non-negligible flux. The calculation was done by fixing the proton density, both molecular and atomic, to the value n = 6.4 cm −3 averaged over Region II. We note that, as in the IC model, it is possible to reduce even more the Bremsstrahlung effects diminishing the value of the cut-off energy of the accelerated electrons of the shell region. It is worth mentioning that the magnetic field intensity of the two regions required to describe the data is much larger than the field corresponding to the interstellar medium, for which its typical strength is 1-2 µG (Foster et al. 2013, and references therein). This is typical in hadronic models of the γ-ray emission (see e.g. Cardillo et al. 2014). We determined the hadronic contribution of the γ-ray flux resulting from the neutral pion decay of accelerated protons (il-lustrated by a solid line labelled as π 0 in Fig. 8). Since neither the H.E.S.S. data nor the Fermi-LAT upper limits are enough to constrain the spectral index of the proton component, we used the spectral index of the electron component fitted for the synchrotron emission in Region II, that is Γ II p = 2.16. Taking into account the normalization constant and the cut-off energy as free fit parameters, we obtained for the proton component a cut-off energy log(E cut,p /eV) = 14.6 ± 0.5. On the other hand, we revisited the one-zone model presented in Abramowski et al. (2014) replacing the upper limits on the radio fluxes considered in that work by reliable flux density measurements of the radio synchrotron emission in the interior of G15.4+0.1 (called Region I in the current work). This region is spatially coincident with the X-ray emission attributed by those authors to the same electron population responsible for the γ-ray emission through the IC process. In this case as well, a power law with an exponential cut-off was assumed to model the spectral energy distribution of electrons. As was done for the hadronic scenario, we imposed the condition (E cut,e ) 2 × B = 1.1536 × 10 21 eV 2 G, which ensures an adequate description of the X-ray data. We would like to note that also in this case the Fermi-LAT upper limits do not constrain the model. The corresponding spectrum is shown in Fig. 9. For this simple model, we obtain a spectral index Γ e = 2.41 ± 0.09, a A&A proofs: manuscript no. castelletti-astroph Brogan et al. 2006, and this work). The shadowed region encompassing a solid line corresponds to the fitted power law and the one sigma region of the X-ray data. The HESS TeV data points are indicated by filled circles (Abramowski et al. 2014), while the arrows pointing downwards show the upper limits in the Fermi-LAT spectrum (Acero et al. 2013). The dashed lines labelled I and II mark the spectra from Region I, assuming non-thermal X-ray and radio emission originating in the interior of G15.4+0.1, and from Region II in the shell of the remnant, respectively. The modelled spectra from the synchrotron radiation by electrons, inverse Compton on the CMB, Bremsstrahlung, and γ rays produced by neutral pions generated in proton-proton interactions are indicated by the curves labelled S, IC, B, and π 0 , respectively. magnetic field B = 3.9 ± 1.4 µG, and a cut-off energy of electrons log(E cut,e /eV) = 13.303 ± 0.071. The fit corresponding to this scenario is as good as the one obtained in Abramowski et al. (2014). However, the fitted parameters derived in this work are restricted to smaller intervals. This is due to the radio data points of the internal region used to fit the SED. Fig. 9. Broadband fit to radio (filled triangles) and γ-ray observations obtained with H.E.S.S. (filled circles). Upper limits to the flux densities from Fermi-LAT (arrows pointing downwards) and the best fit to the Xray data (green hatching) are also included. The radio data correspond to the internal region of SNR G15.4+0.1. The curves labelled S and IC show the model of the synchrotron radiation and inverse Compton mechanism on the CMB by accelerated electron, respectively. The observational data is compatible with both scenarios analyzed here; more data is needed, especially in the energy region relevant to the Fermi-LAT observatory, in order to make progress in the understanding of the most important physical processes taking place in this source. In any case, it is important to emphasize that a hadronic scenario is compatible with the observational data. Conclusions We presented new full-synthesis imaging of the SNR G15.4+0.1 obtained from observations at 624 MHz carried out with the GMRT. We measured an integrated flux density S = 8.0 ± 1.1 Jy at 624 MHz. Based on the combination of our estimate with those previously published, after bringing all values to the same absolute flux density scale, we derive a spectral index α=−0.62± 0.03 for the whole remnant. The estimated global spectral index is consistent with measurements for other typical SNR shells (Kothes et al. 2006). The new GMRT data were used to analyze the correspondences with FIR and X-ray observations towards the remnant. We found an impressive correlation between the infrared emission detected by Herschel at 70 and 250 µm and the cloud of 13 CO colliding with the SNR shock reported in C13. In other regions of G15.4+0.1, however, no IR counterpart is observed to the radio emission. On the other hand, from the comparison of the new 624 MHz image with re-processed XMM-Newton observations between 1 and 8 keV, we can confirm that the synchrotron radio emission surrounds the low surface brightness X-ray radiation. No counterparts were observed in radio to the five point-like sources detected in the XMM-Newton field. We also conducted a search for a neutron star associated with G15.4+0.1 through time-series observations performed with the GMRT, which provided negative results. The analysis of the spatial spectral index variations made between existing observations of G15.4+0.1 at 330 and the new GMRT data at 624 MHz revealed a radio spectrum steepening from the weak interior to the brighter periphery of G15.4+0.1. A similar spectral behaviour is also found in other young SNRs. The overall distribution of the local spectral radio index could be described by diffusive shock acceleration by considering the effect of the ISM gas. We did not recognize any distinct feature in the radio brightness at 624 MHz or in the 330/624 spectral distribution that could be considered a synchrotron nebula, powered by the wind of a yet undetected pulsar. Furthermore, our search for radio pulsations inside the synchrotron shell did not reveal a pulsar, with the upper limits on the mean flux density of any associated putative pulsar of 250 and 300 µJy at 624 and 1404 MHz, respectively, towards the centroid of the TeV source. In addition, the flux density upper limits that we estimated for any putative pulsar morphologically coincident with SNR G15.4+0.1, including the five point-like X-ray sources reported by Abramowski et al. (2014), were of about 700 and 500 µJy at 624 and 1404 MHz, respectively . Our study of the neutral ambient gas in the direction of G15.4+0.1 unveiled a large (about 13.7 pc in radius at a distance of 4.8 kpc) and incomplete HI shell-type structure. We determined a mass for this shell of 1.9 × 10 3 M ⊙ and a kinetic energy of 7.6 × 10 48 erg. On the basis of this study and the new SNR image at 624 MHz, we estimated that G15.4+0.1 was created as the result of 2.6 × 10 51 erg released in a SN explosion that occurred about 8.2 × 10 3 yr ago. Finally, we discussed different models to fit the broadband emission from the SNR G15.4+0.1/HESS J1818−154 system. In particular, we demonstrated that taking into consideration the multiwavelength data known to date, a purely hadronic picture is sufficient to account for the γ-ray data. This is in contrast to the explanation given in Abramowski et al. (2014) for which the X-ray and TeV emission have the same origin. We recognize that more comprehensive X-ray and γ-ray data (especially in the MeV-GeV band) are required to improve the constraints on the observed SED of the system and distinguish between leptonic and hadronic explanations for the origin of HESS J1818−154. Fig. 1 . 1Total intensity image of SNR G15.5+0.1 at 624 MHz obtained with the GMRT. It does not include primary beam correction. The image was smoothed to an angular resolution of 10 ′′ , resulting in an rms noise level of 0.48 mJy beam −1 . The unit of the values in the colour bar is mJy beam −1 . Fig. 2 . 2Visual representation of a pulsar search candidate. The plot shows the detection of the known pulsar PSR J1901−0906, following the procedure described in the text. The bottom left panel shows the peak in the S/N curve over a large range of dispersion measures (DM) values, whereas the top left panel indicates the peak in the S/N with a refined period and DM. The top right plot shows a greyscale representation of intensity as a function of pulse phase in abscissa and the bandpass in ordinate at the refined period and DM, while the bottom right plot shows the averaged profile of the pulsar. Two rotations are shown for clarity in this plot. This representation helps distinguish mistaken RFI candidates from genuine candidate pulsars. Fig. 3 . 3Sensitivity curves for the pulsational search. The plot depicts the 8 standard deviation upper limits obtained from our observations for incoherent array at 1404 MHz (solid black), phased array at 1404 MHz (dashed-dot black), incoherent array at 624 MHz (solid blue), and phased array at 624 MHz (dashed-dot blue) as a function of duty cycle for a possible pulsar. Fig. 5 . 5Integrated radio continuum spectrum of SNR G15.4+0.1. The open symbols correspond to GMRT flux densities measurements at 624 and 1404 MHz. The rest of the data were taken from Brogan et al. (2006). All the data were tied to a common flux density scale (Perley & Butler 2013). The solid line shows the best fit obtained with a single power law to all the values, which yields a global spectral index α=−0.62 ± 0.03. Basically, the procedure involves the generation of a series of residual images S t according to the expression S t = S 330 − 330 624 α t S 624 , where S 330 and S 624 are the maps at 330 and 624 Fig. 4 . 4Multiwavelength view of SNR G15.4+0.1. a) A color composite image showing the new radio image at 624 MHz of G15.4+0.1 (in blue) and FIR emission as observed by Herschel at 70 µm (in red), and 250 µm (in green); b) Comparison of the radio continuum emission at 624 MHz of the SNR (in blue) and the X-ray emission (in yellow) as observed by XMM-Newton in the 1-8 keV energy range. The dashed white circle indicates the size and location of the TeV source HESS J1818−154. In both panels, the yellow contours delineate the 13 CO line emission in the 46-50.3 km s −1 range (C13). Fig. 7 . 7Images of 21 cm line emission in the direction of SNR G15.4+0.1. Each map corresponds to the integration of the neutral gas every 8.2 km s −1 . The range of velocities is indicated at the top right corner of the panels. The colour display was kept for all images between 633 and 1210 K km s −1 . The 0.7 mJy beam −1 contour from the 35 ′′ resolution 624 MHz image is included in each panel to facilitate the comparison between the radio continuum emission from the remnant and its surroundings. Fig. 8 . 8Spectral energy distribution of the SNR G15.4+0.1. The filled squares and triangles with error bars correspond to the emission from the whole remnant and from the interior of the shell (Region I), respectively (data from Table 1 . 1Integrated flux densities for G15.4+0.1.Frequency Scaled References (MHz) flux density (Jy) 330 11.3 ± 0.3 Brogan et al. (2006) 624 8.0 ± 1.1 This work 1400 4.7 ± 0.8 Brogan et al. (2006) 1420 4.7 ± 0.2 C13 2700 2.9 ± 0.3 Brogan et al. (2006) For more detailed information regarding the flux density calibrators at VLA see the explanation of AIPS task SETJY or the website at http://www.nrao.edu/doc/vla/html/calib.shtml XMM-NewtonScience Archive (XSA) http://xmm.esac.esa.int/xsa/index.shtml 3 More information available at http://cxc.harvard.edu/ciao/ 4 Although we do not utilize the equatorial coordinate system, we refer hereafter to north and south to set out locations around the Galactic plane.Article number, page 2 of 11 L.Supan et al.: A complete radio study of SNR G15.4+0.1 from new GMRT observations NED (the NASA/IPAC Extragalactic Database) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. http://sigproc.sourceforge.net 7 The integrated electron column density along the line of sight in units of pc cm −3 . Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. The last version of the QGSJET model was updated with recent Large Hadron Collider data. Acknowledgements. We wish to acknowledge the referee for his very fruitful comments that improved our work. We thank C. Brogan for supplying us with the VLA 330 MHz image. This research has been funded by Argentina grants ANPCYT-PICT 0795/08 and 0571/11. G. C. is Member of the Carrera of Investigator Científico of CONICET, Argentina. L. S. is a Ph.D. Fellow of CON-ICET, Argentina. The GOODS-Herschel data was accessed through the Herschel Database in Marseille (HeDaM -http://hedam.lam.fr) operated by CeSAM and hosted by the Laboratoire d'Astrophysique de Marseille. We thank the staff of the GMRT, who have made these observations possible. GMRT is operated by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research. This work made use of the High Performance Computing facility, funded by grant 12-R&D-TFR-5.02-0711, at National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research. . A Abramowski, F Aharonian, F Ait Benkhali, A&A. 56240Abramowski, A., Aharonian, F., Ait Benkhali, F., et al. 2014, A&A, 562, A40 . F Acero, M Ackermann, M Ajello, ApJ. 77377Acero, F., Ackermann, M., Ajello, M., & et al. 2013, ApJ, 773, 77 . F A Aharonian, S R Kelner, A Y Prosekin, Phys. Rev. D. 8243002Aharonian, F. A., Kelner, S. R., & Prosekin, A. Y. 2010, Phys. Rev. D, 82, 043002 L D Anderson, T M Bania, D S Balser, R T Rood, VizieR Online Data Catalog. 21940032Anderson, L. D., Bania, T. M., Balser, D. S., & Rood, R. T. 2011, VizieR Online Data Catalog, 219, 40032 . M C Anderson, L Rudnick, ApJ. 408514Anderson, M. C. & Rudnick, L. 1993, ApJ, 408, 514 . M G Baring, D C Ellison, S P Reynolds, I A Grenier, P Goret, ApJ. 513311Baring, M. G., Ellison, D. C., Reynolds, S. P., Grenier, I. A., & Goret, P. 1999, ApJ, 513, 311 C L Brogan, B M Gaensler, Y Gelfand, X-Ray and Radio Connections. L. O. Sjouwerman & K. K. Dyer4Brogan, C. L., Gaensler, B. M., Gelfand, Y., et al. 2005a, in X-Ray and Radio Connections, ed. L. O. Sjouwerman & K. K. Dyer, 4 . C L Brogan, J D Gelfand, B M Gaensler, N E Kassim, T J W Lazio, ApJ. 63925Brogan, C. L., Gelfand, J. D., Gaensler, B. M., Kassim, N. E., & Lazio, T. J. W. 2006, ApJ, 639, L25 . C L Brogan, T J Lazio, N E Kassim, K K Dyer, AJ. 130148Brogan, C. L., Lazio, T. J., Kassim, N. E., & Dyer, K. K. 2005b, AJ, 130, 148 . A Cañellas, B C Joshi, J M Paredes, A&A. 543122Cañellas, A., Joshi, B. C., Paredes, J. M., et al. 2012, A&A, 543, A122 . M Cardillo, M Tavani, A Giuliani, A&A. 56574Cardillo, M., Tavani, M., Giuliani, A., et al. 2014, A&A, 565, A74 . G Castelletti, G Dubner, T Clarke, N E Kassim, A&A. 53421Castelletti, G., Dubner, G., Clarke, T., & Kassim, N. E. 2011, A&A, 534, A21 . G Castelletti, L Supan, G Dubner, B C Joshi, M P Surnis, A&A. 557C1315Castelletti, G., Supan, L., Dubner, G., Joshi, B. C., & Surnis, M. P. 2013, A&A, 557, L15 (C13) . R A Chevalier, ApJ. 188501Chevalier, R. A. 1974, ApJ, 188, 501 J M Cordes, T J W Lazio, astro- ph/0207156ArXiv Astrophysics e-prints. Cordes, J. M. & Lazio, T. J. W. 2002, ArXiv Astrophysics e-prints, astro- ph/0207156 . G Dubner, E Giacani, E Reynoso, AJ. 118930Dubner, G., Giacani, E., Reynoso, E., et al. 1999, AJ, 118, 930 . T Foster, R Kothes, J C Brown, ApJ. 77311Foster, T., Kothes, R., & Brown, J. C. 2013, ApJ, 773, L11 . T Foster, D Routledge, R Kothes, A&A. 41779Foster, T., Routledge, D., & Kothes, R. 2004, A&A, 417, 79 . E Giacani, M J S Smith, G Dubner, A&A. 507841Giacani, E., Smith, M. J. S., Dubner, G., et al. 2009, A&A, 507, 841 . M J Griffin, A Abergel, A Abreu, A&A. 5183Griffin, M. J., Abergel, A., Abreu, A., et al. 2010, A&A, 518, L3 . D J Helfand, R H Becker, R L White, A Fallon, S Tuttle, AJ. 1312525Helfand, D. J., Becker, R. H., White, R. L., Fallon, A., & Tuttle, S. 2006, AJ, 131, 2525 . D C Hines, G H Rieke, K D Gordon, ApJS. 154290Hines, D. C., Rieke, G. H., Gordon, K. D., et al. 2004, ApJS, 154, 290 P Hofverberg, R C G Chaves, J Méhault, M De Naurois, International Cosmic Ray Conference. 7247Hofverberg, P., Chaves, R. C. G., Méhault, J., & de Naurois, M. 2011, in Interna- tional Cosmic Ray Conference, Vol. 7, International Cosmic Ray Conference, 247 . F C Jones, Physical Review. 1671159Jones, F. C. 1968, Physical Review, 167, 1159 . M Kachelrie S, S Ostapchenko, Phys. Rev. D. 8643004Kachelrie s , M. & Ostapchenko, S. 2012, Phys. Rev. D, 86, 043004 . T Kamae, N Karlsson, T Mizuno, T Abe, T Koi, ApJ. 662779Kamae, T., Karlsson, N., Mizuno, T., Abe, T., & Koi, T. 2007, ApJ, 662, 779 . D M Katz-Stone, L Rudnick, ApJ. 479258Katz-Stone, D. M. & Rudnick, L. 1997, ApJ, 479, 258 . H W Koch, J W Motz, Reviews of Modern Physics. 31920Koch, H. W. & Motz, J. W. 1959, Reviews of Modern Physics, 31, 920 . R Kothes, K Fedotov, T J Foster, B Uyanıker, A&A. 4571081Kothes, R., Fedotov, K., Foster, T. J., & Uyanıker, B. 2006, A&A, 457, 1081 . R Kothes, B Uyaniker, S Pineault, ApJ. 560236Kothes, R., Uyaniker, B., & Pineault, S. 2001, ApJ, 560, 236 . N M Mcclure-Griffiths, J M Dickey, B M Gaensler, ApJS. 158178McClure-Griffiths, N. M., Dickey, J. M., Gaensler, B. M., et al. 2005, ApJS, 158, 178 . C F Mckee, J P Ostriker, ApJ. 218148McKee, C. F. & Ostriker, J. P. 1977, ApJ, 218, 148 . S Ostapchenko, Phys. Rev. D. 8314018Ostapchenko, S. 2011, Phys. Rev. D, 83, 014018 . R A Perley, B J Butler, ApJS. 20419Perley, R. A. & Butler, B. J. 2013, ApJS, 204, 19 . A Sandberg, J Sollerman, A&A. 504525Sandberg, A. & Sollerman, J. 2009, A&A, 504, 525 . P Slane, D J Helfand, S P Reynolds, ApJ. 67633Slane, P., Helfand, D. J., Reynolds, S. P., et al. 2008, ApJ, 676, L33 . S J Sturner, J G Skibo, C D Dermer, J R Mattox, ApJ. 490619Sturner, S. J., Skibo, J. G., Dermer, C. D., & Mattox, J. R. 1997, ApJ, 490, 619 . L Supan, G Castelletti, G Dubner, M P Surnis, B C Joshi, 56315Boletin de la Asociacion Argentina de Astronomia La Plata ArgentinaSupan, L., Castelletti, G., Dubner, G., Surnis, M. P., & Joshi, B. C. 2014, Boletin de la Asociacion Argentina de Astronomia La Plata Argentina, 56, 315 . M J L Turner, A Abbey, M Arnaud, A&A. 36527Turner, M. J. L., Abbey, A., Arnaud, M., et al. 2001, A&A, 365, L27 . P F Velázquez, G M Dubner, W M Goss, A J Green, AJ. 1242145Velázquez, P. F., Dubner, G. M., Goss, W. M., & Green, A. J. 2002, AJ, 124, 2145 . A Zajczyk, Y A Gallant, P Slane, A&A. 54212Zajczyk, A., Gallant, Y. A., Slane, P., et al. 2012, A&A, 542, A12 . S V Zharikov, D A Zyuzin, Y A Shibanov, R E Mennickent, A&A. 554120Zharikov, S. V., Zyuzin, D. A., Shibanov, Y. A., & Mennickent, R. E. 2013, A&A, 554, A120
[]
[ "Intelligent Traffic Light Control Using Distributed Multi-agent Q Learning", "Intelligent Traffic Light Control Using Distributed Multi-agent Q Learning" ]
[ "Ying Liu \nFujitsu Laboratories of America, Inc\nSunnyvaleCAUSA\n\nElectrical and Computer Engineering\nRutgers University\nNew BrunswickNJUSA\n", "Lei Liu \nFujitsu Laboratories of America, Inc\nSunnyvaleCAUSA\n", "Wei-Peng Chen \nFujitsu Laboratories of America, Inc\nSunnyvaleCAUSA\n" ]
[ "Fujitsu Laboratories of America, Inc\nSunnyvaleCAUSA", "Electrical and Computer Engineering\nRutgers University\nNew BrunswickNJUSA", "Fujitsu Laboratories of America, Inc\nSunnyvaleCAUSA", "Fujitsu Laboratories of America, Inc\nSunnyvaleCAUSA" ]
[]
The combination of Artificial Intelligence (AI) and Internet-of-Things (IoT), which is denoted as AI powered Internet-of-Things (AIoT), is capable of processing huge amount of data generated from large number of devices and handling complex problems in social infrastructures. As AI and IoT technologies are becoming mature, in this paper, we propose to apply AIoT technologies for traffic light control, which is an essential component for intelligent transportation system, to improve the efficiency of smart city's road system. Specifically, various sensors such as surveillance cameras provide real-time information for intelligent traffic light control system to observe the states of both motorized traffic and non-motorized traffic. In this paper, we propose an intelligent traffic light control solution by using distributed multi-agent Q learning, considering the traffic information at the neighboring intersections as well as local motorized and non-motorized traffic, to improve the overall performance of the entire control system. By using the proposed multi-agent Q learning algorithm, our solution is targeting to optimize both the motorized and non-motorized traffic. In addition, we considered many constraints / rules for traffic light control in the real world, and integrate these constraints in the learning algorithm, which can facilitate the proposed solution to be deployed in real operational scenarios. We conducted numerical simulations for a real-world map with realworld traffic data. The simulation results show that our proposed solution outperforms existing solutions in terms of vehicle and pedestrian queue lengths, waiting time at intersections, and many other key performance metrics.
10.1109/itsc.2017.8317730
[ "https://arxiv.org/pdf/1711.10941v1.pdf" ]
602,011
1711.10941
5720a2170a31b70eacd66721ab18c0a9d944dd1f
Intelligent Traffic Light Control Using Distributed Multi-agent Q Learning Ying Liu Fujitsu Laboratories of America, Inc SunnyvaleCAUSA Electrical and Computer Engineering Rutgers University New BrunswickNJUSA Lei Liu Fujitsu Laboratories of America, Inc SunnyvaleCAUSA Wei-Peng Chen Fujitsu Laboratories of America, Inc SunnyvaleCAUSA Intelligent Traffic Light Control Using Distributed Multi-agent Q Learning Index Terms-Reinforcement learningQ learningtraffic light controlnon-motorized traffic The combination of Artificial Intelligence (AI) and Internet-of-Things (IoT), which is denoted as AI powered Internet-of-Things (AIoT), is capable of processing huge amount of data generated from large number of devices and handling complex problems in social infrastructures. As AI and IoT technologies are becoming mature, in this paper, we propose to apply AIoT technologies for traffic light control, which is an essential component for intelligent transportation system, to improve the efficiency of smart city's road system. Specifically, various sensors such as surveillance cameras provide real-time information for intelligent traffic light control system to observe the states of both motorized traffic and non-motorized traffic. In this paper, we propose an intelligent traffic light control solution by using distributed multi-agent Q learning, considering the traffic information at the neighboring intersections as well as local motorized and non-motorized traffic, to improve the overall performance of the entire control system. By using the proposed multi-agent Q learning algorithm, our solution is targeting to optimize both the motorized and non-motorized traffic. In addition, we considered many constraints / rules for traffic light control in the real world, and integrate these constraints in the learning algorithm, which can facilitate the proposed solution to be deployed in real operational scenarios. We conducted numerical simulations for a real-world map with realworld traffic data. The simulation results show that our proposed solution outperforms existing solutions in terms of vehicle and pedestrian queue lengths, waiting time at intersections, and many other key performance metrics. I. INTRODUCTION In the upcoming Internet-of-Things (IoT) era, there will be many complex systems, networks or social infrastructures, connecting huge number of devices which will generate huge amount of data. To manage these devices and data, as well as provide intelligent control in such complex scenarios, artificial intelligence (AI), for its capability and potential for handling complicated tasks, has been widely investigated in recent years. With the maturity of AI and IoT, it is essential to evaluate the feasibility and efficiency of AI based intelligent transportation system (ITS) because the ITS is closely related to people's daily life and is one important aspect of building a smart city. According to a recent competition report [1] made by U.S. Department of Transportation, a lot of cities have a common challenge to manage the traffic flow and reduce the traffic congestion. Considering the fact that the traffic light control system is one of the most straightforward approaches to address the above challenge, in recent years, many solutions and algorithms for traffic light control have been proposed in order to improve the traffic congestion. However, most previous solutions and algorithms only focused on motorized traffic (e.g. vehicles). Non-motorized traffic (such as pedestrian and bicyclist) is rarely considered, and in most cases, non-motorized users have to manually activate the timing system by pushing a button, which may affect the overall efficiency of the entire traffic light control system. Therefore, to improve the overall efficiency of traffic light control, dynamic coordination of vehicular traffic with non-motorized traffic should be taken into account. However, modeling the correlation between vehicular and pedestrian's mobility is a challenging task since it is influenced by many factors which are uncertain and dynamic with time. Any preset control mechanism which is based on a particular rule may not be able to address such a dynamic problem. In this paper, by leveraging the notion of AI and IoT, we propose an intelligent traffic light control solution by using distributed multi-agent Q learning. The solution considers not only motorized traffic, but also non-motorized traffic, by dynamically monitoring and collecting vehicle and pedestrian queue lengths at each intersection. The proposed Q learning algorithm is implemented at each intersection, where the Q learning agent interacts with environment to learn the optimal control actions to minimize the length of waiting queues for both vehicle and pedestrian traffic. The observations at individual intersection are exchanged with its neighboring intersections through the network in a distributed manner to achieve the global optimal schedule for the entire system. Moreover, we considered many constraints / rules for traffic light control in the real world, and integrate them in the learning algorithm, which can facilicate the proposed solution to be deployed in real operational scenarios. To validate the efficiency of the proposed solution, we conducted numerical simulations by using a real-world map (from the OpenStreetMap [2], [3]) and real-world traffic data (from California Department of Transportation [4]). The simulation results show that our proposed solution outperforms existing solutions in terms of vehicle and pedestrian queue lengths, waiting time at intersections, and many other performance metrics. The remainder of this paper is structured as follows: In Section II we briefly review some related works. In Section III a brief introduction for single agent and multi-agent Q-learning is presented. In Section IV, we illustrate details for our proposed traffic light system and algorithms. Section V shows simulation results by using real world map and traffic data. Finally, we summarize the paper and present our future works in Section VI. II. RELATED WORKS Traffic lights are signalling devices which have been widely deployed in the world at road intersections, pedestrian crossings, and other locations to control traffic flows. Traffic lights can greatly affect the traffic condition. A well-designed control algorithm can increase the traffic handling capacity of roads, reduce collisions, waiting and traveling time for both vehicles and pedestrians. On the contrary, an inefficient control algorithm may cause significant traffic congestion, resulting in longer waiting time at intersections. To this end, various traffic signal control techniques have been proposed in the recent years. The commonly used one is fixed-time control where traffic signals are changed after a fixed time period (aka. threshold). However, this threshold can be pre-configured to different values based on different time periods in a day. The other widely deployed solution is dynamic control. To support this solution, detectors such as sensors or surveillance cameras [5] are deployed in the intersections to detect whether vehicles are present or not. The traffic light control module, based on this information, can adjust signal timing and phasing within the pre-determined limits. Moreover, there are some advanced solutions proposed in recent years, which are referred to as adaptive control. The adaptive control solutions are usually associated with centralized or distributed coordination among multiple intersections, and try to change or adapt traffic signal timing based on actual traffic demand. Although the adaptive traffic control is superior to the fixed-time and dynamic control, adaptive traffic light control systems have been deployed on less than one percent of existing traffic signals according to a recent report [6]. Some of the existing systems of adaptive traffic signal include Split Cycle Offset Optimization Technique (SCOOT), Sydney Coordinated Adaptive Traffic System (SCATS) which are both centralized, and Real Time Hierarchical Optimized Distributed Effective System (RHODES) which is decentralized with complex computation. The report [7] compares the performance of these solutions, and investigates their advantages and disadvantages. Considering the challenges for modeling the correlation between vehicular and pedestrian's mobility for its uncertain and dynamic nature, machine learning which arises recently is applied to traffic light control by researchers and shows proven performance [8]- [11]. Among all the machine learning algorithms, reinforcement learning or Q learning, due to its advantage of making decision in a model-free online fashion, has been adopted by several works to design a traffic light system. The work [12] applies a Q-learning and neural network method to decide green light periods in each intersection based on traffic information. However, each intersection only calculates based on its local information, and only tries to optimize local performance. The other work in [8] applies multi-agent reinforcement learning, but each Q learning agent works independently and has not considered other intersections' status. Thus, it is hard to achive the global optimization. Although the traffic light control solutions have been widely studied, most of the previously works haven't considered nonmotorized traffic. Pedestrians and bicyclists have to manually activate the timing system by pushing a button, affecting the overall efficiency of the traffic light control algorithms. Only a limited number of exceptions, for example, the work [13] considers pedestrian crossing in their traffic light control by using a genetic algorithm. In this algorithm, pedestrian metric is expressed in fitness function to evaluate effectiveness of candidate chromosome. However, this work only considers one intersection for local optimization only. Compared with the previous works, the novelties and contributions of this work include: 1) the proposed solution takes both the motorized and non-motorized traffic into consideration; 2) the proposed considers not only local information but also neighboring intersections for global optimization; and 3) multiple realworld constraints and traffic rules are included in the proposed algorithm, which can be flexibly extended and easily applied to real operational scenarios. III. PRELIMINARY Reinforcement learning [14], which is an online learning method, assists an agent to take a series of optimal actions to the environment, then obtains an instantaneous reward to maximize the cumulative benefit over time. The agent's knowledge is reinforced during the learning process. The idea of reinforcement learning has been widely used in robot control, advertising, stock investment and game theory. Q-learning, as one type of reinforcement learning, gains its popularity due to no requirement of knowledge for transition probabilities in Markov decision process (MDP). Due to this reason, it is a model-free learning. For one agent control, Q-learning is expressed as follows: Q t (s, a) = (1 − α)Q t−1 (s, a) + α(R t + γ max a Q t−1 (s, a)) where α is learning rate, γ is discounted. R t is instantaneous reward at time t. Regarding the exploitation and exploration in Q learning, there are many existing strategies, and the most commonly used three strategies are: 1) ε greedy selection, 2) Boltzmann exploration which chooses an action proportional to the probability, and 3) action selection based on tightening upper confidence bound (UCB) in [15] which adjusts the rank of Q values for action and state pairs. In a network, system or social infrastucture, an Q learning agent can be deployed on top of each entity or node, and executes Q-learning computation in a distributed manner. The agent can either work independently to optimize the local cost or collaborate with other agents (i.e. multi-agent Q learning). In the multi-agent Q learning, neighboring information can be exchanged through network connections. By exchanging messages through one hop connections, information that is multihop away can finally propagate across the network to achieve approximate global optimization. The collaboration among multiple agents is important since the advantage obtained by working cooperatively is usually more significant than that obtained by working independently. For transportation system, such kind of multi-agent cooperation is essentially important becuase it is not helpful by just improving traffic locally and moving congestions to different intersections. IV. SYSTEM AND ALGORITHM DESIGN A. System architecture The system architecture is shown in Fig. 1. The proposed distributed multi-agent Q-learning, as will be detailed next, is deployed in the traffic light system. At each intersection, an Q learning agent is deployed to control local traffic lights including both vehicles and pedestrians' lights in all directions. Surveillance cameras are deployed for each direction to detect queue lengths of pedestrians and vehicles. In this work, we consider the queue length is the actual number of vehicles and pedestrians waiting at the intersections for passing through. An agent collects local traffic data that are monitored by these surveillance cameras and stores the data into a local information database. Agents also collects neighborhood data by exchanging information through available network connections. These neighborhood data are also stored in the information database. Based on the database, Q computation module calculates an optimal control action, which is, in turn, executed by a control module which is a pre-programmed hardware controlling traffic light timing, as depicted in Fig. 1. B. State For the state design for the proposed Q learning algorithm, we use length of waiting queues to be states for both vehicles and pedestrians in each traffic direction at an intersection, as expressed by: S t i,d = q t 1i,d , q t 2i,d , ..., q t ji,d , m t 1i,d,L , m t 1i,d,R , ..., m t ji,d,L , m t ji,d,R where i, j are IDs of intersections and j N i ; N i is neighborhood intersections of i; S t i,d is the state at intersection i, at day d and time t; q t ji,d is the queue length from intersection j to intersection i, at day d and time t; m t ji,d,L is the queue length for pedestrians at the left side from intersection j to i, at day d and time t; and m t ji,d,R is the queue length for pedestrians at the right side from intersection j to i, at day d and time t. In this work, as aforementioned, we assume the surveillance cameras are deployed in the intersections to detect vehicles and pedestrians [5]. However, it should be noted that such state data can be also obtained by using other approaches and different type of detectors such as sensors, crosswalk buttons, Fastack/EasyPass, cell phones and GPS. For a large intersection which contains several straight and left lanes, state tuple could be shortened to the following expression to reduce the impact of dimensionality: S t i,d = b D q tb r,i,d , q tb s,i,d , C. Action We design actions for our Q-learning algorithm according to current traffic light rules. In each time slot, only one action can be executed. This action is calculated and selected from the action sets by the Q learning agent which can maximize the rewards (as will be detailed next). The action sets could be different for each intersection and they are configured based on the general practice. For example, for the most common "+" shape intersections, possible action sets are shown in Fig. 2, which include 4, 6, and 8 actions respectively. Note that the "+" shape intersection and the actions sets depicted in Fig. 2 are only examples to illustrate our design, and our solution can be applied to any shape of intersections. D. Reward Utilizing different rewards of Q-learning achieves corresponding different control or optimization purposes. For example, a reward can be the negative value of vehicle and non-motorized queue lengths, emission utility or traffic flows at crossing, etc. In this paper, the objective function is to minimize total average of queue length of both motorized and non-motorized traffic which is a metric that directly reflects congestion conditions. In this sense, we design the local instantaneous reward as follows: R t i,d (a t i,d , a t j,d , S t i,d , S t j,d , W t i,d ) = −( w t 1,d |N i | j Ni q t ji,d + w t 2,d |N i N j | j Ni k Nj q t kj,d + w t 3,d 2 |N i | j Ni (m t ji,d,L + m t ji,d,R ))(1) where R t i,d is the reward at intersection i, at day d and time t; a t i,d is the action at intersection i, at day d and time t; w t 1,d is the weight to present the local vehicular queues at intersection i; w t 2,d is the weight to present the neighborhood vehicular queues at the neighbors of intersection i; w t 3,d is the weight to present the total pedestrian queues at intersection i. W t d = w t 1,d , w t 2,d , w t 3,d . The sum of these weights equals to 1. In (1), j Ni q t ji,d is the incoming vehicular queues from intersection j to intersection i; 1 |Nj | j Ni k Nj q t kj,d is the total vehicular queues at all neighboring intersection js, including the outgoing vehicular traffic from intersection i to intersection j; and k Nj q t kj,d is the total pedestrian queues at intersection i. When exchanging observations, intersection j encapsulates the information of 1 Nj k Nj q t kj,d to a status update message and broadcast it to all neighboring intersections. |N i | is decided by the shape of intersections. For example, |N i | is 4 for a four-way, "+" shaped intersection, and |N i | is 3 for a three-way, "T" shaped intersection. In (1), the weights, w t 1,d , w t 2,d , w t 3,d correlate to the priority assigned to the additive term. That is, the higher the priority, the higher the weight. Since the reward R t i,d is expressed as the negative value of queue lengths at intersection i, at day d and time t, accordingly, the objective of Q-learning is to maximize the reward. In addition, due to neighboring traffic condition is considered in (1), as a result, for example, if neighboring intersections are congested, the action that causes more jams to the neighbors is less likely to be selected due to the smaller R t i,d . Thus, the traffic condition across multiple intersections can be achieved. Moreover, historical traffic data can be incorporated in the determination of actions. For example, data that are days away can be used to calculate estimated instantaneous rewards by autoregressive integrated moving average (ARIMA) model [16] because traffic data in the same time period at different days may have similar statistics and patterns. E. Integrate real-world constraints and traffic rules In a traffic light control system, there are a lot of rules or real-world constraints. The proposed Q learning solution takes these rules and constraints into account, which can facilicate its deployment in real operational scenarios. In this sub-section, two important constraints are presented. 1) Constraint for action selection: when Q-learning chooses an action from the action set for a given time slot, the selection of the next action is constrained by traffic rules. In the current real-world traffic light systems, the sequence of the actions follow some particular rules and it is unrealistic to freely choose an action. For example, in the 8 actions in Fig. 2, the constraints on the action selection could be: a 1 is not directly followed by a 5 , a 6 , a 7 , a 8 ; a 5 is not directly followed by a 1 , a 2 , a 3 , a 4 ; a 2 and a 3 are not directly followed by a 5 ; and a 5 and a 7 are not directly followed by a 1 . To satisfy these constraints in Q-learning operations, we can either greedily choose another action that satisfies these conditions; or for the given action, we can assign it with a very small reward value so that Q-learning rank it low in the priority queue. 2) Constraint for pedestrian protection: another constraint is the pedestrian protection which is related to the safety of pedestrians. The constraint is that a traffic signal that is directing pedestrians should not turn red while pedestrians are crossing the intersection. Our solution to address this constraint is to set a short duration for the default pedestrian green light. However, if pedestrians are crossing the intersection, the Q learning will not change the control actions, which is in turn, can protect crossing pedestrians. The presence of a pedestrian can be detected by surveillance cameras at the intersections which identify walking people. On the other hand, if there is no pedestrian detected, such protection will not be enabled. Moreover, we also set a sufficient time interval for a pedestrian safely crosses an intersection if a pedestrian button is pressed in the previous red light time cycle. In this case, system allocates at least 13 time slots to the pedestrians in the next green cycle to guarantee pedestrians can finish crossing. The pseudo code of traffic light control algorithm is shown in Algorithm 1. We have integrated the above constraints when choosing a proper action. In the pseudo code, N (a) is the occurrence of action a. Eq. (2) represents an action selection based on USB criterion in [15]. According to Algorithm 1, the actual action selection only happens when there is no pedestrian crossing an intersection and when Q-learning computation module is scheduled another action. V. PERFORMANCE EVALUATION We implement our algorithm of traffic light control in the Simulation of Urban MObility (SUMO) [17] which can model Nj k Nj q t kj,d and a t j,d to all its neighbors calculate reward according to (1) update Q t+1 i,d (a t i,d ) if (execute action == Y) and (no pedestrian == Y) then / * exploitation and exploration * / / * using either ε-greedy exploration strategy, Boltzmann exploration strategy or UCB * / do / * in the case of using UCB * / a t+1 i,d = arg max a A Q t+1 i,d (a) + 2 log t N (a)(2) while action constraint in Section IV-E is not met; end end Algorithm 1: Traffic light control microscopic traffic conditions and has a well-designed API for controlling status of traffic light through online interaction. We export the map of an area in Sunnyvale, CA from Open-StreetMap [2], which is a rectangle area that has longitude between -121.964019 and 121.997997, and latitude between 37.322300, and 37.353056 shown in Fig. 3a. The map is converted into a SUMO compliant network topology illustrated in Fig. 3b by the netconvert tool. After convertion, the topology has a total number of 3811 edges, and 33 intersections with traffic lights. We deploy our proposed algorithm into each traffic light in Fig. 3b. In addition to the real world map, we also used real world traffic data which are obtained from the California Department of Transportation [4]. As most of the traffic statistics are related to freeways, in order to estimate the traffic for the simulation area, we firstly obtain the traffic statistics for the freeways surrounding the simulation area, and then we calculate the proportional traffic load for the simulation area for each hour in a day. These estimated values are used to model the vehicle arrival rate in the simulation. On the other hand, as there are no appropriate open data regarding pedestrians, and considering the fact that different cities may have completely different load for pedestrian traffic, in the simulation, we model 3 different arrival rates for pedestrians, which are referred to as high pedestrian rate, medium pedestrian rate, and low pedestrian rate respectively. For those 3 scenarios, the ratio between pedestrian arrival rate and vehicle arrival rate is set to 1:1, 3:5, and 1:10 respectively. The routes for both vehicle and pedestrain traffic are randomly generated by the randomT rips module provided by SUMO. For the configuration of the proposed multi-agent Q learning algorithm, we set equal weights for pedestrian, vehicle, and neighboring queues, that is, w t h,d = 1/3 where h = 1, 2, 3. Ovreall simulation time is set to be 5400 timeslots. Initial learning rate, α is set to be 0.5, and gradually decreases with time. Discounted factor, γ is set to be 0.5. In the simulation, Q-learning does exploration using the ε-greedy exploration strategy, with ε equals to 0.3. In order to evaluate the overall efficiency of the proposed multi-agent Q learning solution, we compare our proposed The real-world solutions, as aforementioned in section II, include: 1) fixed time control with four actions, 2) dynamic control with four actions, 3) dynamic control with six actions, and 4) dynamic control with eight actions, which can represent most cases in reality. Here, we assume sensors are used to detect vehicles and trigger corresponding dynamic control. However, the simulation results are also applicable to other detection methods as mentioned in section II. The examples of four, six and eight action sets are illustrated in Fig. 2. Note that, these real-world solutions haven't considered pedestrians and they need to manually press a button to activate the timing system. For example, in the case of dynamic control with no pedestrian pushing the button, a traffic light may turn to red if no vehicles are waiting in the green light direction, and it may turn to green in a direction where at least one vehicle is waiting. The green traffic light in each direction is programmed with a default sequence which follows a state diagram if all four directions have waiting vehicles. Furthermore, we also compare our proposed solution with the state-of-the-art mechine learning solution for traffic light control [8]. In [8], a multi-agent Q learning algorithm, which is referred to as MARL, is proposed to improve the traffic light performance. Similarly, it also does not consider the pedestrian Figure 4 shows that total cumulated queue length for vehicles and pedestrians. Note that the total queue length in Fig. 4 represents the total number of vehicles or pedestrians that are waited at all the intersections due to red lights. If a vehicle or a pedestrian is not waited at an intersection (e.g. green lights), the vehicle or pedestrian is not taken into account when the total queue length is calculated at this intersection. It can be seen that, compared with the fixed-time and sensor-based dynamic control solutions, the Q learning based solutions, including both the proposed solution in this paper and the MARL solution can greatly reduce the vehicle and pedestrian queue lengths. It is because of the nature of reinforcement learning which is capable of optimizing the control actions, and thus, more vehicles or pedestrians do not wait at intersections, resulting in smaller total queue length. If we compare the performance between the Q learning proposed in this paper and MARL, we can observe that our solution is better than MARL. The major reason is that our solution considers both the vehicle and pedestrian information, while MARL only considers vehicles. Moreover, our solution exchanges information among multiple Q learning agents, which is useful to achieve the optimization for the entire system. Through the GUI of SUMO, we observed that our solution can achieve a smoother traffic flow during the simulation, compared with other solutions. Figure 5 shows the waiting and traveling time comparison results. Similarly, our solution outperforms others. Among the results, the vehicle and pedestrian waiting time is the most straightforward performance metric because an efficient we can see that the proposed distributed multi-agent Q learning is better than exiting solutions and MARL in term of many key performance metrics, which validated the overall feasibility and efficiency of the proposed solution in terms of traffic light control. Figures 8-11 show the simulation results in the case of medium pedestrian rate, and Figures 12-15 show the simulation results in the case of low pedestrian rate. In detail, Figures 8, 9, 10 and 11 depict the simulation results regarding vehicle and pedestrian queue lengths, vehicle and pedestrian waiting and travelling time, CO 2 and CO emission, and fuel consumption and noise pollution in the case of medium pedestrian rate respectively. Figures 12, 13, 14 and 15 depict the simulation results in terms of vehicle and pedestrian queue lengths, vehicle and pedestrian waiting and travelling time, CO 2 and CO emission, and fuel consumption and noise pollution in the case of low pedestrian rate respectively. From the results, we can see that, similar to the case with high pedestrian rate, our proposed solution also outperforms other VI. CONCLUSIONS AND FUTURE WORKS This paper presents an intelligent traffic light control system which takes pedestrians into account in order to achieve optimization for both motorized and non-motorized traffic. The system is empowered by a distributed multi-agent Q learning, which is able to collaboratively calculate the optimal control actions, based on the traffic information not only from the local intersection, but also from neighboring intersections. Moreover, many real-world constraints / rules for traffic light control are integrated in the Q learning algorithm, which can facilicate the proposed solution to be deployed in real operational scenarios. Numerical simulations are carried out based on a real-world map with real-world traffic data. The simulation results show that our proposed solution outperforms Our future works are twofold. Firstly, we will further improve the algorithm performance, for example, the Q learning convergence time, in order to handle the scenario where sharp change of traffic pattern occurs. Secondly, we will investigate different deployment models for the proposed system, in addition to the fully distributed model used in ths paper, to evaluate whether the performance of the entire system can be further improved. As the AI and machine learning technology has proven to be useful in many use cases, we hope the work presented in this paper can shed light on the future real deployment of AI based traffic light control system. Fig. 1 : 1System architecture for distributed multi-agent Q learning Fig. 2 : 2Action sets (green lights) for "+" shape intersection Fig. 3 : 3(a) A map of Sunnyvale, CA downloaded from OpenStreetMap; (b) the topology of the map converted by SUMO Data: both motorized and non-motorized traffic data Result: get optimal green time of traffic light Initialization: Q 0 i,d = 0, action set |A|, an action, a 0 i,d , from action set A for t do observe queue length in all directions, and get q t ji,d , Fig. 4 :Fig. 5 : 45Total cumulated queue lengths for vehicles and pedestrians in the case of high pedestrian rate Total waiting and traveling time for vehicles and pedestrians in the case of high pedestrian rate algorithm with four real-world solutions and one state-of-theart mechine learning based solution for traffic light control. Fig. 6 : 6CO 2 and CO emission of vehicles in the case of high pedestrian rate Q Learning (This paper) Fixed (4 actions) Sensor (4 actions) Sensor (6 actions) Sensor ( Fig. 7 : 7Fuel consumption and noise pollution of vehicles in the case of high pedestrian rate traffic for Q learning computation. Figures 4-7 show the simulation results in the case of high pedestrian rate. Fig. 8 :Fig. 9 : 89Total cumulated queue lengths for vehicles and pedestrians in the case of medium pedestrian rate Total waiting and traveling time for vehicles and pedestrians in the case of medium pedestrian rate traffic light control algorithm should always reduce the waiting time of vehicles and pedestrians at intersections. Due to the shorter waiting time and reduction of unnecessary stopping and starting of traffic, our solution can, in turn, reduce CO 2 and CO emissions, fuel consumption, and noise pollution of vehicles, as depicted in Figs. 6 and 7 respectively. From all the results depicted in Figs. 4-7, Fig. 10 : 10CO 2 and CO emission of vehicles in the case of medium pedestrian rate Q Learning (This paper) Fixed (4 actions) Sensor (4 actions) Sensor (6 actions) Sensor ( ×10 Fig. 13 : 13Total waiting and traveling time for vehicles and pedestrians in the case of low pedestrian rate existing solutions in terms of vehicle and pedestrian queue length, waiting time at intersections, and many other key performance metrics such as emissions and fuel consumptions. Fig. 14 :Fig. 15 : 1415CO 2 and CO emission of vehicles in the case of low pedestrian rate Fuel consumption and noise pollution of vehicles in the case of low pedestrian rate Where means concatenate the set in {}. D is a directional set which contains all directions such as north, south, east, and west. r means right turn. s means going straight. l means left turn. q tb r,i,d and q tb s,i,d can be combined if there is no right turn traffic light. Then, Q-learning maintains a state-action table to keep track of old Q values for new Q value computation.q tb li,d , m tb i,d Fig. 11: Fuel consumption and noise pollution of vehicles in the case of medium pedestrian rate solutions when the pedestrian rate is medium or low. In other words, the proposed solution performs best in all the simulated scenarios, which validated that the proposed solution can be deployed in different type of cities with different traffic patterns. If we further compare the results for the high, medium and low pedestrian rates, we can observe that our solution has larger performance improvement compared with MARL when there are more pedestrians. Take the vehicle waiting time as examples, in the case of high pedestrian rate, our solution can achieve 16.7% improvements for vehicle waiting time reduction compared with MARL. In the case of medium and low pedestrian rates, the number is reduced to 12.2% and 7.0% respectively. This observation indicates that our solution is more efficient than MARL when there are more pedestrians. It is reasonable because our solution jointly considers the vehicle and pedestrian traffic for optimization while MARL only considers the vehicle traffic.4 Fuel Consumption (ml/s) Noise (dB) Smart city challenge: Lessons for building cities of the future. U S , Department of TransportationU. S. Department of Transportation. Smart city challenge: Lessons for building cities of the future. [Online]. . Openstreetmap, Openstreetmap. [Online]. Available: https://www.openstreetmap.org Openstreetmap: User-generated street maps. M Haklay, P Weber, IEEE Pervasive Computing. 74M. Haklay and P. Weber, "Openstreetmap: User-generated street maps," IEEE Pervasive Computing, vol. 7, no. 4, pp. 12-18, 2008. Department of Transportation. Caltrans performance measurement system (pems). C , C. Department of Transportation. Caltrans performance measurement system (pems). [Online]. Available: http://pems.dot.ca.gov/ Vehicle detection using image processing for traffic control and surveillance system. R A Rahmat, K Jumari, Proceedings of the 7th World Congress on Intelligent Transport System. the 7th World Congress on Intelligent Transport SystemR. A. Rahmat and K. Jumari, "Vehicle detection using image processing for traffic control and surveillance system," in Proceedings of the 7th World Congress on Intelligent Transport System, 2000. Department of Transportation, United States. F H A , Adaptive signal control technologyF. H. A. Department of Transportation, United States. Adaptive signal control technology. [Online]. Available: https://www.fhwa.dot. gov/innovation/everydaycounts/edc-1/asct.cfm Adaptive traffic signals, comparison and case studies. K Fehon, J Peters, Institute of Transportation Engineers Western ITE Meeting. San FranciscoK. Fehon and J. Peters, "Adaptive traffic signals, comparison and case studies," in Institute of Transportation Engineers Western ITE Meeting. San Francisco, 2010. Multi-agent reinforcement learning for traffic signal control. K Prabuchandran, H K An, S Bhatnagar, Intelligent Transportation Systems (ITSC). IEEEIEEE 17th International Conference onK. Prabuchandran, H. K. AN, and S. Bhatnagar, "Multi-agent reinforce- ment learning for traffic signal control," in Intelligent Transportation Systems (ITSC), 2014 IEEE 17th International Conference on. IEEE, 2014, pp. 2529-2534. Adaptive traffic signal control with vehicular ad hoc networks. K Pandit, D Ghosal, H M Zhang, C.-N Chuah, IEEE Transactions on Vehicular Technology. 624K. Pandit, D. Ghosal, H. M. Zhang, and C.-N. Chuah, "Adaptive traffic signal control with vehicular ad hoc networks," IEEE Transactions on Vehicular Technology, vol. 62, no. 4, pp. 1459-1471, 2013. Distributed and adaptive traffic signal control within a realistic traffic simulation. D Mckenney, T White, Engineering Applications of Artificial Intelligence. 261D. McKenney and T. White, "Distributed and adaptive traffic signal control within a realistic traffic simulation," Engineering Applications of Artificial Intelligence, vol. 26, no. 1, pp. 574-583, 2013. A real-time adaptive signal control in a connected vehicle environment. Y Feng, K L Head, S Khoshmagham, M Zamanipour, Transportation Research Part C: Emerging Technologies. 55Y. Feng, K. L. Head, S. Khoshmagham, and M. Zamanipour, "A real-time adaptive signal control in a connected vehicle environment," Transportation Research Part C: Emerging Technologies, vol. 55, pp. 460-473, 2015. Intelligent traffic light control of isolated intersections using machine learning methods. S Araghi, A Khosravi, M Johnstone, D Creighton, IEEE International Conference on Systems, Man, and Cybernetics (SMC). IEEES. Araghi, A. Khosravi, M. Johnstone, and D. Creighton, "Intelligent traffic light control of isolated intersections using machine learning methods," in IEEE International Conference on Systems, Man, and Cybernetics (SMC). IEEE, 2013, pp. 3621-3626. The use of genetic algorithm for traffic light and pedestrian crossing control. A M Turky, M S Ahmad, M Z M Yusoff, International Journal of Computer Science and Network Security. 92A. M. Turky, M. S. Ahmad, and M. Z. M. Yusoff, "The use of genetic algorithm for traffic light and pedestrian crossing control," International Journal of Computer Science and Network Security, vol. 9, no. 2, pp. 88-96, 2009. Reinforcement learning: an introduction. R S Sutton, A Barto, LLCR. S. Sutton and A. Barto, Reinforcement learning: an introduction. Springer Science+Business Media, LLC, 1998. Finite-time analysis of the multiarmed bandit problem. P Auer, N Cesa-Bianchi, P Fischer, Machine Learning. 47P. Auer, N. Cesa-Bianchi, and P. Fischer, "Finite-time analysis of the multiarmed bandit problem," Machine Learning, vol. 47, no. 2-3, pp. 235-256, 2002. Time series analysis. W W , - S Wei, Addison-Wesley publ ReadingW. W.-S. Wei, Time series analysis. Addison-Wesley publ Reading, 1994. Sumo (simulation of urban mobility)-an open-source traffic simulation. D Krajzewicz, G Hertkorn, C Rössel, P Wagner, Proceedings of the 4th middle East Symposium on Simulation and Modelling (MESM20002). the 4th middle East Symposium on Simulation and Modelling (MESM20002)D. Krajzewicz, G. Hertkorn, C. Rössel, and P. Wagner, "Sumo (simula- tion of urban mobility)-an open-source traffic simulation," in Proceed- ings of the 4th middle East Symposium on Simulation and Modelling (MESM20002), 2002, pp. 183-187.
[]
[ "Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy", "Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy" ]
[ "H Aratyn \nDepartment of Physics\nUniversity of Illinois at Chicago\n845 W. Taylor St60607-7059ChicagoIL\n", "L A Ferreira \nInstituto de Física Teórica -IFT/UNESP Rua\nPamplona 14501405-900São Paulo -SPBrazil\n", "J F Gomes \nInstituto de Física Teórica -IFT/UNESP Rua\nPamplona 14501405-900São Paulo -SPBrazil\n", "A H Zimerman \nInstituto de Física Teórica -IFT/UNESP Rua\nPamplona 14501405-900São Paulo -SPBrazil\n" ]
[ "Department of Physics\nUniversity of Illinois at Chicago\n845 W. Taylor St60607-7059ChicagoIL", "Instituto de Física Teórica -IFT/UNESP Rua\nPamplona 14501405-900São Paulo -SPBrazil", "Instituto de Física Teórica -IFT/UNESP Rua\nPamplona 14501405-900São Paulo -SPBrazil", "Instituto de Física Teórica -IFT/UNESP Rua\nPamplona 14501405-900São Paulo -SPBrazil" ]
[]
The algebraic matrix hierarchy approach based on affine Lie sl(n) algebras leads to a variety of 1 + 1 soliton equations. By varying the rank of the underlying sl(n) algebra as well as its gradation in the affine setting, one encompasses the set of the soliton equations of the constrained KP hierarchy.The soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine sl(n) algebras realized in the unconventional gradations. Such soliton solutions exhibit non-trivial dependence on the KdV (odd) time flows and KP (odd and even) time flows which distinguishes them from the conventional structure of the Darboux-Bäcklund Wronskian solutions of the constrained KP hierarchy.
10.1088/0305-4470/31/47/009
[ "https://export.arxiv.org/pdf/solv-int/9709004v1.pdf" ]
17,720,731
solv-int/9709004
bb15629344a7a2b47beddf8ad973bf5a44e4e46b
Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy arXiv:solv-int/9709004v1 11 Sep 1997 August, 1997 H Aratyn Department of Physics University of Illinois at Chicago 845 W. Taylor St60607-7059ChicagoIL L A Ferreira Instituto de Física Teórica -IFT/UNESP Rua Pamplona 14501405-900São Paulo -SPBrazil J F Gomes Instituto de Física Teórica -IFT/UNESP Rua Pamplona 14501405-900São Paulo -SPBrazil A H Zimerman Instituto de Física Teórica -IFT/UNESP Rua Pamplona 14501405-900São Paulo -SPBrazil Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy arXiv:solv-int/9709004v1 11 Sep 1997 August, 1997 The algebraic matrix hierarchy approach based on affine Lie sl(n) algebras leads to a variety of 1 + 1 soliton equations. By varying the rank of the underlying sl(n) algebra as well as its gradation in the affine setting, one encompasses the set of the soliton equations of the constrained KP hierarchy.The soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine sl(n) algebras realized in the unconventional gradations. Such soliton solutions exhibit non-trivial dependence on the KdV (odd) time flows and KP (odd and even) time flows which distinguishes them from the conventional structure of the Darboux-Bäcklund Wronskian solutions of the constrained KP hierarchy. Introduction. The Algebraic cKP Model A large class of 1 + 1 soliton equations belongs to the so-called constrained KP (cKP) hierarchy. Some of the most prominent members of this group are the KdV and the nonlinear Schrödinger equations of the AKNS model. The cKP evolution equations possess the familiar Lax pair representations with generally pseudo-differential Lax operators which emerge naturally as reductions of the complete KP hierarchy Lax operators ( [1]). Conventionally, the constrained KP hierarchy is obtained from the KP hierarchy by a process of reduction involving the so-called eigenfunctions. The eigenfunctions appear in the constraint relations introducing a functional dependence between initially infinitely many coefficients of the KP Lax operator. This scheme results in the pseudo-differential cKP Lax operator of the type L = L K+1 + M i=1 Φ i ∂ −1 Ψ i , where L K+1 is the differential operator of the K + 1-order, while Φ i , Ψ i are the eigenfunctions of L. In general, L possesses a finite number of coefficients which enter the soliton equations and depend on all (t 1 , t 2 , t 3 , . . .) isospectral time flows of the KP hierarchy. In the special M = 0 case with the cKP Lax operator being a purely differential operator L = L K+1 we encounter dependence on only some of the original timeflows of the KP hierarchy. The most simple example (K = 1, M = 0) is the KdV hierarchy with only odd time flows present. The soliton solutions for the cKP models have been found in ( [2]) for the arbitrary K and M = 1 case using the Darboux-Bäcklund technique. Generalization to an arbitrary M is simple and was given in ( [3]) (see also ([4])). These solutions appeared in the Wronskian form in terms of the eigenfunctions of the "undressed" L = ∂ K+1 Lax operator. Here, we will present an alternative algebraic viewpoint of the constrained KP hierarchy. In this setting the algebraic dressing methods will provide new soliton solutions which appear to differ from the conventional form of the Darboux-Backlund Wronskian solutions due to a non-trivial mixing of the KdV-like versus KP time flows. This will be shown explicitly in the example characterized by K = M = 1. In an algebraic approach to the constrained KP hierarchy ( [5]) the soliton evolution equations emerge as integrability conditions of the following matrix eigenvalue problem: LΨ = 0 ; L ≡ (D − A − E) ; D ≡ I ∂ ∂x(1) with the matrix Lax operator L belonging to Kac-Moody algebraĜ = sl(M + K + 1). The integrable hierarchy is determined by the choice of gradation ofĜ. By varying the Kac-Moody algebras together with their gradations one is able to reproduce from the matrix hierarchy of eq. (1) the nonlinear evolution equations of the cKP hierarchy. We will be working with a simple setting in which the matrix E in eq.(1) has gradation 1 with respect to gradation specified by the vector ([6]): s = (1, 0, . . . , 0 M , 1, . . . , 1 K )(2) We call this gradation an intermediate gradation as it interpolates between the principal s principal = (1, 1, . . . , 1) and the homogeneous one s homogeneous = (1, 0 . . . , 0). As it is wellknown the Wilson-Drinfeld-Sokolov ([7, 8, 9, 10, 11]) procedure gives, respectively, the (m-)KdV ( [9]) and AKNS ( [12,13]) hierarchies in these two limits. Alternatively, the gradation s can be specified by an operator: Q s ≡ K a=1 λ M +a · H 0 + (K + 1)d(3) Here λ j are the fundamental weights and d is the standard loop algebra derivation. Correspondingly, E stands for E = K a=1 E (0) α M +a + E (1) −(α M +1 +···+α M +K )(4) It is a non-regular (for M > 0) and semisimple element ofĜ. The matrix A in eq. (1) contains the dynamical variables of the model. A is such that it has gradation zero and is parametrized in terms of the dynamical variables q i , r i , U a and ν as follows: A = M i=1 (q i P i + r i P −i ) + K a=1 U M +a (α M +a · H (0) ) + νĉ(5) where P ±i = E The Heisenberg Subalgebra and the Vertex Operator For a regular element E in the conventional Drinfeld-Sokolov approach the isospectral flows are associated with the Heisenberg subalgebra which can be identified with Ker (ad E). Here, due to non-regularity of E the Heisenberg algebra is associated to the center of Ker (ad E). It consists of the following three separate sets of operators: 1) A homogeneous part ofŝl(M) , i = 1, 2, . . . , M − 1 K (n) i = i p=1 p α p · H (n) N i ; N i ≡ i(i + 1)(6) 2) A principal part ofŝl(K + 1) , a = 1, 2, . . . , K A a a+n(K+1) = K+1−a i=1 E (n) α i+M +α i+M +1 +···+α i+M +a−1 + a i=1 E (n+1) −(α i+M +α i+M +1 +···+α i+M +K−a )(7) 3) "A border term" A 0 n(K+1) = M + K + 1 M λ M .H (n) − K 2 M M + K + 1ĉ δ n,0(8) These relations provide a parametrization of the Heisenberg subalgebra in terms of ele- ments: b N,a ≡ A a N =a+n(K+1) , b N,0 ≡ A 0 N =n(K+1) , b N,i ≡ K (N ) i(9) where a = 1, 2, . . . , K, i = 1, 2, . . . , M − 1. The Heisenberg subalgebra elements from (9) enter the oscillator algebra relations (we put c = 1): [ b N,a , b N ′ ,b ] = Nδ N +N ′ δ a,K+1−b ; a, b = 1, 2, . . . , K (10) [ b N,0 , b N ′ ,0 ] = Nδ N +N ′ (11) [ b N,i , b N ′ ,j ] = Nδ N +N ′ δ ij ; i, j = 1, 2, . . . , M − 1(12) Define next the Fubini-Veneziano operators: Q 1≤i≤M −1 (z) = i ∞ n=1 K (n) i z −n n ; Q M (z) = i ∞ n=1 A 0 n(K+1) z −n(K+1) n(K + 1) (13) Q M +a (z) = i ∞ n=0 A a a+n(K+1) z a+n(K+1) a + n(K + 1) ; a = 1, 2, . . . , K(14) The corresponding conjugated Fubini-Veneziano operators Q † (z) are obtained from (13)-(14) by taking into consideration rules K (n) † i = K (−n) i , A 0 n(K+1) † = A 0 −n(K+1) , A a a+n(K+1) † = A a a−n(K+1) as well as z † = z −1 . In reference ( [14]) we found the step operators of sl(M + K + 1) associated with the Cartan subalgebra defined by the Heisenberg subalgebra (9). That in turn enabled us to find the corresponding simple root structure for sl(M +K +1) with the intermediate grading. Knowledge of roots and the Fubini-Veneziano operators is all what is needed to write down a compact expression for the general vertex operator in the normal ordered form: V α (z) ≡ z 1 2 M j=1 (α j ) 2 exp i α * · Q † (z) exp (i α · q) exp ( α · p ln z) exp i α · Q(z)(15) with M + K-component root vector α described in ( [14]) and M + K-component vector Q having components described in (13)- (14). The zero-mode vectors p and q have only first M components different from zero according to ( (8). An explicit example of the sl(3) vertex will be given below in section 4. p) i = p i θ(M − i) and ( q) i = q i θ(M − i). They satisfy relations [ p i , q j ] = −iδ ij . Furthermore, p M is equal to A 0 n=0 from expression The Dressing Technique and the Tau-function The dressing technique ( [15]) deals with reproducing of the nontrivial part E + A of the Lax matrix operator from eq. (1) by the gauge transformations involving generators of positive and negative gradings applied to the semisimple element E: E + A = Θ E Θ −1 + (∂ x Θ) Θ −1 (16) E + A = B −1 Γ E Γ −1 B + ∂ x B −1 Γ Γ −1 B(17) where B −1 Γ contains positive terms and Θ is an expansion in the terms of negative grading such that Θ = exp l<0 θ (l) = 1 + θ (−1) + . . .. From expressions (16) and (17) we obtain two alternative formulas for the same term A of grade 0: A = − E , θ (−1) or A = −B −1 (∂ x B)(18) The term θ (−1) of grade −1 can be expanded as θ (−1) = M +K a=M +1 θ (−1) a E (0) −αa + θ (−1) ψ E (−1) α M +1 +...+α M +K + M l=1 θ (−1) l E (0) −(α l +...+α M +1 ) + M l=1θ (−1) l E (−1) α l +...+α M +K(19) where we included all possible terms of grade −1 according to (3). Therefore E , θ (−1) = M +K a=M +1 θ (−1) a α a · H (0) + θ (−1) ψ −(α M +1 + . . . + α M +K ) · H (0) + c + M l=1 θ (−1) l ǫ(α M +1 , −α l −...−α M +1 ) E (0) −(α l +...+α M ) + M l=1θ (−1) l ǫ(−α M +1 −...−α M +K , α l +...+α M +K ) E (0) α l +...+α M(20) Comparing the last expression with the field content of A as given by (5) we obtain relations for expansions parameters used in (19): ν = −θ (−1) ψ , U a = −θ (−1) a + θ (−1) ψ , r l = −θ (−1) l ǫ(α M +1 , −α l −...−α M +1 ) q l = −θ (−1) l ǫ(−α M +1 −...−α M +K , α l +...+α M +K )(21) We now work with representation of A as given in eq. (18). We split the grade zero element B in a product B = B 1 B 2 with B 1 containing the grade zero sl(M) elements and B 2 ≡ exp K a=1 φ M +a α M +a · H (0) + ρ ·ĉ (22) Accordingly, eq.(18) becomes A = −B −1 2 B −1 1 (∂ x B 1 )B 2 − B −1 2 (∂ x B 2 ) and A can be rewritten as A = − K a=1 ∂ x φ M +a α M +a · H (0) − ∂ x ρ ·ĉ + O(sl(M))(23) where O(sl(M)) contains all possible terms belonging to the sl(M) algebra. Comparison with (5) yields U M +a = −∂ x φ M +a ; ν = −∂ x ρ(24) We define a family of the first order differential matrix operators L N = ∂/∂t L N = Ψ ∂ ∂t N Ψ −1(26) The starting point of the dressing method ( [15]) is the vacuum solution ν = U = r = q = 0. The corresponding L (vac) = D − E matrix Lax operator together with higher flows operators L N , N > 1 for the vacuum solutions are expected to be recovered via (26) from Ψ, which is expressed entirely by the Heisenberg algebra associated with the center of Ker (ad E). Explicitly, for our model Ψ = Ψ (vac) = exp ∞ N =1 t N b (N )(27) with b N given in (9). We define the tau-function vectors as: |τ 0 = Ψ (vac) h Ψ (vac) −1 |λ 0 ; |τ M +a = Ψ (vac) h Ψ (vac) −1 |λ M +a(28) They are associated with the constant group element h and the highest-weight vectors |λ 0 , |λ M +a such that α M +a · H (0) |λ 0 = 0 ; α M +b · H (0) |λ M +a = δ a,b |λ M +a ; a, b = 1, . . . , K.(29) Assuming that h allows the "Gauss" decomposition of Ψ (vac) h Ψ (vac) −1 in positive, negative and zero grade elements we get for the tau-function vectors from (28) an alternative expression: ; |τ 0 = Θ −1 B −1 |λ 0 ; |τ M +a = Θ −1 B −1 |λ M +a Θ −1 = Ψ (vac) h Ψ (vac) −1 − ; B −1 = Ψ (vac) h Ψ (vac) −1 0(30)|τ M +a = Θ −1 B −1 |λ M +a e (−ρ−φ M +a )(31) Denote τ (0) 0 ≡ exp (−ρ) ; τ (0) M +a ≡ exp (−ρ − φ M +a )(32) Accordingly, expanding Θ −1 as below eq.(17) we find |τ M +a τ (0) M +a = 1 − θ (−1) − . . . |λ M +a(33) and similarly for |τ 0 /τ U M +a = −∂ x ln τ (0) 0 / τ (0) M +a ; ν = −∂ x ln τ (0) 0(36) The multi-soliton tau functions are defined in terms of the constant group elements h which are the product of exponentials of eigenvectors of the Heisenberg subalgebra elements h = e F 1 e F 2 · · · e Fn , [b N , F k ] = ω (k) N F k , k = 1, 2, . . . , n . As seen from eq. (37) for such group elements the dependence of the tau-vectors upon the times t N can be made quite explicit |τ a = n k=1 exp(e N ω (k) N t N F k ) |λ a(38) The multi-soliton solutions are conveniently obtained in terms of representations of the "vertex operator" type where the corresponding eigenvectors are nilpotent. The sl(3) Example: Solitons of the Yaijma-Oikawa Hierarchy We apply the above method to the particular case of sl(3) with M = K = 1. From eq.(9) the surviving elements of the Heisenberg subalgebra are in this case: b (2n+1) ≡ b N =2n+1,a=1 = A 1 N =1+n·2 = E (n) α 2 + E (n+1) −α 2 (39) b (2n) ≡ b N =2n,0 = √ 3λ 1 · H (n) −ĉ 2 √ 3 δ n,0(40) and they satisfy the usual Heisenberg subalgebra b (k) , b (k ′ ) = kδ k+k ′ for both even and odd k. The structure of eigenvectors of Heisenberg subalgebra facilitates construction of multisoliton solutions according to (37) and (38). In the current example we find that the eigenvectors and their corresponding eigenvalues (in notation of (37)) are Eα 1 = √ 2 n∈ Z Z [z −2n E (n) α 1 − z −2n−1 E (n) α 1 +α 2 ] ; ω (2n+1) α 1 = z 2n+1 ; ω (2n) α 1 = √ 3 z 2n (41) Eα 2 = n∈ Z Z z −2n−1 E (n) α 2 − E (n+1) −α 2 + z −2n α 2 · H (n) −ĉ 2 δ n,0 (42) ω (2n+1) α 2 = −2z 2n+1 ; ω (2n) α 2 = 0 Eα 1 +α 2 = √ 2 n∈ Z Z [z −2n E (n) α 1 + z −2n−1 E (n) α 1 +α 2 ] ; ω (2n+1) α 1 +α 2 = −z 2n+1 ; ω (2n) α 1 +α 2 = √ 3 z 2n (43) We now realize the above eigenvectors by the nilpotent vertex operators. The construction involves the Fubini-Veneziano operators defined in terms of the Heisenberg elements as in eqs. (13)- (14): Q 1 (z) ≡ i n∈ Z Z b (2n+1) z −2n−1 2n + 1 ; Q 2 (z) ≡ q − ip ln z + i n =0 b (2n) z −2n 2n(44) where the zero mode momentum p = b (0) = √ 3λ 1 · H (0) −ĉ/2 √ 3 satisfies [ q , p ] = i. The step operators from (41)-(43) are then realized, from the algebra point of view, as vertex operators via: Eα 1 ↔ E (1, √ 3) (z) = √ 2 z 3/2 : exp iQ 1 (z) + i √ 3 Q 2 (z) : (45) Eα 2 ↔ E (−2,0) (z) = − 1 2 : exp (−2iQ 1 (z)) : e iπp (46) Eα 1 +α 2 ↔ E (−1, √ 3) (z) = √ 2 z 3/2 : exp −iQ 1 (z) + i √ 3 Q 2 (z) :(47) and similarly for the negative root step operators, with change of sign of i in exponentials. A care has to be exercised in applying this correspondence within the setting of the Fock space with the vacuum vector being |λ 2 , since λ 0 |Eα 2 |λ 0 = −1/2 while λ 2 |Eα 2 |λ 2 = 1/2 as seen from expression (42). Similar consideration applies for the products of E α i 's vertex operators like E (−1,− √ 3) E (−1,√ 3) which produce E (−2,0) . Introduce the notation: V c i ,d i (z) ≡ z d 2 i /2 : exp (ic i Q 1 (z) + id i Q 2 (z)) :(48) It is not difficult to establish the following correlation function: λ σ Ψ (vac) V c 1 ,d 1 (z 1 ) · · · V cn,dn (z n )Ψ (vac) −1 λ σ = δ n j=1 d j ,0 e n j=1 Γ c j ,d j (z j ) n j=1 z (−1) (σ+2)/2 d j 2 √ 3 + d 2 j 2 j 1≤i<j≤n z i − z j z i + z j c i c j /2 [(z i − z j ) (z i + z j )] d i d j /2(49) for σ = 0, 2 and with Γ c j ,d j (z j ) = ∞ n=0 c j t 2n+1 z 2n+1 j + ∞ n=1 d j t 2n z 2n j(50) When substituting V c j ,d j (z j ) by E c j ,d j (z j ) one encounters extra phases originating from the Klein factor in eq.(46) and from the character of the |λ 2 vacuum as discussed below eq. (47). The latter gives rise to the factor exp (iπ/2) n j=1 c j for the λ 2 correlation function as verified on several examples. Recall that for the problem in hand the Lax matrix operator from (1) with A from (5) and E from (4) specifies to L = D −    0 q 0 r U 2 1 0 λ −U 2    − νĉ(51) where λ is the usual loop parameter. In terms of the tau-vectors we have from (34)-(36) the following n-soliton representation of the components of the Lax operator r = 1 τ (0) 2 λ 2 |E (0) α 1 +α 2 |τ 2 = 1 τ (0) 2 λ 2 |E (0) α 1 +α 2 Ψ (vac) n j=1 1 + E c j ,d j (z j ) Ψ (vac) −1 |λ 2 (52) q = 1 τ (0) 0 λ 0 |E (1) −α 1 −α 2 |τ 0 = 1 τ (0) 0 λ 0 |E (1) −α 1 −α 2 Ψ (vac) n j=1 1 + E c j ,d j (z j ) Ψ (vac) −1 |λ 0 (53) U 2 = −∂ x ln τ (0) 0 / τ (0) 2 ; ν = −∂ x ln τ (0) 0(54) where τ (0) σ = λ σ |Ψ (vac) n j=1 1 + E c j ,d j (z j ) Ψ (vac) −1 |λ σ ; σ = 0, 2(55) Using association between the step operators (41)-(43) and the vertex operators (45)-(47) we can rewrite the step operators appearing in (52) and (53) as E (0) α 1 +α 2 = − 1 2iπ dz 0 V 1, √ 3 (z 0 ) ; E (1) −α 1 −α 2 = 1 2iπ dz 0 V 1,− √ 3 (z 0 )(56) We now calculate the zero-curvature equations (25) [ D − E − A , ∂ tn − A n ] = 0 for the first two non-trivial cases of n = 2, 3. We expand A n = n i=0 A n (i) where the index i in the parenthesis equals grading with respect to Q s = λ 2 · H 0 + 2d. We choose A 3 (3) = E (1) α 2 + E (2) −α 2 and A 2 (2) = √ 3λ 1 · H (1) in order to ensure truncation of the expansion. This method yields for the first non-trivial case (n = 2) the evolution equations 0 = ∂ t r + ∂ 2 x r + r∂ x U 2 − qr 2 − U 2 2 r ; 0 = ∂ t U 2 + ∂ x (rq) (57) 0 = ∂ t q − ∂ 2 x q + q∂ x U 2 + rq 2 + U 2 2 q(58) where we defined for simplicity t = √ 3t 2 . The evolution equations for t 3 are 0 = ∂ t 3 U 2 − 1 4 ∂ 3 x U 2 + 1 2 ∂ x U 3 2 + 3 4 ∂ x (r∂ x q − q∂ x r) (59) 0 = ∂ t 3 r − ∂ 3 x r − 3 2 ∂ x r∂ x U 2 − 3 4 r∂ 2 x U 2 + 3 2 rU 2 ∂ x U 2 + 3 2 U 2 2 ∂ x r + 3 2 qr 2 U 2 + 9 4 rq∂ x r − 3 4 r 2 ∂ x q (60) 0 = ∂ t 3 q − ∂ 3 x q + 3 2 ∂ x q∂ x U 2 + 3 4 q∂ 2 x U 2 + 3 2 qU 2 ∂ x U 2 + 3 2 U 2 2 ∂ x q − 3 2 q 2 rU 2 + 9 4 rq∂ x q − 3 4 q 2 ∂ x r(61) These equations follow also from the conventional Sato equations ∂ tn L = (L) (n/2) + , L applied to the scalar cKP Lax operator L = (∂ − U 2 )(∂ + U 2 − q∂ −1 r). We note here that the simple reduction of the matrix Lax operator from (51) yields the scalar spectral problem L 1 χ = λχ with the scalar Lax operator L 1 = (∂ + U 2 )(∂ − U 2 − r∂ −1 q). Both scalar Lax operators are related by a conjugation and the Darboux-Bäcklund transformation: L 1 = (∂ + U 2 )L * (∂ + U 2 ) −1 . We now present few examples of the soliton solutions (52)-(55) satisfying the above evolution equations. 1) One-soliton solution, n = 1. With h = (1 + E −2,0 (z 1 )) we recover the standard m-KdV one-soliton configuration with r = q = 0 and τ (0) 0 = 1 − 1 2 e ( −2 x z 1 −2 t 3 z 1 3 ) ; τ (0) 2 = 1 + 1 2 e ( −2 x z 1 −2 t 3 z 1 3 )(62) 2) Two-soliton solutions, n = 2. For h = (1 + E −2,0 (z 1 ))(1 + E 1, √ 3 (z 2 )) we find τ q = − √ 2 z 2 e ( t 3 z 2 3 +t z 2 2 +x z 2 ) 1 + 1 2 e ( −2 x z 1 −2 t 3 z 1 3 ) z 1 + z 2 z 1 − z 2 /τ (0) 0(63) while r = 0. Similarly, for h = (1 + E −2,0 (z 1 ))(1 + E 1,− √ 3 (z 2 )) we find r = 0 but q = 0. For h = (1 + E 1, √ 3 (z 1 ))(1 + E 1,− √ 3 (z 2 )) we find that both q = 0 and r = 0: τ (0) σ = 1 + (−1) (σ/2) 2 z 1 1+σ/2 z 2 2−σ/2 e ( x z 1 +t z 1 2 +t 3 z 3 1 +x z 2 −t z 2 2 +t 3 z 3 2 ) ( z 1 − z 2 ) ( z 1 + z 2 ) 2 ; σ = 0, 2 (64) r = √ 2 z 2 e ( −t z 2 2 +x z 2 +t 3 z 3 2 ) τ (0) 2 ; q = √ 2 z 1 e ( t z 1 2 +x z 1 +t 3 z 3 1 ) τ (0) 0(65) 3) Three-soliton solutions, n = 3. As an example we take here h = (1 + E −2,0 (z 1 ))(1 + E 1, √ 3 (z 2 ))(1 + E 1,− √ 3 (z 3 ) ). We find τ (0) σ = 1 + (−1) (σ/2) 1 2 e ( −2 x z 1 −2 t 3 z 1 3 ) + (−1) (σ/2) 2 z 3 2−σ/2 z 2 1+σ/2 ( z 2 − z 3 ) ( z 2 + z 3 ) 2 e ( t 3 z 2 3 +t z 2 2 +x z 2 +t 3 z 3 3 −t z 3 2 +x z 3 )(66)+ z 3 2−σ/2 z 2 1+σ/2 ( z 1 + z 2 ) ( z 1 + z 3 ) ( z 2 − z 3 ) ( z 2 + z 3 ) 2 ( z 1 − z 2 )( z 1 − z 3 ) e ( −2 x z 1 −2 t 3 z 1 3 +t z 2 2 +x z 2 +t 3 z 2 3 −t z 3 2 +x z 3 +t 3 z 3 3 ) r = √ 2 z 3 e ( −t z 3 2 +x z 3 +t 3 z 3 3 ) 1 + 1 2 e ( −2 x z 1 −2 t 3 z 1 3 ) ( z 1 + z 3 ) z 1 − z 3 /τ (0) 2 (67) q = √ 2 z 2 e ( t z 2 2 +x z 2 +t 3 z 2 3 ) 1 − 1 2 e ( −2 x z 1 −2 t 3 z 1 3 ) ( z 1 + z 2 ) z 1 − z 2 /τ (0) 0(68) In the above examples U 2 and ν can be obtained from (54). We note that we only kept the explicit time dependence on times t n with n ≤ 3, which was enough to verify the evolution equations (57)-(61). The novel feature of the above soliton solutions is that they mix exponentials exp ∞ n=1 t n z n j which represent a typical time dependance for the KP solutions with pure KdV-like time dependance of the type exp ∞ n=0 t 2n+1 z 2n+1 j involving only odd times. The exception is provided by the pure KP type of solution in eq.(64)-(65), which can also be obtained as a Wronskian arising from the Darboux -Bäcklund transformations. Conclusions We presented here new soliton solutions for the cKP hierarchy. They exhibit a nontrivial mixing of KP times t n with all indices n's and KdV-like times t 2n−1 with only odd indices. From the point of view of intermediate gradation, such a mixture is indeed very natural. In fact, we are able to obtain pure KdV solutions, pure KP solutions (meaning all times) and the arbitrary mixtures thereof, just by varying the constant group elements h of the dressing orbit. We were not able to obtain these solutions from the conventional Darboux-Bäcklund method involving Wronskian representation. Recall, that such Wronskian is given in terms of the eigenfunctions f i satisfying ∂ n f i = ∂ n x f i for all n. This type of time dependence fits only a subclass of our soliton solutions, namely those which are of the pure KP type (meaning that all t n 's with all n's are present without the nontrivial mixture with KdV times). Based on studying many examples of these solutions, we believe that the Wronskian method works equally well in these cases. α i +α i+1 +...+α M ) , i = 1, 2, . . . , M, andĉ is a central element ofĜ. N − A N , N = 1, . . .. The hierarchy is then formulated in terms of the zero-curvature equations for the Lax operators [ L N , L M ] = 0 (25) expressing commutativity of the higher time flows. The zero-curvature equations imply the pure gauge solutions for the potentials A N : for a = 1 , 1. . . , K. Like before, we make a splitting B −1 = B −1 2 B −1 1 in (30) and notice that B −1 1 |λ M +a = |λ M +a for B 1 being an exponential of sl(M) generators. Inserting B 2 from (22) we find |τ 0 = Θ −1 |λ 0 e (−ρ) We find by comparing with relation (21) thatr l = ǫ(α M +1 , −α l −...−α M +1 ) λ M +1 |E(0) α l +...+α M +1 |τ M +1 /τ = ǫ(−α M +1 −...−α M +K , α l +...+α M +K ) λ 0 |E − (α l +...+α M +K ) |τ 0 /τ in (62) but now with q = 0 and equal to . B Konopelchenko, W Strampp, J. Math. Phys. 333676B. Konopelchenko and W. Strampp, J. Math. Phys. 33 (1992) 3676; . Y Cheng, J. Math. Phys. 333774Y. Cheng, J. Math. Phys. 33 (1992) 3774; . L A Dickey, hep-th/9407038Letters in Math. Phys. 34379L.A. Dickey, Letters in Math. Phys. 34 (1995) 379 (hep-th/9407038); . W Oevel, W Strampp, Commun. Math. Phys. 15751W. Oevel and W. Strampp, Commun. Math. Phys. 157 (1993) 51; . H Aratyn, E Nissimov, S Pacheva, I Vaysburd, hep-th/9209006Phys. Lett. 294167H. Aratyn, E. Nissimov, S. Pacheva and I. Vaysburd, Phys. Lett. 294B (1992) 167 (hep-th/9209006) . H Aratyn, E Nissimov, S Pacheva, hep-th/9501018Phys. Lett. 201293H. Aratyn, E. Nissimov and S. Pacheva, Phys. Lett. 201A (1995) 293 (hep-th/9501018) . H Aratyn, E Nissimov, S Pacheva, hep-th/9602068Phys. Lett. 228H. Aratyn, E. Nissimov and S. Pacheva, Phys. Lett. 228A (1997) 164 (hep-th/9602068) . Y Cheng, Zhang , J. Physics. 274581Y. Cheng and Zhang, J. Physics A27 (1994) 4581; . W Oevel, W Strampp, J. Math. Phys. 376213W. Oevel and W. Strampp, J. Math. Phys. 37 (1996) 6213; . I Loris, R Willox, J. Math. Phys. L.-L. Chau, J.C. Shaw and M.-H. Tu384128J. Math. Phys.I. Loris and R. Willox, J. Math. Phys. 38 (1997) 283; L.-L. Chau, J.C. Shaw and M.-H. Tu, J. Math. Phys. 38 (1997) 4128 . H Aratyn, L A Ferreira, J F Gomes, A H Zimerman, hep-th/9509096J. Math. Phys. 381559H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, J. Math. Phys. 38 (1997) 1559 (hep-th/9509096) V G Kac, D H Peterson, Symposium on Anomalies, Geometry and Topology. W.A. Bardeen and A.R. WhiteSingaporeWorld ScientificV.G. Kac and D.H. Peterson, in Symposium on Anomalies, Geometry and Topology, W.A. Bardeen and A.R. White (eds.), Singapore, World Scientific (1985) 276-298; V G Kac, Infinite Dimensional Lie Algebras (3 rd ed.). CambridgeCambridge University PressV.G. Kac, Infinite Dimensional Lie Algebras (3 rd ed.), Cambridge University Press, Cambridge (1990). . V G , V V Sokolov, Soviet. Math. Dokl. 30457J. Soviet Math.V. G. Drinfel'd and V. V. Sokolov, J. Soviet Math. 30 (1985) 1975; Soviet. Math. Dokl. 23 (1981) 457. . G Wilson, Ergod. Th. and Dynam. Sys. 1361G. Wilson, Ergod. Th. and Dynam. Sys. 1 (1981) 361; . Phil. Trans. R. Soc. Lond. A. 315383Phil. Trans. R. Soc. Lond. A 315 (1985) 383 . M F De Groot, T J Hollowood, J L Miramontes, Commun. Math. Phys. 14557M.F. de Groot, T.J. Hollowood, and J.L. Miramontes, Commun. Math. Phys. 145 (1992) 57 . N J Burroughs, M F De Groot, T J Hollowood, J L Miramontes, Commun. Math. Phys. 153187also in hep-th/9109014N.J. Burroughs, M.F. de Groot, T.J. Hollowood, and J.L. Miramontes, Commun. Math. Phys. 153 (1993) 187 (also in hep-th/9109014); . Phys. Lett. 27789also in hep-th/9110024Phys. Lett. 277B (1992) 89 (also in hep-th/9110024) . I Mcintosh, J. Math. Phys. 345159I. McIntosh, J. Math. Phys. 34 (1993) 5159 . A P Fordy, P P Kulish, Commun. Math. Phys. 89427A.P. Fordy and P.P. Kulish, Commun. Math. Phys. 89 (1983) 427; A P Fordy, pg. 315Soliton Theory: a Survey of Results. A.P. FordyManchesterUniversity PressA.P. Fordy, in Soliton Theory: a Survey of Results, (ed. A.P. Fordy) University Press, Manchester (1990), pg. 315 . H Aratyn, J F Gomes, A H Zimerman, J. Math. Phys. 363419also in hepth/9408104H. Aratyn, J.F. Gomes and A.H. Zimerman, J. Math. Phys. 36 (1995) 3419 (also in hep- th/9408104) H Aratyn, L A Ferreira, J F Gomes, A H Zimerman, proceedings of "Integrable Models and Supersymmetry. "Integrable Models and SupersymmetryH. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, in proceedings of "Integrable Models and Supersymmetry" . L A Ferreira, J L Miramontes, J Sánchez Guillén, hep- th/9606066J. Math. Phys. 38L.A. Ferreira, J.L. Miramontes and J. Sánchez Guillén, J. Math. Phys. 38 (1997) (hep- th/9606066)
[]
[ "Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements", "Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements", "Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements", "Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements" ]
[ "Zhiqiang Cai ", "Cuiyu He ", "Shun Zhang ", "Zhiqiang Cai ", "Cuiyu He ", "Shun Zhang " ]
[]
[]
The well-known Prager-Synge identity is valid in H 1 (Ω) and serves as a foundation for developing equilibrated a posteriori error estimators for continuous elements. In this paper, we introduce a new inequality, that may be regarded as a generalization of the Prager-Synge identity, to be valid for piecewise H 1 (Ω) functions for diffusion problems. The inequality is proved to be identity in two dimensions.For nonconforming finite element approximation of arbitrary odd order, we propose a fully explicit approach that recovers an equilibrated flux in H(div; Ω) through a local element-wise scheme and that recovers a gradient in H(curl; Ω) through a simple averaging technique over edges. The resulting error estimator is then proved to be globally reliable and locally efficient. Moreover, the reliability and efficiency constants are independent of the jump of the diffusion coefficient regardless of its distribution.
null
[ "https://arxiv.org/pdf/2001.09102v1.pdf" ]
210,911,814
2001.09102
d47ddcc9b25ff9728a780b90ed9988bb3dd2186c
Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements January 27, 2020 Zhiqiang Cai Cuiyu He Shun Zhang Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements January 27, 2020 The well-known Prager-Synge identity is valid in H 1 (Ω) and serves as a foundation for developing equilibrated a posteriori error estimators for continuous elements. In this paper, we introduce a new inequality, that may be regarded as a generalization of the Prager-Synge identity, to be valid for piecewise H 1 (Ω) functions for diffusion problems. The inequality is proved to be identity in two dimensions.For nonconforming finite element approximation of arbitrary odd order, we propose a fully explicit approach that recovers an equilibrated flux in H(div; Ω) through a local element-wise scheme and that recovers a gradient in H(curl; Ω) through a simple averaging technique over edges. The resulting error estimator is then proved to be globally reliable and locally efficient. Moreover, the reliability and efficiency constants are independent of the jump of the diffusion coefficient regardless of its distribution. Introduction Equilibrated a posteriori error estimators have attracted much interest recently due to the guaranteed reliability bound with the reliability constant being one. This property implies that they are perfect for discretization error control on both coarse and fine meshes. Error control on coarse meshes is important but difficult for computationally challenging problems. For the conforming finite element approximation, a mathematical foundation of equilibrated estimators is the Prager-Synge identity [31] that is valid in H 1 (Ω) (see Section 3). Based on this identity, various equilibrated estimators have been studied recently by many researchers (see, e.g., [27,22,29,20,21,6,3,33,10,12,13,34,17,14]). The key ingredient of the equilibrated estimators for the continuous elements is local recovery of an equilibrated (locally conservative) flux in the H(div; Ω) space through the numerical flux. By using a partition of unity, Ladevèze and Leguillon [27] initiated a local procedure to reduce the construction of an equilibrated flux to vertex patch based local calculations. For the continuous linear finite element approximation to the Poisson equation in two dimensions, an equilibrated flux in the lowest order Raviart-Thomas space was explicitly constructed in [10,12]. This explicit approach does not lead to robust equilibrated estimator with respect to the coefficient jump without introducing a constraint minimization (see [17]). The constraint minimization on each vertex patch may be efficiently solved by first computing an equilibrated flux and then calculating a divergence free correction. For recent developments, see [14] and references therein. The purpose of this paper is to develop and analyze equilibrated a posteriori error estimators for discontinuous elements including both nonconforming and discontinuous Galerkin elements. To do so, the first and the essential step is to extend the Prager-Synge identity to be valid for piecewise H 1 (Ω) functions. This will be done by establishing a generalized Prager-Synge inequality (see Theorem 3.1) that contains an additional term measuring the distance between H 1 (Ω) and piecewise H 1 (Ω). Moreover, by using a Helmholtz decomposition, we will be able to show that the inequality becomes an identity in two dimensions (see Lemma 3.4). A nonoptimal inequality similar to ours was obtained earlier by Braess, Fraunholz, and Hoppe in [11] for the Poisson equation with pure Dirichlet boundary condition. Based on the generalized Prager-Synge inequality and an equivalent form (see Corollary 3.2), the construction of an equilibrated a posteriori error estimator for discontinuous finite element solutions is reduced to recover an equilibrated flux in H(div; Ω) and to recover either a potential function in H 1 (Ω) or a curl free vector-valued function in H(curl; Ω). Recovery of equilibrated fluxes for discontinuous elements has been studied by many researchers. For discontinuous Garlerkin (DG) methods, equilibrated fluxes in Raviart-Thomas (RT) spaces were explicitly reconstructed in [2] for linear elements and in [23] for higher order elements. For nonconforming finite element methods, existing explicit equilibrated flux recoveries in RT spaces seem to be limited to the linear Crouzeix-Raviart (CR) and the quadratic Fortin-Soulie elements by Marini [28] (see [1] in the context of estimator) and Kim [26], respectively. For higher order nonconforming elements, a local reconstruction procedure was proposed by Ainsworth and Rankin in [4] through solving element-wise minimization problems. The recovered flux is not in the RT spaces. Nevertheless, the resulting estimator provides a guaranteed upper bound. In this paper, we will introduce a fully explicit post-processing procedure for recovering an equilibrated flux in the RT space of index k − 1 for the nonconforming elements of odd order of k ≥ 1. Currently, we are not able to extend our recovery technique to even orders. This is because our recovery procedure heavily depends on the finite element formulation and the properties of the nonconforming finite element space; moreover, structure of the nonconforming finite element spaces of even and odd orders are fundamentally different. Recovery of a potential function in H 1 (Ω) for discontinuous elements was studied by some researchers (see, e.g., [4,2,11]). Local approaches for recovering equilibrated flux in [10,12,17,13, 14] may be directly applied (at least in two dimensions) for computing an approximation to the gradient in the curl-free space. (As mentioned previously, this approach requires solutions of local constraint minimization problems over vertex patches.) The resulting a posteriori error estimator from either the potential or the gradient recoveries may be proved to be locally efficient. Nevertheless, to show independence of the efficiency constant on the jump, we have to assume that the distribution of the diffusion coefficient is quasi-monotone (see [30]). In this paper, we will employ a simple averaging technique over edges to recover a gradient in H(curl; Ω). Due to the fact that the recovered gradient is not necessarily curl free, the reliability constant of the resulting estimator is no longer one. However, it turns out that the curl free constraint is not essential and, theoretically we are able to prove that the resulting estimator has the robust local reliability as well as the robust local efficiency without the quad-monotone assumption. This is compatible with our recent result in [15] on the residual error estimator for discontinuous elements. This paper is organized as follows. The diffusion problem and the finite element mesh are introduced in Section 2. The generalized Prager-Synge inequality for piecewise H 1 (Ω) functions are established in Section 3. Explicit recoveries of an equilibrated flux and a gradient and the resulting a posteriori error estimator for discontinuous elements are described in Section 4. Global reliability and local efficiency of the estimator are proved in Section 5. Finally, numerical results are presented in Section 6. Model problem Let Ω be a bounded polygonal domain in where ∇· and ∇ are the respective divergence and gradient operators; n is the outward unit vector normal to the boundary; f ∈ L 2 (Ω) and g ∈ H −1/2 (Γ N ) are given scalar-valued functions; and the diffusion coefficient A(x) is symmetric, positive definite, and piecewise constant full tensor with respect to the domain Ω = ∪ n i=1 Ω i . Here we assume that the subdomain, Ω i for i = 1, · · · , n, is open and polygonal. R d , d = 2, 3, with Lipschitz boundary ∂Ω = Γ D ∪ Γ N , where Γ D ∩ Γ N = ∅. We use the standard notations and definitions for the Sobolev spaces. Let H 1 D (Ω) = v ∈ H 1 (Ω) : v = 0 on Γ D . Then the corresponding variational problem of (2.1) is to find u ∈ H 1 D (Ω) such that a(u, v) := (A∇u, ∇v) = (f, v) − g, v Γ N , ∀ v ∈ H 1 D (Ω), (2.2) where (·, ·) ω is the L 2 inner product on the domain ω. The subscript ω is omitted when ω = Ω. Triangulation Let T = {K} be a finite element partition of Ω that is regular, and denote by h K the diameter of the element K. Furthermore, assume that the interfaces, Γ = {∂Ω i ∩ ∂Ω j : i = j and i, j = 1, · · · , n}, do not cut through any element K ∈ T . Denote the set of all edges of the triangulation T by E := E I ∪ E D ∪ E N , where E I is the set of interior element edges, and E D and E N are the sets of boundary edges belonging to the respective Γ D and Γ N . For each F ∈ E, denote by h F the length of F and by n F a unit vector normal to F . Let K + F and K − F be the two elements sharing the common edge F ∈ E I such that the unit outward normal of K − F coincides with n F . When F ∈ E D ∪ E N , n F is the unit outward normal to ∂Ω and denote by K − F the element having the edge F . Generalized Prager-Synge inequality For the conforming finite element approximation, the foundation of the equilibrated a posteriori error estimator is the Prager-Synge identity [31]. That is, let u ∈ H 1 D (Ω) be the solution of (2.1), then A 1/2 ∇ (u − w) 2 + A −1/2 τ + A 1/2 ∇ u 2 = A −1/2 τ + A 1/2 ∇ w 2 for all w ∈ H 1 D (Ω) and for all τ ∈ Σ f (Ω), where Σ f (Ω) is the so-called equilibrated flux space defined by Σ f (Ω) = τ ∈ H(div; Ω) : ∇ · τ = f in Ω and τ · n = g N . Here, H(div; Ω) ⊂ L 2 (Ω) d denotes the space of all vector-valued functions whose divergence are in L 2 (Ω). The Prager-Synge identity immediately leads to A 1/2 ∇ (u − w) 2 ≤ inf τ ∈Σ f (Ω) A −1/2 τ + A 1/2 ∇ w 2 . (3.1) Choosing w ∈ H 1 D (Ω) to be the conforming finite element approximation, then (3.1) implies that η τ := A −1/2 τ + A 1/2 ∇ w , ∀ τ ∈ Σ f (Ω) (3.2) is a reliable estimator with the reliability constant being one. We now proceed to establish a generalization of (3.1) for piecewise H 1 (Ω) functions with applications to nonconforming and discontinuous Galerkin finite element approximations. To this end, denote the broken H 1 (Ω) space with respect to T by H 1 (T ) = v ∈ L 2 (Ω) : v| K ∈ H 1 (K), ∀ K ∈ T . Define ∇ h be the discrete gradient operator on H 1 (T ) such that for any v ∈ H 1 (T ) (∇ h v)| K = ∇(v| K ), ∀K ∈ T . Theorem 3.1. Let u ∈ H 1 D (Ω) be the solution of (2.1). In both two and three dimensions, for all w ∈ H 1 (T ), we have A 1/2 ∇ h (u − w) 2 ≤ inf τ ∈Σ f (Ω) A −1/2 τ + A 1/2 ∇ h w 2 + inf v∈H 1 D (Ω) A 1/2 ∇ h (v − w) 2 . (3.3) Proof. Let w ∈ H 1 (T ), for all τ ∈ Σ f (Ω) and for all v ∈ H 1 D (Ω), it follows from integration by parts and the Cauchy-Schwarz and Young's inequalities that 2 (∇ h (u − w), A∇u + τ ) = 2 (∇(u − v), A∇u + τ ) + 2 (∇ h (v − w), A∇u + τ ) = 2 (∇ h (v − w), A∇u + τ ) ≤ A 1/2 ∇ h (v − w) 2 + A 1/2 ∇u + A −1/2 τ 2 . (3.4) It is easy to see that A 1/2 ∇ h w + A −1/2 τ 2 = A 1/2 ∇ h (u − w) 2 + A 1/2 ∇u + A −1/2 τ 2 − 2(∇ h (u − w), A∇u + τ ), which, together with (3.4), implies A 1/2 ∇ h (u − w) 2 = A 1/2 ∇ h w + A −1/2 τ 2 − A 1/2 ∇u + A −1/2 τ 2 + 2(∇ h (u − w), A∇u + τ ) ≤ A 1/2 ∇ h w + A −1/2 τ 2 + A 1/2 ∇ h (v − w) 2 for all τ ∈ Σ f (Ω) and all v ∈ H 1 D (Ω). This implies the validity of (3.3) and, hence, the theorem. A suboptimal result for the Poisson equation (A = I) with pure Dirichlet boundary condition is proved in [11] by Braess, Fraunholz, and Hoppe: ∇ h (u − w) ≤ inf τ ∈Σ f (Ω) ∇w + τ + 2 inf v∈H 1 0 (Ω) ∇ h (v − w) . Let H(curl; Ω) ⊂ L 2 (Ω) d be the space of all vector-valued functions whose curl are in L 2 (Ω), and denote its curl free subspace bẙ where t denotes the tangent vector(s). Corollary 3.2. Let u ∈ H 1 D (Ω) be the solution of (2.1). In both two and three dimensions, for all w ∈ H 1 (T ), we have A 1/2 ∇ h (u − w) 2 ≤ inf τ ∈Σ f (Ω) A −1/2 τ + A 1/2 ∇ h w 2 + inf γ∈H D (curl;Ω) A 1/2 (γ − ∇ h w) 2 . (3.5) Proof. The result of (3.5) is an immediate consequence of (3.3) and the fact that ∇H 1 D (Ω) = H D (curl; Ω). In the remaining section, we prove that, in two dimensions, the inequality (3.3) in Theorem 3.1 is indeed an equality. For each F ∈ E, in two dimensions, assume that n F = (n 1,F , n 2,F ), then denote by t F = (−n 2,F , n 1,F ) the unit vector tangent to F and by s F and e F the start and end points of F , respectively, such that e F − s F = h F t F . Let H = v ∈ H 1 (Ω) : Ω v dx = 0 and ∂v ∂t = 0 on Γ N . For a vector-valued function τ = (τ 1 , τ 2 ) ∈ H(curl; Ω), define the curl operator by ∇×τ = ∂τ 2 ∂x − ∂τ 1 ∂y . For a scalar-valued function v ∈ H 1 (Ω), define the formal adjoint operator of the curl by ∇ ⊥ v = ∂v ∂y , − ∂v ∂x . For a fixed w ∈ H 1 (T ), there exist unique φ ∈ H 1 D (Ω) and ψ ∈ H for the following Helmholtz decomposition (see, e.g., [4]) such that A∇ h (u − w) = A∇φ + ∇ ⊥ ψ,(3.6) and φ and ψ satisfy (A∇φ, ∇v) = (A∇ h (u − w), ∇v) ∀v ∈ H 1 D (Ω), and (A −1 ∇ ⊥ ψ, ∇ ⊥ w) = (∇ h (u − w), ∇ ⊥ w) ∀w ∈ H, respectively. It is easy to see that ∇φ and ∇ ⊥ ψ are orthogonal with respect to the L 2 inner product, which yields A 1/2 ∇ h (u − w) 2 = A 1/2 ∇φ 2 + A −1/2 ∇ ⊥ ψ 2 . (3.7) Lemma 3.3. Let w be a fixed function in H 1 (T ) and φ and ψ be the corresponding Helmholtz decomposition of w given in (3.6). We have inf τ ∈Σ f (Ω) A −1/2 τ + A 1/2 ∇ h w = A 1/2 ∇φ and inf v∈H 1 D (Ω) A 1/2 ∇ h (v − w) = A −1/2 ∇ ⊥ ψ . (3.8) Proof. For any τ ∈ Σ f (Ω), (3.6) and integration by parts give A 1/2 ∇φ 2 = (A∇ h (u − w), ∇φ) = (A∇u + τ , ∇φ) − (τ + A∇ h w, ∇φ) = −(τ + A∇ h w, ∇φ), which, together with the Cauchy-Schwarz inequality and the choice τ = ∇ ⊥ ψ − A∇u ∈ Σ f (Ω), yields the first equality in (3.8) as follows: A 1/2 ∇φ ≤ inf τ ∈Σ f (Ω) A −1/2 τ + A 1/2 ∇ h w ≤ A 1/2 ∇ h (u − w) − A −1/2 ∇ ⊥ ψ = A 1/2 ∇φ . Now we proceed to prove the second equality in (3.8). For any v ∈ H 1 D (Ω), by (3.6) and integration by parts, we have A −1/2 ∇ ⊥ ψ 2 = (∇ h (u − w), ∇ ⊥ ψ) = (∇ h (v − w), ∇ ⊥ ψ). The second equality in (3.8) is then a consequence of the Cauchy-Schwartz inequality and the choice of v = u − φ ∈ H 1 D (Ω): A −1/2 ∇ ⊥ ψ ≤ inf v∈H 1 D (Ω) A 1/2 ∇ h (v − w) ≤ A 1/2 ∇ h (u − φ − w = A −1/2 ∇ ⊥ ψ . This completes the proof of the lemma. Lemma 3.4. Let u ∈ H 1 D (Ω) be the solution of (2.1). In two dimensions, for all w ∈ H 1 (T ), we have A 1/2 ∇ h (u − w) 2 = inf τ ∈Σ f (Ω) A −1/2 τ + A 1/2 ∇ h w 2 + inf v∈H 1 D (Ω) A 1/2 ∇ h (v − w) 2 . (3.9) Proof. The identity (3.9) is a direct consequence of (3.7) and Lemma 3.3. Remark 3.5. It is easy to see that if w ∈ H 1 D (Ω) in Lemma 3.4, i.e., w is conforming, the second part on the right of (3.9) vanishes. It is thus natural to refer inf τ ∈Σ f (Ω) A −1/2 τ + A 1/2 ∇ h w 2 or A 1/2 ∇φ as the conforming error and inf v∈H 1 D (Ω) A 1/2 ∇ h (v − w) 2 or A −1/2 ∇ ⊥ ψ as the nonconforming error. For each K ∈ T , denote by Λ K and λ K the maximal and minimal eigenvalues of A K = A| K , respectively. For each F ∈ E, let Λ ± F = Λ K ± F , λ ± F = λ K ± F , and λ F = min{λ + F , λ − F } if F ∈ E I and λ F = λ − F if F ∈ E D ∪ E N . To this end, let Λ T = max K∈T Λ K and λ T = min K∈T λ K . Assume that each local matrix A K is similar to the identity matrix in the sense that its maximal and minimal eigenvalues are almost of the same size. More precisely, there exists a moderate size constant κ > 0 such that Λ K λ K ≤ κ, ∀ K ∈ T . Nevertheless, the ratio of global maximal and minimal eigenvalues, Λ T /λ T , is allowed to be very large. For a function w ∈ H 1 (T ) , denote its traces on F by w| − F := (w| K − F )| F and w| + F := (w| K + F )| F and the jump of w across the edge F by [[w]]| F = w| − F − w| + F , ∀ F ∈ E I , w| − F , ∀ F ∈ E D ∪ E N . In the following lemma, we show the relationship between the nonconforming error and the residual based error of solution jump on edges. It is noted that the constant is robust with respect to the coefficient jump. Lemma 3.6. Let w be a fixed function in H 1 (T ). In two dimensions, there exists a constant C r that is independent of the jump of the coefficient such that inf τ ∈H D (curl;Ω) A 1/2 (τ − ∇ h w) ≤ C r   F ∈E I ∪E D λ F h −1 F [[w]] 2 0,F   1/2 . (3.10) Proof. Let ψ be given in the Helmholtz decomposition in (3.6), then integration by parts gives A −1/2 ∇ ⊥ ψ 2 = (∇ h (u − w), ∇ ⊥ ψ) = − F ∈E I ∪E D F [[w]] ∇ ⊥ ψ · n F ds. Without loss of generality, assume that λ − F ≤ λ + F for each F ∈ E I . It follows from Lemma 2.4 in [15] and the Cauchy-Schwarz inequality that F ∈E I ∪E D F [[w]] ∇ ⊥ ψ · n F ds ≤ C F ∈E I ∪E D h −1/2 F [[w]] 0,F ∇ ⊥ ψ 0,K − F ≤ C   F ∈E I ∪E D λ F h −1 F [[w]] 2 0,F   1/2 A −1/2 ∇ ⊥ ψ , which, together with the above equality, yields A −1/2 ∇ ⊥ ψ ≤ C   F ∈E I ∪E D λ F h −1 F [[w]] 2 0,F   1/2 . This completes the proof of the lemma. Error estimators and indicators NC finite element approximation For the convenience of readers, in this subsection we introduce the nonconforming finite element space and its properties. Let P k (K) and P k (F ) be the spaces of polynomials of degree less than or equal to k on the element K and F , respectively. Define the nonconforming finite element space of order k(k ≥ 1) on the triangulation T by U k (T ) = v ∈ L 2 (Ω) : v| K ∈ P k (K), ∀ K ∈ T and F [[v]] p ds = 0, ∀ p ∈ P k−1 (F ), ∀ F ∈ E I (4.1) and its subspace by U k D (T ) = v ∈ U k (T ) : F v p ds = 0, ∀ p ∈ P k−1 (F ) and ∀ F ∈ E D . The spaces defined above are exactly the same as those defined in [19] for k = 1, [24] for k = 2, [18] for k = 4 and 6, [4] for general odd order, and [32,5] for general order. Then the nonconforming finite element approximation of order k is to find u T ∈ U k D (T ) such that a h (u T , v) := (A∇ h u T , ∇ h v) = (f, v) − g, v Γ N , ∀ v ∈ U k D (T ). (4.2) Below we describe basis functions of U k (T ) and their properties. To this end, for each K ∈ T , let m k = dim(P k−3 (K)) for k > 3 and m k = 0 for k ≤ 3. Denote by {x j , j = 1, · · · , m k } the set of all interior Lagrange points in K with respect to the space P k (K) and by P j,K ∈ P k−3 (K) the nodal basis function corresponding to x j , i.e., P j,K (x i ) = δ ij for i = 1, · · · , m k , where δ ij is the Kronecker delta function. For each 0 ≤ j ≤ k − 1, let L j,F be the jth order Gauss-Legendre polynomial on F such that L j,F (e F ) = 1. Note that L j,F is an odd or even function when j is odd or even. Hence, L j,F (s F ) = −1 for odd j and L j,F (s F ) = 1 for even j. For odd k, the set of degrees of freedom of U k (T ) (see Lemma 2.1 in [4]) can be given by K v P j,K dx, j = 1, · · · , m k (4.3) for all K ∈ T and F v L j,F ds, j = 0, · · · , k − 1 (4.4) for all F ∈ E. Define the basis function φ i,K ∈ U k (T ) satisfying    K φ i,K P j,K dx = δ ij δ KK , ∀ j = 1, · · · , m k , ∀ K ∈ T , F φ i,K L j,F ds = 0, ∀ j = 0, · · · , k − 1, ∀ F ∈ E,(4.5) for i = 1, · · · , m k and K ∈ T , and the basis function φ i,F ∈ U k (T ) satisfying    K φ i,F P j,K dx = 0, ∀ j = 1, · · · , m k , ∀ K ∈ T , F φ i,F L j,F ds = δ ij δ F F , ∀ j = 0, · · · , k − 1, ∀ F ∈ E,(4.6) for i = 0, · · · , k −1 and F ∈ E. Then the nonconforming finite element space is the space spanned by all these basis functions, i.e., Proof. Obviously, (4.5) implies that support{φ j,K } ∈ K. To show that φ j,K | ∂K ≡ 0, considering each edge F ∈ E K , the second equation of (4.5) indicates that there exists a F ∈ R such that U k (T ) = span {φ i,K : K ∈ T } m k i=1 ⊕ span {φ i,F : F ∈ E} k−1 i=0 .Lemmaφ j,K | F = a F L k,F . Note that L k,F is an odd function on F and that values of L k,F at two end-points of F are −1 and 1, respectively. Now the continuity of φ j,K in K implies that a F = 0 and, hence, φ j,K ≡ 0 on ∂K. For each K, denote by E K the set of all edges of K. For each F ∈ E, denote by ω F the union of all elements that share the common edge F ; and define a sign function χ F on the set E K + F ∪ E K − F \ {F } (when F is a boundary edge, let E K + F = ∅) such that χ F (F ) = 1, if e F =F ∩F , −1, if s F =F ∩F .φ j,F =      1 L j,F 2 0,F (L j,F − L k,F ) , on F, 0, on E K + F ∪ E K − F \ {F } (4.7) when j is odd, and φ j,F =              1 L j,F 2 0,F L j,F , on F, χ F (F ) L j,F 2 0,F L k,F , on F ∈ E K + F ∪ E K − F \ {F } (4.8) when j is even. Proof. By (4.6), it is easy to see that support of φ j,F is ω F . Since φ j,F | ± F ∈ P k (F ), there exist constants a ± i,F such that φ j,F | ± F = k i=0 a ± i,F L i,F . Using (4.6) and the orthogonality of {L i,F } k i=0 , it is obvious that a ± i,F = L j,F −2 0,F , for i = j, 0, for 0 ≤ i ≤ k − 1 and i = j and, hence, φ j,F | ± F = 1 L j,F 2 0,F L j,F + a ± k,F L k,F . (4.9) By (4.6), it is also easy to see that there exists constant a j,F,F for each F ∈ E K + F ∪ E K − F \ {F } such that φ j,F | F = a j,F,F L k,F . (4.10) Since L k,F is an odd function for all F ∈ E K + F ∪ E K − F \ {F } and φ j,F is continuous in K + F and K − F , (4.10) implies that φ j,F | K (s F ) = φ j,F | K (e F ), K ∈ {K + F , K − F }. (4.11) Combining the facts that L j,F (e F ) = −L j,F (s F ) = 1 for odd j and that L j,F (e F ) = L j,F (s F ) = 1 for even j, (4.9), and (4.11), we have a ± k,F =      − 1 L j,F 2 0,F , for odd j, 0, for even j, which, together with (4.9), leads to the formulas of φ j,F | F in (4.7) and (4.8). Finally, for each F ∈ E K + F ∪ E K − F \ {F }, a j, Equilibrated flux recovery In this subsection, we introduce a fully explicit post-processing procedure for recovering an equilibrated flux. To this end, define f k−1 ∈ L 2 (Ω) by f k−1 | K = Π K (f ), ∀ K ∈ T , where Π K is the L 2 projection onto P k−1 (K). For simplicity, assume that the Neumann data g is a piecewise polynomial of degree less than or equal to k − 1, i.e., g| F ∈ P k−1 (F ) for all F ∈ E N . Denote the H(div; Ω) conforming Raviart-Thomas (RT) space of index k − 1 with respect to T by RT k−1 (T ) = τ ∈ H(div; Ω) : τ | K ∈ RT k−1 (K), ∀ K ∈ T , where RT k−1 (K) = P k−1 (K) d + x P k−1 (K). Let Σ k−1 f (T ) = τ ∈ RT k−1 : ∇ · τ = f k−1 in Ω and τ · n F = g on Γ N . On a triangular element K ∈ T , a vector-valued function τ in RT k−1 (K) is characterized by the following degrees of freedom (see Proposition 2.3.4 in [9]): K τ · ζ dx, ∀ ζ ∈ P k−2 (K) d , and F (τ · n F ) p ds, ∀ p ∈ P k−1 (F ) and ∀ F ∈ E K . For each K ∈ T , define a sign function µ K on E K such that µ K (F ) = 1, if n K | F = n F , −1, if n K | F = −n F . Define the numerical flux σ T = −A∇ h u T andσ K = −A∇(u T | K ), ∀ K ∈ T . (4.13) With the numerical fluxσ T given in (4.13), for each element K ∈ T , we recover a fluxσ K ∈ RT k−1 (K) such that: Kσ K · τ dx = Kσ T · τ dx, ∀ τ ∈ P k−2 (K) d (4.14) and that Fσ K · n F L i,F ds =            µ K (F ) L i,F 2 0,F Kσ T · ∇φ i,F dx + K f φ i,F dx , ∀ F ∈ E K \ E N , µ K (F ) L i,F 2 0,F F g φ i,F ds , ∀ F ∈ E K ∩ E N (4.15) for i = 0, · · · , k − 1. Now the global recovered fluxσ T is defined bŷ σ T K =σ K , ∀ K ∈ T . (4.16) Lemma 4.5. Let u T be the finite element solution in (4.2) andσ T be the recovered flux defined in (4.16). Then for any K ∈ T , the following equality ∂Kσ T · n K q dx = Kσ T · ∇q dx + K f q dx (4.17) holds for all q ∈ P k (K). Proof. Without loss of generality, assume that K ∈ T is an interior element. For each q ∈ P k (K), there exist a j,F and a j,K such that q = F ∈E K k−1 j=0 a j,F φ j,F + m k j=1 a j,K φ j,K ≡ F ∈E K q F + q K . It follows from Lemma 4.1, (4.12), Lemma 4.2, and the definition of the recovered fluxσ T in (4.15) that ∂Kσ K · n K q ds = F ∈E K k−1 j=0 a j,F Fσ K · n K φ j,F ds = F ∈E K k−1 j=0 a j,F µ K (F ) L j,F 2 F Fσ K · n F L j,F ds = F ∈E K k−1 j=0 a j,F Kσ T · ∇φ j,F dx + K f φ j,F dx = F ∈E K Kσ T · ∇q F dx + K f q F dx . (4.18) Choosing v = φ j,K in (4.2) gives Kσ T · ∇φ j,K dx + K f φ j,K dx = 0 for j = 1, · · · , m k . Multiplying the above equality by a j,K and summing over j imply Kσ T · ∇q K dx + K f q K dx = 0. (4.19) Now (4.17) is the summation of (4.18) and (4.19). This completes the proof of the lemma. Proof. First we prove thatσ T ∈ H(div; Ω). For each F ∈ E I , note thatσ T | ± F ∈ P k−1 (F ). Then it follows from Lemma 4.2, (4.15), the assumption that g| F ∈ P k−1 (F ), and (4. 2) with v = φ j,F that F [[σ · n F ]] φ j,F ds = K∈{K + F ,K − F } µ K (F ) L k,F 2 F Fσ K · n F L j,F ds = K∈{K + F ,K − F } Kσ T · ∇φ j,F ds + K f φ j,F ds = ω Fσ T · ∇φ j,F ds + ω F f φ j,F ds − Γ N ∩∂ω F g φ j,F ds = 0 for j = 0, · · · , k − 1. Now Lemma 4.4 implies that [[σ T · n F ]]| F = 0 and, hence,σ T ∈ H(div, Ω). Second, for each K ∈ T and for any p ∈ P k−1 (K), note that ∇p ∈ P k−2 (K) d . By integration by parts, (4.14), and Lemma 4.5, we have which implies that ∇ ·σ T = f k−1 in Ω. Finally, for F ∈ E N , Lemma 4.4 and (4.15) gives Fσ T · n F φ j,F ds = L j,F −2 0,F Fσ T · n F L j,F ds = F g φ j,F ds, for j = 0, · · · , k − 1, which, together with Lemma 4.4, implies thatσ T · n F = g| F for all F ∈ E N . This completes the proof of the theorem. Gradient recovery In this subsection, we recover a gradient in the space of H(curl; Ω) for the nonconforming finite element solutions of odd orders in the two dimensions. We note that such recovery is fully explicit through a simple weighted average on each edge. Such recovery technique can be easily extended to three dimensional finite element problems with the average on facets. For the first order nonconforming Crouzeix-Raviart element, the weighted average approach is first introduced in [16]. Define index k − 1 with respect to T by NE k−1 (T ) = τ ∈ H D (curl; Ω) : τ | K ∈ NE k−1 (K), ∀ K ∈ T , where NE k−1 (K) = P k−1 (K) 2 +(−y, x) P k−1 (K) . On a triangular element K ∈ T , a vector valued function τ ∈ NE k−1 (K) is characterized by the following degrees of freedom (see Proposition 2.3.1 in [9]): K τ · ζ dx, ∀ ζ ∈ P k−2 (K) 2 and F τ · t p dx, ∀ p ∈ P k−1 (F ) and ∀F ∈ E K . Define the numerical gradient ρ T = ∇ h u T andρ K = ∇u T | K , ∀ K ∈ T . (4.20) For each edge F ∈ E, denote the i-th moment of a weighted average of the tangential components of the numerical gradient by S i,F =                  θ F F ρ K − F · t F L i,F ds + (1 − θ F ) F ρ K + F · t F L i,F ds, if F ∈ E I , 0, if F ∈ E D , F ρ K − F · t F L i,F ds, if F ∈ E N with the weight θ F = Λ − F Λ − F + Λ + F for i = 0, · · · , k − 1. For each K ∈ T , defineρ K ∈ NE k−1 (K) by          F ρ K · t F L i,F ds = S i,F , for i = 0, · · · , k − 1 and ∀ F ∈ E K , Kρ K · ζ dx = Kρ K · ζ dx, ∀ ζ ∈ P k−2 (K) 2 . (4.21) Then the recovered gradientρ T is defined in NE k−1 (T ) such that ρ T K =ρ K , ∀ K ∈ T . Equilibrated a posteriori error estimation for nonconforming solutions In section 4.2, we introduce an equilibrated flux recovery for the nonconforming elements of odd order. The construction is fully explicit. Letσ T ∈ Σ f (Ω) be the recovered flux defined in (4.16), then the local indicator and the global estimator for the conforming error are defined by η σ,K = A −1/2 (σ T −σ T ) 0,K , ∀ K ∈ T (4.23) and η σ = K∈T η 2 σ,K 1/2 = A −1/2 (σ T −σ T ) , (4.24) respectively. In section 4.3, we recover the gradient in H D (curl; Ω) through averaging on each edge. This post-process procedure is also fully explicit. Letρ T ∈ H D (curl; Ω) be the recovered gradient defined in (4.22), then the local indicator and the global estimator for the nonconforming error are defined by η ρ,K = A 1/2 (ρ T −ρ T ) 0,K , ∀ K ∈ T (4.25) and η ρ = K∈T η 2 ρ,K 1/2 = A 1/2 (ρ T −ρ T ) , (4.26) respectively. The local indicator and the global estimator for the nonconforming elements are then defined by η K = η 2 σ,K + η 2 ρ,K 1/2 and η = K∈T η 2 K 1/2 = η 2 σ + η 2 ρ 1/2 , (4.27) respectively. Equilibrated a posteriori error estimation for DG solutions We first introduce the DG finite element method. For any K ∈ T and some α > 0, let V 1+α (K) = {v ∈ H 1+α (K) : ∆ v ∈ L 2 (K)} and let V 1+α (T ) := {v : v| K ∈ V 1+α (K), ∀ K ∈ T }. We also denote the discontinuous finite element space D k of order k (for k ≥ 0) by D k = {v ∈ L 2 (Ω) : v| K ∈ P k (K), ∀K ∈ T }. For each F ∈ E I , we define the following weights: ω ± F = λ ∓ F λ − F + λ + F . In the weak formulation, we use the following weighted average: {v} F w = w + F v + F + w − F v − F , F ∈ E I , v, F ∈ E D ∪ E N . It is noted that the weighted average defined in the above way guarantees the robustness of the error estimation, see [15]. Similar to [15] we introduce the following DG formulation for (2.1): find u ∈ V 1+ (T ) with > 0 such that a dg (u, v) = (f, v) − g N , v Γ N , ∀ v ∈ V 1+ (T ),(4.28) where the bilinear form a dg (·, ·) is given by a dg (u, v) = (A∇ h u, ∇ h v) + F ∈E\E N F γ α H h F [[u]][[v]] ds − F ∈E\E N F {A∇u · n F } F w [[v]]ds − F ∈E\E N F {A∇v · n F } F w [[u]]ds. Here, α H is the harmonic average of λ over F , i.e., α H = λ + F λ − F λ + F + λ − F and γ is a positive constant only depending on the shape of elements. The discontinuous Galerkin finite element method is then to seek u dg k ∈ D k such that a dg (u dg k , v) = (f, v) ∀ v ∈ D k . (4.29) For simplicity, we consider only this symmetric version of the interior penalty discontinuous Galerkin finite element method since its extension to other versions of discontinuous Galerkin approximations is straightforward. Thanks to the complete discontinuity of the space D k , an equilibrate flux for the DG solution u dg k can be easily obtained. Here we present a formula similar to those introduced in [2, 23,8]. Recovering an equilibrate flux,σ dg k ∈ RT k−1 (K), such that (σ dg k , τ ) K = −(A∇u dg k , τ ) K − F ∈E K ∩E I 1 2 µ K Aτ · n F , [[u dg k ]] F − F ∈E K ∩E D Aτ · n F , u dg k F (4.30) for all K ∈ T and for all τ ∈ P k−2 (K) d , and that σ dg k · n F =        −{A∇u h · n F } + γh −1 F [[u h ]], ∀F ∈ E I , −A∇u h · n F + γh −1 F u h , ∀F ∈ E D , g N , ∀F ∈ E N . (4.31) It is easy to verify that the flux defined in (4.30) is equilibrate, i.e., ∇ ·σ dg k = f k where f k is the L 2 projection of f onto the space of D k . The recovery of the DG solution in the H 1 (Ω) or theH D (curl; Ω) spaces, again, suffers the lack of robustness. Similar to the nonconforming method, we also recover a gradient in the H D (curl; Ω) space. Let ρ dg k be the recovered gradient for u dg k based on the formulas in section 4.3. The error indicators and estimators for u dg k can then be similarly defined as in (4.25)-(4.27). Global reliability and local efficiency In this section, we establish the global reliability and efficiency for the error indicators and estimator defined in in (4.25)-(4.27) for the NC elements of the odd orders. Similar robust results for DG solutions can be proved in the same way. Let There exist constants C r and C that is independent of the jump of the coefficient such that osc (f, K) = h K √ λ K f − f kA 1/2 ∇ h (u − u T ) 0,Ω ≤ η σ + C r η ρ + C osc (f, T ). (5.32) Proof. The theorem is a direct result of Lemmas 5.2 and 5.3. Note that the global reliability bound in (5.32) does not require the quasi-monotonicity assumption on the distribution of the diffusion coefficient A(x). The reliability constant C r for the nonconforming error is independent of the jump of A(x), but not equal to one. This is due to the fact that the explicitly recovered gradientρ T is not curl free. In the following, we bound the conforming error above by the estimator η σ given in (4.24). inf τ ∈Σ f (Ω) A 1/2 (τ −σ T ) ≤ η σ + C osc (f, T ). (5.33) Proof. Let φ ∈ H 1 D (Ω) be the conforming part of the Helmholtz decomposition of u − u T . By (3.8), integration by parts, and the assumption that g| F ∈ P k−1 (F ), we have inf τ ∈Σ f (Ω) A −1/2 τ + A 1/2 ∇ h u T 2 0,Ω = A 1/2 ∇φ 2 = (A∇(u − u T ), ∇φ) = (A∇u +σ T , ∇φ) − (σ T −σ T , ∇φ) = (f − f k−1 , φ) − (σ T −σ T , ∇φ). Letφ K = 1 |K| K φ dx. It follows from the definitions of f k−1 and the Cauchy-Schwarz and the Poincaré inequalities that K∈T (f − f k−1 , φ) K = K∈T (f − f k−1 , φ −φ K ) K ≤ C K∈T h K λ 1/2 K f − f k−1 0,K A 1/2 ∇φ 0,K ≤ C osc (f, T ) A 1/2 ∇φ , which, together with (5.34) and the Cauchy-Schwartz inequality, leads to (5.33). This completes the proof of the lemma. Since our recovered gradient is not inH D (curl; Ω), it is not straightforward to verify the reliability bound by Theorem 3.1. However, it still plays a role in our reliability analysis. A 1/2 (τ − ∇ h u T ) ≤ C r η ρ . (5.34) Proof. By Lemma 3.6, to show the validity of (5.34), it then suffices to prove that λ 1/2 F h −1/2 F [[u T ]] 0,F ≤ C A 1/2 (ρ T −ρ T ) 0,ω F (5.35) for all F ∈ E I ∪ E D . Note that [[u T ]]| F is an odd function for all F ∈ E I . Hence, [[ρ T · t F ]] 0,F = 0 implies [[u T ]] 0,F = 0. By the equivalence of norms in a finite dimensional space and the scaling argument, we have that h −1/2 F [[u T ]] 0,F ≤ C h 1/2 F [[ρ T · t F ]] 0,F . (5.36) Sinceρ T ∈ H D (curl; Ω), it then follows from the triangle, the trace, and the inverse inequalities that [[ρ T · t F ]] 0,F = [[(ρ T −ρ T ) · t F ]] 0,F ≤ (ρ T −ρ T )| K + F · t F 0,F + (ρ T −ρ T )| K − F · t F 0,F ≤ C h −1/2 F ρ T −ρ T 0,ω F + h F ∇×(ρ T −ρ T ) 0,ω F ≤ C h −1/2 F ρ T −ρ T 0,ω F ≤ C λ −1/2 F h −1/2 F A 1/2 ρ T −ρ T 0,ω F for all F ∈ E I , which, together with (5.36), implies (5.35) and, hence, (5.34). In the case that F ∈ E D , (5.35) can be proved in a similar fashion. This completes the proof of the lemma. Local Efficiency In this section, we establish local efficiency of the indicators η σ,K and η ρ,K defined in (4.23) and (4.25), respectively. η K ≤ C e A 1/2 ∇ h (u − u T ) 0,ω K + osc (f, K) , (5.37) where ω K is the union of all elements that shares at least an edge with K. Proof. (5.37) is a direct consequence of Lemmas 5.6 and 5.7. Note that the local efficiency bound in (5.37) holds regardless the distribution of the diffusion coefficient A(x). Local Efficiency for η σ,K To establish local efficiency bound of η σ,K , we introduce some auxiliary functions defined locally in K. To this end, for each edge F ∈ E K , denote by F and F the other two edges of K such that F, F , and F form counter-clockwise orientation. Without loss of generality, assume that µ K ≡ 1 on E K . Let w F = σ K −σ K · n K | F ∈ P k−1 (F ), a F = w F (s F ), and b F = w F (e F ). (5.38) Define the auxiliary function corresponding to F ,w F ∈ P k (K), such that Kw F P j,K dx = 0, ∀ j = 1, · · · , m k andw F | F = w F + γ F L k,F ,w F | F = −β F L k,F , andw F | F = β F L k,F , where γ F = a F − b F 2 and β F = a F + b F 2 . Lemma 5.5. For each F ∈ E K , there exists a positive constant C such that w F 0,K ≤ C h 1/2 F w F 0,F . (5.39) Proof. By the Cauchy-Schwarz and the inverse inequalities, we have γ F = 1 2 F w F ds ≤ h 1/2 F 2 w F 0,F ≤ Ch −1/2 F w F 0,F . (5.40) Approximation property and the inverse inequality give w F − β F 0,F ≤ Ch F w F 0,F ≤ C w F 0,F , which, together with the triangle inequality, gives |β F | = h −1/2 F β F 0,F ≤ h −1/2 F w F − β F 0,F + w F 0,F ≤ C h −1/2 F w F 0,F . (5.41) Since L k,F 0,F ≤ h 1/2 F for all F ∈ E K , by (5.40) and (5.41), we have that w F 0,F = w F 2 0,F + γ 2 F L k,F 2 0,F 1/2 ≤ C w F 0,F and that w F 0,F ≤ h 1/2 F |β F | ≤ C w F 0,F and w F 0,F ≤ h 1/2 F |β F | ≤ C w F 0,F . Now (5.39) is a direct consequence of the fact that w F 0,K ≤ C F ∈E K h 1/2 F w F 0,F which follows from the equivalence of norms in a finite dimensional space, and the fact that w F ∂K = 0 implies w F K = 0. This completes the proof of the lemma. Lemma 5.6. There exists a positive constant C such that η σ,K ≤ C A 1/2 ∇ h (u − u T ) 0,K + osc (f, K) , ∀ K ∈ T . (5.42) Proof. According to (4.14), it is easy to see that (σ K −σ K ) · n F 0,F = 0 for all F ∈ E K implies that σ K −σ K 0,K = 0. Hence, by the equivalence of norms in a finite dimensional space, we have that σ K −σ K 0,K ≤ C F ∈E K h 1/2 F (σ K −σ K ) · n F 0,F ≤ C F ∈E K h 1/2 F w F 0,F ,(5.43) where w F is defined in (5.38). By the orthogonality property of {L j,F } k j=0 and the definition of w F , we have w F 2 0,F = ∂K (σ K −σ K ) · nw F ds. It then follows from (4.17), integration by parts, the Cauchy-Schwarz inequality, and (5.39) that w F 2 0,F = Kσ K · ∇w F dx + K fw F dx − Kσ K · ∇w F dx − K (∇ ·σ K )w F dx = K (f − ∇ ·σ K )w F dx ≤ C h 1/2 F f − ∇ ·σ K 0,K w F 0,F , which implies w F 0,F ≤ Ch 1/2 F f − ∇ ·σ K 0,K . Together with (5.43), we have η σ,K ≤ λ −1/2 K σ K −σ K 0,K ≤ C h K √ λ K f − ∇ ·σ K 0,K . Now (5.42) is a direct consequence of the following efficiency bound of the element residual (see, e.g., [7]): h K √ λ K f − ∇ ·σ K K ≤ C A 1/2 ∇(u − u T ) 0,K + h K √ λ K f − f k−1 0,K . This completes the proof of the theorem. Local Efficiency for η ρ,K In this section, we establish local efficiency bound for the nonconforming error indicator η ρ,K defined in (4.25). Lemma 5.7. There exists a positive constant C that is independent of the mesh size and the jump of the coefficient such that η ρ,K ≤ C A 1/2 ∇ h (u − u T ) 0,ω K , ∀ K ∈ T . (5.44) Proof. By (4.21), it is easy to see that (ρ K −ρ K ) · t F 0,F = 0 for all F ∈ E K implies that ρ K −ρ K 0,K = 0. By the equivalence of norms in a finite dimensional space and the scaling argument, we have ρ K −ρ K 0,K ≤ C F ∈E K h 1/2 F (ρ K −ρ K ) · t F 0,F . (5.45) Without loss of generality, assume that K is an interior element. By (4.21), a direct calculation gives (ρ K −ρ K ) F · t F =    (θ F − 1)[[ρ · t F ]]| F , if K = K − F , θ F [[ρ · t F ]]| F , if K = K + F (5.46) for all F ∈ E K . It is also easy to verify that Λ − F 1/2 (1 − θ F ) ≤ Λ − F Λ + F Λ − F + Λ + F 1/2 and Λ + F 1/2 θ F ≤ Λ − F Λ + F Λ − F + Λ +η ρ,K ≤ Λ 1/2 K ρ K −ρ K K ≤ C F ∈E K Λ − F Λ + F Λ − F + Λ + F 1/2 h 1/2 F [[ρ T · t F ]] 0,F . (5.48) Now, (5.44) is a direct consequence of (5.48) and the following efficiency bound for the jump of tangential derivative on edges Λ − F Λ + F Λ − F + Λ + F 1/2 h 1/2 F [[ρ · t F ]] 0,F ≤ C A 1/2 ∇(u − u T ) 0,ω F for all F ∈ E I . This completes the proof of the lemma. Numerical Result In this section, we report numerical results on two test problems. The first one is on the Crouziex-Raviart nonconforming finite element approximation to the Kellogg benchmark problem [25]. This is an interface problem in ( Starting with a coarse mesh, Figure 1 depicts the mesh when the relative error is less than 10%. Here the relative error is defined as the ratio between the energy norm of the true error and the energy norm of the exact solution. Clearly, the mesh is centered around the singularity (the origin) and there is no over-refinement along interfaces. Figure 2 is the log-log plot of the energy norm of the true error and the global error estimator η versus the total number of degrees of freedom. It can be observed that the error converges in an optimal order (very close to −1/2) and that the efficiency index, i.e., η A 1/2 ∇ h (u − u T ) is close to one when the mesh is fine enough. With f = 0 for the Kellogg problem, we note that η σ = 0, therefore, η = η ρ . Even though for the nonconforming error we recover a gradient that is not curl free, (thus we were not be able to prove that the reliability constant is 1 for the nonconforming error) the numerics still shows the behavior of asymptotic exactness, i.e., when the mesh is fine enough the efficiency index is close to 1. For the second test problem, we consider a Poisson L-shaped problem that has a nonzero conforming error η σ . On the L-shaped domain Ω = [−1, 1] 2 \ [0, 1] × [−1, 0], the Poisson problem (A = I) has the following exact solution u(r, θ) = r 2/3 sin((2θ + π)/3) + r 2 /2. The numerics is based on the Crouziex-Raviart finite element approximation. With the relative error being less than 0.75%, the final mesh generated the adaptive mesh refinement algorithm is depicted in Figure 3. Clearly, the mesh is relatively centered around the singularity (origin). Comparison of the true error and the estimator is presented in Figure 4. It is obvious that the error converges in an optimal order (very close to −1/2) and that the efficiency index is very close to 1 for all iterations. For simplicity, assume that meas d−1 (Γ D ) = 0. Considering the diffusion problem: −∇ · (A∇u) = f in Ω, (2.1) with boundary conditions u = 0 on Γ D and − A∇u · n = g on Γ N , H D (curl; Ω) = {τ ∈ H(curl; Ω) : ∇×τ = 0 in Ω and τ · t = 0 on Γ D } , 4. 1 . 1For all K ∈ T , the basis functions {φ j,K } m k j=1 have support on K and vanish on the boundary of K, i.e., φ j,K ≡ 0 on ∂K. Lemma 4. 2 . 2For all F ∈ E, the basis functions {φ j,F } k−1 j=0 have support on ω F , and their restrictions on E K + F ∪ E K − F has the following representation: F,F in (4.10) can be directly computed based on the continuity of φ j,F in K + F and K − F . This completes the proof of the lemma. Remark 4. 3 . 3As a consequence of Lemma 4.2, the basis function φ j,F is continuous on the edge F , i.e., [[φ j,F ]] F = 0 for all j = 0, · · · , k − 1; moreover, φ j,F vanishes at end points of F , i.e., φ j,F (s F ) = φ j,F (e F ) = 0, for odd j. Lemma Theorem 4. 6 . 6Let u T be the finite element solution in(4.2). Then the recovered fluxσ T defined in (4.16) belongs to Σ k−1 f (T ). H D (curl; Ω) = {τ ∈ H(curl; Ω) : τ · t = 0 on Γ N .} To this end, denote the H D (curl; Ω) conforming Nédélec (NE) space of . Theorem 5.1. (Global Reliability) Let u T be the nonconforming solution to (4.2). Theorem 5. 4 . 4(Local Efficiency) For each K ∈ T , there exists a positive constant C e that is independent of the mesh size and the jump of the coefficient such that 2.1) with Ω = (−1, 1) 2 , Γ N = ∅, f = 0, exact solution in the polar coordinates is given by u(r, θ) = r 0.1 µ(θ), where µ(θ) is a smooth function of θ. Figure 1 : 1Kellogg problem: final mesh. Figure 2 : 1 Figure 3 : 213Error comparison. [13] D. Braess, V. Pillwein, and J. Schöberl, Equilibrated residual error estimates are p-robust, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1189-1197. 1 [14] D. Cai, Z. Cai, and S. Zhang, Robust equilibrated a posteriori error estimator for higher order finite element approximations to diffusion problems, Numer. Math., 144:1 (2020), 1-21. L-shape problem: final mesh. Figure 4 : 4Error comparison. 4.4.Let F be an edge of K. Assume that p ∈ P k−1 (F ). Then we have that∂K p φ j,F ds = F p φ j,F ds. (4.12) Moreover, if F p φ j,F ds = 0 for all j = 0, · · · , k − 1, then p ≡ 0 on F . Proof. Since {L j,F } k j=0 are orthogonal polynomials on F , Lemma 4.4 is a direct consequence of Lemma 4.2. Lemma 5.3. The global nonconforming error estimator, η ρ , given in (4.26) is reliable, i.e., there exists a constant C r such thatinf τ ∈H D (curl;Ω) Robust a posteriori error estimation for nonconforming finite element approximation. M Ainsworth, SIAM J. Numer. Anal. 421M. Ainsworth, Robust a posteriori error estimation for nonconforming finite element ap- proximation, SIAM J. Numer. Anal., 42:6 (2005), 2320-2341. 1 A posteriori error estimation for discontinuous Galerkin finite element approximation, SIAM. M Ainsworth, J. Sci. Comput. 301, 4.5M. Ainsworth, A posteriori error estimation for discontinuous Galerkin finite element approximation, SIAM J. Sci. Comput., 30 (2007), 189-204. 1, 4.5 A unified approach to a posteriori error estimation using element residual methods. M Ainsworth, J T Oden, Numer. Math. 651M. Ainsworth and J. T. Oden, A unified approach to a posteriori error estimation using element residual methods, Numer. Math., 65 (1993), 23-50. 1 Fully computable bounds for the error in nonconforming finite element approximations of arbitrary order on triangular elements. M Ainsworth, R Rankin, SIAM J. Numer. Anal. 461, 3, 4.1, 4.1M. Ainsworth and R. Rankin, Fully computable bounds for the error in nonconforming finite element approximations of arbitrary order on triangular elements, SIAM J. Numer. Anal., 46 (2008), 3207-3232. 1, 3, 4.1, 4.1 Gauss-Legendre elements: A stable, higher order nonconforming finite element family, Computing. Á Baran, G Stoyan, 79Á. Baran and G. Stoyan, Gauss-Legendre elements: A stable, higher order non- conforming finite element family, Computing, 79 (2007), 1-21. 4.1 A Posteriori Error Estimation in Finite Element analysis. M Ainsworth, J T Oden, John Wiley & Sons, IncM. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element analysis, John Wiley & Sons, Inc., 2000. 1 Adaptive finite element methods for elliptic equations with non-smooth coefficients. C Bernardi, R Verfürth, Numer. Math. 852C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients, Numer. Math., 85 (2000), 579-608. 5.2 Local flux reconstructions for standard finite element methods on triangular meshes. R Becker, D Capatina, R Luce, SIAM Journal on Numerical Analysis. 544.5R. Becker, D. Capatina, and R. Luce, Local flux reconstructions for standard finite element methods on triangular meshes, SIAM Journal on Numerical Analysis, 54 (2016), 2684-2706. 4.5 D Boffi, F Brezzi, M Fortin, Mixed Finite Element Methods and Applications. HeidelbergSpringer4.2, 4.3D. Boffi, F. Brezzi, and M. Fortin, Mixed Finite Element Methods and Applications, Springer, Heidelberg, 2013. 4.2, 4.3 D Braess, Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics. Cambridge, UKCambridge University Press3rd ed.D. Braess, Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, 3rd ed., Cambridge University Press, Cambridge, UK, 2007. 1 An equilibrated a posteriori error estimator for the Interior Penalty Discontinuous Galerkin method. D Braess, T Fraunholz, R H Hoppe, SIAM J. Numer. Anal. 523D. Braess, T. Fraunholz, and R. H. Hoppe, An equilibrated a posteriori error estimator for the Interior Penalty Discontinuous Galerkin method, SIAM J. Numer. Anal., 52 (2014), 2121-2136. 1, 3 Equilibrated residual error estimator for edge elements. D Braess, J Schöberl, Math. Comp. 771D. Braess and J. Schöberl, Equilibrated residual error estimator for edge elements, Math. Comp., 77 (2008), 651-672. 1 Discontinuous finite element methods for interface problems: robust a priori and a posteriori error estimate. Z Cai, C He, S Zhang, SIAM. J. Numer. Anal. 551, 3, 4.5Z. Cai, C. He, and S. Zhang, Discontinuous finite element methods for interface problems: robust a priori and a posteriori error estimate, SIAM. J. Numer. Anal., 55 (2017), 400-418. 1, 3, 4.5 Recovery-based error estimator for interface problems: Mixed and nonconforming elements. Z Cai, S Zhang, SIAM. J. Numer. Anal. 484.3Z. Cai and S. Zhang, Recovery-based error estimator for interface problems: Mixed and nonconforming elements, SIAM. J. Numer. Anal., 48:1 (2010), 30-52. 4.3 Robust equilibrated residual error estimator for diffusion problems: conforming elements. Z Cai, S Zhang, SIAM J. Numer. Anal. 501Z. Cai and S. Zhang, Robust equilibrated residual error estimator for diffusion problems: conforming elements, SIAM J. Numer. Anal., 50 (2012), 151-170. 1 Stable nonconforming methods for the Stokes problem. Y Cha, M Lee, S Lee, Appl. Math. Comput. 1144.1Y. Cha, M. Lee, and S. Lee, Stable nonconforming methods for the Stokes problem, Appl. Math. Comput., 114 (2000), 155-174. 4.1 Conforming and non-conforming finite element methods for solving the stationary Stokes equations. M Crouzeix, P A Raviart, 33-75. 4.1RAIRO Anal. Numer. 7M. Crouzeix and P. A. Raviart, Conforming and non-conforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numer., 7 (1977), 33-75. 4.1 Explicit error bounds for a nonconforming finite element method. P Destuynder, B Métivet, SIAM J. Numer. Anal. 351P. Destuynder and B. Métivet, Explicit error bounds for a nonconforming finite element method, SIAM J. Numer. Anal., 35:5 (1998), 2099-2115. 1 Explicit error bounds in a conforming finite element method. P Destuynder, B Métivet, Math. Comp. 681P. Destuynder and B. Métivet, Explicit error bounds in a conforming finite element method, Math. Comp., 68 (1999), 1379-1396. 1 An adaptive finite element method for a class of variational inequalities. L Demkowicz, M Swierczek, Selected Problems of Modern Continuum Theory. BolognaL. Demkowicz and M. Swierczek, An adaptive finite element method for a class of variational inequalities, in Selected Problems of Modern Continuum Theory, Bologna, June 3-6, 1987, 11-28. 1 An accurate H (div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. E Alexandre, N Serge, V Martin, Comptes Rendus Mathematique. 3451, 4.5E. Alexandre, N. Serge and V. Martin, An accurate H (div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems, Comptes Rendus Mathematique, 345 (2007), 709-712. 1, 4.5 A non-conforming piecewise quadratic finite element on triangles. M Fortin, M Soulie, 505-520. 4.1International Journal for Numerical Methods in Engineering. 19M. Fortin and M. Soulie, A non-conforming piecewise quadratic finite element on tri- angles, International Journal for Numerical Methods in Engineering, 19 (1983), 505-520. 4.1 On the Poisson equation with intersecting interfaces. R B Kellogg, Appl. Anal. 46R.B. Kellogg, On the Poisson equation with intersecting interfaces, Appl. Anal., 4 (1975), 101-129. 6 Flux reconstruction for the P 2 nonconforming finite element method with application to a posteriori error estimation. K Y Kim, Appl. Numer. Math. 621K.Y. Kim, Flux reconstruction for the P 2 nonconforming finite element method with appli- cation to a posteriori error estimation, Appl. Numer. Math., 62 (2012), 1701-1717. 1 Error estimate procedure in the finite element method and applications. P Ladevèze, D Leguillon, SIAM J. Numer. anal. 201P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM J. Numer. anal., 20:3 (1983), 485-509. 1 An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. L D Marini, SIAM J. Numer. Anal. 221L. D. Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method, SIAM J. Numer. Anal., 22 (1985), 493-496. 1 Toward a universal h-p adaptive finite element strategy, Part 2. A posteriori error estimation. J T Oden, L Demkowicz, W Rachowicz, T A Westermann, Comput. Methods Appl. Mech. Engrg. 771J. T. Oden, L. Demkowicz, W. Rachowicz and T. A. Westermann, Toward a uni- versal h-p adaptive finite element strategy, Part 2. A posteriori error estimation, Comput. Methods Appl. Mech. Engrg., 77 (1989), 113-180. 1 a posteriori error estimators for elliptic equations with discontinuous coefficients. M Petzoldt, Adv. Comp. Math. 16M. Petzoldt, a posteriori error estimators for elliptic equations with discontinuous coeffi- cients, Adv. Comp. Math., 16 (2002), 47-75. Approximations in elasticity based on the concept of function space. W Prager, J L Synge, Quart. Appl. Math. 513W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math., 5 (1947), 286-292. 1 1, 3 Crouzeix-Velte decompositions for higher-order finite elements. G Stoyan, Á Baran, Comput. Math. Appl. 511G. Stoyan and Á. Baran, Crouzeix-Velte decompositions for higher-order finite elements, Comput. Math. Appl., 51 (2006), 967-986. 4.1 Guaranteed and locally computable a posteriori error estimate. T Vejchodský, IMA J. Numer. Anal. 261T. Vejchodský, Guaranteed and locally computable a posteriori error estimate, IMA J. Numer. Anal., 26 (2006), 525-540. 1 A note on constant-free a posteriori error estimates. R Verfürth, SIAM J. Numer. Anal. 471R. Verfürth, A note on constant-free a posteriori error estimates, SIAM J. Numer. Anal., 47 (2009), 3180-3194. 1
[]
[ "Ultrafast X-ray Phase Contrast Imaging of High Repetition Rate Shockwaves", "Ultrafast X-ray Phase Contrast Imaging of High Repetition Rate Shockwaves" ]
[ "Christopher S Campbell \nDepartment of Mechanical Engineering\nTexas A&M University\n77843College StationTexasUSA\n\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "Mirza Akhter \nDepartment of Mechanical Engineering\nTexas A&M University\n77843College StationTexasUSA\n", "Samuel Clark \nX-ray Science Division\nAdvanced Photon Source\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n", "Kamel Fezzaa \nX-ray Science Division\nAdvanced Photon Source\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n", "Zhehui Wang \nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n", "David Staack \nDepartment of Mechanical Engineering\nTexas A&M University\n77843College StationTexasUSA\n" ]
[ "Department of Mechanical Engineering\nTexas A&M University\n77843College StationTexasUSA", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Department of Mechanical Engineering\nTexas A&M University\n77843College StationTexasUSA", "X-ray Science Division\nAdvanced Photon Source\nArgonne National Laboratory\n60439ArgonneIllinoisUSA", "X-ray Science Division\nAdvanced Photon Source\nArgonne National Laboratory\n60439ArgonneIllinoisUSA", "Los Alamos National Laboratory\n87545Los AlamosNew MexicoUSA", "Department of Mechanical Engineering\nTexas A&M University\n77843College StationTexasUSA" ]
[]
High-repetition-rate plasma-induced shockwaves in liquid have been observed using ultrafast X-ray phase contrast imaging (PCI) for the first time. Using a laser-triggered nanosecond-pulsed plasma device in heptane at ambient conditions, it is demonstrated that these well-timed weak shocks can be generated at an unprecedented repetition rate (>3 per minute), significantly faster than that of more commonly-used dynamic targets (exploding wire, gas gun). This simple portable target can easily be adapted to study discharges in different media (water, oils, solids) at comparably high repetition rates and over a wide range of possible input energies. Compared to previously PCI-imaged shocks, these shocks are relatively weak (1 < Mach number < 1.4), which advances the resolution and sensitivity limits of this high-speed imaging diagnostic. Numeric solutions of a Fresnel-Kirchhoff diffraction model are used to estimate post-shock thermodynamic conditions, the results of which show good agreement with expectations based on Rankine-Hugoniot normal shock thermodynamic relations.
null
[ "https://export.arxiv.org/pdf/2303.13690v2.pdf" ]
257,757,260
2303.13690
c491aee810a070c1db97fa6fc523e3d0f96775b4
Ultrafast X-ray Phase Contrast Imaging of High Repetition Rate Shockwaves 10 Apr 2023 Christopher S Campbell Department of Mechanical Engineering Texas A&M University 77843College StationTexasUSA Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Mirza Akhter Department of Mechanical Engineering Texas A&M University 77843College StationTexasUSA Samuel Clark X-ray Science Division Advanced Photon Source Argonne National Laboratory 60439ArgonneIllinoisUSA Kamel Fezzaa X-ray Science Division Advanced Photon Source Argonne National Laboratory 60439ArgonneIllinoisUSA Zhehui Wang Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA David Staack Department of Mechanical Engineering Texas A&M University 77843College StationTexasUSA Ultrafast X-ray Phase Contrast Imaging of High Repetition Rate Shockwaves 10 Apr 2023(Dated: April 11, 2023) High-repetition-rate plasma-induced shockwaves in liquid have been observed using ultrafast X-ray phase contrast imaging (PCI) for the first time. Using a laser-triggered nanosecond-pulsed plasma device in heptane at ambient conditions, it is demonstrated that these well-timed weak shocks can be generated at an unprecedented repetition rate (>3 per minute), significantly faster than that of more commonly-used dynamic targets (exploding wire, gas gun). This simple portable target can easily be adapted to study discharges in different media (water, oils, solids) at comparably high repetition rates and over a wide range of possible input energies. Compared to previously PCI-imaged shocks, these shocks are relatively weak (1 < Mach number < 1.4), which advances the resolution and sensitivity limits of this high-speed imaging diagnostic. Numeric solutions of a Fresnel-Kirchhoff diffraction model are used to estimate post-shock thermodynamic conditions, the results of which show good agreement with expectations based on Rankine-Hugoniot normal shock thermodynamic relations. In the fields of high-speed X-ray science and synchrotron radiation, the maximum achievable repetition rate of a dynamic target of interest is an important figure of merit when pursuing efficient use of limited beamtime. However, many of the common dynamic processes of most interest feature destructible devices, requiring complete or partial reassembly of the target after each imaging event [1][2][3] which severely limits repetition rate. It would therefore be beneficial to develop a target which requires minimal to no maintenance between events, without compromising phenomena of interest such as high instantaneous power density, high mass density gradients, high pressure and temperature gradients, and supersonic behavior/shockwaves. In this letter we present such a target, a pulsed power device submerged in ambient liquid heptane which can produce well-timed nanosecondpulsed spark discharges. This target was taken to the Advanced Photon Source (APS) for ultrafast phase-contrast imaging (PCI) experiments, the results of which are presented herein. Of particular interest in this subset of imaging results is the presence of a visible expanding shock front generated by the spark discharge event, which to the best of our knowledge represents one of the weakest shock fronts ever imaged using PCI (Ma ≈ 1.2). While a sufficiently strong shock would be easily visible using less sensitive imaging techniques due to its relatively high mass density ratio, the fact that such a weak shock front is still observable in this work highlights the superior sensitivity of this implementation of PCI to very subtle dynamic phenomena, while still revealing the limits of current techniques and the path forward for the next generation of ultrafast imaging. The high repetition rate (>3 events/minute), low cost (<US$100k), and portability of this imaging target makes it quite attractive to those fields interested in events of similar timescales and power densities (e.g. ICF, dynamic compression, shock physics), but which rely on apparatuses which are either immovable or have a prohibitively slow event repetition rate. This target has the potential to open new opportunities for such fields to benefit from the superior imaging capabilities of the APS and similar user facilities. The pulsed power device and high-voltage circuit used in this work to generate the submerged spark discharge event is similar to those used in our prior work [4,5], with this implementation consisting of two electrodes between which a well-timed submerged spark discharge occurs ( Figure 1). The event of interest dissipates approximately 100mJ of nanosecond-timescale plasma processes (light, sound, chemistry, shockwaves) in the target over a pulse duration of 100ns, implying an instantaneous power of roughly 1MW. Assuming an approximate discharge cross section of 5µm during peak current across a gap of 0.5mm, we estimate a peak energy density of 15GJ/kg, within two orders of magnitude (albeit at a lower instantaneous power) of the 1TJ/kg implied by recent hotspot energy and mass results from the National Ignition Facility [6]. The X-ray imaging method consists of a 128frame Shimadzu HPV-X2 camera (3µm/pixel), used to image a scintillator placed 46cm from the imaging target (see Figure 1). This setup is capable of a 6.5MHz X-ray framerate, made possible by the APS's 24-singlet standard operating mode [7]. See Figure 2 for selected sequential PCI frames from a single spark discharge event. Additionally, see Figure 3 for a compilation of frames from multiple similar events in which the shock is visible in frame, sorted by frame time relative to the instant of plasma initiation. The estimated speed of this shock is shown in Figure 4 to be 1.45 ± 0.13km/s for this dataset, corresponding to a Mach number in ambient heptane (v sound = 1.129km/s) of 1.28 ± 0.13. The transverse profile of these shock images is consistent with expected PCI for a step discontinuity in density. Also apparent via comparison to this linear trend is the slight negative concavity of the data. This suggests a shock speed which decreases with time, which is consistent which the time-dependent shock speed found by linearly fitting to subsamples of the full dataset (within 100ns of a given instant in time). While at first glance the time dependency from Taylor-von Neumann-Sedov blast wave theory (proportional to t 0.4 for spherically expanding shocks [8] and t 0.5 for cylindrically expanding shocks [9]) would presumably serve as a physically-grounded model to fit to this data, the plasma-induced shock front imaged here violates two of the main assumptions required by the Taylor-von Neumann-Sedov theory: instantaneous energy input with the shock originating from a zero-radius point or line (compare to Section SM.II), and negligible ambient pressure (p post-shock p ambient ). By the time that the shock becomes visible to this diagnostic (earliest measured shock image at 45ns after initiation), the post-shock pressure has decreased drastically, resulting in a near-linear position vs. time data. For this analysis it was decided that a phenomenological quadratic fit (green curve on Figure 4) would be appropriate, since it requires the least amount of assumptions but still captures this apparent negative concavity. The radiographic attenuation contrast for such a weak shock is quite low; in our imaging target, the ambient X-ray path consists of a 6mm-thick layer of heptane, the shock has a characteristic size of about 100µm within the field of view, and the expected density within the postshock region is approximately 1.2 times that of ambient, implying a maximum possible attenuation contrast of 0.02% in this case, which is well below the the level of detectability in this experiment. However, the diffractioninduced contrast enhancement and edge detection features of PCI cause this shock to be visible above background noise as localized maxima and minima in brightness, with the maxima occurring on the side of the discontinuity with lower density, and the minima on the side FIG. 2. Selected frames from a single spark discharge event in heptane for which the plasma-induced shock is visible, with timestamps measured relative to spark initiation. Left and right columns show contrast-enhanced raw images and corresponding background-subtracted frames respectively; the average of all 128 frames from the event was used as the background. Note the location of the shock front visible at t=79ns and t=232ns, denoted by red arrows. with higher density. This type of diffraction is the essence of PCI and is governed by the Fresnel-Kirchhoff integral [10], modified for cylindrically symmetric geometries in FIG. 3. Frames from selected PCI heptane spark events in which a shock front was visible, sorted by frame time relative to spark initiation. Each frame is duplicated across both columns, the right column includes annotations which indicate the contour of the shock using red splines. FIG. 4. Plot of cylindrical shock positions/radii, relative to the axis of the plasma, compiling data from eighty-five different PCI frames for which the shock was visible (Figure 3), some of which came from more than one frame of a single event. Shock position measurement was performed manually for each of the 85 frames in which a shock was visible. The twenty measurements for each frame were then used to determine uncertainty (two standard deviations away from the average). The solid line (red) shows the least-squares linear fit to these points (1.45 km/s), with the two red dotted lines assume a shock speed 90% and 110% of that linear fit to roughly illustrate inherent shock speed uncertainty. Blue circles indicate shock speed subestimates calculated by fitting to subsamples of the full set (within 100ns of that subestimate's position on the horizontal axis). The green curve shows a quadratic fit to the position vs. time data, used for later analysis. Equation 2 to ease computation: g out (x , y ) = e 2πiz/λ iλz g in (x, y)e iπ λz ((x −x) 2 +(y −y) 2 ) dxdy (1) = e 2πiz/λ √ iλz g in (x)e iπ λz (x −x) 2 dx(2) where g in and g out represent the complex-valued electric field at the target and the imaging plane respectively. In this work, z = 46cm, and all complex index of refraction data is from [11]. This model can now be fit to experiment ( Figure 5, analyzing the fifth frame from Figure 3), constituting a measurement technique to estimate postshock density. See Section SM.III for a more complete derivation of Equation 2 and further explanation of how the computational model was implemented, and also refer to the similar model from our prior work [4]. Alongside this X-ray diffraction method for estimating the density change across the shock from experimental data, it is also possible to relate ρ 2 and v shock by solving the system of Rankine-Hugoniot thermodynamic shock relations in heptane: ρ 1 v 1 = ρ 2 v 2 (mass), p 1 + ρ 1 v 2 1 = p 2 + ρ 2 v 2 2 (momentum), h 1 + 1 2 v 2 1 = h 2 + 1 2 v 2 2 FIG. 5. Illustration of a cutline extraction algorithm which uses a spline fit to convert the two-dimensional X-ray frame of the shock front (a) into a one-dimensional plot (b). The image analyzed here corresponds with the fifth image from Figure 3 (energy), and a tabulated equation of state for heptane [12]. See Section SM.I for a derivation of this system of equations. Implicitly solving this system results in a unique relationship between ρ 2 and v shock , shown in Figure 6 with a red curve. However, measured values for ρ 2 and v shock (from the diffraction model and quadratic fit to Figure 4) tend to reside above this red curve, suggesting a higher-density post-shock region than what is implied by this thermodynamic model. We attribute this to the fact that the thermodynamic model ignores the cavitation bubble which follows the expanding shock, clearly visible in Figures 2 and 3. This bubble interface is generated at a relatively small radius (< 10µm) and high energy density, and expands outward with a large amount of inertia, compressing the post-shock region to a higher density. This is shown with green asterisks in Figure 6, generated by assuming that this compression has no effect on the speed of the shock; although this still does not result in a model that sufficiently matches experiment, it represents the maximum post-shock density ρ 2 which could be justified by the data. The original thermodynamic model (red curve) represents the minimum bound on ρ 2 , since it completely neglects this compression effect. In reality, the higher ρ 2 caused by this compression effect should increase the shock speed, though the full nature of this effect is not explored here; to fully investigate this effect, a multiphysics model would be necessary which couples together the dynamics of the cavitation bubble (Rayleigh-Plesset), the liquid post-shock region (Navier-Stokes), and the jump conditions which govern the shock (Rankine-Hugoniot). Similarity of this work to the submerged exploding wire PCI experiments by Yanuka [13] at the European Synchrotron Radiation Facility (ESRF) allows us to directly compare figures of merit for the two different implementations of PCI and of cylindrically expanding shock events, of which the most dramatic difference is peak instantaneous power and total event energy; for Yanuka, these values were approximately 1GW and 300J, respectively (recall 1MW and 100mJ for this work). The fact that the shock front images presented in this work are still visible with such a small event energy demonstrates the superior sensitivity and utility of PCI as a diagnostic for observing and analyzing propagating shocks. In summary, we present successful observation of weak shocks in liquid heptane; this work constitutes a feat in imaging sensitivity and resolution and serves as strong evidence that further utilization of PCI for shock imaging in different media and phases of matter may likely prove fruitful. The plasma-based method of shock generation used here exhibits an order of magnitude increase in repetition rate (multiple events per minute) over conventional dynamic targets in similar experiments (e.g. exploding wire), and can easily be increased to well over 1Hz given a sufficient data acquisition scheme. The ability to quickly generate large datasets is potentially useful for machine learning applications; the eighty-five shock fronts cataloged in Figure 4 could conceivably be used to train a deep learning model which could then rapidly compute shock parameters such as position and density. A wide range of possible parameter sweeps is achievable using this target, either by changing the input energy via choice of charging capacitor or simply taking advantage of the stochasticity of the phenomena of interest to automatically vary parameters such as imaging delay time, shock shape, or breakdown voltage. By exchanging the heptane for alternate discharge media (e.g. water, mineral oil, ice, plastics, rock), shock propagation can be studied in these materials without compromising repetition rate. Future work will push the PCI sensitivity limit further, while at the same time continuing to develop a quantitative analysis toolkit for shock imaging as well as PCI in general. FIG. 1 . 1Simplified schematic of the target and field of view for X-ray imaging. (t=234ns). The black solid line in represents the average cutline (dotted lines show upper and lower quartiles), and the red line shows the best simulated shock front PCI profile with a post-shock density of ρ2 = 0.799 +0.116 −0.057 g/cc (ρ2/ρ1 = 1.176 +0.171 −0.084 ). FIG. 6. Hugoniot states in the ρ2-v shock space, showing how values estimated from this work's PCI data and diffraction model (black) compare to normal shock thermodynamic relations in heptane both with (green) and without (red) the cavitation bubble compression effect. The dashed portion of the red curve indicates extrapolation of the heptane equation of state. The blue datapoint corresponds with the particular X-ray diffraction model fit result from Figure 5. Implementation of the exploding wire technique to study blast-wave-structure interaction. O Ram, O Sadot, Experiments in Fluids. 532012O. Ram and O. Sadot, "Implementation of the explod- ing wire technique to study blast-wave-structure interac- tion," Experiments in Fluids, vol. 53, pp. 1335-1345, 11 2012. Shockwave dissipation by interface-dominated porous structures. D M Dattelbaum, A Ionita, B M Patterson, B A Branch, L Kuettner, AIP Advances. 107D. M. Dattelbaum, A. Ionita, B. M. Patterson, B. A. Branch, and L. Kuettner, "Shockwave dissipation by interface-dominated porous structures," AIP Advances, vol. 10, no. 7, 2020. High-speed imaging of transition from fluid breakup to phase explosion in electric explosion of tungsten wires in air. Y Sechrest, C Campbell, X Tang, D Staack, Z Wang, Applied Physics Letters. 11712Y. Sechrest, C. Campbell, X. Tang, D. Staack, and Z. Wang, "High-speed imaging of transition from fluid breakup to phase explosion in electric explosion of tung- sten wires in air," Applied Physics Letters, vol. 117, no. 12, pp. 10-15, 2020. Ultrafast x-ray imaging of pulsed plasmas in water. C Campbell, X Tang, Y Sechrest, K Fezzaa, Z Wang, D Staack, Physical Review Research. 32021C. Campbell, X. Tang, Y. Sechrest, K. Fezzaa, Z. Wang, and D. Staack, "Ultrafast x-ray imaging of pulsed plas- mas in water," Physical Review Research, vol. 3, 6 2021. Underwater plasma breakdown characteristics with respect to highly pressurized drilling applications. M Akhter, J Mallams, X Tang, D Staack, Journal of Applied Physics. 1292021M. Akhter, J. Mallams, X. Tang, and D. Staack, "Un- derwater plasma breakdown characteristics with respect to highly pressurized drilling applications," Journal of Applied Physics, vol. 129, 5 2021. . A B Zylstra, A L Kritcher, O A Hurricane, D A Callahan, J E Ralph, D T Casey, A Pak, O L Landen, B Bachmann, K L Baker, L Hopkins, S D Bhandarkar, J Biener, R M Bionta, N W Birge, T Braun, T M Briggs, P M Celliers, H Chen, C Choate, D S Clark, L Divol, T Döppner, D Fittinghoff, M J Edwards, M Johnson, N Gharibyan, S Haan, K D Hahn, E Hartouni, D E Hinkel, D D Ho, M Hohenberger, J P Holder, H Huang, N Izumi, J Jeet, O Jones, S M Kerr, S F Khan, H Kleinrath, V Kleinrath, C Kong, K M Lamb, S Le Pape, N C Lemos, J D Lindl, B J Macgowan, A J Mackinnon, Macphee, E. V. Marley, K. Meaney, M. Millot, A. SA. B. Zylstra, A. L. Kritcher, O. A. Hurricane, D. A. Callahan, J. E. Ralph, D. T. Casey, A. Pak, O. L. Landen, B. Bachmann, K. L. Baker, L. Berzak Hop- kins, S. D. Bhandarkar, J. Biener, R. M. Bionta, N. W. Birge, T. Braun, T. M. Briggs, P. M. Celliers, H. Chen, C. Choate, D. S. Clark, L. Divol, T. Döppner, D. Fittinghoff, M. J. Edwards, M. Gatu Johnson, N. Gharibyan, S. Haan, K. D. Hahn, E. Hartouni, D. E. Hinkel, D. D. Ho, M. Hohenberger, J. P. Holder, H. Huang, N. Izumi, J. Jeet, O. Jones, S. M. Kerr, S. F. Khan, H. Geppert Kleinrath, V. Geppert Klein- rath, C. Kong, K. M. Lamb, S. Le Pape, N. C. Lemos, J. D. Lindl, B. J. Macgowan, A. J. Mackinnon, A. G. Macphee, E. V. Marley, K. Meaney, M. Millot, A. S. Experimental achievement and signatures of ignition at the National Ignition Facility. K Moore, J M Newman, A Di Nicola, R Nikroo, P K Nora, N G Patel, M S Rice, J Rubery, D J Sater, S M Schlossberg, K Sepke, S J Sequoia, M Shin, S Stadermann, D J Stoupin, C A Strozzi, R Thomas, C Tommasini, E R Trosseille, P L Tubman, C R Volegov, C Weber, D T Wild, S T Woods, C V Yang, Young, Physical Review E. 1062022Moore, K. Newman, J. M. Di Nicola, A. Nikroo, R. Nora, P. K. Patel, N. G. Rice, M. S. Rubery, J. Sater, D. J. Schlossberg, S. M. Sepke, K. Sequoia, S. J. Shin, M. Sta- dermann, S. Stoupin, D. J. Strozzi, C. A. Thomas, R. Tommasini, C. Trosseille, E. R. Tubman, P. L. Vole- gov, C. R. Weber, C. Wild, D. T. Woods, S. T. Yang, and C. V. Young, "Experimental achievement and signatures of ignition at the National Ignition Facility," Physical Re- view E, vol. 106, 8 2022. APS Storage Ring Parameters. M Borland, G Decker, L Emery, W Guo, K Harkay, V Sajaev, C.-Y Yao, 9M. Borland, G. Decker, L. Emery, W. Guo, K. Harkay, V. Sajaev, and C.-Y. Yao, "APS Storage Ring Parame- ters," https://ops.aps.anl.gov/SRparameters, 9 2010. The formation of a blast wave by a very intense explosion I. Theoretical discussion. G I Taylor, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 2011065G. I. Taylor, "The formation of a blast wave by a very intense explosion I. Theoretical discussion," Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, vol. 201, no. 1065, pp. 159-174, 1950. Cylindrical shock waves produced by instantaneous energy release. S C Lin, Journal of Applied Physics. 251S. C. Lin, "Cylindrical shock waves produced by in- stantaneous energy release," Journal of Applied Physics, vol. 25, no. 1, pp. 54-57, 1954. Massachusetts Institute of Technology: MIT OpenCourseWare. G Barbastathis, C Sheppard, S B , License: Creative Commons BY-NC-SA. Oh, 2.71 Optics, Lecture 15G. Barbastathis, C. Sheppard, and S. B. Oh, 2.71 Op- tics, Lecture 15. Massachusetts Institute of Technology: MIT OpenCourseWare, \emph{https://ocw.mit.edu}. License: Creative Commons BY-NC-SA. Xray interactions: photoabsorption, scattering, transmission, and reflection at {E}=50-30000 e{V}, {Z}=1-92. B L Henke, E M Gullikson, J C Davis, Atomic Data and Nuclear Data Tables. 2B. L. Henke, E. M. Gullikson, and J. C. Davis, "X- ray interactions: photoabsorption, scattering, transmis- sion, and reflection at {E}=50-30000 e{V}, {Z}=1-92," Atomic Data and Nuclear Data Tables, vol. 2, 7 1993. Equations of State for Technical Applications. II. Results for Nonpolar Fluids. R Span, W Wagner, Tech. Rep. 1R. Span and W. Wagner, "Equations of State for Techni- cal Applications. II. Results for Nonpolar Fluids," Tech. Rep. 1, 2003. Multi frame synchrotron radiography of pulsed power driven underwater single wire explosions. D Yanuka, A Rososhek, S Theocharous, S N Bland, Y E Krasik, M P Olbinado, A Rack, Journal of Applied Physics. 124D. Yanuka, A. Rososhek, S. Theocharous, S. N. Bland, Y. E. Krasik, M. P. Olbinado, and A. Rack, "Multi frame synchrotron radiography of pulsed power driven under- water single wire explosions," Journal of Applied Physics, vol. 124, 10 2018.
[]
[ "Online and Dynamic Algorithms for Geometric Set Cover and Hitting Set", "Online and Dynamic Algorithms for Geometric Set Cover and Hitting Set" ]
[ "Arindam Khan \nIndian Institute of Science\nBengaluruIndia\n", "Aditya Lonkar \nIndian Institute of Science\nBengaluruIndia\n", "Saladi Rahul \nIndian Institute of Science\nBengaluruIndia\n", "Aditya Subramanian \nIndian Institute of Science\nBengaluruIndia\n", "Andreas Wiese \nTechnical University of Munich\nGermany\n" ]
[ "Indian Institute of Science\nBengaluruIndia", "Indian Institute of Science\nBengaluruIndia", "Indian Institute of Science\nBengaluruIndia", "Indian Institute of Science\nBengaluruIndia", "Technical University of Munich\nGermany" ]
[]
Set cover and hitting set are fundamental problems in combinatorial optimization which are well-studied in the offline, online, and dynamic settings. We study the geometric versions of these problems and present new online and dynamic algorithms for them. In the online version of set cover (resp. hitting set), m sets (resp. n points) are given and n points (resp. m sets) arrive online, one-by-one. In the dynamic versions, points (resp. sets) can arrive as well as depart. Our goal is to maintain a set cover (resp. hitting set), minimizing the size of the computed solution.For online set cover for (axis-parallel) squares of arbitrary sizes, we present a tight O(log n)competitive algorithm. In the same setting for hitting set, we provide a tight O(log N )-competitive algorithm, assuming that all points have integral coordinates in [0, N ) 2 . No online algorithm had been known for either of these settings, not even for unit squares (apart from the known online algorithms for arbitrary set systems).For both dynamic set cover and hitting set with d-dimensional hyperrectangles, we obtain (log m) O(d) -approximation algorithms with (log m) O(d) worst-case update time. This partially answers an open question posed by Chan et al. [SODA'22]. Previously, no dynamic algorithms with polylogarithmic update time were known even in the setting of squares (for either of these problems). Our main technical contributions are an extended quad-tree approach and a frequency reduction technique that reduces geometric set cover instances to instances of general set cover with bounded frequency.
10.48550/arxiv.2303.09524
[ "https://export.arxiv.org/pdf/2303.09524v1.pdf" ]
257,557,670
2303.09524
666ace83d0f20b3a2a8e7ca014b3481f0f3da723
Online and Dynamic Algorithms for Geometric Set Cover and Hitting Set March 17, 2023 Arindam Khan Indian Institute of Science BengaluruIndia Aditya Lonkar Indian Institute of Science BengaluruIndia Saladi Rahul Indian Institute of Science BengaluruIndia Aditya Subramanian Indian Institute of Science BengaluruIndia Andreas Wiese Technical University of Munich Germany Online and Dynamic Algorithms for Geometric Set Cover and Hitting Set March 17, 2023 Set cover and hitting set are fundamental problems in combinatorial optimization which are well-studied in the offline, online, and dynamic settings. We study the geometric versions of these problems and present new online and dynamic algorithms for them. In the online version of set cover (resp. hitting set), m sets (resp. n points) are given and n points (resp. m sets) arrive online, one-by-one. In the dynamic versions, points (resp. sets) can arrive as well as depart. Our goal is to maintain a set cover (resp. hitting set), minimizing the size of the computed solution.For online set cover for (axis-parallel) squares of arbitrary sizes, we present a tight O(log n)competitive algorithm. In the same setting for hitting set, we provide a tight O(log N )-competitive algorithm, assuming that all points have integral coordinates in [0, N ) 2 . No online algorithm had been known for either of these settings, not even for unit squares (apart from the known online algorithms for arbitrary set systems).For both dynamic set cover and hitting set with d-dimensional hyperrectangles, we obtain (log m) O(d) -approximation algorithms with (log m) O(d) worst-case update time. This partially answers an open question posed by Chan et al. [SODA'22]. Previously, no dynamic algorithms with polylogarithmic update time were known even in the setting of squares (for either of these problems). Our main technical contributions are an extended quad-tree approach and a frequency reduction technique that reduces geometric set cover instances to instances of general set cover with bounded frequency. Introduction Geometric set cover is a fundamental and well-studied problem in computational geometry [21,18,34,28,30]. Here, we are given a universe P of n points in R d , and a family S of m sets, where each set S ∈ S is a geometric object (we assume S to be a closed set in R d and S covers all points in P ∩ S), e.g., a hyperrectangle. Our goal is to select a collection S ⊆ S of these sets that contain (i.e., cover) all elements in P , minimizing the cardinality of S (see Figure 1 for an illustration). The frequency f of the set system (P, S) is defined as the maximum number of sets that contain an element in P . In the offline setting of some cases of geometric set cover, better approximation ratios are known than those for the general set cover, e.g., there is a polynomial-time approximation scheme (PTAS) for (axis-parallel) squares [31]. However, much less is understood in the online and in the dynamic variants of geometric set cover. In the online setting, the sets are given offline and the points arrive one-by-one, and for an uncovered point, we have to select a (covering) set in an immediate and irrevocable manner. To the best of our knowledge, even for 2-D unit squares, there is no known online algorithm with an asymptotically improved competitive ratio compared to the O(log n log m)-competitive algorithm for general online set cover [4,16]. In the dynamic case, the sets are again given offline and at each time step a point is inserted or deleted. Here, we are interested in algorithms that update the current solution quickly when the input changes. In particular, it is desirable to have algorithms whose update times are polylogarithmic. Unfortunately, hardly any such algorithm is known for geometric set cover. Agarwal et al. [2] initiated the study of dynamic geometric set cover for intervals and 2-D unit squares and presented (1 + ε)and O(1)-approximation algorithms with polylogarithmic update times, respectively. To the best of our knowledge, for more general objects, e.g., rectangles, three-dimensional cubes, or hyperrectangles in higher dimensions, no such dynamic algorithms are known. Note that in dynamic geometric set cover, the inserted points are represented by their coordinates, which is more compact than for general (dynamic) set cover (where for each new point p we are given a list of the sets that contain p, and hence, already to read this input we might need Ω(f ) time). Related to set cover is the hitting set problem (see Figure 1 for an illustration) where, given a set of points P and a collection of sets S, we seek to select the minimum number of points P ⊆ P that hit each set S ∈ S, i.e., such that P ∩ S = ∅. Again, in the offline geometric case, there are better approximation ratios known than for the general case, e.g., a PTAS for squares [31], and an O(log log OPT)-approximation for rectangles [5]. However, in the online and the dynamic cases, only few results are known that improve on the results for the general case. In the online setting, there is an O(log n)-competitive algorithm for d = 1, i.e., intervals, and an O(log n)-competitive algorithm for unit disks [23]. In the dynamic case, the only known algorithms are for intervals and unit squares (and thus, also for quadrants), yielding approximation ratios of (1 + ε) and O(1), respectively [2]. Our results In this paper, we study online algorithms for geometric set cover and hitting set for squares of arbitrary sizes, while previously no improved results were known even for unit squares. Also, we present dynamic algorithms for these problems for hyperrectangles of constant dimension d (also called d-boxes or orthotopes) which are far more general geometric objects than those which were previously studied, e.g., intervals [2] or (2-D) squares [20]. Online set cover for squares In Section 2 we study online set cover for axis-parallel squares of arbitrary sizes and provide an online O(log n)-competitive algorithm. We also match (asymptotically) the lower bound of Ω(log n), and hence, our competitive ratio is tight. In our online model (as in [4]), we assume that the sets (squares) are given initially and the elements (points) arrive online. Our online algorithm is based on a new offline algorithm that is monotone, i.e., it has the property that if we add a new point p to P , the algorithm outputs a superset of the squares that it outputs given only P without p. The algorithm is based on a quad-tree decomposition. It traverses the tree from the root to the leaves, and for each cell C in which points are still uncovered, it considers each edge e of C and selects the "most useful" squares containing e, i.e., the squares with the largest intersection with C. We assume (throughout this paper) that all points and all vertices of the squares have integral coordinates in [0, N ) 2 for a given N , and we obtain a competitive ratio of O(log N ). If we know that all the inserted points come from an initially given set of n candidate points P 0 (as in, e.g., Alon et al. [4]), we improve our competitive ratio to O(log n). For this case, we use the BBD-tree data structure due to Arya et al. [7] which uses a more intricate decomposition into cells than a standard quad-tree, and adapt our algorithm to it in a non-trivial manner. Due to the monotonicity of our offline algorithm, we immediately obtain an O(log n)-competitive online algorithm. Online hitting set for squares In Section 3 we present an O(log N )-competitive algorithm for online hitting set for squares of arbitrary sizes, where the points are given initially and the squares arrive online. This matches the best-known O(log N )-competitive algorithm for the much simpler case of intervals [23]. Also, there is a matching lower bound of Ω(log N ), even for intervals. In a nutshell, if a new square S is inserted by the adversary, we identify O(log N ) quad-tree cells for which S contains one of its edges. Then, we pick the most useful points in these cells to hit such squares: those are the points closest to the four edges of the cell. We say that this activates the cell. In our analysis, we turn this around: we show that for each point p ∈ OPT there are only O(log N ) cells that can possibly get activated if a square S is inserted that is hit by p. This yields a competitive ratio of O(log N ). Dynamic set cover and hitting set for d-D hyperrectangles Then, in Section 4 and 5 we present our dynamic algorithms for set cover and hitting set for hyperrectangles in d dimensions. Note that no dynamic algorithm with polylogarithmic update time and polylogarithmic approximation ratio is known even for set cover for rectangles and it was asked explicitly by Chan et al. [20] whether such an algorithm exists. Thus, we answer this question in the affirmative for the case when only points are inserted and deleted. Note that this is the relevant case when we seek to store our solution explicitly, as discussed above. Even though our considered objects are very general, our algorithms need only polylogarithmic worst-case update time. In contrast, Abboud et al. [1] showed that under Strong Exponential Time Hypothesis any general (dynamic) set cover algorithm with an amortized update time of O(f 1−ε ) must have an approximation ratio of Ω(n α ) for some constant α > 0, and f can be as large as Θ(m). We first discuss our algorithm for set cover. We start with reducing the case of hyperrectangles in d dimensions to 2d-dimensional hypercubes with integral corners in [0, 4m] 2d . Then, a natural approach would be to adapt our algorithm for squares from above to 2d-dimensional hypercubes. A canonical generalization would be to build a quad-tree, traverse it from the root to the leaves, and to select for each cell C and for each facet F of C the most useful hypercube S containing F , i.e., the hypercube S with maximal intersection with C. Unfortunately, this is no longer sufficient, not even in three dimensions: it might be that there is a cell C for which it is necessary that we select cubes that contain only an edge of C but not a facet of C (see Figure 2). Here, we introduce a crucial new idea: for each cell C of the (standard) quad-tree and for each dimension i ∈ [2d], consider the hypercubes which are "edge-covering" C along dimension i. Based on these hypercubes a (2d−1)dimensional recursive secondary structure is built on all the dimensions except the i-th dimension (see Figure 14). Figure 2: The red cube is the only cube that covers a facet of the (uncolored) cell. The green cube (from OPT) only covers an edge of the cell. Note that there is no corner of a cube from OPT in the cell. Picking the red cube does not cover the the intersection of the green cube with the cell. We call the resulting tree the extended quad-tree. Even though it is much larger than the standard quad-tree, we show that each point is contained in only (log m) O(d) cells. Furthermore, we use it for our second crucial idea to reduce the frequency of the set cover instance: we build an auxiliary instance of general set cover with bounded frequency. It has the same points as the given instance of geometric set cover, but different sets: for each node corresponding to a one-dimensional cell C of the extended quadtree , we consider each of its endpoints p and introduce a set that corresponds to the "most useful" hypercube covering p, i.e., the hypercube covering p with maximal intersection with C. Since each point is contained in only (log m) O(d) cells, the resulting frequency is bounded by (log m) O(d) . Also, we show that our auxiliary set cover instance admits a solution with at most OPT· (log m) O(d) sets. Then we use a dynamic algorithm from [13] for general set cover to maintain an approximate solution for our auxiliary instance, which yields a dynamic (log m) O(d) -approximation algorithm. We further adapt our dynamic set cover algorithm mentioned above to hitting set for d-dimensional hyperrectangles with an approximation ratio of (log n) O(d) . Finally, we extend our algorithms for set cover and hitting set for d-dimensional hyperrectangles even to the weighted case, at the expense of only an extra factor of (log W ) O(1) in the update time and approximation ratio, assuming that all sets/points in the input have weights in [1, W ]. See the following tables for a summary of our results. Other related work The general set cover is well-studied in both online and dynamic settings. Several variants and generalizations of online set cover have been considered, e.g., online submodular cover [27], online set cover under random-order arrival [24], online set cover with recourse [25], etc. For dynamic setting, Gupta et al. [25] initiated the study and provided O(log n)-approximation O(1) [2] (log n) O(1) d-D hyperrectangles O(log 4d−1 m) log W [Thm 5] O(log 2d m) log 3 (W n) Hitting set unit squares algorithm with O(f log n)-amortized update time, even in the weighted setting. Similar to our model, in their model sets are given offline and only elements can appear or depart. After this, there has been a series of works [1,10,12,11,13,25,26,8]. O(1) [2] (log n) O(1) d-D hyperrectangles O(log 4d−1 n) log W [Thm 6] O(log 2d−1 n) log 3 (W m) Bhattacharya et al. [13] have given deterministic (1 + ε)f -approximation in O (f 2 /ε 3 ) + (f /ε 2 ) log(W ) -amortized update time and O(f log 2 (W n)/ε 3 )-worst-case update time, where W denotes the ratio of the weights of the highest and lowest weight sets. Assadi and Solomon [8] have given a randomized f -approximation algorithm with O(f 2 )-amortized update time. Agarwal et al. [2] studied another dynamic setting for geometric set cover, where both points and sets can arrive or depart, and presented (1 + ε)and O(1)-approximation with sublinear update time for intervals and unit squares, respectively. Chan and He [19] extended it to set cover with arbitrary squares. Recently, Chan et al. [20] gave (1 + ε)-approximation for the special case of intervals in O(log 3 n/ε 3 )-amortized update time. They also gave O(1)-approximation for dynamic set cover for unit squares, arbitrary squares, and weighted intervals in amortized update time of 2 O( √ log n) , n 1/2+ε , and 2 O( √ log n log log n) , respectively. Dynamic algorithms are also well-studied for other geometric problems such as maximum independent set of intervals and hyperrectangles [29,14,17], and geometric measure [22]. Set cover for squares In this section we present our online and dynamic algorithms for set cover for squares. We are given a set of m squares S such that each square S ∈ S has integral corners in [0, N ) 2 . W.l.o.g. assume that N is a power of 2. We first describe an offline O(log N )-approximate algorithm. Then we construct an online algorithm and a dynamic algorithm based on it, such that both of them have approximation ratios of O(log N ) as well. For our offline algorithm, we assume that in addition to S and N , we are given a set of points P that we need to cover, such that P ⊆ [0, N ) 2 and each point p ∈ P has integral coordinates. Quad-tree We start with the definition of a quad-tree T = (V, E), similarly as in, e.g., [6,9]. In T each node v ∈ V corresponds to a square cell C v ⊆ [0, N ) 2 whose vertices have integral coordinates. The root r ∈ V of T corresponds to the cell C r := [0, N ) 2 . Recursively, consider a node v ∈ V , corresponding to a cell C v and assume that C v = [x (1) 1 , x (1) 2 ) × [x (2) 1 , x(2) 2 ). If C v is a unit square, i.e., |x (1) 2 − x (1) 1 | = |x (2) 2 − x(2) 1 | = 1, then we define that v is a leaf. Otherwise, we define that v has four children v 1 , v 2 , v 3 , v 4 that correspond to the four cells that we obtain if we partition C v into four equal sized smaller cells, i.e., define x (1) mid := (x (1) 2 − x (1) 1 )/2 and x (2) mid := (x (2) 2 − x (2) 1 )/2 and C v 1 = [x (1) 1 , x (1) mid ) × [x (2) 1 , x (2) mid ), C v 2 = [x (1) 1 , x (1) mid ) × [x (2) mid , x (2) 2 ), C v 3 = [x (1) mid , x (1) 2 ) × [x (2) 1 , x (2) mid ), and C v 4 = [x (1) mid , x (1) 2 ) × [x (2) mid , x(2) 2 ). Note that the depth of this tree is log N , where depth of a node in the tree is its distance from the root of T , and depth of T is the maximum depth of any node in T . By the construction, each leaf node contains at most one point and it will lie on the bottom-left corner of the corresponding cell. Offline algorithm In the offline algorithm A off , we traverse T in a breadth-first-order, i.e., we order the nodes in V by their distances to the root r and consider them in this order (breaking ties arbitrarily but in a fixed manner). Suppose that in one iteration we consider a node v ∈ V , corresponding to a cell C v . We check whether the squares selected in the ancestors of v cover all points in P ∩ C v . If this is the case, we do not select any squares from S in this iteration (corresponding to v). Observe that hence we also do not select any squares in the iterations corresponding to the descendants of v in T (so we might as well skip the whole subtree rooted at v). Suppose now that the squares selected in the ancestors of v do not cover all points in P ∩ C v . We call such a node to be explored by our algorithm. Let e be an edge of C v . We say that a square containing e is edge-covering for e. We select a square from S that is edge-covering for e and that has the largest intersection with C v among all such squares in S (we call such a square maximum area-covering for C v for edge e). We break ties in an arbitrary but fixed way, e.g., by selecting the square with smallest index according to an arbitrary ordering of S. If there is no square in S that is edge-covering for e then we do not select a square corresponding to e. We do this for each of the four edges of C v . See Figure 3. If we reach a leaf node, and if there is an uncovered point (note that it must be on the bottom-left corner of the cell), then we select any arbitrary square that covers the point (the existence of such a square is guaranteed as some square in OPT covers it). See Figure 4. Lemma 1. A off outputs a feasible set cover for the points in P . Proof. Assume for contradiction that no square in ALG covered some point p ∈ P . Since OPT is a feasible set cover, there is a square S ∈ OPT which covered p. There are two cases to consider here: either p is exactly at one of the corners of S, or not. In the latter case, note that S is edgecovering for at least one quad-tree cell containing p. Let C v be such a cell (which contains p and its edge e is contained in S) with minimum depth. Now the algorithm will traverse T till we reach the node v (corresponding to cell C v ) containing p. As the squares selected by the algorithm for the ancestors of v do not cover p, we will select the maximum area-covering square S (for e) in ALG. As (S ∩ C v ) ⊆ (S ∩ C v ), S will cover p. This is a contradiction. Now in the first case, i.e., where p is at one of the corners of S ∈ OPT, either there is a leaf v ∈ T which contains it and S is edge-covering for C v , or for such a leaf v, S is corner-covering. In both the cases, A off will pick a square for v or one of its ancestors such that this square covers p. Approximation ratio Let ALG ⊆ S denote the selected set of squares and let OPT denote the optimal solution. To prove the O(log N )-approximation guarantee, the main idea is the following: Figure 4: Point p lies in a leaf cell C (which may not even have any edge-covering squares). In this case, we pick an arbitrary square S to cover the point (since one such square always exists). consider a node v ∈ V and suppose that we selected at least one square in the iteration corresponding to v. If C v contains a corner of a square S ∈ OPT, then we charge the (at most four) squares selected for v to S. Otherwise, we argue that the squares selected for v cover at least as much of C v as the squares in OPT, and that they cover all the remaining uncovered points in P ∩ C v . In particular, we do not select any further squares in the descendants of v. The squares selected for v are charged to the parent of v (which contains a corner of a square S ∈ OPT). Since each corner of each square S ∈ OPT is contained in O(log N ) cells, we show that each square S ∈ OPT receives a total charge of O(log N ). Thus, we obtain the following lemma. Lemma 2. We have that |ALG| = O(log N ) · |OPT|. Proof. We will charge each square picked in ALG to some square in OPT. A cell C v with its corresponding node v, can either contain (at least) a corner of some square in OPT, or be edge-covered by (at least) a square in OPT, or not intersect any square from OPT at all. • C v contains a corner of S ∈ OPT: In this case, A off picks at most four squares for the cell, and we charge these squares to a corner of S in the cell. If there are multiple squares from OPT with a corner in the cell, pick one arbitrarily. This claim is true even when C v corresponds to a leaf node. • Some square S ∈ OPT is edge-covering for C v (and C v has no corner of a square in OPT): If A off picks no edge-covering squares for such a cell, then we are fine. Otherwise, if A off picks squares for such a cell, we claim that it covers all points in the cell. This is due to the fact that any point in this cell is covered by a square in OPT that is edge-covering for C v , due to the absence of corners of squares of OPT. So when A off picks edge-covering squares with the largest intersection with the cell, the intersection of any square S ∈ OPT with C v will also get covered). So, no child node of v will be further explored by the algorithm. This also means that the parent v of v in the tree will contain a corner of S (because C v intersects S, but cannot be edge-covered by it). We charge any squares picked by A off at C v (at most four times) to this particular corner in the parent node. If there are multiple such corners, pick one arbitrarily. • No squares from OPT intersect C v : In this case, C v does not contain any points in P . Thus, A off will not pick any squares for such a cell. Now we note that a corner of any square in OPT, will lie in at most log N cells of the quad-tree. For each of these cells, a corner is charged at most four times for the squares picked at the cell, and at most four times for each of its four child nodes. This amounts to a total charge of at most 20 per corner per cell. So each square in OPT is charged at most 20 (per corner, per cell) × 4 (corners) × log N (cells) = 80 log N times. Therefore, there are at most 80 log N · |OPT| squares in ALG. Online set cover for squares O(log N )-approximate online algorithm We want to turn our offline algorithm A off into an online algorithm A on , assuming that in each round a new point is introduced by the adversary. The key insight for this is that the algorithm above is monotone, i.e., if we add a point to P , then it outputs a superset of the squares from S that it had output before (when running it on P only). For a given set of points P , let ALG(P ) ⊆ S denote the set of squares that our (offline) algorithm outputs. Proof. Assume towards contradiction that there exists some square S in ALG(P ) which did not belong to ALG(P ∪ {p}). According to the description of A off , we can infer that S was picked by the algorithm in some iteration because it was maximum area-covering for some cell C v (corresponding to node v in T ) that contained a point p ∈ P introduced by the adversary. Also, A off in its run must have explored all the ancestors of v in T . Note that any such point p could be covered in a run of the algorithm only when it traverses cells that contain p . This is due to the fact that once we pick some squares associated with a cell in the quad-tree, we only account for the area inside this cell that the squares cover. In light of this fact, if A off did not explore C v in this time step, then it also would not have explored the children of v in T . Hence, the point p would not have been covered which is a contradiction. Hence, it is easy now to derive an online algorithm for set cover for squares. Initially, P = ∅. If a point p is introduced by the adversary, then we compute ALG(P ) (where P denotes the set of previous points, i.e., without p) and ALG(P ∪{p}) and we add the squares in ALG(P ∪{p})\ALG(P ) to our solution. Therefore, due to Lemma 2 and Lemma 3 we obtain an O(log N )-competitive online algorithm. O(log n)-approximate online set cover for squares We assume now that we are given a setP ⊆ R 2 with |P | = n such that in each round a point from P is inserted to P , i.e., P ⊆P after each round. We want to get a competitive ratio of O(log n) in this case. If N = n O(1) then this is immediate. Otherwise, we extend our algorithm such that it uses the balanced box-decomposition tree (or BBD-tree) data structure due to Arya et al. [7], instead of the quad-tree. Before the first round, P = ∅ and we initialize the BBD-tree which yields a treẽ T = (Ṽ ,Ẽ) with the following properties: • each node v ∈Ṽ corresponds to a cellC v ⊆ [0, N ) 9 • the aspect ratio of b O , i.e., the ratio between the length of the longest edge to the length of the shortest edge of b O , is bounded by 3. • if b I = ∅, then b I is sticky which intuitively means that in each dimension, the distance of b I to the boundary of b O is either 0 or at least the width of b I . Formally, assume that b O = [x (1) O , x (2) O ] × [y (1) O , y(2)O ] and b I = [x (1) I , x(2)I ] × [y (1) I , y(2)I ]. Then x (1) O = x (1) I or x (1) I − x (1) O ≥ x (2) I − x (1) I . Also x (2) O = x (2) I or x (2) O − x (2) I ≥ x (2) I − x (1) I . Analogous conditions also hold for the y-coordinates. • each node v ∈Ṽ is a leaf or it has two children v 1 , v 2 ∈Ṽ ; in the latter caseC v =C v 1∪C v 2 . • the depth ofT is O(log n) and each point q ∈ [0, N ) 2 is contained in O(log n) cells. • each leaf node v ∈Ṽ contains at most one point inP . In the construction of the BBD-tree, we make the cells at the same depth disjoint so that a point p may be contained in exactly one cell at a certain depth. Hence, for a cellC v = b O \ b I we assume both b O and b I to be closed set, i.e., the boundary of the outer box b O is part of the cell and the boundary of the inner box b I is not part of the cell. We now describe an adjustment of our offline algorithm from Section 2, working withT instead of T . Similarly, as before, we traverseT in a breadth-first-order. Suppose that in one iteration we consider a node v ∈Ṽ corresponding to a cell C v . We check whether the squares selected in the ancestors of v cover all points in P ∩C v . If this is the case, we do not select any squares from S in this iteration corresponding to v. Suppose now that the squares selected in the ancestors of v do not cover all points in P ∩C v . Similar to Section 2, we want to select O(1) squares forC v such that ifC v contains no corner of a square S ∈ OPT, then the squares we selected forC v should cover all points in P ∩C v . Similarly as before, for each edge e of b O we select a square from S that contains e and that has the largest intersection with b O among all such squares in S. We break ties in an arbitrary but fixed way. However, asC v may not be a square and can have holes (due to b I ), apart from the edge-covering squares, we need to consider two additional types of squares in OPT with nonempty overlap with C v : (a) crossingC v , i.e., squares that intersect two parallel edges of b O ; (b) has one or two corners inside b I . The following greedy subroutine G will be useful in our algorithm to handle such problematic cases. Let R be a box of width w and height h such that w/h ≤ B, for some constant B ∈ N; and P R be a set of points inside R that can be covered by a collection of vertically-crossing (i.e., they intersect both horizontal edges of R) squares S . Then, the set of squares picked according to G covers P R in the following way: • While there is an uncovered point p ∈ P R : -Consider the leftmost such uncovered point p ∈ P R . -Select the vertically-crossing square intersecting p (by assumption, such a square exists) with the rightmost edge. (The above subroutine is for finding vertically-crossing squares. For finding horizontally-crossing squares, we can appropriately rotate the input 90 • anti-clockwise, and apply the same subroutine.) Then, we have the following claim about the aforementioned subroutine. Proof. Consider each iteration of the greedy subroutine G. We call pivot to be the leftmost point p ∈ P R that is not already covered by a square selected by G so far. Then all selected verticallycrossing squares for R will contain exactly one point that was identified as a pivot point at some point during the execution of the algorithm. As the aspect ratio is bounded by B and the squares are vertically-crossing (i.e., their vertical length is more than the vertical length of R), there can be at most B + 1 pivot points. Hence, we select at most B + 1 crossing squares due to R. This produces a feasible set cover. Now we describe our algorithm. First, we take care of the squares that can cross b O . So, we apply the greedy subroutine G on b O . As b O has bounded aspect ratio of 3, from Claim 1, we obtain at most (3 + 1) + (1 + 1) = 6 squares that can cross C v vertically or horizontally. If b I = ∅, we do not select any more squares. Otherwise, we need to take care of the squares that can have one or two corners inside b I . Let 1 , 2 , 3 , 4 denote the four lines that contain the four edges of b I . Observe that 1 , 2 , 3 , 4 partition b O into up to nine rectangular regions, one being identical to b I . For each such rectangular region R, if it is sharing a horizontal edge with b I , we again use G to select vertically-crossing squares. Otherwise, if R is sharing a vertical edge with b I , we use the subroutine G appropriately to select horizontally-crossing squares. This takes care of squares having two corners inside b I . Otherwise, if the rectangular region R does not share an edge with b I , then we check if there is a square S ∈ S with a corner within b I that completely contains R. We add S to our solution too. This finally takes care of the case when a square has a single corner inside b I . Finally, to complete our algorithm, before its execution, we do the following: for every leaf v for which C v contains at most one point p ∈P , we associate a fixed square which covers p. Then, if our algorithm reaches a leaf v while traversing that has an uncovered point p, we pick the associated square with this leaf that covers it. This condition in our algorithm guarantees feasibility. Similarly, by our selected squares (that vertically or horizontally crosses b O ) in the greedy subroutine G, we have covered all points that can be covered by such crossing squares in OPT that crosses b O . For squares that have (at least) a corner inside b I , note that they have to cross one of the rectangular regions that came from partitioning of b O and shares an edge with b I . In fact, for such a square S with a corner inside b I , there is a rectangular region R (say, with width w and height h) among these four rectangular regions such that either S is vertically-crossing for R and R shared a horizontal edge with b I (then w/h ≤ 3) or S is horizontally-crossing for R and R shared a vertical edge with b I (then h/w ≤ 3). But then using Claim 1, we cover all points covered by such squares. Additionally, a square that has exactly one corner inside b I may completely contain another rectangular region from partitioning of b O (that do not share an edge with b I ). For them, again we have covered them by selecting one square, if it exists. Thus, we can establish a similar charging scheme as in Section 2. To pay for our solution, we charge each corner q of a square S ∈ OPT at most O(log n) times. Hence, our approximation ratio is O(log n). Similarly as in Section 2, we can modify the above offline algorithm to an online algorithm with an approximation ratio of O(log n) each. Theorem 1. There is a deterministic O(log n)-competitive online algorithm for set cover for axisparallel squares of arbitrary sizes. Lower bounds It is a natural question whether algorithms having a competitive factor better than O(log n) are possible for online set cover for squares. We answer this question in the negative, even for the case of unit squares and even for randomized algorithms. We also remark here that there exists a tight 2-competitive online algorithm for set cover for intervals (see Section A). Unit squares and quadrants Given a set cover instance (P, F), for each F ∈ F define a variable x F which takes values ∈ [0, 1]. For a point p ∈ P , let F p ⊆ F be the sets that cover it. In the fractional set cover problem, the aim is to assign values to the variables x F such that for all points p ∈ P , F ∈Fp x F ≥ 1. . Also p i,j does not intersect any of the last m − j + 1 quadrants because p(y) > m − j. Now consider an adversary that introduces points as follows: P 1 = p 1,m . If the algorithm assigns values to the variables such that m/2 i=1 x i ≥ i=m i=m/2+1 x i , then the point P 2 = p m/2+1,m is given, otherwise the point P 2 = p 1,m/2 is given. The adversary repeats this process of halving the set of quadrants intersected, and puts the next point in the range with the lower sum, till only one quadrant, say quadrant j is left. The optimal solution would have been to assign only x j to 1 and the remaining variables to zero. But any online algorithm can only halve the set of the potential optimal solution in each step, while assigning at least 1/2 "cost" to the non-optimal quadrants. Hence, the cost of any online algorithm is Ω(log m). Corollary 1. There is an instance of the online fractional covering problem on m unit squares such that any deterministic online algorithm is Ω(log m)-competitive on this instance. Proof. We can appropriately extend the quadrants from Lemma 5 in the bottom-left direction to obtain a similar instance on squares with side-length of m. More precisely, let the bottom left corner of the square i corresponding to quadrant i be (i − m, 1 − i). The points introduced by the adversary are the same as in the quadrants instance. Now scale down this instance on squares appropriately, by a factor of m, to get the required unit square instance. Using standard techniques, as in [15], we can extend the lower bound for deterministic algorithms for the fractional variant to the lower bound for randomized algorithms for the integral variant. Corollary 2. There is an instance of the online (integral) set cover problem on m unit squares such that any randomized online algorithm is Ω(log m)-competitive on this instance. Since in our lower bound construction, n = Θ(m 2 ), log n = Θ(log m) and hence, we have the theorem as stated below. Theorem 2. Any deterministic or randomized online algorithm for set cover for unit squares has a competitive ratio of Ω(log n), even if all squares contain the origin and all points are contained in the same quadrant. Online hitting set for squares We present our online algorithm for hitting set for squares. We assume that we are given a fixed set of points P ⊆ [0, N ) 2 with integral coordinates. We maintain a set P of selected points such that initially P := ∅. In each round, we are given a square S ⊆ [0, N ) 2 whose corners have integral coordinates. We assume w.l.o.g. that N is a power of 2. Let Q be all points with integral coordinates in [0, N ) 2 , i.e., P ⊆ Q. For each point q ∈ Q we say that q = (q x , q y ) is of level if both q x and q y are integral multiples of N/2 . We build the same quad-tree as in Section 2. We say that a cell C v is of level if its height and width equal N/2 . We present our algorithm now. Suppose that in some round a new square S is given. If S∩P = ∅ then we do not add any point to P . Suppose now that S ∩ P = ∅. Let q be a point of smallest level among all points in Q ∩ S (if there are many such points, then we select an arbitrary point in Q ∩ S of smallest level). Intuitively, we interpret q as if it were the origin and partition the plane into four quadrants. We define O T R := {(p x , p y ) | p x ≥ q x , p y ≥ q y }, and S T R := O T R ∩ S, and define similarly O T L , O BR , O BL , and S T L , S BR , S BL . Consider O T R and S T R . For each level = 0, 1, . . . , log N , we do the following. Consider each cell C of level in some fixed order such that C ⊆ O T R and S T R is edge-covering for some edge e of C. Then, for each edge identify the point p b (p t , p l , p r , resp.) in P ∩ C that is closest to its bottom (top, left, and right, resp.) edge. We add these (at most 4) points to our solution if at least one of p b , p t , p l , p r is contained in S T R (see Figure 9). If we add at least one such point p of the cell C to P in this way, we say that C gets activated. Note that we add possibly all of the points p b , p t , p l , p r to P even though only one may be contained in S T R . This is to ensure that C gets activated at most once during a run of the online algorithm. This will be proved in Claim 2, which will ultimately help us prove that our algorithm is O(log N )-competitive. If for the current level we activate at least one cell C of level , then we stop the loop and do not consider the other levels + 1, . . . , log N . Otherwise, we continue with level + 1. We do a symmetric operation for the pairs (O T L , S T L ), (O BR , S BR ), and (O BL , S BL ). We now prove the correctness of the algorithm and that its competitive ratio is O(log N ). Lemma 6. After each round, the set P is a hitting set for the squares that have been added so far. Proof. We will prove that in case S ∩ P = ∅ when a square S is inserted, at least one cell C gets activated. Note that there exists a point p in an optimum hitting set such that p ∈ S. Assume w.l.o.g. that p belongs to S T R . Then, consider the set of cells C p that contain the point p. Since S has side-length at least 1 (it has integral coordinates for the corners) there exists a cell C ∈ C p such that S T R covers an edge e of C . Hence, there will exist one level in {0, 1, ..., log N } such that a cell C ⊆ O T R of level exists for which S covered its edge (say, the bottom edge e ) and p ∈ C ∩ S. Then, our algorithm picked p b (point closest to the bottom edge) for C , such that p b ∈ C ∩ S. Now we show that in each round O(1) points are added to P . Lemma 7. In each round we add O(1) points to P . Proof. We show that given a square S such that S ∩ P = ∅ (P is the hitting set maintained by our algorithm), our algorithm activates at most 4 cells. For this, we just observe that in each of the four quadrants O T R , O BR , O T L , O BL , we activate at most 1 cell. For each of these cells, we pick at most 4 points and hence, we add at most 16 points in any round. Denote by OPT the optimal solution after the last round of inserting a square. Proof. First, define horizontal distance between two cells of level to be the distance between the x-coordinates of their left edges. Analogously, define the vertical distance. Now, we define a set of cells C p corresponding to the point p, initialized to ∅. We will show later that in a certain round if a square S introduced by the adversary contains p, and our algorithm activates at least one cell, then one cell in C p is also activated. For each level ∈ {0, 1, . . . , log N }, include in the set C p : the cell C of level containing p and the other cells of level if they exist which have the same parent as C. We call these cells to be primary cells. Further, for level ∈ {0, 1, . . . , log N } consider the cell C of level which contains p. Then, consider all the cells of level which are at a distance of N/2 horizontally but at a distance of 0 vertically; also, consider cells which are at a distance of N/2 vertically but at a distance of 0 horizontally from C. There can be at most 4 such cells. For each such cell, include all its children in C p (which are 4 in number). We call these cells to be secondary cells. Hence, per level we select at most 4 + 4 × 4 = 20 cells. Therefore, |C p | = O(log N ). We first observe that once a cellĈ gets activated, it does not get activated again. Claim 2. A cellĈ ∈ T does not get activated more than once by our algorithm. We now want to show that if a square S is inserted in some round where S ∩P = ∅ but p ∈ S T R , then one cell in C p gets activated but no cellĈ / ∈ C p withĈ ⊆ O T R gets activated. Once we prove this, observe that in each such round we activate one cell in C p . Then, by using the above claim that no cell in C p gets activated again in such a round, we are guaranteed that after |C p | rounds, every square S introduced by the adversary which contained p was hit by at least one point in the hitting set that the algorithm maintained. Then we do a symmetric argumentation for the cases that p ∈ S T L , p ∈ S BR , and p ∈ S BL , each of them yielding the fact that if a square S is added with S ∩ P = ∅ but p ∈ S, some cell among the cells in C p gets activated. Thus, there can be only |C p | = O(log N ) such rounds. Therefore, finally it remains to prove the claim. Claim 4. If a square S is inserted in some round where S ∩ P = ∅ but p ∈ S T R , then one cell in C p gets activated but no cellĈ / ∈ C p withĈ ⊆ O T R gets activated. Claim 5. Denote the level of q which was one of the points at the smallest level among points in Q ∩ S to be . Assume by contradiction that a cellĈ / ∈ C p withĈ ⊆ O T R gets activated and S w.l.o.g. covered its bottom edge. Let be the level ofĈ. Let C be the cell of level containing p and let C be its parent (which is at level − 1). By the construction, we know thatĈ ∩ C = ∅ and hence, the parent ofĈ is not C . Then, the parent ofĈ (denote by C ) and C are level − 1 cells at a distance at least N/2 −1 , either horizontally or vertically. Assume w.l.o.g. that C is to the right side of C . By our assumption, right edge ofĈ does not lie to the right side of the left edge of C (could coincide). Any of the corners ofĈ are points at level at most . Then, we know that the level of q, which was is at most . Now there are two cases. In the first case, the horizontal distance between q and the left edge of C is at least N/2 −1 (see Figure 11(a)). Then ≤ − 1 since then S T R covers the left bottom corner of C . In this case, by our assumption S T R covers the bottom edge of C which also contains at least one point that hits it. This is a contradiction on the level of the activated cell in this round since C has level − 1. In the other case, the horizontal distance between q and the left edge of C is strictly less than N/2 −1 (see Figure 11(b)). In this case, q is again at level exactly = − 1. Then, the right edge of C coincides with left edge of C . Therefore, C is at a distance of exactly N/2 −1 to the left of C and should have been added as a secondary cell. Hence,Ĉ ∈ C p . This is a contradiction. This completes the proof of the lemma. Figure 11: Proof of Claim 4. (a) Either, the distance between C and C is large, in which case S is edge-covering for C , or (b)Ĉ ∈ C p . O T L O BL O T R O BR C ′′ C ′ C C p q S O T L O BL O T R O BR C ′′ C ′ C C p q S (a) (b) Hence, Lemma 7 and Lemma 8 imply that for each point p ∈ OPT we add O(log N ) points to P . Thus, our competitive ratio is O(log N ). Theorem 3. There is an O(log N )-competitive deterministic online algorithm for hitting set for axisparallel squares of arbitrary sizes. This is tight, as even for intervals, Even et al. [23] have shown an Ω(log N ) lower bound. Dynamic set cover for d-dimensional hyperrectangles In this section, we will design an algorithm to dynamically maintain an approximate set cover for d-dimensional hyperrectangles.The main result we prove in this section is the following. Our goal is to adapt the quad-tree based algorithms designed in the previous sections of the paper. As a first step towards that, we transform the problem such that the points and hyperrectangles in R d get transformed to points and hypercubes in R 2d , and the new problem is to cover the points in R 2d with these hypercubes. As discussed in the introduction, a simple 2d-dimensional quad-tree on the hypercubes does not suffice for our purpose. We augment the quad-tree in two ways: (a) at each node, we collect the hypercubes which are edge-covering w.r.t. that node and "ignore" that dimension in which they are edge-covering, and (b) recursively construct a (2d−1)-dimensional quad-tree on these hypercubes based on the remaining 2d−1 dimensions. We call this new structure an extended quad-tree. The nice feature we obtain is that any point in R 2d will belong to only O(log 2d m) cells in the extended quad-tree. Furthermore, at the 1-dimensional cells of the extended quad-tree, for each cell we will identify O(1) "most useful" hypercubes. This ensures that any point belongs to only O(log 2d m) most useful hypercubes. As a result, a "bounded frequency" set system can be constructed with the most useful hypercubes. The dynamic algorithm from Bhattacharya et al. [13] (for general set cover) works efficiently on bounded frequency set systems and applying it in our setting leads to an O(log 4d−1 m)-approximation algorithm. Transformation to hypercubes in R 2d . Recall that the input is a set P of points and S is a collection of hyperrectangles in R d . The first step of the algorithm is to transform the hyperrectangles in S to hypercubes in R 2d . Consider a hyperrectangle S ∈ S with a = (a 1 , . . . , a d ) and b = (b 1 , . . . , b d ) being the "lower-left" and the "upper-right" corners of S, respectively. Let ∆ = max d j=1 (b j − a j ). Then S is transformed to a hypercube S in R 2d with side-length ∆ and "top-right" corner (−a 1 , −a 2 , . . . , −a d , b 1 , b 2 , . . . , b d ). Let S be the collection of these m transformed hypercubes. Let P be the set of n points in R 2d obtained by transforming each point p = (p 1 , . . . , p d ) ∈ P to p = (−p 1 , . . . , −p d , p 1 , . . . , p d ). Proof. Assume that p lies inside S. Consider the i-th coordinate of p with i ≤ d. p = (p 1 ) a = (a 1 ) b = (b 1 ) (−a 1 , b 1 ) (a) (b) p ′ = (−p 1 , p 1 ) (−b 1 , a 1 ) S ′Since b i − a i ≤ ∆ implies that −a i − ∆ ≤ −b i , we observe that −a i − ∆ ≤ −b i ≤ −p i ≤ −a i . Therefore, for all 1 ≤ i ≤ d, we have −a i − ∆ ≤ −p i ≤ −a i . Now consider the i-th coordinate of p with i > d. Since b i − a i ≤ ∆ implies that b i − ∆ ≤ a i , we observe that b i − ∆ ≤ a i ≤ p i ≤ b i . Therefore, for all d + 1 ≤ i ≤ 2d, we have b i − ∆ ≤ p i ≤ b i . Thus, we claim that if a point p lies inside S, then p will lie inside S . It is easy to prove the other direction. If p lies outside S, then there is at least one coordinate (say i) in which either p i < a i or p i > b i . If p i < a i , then −p i > −a i and hence, p lies outside S . On the other hand, if p i > b i , then again p lies outside S . By a standard rank-space reduction we can assume that the corners of the hyperrectangles in S lie on the grid [0, 2m] d . After applying the above transformation, we note that the coordinates of the corners of the hypercubes in S will lie on the grid [−4m, 0] d × [−2m, 2m] d : trivially, ∆ + a i ≤ 4m, and hence −4m ≤ −a i − ∆ ≤ −a i ≤ 0. Also, −2m ≤ b i − ∆ ≤ b i ≤ 2m. After performing a suitable shifting of the grid, we will assume that all corners of the hypercubes in S will lie on the grid [0, 4m] 2d . Constructing a bounded frequency set system. We will now present a technique to select a setŜ ⊆ S with the following properties: 1. (Bounded frequency) Any point in P lies inside O(log 2d m) hypercubes inŜ. 2. An α-approximation dynamic set cover algorithm for (P ,Ŝ) implies an O(α log 2d−1 m)-approximation dynamic set cover algorithm for (P , S ). 3. The time taken to update the solution for the set system (P ,Ŝ) is O(log 2d m · log 2 n). 4. The time taken to construct the setŜ is O(m log 2d m). Extended quad-tree for 2-dimensional squares. Given a set of squares S , construct a 2-dimensional quad-tree T (as defined in Section 2), such that its root contains all the squares in S . We assume for simplicity that no two input squares in S share a corner. Then, we can perturb the input points slightly so that no point p ∈ P lies on any of the grid points of the quad-tree and each square still contains the same set of points as before. Consider a node v ∈ T and a square S ∈ S . Let C and par(C) be the cell corresponding to node v and the parent node of v, respectively. Let proj i (C), proj i (par(C)) and proj i (S) be the projection of C, par(C) and S, respectively, on to the i-th dimension. Then S is i-long at v if and only if proj i (C) ⊆ proj i (S) but proj i (par(C)) ⊆ proj i (S). See Figure 13(a). For all u ∈ T, let S(u, i) ⊆ S be the squares which are i-long at node u. Intuitively, these are squares that cover the edge of C in the i-th dimension but do not cover any edge of par(C) in the i-th dimension. Now, at each node of T we will construct two secondary structures as follows: the first structure is a 1-dimensional quad-tree built on the projection of the squares in S(u, 1) on to the second dimension , and the second structure is a 1-dimensional quad-tree built on the projection of the squares in S(u, 2) on to the first dimension. In each secondary structure, an interval I (corresponding to a square S ∈ S ) is assigned to a node u if and only if u is the node with the smallest depth (the root is at depth zero) where I intersects either the left endpoint or the right endpoint of the cell C u . See Figure 13(b). By this definition, any interval will be assigned to at most two nodes in the secondary structure. Now we will use T to construct the geometric collectionŜ. Let V sec be the set of nodes in all the secondary structures of T. For any node u ∈ V sec , among its assigned intervals which intersect the left (resp., right) endpoint of the cell C u , identify the maximal interval I (resp., I r ) , i.e., the interval which has maximum overlap with C u . See Figure 13(c). We then do the following set of operations over all the nodes in V sec : For a node u ∈ V sec , denote by S and S the corresponding squares for the assigned intervals I and I r , respectively. Further, let w be the node in T, on which the secondary structure of u was constructed. Then, we include inŜ the rectangles S 1 ∩ C w and S 2 ∩ C w . Figure 13: (a) A square S which is 1-long at node v (corrs. cell C is highlighted in darker orange), (b) I is assigned to the two children of v, and (c) the maximal intervals I and I r at C v . par(C) S I par(u) u (a) (b) (c) dim 2 dim 1 C I r I ℓ C u (b) Extended quad-tree for 2d-dimensional hypercubes. In this section, we need a generalization of the quad-tree defined in Section 2. For d > 2, a d -dimensional quad-tree is defined analogously to the the quad-tree defined in Section 2, where instead of four, each internal node will now have 2 d children. Assume by induction that we have defined how to construct the extended quad-tree for all dimensions less than or equal to 2d−1. (The base case is the extended quad-tree built for 2-dimensional squares). We define now how to construct the structure for 2d-dimensional hypercubes. First construct the regular 2d-dimensional quad-tree T for the set of hypercubes S . Consider any node v ∈ T. Generalizing the previous definition, for any 1 ≤ i ≤ 2d, a hypercube S ∈ S is defined to be i-long at node v if and only if proj i (C) ⊆ proj i (S), but proj i (par(C)) ⊆ proj i (S). For all v ∈ T, let S(v, i) ⊆ S be the hypercubes which are i-long at node v. Now, at each node of T we will construct 2d secondary structures as follows: for all 1 ≤ i ≤ 2d, the i-th secondary structure is a (2d−1)-dimensional extended quadtree built on S(v, i) and all its 2d dimensions except the i-th dimension. Specifically, any hypercube S ∈ S(v, i) of the form 1 × · · · × i × · · · × 2d is projected to a (2d−1)-dimensional hypercube 1 × · · · × i−1 × i+1 × · · · × 2d . LetŜ v be the collection of the (2d−1)-dimensional hyperrectangles that are inductively picked for the secondary structure constructed at v ∈ T using the routine . Define the function g which maps a (2d−1)-dimensional hyperrectangle picked as part of the collectionŜ v (for a v ∈ T) to its corresponding 2d-dimensional hypercube S ∈ S . We now define the collection of setsŜ consisting of 2d-dimensional hyperrectangles: S ← v∈T   S ∈Ŝv (g(S ) ∩ C v )   . Claim 6. (Feasibility) Any point p ∈ P is covered by at least one set in the collectionŜ. Proof. We will prove the claim first for the case of 2-D squares. For any point p ∈ P , we know that at least one square S ∈ S covers it. By our assumption, p is not on any of the grid points of the quad-tree. Then, we claim that S is edge-covering for a cell C v such that v is a leaf node and Figure 14: Extended quad-tree with a 2 × 2 × 2 cube as the root. C v contains p. Then, by the definition of i-long, there exists an ancestor u of v in T (or possibly v itself) such that S is i-long for C u , for some i ∈ [2]. This implies thatŜ consists of a (maximal) square S such that S ∈ S(u, i) and S ∩ C u ⊇ S ∩ C u . Then, by our construction of the extended quad-tree, S ∪ C u is part of the collectionŜ. Hence, the claim holds for 2-D squares. Generalizing this idea, feasibility can be guaranteed for the case of 2d-dimensional hypercubes for d ≥ 1. Lemma 9. (Bounded frequency) Any point in P lies inside O(log 2d m) sets inŜ. Proof. For the extended quad-tree for 2-dimensional squares, let τ (2) be the maximum number of sets inŜ which contain a point p = (p x , p y ) ∈ P . By properties of standard quad-tree, the number of nodes in the 2-dimensional quad-tree whose corresponding cells contain p is O(log m). At any such node, each of the secondary structures will have O(log m) nodes whose corresponding cells contain the projection of p (either p x or p y ). At each cell in the secondary structure which contains p, we select at most two (maximal) hyperrectangles intoŜ. Therefore, τ (2) = O(log 2 m). In general, for an extended quad-tree for d -dimensional hypercubes, let τ (d ) be the maximum number of sets inŜ that contain any given d -dimensional point. Since we construct 2d secondary structures at each node, we obtain the following recurrence: τ (2d) = O((2d) log m) · τ (2d−1) = O(log 2d m), where the constant hidden by the big-O-notation depends on d. Lemma 10. If there is an α-approximation dynamic set cover algorithm for (P ,Ŝ) then there is an O(α log 2d−1 m)-approximation dynamic set cover algorithm for (P , S ). Proof. Let OPT and OPT be the size of the optimal set cover for (P , S ) and (P ,Ŝ). For any hypercube S in S we will show that there exists O(log 2d−1 m) hyperrectangles inŜ whose union will completely cover S. Therefore, OPT = OPT · O(log 2d−1 m). Note that for every set S ∈Ŝ, there exists a corresponding hypercube S ∈ S which covers at least the set of points in P that S covers. For (P , S ), an α-approximation set cover for (P ,Ŝ) implies an approximation factor of: b OPT = b · O(log 2d−1 m) OPT = O(α log 2d−1 m), where the last equation follows from b ≤ α · OPT. Finally, we establish the covering property. We will prove it via induction on the dimension size. As a base case, for squares in 2-D, let µ(2) be the number of sets needed inŜ such that their union completely covers a square S ∈ S . For any S ∈ S , let long(S) be the set of nodes in T where S is 1-long. By standard properties of a quad-tree, we have (a) |long(S)| = O(log m), and (b) S ← v∈long(S) (S ∩ C v ), where C v is the cell corresponding to v. Now consider any node v ∈ long(S). Let I be the interval corresponding to S in the secondary structure of v built on S(v, 1). Via our selection of maximal intervals at the secondary nodes, it is clear that there exist two maximal intervals which cover I. Therefore, µ(2) ≤ 2 · |long(S)| = O(log m). In general, let µ(2d) be the number of sets needed inŜ such that their union completely covers a hypercube S ∈ S . Then we claim that µ(2d) = O(2 d log m) × µ(2d−1), where O(2 d log m) is the number of nodes in T where S is 1-long. Solving the recurrence, we obtain µ(2d) = O(log 2d−1 m). The final algorithm For the general dynamic set cover problem, Bhattacharya et al. [13] gave an O(f )-approximation algorithm with a worst-case update time of O(f log 2 n). Recall that f is the frequency of the set system. We will use their algorithm as a blackbox on the set system (P ,Ŝ). Let ALG be the sets reported by their algorithm. Then our algorithm will also report ALG as a set cover for (P , S ). The solution is feasible since each set inŜ belongs to S as well. Lemma 11. The approximation factor of our algorithm is O(log 4d−1 m). Proof. For the set system (P ,Ŝ) the frequency f = O(log 2d m), and hence, the algorithm of [13] leads to an O(log 2d m) approximation for this set system. Using Lemma 10, this implies an approximation factor of O(log 2d m) · O(log 2d−1 m) = O(log 4d−1 m) for the set system (P , S ). T (m, 2d) = O(m log m) + v T (m v , 2d − 1) = O(m log m) + v m v log 2d−1 m v (by induction) = O(m log m) + v m v log 2d−1 m = O(m log 2d m), since, v m v = O(m log m). Weighted setting We present an easy extension of our algorithm to the setting where each hyperrectangle S ∈ S has a weight w S ∈ [1, W ]. First, we round the weight of each set S to the smallest power of two greater than or equal to w S . This leads to O(log W ) different weight classes. Next, for each weight class, we will build an extended quad-tree based on the hypercubes of that weight class after the reduction to the case of 2d-dimensional hypercubes from d-dimensional hyperrectangles as shown in the previous section. Finally, letŜ be the collection of (maximal) hypercubes obtained from all the O(log W ) extended quad-trees. Run the dynamic set cover algorithm of Bhattacharya et al. [13] on (P ,Ŝ). Lemma 14. The approximation factor of the algorithm is O(log 4d−1 m · log W ). Proof. Consider the optimal solution for (P , S ) and let OPT be the optimal weight. By rounding the weight of each set in the optimal solution, their total weight becomes at most 2·OPT. Therefore, the weight of the optimal solution in the set system after rounding is at most 2 · OPT. Compared to the unweighted setting, now the frequency of the set system (P ,Ŝ) increases by a O(log W ) factor. As a result, we obtain an approximation factor of O(log 4d−1 m · log W ). Theorem 5. There is an algorithm for weighted dynamic set cover for d-dimensional hyperrectangles with an approximation factor of O(log 4d−1 m · log W ) and an update time of O(log 2d m · log 3 (W m)). We also have the following corollary for dynamic set cover for 2d-dimensional hypercubes where all the corners are integral and bounded in [0, cm] 2d for some constant c > 0. Corollary 3. There is an algorithm for weighted dynamic set cover for 2d-dimensional hypercubes with an approximation factor of O(log 4d−1 m · log W ) and an update time of O(log 2d m · log 3 (W m)), when all of their corners are integral and bounded in [0, cm] 2d for a fixed c > 0. Dynamic hitting set for d-dimensional hyperrectangles In this section we present a dynamic algorithm for hitting set for d-dimensional hyperrectangles. We will reduce the problem to an instance of dynamic set cover in 2d-dimensional space and use the algorithm designed in the previous section (Theorem 3). Recall that P is the set of points and S is the set of hyperrectangles. Assume that all the points of P and the hyperrectangles in S lie in the box [0, N ] d . For all p ∈ P , we transform p = (p 1 , . . . , p d ) to a 2d-dimensional hypercube S(p) of side length N and "lower-left" corner p = (−p 1 , . . . , −p d , p 1 , . . . , p d ). Let S(P ) be the transformed hypercubes. Next, we transform each hyperrectangle, say S ∈ S, with "lowerleft" corner a = (a 1 , . . . , a d ) and "top-right" corner b = (b 1 , . . . , b d ) into a 2d-dimensional point P (S) = (−a 1 , . . . , −a d , b 1 , . . . , b d ). Let P (S) be the transformed points. See Figure 15. Proof. We define a point q(q 1 , . . . , q 2d ) to dominate another point q (q 1 , . . . , q 2d ) if and only if q i ≥ q i , for all 1 ≤ i ≤ 2d. Assume that p lies inside S. Then we have p i ≥ a i and p i ≤ b i , for all 1 ≤ i ≤ d. This implies that P (S) = (−a 1 , . . . , −a d , b 1 , . . . , b d ) dominates the point p (−p 1 , . . . , −p d , p 1 , . . . , p d ), which is the lower-left corner of S(p). The coordinates of the top-right corner of S(p) is (−p 1 + N, . . . , −p d + N, p 1 + N, . . . , p d + N ). For all 1 ≤ i ≤ d, since N ≥ p i − a i , it implies that −p i + N ≥ −a i . For all 1 ≤ i ≤ d, since p i + N ≥ N ≥ b i , it implies that p i + N ≥ b i . This finally implies that that the top-right corner of S(p) dominates P (S). Therefore, S(p) contains P (S). Assume that p lies outside S. Then there exists a dimension i such that p i < a i or p i > b i . If p i < a i , then −p i > −a i , which implies that P (S) cannot dominate the lower-left corner of S(p). If p i > b i , again P (S) cannot dominate the lower-left corner of S(p). This implies that P (S) lies outside S(p). We will use the above reduction to transform the points in P into hypercubes in R 2d and transform the hyperrectangles in S into points in R 2d . Therefore, the hitting set problem in R d on hyperrectangles has been reduced to the set cover problem in R 2d on hypercubes. And with a similar rank-space reduction as mentioned in the previous section, all the points as well as the corners of the hypercubes in this instance have integral coordinates. Then, the set cover instance is answered using Corollary 3. The correctness follows from Lemma 16. Noting that for 2d-dimensional hypercubes, O(log n) = O(log m), the performance of the algorithm is summarized below. Theorem 6. After performing a pre-processing step which takes O(n log 2d n) time, there is an algorithm for hitting set for d-dimensional hyperrectangles with an approximation factor of O(log 4d−1 n) and an update time of O(log 2d+2 n). In the weighted setting, the approximation factor is O(log 4d−1 n · log W ) and the update time is O(log 2d n log 3 (W n)). Future work In the first part of this work, we have studied online geometric set cover and hitting set for 2-D squares. This opens up an interesting line of work for the future. We state a few open problems: 1. As a natural extension of 2-D squares, is it possible to design a o(log n log m)-competitive algorithm for 3-D cubes? The techniques used in this paper for 2-D squares do not seem to extend to 3-D cubes. Another setting of interest here is when the geometric objects are 2-D disks. Can we obtain a o(log n log m)-competitive online set cover algorithm for them? 2. Design an online algorithm for set cover (resp., hitting set) for rectangles with competitive ratio o(log 2 n) or show an almost matching lower bound of Ω log 2 n log log n (which holds for the general case of online set cover [4])? 3. As a generalization of the above question, is it possible to obtain online algorithms for set cover and hitting set with competitive ratio o(log 2 n) for set systems with "constant" VCdimension. 4. For the weighted case of online set cover, even in unit squares, can we obtain algorithms with competitive ratio o(log n log m)? 5. Design an online algorithm for hitting set for squares with competitive ratio O(log n), and hence, improving our algorithm's competitive ratio of O(log N ) (where the corners of the squares are integral and contained in [0, N ) 2 ). In the second part of our work, we studied dynamic geometric set cover and hitting set for d-dimensional hyperrectangles. This line of work nicely brings together data structures, computational geometry, and approximation algorithms. We finish with a few open problems in the dynamic setting: Figure 1 : 1(a) A set of squares S and a set of points P , (b) A set cover (in green) S ⊆ S covering P , (c) A hitting set (green points) P ⊆ P for S. Figure 3 : 3Left figure shows a quad-tree cell in purple. The maximum area-covering square (solid black) is picked, while the other edge-covering squares (dashed) are not. Right figure shows the quad-tree cells (level-wise color-coded) containing an uncovered point. In increasing order of depth of these cells, at most 4 maximum-area covering squares (solid black) are picked together per cell, till the point gets covered. Figure 5 : 5Charging picked (red) edge-covering squares to the corner of a (cyan) square in OPT. In the image on the left, the (yellow) cell contains a corner of the square from OPT, and in the image on the right, the parent of the cell contains such a corner. Lemma 3 . 3Consider a set of points P and a point p. Then ALG(P ) ⊆ ALG(P ∪ {p}). 2 which is described by an outer box b O ⊆ [0, N ) 2 and an inner box b I ⊆ b O ; both of them are axis-parallel rectangles andC v = b O \ b I (Note that b I could be the empty set). Figure 6 : 6Outer box b O being partitioned into at most 9 rectangles due to inner box b I . Lemma 4 .Figure 7 : 47LetC v be a cell such that the squares selected in the ancestors of v do not cover all points in P ∩C v . Then (a) we select at most O(1) squares forC v and (b) ifC v contains no corner of a square S ∈ OPT, then the squares we selected forC v cover all points in P ∩C v . Possible intersections of a (cyan) square from OPT with a cell, such that no corner of the square is in the cell. The left image shows edge-covering, and crossing squares. The right image shows squares with one of two corners inside b I .Proof. First, we prove part (a). IfC v corresponds to a leaf node, we select at most one square. Otherwise, We select at most 4 edge-covering squares forC v . From Claim 1, we select O(1) number squares forC v that are horizontally or vertically-crossing b O . We select no more squares if b I = ∅.So consider the other case: b I = ∅. Let R be one of the (at most) four rectangular regions obtained from partitioning of b O (by 1 , 2 , 3 , 4 ) that share an edge with b I . Let w, h be width and height of R, respectively. W.l.o.g. assume R shares a horizontal edge with b I . As b I is sticky, and b I and b O have a bounded aspect ratio of 3, it can be seen that R also has a w/h ≤ 3 (similarly, if R shared a vertical edge with b I , then h/w ≤ 3). Again using Claim 1, we select O(1) vertically-crossing squares for R. We do a similar operation for other such regions. Now consider the remaining (at most four) regions obtained from partitioning of b O (by 1 , 2 , 3 , 4 ) that do not share an edge with b I . We select at most one square for each of them. Thus, in total, we select at most O(1) squares forC v . Now we prove part (b). IfC v contains no corner of a square S ∈ OPT, then all the squares in OPT that intersectC v are either edge-covering b O , or crossingC v , or contain one or two corners inside b I . However, as we have picked maximal edge-covering squares for b O , they contain all points inC v that are covered by the edge-covering squares from OPT. Figure 8 : 8Ω(log m) hard instances for quadrants and unit squares Lemma 5. There is an instance of the online fractional covering problem on m quadrants (quadrant is a rectangle with one of its corners as the origin) such that any online deterministic algorithm is Ω(log m)-competitive on this instance.Proof. Consider the set of m quadrants (SeeFigure 8) with their top-right corners as follows:Q 1 = (1, m), Q 2 = (2, m− 1), . . . , Q k = (k, m − k + 1), . . . , Q m = (m, 1) (and let the corresponding variables for the set cover instance be x 1 , x 2 , . . . , x m , respectively). Now consider a point p i,j = (i − 0.5, m − j + 0.5). We claim that this point intersects exactly the i th to j th indexed quadrants. Since p(x) > i − 1, p i,j does not intersect any of the first (i − 1) quadrants. Additionally since p(y) = m − j + 0.5 < m − i − 1 it does intersect quadrant i and this holds true up to quadrant j (Since, p(y) = m − j + 0.5 < m − i − 1) Figure 9 : 9In cell C lying in O TR the red points are chosen by the algorithm. Lemma 8 . 8Let p ∈ OPT. Then there are O(log N ) rounds in which a square S with p ∈ S was inserted, such that at the beginning of the round P ∩ S = ∅. Figure 10 : 10Cell C contains point p. The primary cells are highlighted in green, while the secondary cells are highlighted in pink. Theorem 4 . 4After performing a pre-processing step which takes O(m log 2d m) time, there is an algorithm for dynamic set cover for d-dimensional hyperrectangles with an approximation factor of O(log 4d−1 m) and an update time of O(log 2d+2 m). Figure 12 : 12(a) A point p in 1-D lying inside an interval S = [a 1 , b 1 ], and (b) the transformation of p into a point p = (−p 1 , p 1 ), and the transformation of S into a square S in 2-D. Observation 1. A point p = (p 1 , . . . , p d ) lies inside S if and only if p = (−p 1 , . . . , −p d , p 1 , . . . , p d ) lies inside S . Lemma 12 . 12The update time is O(log 2d m · log 2 n) = O(log 2d+2 m). Proof. When a point is inserted or deleted, the O(log 2d m) sets inŜ containing the point can be found in O(log 2d m) time by traversing the tree T. The algorithm of [13] has an update time of O(f log 2 n) = O(log 2d m · log 2 n) = O(log 2d+2 m). This is true since d-dimensional hypercubes have dual V C-dimension of O(d) [32, 33] and hence, O(log n) = O(log m). Lemma 13. The time taken to construct the setŜ is O(m log 2d m). Proof. Let T (m, d) be the time taken to build the extended quad-tree on m hypercubes in d dimensions. As a base case, we first compute T (m, 1). Constructing the skeleton structure of the 1-dimensional quad-tree takes O(m) time, since the endpoints of the intervals lie on the integer grid [0, 4m]. Then "assigning" each interval to a node in this quad-tree takes O(m log m) time. For a node v which is assigned m v intervals, finding the two maximal intervals takes O(m v ) time. Therefore, T (m, 1) = O(m log m) + v O(m v ) = O(m log m). Now consider the extended quad-tree on m hypercubes in 2d dimensions. Again constructing the skeleton of the quad-tree takes only O(m) time. For any 1 ≤ i ≤ 2d, finding the nodes in T where a hypercube is "i-long" takes O(2 2d log m) = O(log m) time. Therefore, Lemma 15 . 15The update time of the algorithm is O(log 2d m log 3 (W m)). Proof. When a point is inserted or deleted, the O(log 2d m log W ) sets inŜ containing the point can be found in O(log 2d m log W ) time by traversing the O(log W ) extended quad-trees. The algorithm of [13] has an update time of O(f log 2 W n) = O(log 2d m log W log 2 W n) = O(log 2d m log 3 (W m)). Figure 15 : 15(a) A point p in 1-D lying inside an interval S = [a 1 , b 1 ], and (b) the transformation of p into a square S(p), and transformation of S into a point (−a 1 , b 1 ) in 2-D. Lemma 16. In R d a point p lies inside a hyperrectangle S, if and only if, in R 2d the hypercube S(p) contains the point P (S). Table 2 : 2Dynamic algorithms for geometric set cover and hitting set. Update times in[2] are amortized and for the unweighted case. Our results are for worst-case update times. Claim 1.Let R be a box of width w and height h such that w/h ≤ B, for some constant B ∈ N; and P R be a set of points inside R that can be covered by a collection of vertically-crossing (i.e., they intersect both horizontal edges of R) squares S . Then we can find at most B + 1 squares from S that can cover all points inside R.We have an analogous claim for horizontally-crossing squares when h/w ≤ B. Claim 3 . 3Assume for contradiction thatĈ was already activated in a previous round for a square S . In the current round, the square S was introduced by the adversary such that S ∩ P = ∅. IfĈ gets activated again, S covered an edge ofĈ (assume this is the bottom edge w.l.o.g.). IfĈ was already activated in a previous round, then the algorithm must have picked points closest to the bottom, top, left, and right edges ofĈ and at least one hit the square. Among these points, denote the point picked closest to the bottom edge by p . IfĈ was activated again for S, it clearly contained at least one point that hit S by definition. Then, p would have hit S since it had the lowest y-coordinate inĈ. . Improve the approximation factor for dynamic set cover for the case of 2-D rectangles. Specifically, is it possible to obtain an O(log n) approximation with polylogarithmic update time? In this setting the rectangles are fixed, but the points are dynamic.2. For weighted dynamic set cover for the case of 2-D rectangles, is it possible to obtain approximation and update bounds independent of W (where W is the ratio of the weight of the highest weight rectangle to the lowest weight rectangle in the input)?3. For the (fully) dynamic case of set cover studied in[3,19,20], can we obtain algorithms with sublinear update time and polylogarithmic approximation when the sets are rectangles (as originally asked in[20])? A Online algorithms for interval set coverIn this section, we present a tight 2-competitive algorithm for the case of interval set cover.In the algorithm, we start with an empty set cover. In each iteration, when a new point p arrives, if it is covered then we do nothing. Otherwise, we select among the intervals covering p, the one with the rightmost right end-point and the one with the leftmost left end-point.The correctness of the algorithm follows trivially, since for every new uncovered point we pick an interval covering it. We do not remove intervals from our solution at any later steps in the algorithm, and hence, all points are covered when the algorithm terminates.Theorem 7.There exists a 2-competitive algorithm for the online interval set cover problem.Proof. Consider an interval I in the optimum solution OPT. When the first uncovered point covered by it, arrives in the input, our algorithm picks two intervals and ensures that these two intervals cover all of I. Hence, for each interval in OPT, we pick at most 2 intervals in our solution, giving us a 2-competitive solution.Theorem 8. There is an instance of the set cover problem on intervals such that any online algorithm (without recourse) can at best be 2-competitive on this instance.Proof. Consider the given set of intervals to be A := [0, 1], B :=[1,2], C :=[2,3], D :=[3,4]. The first point to arrive is p 1 := 2. If the algorithm picks two or more sets, then we are done as OPT is of size 1. Otherwise, to cover p, an algorithm can pick either interval B or C. In the former case, the second point should be p 2 := 3; and in the latter p 3 := 1. We see that in both cases OPT is of size one, but an online algorithm is forced to pick two intervals. Dynamic set cover: improved algorithms and lower bounds. Amir Abboud, Raghavendra Addanki, Fabrizio Grandoni, Debmalya Panigrahi, Barna Saha, Proceedings of the 51st. the 51stAmir Abboud, Raghavendra Addanki, Fabrizio Grandoni, Debmalya Panigrahi, and Barna Saha. Dynamic set cover: improved algorithms and lower bounds. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC). Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 114-125, 2019. Dynamic geometric set cover and hitting set. K Pankaj, Hsien-Chih Agarwal, Subhash Chang, Allen Suri, Jie Xiao, Xue, 36th International Symposium on Computational Geometry. 164:15. Schloss Dagstuhl -Leibniz-Zentrum für InformatikPankaj K. Agarwal, Hsien-Chih Chang, Subhash Suri, Allen Xiao, and Jie Xue. Dynamic geo- metric set cover and hitting set. In 36th International Symposium on Computational Geometry, SoCG 2020, volume 164 of LIPIcs, pages 2:1-2:15. Schloss Dagstuhl -Leibniz-Zentrum für Informatik, 2020. Dynamic geometric set cover and hitting set. K Pankaj, Hsien-Chih Agarwal, Subhash Chang, Allen Suri, Jie Xiao, Xue, arXiv:2003.00202arXiv preprintPankaj K Agarwal, Hsien-Chih Chang, Subhash Suri, Allen Xiao, and Jie Xue. Dynamic geo- metric set cover and hitting set. arXiv preprint arXiv:2003.00202, 2020. The online set cover problem. Baruch Noga Alon, Yossi Awerbuch, Azar, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing (STOC). the thirty-fifth annual ACM symposium on Theory of computing (STOC)Noga Alon, Baruch Awerbuch, and Yossi Azar. The online set cover problem. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing (STOC), pages 100-105, 2003. Small-size \eps-nets for axis-parallel rectangles and boxes. Boris Aronov, Esther Ezra, Micha Sharir, SIAM Journal on Computing. 397Boris Aronov, Esther Ezra, and Micha Sharir. Small-size \eps-nets for axis-parallel rectangles and boxes. SIAM Journal on Computing, 39(7):3248-3282, 2010. Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. Sanjeev Arora, Journal of the ACM (JACM). 455Sanjeev Arora. Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. Journal of the ACM (JACM), 45(5):753-782, 1998. An optimal algorithm for approximate nearest neighbor searching fixed dimensions. Sunil Arya, David M Mount, Nathan S Netanyahu, Ruth Silverman, Angela Y Wu, Journal of the ACM (JACM). 456Sunil Arya, David M. Mount, Nathan S. Netanyahu, Ruth Silverman, and Angela Y. Wu. An optimal algorithm for approximate nearest neighbor searching fixed dimensions. Journal of the ACM (JACM), 45(6):891-923, 1998. Fully dynamic set cover via hypergraph maximal matching: An optimal approximation through a local approach. Sepehr Assadi, Shay Solomon, 29th Annual European Symposium on Algorithms, ESA 2021, page 8. Schloss Dagstuhl-Leibniz-Zentrum fur Informatik GmbH. Dagstuhl PublishingSepehr Assadi and Shay Solomon. Fully dynamic set cover via hypergraph maximal matching: An optimal approximation through a local approach. In 29th Annual European Symposium on Algorithms, ESA 2021, page 8. Schloss Dagstuhl-Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2021. Computational geometry. Marc Mark De Berg, Mark Van Kreveld, Otfried Overmars, Schwarzkopf, Computational geometry. SpringerMark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf. Computational geometry. In Computational geometry, pages 1-17. Springer, 1997. Deterministic fully dynamic data structures for vertex cover and matching. Sayan Bhattacharya, Monika Henzinger, Giuseppe F Italiano, SIAM Journal on Computing. 473Sayan Bhattacharya, Monika Henzinger, and Giuseppe F. Italiano. Deterministic fully dynamic data structures for vertex cover and matching. SIAM Journal on Computing, 47(3):859-887, 2018. Dynamic algorithms via the primal-dual method. Sayan Bhattacharya, Monika Henzinger, Giuseppe F Italiano, Information and Computation. 261Sayan Bhattacharya, Monika Henzinger, and Giuseppe F. Italiano. Dynamic algorithms via the primal-dual method. Information and Computation, 261:219-239, 2018. A new deterministic algorithm for dynamic set cover. Sayan Bhattacharya, Monika Henzinger, Danupon Nanongkai, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS). IEEESayan Bhattacharya, Monika Henzinger, and Danupon Nanongkai. A new deterministic algo- rithm for dynamic set cover. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), pages 406-423. IEEE, 2019. Dynamic set cover: Improved amortized and worst-case update time. Sayan Bhattacharya, Monika Henzinger, Danupon Nanongkai, Xiaowei Wu, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA). the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)SIAMSayan Bhattacharya, Monika Henzinger, Danupon Nanongkai, and Xiaowei Wu. Dynamic set cover: Improved amortized and worst-case update time. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2537-2549. SIAM, 2021. Dynamic geometric independent set. Sujoy Bhore, Jean Cardinal, John Iacono, Grigorios Koumoutsos, Japan conference on Discrete and Computational Geometry, Graphs, and Games. Sujoy Bhore, Jean Cardinal, John Iacono, and Grigorios Koumoutsos. Dynamic geometric independent set. In Japan conference on Discrete and Computational Geometry, Graphs, and Games, 2021. Unbounded lower bound for k-server against weak adversaries. Marcin Bienkowski, Jarosław Byrka, Christian Coester, Łukasz Jeż, Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC). the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC)Marcin Bienkowski, Jarosław Byrka, Christian Coester, and Łukasz Jeż. Unbounded lower bound for k-server against weak adversaries. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 1165-1169, 2020. The design of competitive online algorithms via a primaldual approach. Niv Buchbinder, Joseph Naor, Found. Trends Theor. Comput. Sci. 32-3Niv Buchbinder and Joseph Naor. The design of competitive online algorithms via a primal- dual approach. Found. Trends Theor. Comput. Sci., 3(2-3):93-263, 2009. Worst-case efficient dynamic geometric independent set. Jean Cardinal, John Iacono, Grigorios Koumoutsos, 29th Annual European Symposium on Algorithms (ESA 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik. Jean Cardinal, John Iacono, and Grigorios Koumoutsos. Worst-case efficient dynamic geomet- ric independent set. In 29th Annual European Symposium on Algorithms (ESA 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2021. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. Timothy M Chan, Elyot Grant, Jochen Könemann, Malcolm Sharpe, Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms (SODA). the twenty-third annual ACM-SIAM symposium on Discrete Algorithms (SODA)SIAMTimothy M. Chan, Elyot Grant, Jochen Könemann, and Malcolm Sharpe. Weighted capaci- tated, priority, and geometric set cover via improved quasi-uniform sampling. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms (SODA), pages 1576- 1585. SIAM, 2012. More dynamic data structures for geometric set cover with sublinear update time. Timothy M Chan, Qizheng He, 37th International Symposium on Computational Geometry. Schloss Dagstuhl-Leibniz-Zentrum für InformatikTimothy M. Chan and Qizheng He. More dynamic data structures for geometric set cover with sublinear update time. In 37th International Symposium on Computational Geometry (SoCG 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2021. Dynamic geometric set cover, revisited. Timothy M Chan, Qizheng He, Subhash Suri, Jie Xue, Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)SIAMTimothy M. Chan, Qizheng He, Subhash Suri, and Jie Xue. Dynamic geometric set cover, revisited. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3496-3528. SIAM, 2022. Improved approximation algorithms for geometric set cover. L Kenneth, Kasturi Clarkson, Varadarajan, Proceedings of the twenty-first annual symposium on Computational geometry (SoCG). the twenty-first annual symposium on Computational geometry (SoCG)Kenneth L. Clarkson and Kasturi Varadarajan. Improved approximation algorithms for geo- metric set cover. In Proceedings of the twenty-first annual symposium on Computational geom- etry (SoCG), pages 135-141, 2005. Conditional lower bounds for dynamic geometric measure problems. Justin Dallant, John Iacono, arXiv:2112.10095arXiv preprintJustin Dallant and John Iacono. Conditional lower bounds for dynamic geometric measure problems. arXiv preprint arXiv:2112.10095, 2021. Hitting sets online and vertex ranking. Guy Even, Shakhar Smorodinsky, Algorithms -ESA 2011 -19th Annual European Symposium. Springer6942Guy Even and Shakhar Smorodinsky. Hitting sets online and vertex ranking. In Algorithms -ESA 2011 -19th Annual European Symposium, volume 6942 of Lecture Notes in Computer Science, pages 347-357. Springer, 2011. Random order online set cover is as easy as offline. Anupam Gupta, Gregory Kehne, Roie Levin, 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). IEEEAnupam Gupta, Gregory Kehne, and Roie Levin. Random order online set cover is as easy as offline. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 1253-1264. IEEE, 2022. Online and dynamic algorithms for set cover. Anupam Gupta, Ravishankar Krishnaswamy, Amit Kumar, Debmalya Panigrahi, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC). the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC)Anupam Gupta, Ravishankar Krishnaswamy, Amit Kumar, and Debmalya Panigrahi. Online and dynamic algorithms for set cover. In Proceedings of the 49th Annual ACM SIGACT Sympo- sium on Theory of Computing (STOC), pages 537-550, 2017. Fully-dynamic submodular cover with bounded recourse. Anupam Gupta, Roie Levin, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS). IEEEAnupam Gupta and Roie Levin. Fully-dynamic submodular cover with bounded recourse. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 1147- 1157. IEEE, 2020. The online submodular cover problem. Anupam Gupta, Roie Levin, Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)SIAMAnupam Gupta and Roie Levin. The online submodular cover problem. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1525-1537. SIAM, 2020. Weighted geometric set cover problems revisited. Sariel Har-Peled, Mira Lee, Journal of Computational Geometry. 31Sariel Har-Peled and Mira Lee. Weighted geometric set cover problems revisited. Journal of Computational Geometry, 3(1):65-85, 2012. Dynamic approximate maximum independent set of intervals, hypercubes and hyperrectangles. Monika Henzinger, Stefan Neumann, Andreas Wiese, 36th International Symposium on Computational Geometry. 16414. Schloss Dagstuhl -Leibniz-Zentrum für InformatikMonika Henzinger, Stefan Neumann, and Andreas Wiese. Dynamic approximate maximum independent set of intervals, hypercubes and hyperrectangles. In 36th International Sym- posium on Computational Geometry, SoCG 2020, volume 164 of LIPIcs, pages 51:1-51:14. Schloss Dagstuhl -Leibniz-Zentrum für Informatik, 2020. Settling the apx-hardness status for geometric set cover. H Nabil, Rajiv Mustafa, Saurabh Raman, Ray, IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS). IEEENabil H. Mustafa, Rajiv Raman, and Saurabh Ray. Settling the apx-hardness status for geo- metric set cover. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), pages 541-550. IEEE, 2014. PTAS for geometric hitting set problems via local search. H Nabil, Saurabh Mustafa, Ray, Proceedings of the 25th ACM Symposium on Computational Geometry (SoCG). the 25th ACM Symposium on Computational Geometry (SoCG)ACMNabil H. Mustafa and Saurabh Ray. PTAS for geometric hitting set problems via local search. In Proceedings of the 25th ACM Symposium on Computational Geometry (SoCG), pages 17-22. ACM, 2009. On the density of families of sets. Norbert Sauer, Journal of Combinatorial Theory, Series A. 131Norbert Sauer. On the density of families of sets. Journal of Combinatorial Theory, Series A, 13(1):145-147, 1972. A combinatorial problem; stability and order for models and theories in infinitary languages. Saharon Shelah, Pacific Journal of Mathematics. 411Saharon Shelah. A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific Journal of Mathematics, 41(1):247-261, 1972. Weighted geometric set cover via quasi-uniform sampling. Kasturi Varadarajan, Proceedings of the forty-second ACM symposium on Theory of computing (STOC). the forty-second ACM symposium on Theory of computing (STOC)Kasturi Varadarajan. Weighted geometric set cover via quasi-uniform sampling. In Proceedings of the forty-second ACM symposium on Theory of computing (STOC), pages 641-648, 2010.
[]
[ "Simplicially driven simple contagion", "Simplicially driven simple contagion" ]
[ "Maxime Lucas \nMathematics and Complex Systems Research Area\nISI Foundation\nVia Chisola 510126TurinItaly\n\nCENTAI\nCorso Inghilterra 310138TurinItaly\n", "Iacopo Iacopini \nDepartment of Network and Data Science\nCentral European University\n1100ViennaAustria\n", "Thomas Robiglio \nDepartment of Physics\nUniversity of Turin\nVia Pietro Giuria 110125TurinItaly\n", "Alain Barrat \nAix Marseille Univ\nUniversité de Toulon\nCNRS\n13009MarseilleCPTFrance\n", "Giovanni Petri \nMathematics and Complex Systems Research Area\nISI Foundation\nVia Chisola 510126TurinItaly\n\nCENTAI\nCorso Inghilterra 310138TurinItaly\n" ]
[ "Mathematics and Complex Systems Research Area\nISI Foundation\nVia Chisola 510126TurinItaly", "CENTAI\nCorso Inghilterra 310138TurinItaly", "Department of Network and Data Science\nCentral European University\n1100ViennaAustria", "Department of Physics\nUniversity of Turin\nVia Pietro Giuria 110125TurinItaly", "Aix Marseille Univ\nUniversité de Toulon\nCNRS\n13009MarseilleCPTFrance", "Mathematics and Complex Systems Research Area\nISI Foundation\nVia Chisola 510126TurinItaly", "CENTAI\nCorso Inghilterra 310138TurinItaly" ]
[]
Single contagion processes are known to display a continuous transition from an epidemic-free state to an epidemic one, for contagion rates above a critical threshold. This transition can become discontinuous when two simple contagion processes are coupled in a bi-directional symmetric way. However, in many cases, the coupling is not symmetric and the nature of the processes can differ. For example, risky social behaviors-such as not wearing masks or engaging in large gatherings-can affect the spread of a disease, and their adoption dynamics via social reinforcement mechanisms are better described by complex contagion models rather than by simple contagions, more appropriate for disease spreading. Here, we consider a simplicial contagion (describing the adoption of a behavior) that uni-directionally drives a simple contagion (describing a disease propagation). We show, both analytically and numerically, that, above a critical driving strength, such a driven simple contagion can exhibit both discontinuous transitions and bi-stability, absent otherwise. Our results provide a novel route for a simple contagion process to display the phenomenology of a higher-order contagion, through a driving mechanism that may be hidden or unobservable in practical instances.
10.1103/physrevresearch.5.013201
[ "https://export.arxiv.org/pdf/2206.07645v2.pdf" ]
249,674,411
2206.07645
5593a2d7f3831aab88ba6959297c122f8b12e5af
Simplicially driven simple contagion Maxime Lucas Mathematics and Complex Systems Research Area ISI Foundation Via Chisola 510126TurinItaly CENTAI Corso Inghilterra 310138TurinItaly Iacopo Iacopini Department of Network and Data Science Central European University 1100ViennaAustria Thomas Robiglio Department of Physics University of Turin Via Pietro Giuria 110125TurinItaly Alain Barrat Aix Marseille Univ Université de Toulon CNRS 13009MarseilleCPTFrance Giovanni Petri Mathematics and Complex Systems Research Area ISI Foundation Via Chisola 510126TurinItaly CENTAI Corso Inghilterra 310138TurinItaly Simplicially driven simple contagion Single contagion processes are known to display a continuous transition from an epidemic-free state to an epidemic one, for contagion rates above a critical threshold. This transition can become discontinuous when two simple contagion processes are coupled in a bi-directional symmetric way. However, in many cases, the coupling is not symmetric and the nature of the processes can differ. For example, risky social behaviors-such as not wearing masks or engaging in large gatherings-can affect the spread of a disease, and their adoption dynamics via social reinforcement mechanisms are better described by complex contagion models rather than by simple contagions, more appropriate for disease spreading. Here, we consider a simplicial contagion (describing the adoption of a behavior) that uni-directionally drives a simple contagion (describing a disease propagation). We show, both analytically and numerically, that, above a critical driving strength, such a driven simple contagion can exhibit both discontinuous transitions and bi-stability, absent otherwise. Our results provide a novel route for a simple contagion process to display the phenomenology of a higher-order contagion, through a driving mechanism that may be hidden or unobservable in practical instances. I. INTRODUCTION Contagion processes have been widely studied using complex networks as the underlying structure supporting the propagation of diseases, innovation, and opinions [1][2][3][4]. The most studied examples include simple contagion models (where a contagion event can be caused by a single contact), such as the paradigmatic Susceptible-Infectious-Susceptible (SIS), widely used to describe the diffusion of a single pathogen in a population [5,6]. In reality, however, contagion processes often co-exist and affect each other [7]. Infectious diseases can indeed display complex comorbidity interactions, in which the presence of a pathogen impacts the individual susceptibility towards another [8], like HIV increasing susceptibility to other sexually transmitted diseases [9]. Modeling efforts in this direction include both cooperation [10][11][12] and competition [13][14][15] between diseases. However, to date, models of interacting contagion processes have been developed under two main assumptions: (i) the processes are simple contagions, and (ii) their interaction is symmetric, that is, bi-directional and of equal strength. Within these restrictions, cooperative models can display a discontinuous transition to the epidemic state [11], and become indistinguishable at the mean-field level from complex contagion models describing social reinforcement [16] (where exposure to multiple sources presenting the same stimulus is needed for the contagion to occur [17]). Interactions between spreading processes are naturally not restricted to infectious diseases: a social behavior can also dramatically impact the spread of a disease [18][19][20][21][22]. A current and cogent example is the impact of the adoption of risky behaviors (no hand washing, no masks, no self-isolation * I.I. and M.L. contributed equally to this work. † A.B and G.P. jointly supervised this work. or reduction of face-to-face contacts) during the COVID pandemic [22]. Motivated by this example, we challenge both restrictions described above. First, it is known that reinforcement mechanisms influence social behavior so that models of simple contagion-that assume independent pairwise exposures-do not offer the most adequate description [17]. Simplicial contagion has been proposed as an alternative approach to account for simultaneous exposures via group-contagion events [23,24]. Such group ("higher-order") contributions induce discontinuous transitions, bi-stability and critical mass phenomena even for single processes [25][26][27][28][29]. Second, most contagion processes do not interact in a symmetric way. This can happen, e.g., for diseases with very different time scales [30], or when considering interactions between a disease and the adoption of prudent behaviors [21], which is instead driven by a phenomenologically and analytically different social contagion process. Here, we show that a simple contagion (describing infectious disease spreading) can exhibit the characteristics of a simplicial contagion when it is cooperatively driven by a simplicial contagion (describing the spread of a risky social behavior). Namely, a simple contagion in the epidemic-free regime can exhibit an abrupt transition to the epidemic regime, as well as bi-stability, if the cooperative driving by the social process is stronger than a critical value. In particular, in the asymmetrically driven case, discontinuous transitions can only take place when the driving process is simplicial, contrary to the case of symmetric interactions. We describe the phase diagram of the system through a mean-field (MF) approach, complemented by the numerical integration of coupled Markov-chain equations, and provide an analytical expression for the critical value of the cooperation. Finally, we identify effective infectivities as markers of the abrupt driven transition by rewriting the MF equations as a simple contagion with effective parameters. S A AB (a) (b) (c) (d) (e) (f) (g) FIG. 1. The model of interacting simplicial contagions. (a) Transition probabilities between the compartments: susceptible (S, gray), infected exclusively by one disease (A or B, respectively blue/red) or by both (AB, black). (b)-(d) A susceptible node i can acquire A after a contact with an infectious k-hyperedge (this also includes AB individuals). In (d), since i is part of a 2-simplex composed by two other infectious nodes, the infection can come both from each of the two 1-hyperedges (links) with probability βA and from the 2-hyperedge with probability β A . (e)-(g) If i is already infected with B, the probability of getting A for each contact is affected by the coupling factor BA. The same rules symmetrically apply to B instead of A. II. RESULTS A. Model for interacting simplicial contagion processes We consider a model for two interacting spreading processes, denoted as A and B, which also include simplicial contagions [23,25]. Individuals are represented by a set of N nodes that can each be in one of four compartments, following the standard SIS framework [6]: those susceptible to both diseases (S), infected exclusively by one of the two diseases (either A or B), or by both (AB) [see Fig. 1(a)]. The compartment membership of each node i is encoded in three binary variables x γ i ∈ {0, 1}, where γ ∈ {A, B, AB}. If node i is in state γ then x γ i = 1, otherwise it is zero: each node has either one non-zero or all zero variables. The density of nodes in state γ is given, at each time t, by ρ γ (t) = 1 N N i=1 x γ i (t). The densities ρ(t) = {ρ A (t), ρ B (t), ρ AB (t)} serve as macroscopic order parameters (with ρ S (t) = 1 − ρ A (t) − ρ B (t) − ρ AB (t) the density of susceptible individuals). Nodes can interact in pairs or larger groups, so that contagion events, which cause nodes to change compartment, take place on top of a contact structure that allows for higherorder (non-pairwise) interactions [24,[31][32][33]. We mathematically represent a group encounter as a k-hyperedge, a set of k + 1 interacting nodes [34]. For simplicity, we allow for interactions up to dimension k = 2, i.e., on 1-hyperedges (links) and 2-hyperedges (triangles). Six parameters-three for each disease-yield contagion and recovery probabilities (Fig. 1). The infectivity of disease x ∈ {A, B} at order k = 1, β x,1 ≡ β x , is the probability per unit time for a node i susceptible to pathogen x to acquire x from an "infectious" 1hyperedge it is part of [ Fig. 1 (b)-(d)] . Similarly, β x,2 ≡ β x control infections coming from 2-hyperedges [ Fig. 1(d)]. Note that all other nodes in the hyperedge need to be infectious for the hyperedge to be considered so. Finally, µ x ∈ [0, 1] denotes the standard spontaneous recovery probability (from x) per unit time. The interaction between the two contagion processes is controlled via two additional non-negative parameters, the coupling factors AB and BA that multiply the transition probabilities to a double infection (AB) from a single infection (A or B). For example, the transition B → AB occurs with probability BA β A from a pairwise contact with A [see Fig. 1(e)-(g)]. The two processes cooperate if xx > 1 and compete if xx < 1, while they are independent if xx = 1. Note that the symmetry AB = BA does not need to hold. Furthermore, although the model is defined on a generic higher-order structure, we focus here on simplicial complexes, a particular class of hypergraphs [24]. In a simplicial complex K, by definition, groups of nodes are called simplices and respect downward closure: each sub-simplex ν ⊂ σ built from subsets of a simplex σ ⊂ K is also part of the complex K [in an infectious 2-simplex thus, contagion can occur both through the 1-hyperedges contained and through the 2-hyperedge itself, see Fig. 1(d),(g)]. We make this choice for coherence with previous work [23], but it can be relaxed to more general hypergraphs [25][26][27]29] without affecting the MF results. B. Mean-field description We consider the MF description of the model, obtained under a homogeneous mixing hypothesis [35]. For simplicity, we assume identical recovery rates for the two processes, that is µ A = µ B = µ. In fact, for µ A = µ B the equations can be simply refactored in term of a new parameter δ = µ A /µ B leaving the asymptotic dynamics unchanged (see Sup. Mat. I). We also introduce the rescaled infectivity parameters λ x = β x k /µ and λ x = β x k /µ, for x ∈ {A, B}, where k and k respectively denote the average numbers of 1-and 2-hyperedges incident on a node. After rescaling time by µ, the general mean-field equations describing the evolution of the densities are: ρ A = − ρ A + λ A ρ S (ρ A + ρ AB ) + λ A ρ S (ρ A + ρ AB ) 2 + ρ AB − AB λ B ρ A (ρ B + ρ AB ) − AB λ B ρ A (ρ B + ρ AB ) 2 (1a) ρ B = − ρ B + λ B ρ S (ρ B + ρ AB ) + λ B ρ S (ρ B + ρ AB ) 2 + ρ AB − BA λ A ρ B (ρ A + ρ AB ) − BA λ A ρ B (ρ A + ρ AB ) 2 (1b) ρ AB = − 2ρ AB + AB λ B ρ A (ρ B + ρ AB ) + AB λ B ρ A (ρ B + ρ AB ) 2 + BA λ A ρ B (ρ A + ρ AB ) + BA λ A ρ B (ρ A + ρ AB ) 2 (1c) with the additional condition that ρ S = 1 − ρ A − ρ B − ρ AB .(2) In the following, we focus on a simplicial contagion A (representing a risky social behavior) that cooperatively and uni-directionally drives a simple contagion B (representing a disease). We thus set λ A > 0, λ B = 0, AB > 1 and BA = 1. In this scenario, it is convenient to consider the total density of infectious individuals for each contagion, regardless of whether they are also infected by the other one. Formally, we introduce two new variables ρ Atot = ρ A + ρ AB and ρ Btot = ρ B + ρ AB . In other words, ρ Atot is the total density of people with a risky behaviour, having been infected by B (ρ AB ), or not (ρ A ). Similarly, ρ Btot is the total density of people infected by disease B, having a risky behavior (ρ AB ), or not (ρ B ). After introducing these two variables, we end up with the system of coupled equations (see Appendix A): ρ Atot = ρ Atot [−1 + λ A (1 − ρ Atot ) + λ A ρ Atot (1 − ρ Atot )], (3a) ρ Btot = ρ Btot [−1 + λ B (1 − ρ Btot ) +λ B ( AB − 1)(ρ Atot − ρ AB )] , (3b) ρ AB = −2ρ AB + AB λ B (ρ Atot − ρ AB )ρ Btot + λ A (ρ Btot − ρ AB )ρ Atot + λ A (ρ Btot − ρ AB )ρ 2 Atot . (3c) Equations (3) include two known specific cases. First, without any interaction between the processes ( AB = BA = 1), ρ Btot and ρ Atot evolve independently as a simple and simplicial contagion [23], respectively. Second, by considering only pairwise interactions, λ A = 0 = λ B , A and B evolve as interacting simple contagions [8]. In the general case we consider ( BA = 1), the dynamics of ρ Atot is decoupled from the other two variables and drives them. We first study the non-equilibrium stationary state (NESS) reached by the system of Eqs. (3) at large times ρ * Btot = lim t→∞ ρ Btot (t), by numerical integration. Figure 2 shows the resulting ρ * Btot values and their transitions. The most interesting case is given by λ B < 1 as, without a driving process A, the simple contagion process B would be in the epidemicfree absorbing state (ρ * Btot = 0). We thus illustrate how the B NESS ρ * Btot depends on the parameters of the driver A, on the coupling AB , and how it can transition to the epidemic active state, despite λ B < 1. For λ A ≤ 1, we always obtain a continuous transition for ρ * Btot [see Fig. 2(a,b)]. On the other hand, if λ A > 1, the driven process can exhibit a discontinuous transition [see Fig. 2(c,d)]. More precisely, the transition changes from continuous to discontinuous when the coupling parameter AB becomes larger than a critical value c AB . Above this threshold [ AB > c AB , black circles in Fig. 2(c)], there is a discontinuous transition at a critical value λ c A that does not depend on AB . For weaker cooperation [ AB < c AB , white and gray symbols in Fig. 2(c)], a continuous transition occurs from ρ * Btot = 0 to the epidemic state ρ * Btot > 0 when λ A crosses another critical value λ c A ≥ λ c A . In other words, λ c A is the critical value of λ A at which the continuous transition occurs, for AB < c AB , and its value decreases as AB increases-e.g. from λ c A ≈ 1.1 (white symbols) to λ c A ≈ 0.7 (gray symbols). On the contrary, λ c A (yellow label) is where the discontinuous transition occurs, for AB > c AB , and does not depend on AB . Note that λ c A → λ c A in the limit AB → c AB . It is important to note that the epidemic-free absorbing state ρ * Btot = 0 remains stable as long as λ A < 1. As a consequence, there is a region of bi-stability λ c A < λ A < 1 (region shaded with yellow background) for AB > c AB . Bi-stability can also be observed when AB < c AB in the region λ c A < λ A < 1 (gray symbols) as long as λ c A < 1. For λ c A ≥ 1 (white symbols), the continuous transition occurs for values larger than one and there is no region of bi-stability. Finally, when λ c A < 1, the stability of the absorbing state implies the existence of a forward discontinuous transition at λ c A = 1 (upward arrows). In conclusion, the simple contagion B exhibits characteristics of a simplicial contagion-an abrupt transition and bi-stabilitydue to the driving of the simplicial contagion A. To analytically explain this behavior, we need to find the NESS by settingρ x = 0 (see Appendix B for additional details). Solvingρ Atot = 0, as Eq. (3a) exactly maps back to the single simplicial contagion analyzed in Ref. [23], leads to a trivial solution ρ * Atot = 0 and two other NESS ρ * ,± Atot . Similarly, solving the full two-dimensional system (ρ Btot , ρ AB ) leads to the absorbing state (0, 0) and the implicit solutions for ρ * Btot : ρ * ,± Btot = 1 − 1 λ B + (ρ * ,± Atot − ρ * ,± AB )( AB − 1).(4) Equation (4) (see [23]). This implies that ρ AB also goes to the absorbing state if λ A < 1: if nobody is infected by A, nobody can be infected by both A and B. As a consequence, the term coming from the cooperation vanishes: ρ * ,± Atot − ρ * ,± AB = 0. Hence, from Eq. (4), we have lim λ A →1 − ρ * ,± Btot = 1 − 1 λ B ≤ 0 (recall λ B < 1). If negative, it is not a valid solution and only the absorbing state is. As λ A increases above 1, ρ * ,+ Atot increases continuously [23], leading ρ * ,± Btot to also cross continuously 0 at a certain λ A ≥ 1. For λ A ≥ 1 instead, ρ * Atot has a discontinuous transition at λ A = λ c A which implies that ρ * AB has one too. Consequently, from Eq. (4), since λ B < 1, we have lim λ A →λ c,− A ρ * Btot = 0, but lim λ A →λ c,+ A ρ * ,± Btot > 0 above a certain value AB > c AB . Hence, this critical value c AB can be derived analytically by solving ρ * ,+ Btot = 0 at λ c A . In other words, we find the critical driving strength c AB by finding the curve ρ * ,+ Btot , between the gray and black curves in Fig. 2(c), such that it reaches zero at λ c A . This corresponds to the case λ B ≤ 1, λ A > 1 which we denoted region I in Fig. 3. Similarly, if λ B > 1 instead (with λ A > 1, region II), it suffices solving ρ * ,+ Btot = 1 − 1/λ B at λ c A because 1 − 1/λ B is now the pre-transition NESS. In summary, the discontinuity is controlled by the term coming from the cooperation, (ρ * ,± Atot − ρ * ,± AB )( AB − 1) which will be discontinuous if A has a discontinuous transition. In short, the nature of the transition depends on the new second term induced by the driving in Eq. (4) and, in particular, on the λ A value at which it becomes positive. This fact, allow us to obtain the critical c AB at which the discontinuous transition in B becomes possible (Fig. 3): c AB =    √ λ A −λ B √ λ A −1 λ B in region I (λ B ≤ 1, λ A > 1), 1 in region II (λ B > 1, λ A > 1).(5) In region I, increasing λ A or λ B makes c AB decrease, so that discontinuous transitions are obtained for smaller values of the driving strength AB . In fact, c AB → +∞ as λ A → 1 or λ B → 0. In region II, all values of cooperation AB > 1 yield a discontinuous transition. Finally, no critical value of cooperation can be defined in regions III and IV (λ A ≤ 1), where transitions are always continuous. C. Effective formalism In this section, we devise an effective contagion theory to highlight the origins of the observed transitions. In particular, we saw that, for AB > c AB -that is, when simplicial behavior is possible for B-the driven process B will exhibit discontinuous transitions as a function of λ A . We illustrate how this phenomenology emerges by rewriting the dynamics of B as an effective simple contagion following Ref. [16]. First, note that we can rewrite the MF equation of the single simplicial ρ Atot from Eq. (3a) as a simple contagioṅ ρ Atot = −ρ Atot +λ A ρ Atot [1 − ρ Atot ],(6) with effective infectivityλ A = λ A +λ A ρ Atot . As expected, the effective infectivity depends on both the simple and simplicial infectivities. Since Eq. (6) is written as a simple contagion, its well known stationary solutions given by 1 − 1/λ A and the effective infectivity also has a critical value of 1. Similarly, we can also rewrite Eq. (3b) of the driven ρ Btot as a simple contagioṅ ρ Btot = −ρ Btot +λ B ρ Btot [1 − ρ Btot ],(7) with effective infectivitỹ λ B = λ B + λ B ( AB − 1) 1 1 − ρ Btot ρ A .(8) Since we observed characteristics of the driver A in the driven contagion B, we may want to further cast its effective infectivity into a form similar to that of A:λ B = λ B + λ B ρ Btot . This is achieved by defining an effective simplicial infectivitỹ λ B = λ B ( AB − 1) ρ A ρ Btot (1 − ρ Btot ) ,(9) which implicitly depends on λ A and λ A through ρ A . If there is no interaction ( AB = 1), we recoverλ B = λ B because the effective simplicial infectivity vanishes,λ B = 0, as expected. More importantly, since ρ Btot evolves according to the effectively simple contagion of Eq. (7), its stationary solution is given by 1 − 1/λ B which yields a critical valuẽ λ B = 1. This can help distinguish between the transitions Fig. 2. Indeed, below the critical value,λ B < 1, the only stable NESS of ρ Btot is 0, and above it there is also a positive solution. Thus, the driven contagion B has a transition to an epidemic state if and only ifλ B crosses 1 as λ A increases. Finally and most importantly, this transition is discontinuous if and only if the transition ofλ B across the values one is discontinuous. This can be seen by comparing the three curves in Fig. 4(a)-crossing the value one (white background) discontinuously (black) or continuously (gray and white)-with the corresponding curves for ρ * Btot in Fig. 2(c). D. Temporal properties So far, we lack information about the temporal trajectories, which, in practical settings, are often the only data available. Consider observing the spread of B via ρ Btot (t), while the driving social contagion process A remains unobservable. Interestingly, the observed B evolves differently depending on the initial conditions of the hidden process A. We show the phenomenology described beyond the homogeneous mixing hypothesis, by shifting to a Markov-chain formalism [36,37]. With this microscopic approach we can encode any interaction structure between nodes-contrary to MF approaches that assume homogeneous mixing of the population-while keeping the computational cost lower than the one required for Monte Carlo simulations. The complete Markov-chain description of our model can be found in Appendix C. We build a synthetic random simplicial complex up to dimension 2 by means of the generative model introduced in Ref. [23]. This model, a direct extension of Erdös-Rényi-like models for graphs, allows to generate a simplicial complex starting from a number of nodes that get randomly connected to form simplices. We here generate a simplicial complex with N = 2000 nodes having k = 20 and k = 6 and integrate the associated Markov equations to follow the temporal evolution of the system. We consider the scenario of the black curve ( AB = 1.75) of Fig. 2(b), that is, with a simplicial driver A (λ A = 2.5). We fix all other parameters, including the initial condition ρ Btot (0), but vary the initial condition of the driver, ρ Atot (0). As shown in Fig. 5, if the driver contagion A is in the endemic regime but not in the bi-stability region (e.g., λ A = 1.2), ρ Btot reaches the same NESS for all values of ρ Atot (0), but with different transient dynamics and even non-monotonic evolutions [ Fig. 5(a)]. Moreover, if the simplicial driver is in the bi-stability region (λ A = 0.7), it induces bi-stability in B: ρ Btot (t) can reach two different states, depending on the driving initial condition, even though all "visible" B parameters are fixed [ Fig. 5(b)]. Note that this bi-stability emerges only if the driving is simplicial with λ A > 1 (see also Sup. Mat. Fig. S1). III. DISCUSSION In conclusion, our results highlight that an abrupt transition in the observed process can occur as a function of the control parameter of a second-potentially hidden-driver process. Consider an observer of an epidemic process of unknown nature. A natural intervention would try to reduce the intrinsic infectivity of the spreading pathogen, e.g. through pharmaceutical interventions or reduction of social contacts (sanitary lockdowns): this would however lead only to a continuous change in the incidence. However, if the spread is driven by an underlying complex contagion, then acting on the hidden driver process (e.g. trying to reduce the social adoption of risky behaviors) could more effectively lead to an abrupt transition to the epidemic-free state (if the interaction is strong enough AB > c AB ). Finally, different populations could be characterised by different properties of the hidden behavioral contagion process (different values of λ A and λ ∆ A values), thus leading to a large diversity of temporal evolutions, and-potentially-of final outcomes of the pathogen's spread, without the need for different intrinsic infectivity properties of the pathogen across these populations. Results also suggest that other driving spreading processes could yield a similar phenomenology if they exhibit a discontinuous transition (e.g. [38]), inducing a change from a continuous to a discontinuous transition in the driven process. We note in this context that the framework of Ref. [39] suggests a universal route to abrupt transitions, achieved through the addition of a control parameter to a process that displays a continuous phase transition. However, the situation that we have explored here broadens the picture. Indeed, if both spreading processes are simple contagions, it appears that a bi-directional interaction (leading to a feedback loop) is an additional necessary condition for a discontinuous transition to emerge. In the case of a uni-directional coupling, instead, the driving process needs to be itself simplicial with bi-stability. Our results thus provide a different route to the emergence of abrupt transitions in epidemic-like processes due to the asymmetric coupling of the contagion dynamics, as opposed to the addition of a control parameter [39]. This resonates with recent results in synchronization phenomena [40]. The exact conditions under which these routes apply to coupled systems in general would be an interesting direction for future work. Another interesting perspective would consist in the analysis of real-world data and the development of tools to detect the footprints of simple, complex, or coupled processes from observed time series [41] in order to discriminate them or, potentially, perform full reconstruction [42]. A.B. acknowledges support from the Agence Nationale de la Recherche (ANR) project DATAREDUX (ANR-19-CE46-0008). M.L. and G.P. acknowledge partial support from the Intesa Sanpaolo Innovation Center during the preparation of this work. Code availability. The code used in this study is available at https://github.com/iaciac/ interacting-simplagions. Appendix A: Derivation of MF description As explained in the main text, we focus on the case BA = 1, λ B = 0, so that Eqs. (1) becomeṡ ρ A = − ρ A + λ A ρ S (ρ A + ρ AB ) + λ A ρ S (ρ A + ρ AB ) 2 + ρ AB − AB λ B ρ A (ρ B + ρ AB ), (A1a) ρ B = − ρ B + λ B ρ S (ρ B + ρ AB ) + ρ AB − λ A ρ B (ρ A + ρ AB ) − λ A ρ B (ρ A + ρ AB ) 2 , (A1b) ρ AB = − 2ρ AB + AB λ B ρ A (ρ B + ρ AB ) + λ A ρ B (ρ A + ρ AB ) + λ A ρ B (ρ A + ρ AB ) 2 . (A1c) Then, we apply the following change of variables: ρ Atot = ρ A + ρ AB , ρ Btot = ρ B + ρ AB . This yieldṡ ρ Atot = (−ρ A − ρ AB ) + λ A ρ Atot [1 − ρ Btot − ρ A + ρ B ] + λ A ρ 2 Atot [1 − ρ Btot − ρ A + ρ B ],(A2a)ρ Btot = (−ρ B − ρ AB ) + λ B ρ Btot [1 − ρ Atot − ρ B + AB ρ A ],(A2b)ρ AB = − 2ρ AB + AB λ B ρ A ρ Btot + λ A ρ B ρ Atot + λ A ρ B ρ 2 Atot .(A2c) We further rewrite this by replacing all remaining ρ A and ρ B , and using the identity 1 − ρ Btot − ρ A + ρ B = 1 − ρ Atot , ρ Atot = − ρ Atot + λ A ρ Atot [1 − ρ Atot ] + λ A ρ 2 Atot [1 − ρ Atot ],(A3a)ρ Btot = − ρ Btot + λ B ρ Btot [1 − ρ Atot − ρ Btot + ρ AB + AB (ρ Atot − ρ AB )], (A3b) ρ AB = − 2ρ AB + AB λ B (ρ Atot − ρ AB )ρ Btot + λ A (ρ Btot − ρ AB )ρ Atot + λ A (ρ Btot − ρ AB )ρ 2 Atot ,(A3c) which can be refactored to obtain Eqs. (3) from the main text. Appendix B: Derivation of the MF fixed points Equation (3a) is the same as the simplicial contagion from Ref. [23], and its non-trivial solutions are ρ * ,± Atot = (λ A − λ A ) ± (λ A − λ A ) 2 + 4λ A (λ A − 1) 2λ A . (B1) For this isolated case, we know that λ A controls the type of transition to the epidemic state [23]. That is, for λ A ≤ 1, the bifurcation diagram has a continuous transition at λ A = 1 from ρ * Atot = 0 to the epidemic state ρ * ,+ Atot . When instead λ A > 1, a discontinuous transition to ρ * ,+ Atot occurs at λ c A = −λ A +2 λ A ≤ 1. The epidemic-free state remains stable for λ A ≤ 1, but becomes unstable above: This leads to bi-stability in the parameter region {λ A > 1, λ c A ≤ λ A ≤ 1}. The discontinuous transition is therefore the direct consequence of a sufficiently strong three-body (higher-order) interaction in A (λ A > 1). The remaining two-dimensional system (ρ Btot , ρ AB ) can be solved analytically by hand or with the help of software such as Mathematica [43]. As discussed, the implicit solution for Eq. (3b) is given by Eq. (4). To solve for ρ AB , we rewrite Eq. (3c) by factorizing and setting the left-hand side to zero: 0 = − 2ρ AB + AB λ B (ρ Atot − ρ AB )ρ Btot + λ A (ρ Btot − ρ AB )ρ Atot + λ A (ρ Btot − ρ AB )ρ 2 Atot , (B2) =ρ AB −2 − AB λ B ρ Btot − λ A ρ Atot − λ A ρ 2 Atot (B3) +ρ Atot ρ Btot AB λ B + λ A + λ A ρ Atot , from which we already see that ρ * AB = 0 if ρ Atot = 0 or ρ Btot = 0. Now, we inject the expression of ρ * Btot from Eq. (4) and cast the equation into quadratic form in ρ AB : 0 = Aρ 2 AB + Bρ AB + C,(B4) where A = + AB λ B E − AB ,(B5)B = − 2 − AB λ B (Λ − B + E − AB ρ * Atot ) (B6) − (λ A + E − AB K)ρ * Atot − λ A ρ * 2 Atot , C = ρ * Atot K(Λ − B + E − AB ρ * Atot ).(B7) To shorten the notation, we have also defined E − AB = AB − 1,(B8)Λ − i = 1 − 1/λ i ,(B9)K = AB λ B + λ A + λ A ρ * Atot .(B10) The non-zero solutions for ρ AB is the standard quadratic solution ρ * ,± AB = −B ± √ B 2 − 4AC 2A ,(B11) which, unfolded, is an expression in terms of the parameters of the system only. These together with ρ * Atot can be reinjected into Eq. (4) for ρ * Btot to close the system. Appendix C: Markov-chain approach Here, we write a system of coupled Markov-chain equations which govern the microscopic evolution of our model [36,37]. More precisely, we can write down the conditional probability P (x γ i (t + 1) = 1|x(t), θ, A) ≡ p i γ (t) of finding each node i in state γ = {S, A, B, AB} at time t + 1 given the probability vector representing the status of all nodes at time t x(t) = x γ i (t), the model parameters θ = {β A , β A , β B , β B , µ A , µ B , AB , BA }, and the structure A. Using the simplified notation p i γ (t), we impose that, at each time, p i S (t) = 1 − p i A (t) − p i B (t) − p i AB (t).(C1) The Markov-chain equations for the three states are the following: p i AB (t + 1) = + p B i (t)(1 − µ B )(1 − q i A (t)) + p i A (t)(1 − µ A )(1 − q i B (t)) + p i AB (t)(1 − µ A )(1 − µ B ),(C2a)p i A (t + 1) = + p i AB (t)µ B (1 − µ A ) + p i A (t)(1 − µ A )q i B (t) + p i B (t)µ B (1 − q i A (t)) + p i S (t)(1 − q i AB (t))f i A (t),(C2b)p i B (t + 1) = + p i AB (t)µ A (1 − µ B ) + p i B (t)(1 − µ B )q i A (t) + p i A (t)µ A (1 − q i B (t)) + p i S (t)(1 − q i AB (t))f i B (t).(C2c) The different q i x (t) denote the probability of node i not being infected by disease x by any of the simplices it participates in. Considering again only contributions up to D = 2, we have: q i A (t) = j∈V 1 − a ij BA β A [p j A (t) + p j AB (t)] j,l∈V 1 − a ijl BA β A [p j A (t) + p j AB (t)][p l A (t) + p l AB (t)] ,(C3a)q i B (t) = j∈V 1 − a ij AB β B [p j B (t) + p j AB (t)] j,l∈V 1 − a ijl AB β B [p j B (t) + p j AB (t)][p l B (t) + p l AB (t)] ,(C3b)q i AB (t) = j∈V 1 − a ij β A p j A (t) + β B p j B (t) + [β A (1 − β B ) + β B (1 − β A ) + β A β B ]p j AB (t) j,l∈V 1 − a ijl β A [p j A (t)p l A (t) + p j A (t)p l AB (t) + p j AB (t)p l A (t)] + β B [p j B (t)p l B (t) + p j B (t)p l AB (t) + p j AB (t)p l B (t)] + [β A (1 − β B ) + β B (1 − β A ) + β A β B p j AB (t)p l AB (t)] = j∈V 1 − a ij β A [p j A (t) + p j AB (t)] + β B [p j B (t) + p j AB (t)] − β A β B [p j AB (t)] j,l∈V 1 − a ijl β A [p j A (t) + p j AB (t)][p l A (t) + p l AB (t)] + β B [p j B (t) + p j AB (t)][p l B (t) + p l AB (t)]−β A β B p j AB (t)p l AB (t) ,(C3c) where the first product of each equation accounts for the contagion through the links of the simplicial complex K. These links are fully specified by means of the standard adjacency matrix {a ij }, whose elements a ij = 0, 1 denote the absence or presence of a link (i, j). Similarly, the second product accounts for the contagion of i through the 2-simplices of K (triangles), which are analogously specified by the elements of the adjacency tensor {a ijl }. This tensor is the 3-dimensional version of the adjacency matrix, in which a non-zero element denotes the presence of a 2-simplex (i, j, l). Finally, the factors f i A (t) and f i B (t) in Eq. (C2) denote the probability of transitioning from state S to one of the states A or B when exposed simultaneously to both pathogens. As-suming an equal probability for both diseases [37], we can write: In the main text we assumed identical recovery rates. Here, we remove this constraint and allow them to be potentially different, so that µ A = µ B . By rescaling all equations by µ A (instead of µ), we have the following-instead of Eqs. (A1): f i A (t) =q i A (t)(1 − 0.5q i B (t)) q i A (t)(1 − 0.5q i B (t)) +q i B (t)(1 − 0.5q i A (t)) (C4a) f i B (t) =q i B (t)(1 − 0.5q i A (t)) q i A (t)(1 − 0.5q i B (t)) +q i B (t)(1 − 0.5q i A (t)) (C4b) whereq i A (t) andq i B (t) correspond to 1 − q i A (t)ρ A = − 1ρ A + λ A ρ S (ρ A + ρ AB ) + λ A ρ S (ρ A + ρ AB ) 2 + µ B µ A ρ AB − AB λ B µ B µ A ρ A (ρ B + ρ AB ) (S1a) ρ B = − µ B µ A ρ B + λ B µ B µ A ρ S (ρ B + ρ AB ) + 1ρ AB − λ A ρ B (ρ A + ρ AB ) − λ A ρ B (ρ A + ρ AB ) 2 (S1b) ρ AB = − (1 + µ B µ A )ρ AB + AB λ B µ B µ A ρ A (ρ B + ρ AB ) + λ A ρ B (ρ A + ρ AB ) + λ A ρ B (ρ A + ρ AB ) 2 (S1c) which, introducing the total densities, becomeṡ ρ Atot = (−ρ A − ρ AB ) + λ A ρ Atot [1 − ρ Btot − ρ A + ρ B ] + λ A ρ 2 Atot [1 − ρ Btot − ρ A + ρ B ] (S2a) ρ Btot =(−ρ B − ρ AB ) µ B µ A + λ B µ B µ A ρ Btot [1 − ρ Atot − ρ B + AB ρ A ] (S2b) ρ AB = − (1 + µ B µ A )ρ AB + AB λ B µ B µ A ρ A ρ Btot + λ A ρ B ρ Atot + λ A ρ B ρ 2 Atot .(S2c) and thenρ Atot = − ρ Atot 1 + λ A ρ Atot [1 − ρ Atot ] + λ A ρ 2 Atot [1 − ρ Atot ] (S3a) ρ Btot = − ρ Btot µ B µ A + λ B µ B µ A ρ Btot [1 − ρ Atot − ρ Btot + ρ AB + AB (ρ Atot − ρ AB )] (S3b) ρ AB = − (1 + µ B µ A )ρ AB + AB λ B µ B µ A (ρ Atot − ρ AB )ρ Btot + λ A (ρ Btot − ρ AB )ρ Atot + λ A (ρ Btot − ρ AB )ρ 2 Atot (S3c) which, compared to the case of identical recovery rates, contain the additional µ B µ A factors. We denote that dimensionless ratio δ = µ B µ A and the equations become, after refactoring: ρ Atot = ρ Atot [−1 + λ A (1 − ρ Atot ) + λ A ρ Atot (1 − ρ Atot )], (S4a) ρ Btot = ρ Btot δ [−1 + λ B (1 − ρ Btot ) +λ B ( AB − 1)(ρ Atot − ρ AB )] ,(S4b)ρ AB = −(1 + δ)ρ AB + AB λ B δ(ρ Atot − ρ AB )ρ Btot + λ A (ρ Btot − ρ AB )ρ Atot + λ A (ρ Btot − ρ AB )ρ 2 Atot .(S4c) So, the equation for ρ Atot (simplagion) is unchanged, as expected. For ρ Btot , we notice a temporal rescaling by a factor δ, but the implicit solution is unchanged, ρ * ,± Btot = 1 − 1 λ B + (ρ * ,± Atot − ρ * ,± AB )( AB − 1).(S5) We can consider two limits where the timescales for A and B are of different orders. First, in the limit δ 1, which means that B heals much slower than A,ρ Btot ≈ 0, that is process B is quasi-static compared to the timescale of process A. Thus, ρ Atot converges fast to its NESS and ρ Btot is driven by that NESS. Second, in the limit δ 1, B heals much faster than A, it is the opposite. It is possible then to rescale time by δ to see that process A now appears quasi-static compared to the timescale of B. So, ρ Btot converges fast to its NESS which is in fact adiabatically moving towards its asymptotic NESS, driven by ρ Atot that slowly converges to its own NESS. FIG. S1. The temporal evolution of the simple contagion B is affected by the initial conditions of the hidden driver process A. As for Fig. 5 of the main text, we show ρB tot over time (a-c), but together with the temporal dynamics of the driver process, as given by ρA tot (d-f). In (a,d) a simple driver process is used (λ A = 0.8), while in (b,e) and (c,f) the driver process A is truly simplicial (λ A = 2.5). The process A is placed either in the endemic region, λA = 1.2 [(a,d) and (b,e)] or in the bi-stable region (λA = 0.7). Different curves correspond to different initial conditions of the driver process, ρA(0). The other parameters are set to λB = 0.8, λ B = 0, and AB = 2. FIG. 2 . 2Abrupt transition induced by a simplicial driver. A simplicial driver process for A, with λ A = 2.5, can induce a discontinuous transition (c,d), contrary to a simple driver (a,b), with λ A = 0 [λB = 0.8, λ B = 0]. Note the different scales on the horizontal axes. (a,c) Stationary solutions ρ * Btot of the MF Eqs. (3) plotted as a function of the rescaled pairwise infectivity λA for three values of the driving strength AB . In (c), the transition of the simple contagion B becomes discontinuous above a critical value of cooperation c AB . (b,d) Heatmaps of ρ * Btot as a function of λA and AB . Dashed horizontal lines correspond to the selected AB values shown in (a) and (c) respectively. The blue dot in (d) highlights the critical point (λ c A , c AB ). The blue and red crosses represent a visual hint to locate the results within the full phase diagram of Fig. 3. FIG. 3 . 3implicitly contains two solutions ± from ρ * ,± Atot and ρ * ,± AB . Using the implicit solutions Eqs. (4), we can understand the behavior shown inFig. 2. If λ A < 1 [Figs. 2(a,b)], ρ * Atot exhibits a continuous transition at λ A = 1, below which it is zero Phase diagram of the system. The (λB, λ A ) parameter space exhibits four regions. In region I (λB ≤ 1, λ A > 1), ρ * Btot undergoes an abrupt transition if the driving cooperation is strong enough, AB > c AB . The value of c AB is represented by shades of green. For visual clarity, the green scale is truncated at a maximum value of 5, so that larger values are represented by the same color as 5. The red cross corresponds to the case shown in Figs. 2(c,d). In region II (λB > 1, λ A > 1), c AB = 1 and the transition is discontinuous for all AB > 1. For λ A ≤ 1, that is regions III and IV, the transition is always continuous. The blue cross indicates the case shown inFigs. 2(a,b). FIG. 4 . 4The discontinuous nature of the driven contagion B can be determined from its effective infectivityλB. (a) We showλB against the infectivity λA for several values of cooperation AB , corresponding to the curves inFig. 2(c). The full phase diagram as a function of both λA and AB is shown as a heatmap in panel (b), where the dot corresponds to the critical point (λ c A , c AB ). The background color in (a) corresponds to the colorbar in (b). It corresponds to the values of the vertical axis and highlight visually the critical valueλB = 1 observed in FIG. 5 . 5The temporal evolution of the simple contagion B is affected by the initial conditions of the (hidden) simplicial driver A. We show ρB tot over time, resulting from the numerical integration of the Markov-chain equations for a simplicial complex with N = 2000 nodes, k = 20 and k = 6, for A in (a) the endemic region, λA = 1.2, and (b) the bi-stable region, λA = 0.7. Shades of red from dark to light represent a range of initial conditions of the driver ρA(0) from 0.001 to 0.35 [see Sup. Mat.Fig. S1for the temporal evolution of ρA(t)]. In (b), the simple contagion process B can reach one of two stationary states, depending on the initial conditions of the driver A. Other parameters are set to λB = 0.8, λ B = 0, AB = 1.75, and λ A = 2.5. ACKNOWLEDGEMENTS I.I. acknowledges support from the James S. McDonnell Foundation 21 st Century Science Initiative Understanding Dynamic and Multi-scale Systems -Postdoctoral Fellowship Award. and 1 − q i B (t), as given by Eqs. (C3), after setting AB = BA = 1. Supplementary Material: Simplicially driven simple contagion I. CASE OF DIFFERENT RECOVERY RATES: µA = µB B FIG. S2. Effective triangle infectivityλ B of simple contagion B as a function of λA, for several values of the interaction AB (indicated on the curves). The dashed grey curve indicates the value λ c A , where divergeλ B diverges. Epidemics and rumours. D J Daley, D G Kendall, 10.1038/2041118a0Nature. 2041118D. J. Daley and D. G. Kendall, Epidemics and rumours, Nature 204, 1118 (1964). Epidemic spreading in scale-free networks. R Pastor-Satorras, A Vespignani, 10.1103/PhysRevLett.86.3200Phys. Rev. Lett. 863200R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett. 86, 3200 (2001). Diffusion of Innovations. E Rogers, Free Press5th EditionE. Rogers, Diffusion of Innovations, 5th Edition (Free Press, 2003). A Barrat, M Barthélemy, A Vespignani, Dynamical Processes on Complex Networks. Cambridge University PressA. Barrat, M. Barthélemy, and A. Vespignani, Dynamical Pro- cesses on Complex Networks (Cambridge University Press, 2008). Modelling dynamical processes in complex sociotechnical systems. A Vespignani, 10.1038/nphys2160Nat. Phys. 832A. Vespignani, Modelling dynamical processes in complex socio- technical systems, Nat. Phys. 8, 32 (2012). Epidemic processes in complex networks. R Pastor-Satorras, C Castellano, P Van Mieghem, A Vespignani, 10.1103/RevModPhys.87.925Rev. Mod. Phys. 87925R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys. 87, 925 (2015). Dynamics of interacting diseases. J Sanz, C.-Y Xia, S Meloni, Y Moreno, 10.1103/PhysRevX.4.041005Phys. Rev. X. 441005J. Sanz, C.-Y. Xia, S. Meloni, and Y. Moreno, Dynamics of interacting diseases, Phys. Rev. X 4, 041005 (2014). Coevolution spreading in complex networks. W Wang, Q.-H Liu, J Liang, Y Hu, T Zhou, 10.1016/j.physrep.2019.07.001Phys. Rep. 8201W. Wang, Q.-H. Liu, J. Liang, Y. Hu, and T. Zhou, Coevolution spreading in complex networks, Phys. Rep. 820, 1 (2019). A systematic review of the epidemiologie interactions between classic sexually transmitted diseases and hiv: How much really is known?. J.-A Røttingen, D W Cameron, G P Garnett, Sex. Transm. Dis. 579J.-A. Røttingen, D. W. Cameron, and G. P. Garnett, A systematic review of the epidemiologie interactions between classic sexu- ally transmitted diseases and hiv: How much really is known?, Sex. Transm. Dis. , 579 (2001). Avalanche outbreaks emerging in cooperative contagions. W Cai, L Chen, F Ghanbarnejad, P Grassberger, 10.1038/nphys3457Nat. Phys. 11936W. Cai, L. Chen, F. Ghanbarnejad, and P. Grassberger, Avalanche outbreaks emerging in cooperative contagions, Nat. Phys. 11, 936 (2015). Fundamental properties of cooperative contagion processes. L Chen, F Ghanbarnejad, D Brockmann, 10.1088/1367-2630/aa8bd2New J. Phys. 19103041L. Chen, F. Ghanbarnejad, and D. Brockmann, Fundamental properties of cooperative contagion processes, New J. Phys. 19, 103041 (2017). Mutually cooperative epidemics on power-law networks. P.-B Cui, F Colaiori, C Castellano, 10.1103/PhysRevE.96.022301Phys. Rev. E. 9622301P.-B. Cui, F. Colaiori, C. Castellano, et al., Mutually cooperative epidemics on power-law networks, Phys. Rev. E 96, 022301 (2017). Competing epidemics on complex networks. B Karrer, M E Newman, 10.1103/PhysRevE.84.036106Phys. Rev. E. 8436106B. Karrer and M. E. Newman, Competing epidemics on complex networks, Phys. Rev. E 84, 036106 (2011). Characterising two-pathogen competition in spatially structured environments. C Poletto, S Meloni, A Van Metre, V Colizza, Y Moreno, A Vespignani, 10.1038/srep07895Sci. Rep. 51C. Poletto, S. Meloni, A. Van Metre, V. Colizza, Y. Moreno, and A. Vespignani, Characterising two-pathogen competition in spatially structured environments, Sci. Rep. 5, 1 (2015). Competing spreading dynamics in simplicial complex. W Li, X Xue, L Pan, T Lin, W Wang, 10.1016/j.amc.2021.126595Appl. Math. Comput. 412126595W. Li, X. Xue, L. Pan, T. Lin, and W. Wang, Competing spread- ing dynamics in simplicial complex, Appl. Math. Comput. 412, 126595 (2022). Macroscopic patterns of interacting contagions are indistinguishable from social reinforcement. L Hébert-Dufresne, S V Scarpino, J.-G Young, 10.1038/s41567-020-0791-2Nat. Phys. 16426L. Hébert-Dufresne, S. V. Scarpino, and J.-G. Young, Macro- scopic patterns of interacting contagions are indistinguishable from social reinforcement, Nat. Phys. 16, 426 (2020). Complex contagions and the weakness of long ties. D Centola, M Macy, 10.1086/521848Am. J. Sociol. 113702D. Centola and M. Macy, Complex contagions and the weakness of long ties, Am. J. Sociol. 113, 702 (2007). Modelling the influence of human behaviour on the spread of infectious diseases: a review. S Funk, M Salathé, V Jansen, 10.1098/rsif.2010.0142J. R. Soc. Interface. 71247S. Funk, M. Salathé, and V. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: a review, J. R. Soc. Interface 7, 1247 (2010). Towards a characterization of behavior-disease models. N Perra, D Balcan, B Gonçalves, A Vespignani, 10.1371/journal.pone.0023084PLoS One. 623084N. Perra, D. Balcan, B. Gonçalves, and A. Vespignani, Towards a characterization of behavior-disease models, PLoS One 6, e23084 (2011). Dynamical interplay between awareness and epidemic spreading in multiplex networks. C Granell, S Gómez, A Arenas, 10.1103/PhysRevLett.111.128701Phys. Rev. Lett. 111128701C. Granell, S. Gómez, and A. Arenas, Dynamical interplay be- tween awareness and epidemic spreading in multiplex networks, Phys. Rev. Lett. 111, 128701 (2013). The effect of a prudent adaptive behaviour on disease transmission. S V Scarpino, A Allard, L Hébert-Dufresne, 10.1038/nphys3832Nat. Phys. 121042S. V. Scarpino, A. Allard, and L. Hébert-Dufresne, The effect of a prudent adaptive behaviour on disease transmission, Nat. Phys. 12, 1042 (2016). Non-pharmaceutical interventions during the covid-19 pandemic: A review. N Perra, 10.1016/j.physrep.2021.02.001Phys. Rep. 9131N. Perra, Non-pharmaceutical interventions during the covid-19 pandemic: A review, Phys. Rep. 913, 1 (2021). Simplicial models of social contagion. I Iacopini, G Petri, A Barrat, V Latora, 10.1038/s41467-019-10431-6Nat. Commun. 101I. Iacopini, G. Petri, A. Barrat, and V. Latora, Simplicial models of social contagion, Nat. Commun. 10, 1 (2019). Networks beyond pairwise interactions: Structure and dynamics. F Battiston, G Cencetti, I Iacopini, V Latora, M Lucas, A Patania, J.-G Young, G Petri, 10.1016/j.physrep.2020.05.004Phys. Rep. 8741F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lucas, A. Patania, J.-G. Young, and G. Petri, Networks beyond pairwise interactions: Structure and dynamics, Phys. Rep. 874, 1 (2020). Social contagion on higher-order structures. A Barrat, G Ferraz De Arruda, I Iacopini, Y Moreno, 10.1007/978-3-030-91374-8_13Higher-Order Systems. SpringerA. Barrat, G. Ferraz de Arruda, I. Iacopini, and Y. Moreno, Social contagion on higher-order structures, in Higher-Order Systems (Springer, 2022) pp. 329-346. The effect of heterogeneity on hypergraph contagion models. N W Landry, J G Restrepo, 10.1063/5.0020034Chaos. 30103117N. W. Landry and J. G. Restrepo, The effect of heterogeneity on hypergraph contagion models, Chaos 30, 103117 (2020). Phase transitions and stability of dynamical processes on hypergraphs. G Ferraz De Arruda, M Tizzani, Y Moreno, 10.1038/s42005-021-00525-3Commun. Phys. 424G. Ferraz de Arruda, M. Tizzani, and Y. Moreno, Phase tran- sitions and stability of dynamical processes on hypergraphs, Commun. Phys. 4, 24 (2021). Abrupt phase transition of epidemic spreading in simplicial complexes. J T Matamalas, S Gómez, A Arenas, 10.1103/PhysRevResearch.2.012049Phys. Rev. Research. 212049J. T. Matamalas, S. Gómez, and A. Arenas, Abrupt phase transi- tion of epidemic spreading in simplicial complexes, Phys. Rev. Research 2, 012049 (2020). Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs. G St-Onge, I Iacopini, V Latora, A Barrat, G Petri, A Allard, L Hébert-Dufresne, 10.1038/s42005-021-00788-wCommun. Phys. 51G. St-Onge, I. Iacopini, V. Latora, A. Barrat, G. Petri, A. Al- lard, and L. Hébert-Dufresne, Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs, Commun. Phys. 5, 1 (2022). Role of time scale in the spreading of asymmetrically interacting diseases. P C Ventura, Y Moreno, F A Rodrigues, 10.1103/PhysRevResearch.3.013146Phys. Rev. Research. 313146P. C. Ventura, Y. Moreno, and F. A. Rodrigues, Role of time scale in the spreading of asymmetrically interacting diseases, Phys. Rev. Research 3, 013146 (2021). From networks to optimal higher-order models of complex systems. R Lambiotte, M Rosvall, I Scholtes, 10.1038/s41567-019-0459-yNat. Phys. R. Lambiotte, M. Rosvall, and I. Scholtes, From networks to optimal higher-order models of complex systems, Nat. Phys. 10.1038/s41567-019-0459-y (2019). What are higher-order networks?. C Bick, E Gross, H A Harrington, M T Schaub, 10.48550/arXiv.2104.11329arXiv:2104.11329arXiv preprintC. Bick, E. Gross, H. A. Harrington, and M. T. Schaub, What are higher-order networks?, arXiv preprint arXiv:2104.11329 https://doi.org/10.48550/arXiv.2104.11329 (2021). The physics of higher-order interactions in complex systems. F Battiston, E Amico, A Barrat, G Bianconi, G Ferraz De Arruda, B Franceschiello, I Iacopini, S Kéfi, V Latora, Y Moreno, M Murray, T Peixoto, F Vaccarino, G Petri, 10.1038/s41567-021-01371-4Nat. Phys. 171093F. Battiston, E. Amico, A. Barrat, G. Bianconi, G. Ferraz de Arruda, B. Franceschiello, I. Iacopini, S. Kéfi, V. Latora, Y. Moreno, M. Murray, T. Peixoto, F. Vaccarino, and G. Petri, The physics of higher-order interactions in complex systems, Nat. Phys. 17, 1093 (2021). The why, how, and when of representations for complex systems. L Torres, A S Blevins, D Bassett, T Eliassi-Rad, 10.1137/20M1355896SIAM Rev. 63435L. Torres, A. S. Blevins, D. Bassett, and T. Eliassi-Rad, The why, how, and when of representations for complex systems, SIAM Rev. 63, 435 (2021). I Kiss, J Miller, P Simon, Mathematics of Epidemics on Networks: From Exact to Approximate Models. Springer International PublishingI. Kiss, J. Miller, and P. Simon, Mathematics of Epidemics on Networks: From Exact to Approximate Models, Interdisciplinary Applied Mathematics (Springer International Publishing, 2017). Discrete-time markov chain approach to contactbased disease spreading in complex networks. S Gómez, A Arenas, J Borge-Holthoefer, S Meloni, Y Moreno, 10.1209/0295-5075/89/38009Europhys. Lett. 8938009S. Gómez, A. Arenas, J. Borge-Holthoefer, S. Meloni, and Y. Moreno, Discrete-time markov chain approach to contact- based disease spreading in complex networks, Europhys. Lett. 89, 38009 (2010). Markovian approach to tackle the interaction of simultaneous diseases. D Soriano-Paños, F Ghanbarnejad, S Meloni, J Gómez-Gardeñes, 10.1103/PhysRevE.100.062308Phys. Rev. E. 10062308D. Soriano-Paños, F. Ghanbarnejad, S. Meloni, and J. Gómez- Gardeñes, Markovian approach to tackle the interaction of si- multaneous diseases, Phys. Rev. E 100, 062308 (2019). . J Gómez-Gardenes, L Lotero, S Taraskin, F Pérez-Reche, 10.1038/srep19767Explosive contagion in networks, Sci. Rep. 61J. Gómez-Gardenes, L. Lotero, S. Taraskin, and F. Pérez-Reche, Explosive contagion in networks, Sci. Rep. 6, 1 (2016). A universal route to explosive phenomena. C Kuehn, C Bick, 10.1126/sciadv.abe3824Sci. Adv. 73824C. Kuehn and C. Bick, A universal route to explosive phenomena, Sci. Adv. 7, eabe3824 (2021). Explosive synchronization in multilayer networks through partial adaptation. P Khanra, P , 10.1016/j.chaos.2020.110621Chaos Solit. Fractals. 143110621P. Khanra and P. Pal, Explosive synchronization in multilayer networks through partial adaptation, Chaos Solit. Fractals 143, 110621 (2021). Detecting critical slowing down in high-dimensional epidemiological systems. T Brett, M Ajelli, Q.-H Liu, M G Krauland, J J Grefenstette, W G Van Panhuis, A Vespignani, J M Drake, P Rohani, 10.1371/journal.pcbi.1007679PLoS Comput. Biol. 161007679T. Brett, M. Ajelli, Q.-H. Liu, M. G. Krauland, J. J. Grefenstette, W. G. van Panhuis, A. Vespignani, J. M. Drake, and P. Rohani, Detecting critical slowing down in high-dimensional epidemio- logical systems, PLoS Comput. Biol. 16, e1007679 (2020). Network reconstruction and community detection from dynamics. T P Peixoto, 10.1103/PhysRevLett.123.128301Phys. Rev. Lett. 123128301T. P. Peixoto, Network reconstruction and community detection from dynamics, Phys. Rev. Lett. 123, 128301 (2019). . W R Inc, Mathematica, 2022Version 13.3.0.0, champaign, ILW. R. Inc., Mathematica, Version 13.3.0.0, champaign, IL, 2022.
[ "https://github.com/iaciac/" ]
[ "Evaluation of Static Vulnerability Detection Tools with Java Cryptographic API Benchmarks", "Evaluation of Static Vulnerability Detection Tools with Java Cryptographic API Benchmarks" ]
[ "Sharmin Afrose ", "Ya Xiao ", "Sazzadur Rahaman ", "DanfengBarton P Miller ", "Daphne Yao " ]
[]
[]
Several studies showed that misuses of cryptographic APIs are common in real-world code (e.g., Apache projects and Android apps). There exist several open-sourced and commercial security tools that automatically screen Java programs to detect misuses. To compare their accuracy and security guarantees, we develop two comprehensive benchmarks named CryptoAPI-Bench and ApacheCryptoAPI-Bench. CryptoAPI-Bench consists of 181 unit test cases that cover basic cases, as well as complex cases, including interprocedural, field sensitive, multiple class test cases, and path sensitive data flow of misuse cases. The benchmark also includes correct cases for testing false-positive rates. The ApacheCryptoAPI-Bench consists of 121 cryptographic cases from 10 Apache projects. We evaluate four tools, namely, SpotBugs, CryptoGuard, CrySL, and Coverity using both benchmarks. We present their performance and comparative analysis. The ApacheCryptoAPI-Bench also examines the scalability of the tools. Our benchmarks are useful for advancing state-of-the-art solutions in the space of misuse detection.
10.1109/tse.2022.3154717
[ "https://arxiv.org/pdf/2112.04037v1.pdf" ]
244,954,654
2112.04037
ef3e139418289eafa4eb6ba5671901ab426196e3
Evaluation of Static Vulnerability Detection Tools with Java Cryptographic API Benchmarks Sharmin Afrose Ya Xiao Sazzadur Rahaman DanfengBarton P Miller Daphne Yao Evaluation of Static Vulnerability Detection Tools with Java Cryptographic API Benchmarks 1Index Terms-Cryptographic API misusesbenchmarkJava Several studies showed that misuses of cryptographic APIs are common in real-world code (e.g., Apache projects and Android apps). There exist several open-sourced and commercial security tools that automatically screen Java programs to detect misuses. To compare their accuracy and security guarantees, we develop two comprehensive benchmarks named CryptoAPI-Bench and ApacheCryptoAPI-Bench. CryptoAPI-Bench consists of 181 unit test cases that cover basic cases, as well as complex cases, including interprocedural, field sensitive, multiple class test cases, and path sensitive data flow of misuse cases. The benchmark also includes correct cases for testing false-positive rates. The ApacheCryptoAPI-Bench consists of 121 cryptographic cases from 10 Apache projects. We evaluate four tools, namely, SpotBugs, CryptoGuard, CrySL, and Coverity using both benchmarks. We present their performance and comparative analysis. The ApacheCryptoAPI-Bench also examines the scalability of the tools. Our benchmarks are useful for advancing state-of-the-art solutions in the space of misuse detection. INTRODUCTION Various studies have shown that a vast majority of Java and Android applications misuse cryptographic libraries and APIs, causing devastating security and privacy implications. The most pervasive cryptographic misuses include exposed secrets (e.g., secret keys and passwords), predictable random numbers, use of insecure crypto primitives, vulnerable certificate verification [1]- [6]. Several studies showed that the prominent causes for cryptographic misuses are the deficiency in understanding of security API usage [4], [7], complex API designs [7], [8], the lack of cybersecurity training [4], insecure code generation tools [9] and insecure/misleading suggestions in Stack Overflow [4], [10]. The reality is that most developers, with tight project deadlines and short product turnaround time, spend little effort on improving their knowledge or hardening their code for long-term benefits [11]. Recognizing these practical barriers, automatic cryptographic code generation [12], and misuse detection tools [5] play a significant role in assisting developers with writing and maintaining secure code. The security community has produced several impressive static (e.g., CryptoLint [3], CrySL [13], FixDroid [14], MalloDroid [1], CryptoGuard [5]) and dynamic code screening tools (e.g., Crylogger [15], SMV-Hunter [16], and An-droSSL [17]) to detect API misuses in Java. The static analysis does not require a program to execute, rather it is performed on a version of the code (e.g., source code, intermediate representations or binary). Many abstract se-curity rules are reducible to concrete program properties that are enforceable via generic static analysis techniques [5], [18]. Consequently, static analysis tools have the potential to cover a wide range of security rules. In contrast, dynamic analysis tools require one to execute a program and spend a significant effort to trigger and detect specific misuse symptoms at runtime. Hence, dynamic analysis tools may be limited in their coverage. A code screening tool needs to be scalable with wide coverage. Thus, static analysisbased tools are usually more favorable than their dynamic counterparts. However, a major weakness of static analysis tools is their tendency to produce false alerts. False alerts substantially diminished the value of a tool. To reduce the number of false positives, most of the static analysis tools offer a trade-off between completeness and scalability [19]. We define completeness as the ability to detect all the misuse instances and scalability as the ability to induce low computational overhead to analyze large code-bases. Designing tools that would produce fewer false positives and false negatives with smaller computational overhead help the real-world deployment. To advance and monitor the scientific progress of domains to produce effective tools, a mechanism for comparative studies is required. Unfortunately, for automatic detection of cryptographic API misuses, no suitable mechanism or benchmark exists. Such a benchmark needs to have several requirements: i) It should cover a wide range of misuse instances. ii) It should cover interesting program properties (e.g., flow-, context-, field-, path-sensitivity, etc.) [20], [21]. These are different detection capabilities required for capturing certain vulnerabilities. iii) Test cases should be written in easily compilable source codes, so that both source code and binary code analysis tools can be easily evaluated. None of the existing benchmarks follows these criteria (e.g., DroidBench [22], Ghera [23]). For example, Droid-Bench [22] only contains binaries. Ghera [23] has sources of provided Android apps. However, both DroidBench and Ghera barely cover cryptographic API misuses. In this paper, we present two benchmarks for cryptographic API misuses. The first one is CryptoAPI-Bench, a comprehensive benchmark for comparing the quality of cryptographic vulnerability detection tools. It consists of 181 unit test cases covering 18 types of cryptographic misuses. Several test cases include interesting program properties [20], [21]. Flow-sensitive correctly computes and analyzes the order of statements in a program. Path-sensitivity analysis computes different dataflow analysis information dependent on conditional branch statements. Field-sensitive analysis distinguishs two fields containing the same object in a class. A context-sensitive analysis is interprocedural analysis that analyzes the target of a function call. The second one is ApacheCryptoAPI-Bench which is built upon 10 real-world Apache projects. It contains early versions of activemq-artemis, deltaspike, directory-server, manifoldcf, meecrowave, spark, tika, tomee, wicket projects. We identify 121 crypto cases in them, including 79 basic cases and 42 advanced cases. We run CryptoAPI-Bench and ApacheCryptoAPI-Bench on four static analysis tools (i.e., SpotBugs [24], Crypto-Guard, CrySL, and Coverity [25]) and perform a comparative analysis of these tools. These tools are i) capable of detecting cryptographic misuse vulnerabilities and ii) opensourced and/or provide free evaluation license. CrySL and CryptoGuard are open-sourced research prototypes that are actively being maintained to improve their accuracy and coverage. SpotBugs is also an actively maintained opensource project, which is the successor of FindBugs. Coverity is one of the most popular static analysis platforms for decades. Our main technical contributions are summarized as follows. • We provide a benchmark named CryptoAPI-Bench, which consists of 181 test cases covering 16 types of Cryptographic and SSL/TLS API misuse vulnerabilities. CryptoAPI-Bench utilized various interesting program properties (e.g., field-, context-, and pathsensitivity) to produce a diverse set of test cases. Our benchmark is open-sourced and can be found on GitHub [26]. • We provide another benchmark named ApacheCryptoAPI-Bench for checking the scalability property of the cryptographic vulnerability detection tools. We document 121 test cases covering 12 types of Cryptographic and SSL/TLS API misuse vulnerabilities from 10 real-world Apache projects. The detailed information regarding ApacheCryptoAPI-Bench can be found on GitHub [27]. • We evaluate four static analysis tools that are capable of detecting cryptographic misuse vulnerabilities. Our experimental evaluation revealed some interesting insights. For complex cases, specialized tools (e.g., CryptoGuard, CrySL) detect more cryptographic misuses and cover more rules than generalpurpose tools (e.g., SpotBugs, Coverity). Currently, none of these tools supports path-sensitive analysis. A preliminary version of the work appeared in the Proceedings of the 2019 ACM Conference on Computer and Communications Security (CCS) [5] and 2019 IEEE Secure Development Conference (SecDev) [28]. We expanded the conference version by adding a new benchmark ApacheCryptoAPI-Bench (Section 4, Table 2) that contains complex real-world Java programs and we test four static tools' performance in real-world code (Section 6.4, Table 7, Table 8). For CryptoAPI-Bench, We also add two new misuse categories (Section 2.17, Section 2.18), 11 new test cases (Table 1), and update tools' performance evaluation (Table 4, Table 5, Table 6). The remainder of this paper is organized as follows. Section 2 describes cryptographic API misuse categories. Section 3 and Section 4 outlines the design of CryptoAPI-Bench and ApacheCryptoAPI-Bench. Section 5 reviews existing cryptographic vulnerability detection tools. Section 6 presents the evaluation and performance analysis of the tools on the benchmarks. Discussion is given in Section 7. Section 8 describes the related works. Finally, Section 9 concludes this paper. CRYPTO API MISUSE CATEGORIES In this section, we discuss 18 Java cryptographic API misuse categories. We got the insights of these misuse categories from previous literature [5], [13], [14], NIST documents [29]- [31], and other blogs [32]. We describe reasons for vulnerability and possible secure solutions for these misuse categories. 2.1 Cryptographic Keys: For encryption, it is expected to use an unpredictable key using javax.crypto.spec.SecretKeySpec API that takes a byte array as input. If the Byte array is constant or hardcoded inside the code, the adversary can easily read the cryptographic key and may obtain sensitive information. Therefore, an unpredictable byte array should be used as a parameter in SecretKeySpec to generate a secure key. HostnameVerifier class is set to return true by default so that the verification method can quickly get past an exception. However, this arrangement causes a security threat, where URL spoofing [33] attacks can be possible. URL spoofing makes it simpler for numerous cyber-attacks (e.g., identity theft, phishing). Certificate Validation: Empty methods are often implemented in javax.net.ssl.X509TrustManager interface to connect quickly and easily with clients and remote servers without any certificate validation. In that case, the TrustManager accepts and trusts every entity including the entity that is not signed by a trusted certificate authority. It may cause Man-in-the-middle (MitM) attacks [1], [34]. 2.6 SSL Sockets: javax.net.ssl.SSLSocket connects a specific host to a specific port. However, before the connection, the hostname of the server should be verified and authenticated using javax.net.ssl.HostnameVerifier API. However, incorrect implementation omits the hostname verification when the socket is created [2], [35]. Hypertext Transfer Protocol: HyperText Transfer Protocol (HTTP) sends a request to a server to retrieve a web page. However, HTTP allows hackers to intercept and read sensitive information [36]. Therefore, it is recommended to use HyperText Transfer Protocol Secure (HTTPS) that utilizes a secured socket layer to encrypt sensitive information. Pseudorandom Number Generator (PRNG): The generation of a pseudorandom number using java.util.Random is vulnerable as the generated random number is not completely random, because it uses a definite mathematical algorithm (Knuth's subtractive random number generator algorithm [37]) that is proven to be insecure. To solve the problem, java.security.SecureRandom provides nondeterministic and unpredictable random numbers. Seeds in Pseudorandom Number Generator (PRNG) While using java.security.SecureRandom, if a constant or static seed is provided in SecureRandom, then it is possible to have the same outcome on every run. Therefore, developers should use a non-deterministic random seed. Salts in Password-based encryption (PBE): javax.crypto.spec.PBEParameterSpec API takes salt as one of the parameters for Password-based encryption. Using constant or static salts increases the possibility of a dictionary attack. The salt should be a random number that produces a random and unpredictable key. Mode of Operation: The Electronic Codebook (ECB) mode of operation is insecure to use in javax.crypto.Cipher as ECB-encrypted ciphertext can leak information about the plaintext. Instead of ECB, Cipher Block Chaining (CBC) or Galois/Counter Mode (GCM) is more secure to use. Initialization Vector (IV): The initialization vector (IV) is used during encryption and decryption with several modes of operation. Static/constant initialization vector introduces vulnerabilities for CBC mode of operation. Therefore, it is suggested to use an unpredictable random initialization vector in crypto.spec.IvParameterSpec API. Note that, for several modes of operation (e.g., CTR, CBC-MAC), unpredictable random IV is not required. Iteration Count in Password-based Encryption (PBE): In javax.crypto.spec.PBEParameterSpec API, it takes iteration count as one of the parameters for Password-based Encryption (PBE). In PKCS #5 [38], it is suggested that the number of iteration should be more than 1000 to provide a reasonable security level. Symmetric Ciphers: In symmetric cryptography, the same key is used for encryption and decryption. Some symmetric ciphers, e.g., DES, Blowfish, RC4, RC2, IDEA are considered broken, as brute-force attack is possible for 64bit ciphers. To overcome the attack, developers need to use AES which can support a block length of 128 bits and key lengths of 128, 192, and 256 bits [39]. Asymmetric Ciphers: In asymmetric cryptography, two keys, i.e., a public key and a private key are used for encryption and decryption. RSA is considered insecure for 1024-bit ciphers [30]. For this reason, developers are recommended to use RSA with a key size of 2048 bits or higher. Cryptographic Hash Functions: A cryptographic hash function generates a fixed-length alphanumeric hash value or message digest which is commonly used in verifying message integrity, digital signature, and authentication. A cryptographic hash function is contemplated as broken if a collision can be observed, i.e., the same hash value is generated for two different inputs. The list of broken hash functions includes SHA1, MD4, MD5, and MD2. Therefore, developers need to use a strong hash function, e.g., SHA-256. Cryptographic MAC: A MAC algorithm HmacMD5 and HmacSHA1 are considered insecure as these are susceptible to collision attacks [40]. Therefore, the developers need to use a strong MAC algorithm, e.g., HmacSHA256. Credentials in String: Credentials (passwords, secret keys, etc) should not be stored in the String variable. In Java, String is a final and immutable class stored in the heap. More specifically, it exists in the memory until garbage collection. Therefore, sensitive information should not be stored in String [41], [42]. Compared with String, it is highly recommended to use mutable data structures (e.g., byte or char array) for sensitive information and clear it immediately after use. This reduces the window of opportunity for an adversary. [43]. DESIGN OF CRYPTOAPI-BENCH In this section, we present the design of the CryptoAPI-Bench. We manually generate 181 unit test cases guided by 18 types of misuses presented in Section 2. We divide all test cases into two groups, i.e., basic cases and advanced cases. These test cases incorporate the majority of possible variations in the perspective of program analysis to detect cryptographic vulnerability. Basic Cases Basic test cases are simple ones where the probable source of vulnerability for Crypto API exists within the same method. For example, Listing 1 shows that Cipher API Advanced Cases The advanced cases are more complex compared to basic cases where the probable source of vulnerability of a Crypto API appears from other methods, classes, field variables, or conditional statements . In CryptoAPI-Bench, we include 136 advanced cases. The distribution of advanced cases is presented from the fourth to tenth columns of TABLE 1. Interprocedural Cases In interprocedural cases, the probable source of vulnerability in a Crypto API comes from other methods (i.e., procedures). We create two types of interprocedural cases: two-interprocedural (i.e., involving two methods) and three-interprocedural (i.e., involving three methods). In a two-interprocedural test case, the probable source of vulnerability comes from another method as a parameter. Listing 2 shows the code snippet of a two-interprocedural test case. In method2, Cipher API takes cryptoAlgo as an argument, and cryptoAlgo is not defined in method2, rather, it comes from another method method1. The assigned value of cryptoAlgo in method1 shows that the test case is insecure. In three-interprocedural test cases, the probable source of vulnerability comes from two consecutive methods (i.e., source defined in one method, passes to another method, and then passes again to be used in Cipher API Field Sensitive Cases In field-sensitive cases, the probable source of cryptographic vulnerabilities can be detected by the analysis tools if the tools are capable of performing field-sensitive data flow analysis. Field-sensitive refers to an analysis that is able to differentiate multiple fields or variables with the same object [21]. In Listing 3, algo is an instance or field variable in the Crypto class. The constructor Crypto() stores algo with defAlgo object. A class member function encrypt() use this algo value in Cipher API. Both algo and defAlgo contain the same object, i.e., a secure or insecure cipher algorithm. This is a field-sensitive case as the tools need to trace the field variable algo as the probable source of vulnerability. CryptoAPI-Bench contains 20 field-sensitive test cases. DESIGN OF APACHECRYPTOAPI-BENCH We include the early version of real-world large 10 Apache projects to check the scalability property of different tools. The second and third columns of TABLE 2 show the number of Java files and lines of Java Code in Apache projects. The spark project is the largest among 10 considered projects containing 2,005 Java files with 311,856 lines of code. The meecrowave project contains the lowest number of Java files (40 Java files) and deltaspike contains the lowest number of lines of code (i.e., 5,116 LoC). We enlist 121 test cases in ApacheCryptoAPI-Bench. Among them, 79 test cases are basic cases, i.e., the vulnerability rise within the same method. There are 42 advanced test cases where probable source vulnerability comes from other methods (interprocedural cases), other classes (multiple class cases), class variables (field sensitive cases), etc. We detect 88 cryptographic misuses, i.e., true positive alerts. Regarding true negatives, we consider only the cases where a tool shows the case as a false alert. With this consideration, we show 33 true negative cases. We look into the Apache projects in the Benchmark and made detailed documentation. The documentation consists of cryptographic vulnerabilities the project contains, an explanation of the error, the location (file name, method name, line number) of the vulnerabilities. The documentation and corresponding ApacheCryptoAPI-Bench benchmark are available in the GitHub repository [27]. EXISTING CRYPTOGRAPHIC VULNERABILITY DETECTION TOOLS In this section, we summarize the vulnerability detection tools that we choose to run on CryptoAPI-Bench and ApacheCryptoAPI-Bench. We consider three criteria while choosing the analysis tools. (1) Open-sourced tools: The open-sourced vulnerability detection tools, i.e., CrySL [13], CryptoGuard [5], SpotBugs [24] are convenient to use as we are able to analyze their codes and understand the reason for their lack of performance. (2) Static analysis tools: We choose static analysis tools that can examine and detect vulnerability without executing the code. SpotBugs, CryptoGuard, CrySL, and Coverity [25] are static analysis tools. (3) Free cryptographic vulnerability detection services: We consider Coverity as a provider of free cryptographic vulnerability detection service. Coverity is not open-sourced. However, Coverity provides online services to detect vulnerability. We also consider GrammaTech [44], QARK [45] and FixDroid [14]. However, GrammaTech is a commercial tool. We were unable to access its trial version. The online SWAMP [46] contains GrammaTech tool to use that only 2 ApacheCryptoAPI-Bench: Summary of unit test cases. Contents (number of Java file and lines of code) of the considered Apache projects are summarized here. There are total 121 unit test cases with 79 basic cases and 42 advanced cases. Details information are presented in Section 4. Code Total Case Basic Case Advanced Cases True Positive True Negative deltaspike 87 5116 5 2 3 2 3 directory-server 468 20780 36 15 21 19 17 incubator-taverna-workbench 45 9919 8 5 3 5 3 manifoldcf 126 16998 7 4 3 5 2 meecrowave 40 5646 3 3 0 3 0 spark 2005 311856 27 27 0 27 0 tika 225 16558 3 0 3 0 3 tomee 1029 118661 9 6 3 7 2 wicket 204 13442 8 4 4 5 3 artemis-commons 126 8915 15 13 2 15 0 Total 121 79 42 88 33 supports vulnerability detection for C and C++. Therefore, we excluded GrammaTech from our list of tools. QARK is a tool that is mainly designed to capture security vulnerabilities in Android applications. FixDroid is built as a research prototype that is embedded as a plugin in Android Studio to conduct a usability study. Our investigation shows that the detection capability of FixDroid and QARK is limited. Though QARK has been maintained and updated, FixDroid has not been updated since 2017. Therefore, we mainly focus on four tools, i.e., Spot-Bugs, CryptoGuard, CrySL, and Coverity to evaluate on CryptoAPI-Bench. Test Cases Apache Project Number of Java Files Lines of SpotBugs SpotBugs is a static analysis tool also for capturing deficiencies in Java code. The tool is built based on a plugin structure. The tools detect defects by utilizing visitor patterns in class files or bytecodes of Java, state machine, flags. We use the SpotBugs tool (version 3.1.12) available online in SWAMP [46]. However, currently, SWAMP is in the transition to a new host service [47]. CryptoGuard CryptoGuard [5] is a static analysis tool that is operated based on program slicing with novel language-based refinement algorithms. It significantly reduces the false positive rate which is a typical problem for static analysis. Furthermore, CryptoGuard covers 16 cryptographic rules and achieves high precision. The authors showed screening a large number of Apache projects and Android apps to present their high precision rate and low false positive rate. We run the experiment on CryptoGuard (commitID: 97b220) available on GitHub [48]. CrySL CrySL [13] is a domain-specific language for cryptographic libraries. The static analysis CogniCrypt SAST takes the rules provided in the specification language CrySL as input, and performs a static analysis based on the specification of the rules. CrySL is open-sourced and we run the experiment on CrySL (commit ID: 004cd2) available on GitHub [49]. Coverity Coverity is a commercial tool that analyzes the vulnerabilities of codes. Unlike other tools, it takes the source code and performs its analysis. The Coverity analysis tool is available to use online [25]. We perform the latest analysis using Coverity around September 2020. EVALUATION AND ANALYSIS In this section, we evaluate the results for four cryptographic misuse detection tools, i.e., SpotBugs, CryptoGuard, CrySL and Coverity. We show the experimental setup, evaluation criteria, and analysis results using both benchmarks. Experimental Setup We evaluate mainly four cryptographic analysis tools, i.e., SpotBugs [24], CryptoGuard [5], CrySL [13], Coverity [25] on both Benchmarks. We follow the instructions from GitHub to set up the environment of CryptoGuard and (15 in basic and 22 in advanced), which a tool should not raise any alerts on. GTP stands for ground truth positive, which is the number of insecure API use cases in the benchmark. Findings of the table are reported in Section 6.3. CryptoGuard CrySL Coverity TP FP TP FP TP FP TP FP 1 Cryptographic Key 7 0 3 5 1 0 8 5 1 2 Password in PBE 8 2 0 7 1 0 10 7 1 3 Password in KeyStore 7 1 1 7 1 0 10 5 1 4 Hostname Verifier 1 --1 0 --1 0 5 Certificate Validation 3 3 0 3 0 --3 0 6 SSL Socket 1 - No. Misuse Categories GTP SpotBugs - 1 0 - - 1 0 7 HTTP Protocol 6 - - 6 1 - - - - 8 PRNG 1 1 0 1 0 1 0 - - 9 Seed in PRNG 14 -- CrySL in our machine to perform the analysis. We upload JAR files from CryptoAPI-Bench and Apache projects into SpotBugs tool available in SWAMP. Coverity is an online commercial tool [25] that takes GitHub link and compressed code files in order to start analysis execution. Evaluation Criteria We evaluate the vulnerability detection tools by running these tools on our benchmarks. After performing the analysis, we capture true positives, false positives, and false negatives from the corresponding tool's result log. As our purpose is to detect cryptographic vulnerability detection, we consider only cryptographic misuse alerts and discard others. In TABLE 3, we present the alert keywords that detection tools use while showing a specific cryptographic misuse. This can assist developers to understand which keyword they should search in the result log to find a specific type of vulnerability. In the following, we provide a brief description of our process of identification of true positive, false positive, and false negative alerts. True positive (TP) If a tool generates an alert due to the correct reason while screening any specific vulnerable unit test case in CryptoAPI-Bench, then the event is considered as a true positive. False positive (FP) The false positive alert can be captured from two different scenarios. If an alert raised by a tool is unexpected (i.e., does not exist in a specific unit test case), then the alert is a false positive. In addition, if a tool gives an inaccurate reason for an expected alert, then it is also considered a false positive. False negative (FN) A vulnerable test case may not be detected by the evaluation tools. This missed detection is characterized as a false negative. After analyzing the results by determining the true positive (TP), false positive (FP), and false negative (FN) values, we compute the recall and precision to determine the performance of the tools. CryptoAPI-Bench: Analysis of Results In this section, we describe CryptoAPI-Bench evaluation findings on each detection tool based on the result log and performance analysis. TABLE 4 presents the number of true positive and false positive vulnerability threat detection captured by the tools for 18 cryptographic misuse categories. There are only 6 common cryptographic misuse categories detected by all tools. To ensure fairness in comparison, we consider only these 6 common cryptographic misuses while finding the comparative analysis results of tools based on the basic and advanced benchmark in TABLE 5 and TABLE 6, respectively. The analysis results are presented in terms of false positive rate (FPR), false negative rate (FNR), recall, and precision. Analysis Overview: TABLE 4 shows that among the 18 specified high impact cryptographic misuse categories in Section 2, the cryptographic vulnerability detection tools are able to detect a subset of rules. • SpotBugs, CryptoGuard, CrySL, Coverity covers 9, 16, 14, 10 cryptographic misuse categories, respectively. • In total, the benchmark contains 144 vulnerable test cases and among these true positive cases, SpotBugs, CryptoGuard, CrySL, Coverity detects 20, 124, 76, 32 cases, respectively. • In addition, SpotBugs, CryptoGuard, CrySL, Coverity also generate 32, 20, 67, 14 false alarms, respectively that are included as false positive cases. TABLE 5 shows the performance analysis result of four detection tools on six common cryptographic misuse categories based on the basic benchmark. We capture the following findings based on TABLE 5. Analysis on Basic Benchmark • SpotBugs does not produce any false positive errors. It detects all cases except one. SpotBugs is not designed to capture threats in the basic case of the vulnerable cryptographic key misuse. True Negative Count TP FP FN TP FP FN TP FP FN TP FP FN IntraProcedural 14 6 13 3 1 14 0 0 10 7 4 Count TP FP FN TP FP FN TP FP FN TP FP FN Two-Interprocedural 13 0 0 0 13 12 0 1 10 3 3 3 0 10 Three-Interprocedural 13 0 0 0 13 12 0 1 10 3 3 3 0 10 Field Sensitive 13 0 0 0 13 13 0 0 10 2 3 1 0 In summary, for all basic cases, CryptoGuard and Coverity generate a precision of 100%. For SpotBugs and CrySL, it produces some false positives and hence generates precision of 81.25%, 58.82% respectively. TABLE 6 shows the performance analysis result of four detection tools on six common cryptographic misuse categories based on the advanced benchmark. We capture the following findings based on TABLE 6. Analysis on Advanced Benchmark • In the prospect of path sensitivity, it is obvious that none of the cryptographic vulnerability detection tools is path-sensitive in their static analysis. The tools generate 10, 13, 13, 12 false positive alerts for path sensitive cases, respectively. The possible reason for the false positive alert is that for the concerned variable, a container is defined to store all values of the concerned variable. There is no ordered list that shows the latest assignment. Therefore, alerts will be raised if the container contains any vulnerable value that is intended to be used in the Crypto API. A significant reason for having a high false positive rate because of the tools being path insensitive. • SpotBugs is not designed to capture vulnerability threats in advanced cases. Therefore, it shows 0% precision and recall. • SpotBugs produces 12 false positives for combined cases. In combined cases, SpotBugs failed to detect the source of vulnerability using both interprocedural and field sensitive analysis. For example, in Symmetric Cipher cases, instead of showing the correct "CIPHER INTEGRITY" alert, it produces an incorrect "HARD CODE PASSWORD" alert. • CryptoGuard performs better than other tools in terms of both precision and recall. The reason behind this is 1) Cryptoguard performs dataflow analysis based on forward slicing and backward slicing that efficiently handles the advanced cases, 2) Crypto-Guard follows several refinement insights that systematically remove irrelevant constants, hence reduced false positives. However, as being a static analysis tool, CryptoGuard cannot handle path-sensitive cases. In addition, CryptoGuard missed 3 vulnerabilities due to clipping orthogonal method invocation (i.e., limiting the depth to visit callee method). • CrySL produces incorrect "RequiredPredicateError" alerts for the cryptographic key, password in PBE, password in KeyStore misuse test cases that contribute to generate a high false positive rate. The reason is that the cryptographic APIs used in these cases follow strict rules in CrySL. Therefore, even if we use a secure unpredictable byte array as an argument for crypto APIs, it still generates incorrect alerts. • Coverity is not designed to detect vulnerable ciphers and cryptographic hash functions in advanced cases. That is the reason for having high false negative values and generating high FNR in Coverity. Coverity is a closed sourced detection tool. Therefore, we are unable to confirm the reason for the incorrect detection cases. In summary, for all of the advanced cases, SpotBugs is not designed to identify the advanced vulnerability threats correctly. Therefore, the precision rate is 0%. CryptoGuard detects fairly well (missed only 3 cases) among all detection tools with a precision of 83.33%. For CrySL produce precision of 56.34%. Coverity generates a precision of 52.00%. TABLE 7 ApacheCryptoAPI-Bench comparison of SpotBugs, CryptoGuard, CrySL and Coverity on 10 Apache projects. GTP stands for ground truth positive, which is the number of insecure API use cases in the Apache codes. CrySL Coverity Apache Project GTP TP FP FN TP FP FN TP FP FN TP FP FN deltaspike 2 2 0 0 2 0 0 2 3 0 2 0 0 directory-server 19 11 0 8 5 0 14 18 17 1 5 0 14 incubator-taverna-workbench 5 2 0 3 4 0 1 4 3 1 3 0 2 manifoldcf 5 3 0 2 3 0 2 3 2 2 3 0 2 meecrowave 3 3 0 0 2 0 1 2 0 1 2 0 1 spark 27 23 0 4 27 0 0 ---4 0 23 tika 0 0 0 0 0 0 0 0 3 0 0 0 0 tomee 9 6 0 3 6 0 3 4 2 5 4 0 5 wicket 5 2 0 3 5 0 0 2 3 3 0 0 5 artemis-commons 15 13 0 2 13 0 2 ------Total 88 63 0 25 67 0 21 35 33 11 23 0 50 fails to analyze spark and artemis-commons project. Coverity fails to analyze artemis-commons project. SpotBugs and CryptoGuard successfully analyze all 10 projects. Overall, we capture the following findings. • After analyzing ten Apache projects, we find that there are 79 basic cases, whereas, the number of advance cases is only 43. Therefore, in the real world codes, the number of basic cases is much higher than advanced cases. Vulnerability detection tools should consider expanding their coverage to detect more categories of vulnerabilities. ApacheCryptoAPI-Bench: Analysis of Results SpotBugs CryptoGuard • From DISCUSSION Tool insights. No tool can cover all categories of vulnerabilities (TABLE 4). However, their methodologies can be extended to cover most of these vulnerabilities. For example, the technique that Coverity uses to detect constant cryptographic keys can be transferred to detect static IVs or fewer iteration counts. The main differences among different tools are within their approach to trade-offs among false positives, false negatives. Our experimental evaluation reveals that all of these tools produce a number of false positives and false negatives. CryptoGuard performs on-demand inter-procedural dataflow analysis. Its backward data flow analysis starts from the slicing criteria and explores upward (↑) and orthogonally (→) on-demand. Orthogonal method invocation chains always return to the call sites. By leveraging this insight, CryptoGuard offers a performance vs scalability tradeoff by limiting the depth of the orthogonal invocations (which is "clipping of orthogonal method invocations"). In the current implementation, the depth is set to be 1. That means CryptoGuard will skip deeper orthogonal callee methods, which may result in false negatives. However, the advantage of the orthogonal method invocation technique is that it helps to improve precision. The main focus of CrySL is to provide a language to specify a class of cryptographic misuse vulnerabilities that can be detected using a generic detection engine. A prime reason behind the false positives can be the strictness of the rule definitions that is inherited from the language itself. For example, CrySL raises an alert if a cryptographic key is not generated using a key generator. However, one can legitimately reuse a previously generated key, which CrySL would mistakenly detect as a vulnerability. An impressive aspect of CrySL is that it is constantly being maintained and updated to improve its accuracy. The methodology of SpotBugs is inherently limited to detect advanced cases as they use patterns to detect most of the vulnerabilities. None of these tools are path-sensitive, i.e., all raising false alerts in path sensitive cases. A possible reason for this failure is that the existing path-sensitive analysis techniques are usually costly, i.e., high runtime complexity. CryptoAPI-Bench cannot be used to evaluate scalability property. All of our test cases are lightweight by design. The primary focus is to produce easily readable test cases that demand minimal code to express complex program properties. On the other hand, all of the projects in ApacheCryptoAPI-Bench are complex programs including a lot of files and lines of code. The primary focus is to test the vulnerability detection tool's scalability property and extrapolation to applications on real-world code. Our limitation. Currently, our benchmark does not contain cryptographic cases, e.g., digital signature, CBC-MAC misuses in MAC, other modes of operations (e.g., CTR). We plan to include test cases based on these cryptographic vulnerabilities in our CryptoAPI-Bench benchmark. Furthermore, our benchmark does not have any cases that involve Java reflection APIs. The primary reason is that the use of Java reflection during cryptographic coding is highly unlikely. Consequently, none of the existing open-sourced tools is designed to detect such cases. However, we plan to include new cases that leverage Java reflection APIs to induce cryptographic misuse vulnerabilities. RELATED WORK Vulnerability detection benchmarks. AndroZoo++ [50] is a collection of over eight million Android apps [51] that drives a lot of security, software engineering, and malware analysis research. However, vulnerabilities in these apps are not documented, hence not suitable for vulnerability detection benchmarking purposes. DroidBench [22], a benchmark containing vulnerable android apps, fills the gap by providing specific vulnerability locations within the benchmark. Till date, DroidBench is one of the most popular benchmarks to evaluate the performance of vulnerability detection tools in Android literature. In total, DroidBench has 119 APKs from 13 categories (Commit id 0fe281b). Categories include vulnerabilities that use field and object sensitivity, inter-app communication, intercomponent communication, android life-cycle, reflection, etc. However, DroidBench i) does not cover cryptographic misuse vulnerabilities and ii) does not have source code. To the best of our knowledge, Ghera [23] is the only Android app benchmark that contains app source code. Like Droid-Bench, most of the vulnerabilities in Ghera are specific to Android apps and barely contain any cryptographic misuse vulnerabilities. To be specific, CryptoAPI-Bench and Ghera have only 2 types of vulnerabilities in common. OWASP Benchmark [52] is fundamentally designed to capture eleven cybersecurity vulnerabilities. However, among the detected vulnerabilities, it builds to address only three Java cryptographic vulnerabilities, i.e., weak encryption algorithm, weak hash algorithm, and a weak random number. SonarSource [32] released a set of vulnerability samples that can be useful to check for coverage of vulnerability categories. A verification tool for five common audit controls is proposed for ensuring continuous compliance [53]. Other benchmarks. The DaCapo benchmarks [54] are designed to evaluate the performance of various components of Java virtual machine (JVM), Garbage collection (GC), Just-in-time (JIT) compiler itself. BugBench [55] is a benchmark to find C/C++ bugs that contains 17 realworld applications. BugBench mostly covers various memory, concurrency, and semantic bugs. To detect bugs in the multi-threaded Java programs, a benchmark and framework have been proposed [56], [57]. For dynamic software updating system, a standardized benchmark system is proposed to check the system's practicality, flexibility, and usability [58]. Coding practice and recommendations are provided for 28 enterprise applications that use Spring security framework [59]. ManyBugs and IntroClass benchmarks are designed to evaluate various C/C++ code repair techniques [60]. Most of the defects in ManyBugs and IntroClass do not impact security, e.g., in the ManyBugs benchmark, more than half of the instances impact correctness, not necessary security. CONCLUSION AND FUTURE WORK We believe that for scientific, in-depth, and reproducible comparisons benchmark is an important component. In this paper, we present CryptoAPI-Bench and ApacheCryptoAPI-Bench to evaluate the detection accuracy, scalability, and security guarantees of various cryptographic misuse detection tools. Our benchmarks are open-sourced and are available on GitHub. We evaluated four static analysis tools that are capable of detecting cryptographic misuses. Our evaluation revealed some interesting insights, i.e., i) tools that are specialized to detect cryptographic misuses (e.g., CryptoGuard, CrySL) cover more rules and higher recall than general purpose tools (e.g., SpotBugs, Coverity), ii) none of the existing tools is path-sensitive. We are actively working on expanding CryptoAPI-Bench by adding new rules, test cases, and covering new cryptographic APIs. In the future, we plan to achieve the following goals. • To motivate the research of cryptographic misuse detection tools for other platforms, we plan to extend CryptoAPI-Bench to cover other popular languages, e.g., Python. • Other non-cryptographic API misuses (e.g., Android APIs to access sensitive information (location, IMEI, passwords, etc.) [61], [62], fingerprint protection [63], cloud service APIs for information storage [64]) are also proven to cause catastrophic security consequences. We also plan to include the misuses of these critical non-cryptographic APIs. • CrySL produces 7 false positive errors due to maintaining strict rules in Crypto APIs of the cryptographic key, password in PBE, password in KeyStore.• Coverity does not generate any false positive errors. It can successfully detect every vulnerability except one. Coverity is not designed to capture IDEA as a vulnerable cryptographic algorithm.• For insecure uses of pseudo-random number generators, SpotBugs and CryptoGuard flag all uses of java.util.Random. However, CrySL flags the insecure random variable when they use in crypto contexts. TABLE 1 1CryptoAPI-Bench: Summary of unit test cases. There are 181 unit test cases with 45 basic cases and 136 advanced cases (interprocedural, field sensitive, combined case, path sensitive, miscellaneous, and multiple class test cases). Total test cases per group and misuse categories are summarized here. Details information are presented in Section 3. No. Misuse Categories Basic Cases Two- Interproc. Three- Interproc. Field Sensitive Combined Case Path Sensitive Misc. Multiple Class 16 Cryptographic Hash 5 4 4 4 4 4 0 4 29 17 Cryptographic MAC 3 0 0 0 0 0 0 0 3 18 Credentials in String 2 1 1 1 1 0 1 1 8 Total Cases per Group 45 21 21 20 21 20 12 21 181 takes cryptoAlgo as an argument. Note that, cryptoAlgo contains an insecure cipher algorithm that is defined within the same method method1. In CryptoAPI-Bench, we create 45 basic test cases covering all 18 misuse categories. Among these test cases, 30 test cases contain cryptographic vulnera- bility (i.e., true positive), and 15 test cases do not contain any cryptographic vulnerability (i.e., true negative). These test cases identify a tool's capability to detect a specific misuse category. 1 public void method1 () 2 { ... 3 cryptoAlgo = "DES/ECB/PKCS5Padding" 4 Cipher cipher = Cipher.getInstance(cryptoAlgo) 5 ... 6 } Listing 1. Example code snippet of a basic test case ). CryptoAPI-Bench contains a total of 42 interprocedural test cases. Among them, 21 are twointerprocedural test cases, and 21 are three-interprocedural test cases. The purpose of having the interprocedural test cases is to check the detection tool's interprocedural data flow handling capability. Listing 2. Example code snippet of a two-interprocedural test case1 public void method1 () 2 { ... 3 cryptoAlgo = "DES/ECB/PKCS5Padding" 4 method2(cryptoAlgo) 5 ... 6 } 7 public void method2 (String cryptoAlgo) 8 { ... 9 Cipher cipher = Cipher.getInstance(cryptoAlgo) 10 ... 11 } Listing 3. Example code snippet of a field sensitive test case3.2.3 Combined CasesThe combined cases are a bit more complex where both interprocedural and field sensitivity properties are combined, i.e., both Listing 2 and Listing 3 are incorporated to generate complicated test cases. CryptoAPI-Bench has 21 combined test cases.3.2.4 Path-Sensitive CasesIn path-sensitive test cases, conditional branch instructions are included in the test cases containing the definition of the probable source of a vulnerability. In Listing 4, an example code snippet of a path sensitivity case is given. Depending on the choice variable, the Cipher is getting the instance from a secure or an insecure cryptographic algorithm. There are 20 path-sensitive test cases in CryptoAPI-Bench.Cipher ch = Cipher.getInstance ("DES/ECB/...") ; ch = Cipher.getInstance ("AES/CBC/...") ; ch.init (Cipher.ENCRYPT_MODE, key) ; Listing 4. Example code snippet of a path sensitive test case3.2.5 Miscellaneous CasesMiscellaneous test cases evaluate the tool's abilities to recognize irrelevant constraints and other interfaces, e.g., Map. In Listing 5, the Map interface of Line 3-6 provides a secure key or insecure key depending on the choice variable. The Map indices (e.g., "a", "b") represent only index values, not security-relevant values. Similarly, in Line 8, the "UTF-8" represents byte encoding, not any constant or hardcoded value. CryptoAPI-Bench contains 12 miscellaneous test cases. Multiple Class Cases In multiple class test cases, the probable source of vulnerabilities comes from another Java class. An example code snippet of multiple class case is presented in Listing 6. It is necessary to detect whether a secure or an insecure algorithm is passed in Line 4 in MultipleClass1 and used in Line 9 in MultipleClass2. CryptoAPI-Bench has 21 multiple class test cases.public void method2 (String cryptoAlgo) { Cipher c = Cipher.getInstance (cryptoAlgo); Listing 6. Example code snippet of a multiple class test case1 class Crypto { 2 String algo 3 public Crypto (String defAlgo) { 4 algo = defAlgo; 5 } 6 public void encrypt(... ) { 7 ... 8 Cipher cipher = Cipher.getInstance(algo); 9 ... 10 } 11 } 1 public void method1 (int choice) { 2 ... 3 4 if (choice > 1) { 5 6 } 7 8 ... 9 } 1 public void method1 (String choice) { 2 ... 3 Map<String,String> hm = new HashMap<String, String>(); 4 hm.put("a", secureKeyString); 5 hm.put("b", insecureKeyString); 6 String keyString = hm.get(choice); 7 8 byte [] b = secureKeyString.getBytes("UTF-8"); 9 IvParameterSpec ivSpec = new IvParameterSpec(b); 10 ... 11 } Listing 5. Example code snippet of a miscellaneous test case 3.2.6 1 public class MultipleClass1 { 2 public void method1 (String passedAlgo) { 3 MultipleClass2 mc = new MultipleClass2 (); 4 mc.method2 (passedAlgo); 5 } 6 } 7 public class MultipleClass2 { 8 9 10 } 11 } TABLE TABLE 3 3Generated alert keywords for each misuse category from cryptographic vulnerability detection tools (SpotBugs, CryptoGuard, CrySL, and Coverity). For example, for misuse category 16 (i.e., Cryptographic Hash), the generated alert keywords in tools are WEAK MESSAGE DIGEST, broken hash scheme, ConstraintError, RISKY CRYPTO, respectively.Misuse Categories SpotBugs CryptoGuard CrySL Coverity 1 HARD CODE PASSWORD Constant keys RequiredPredicateError HARDCODED CREDENTIALS 2 HARD CODE PASSWORD Constant keys HardCodedError HARDCODED CREDENTIALS 3 HARD CODE PASSWORD Predictable password HardCodedError HARDCODED CREDENTIALS 4 WEAK HOSTNAME VERIFIER Manually verify hostname - BAD CERT VERIFICATION 5 WEAK TRUST MANAGER Untrusted TrustManager - BAD CERT VERIFICATION 6 - Does not manually verify socket - RESOURCE LEAK 7 - HTTP protocol - - 8 PREDICTABLE RANDOM Untrusted PRNG RequiredPredicateError - 9 - Predictable Seed RequiredPredicateError PREDICTABLE RANDOM SEED 10 - Constant Salt RequiredPredicateError - 11 CIPHER INTEGRITY Broken crypto scheme ConstraintError RISKY CRYPTO 12 STATIC IV Constant IV RequiredPredicateError - 13 - <1000 iteration ConstraintError - 14 CIPHER INTEGRITY Broken crypto scheme ConstraintError RISKY CRYPTO 15 - Export grade public key ConstraintError - 16 WEAK MESSAGE DIGEST Broken hash scheme ConstraintError RISKY CRYPTO 17 - - ConstraintError - 18 - - RequiredPredicateError - TABLE 4 4CryptoAPI-Bench comparison of SpotBugs, CryptoGuard, CrySL and Coverity on all 18 rules with CryptoAPI-Bench's 181 test cases. There are 37 secure API use cases TABLE 5 5CryptoAPI-Bench comparison of SpotBugs, CryptoGuard, CrySL and Coverity on six common misuse categories with CryptoAPI-Bench's common 21 basic cases. TP, FP, FN stand for true positive, false positive, false negative, respectively. Findings of the table are reported in Section 6.3.1. SpotBugs CryptoGuard CrySL Coverity Basic Test Cases True Positive Count TABLE 6 6CryptoAPI-Bench comparison of SpotBugs, CryptoGuard, CrySL and Coverity on six common misuse categories with CryptoAPI-Bench's common 84 advanced cases. TP, FP, FN stand for true positive, false positive, false negative, respectively. Findings of the table are reported in Section 6.3.2.SpotBugs CryptoGuard CrySL Coverity Advanced Test Cases True Positive Count True Negative TABLE 7 7presents the number of true positive and false positive vulnerability threats detected by the tools. CrySL TABLE 7 , 7we observe that CrySL fails to analyze two Apache projects: spark and artemis- commons. CrySL throws StackOverFlowError (i.e., memory error) during analyzing objects for spark. The probable reason is the larger number of files and lines of code Spark contains for analysis. For artemis- commons, CrySL throws NullPointerErrorException during analysis due to the reference variable not pointing to any object. Coverity fails to analyze only the artemis-commons project. Coverity is closed source, therefore, we are unable to confirm the rea- son for this failure. TABLE 8 shows the runtime on Apache projects for only CryptoGuard and CrySL. For Coverity and SpotBugs, we use the web version that takes all scan requests for users and reports results after complete scanning. Therefore, we can- not calculate their original runtime for comparison. Among the 8 successful analyzed projects, we ob- serve average runtime for CrySL is 14.64 seconds and CryptoGuard is 11.46 seconds. For the largest Apache project Spark (LoC: 311,856), CryptoGuard TABLE 8 8Runtime for analyzing Apache projects. Star (*) symbol indicates that the analysis was unsuccessful.successfully analyzes in 88.68 seconds and CrySL shows the failure of analysis report after 46.84 seconds. Overall, SpotBugs and CryptoGuard successfully analyze all 10 Apache projects. Therefore, Spot-Bugs, CryptoGuard are scalable for large projects.Our benchmarks are open-sourced and are available on GitHub [26],[27]. It contains the Java cryptographic API test cases. The detailed documentation and explanation are provided there.Runtime (sec) Apache Projects LoC CryptoGuard CrySL deltaspike 5.1K 4.31 6.95 directory-server 20.8K 8.96 23.03 incubator-taverna-workbench 9.9K 12.69 7.94 manifoldcf 17K 7.07 8.20 meecrowave 5.6K 4.67 7.24 spark 311.9K 88.68 46.84* tika 16.6K 7.46 8.15 tomee 118.7K 40.52 34.81 wicket 13.4K 5.99 20.83 artemis-commons 8.9K 5.63 19.82* 6.5 Verifiability ACKNOWLEDGMENTSThis work has been supported by the National Science Foundation under Grant No. CNS-1929701 and the Virginia Commonwealth Cyber Initiative (CCI). Why Eve and Mallory Love Android: An Analysis of Android SSL (in) Security. S Fahl, M Harbach, T Muders, the ACM Conference on Computer and Communications Security, CCS'12. Raleigh, NC, USAS. Fahl, M. Harbach, T. Muders et al., "Why Eve and Mallory Love Android: An Analysis of Android SSL (in) Security," in the ACM Conference on Computer and Communications Security, CCS'12, Raleigh, NC, USA, October 16-18, 2012, 2012, pp. 50-61. The Most Dangerous Code in the World: Validating SSL Certificates in Non-Browser Software. M Georgiev, S Iyengar, S Jana, the ACM Conference on Computer and Communications Security, CCS'12. Raleigh, NC, USAM. Georgiev, S. Iyengar, S. Jana et al., "The Most Dangerous Code in the World: Validating SSL Certificates in Non-Browser Software," in the ACM Conference on Computer and Communications Security, CCS'12, Raleigh, NC, USA, October 16-18, 2012, pp. 38-49. An Empirical Study of Cryptographic Misuse in Android Applications. M Egele, D Brumley, Y Fratantonio, ACM Conference on Computer and Communications Security, CCS'13. M. Egele, D. Brumley, Y. Fratantonio et al., "An Empirical Study of Cryptographic Misuse in Android Applications," in ACM Con- ference on Computer and Communications Security, CCS'13, 2013, pp. 73-84. Secure Coding Practices in Java: Challenges and Vulnerabilities. N Meng, S Nagy, D Yao, International Conference on Software Engineering, ICSE'18. Gothenburg, SwedenN. Meng, S. Nagy, D. Yao et al., "Secure Coding Practices in Java: Challenges and Vulnerabilities," in International Conference on Software Engineering, ICSE'18, Gothenburg, Sweden, May 2018. CryptoGuard: High Precision Detection of Cryptographic Vulnerabilities in Massive-sized Java Projects. S Rahaman, Y Xiao, S Afrose, ACM Conference on Computer and communications security, CCS'19. London, UKS. Rahaman, Y. Xiao, S. Afrose et al., "CryptoGuard: High Preci- sion Detection of Cryptographic Vulnerabilities in Massive-sized Java Projects," in ACM Conference on Computer and communications security, CCS'19, London, UK, Nov. 2019. Program Analysis of Cryptographic Implementations for Security. S Rahaman, D Yao, 10.1109/SecDev.2017.23IEEE Cybersecurity Development, SecDev'17. Cambridge, MA, USAS. Rahaman and D. Yao, "Program Analysis of Cryptographic Implementations for Security," in IEEE Cybersecurity Development, SecDev'17, Cambridge, MA, USA, September 24-26, 2017, pp. 61-68. [Online]. Available: https://doi.org/10.1109/SecDev.2017.23 Comparing the Usability of Cryptographic APIs. Y Acar, M Backes, S Fahl, IEEE Symposium on Security and Privacy, SP'17. San Jose, CA, USAY. Acar, M. Backes, S. Fahl et al., "Comparing the Usability of Cryptographic APIs," in IEEE Symposium on Security and Privacy, SP'17, San Jose, CA, USA, May 22-26, 2017, pp. 154-171. Jumping Through Hoops: Why Do Java Developers Struggle with Cryptography APIs. S Nadi, S Krüger, M Mezini, International Conference on Software Engineering, ICSE'16. S. Nadi, S. Krüger, M. Mezini et al., "Jumping Through Hoops: Why Do Java Developers Struggle with Cryptography APIs?" in International Conference on Software Engineering, ICSE'16, 2016, pp. 935-946. The Rise of the Citizen Developer: Assessing the Security Impact of Online App Generators. M Oltrogge, E Derr, C Stransky, IEEE Symposium on Security and Privacy, SP'18. San Francisco, California, USAM. Oltrogge, E. Derr, C. Stransky et al., "The Rise of the Citizen Developer: Assessing the Security Impact of Online App Gen- erators," in IEEE Symposium on Security and Privacy, SP'18, San Francisco, California, USA, 21-23 May, 2018, pp. 634-647. You Get Where You're Looking for: The Impact of Information Sources on Code Security. Y Acar, M Backes, S Fahl, IEEE Symposium on Security and Privacy, SP'16. San Jose, CA, USAY. Acar, M. Backes, S. Fahl et al., "You Get Where You're Looking for: The Impact of Information Sources on Code Security," in IEEE Symposium on Security and Privacy, SP'16, San Jose, CA, USA, May 23-25, 2016, pp. 289-305. Security in the Software Development Lifecycle. H Assal, S Chiasson, Fourteenth Symposium on Usable Privacy and Security, SOUPS'18. Baltimore, MD, USAH. Assal and S. Chiasson, "Security in the Software Development Lifecycle," in Fourteenth Symposium on Usable Privacy and Security, SOUPS'18, Baltimore, MD, USA, August 12-14, 2018, pp. 281-296. CogniCrypt: Supporting Developers in using Cryptography. S Krüger, IEEE/ACM International Conference on Automated Software Engineering, ASE'17. S. Krüger et al., "CogniCrypt: Supporting Developers in using Cryptography," in IEEE/ACM International Conference on Automated Software Engineering, ASE'17, 2017, pp. 931-936. CrySL: An Extensible Approach to Validating the Correct Usage of Cryptographic APIs. S Krüger, J Späth, K Ali, European Conference on Object-Oriented Programming, ECOOP'18. 1027S. Krüger, J. Späth, K. Ali et al., "CrySL: An Extensible Approach to Validating the Correct Usage of Cryptographic APIs," in European Conference on Object-Oriented Programming, ECOOP'18, 2018, pp. 10:1-10:27. A Stitch in Time: Supporting Android Developers in Writing Secure Code. D C Nguyen, ACM Conference on Computer and Communications Security, CCS'17. D. C. Nguyen et al., "A Stitch in Time: Supporting Android De- velopers in Writing Secure Code," in ACM Conference on Computer and Communications Security, CCS'17, 2017, pp. 1065-1077. Crylogger: Detecting Crypto Misuses Dynamically. L Piccolboni, G Di Guglielmo, L P Carloni, arXiv:2007.01061arXiv preprintL. Piccolboni, G. Di Guglielmo, L. P. Carloni et al., "Cry- logger: Detecting Crypto Misuses Dynamically," arXiv preprint arXiv:2007.01061, 2020. SMV-Hunter: Large Scale, Automated Detection of SSL/TLS Man-in-the-Middle Vulnerabilities in Android Apps. D Sounthiraraj, J Sahs, G Greenwood, The Network and Distributed System Security Symposium, NDSS'14. D. Sounthiraraj, J. Sahs, G. Greenwood et al., "SMV-Hunter: Large Scale, Automated Detection of SSL/TLS Man-in-the-Middle Vulnerabilities in Android Apps," in The Network and Distributed System Security Symposium, NDSS'14, 2014. AndroSSL: A Platform to Test Android Applications Connection Security. F Gagnon, M Ferland, M Fortier, International Symposium on Foundations and Practice of Security, FPS'15. F. Gagnon, M. Ferland, M. Fortier et al., "AndroSSL: A Platform to Test Android Applications Connection Security," in International Symposium on Foundations and Practice of Security, FPS'15, 2015, pp. 294-302. Using Programmer-Written Compiler Extensions to Catch Security Holes. K Ashcraft, D R Engler, IEEE Symposium on Security and Privacy. Berkeley, California, USASP'02)K. Ashcraft and D. R. Engler, "Using Programmer-Written Com- piler Extensions to Catch Security Holes," in IEEE Symposium on Security and Privacy, (SP'02), Berkeley, California, USA, May 12-15, 2002, pp. 143-159. DR. CHECKER: A Soundy Analysis for Linux Kernel Drivers. A Machiry, C Spensky, J Corina, 26th USENIX Security Symposium, USENIX Security'17. Vancouver, BC, CanadaA. Machiry, C. Spensky, J. Corina et al., "DR. CHECKER: A Soundy Analysis for Linux Kernel Drivers," in 26th USENIX Security Symposium, USENIX Security'17, Vancouver, BC, Canada, August 16- 18, 2017, 2017, pp. 1007-1024. Data-flow Analysis -Wikipedia, The Free Encyclopedia. Wikipedia contributors. last accessedWikipedia contributors, "Data-flow Analysis -Wikipedia, The Free Encyclopedia," last accessed: September 9, 2021. [Online]. Context-, Flow-, and Fieldsensitive Data-flow Analysis Using Synchronized Pushdown Systems. J Späth, K Ali, E Bodden, 10.1145/3290361Proc. ACM Program. Lang. 3POPLJ. Späth, K. Ali, and E. Bodden, "Context-, Flow-, and Field- sensitive Data-flow Analysis Using Synchronized Pushdown Systems," Proc. ACM Program. Lang., vol. 3, no. POPL, Jan. 2019. [Online]. Available: https://doi.org/10.1145/3290361 FlowDroid: Precise Context, Flow, Field, Object-Sensitive and Lifecycle-aware Taint Analysis for Android Apps. S Arzt, S Rasthofer, C Fritz, ACM SIGPLAN Conference on Programming Language Design and Implementation, PLDI'14. S. Arzt, S. Rasthofer, C. Fritz et al., "FlowDroid: Precise Context, Flow, Field, Object-Sensitive and Lifecycle-aware Taint Analysis for Android Apps," in ACM SIGPLAN Conference on Programming Language Design and Implementation, PLDI'14, 2014, pp. 259-269. Ghera: A Repository of Android App Vulnerability Benchmarks. J Mitra, V Ranganath, Proceedings of the 13th International Conference on Predictive Models and Data Analytics in Software Engineering, PROMISE'17. the 13th International Conference on Predictive Models and Data Analytics in Software Engineering, PROMISE'17Toronto, CanadaJ. Mitra and V. Ranganath, "Ghera: A Repository of Android App Vulnerability Benchmarks," in Proceedings of the 13th International Conference on Predictive Models and Data Analytics in Software Engi- neering, PROMISE'17, Toronto, Canada, November 8, 2017, 2017, pp. 43-52. SpotBugs: Find Bugs in Java Programs. Last accessed"SpotBugs: Find Bugs in Java Programs," https://spotbugs.github.io/, online; Last accessed: December 3, 2020. CryptoAPI-Bench. S Afrose, Coverity Static Application Security Testing (SAST). 27Last accessed"Coverity Static Application Security Testing (SAST)," https://www.synopsys.com/software-integrity/security-testing/ static-analysis-sast.html, online; Last accessed: December 3, 2020. [26] S. Afrose, "CryptoAPI-Bench," https://github.com/ CryptoAPI-Bench/CryptoAPI-Bench, 2019. [27] --, "ApacheCryptoAPI-Bench," https://github.com/ CryptoAPI-Bench/ApacheCryptoAPI-Bench, 2020. CryptoAPI-Bench: A Comprehensive Benchmark on Java Cryptographic API Misuses. S Afrose, S Rahaman, D Yao, 2019 IEEE Cybersecurity Development (SecDev). IEEES. Afrose, S. Rahaman, and D. Yao, "CryptoAPI-Bench: A Compre- hensive Benchmark on Java Cryptographic API Misuses," in 2019 IEEE Cybersecurity Development (SecDev). IEEE, 2019, pp. 49-61. Transitioning the Use of Cryptographic Algorithms and Key Lengths. E Barker, A Roginsky, Special Publication (NIST SP). Gaithersburg, MDNational Institute of Standards and TechnologyE. Barker and A. Roginsky, "Transitioning the Use of Crypto- graphic Algorithms and Key Lengths," in Special Publication (NIST SP), National Institute of Standards and Technology, Gaithersburg, MD, 2019. Recommendation for Pairwise Key Establishment Using Integer Factorization Cryptography. E Barker, L Chen, A Roginsky, Special Publication (NIST SP). Gaithersburg, MDNational Institute of Standards and TechnologyE. Barker, L. Chen, A. Roginsky et al., "Recommendation for Pair- wise Key Establishment Using Integer Factorization Cryptogra- phy," in Special Publication (NIST SP), National Institute of Standards and Technology, Gaithersburg, MD, 2019. SonarSource Static Code Analysis. Last accessed"SonarSource Static Code Analysis," https://rules.sonarsource.com/, online; Last accessed: December 3, 2020. . &quot; Url Spoofing, Last accessed"URL Spoofing," http://www.securitysupervisor.com/ security-q-a/network-security/262-what-is-url-spoofing.html, online; Last accessed: December 3, 2020. Find Security Bugs. Last accessed"Find Security Bugs," https://find-sec-bugs.github.io/, online; Last accessed: December 3, 2020. Hostname Verification to SSL Socket. "Hostname Verification to SSL Socket," https://www.thecodingforums.com/threads/ adding-hostname-verification-to-sslsocket.958287/, online; Transferring Data via a Secure Network Connection. C A Barton, G A Clarke, S Crowe, US Patent. 7121C. A. Barton, G. A. Clarke, and S. Crowe, "Transferring Data via a Secure Network Connection," 2006, US Patent 7,093,121. D E Knuth, Seminumerical Algorithms. Addison-Wesley Professional2Art of Computer ProgrammingD. E. Knuth, Art of Computer Programming, volume 2: Seminumerical Algorithms. Addison-Wesley Professional, 2014. PKCS #5: Password-Based Cryptography Specification Version 2.1. K Moriarty, B Kaliski, A Rusch, K. Moriarty, B. Kaliski, and A. Rusch, "PKCS #5: Password-Based Cryptography Specification Version 2.1," 2017. . AES Encryption. Last accessed"AES Encryption," https://aesencryption.net/, online; Last ac- cessed: December 3, 2020. New Proofs for NMAC and HMAC: Security Without Collision Resistance. M Bellare, Journal of Cryptology. 284M. Bellare, "New Proofs for NMAC and HMAC: Security Without Collision Resistance," Journal of Cryptology, vol. 28, no. 4, pp. 844- 878, 2015. Oracle. Last accessed"Oracle," https://docs.oracle.com/javase/8/docs/api/java/ secur-ity/Permission.html, online; Last accessed: September 3, 2021. Secure Code Review: 8 Security Code Review Best Practices. Last accessed"Secure Code Review: 8 Security Code Review Best Practices," https://snyk.io/blog/secure-code-review/, online; Last accessed: September 3, 2021. Secure Coding Guidelines for Java SE. Last accessed"Secure Coding Guidelines for Java SE," https://www.oracle. com/java/technologies/javase/seccodeguide.html, online; Last accessed: September 3, 2021. GRAMMATECH. Quick Android Review Kit. Last accessed. Last accessed"GRAMMATECH," https://www.grammatech.com/, online; Last accessed: December 3, 2020. [45] "Quick Android Review Kit (QARK)," https://github.com/linkedin/qark, online; Last accessed: December 3, 2020. Welcome to the SWAMP. "Welcome to the SWAMP," https://continuousassurance.org, 2018. Transition of SWAMP software. Last accessed"Transition of SWAMP software," https://continuousassurance.org/blog/, online; Last accessed:December 3, 2020. CryptoGuard. Last accessed"CryptoGuard," https://github.com/CryptoGuardOSS/cryptoguard, online; Last accessed: December 3, 2020. Cognicrypt SAST: CrySLtoStatic Analysis Compiler. "Cognicrypt SAST: CrySLtoStatic Analysis Compiler," https://github.com/CROSSINGTUD/CryptoAnalysis, online; AndroZoo++: Collecting Millions of Android Apps and Their Metadata for the Research Community. L Li, J Gao, M Hurier, arXiv:1709.05281arXiv preprintL. Li, J. Gao, M. Hurier et al., "AndroZoo++: Collecting Millions of Android Apps and Their Metadata for the Research Community," arXiv preprint arXiv:1709.05281, 2017. AndroZoo. Last accessed: December 3"AndroZoo," https://androzoo.uni.lu/, online; Last accessed: De- cember 3, 2020. Security Analysis of the OWASP Benchmark with Julia. E Burato, P Ferrara, F Spoto, The Italian Conference on Cyber-Security (ITASEC). 17E. Burato, P. Ferrara, and F. Spoto, "Security Analysis of the OWASP Benchmark with Julia," The Italian Conference on Cyber- Security (ITASEC), vol. 17, 2017. Continuous Compliance. M Kellogg, M Schäf, S Tasirans, 35th IEEE/ACM International Conference on Automated Software Engineering (ASE). M. Kellogg, M. Schäf, S. Tasirans et al., "Continuous Compliance," in 35th IEEE/ACM International Conference on Automated Software Engineering (ASE), 2020, pp. 511-523. The DaCapo Benchmarks: Java Benchmarking Development and Analysis. S M Blackburn, Proceedings of the 21th. the 21thS. M. Blackburn et al., "The DaCapo Benchmarks: Java Bench- marking Development and Analysis," in Proceedings of the 21th Annual ACM SIGPLAN Conference on Object-Oriented Programming, Systems, Languages, and Applications, OOPSLA'06. Portland, Oregon, USAAnnual ACM SIGPLAN Conference on Object-Oriented Programming, Systems, Languages, and Applications, OOPSLA'06, Portland, Oregon, USA, October 22-26, 2006, pp. 169-190. BugBench: Benchmarks for Evaluating Bug Detection Tools. S Lu, Z Li, F Qin, Workshop on the Evaluation of Software Defect Detection Tools. 5S. Lu, Z. Li, F. Qin et al., "BugBench: Benchmarks for Evaluating Bug Detection Tools," in Workshop on the Evaluation of Software Defect Detection Tools, vol. 5, 2005. Benchmark and Framework for Encouraging Research on Multi-Threaded Testing Tools. K Havelund, S D Stoller, S Ur, Proceedings International Parallel and Distributed Processing Symposium, IPDPS'03. International Parallel and Distributed Processing Symposium, IPDPS'03IEEE286K. Havelund, S. D. Stoller, and S. Ur, "Benchmark and Framework for Encouraging Research on Multi-Threaded Testing Tools," in Proceedings International Parallel and Distributed Processing Sympo- sium, IPDPS'03. IEEE, 2003, p. 286. Towards a Framework and a Benchmark for Testing Tools for Multi-Threaded Programs. Y Eytani, K Havelund, S D Stoller, Concurrency and Computation: Practice and Experience. 193Y. Eytani, K. Havelund, S. D. Stoller et al., "Towards a Framework and a Benchmark for Testing Tools for Multi-Threaded Programs," Concurrency and Computation: Practice and Experience, vol. 19, no. 3, pp. 267-279, 2007. Towards Standardized Benchmarks for Dynamic Software Updating Systems. E K Smith, M Hicks, J S Foster, 4th International Workshop on Hot Topics in Software Upgrades (HotSWUp'12). IEEEE. K. Smith, M. Hicks, and J. S. Foster, "Towards Standard- ized Benchmarks for Dynamic Software Updating Systems," in 4th International Workshop on Hot Topics in Software Upgrades (HotSWUp'12). IEEE, 2012, pp. 11-15. Coding Practices and Recommendations of Spring Security for Enterprise Applications. M Islam, S Rahaman, N Meng, IEEEin 2020 IEEE Secure Development (SecDevM. Islam, S. Rahaman, N. Meng et al., "Coding Practices and Recommendations of Spring Security for Enterprise Applications," in 2020 IEEE Secure Development (SecDev). IEEE, 2020, pp. 49-57. The ManyBugs and IntroClass Benchmarks for Automated Repair of C Programs. C Le Goues, N Holtschulte, E K Smith, IEEE Transactions on Software Engineering. 4112C. Le Goues, N. Holtschulte, E. K. Smith et al., "The ManyBugs and IntroClass Benchmarks for Automated Repair of C Programs," IEEE Transactions on Software Engineering, vol. 41, no. 12, pp. 1236- 1256, 2015. Collusive Data Leak and More: Large-Scale Threat Analysis of Inter-app Communications. A Bosu, F Liu, D Yao, ACM ASIA Conference on Computer and Communications Security, AsiaCCS'17. A. Bosu, F. Liu, D. Yao et al., "Collusive Data Leak and More: Large-Scale Threat Analysis of Inter-app Communications," in ACM ASIA Conference on Computer and Communications Security, AsiaCCS'17, 2017, pp. 71-85. Finding Clues for Your Secrets: Semantics-Driven, Learning-Based Privacy Discovery in Mobile Apps. Y Nan, Z Yang, X Wang, 25th Annual Network and Distributed System Security Symposium, NDSS'18. San Diego, California, USAY. Nan, Z. Yang, X. Wang et al., "Finding Clues for Your Secrets: Semantics-Driven, Learning-Based Privacy Discovery in Mobile Apps," in 25th Annual Network and Distributed System Security Symposium, NDSS'18, San Diego, California, USA, February 18-21, 2018. Broken Fingers: On the Usage of the Fingerprint API in Android. A Bianchi, Y Fratantonio, A Machiry, 25th Annual Network and Distributed System Security Symposium, NDSS'18. San Diego, California, USAA. Bianchi, Y. Fratantonio, A. Machiry et al., "Broken Fingers: On the Usage of the Fingerprint API in Android," in 25th Annual Network and Distributed System Security Symposium, NDSS'18, San Diego, California, USA, February 18-21, 2018. Why Does Your Data Leak? Uncovering the Data Leakage in Cloud from Mobile Apps. C Zuo, Z Lin, Y Zhang, IEEE Symposium on Security and Privacy. London, UKSP'19C. Zuo, Z. Lin, and Y. Zhang, "Why Does Your Data Leak? Uncovering the Data Leakage in Cloud from Mobile Apps," in IEEE Symposium on Security and Privacy, (SP'19), London, UK, 2019.
[ "https://github.com/linkedin/qark,", "https://github.com/CryptoGuardOSS/cryptoguard,", "https://github.com/CROSSINGTUD/CryptoAnalysis," ]
[ "N = 2 supersymmetric pseudodifferential symbols and super W-algebras", "N = 2 supersymmetric pseudodifferential symbols and super W-algebras" ]
[ "Stéphane Gourmelen \nInstitut de Physique Nucléaire de Lyon\nIN2P3/CNRS\nUniversité Claude\nBernard 43, boulevard du 11 novembre 1918 F -69622 -Villeurbanne Cedex\n" ]
[ "Institut de Physique Nucléaire de Lyon\nIN2P3/CNRS\nUniversité Claude\nBernard 43, boulevard du 11 novembre 1918 F -69622 -Villeurbanne Cedex" ]
[]
We study the superconformally covariant pseudodifferential symbols defined on N = 2 super Riemann surfaces. This allows us to construct a primary basis for N = 2 super W (n) KP -algebras and, by reduction, for N = 2 super W n -algebras.(n) KP -algebra[8]. The latter is obtained by applying the second hamiltonian structure of W n to pseudodifferential symbols rather than
null
[ "https://export.arxiv.org/pdf/hep-th/9812151v1.pdf" ]
15,831,592
hep-th/9812151
f0be503354918a8d6c28924de032395d4b4e71d0
N = 2 supersymmetric pseudodifferential symbols and super W-algebras arXiv:hep-th/9812151v1 17 Dec 1998 Stéphane Gourmelen Institut de Physique Nucléaire de Lyon IN2P3/CNRS Université Claude Bernard 43, boulevard du 11 novembre 1918 F -69622 -Villeurbanne Cedex N = 2 supersymmetric pseudodifferential symbols and super W-algebras arXiv:hep-th/9812151v1 17 Dec 1998 We study the superconformally covariant pseudodifferential symbols defined on N = 2 super Riemann surfaces. This allows us to construct a primary basis for N = 2 super W (n) KP -algebras and, by reduction, for N = 2 super W n -algebras.(n) KP -algebra[8]. The latter is obtained by applying the second hamiltonian structure of W n to pseudodifferential symbols rather than Introduction W-symmetry plays an important rôle in the context of two-dimensional conformal field theories [1,2] and their applications to critical phenomena and to string theory [3]. Classical W-algebras first appeared in the study of integrable systems, namely of generalized KdV hierarchies [4]. More precisely, the W n -algebra arises as the second hamiltonian structure of the n-th KdV hierarchy whose Poisson brackets are defined on the manifold of differential operators of order n ≥ 2. In the simplest case (n = 2), this algebra coincides with the Virasoro algebra, as was first noticed in ref. [5]. This fact exhibits the connection between W n -algebras and conformal field theory and suggests to apply conformal symmetry to the formulation of classical W n -algebras. This was done in ref. [6] (see also references therein) where it was shown that the W n -algebra possesses a 'primary basis' of generators consisting of a projective connection (the Virasoro generator) and n−2 primary fields which transform like k-forms (k = 3, .., n) under the Virasoro flow. Besides W n -algebras there are other W-algebras which are said to be 'infinite' because they contain an infinity of independent generators. These algebras are related to each other by reductions, truncations or contractions [7]. In particular, every W n -algebra can be obtained by reduction from the infinite W differential operators. It was shown in ref. [9] that such a symbol can be parametrized by a projective connection and an infinity of primary fields, thus providing a primary basis for every W (n) KP -algebra (n ≥ 2). W-algebras admit supersymmetric extensions which manifest themselves in the context of superstring or super-Toda field theories. The second hamiltonian structure of N = 1 super W-algebras was constructed in superspace in ref. [10]. Their primary basis was determined in ref. [11] for super W n and generalized to super W (n) KP [12]. In this paper, we are interested in N = 2 super W-algebras. They have been extensively studied in the N = 1 formalism (see [13,14,15,16] and references therein) until their formulation in N = 2 superspace was discovered [17,18]. N = 2 super W n -algebras were shown to admit a primary basis which was constructed in ref. [19]. The aim of this paper is to study the N = 2 supersymmetric pseudodifferential symbols and to apply them to the determination of a primary basis for N = 2 super W KP -algebras in N = 2 superspace. In section 2, we summarize some concepts and tools of N = 2 superconformal symmetry and superconformally covariant operators. These are used in section 3 to study N = 2 supersymmetric pseudodifferential symbols. We pay a particular attention to the so-called Bol symbols (parametrized by a superprojective connection) which are studied systematically. A particular class of them is generalized in section 4 and applied to the formulation of N = 2 super W KP -algebras in a primary basis. 2 N = 2 supersymmetric differential operators For further details concerning the notions summarized in this section, we refer to ref. [19]. Geometric framework N = 2 supersymmetry In order to make N = 2 supersymmetry manifest, all considerations will be carried out on a compact two-dimensional N = 2 supermanifold Σ [20] with local coordinates z ≡ (z, θ,θ) and their complex conjugates (c.c.)z ≡ (z, θ − ,θ − ). Here, z,z are even and θ,θ, θ − ,θ − are odd Grassmann numbers. The tangent space is spanned by the derivatives (∂, D,D) (and their c.c. (∂, D − ,D − )) defined by ∂ = ∂ ∂z , D = ∂ ∂θ + 1 2θ ∂ ,D = ∂ ∂θ + 1 2 θ∂ . Their graded Lie brackets {D,D} = ∂ , D 2 = 0 =D 2 are those of the N = 2 supersymmetry algebra. N = 2 superconformal symmetry Since we will be interested in N = 2 superconformal symmetry, we require the supermanifold Σ to be a super Riemann Surface (SRS). This means that local coordinate systems on Σ are related by superconformal transformations. By definition [21], a superconformal transformation of local coordinates of Σ is a superdiffeomorphism (z;z) −→ (z ′ ;z ′ ) satisfying the following three properties (as well as the c.c. relations): (i) z ′ = z ′ (z) ⇐⇒ D − z ′ = 0 =D − z ′ (ii) Dθ ′ = 0 =Dθ ′ (iii) Dz ′ = 1 2θ ′ Dθ ′ ,Dz ′ = 1 2 θ ′Dθ′ . These relations imply that D andD transform homogeneously, D ′ = e w D ,D ′ = ewD , where e −w ≡ Dθ ′ , Dw = 0 and e −w ≡Dθ ′ ,Dw = 0. Superprojective connection The super Schwarzian derivative S(z ′ , z) associated to a superconformal change of coordinates z −→ z ′ (z) is defined by −S(z ′ , z) = ∂Dθ ′ Dθ ′ − ∂Dθ ′ Dθ ′ + ∂θ ′ Dθ ′ ∂θ ′ Dθ ′ = 2 e − 1 2 (w+w) [D,D] e 1 2 (w+w) . The following study will make an extensive use of a superprojective connection. This is a superfield R ≡ R θθ (z) which is locally superanalytic (i.e. D − R = 0 =D − R) and which transforms under a superconformal transformation of coordinates according to R ′ (z ′ ) = e w+w [R(z) − S(z ′ , z)] . Such a field can be globally defined on compact SRS's of arbitrary genus [22,21]. Superconformally covariant differential operators A superconformal (or primary) field of superconformal weight (p, q) is a function C p,q ∈ C ∞ (Σ) (i.e. the space of supersmooth functions on Σ [20]) which transforms under a superconformal change of local coordinates according to C ′ p,q (z ′ ;z ′ ) = e pw+qw C p,q (z;z) ( p, q ∈ Z/2 , p + q ∈ Z ) .(1) The space of these fields will be denoted by F p,q . Definition 1 A generic superdifferential operator on Σ is locally defined by L = nmax n=0 a n + α n D + β nD + b n [D,D] ∂ n ,(2) where a n , b n and α n , β n are, respectively, even and odd superfields belonging to C ∞ (Σ). Such an operator is called superconformally covariant (or covariant for short) if it maps primary fields of some weight (p, q) to primary fields of some weight (p ′ , q ′ ): L ≡ L p,q : F p,q −→ F p ′ ,q ′ .(3) Note that the correspondence (3) is equivalent to the following transformation law under superconformal changes of local coordinates : (L p,q ) ′ = e p ′ w+q ′w L p,q e −pw−qw .(4) In order to construct such covariant operators, it is convenient to use a superaffine connection. This is a collection of superfields B ≡ B θ ,B ≡Bθ which are locally defined on Σ and which satisfy the following three conditions : they are locally superanalytic, they satisfy the chirality conditions D B = 0 =DB and they transform under a superconformal change of local coordinates according to B ′ (z ′ ) = e w [B(z) + Dw] ,B ′ (z ′ ) = ew B (z) +Dw . Using an affine connection one can introduce supercovariant derivatives ∇ ≡ ∇ p,q = D − qB : F p,q −→ F p+1,q ∇ ≡∇ p,q =D − pB : F p,q −→ F p,q+1 . By construction, these are nilpotent: ∇ 2 = 0 =∇ 2 . The most general covariant operator that is locally defined on Σ is simply obtained by replacing the two fermionic derivatives D andD in expression (2) by supercovariant ones, ∇ and∇: L p,q = nmax n=0 a n + α n ∇ + β n∇ + b n [∇,∇] {∇,∇} n .(5) Bol differential operators The only compact SRS's which admit a globally defined affine connection are those of genus one [22,21]. On the other hand, affine and projective connections are locally related by the super Miura transformation 1 R = (DB) − (D B) − BB .(6) The fact that a projective connection can always be defined globally on a SRS motivates the following definition. Definition 2 A Bol operator is a covariant differential operator on Σ which depends on an unique superfield, namely a projective connection. It follows directly from this definition that a Bol operator is globally defined on the SRS. In order to construct a Bol operator, we require that the operator L p,q of eq. (5) only depends on the affine connections B,B through the combination R = DB −DB − BB given by the Miura transformation. By using a variational argument (one imposes that δL p,q = 0 while varying B andB subject to the condition that R is fixed), one is led to the following result [19] : Theorem 1 For each superconformal weight (p, q) ∈ (Z/2, Z/2) such that −(p + q) ∈ N * , there exists a Bol operator defined on the whole space F p,q . This operator is unique up to a global factor and is of order n = −(p + q). It reads L p,q (R) = q(∇∇) n − p(∇∇) n : F p,q −→ F p+n,q+n .(7) Note that there are other Bol operators which are only defined on appropriate subspaces of F p,q [19]. 3 N = 2 supersymmetric pseudodifferential symbols Basic definitions and relations We aim to extend the analysis of sections 2.2 and 2.3 to the pseudodifferential case. A generic pseudodifferential symbol (or symbol for short) is locally defined on Σ by [17] L = nmax n=−∞ a n + α n D + β nD + b n [D,D] ∂ n ,(8) where we have introduced the inverse ∂ −1 of the usual derivative : ∂ ∂ −1 = ∂ −1 ∂ = 1 .(9) The symbol L can be divided into its differential part (the summation going from n = 0 to n max ) and its integral part (the summation going from n = −∞ to −1) which will be denoted by (L) + and (L) − , respectively. By using the identity [D,D] 2 = ∂ 2 ,(10) one immediately verifies that [D,D] −1 = ∂ −2 [D,D] .(11) From (9) and (11), it then follows that for all α, β ∈ R, α∂ − β[D,D] −1 = 1 α 2 − β 2 α∂ −1 + β[D,D] −1 if α = ± β .(12) Since ∂ = {D,D}, the condition α = ± β reflects the fact that the operators DD andDD are not invertible. Superconformal covariance of a pseudodifferential symbol Since we are now dealing with (pseudodifferential) symbols rather than (differential) operators, we have to generalize definition 1. In fact, the correspondence (3) does not make sense in the present case (because a symbol does not transform a field into another field). However, we can postulate the transformation law (4). Definition 3 A pseudodifferential symbol L p,q , locally defined on Σ by (8), is superconformally covariant (or covariant for short) if it transforms (under a superconformal change of local coordinates) according to (L p,q ) ′ = e p ′ w+q ′w L p,q e −pw−qw ,(13) where p, q, p ′ , q ′ ∈ Z/2 and p + q, p ′ + q ′ ∈ Z . If L p,q is a covariant symbol, then its differential part (L p,q ) + and its integral part (L p,q ) − are separately covariant. In particular, for the differential part, definition 3 reduces to definition 1 and the analysis of sections 2.2 and 2.3 applies. Once again superconformal covariance can be ensured locally by introducing supercovariant derivatives in expression (8) 2 : L p,q = nmax n=−∞ a n + α n ∇ + β n∇ + b n [∇,∇] {∇,∇} n .(14) For later reference, we note that the nilpotency of the supercovariant derivatives allows to obtain the covariant analogon of relations (10)- (12) : [∇,∇] 2 = {∇,∇} 2 [∇,∇] −1 = {∇,∇} −2 [∇,∇] α{∇,∇} − β[∇,∇] −1 = 1 α 2 − β 2 α{∇,∇} −1 + β[∇,∇] −1 if α = ± β . Bol pseudodifferential symbols As in the differential case, we require a covariant pseudodifferential symbol to be globally defined on any compact SRS: Definition 4 A Bol symbol is a superconformally covariant pseudodifferential symbol which depends on an unique superfield, namely a projective connection. Bol symbols can be determined by using the same variational method as the one used for Bol operators. This leads to the following result : Theorem 2 For each superconformal weight (p, q) ∈ (Z/2, Z/2) such that −(p + q) ∈ Z * , there exists a Bol symbol which is unique up to a global factor. The latter is of order n = −(p + q) and reads L p,q (R) = {∇,∇} n−1 (q∇∇ p,q − p∇∇ p,q ) . For n > 0, this expression reduces to the Bol operator (7). The inverse of a Bol symbol (which exists if and only if p = 0 and q = 0) is also a Bol symbol. In fact, it follows from expression (15) that Bol operators and Bol symbols are related by the following inversion property: L −1 p,q = − 1 p q L −q,−p if p = 0 and q = 0.(16) This relation can be used to determine explicitly purely integral Bol symbols in terms of the projective connection R by inverting Bol differential operators. Before discussing the singular cases p = 0 or q = 0, we briefly consider the symmetric case (p = q). Symmetric Bol symbols For p = q, theorem 2 states that L sym n (R) ≡ {∇,∇} n−1 [∇,∇] − n 2 ,− n 2 (n = 0)(17) is a Bol symbol with inverse (L sym n ) −1 = L sym −n . Interestingly enough, these properties can be generalized to the case n = 0. The corresponding symbol as given by eq.(17) reads L sym 0 = ∂ −1 [D,D] so that it is a Bol symbol which coincides with its own inverse according to eq.(10) : (L sym 0 ) 2 = 1. For n > 0, the first symmetric Bol operators have been calculated in [19] and have been shown to appear in the commutation relations of N = 2 super W-algebras (see also equations (31) below). Although they are not invertible (as stated above), they nevertheless are related by a kind of inversion relation, which we will now discuss. We first note that expressions (18) allow us to define the following equivalence classes of symbols, DJ n ≡ D{∇,∇} n modulo DλD DK n ≡D{∇,∇} n moduloDµD (n ∈ Z),(19) where λ and µ are arbitrary pseudodifferential symbols. In the differential case (n ≥ 0), this amounts to restricting the domains of definition of the operators DJ n andDK n to antichiral and chiral superfields, respectively [19]. Obviously the representatives of the equivalence classes (19) are related to the Bol symbols (18) and their interest consists of the fact that they satisfy the two (equivalent) inversion relations (DJ n ) −1 =DK −n−1 (DK n ) −1 = DJ −n−1 .(20) The latter can be used to determine in a simple way the purely pseudodifferential chiral or antichiral Bol symbols by starting from the differential ones. The simplest examples of symbols given by eqs. (19) read DK −1 =D∂ −1 DJ −1 = D∂ −1 DK 0 =D DJ 0 = D DK 1 =D[∂ + R] DJ 1 = D[∂ − R] DK 2 =D[∂ 2 + 3R∂ + (DDR) + 2(DDR) + 2R 2 ](21) and one can explicitly verify that they satisfy the relations (20). For later reference, we note that the leading terms of the generic antichiral Bol symbol read (n ∈ Z) DK n D =D ∂ n + c (n) R∂ n−1 + ... D with c (n) = n(n + 1) 2 .(22) 4 Covariant symbols and their applications to N = 2 super W -algebras The antichiral Bol symbol L anti n (R) given by (18) is a symbol of the generic form DL (n) D =D ∂ n + ∞ k=1 a (n) k ∂ n−k D (n ∈ Z) .(23) It is covariant and has the property that it only depends on a projective connection R. This suggests that, more generally, by requiring a generic symbol of the form (23) to be covariant and globally defined on any SRS, one should obtain a reparametrization of this symbol in terms of a projective connection and some superconformal fields of appropriate weight. This has been worked out in ref. [19] for differential operators and the extension to the pseudodifferential case is the following. Given a superconformal field W k of weight (k, k) (with k ∈ N * ), the covariant symbol DM (23) to span the phase space of superfields, we can use symbols of the form (24) to obtain a parametrization which relies on a projective connection R and of superconformal fields. (n) W k D is defined for n ∈ Z bȳ DM (n) W k D =∇ ∞ l=0 A (n) k,l [(∇∇) l W k ] + B Theorem 3 The most general symbol of the form (23) which is covariant and globally defined on any SRS is parametrized by a projective connection R and an infinity of superconformal fields W k of weight (k, k) -one for each value of k ∈ {2, .., ∞} -according toD K (n) D =D K n + ∞ k=2 M (n) W k D (n ∈ Z) .(26) HereDK n D is the antichiral Bol symbol (18) andDM (n) W k D is given by eq. (24). The superfields R and W 2 , W 3 , ... are related to the former by invertible differential polynomials which can be explicitly determined by identifying the expressions (23) and (26) : a (n) 1 = c (n) R (27) a (n) i = i k=1 A (n) k,i−k (DD) i−k W k + B (n) k,i−k (DD) i−k W k + nonlinear terms. (28) The factor c (n) was given in eq. (22) and we have used the notation W 1 ≡ a (n) 1 in the last relation. Thus we have achieved a superconformal (or primary) parametrization of the symbol (23), reflecting its superconformal covariance property. Of course, by starting from the chiral rather than the anti-chiral Bol symbols, one can repeat the whole analysis in order to achieve an analogous parametrization for the symbols of the form DJ (n)D . N = 2 super W (n) KP -algebras We denote by M n the manifold of covariant symbols of the form (23) and, from now on, we restrict our study to the case n ≥ 1 for which the Bol symbolDK n D is purely differential and depends on the projective connection R. Following ref. [18], one introduces the residue and the trace of a symbol L ∈ M n by res L = a J L : T * L (M n ) → T L (M n ) U → (LU) + L − L (UL) + + L Φ U (L) +Φ U (L) L(30) determines a hamiltonian structure on the manifold M n [18] 4 . This hamiltonian map is an N = 2 extension of the usual Adler map [23,4]; it defines a Poisson algebra, namely the N = 2 super W (n+1) KP -algebra. The superfields a (n) 1 and W i (i ≥ 2) constitute a primary basis for this superalgebra. In this basis, the Poisson brackets take a particular form which reflects the superconformal covariance property of these superfields : a (n) 1 (z 2 ), a (n) 1 (z 1 ) = c (n) L sym 2 (R) δ (3) (z 2 , z 1 ) a (n) 1 (z 2 ), W k (z 1 ) = k V∂ − (DV)D − (DV)D + (∂V) δ (3) (z 2 , z 1 ) (31) {W k (z 2 ), W l (z 1 )} = c (n) kl L sym k+l (R) + ... δ (3) (z 2 , z 1 ) The first relation represents the N = 2 Virasoro superalgebra. The differential operator L sym 2 (R) is the symmetric Bol operator given by (17) and explicitly reads L sym N = 2 super W n -algebras Interestingly enough, the primary parametrization (26) allows us to split automatically the differential and integral parts of the symbolDK (n) D. In fact, if k > n, the symbol (24) is purely integral:DM (n) W k D = (DM (n) W k D) − . On the contrary, if k ≤ n, it is a differential operator (the summation over l is going from 0 to n − k since the coefficients (25) vanish for larger values of l):DM (n) W k D = (DM (n) W k D) + . Thus a symbol L ∈ M n can be easily divided into its differential and integral parts : L + =D K n + n k=2 M (n) W k D , L − =D ∞ k=n+1 M (n) W k D .(32) According to eqs.(30), L − = 0 implies J L (U) − = 0 for all U. Thus, one can impose the constraint L − = 0. Hence the superfields W k with k > n which parametrize L − generate an ideal I n of W (n+1) KP [7]. This ideal is centerless so that the central charges c (n) kl in (31) vanish if both k > n and l > n. Moreover, the constraint L − = 0 allows for a hamiltonian reduction : the quotient of W (n+1) KP by its ideal I n is isomorphic to the N = 2 super W n+1 -algebra generated by the differential operator L + . According to eq.(26), the latter is parametrized by the n superfields R and W 2 , .., W n which thus span a primary basis of the N = 2 super W n+1algebra as shown in ref. [19]. Conclusion By studying N = 2 covariant symbols, we have achieved a classification of the Bol symbols which are characterized by their dependence on a superprojective connection. Among them, the antichiral Bol symbolsDK n D are of particular interest since they are a special case of symbolsDL (n) D of the form (23) whose manifold can be endowed with a hamiltonian structure leading to the N = 2 super W (n+1) KP -algebra. We have parametrized such symbols by using a superprojective connection and an infinity of primary superfields W k of weights k = 2, .., ∞. This provides us with a primary basis of generators for the N = 2 super W (n+1) KP -algebra and, by reduction, also one for the super W n+1 -algebra. If expressed in this basis, the Poisson brackets take a form which reflects their N = 2 superconformal symmetry. This allowed us to study the central terms: the Virasoro central charge c (n) of W (n+1) KP can be explicitly determined. In principle, the other charges c (n) kl could also be computed by inverting relation (28) and proceeding along the lines of ref. [6]. We conjecture that, due to the choice of the primary basis, these central charges turn out to be diagonal (c ( Anti-)chiral Bol symbols Among the Bol symbols given by expression(15), the noninvertible ones correspond to the so-called chiral (p = 0) and anti-chiral (q = 0) solutions : L chir n (R) ≡ ∇{∇,∇} n−1∇ 0,−n = D{∇,∇} n−1D (18) L anti n (R) ≡∇{∇,∇} n−1 ∇ −n,0 =D{∇,∇} n−1 D . = DJ n−1D L anti n =DK n−1 D l [(∇∇) l W k ] {∇,∇} n−k−l ∇ −n−1,0 . (24)It depends linearly on W k and it depends on the projective connection R given by the Miura transformation (6) provided the coefficients are chosen to ... of expression 3 The binomial coefficients n p are extended to n ∈ Z by n p = n(n − 1)...(n − p + 1) p! if p ∈ N * and n p = 1 if p = 0. , Tr L = d 3 z res L (29) where d 3 z = dzdθdθ. This trace can be used to define the pairing of two symbols by A, B = Tr(AB). Let T L (M n ) and T * L (M n ) denote the tangent and cotangent spaces of M n at the point L and Φ U (L) andΦ U (L) denote, respectively, the chiral and antichiral parts of the trace Tr(res[L, U]). Then the map 2 ( 2R) = ∂[D,D] + R∂ − (DR)D − (DR)D + (∂R) ; hence the Virasoro central term reads 1 2 n(n + 1) ∂[D,D]δ (3) (z 2 , z 1 ). The second relation reflects the fact that W k is a superconformal field of weight (k, k). In the last relation, we have only written the terms which do not depend on the primary fields W i (i ≥ 2). Since the leading term of the operator L sym k+l (R) = ∂ k+l−1 [D,D] + ... does not depends on any field, the central term reads c (n) kl ∂ k+l−1 [D,D]δ (3) (z 2 , z 1 ). δ kl ) as in the nonsupersymmetric case. In order to avoid ambiguities, we adopt from now on the following notation for the action of derivatives on a field C: (∂C), (DC), (DC), ([D,D]C),... denote derivatives of the field C while a derivative acts operatorially otherwise, e.g. ∂C = (∂C) + C∂. Note that {∇,∇} locally reads ∇∇ p,q +∇∇ p,q = ∂ − BD −BD − p(DB) − q(DB) + (p − q)BB and that it is invertible because its leading term ∂ is invertible. The map (30) has been determined in ref.[18] for the case of chiral operators of the form DLD and is easily transposed to the antichiral case. As pointed out in this reference, there exists a second quadratic hamiltonian map in the N = 2 supersymmetric case. Acknowledgements The author would like to thank François Gieres for his careful reading of the manuscript and his suggestions. . P Bouwknegt, K Schoutens, Phys. Rep. 223183P. Bouwknegt and K. Schoutens. Phys. Rep., 223:183, 1993. S V Ketov, Conformal Field Theory. World ScientificS.V. Ketov. Conformal Field Theory. (World Scientific, 1995). . P Di Francesco, P Ginsparg, J Zinn-Justin, Phys. Rep. 2541P.Di Francesco, P. Ginsparg and J. Zinn-Justin. Phys. Rep., 254:1, 1995. Soliton Equations and Hamiltonian Systems. L A Dickey, Advanced Series in Mathematical Physics. 12World ScientificL.A. Dickey. Soliton Equations and Hamiltonian Systems, volume 12 of Advanced Series in Mathematical Physics. (World Scientific, 1991). . J.-L Gervais, A Neveu, Nucl. Phys. 209125J.-L. Gervais and A. Neveu. Nucl. Phys., B209:125, 1982. . P Di Francesco, C Itzykson, J.-B Zuber, Commun. Math. Phys. 140543P. Di Francesco, C. Itzykson and J.-B. Zuber. Commun. Math. Phys., 140:543, 1991. . J M Figueroa-O&apos;farrill, J Mas, E Ramos, Phys. Lett. 29941J.M. Figueroa-O'Farrill, J. Mas and E. Ramos. Phys. Lett., B299:41, 1993. . J M Figueroa-O&apos;farrill, J Mas, E Ramos, Commun. Math. Phys. 15817J.M. Figueroa-O'Farrill, J. Mas and E. Ramos. Commun. Math. Phys., 158:17, 1993. . W.-J Huang, J. Math. Phys. 35993W.-J. Huang. J. Math. Phys., 35:993, 1994. . J M Figueroa-O&apos;farrill, E Ramos, Phys. Lett. 262265J.M. Figueroa-O'Farrill and E. Ramos. Phys. Lett., B262:265, 1991. . F Gieres, S Theisen, J.Math.Phys. 345964F.Gieres and S.Theisen. J.Math.Phys., 34:5964, 1993. . W.-J Huang, J. Math. Phys. 352570W.-J. Huang. J. Math. Phys., 35:2570, 1994. . J M Figueroa-O&apos;farrill, E Ramos, Nucl. Phys. 368361J.M. Figueroa-O'Farrill and E. Ramos. Nucl. Phys., B368:361, 1992. . K Huitu, D Nemeschansky, Mod. Phys. Lett. 63179K. Huitu and D. Nemeschansky. Mod. Phys. Lett., A6:3179, 1991. . F Gieres, S Theisen, Int. J. Mod. Phys. 9383F. Gieres and S. Theisen. Int. J. Mod. Phys., A9:383, 1994. . W.-J Huang, J C Shaw, H C Yen, J. Phys. 28165W.-J. Huang, J.C. Shaw and H.C. Yen. J. Phys., A28:165, 1995. . Z. Popowicz. Phys.Lett. 319478Z. Popowicz. Phys.Lett., B319:478, 1993. . F Delduc, L Gallot, Commun. Math. Phys. 190395F. Delduc and L. Gallot. Commun. Math. Phys., 190:395, 1997. . F Gieres, S Gourmelen, J. Math. Phys. 393453F. Gieres and S. Gourmelen. J. Math. Phys., 39:3453, 1998. . B Dewitt, Supermanifolds, Cambridge University Press2nd editionB. DeWitt . Supermanifolds. 2nd edition (Cambridge University Press, 1992). . F Delduc, F Gieres, S Gourmelen, Class. Quantum Grav. 141623F. Delduc, F. Gieres and S. Gourmelen. Class. Quantum Grav., 14:1623, 1997. . F Gieres, Int. J. Mod. Phys. 81F. Gieres. Int. J. Mod. Phys., A8:1, 1993. . M Adler, Invent. Math. 50403M. Adler. Invent. Math., 50:403, 1979.
[]
[ "SALSA: A Sequential Alternating Least Squares Approximation Method For MIMO Channel Estimation", "SALSA: A Sequential Alternating Least Squares Approximation Method For MIMO Channel Estimation" ]
[ "Sepideh Gherekhloo ", "Khaled Ardah ", "Martin Haardt " ]
[]
[]
In this paper, we consider the channel estimation problem in sub-6 GHz uplink wideband MIMO-OFDM communication systems, where a user equipment with a fully-digital beamforming structure is communicating with a base station having a hybrid analog-digital beamforming structure. A novel channel estimation method called Sequential Alternating Least Squares Approximation (SALSA) is proposed by exploiting a hidden tensor structure in the uplink measurement matrix. Specifically, by showing that any MIMO channel matrix can be approximately decomposed into a summation of R factor matrices having a Kronecker structure, the uplink measurement matrix can be reshaped into a 3-way tensor admitting a Tucker decomposition. Exploiting the tensor structure, the MIMO channel matrix is estimated sequentially using an alternating least squares method. Detailed simulation results are provided showing the effectiveness of the proposed SALSA method as compared to the classical least squares method.
null
[ "https://export.arxiv.org/pdf/2304.06643v1.pdf" ]
258,107,940
2304.06643
deb038de1fa6427bfaf34d2ff113b1c1d1c63367
SALSA: A Sequential Alternating Least Squares Approximation Method For MIMO Channel Estimation Sepideh Gherekhloo Khaled Ardah Martin Haardt SALSA: A Sequential Alternating Least Squares Approximation Method For MIMO Channel Estimation 1Index Terms-Channel estimationmassive MIMOTucker tensor decompositionalternating least squares In this paper, we consider the channel estimation problem in sub-6 GHz uplink wideband MIMO-OFDM communication systems, where a user equipment with a fully-digital beamforming structure is communicating with a base station having a hybrid analog-digital beamforming structure. A novel channel estimation method called Sequential Alternating Least Squares Approximation (SALSA) is proposed by exploiting a hidden tensor structure in the uplink measurement matrix. Specifically, by showing that any MIMO channel matrix can be approximately decomposed into a summation of R factor matrices having a Kronecker structure, the uplink measurement matrix can be reshaped into a 3-way tensor admitting a Tucker decomposition. Exploiting the tensor structure, the MIMO channel matrix is estimated sequentially using an alternating least squares method. Detailed simulation results are provided showing the effectiveness of the proposed SALSA method as compared to the classical least squares method. I. INTRODUCTION M Assive MIMO [1] is one of the key enabling technologies of 5G-NR mobile communications [2] and it shall remain relevant in future 6G wireless systems. By employing a large number of antennas at the base station (BS) relative to the number of scheduled users, massive MIMO systems increase the data throughput relative to legacy systems by providing a large beamforming gain and an improved multi-user interference suppression owing to its high spatial resolution [3]. Recently, massive MIMO communications have received a special attention with the introduction of millimeterwave (mm-wave)-based wireless communications [4], since the use of massive MIMO in such systems becomes a requirement rather than an option to compensate the high pathloss encountered in the wireless communication systems at higher frequencies. However, it is well-known that the promised theoretical massive MIMO gains heavily rely on the availability of accurate channel state information (CSI) and the considered beamforming structure. On the one hand, classical fully-digital (FD) beamforming structures, which generally provide the maximum beamforming gain, require a dedicated radio frequency (RF) chain for each antenna element. This increases not only the implementation cost and complexity of massive MIMO systems, but also the circuit energy consumption. A promising solution to these issues relies on the recently introduced hybrid analog-digital (HAD) beamforming structures [5], [6], [4], [7], [8], which use a combination of analog beamforming in the RF domain and digital beamforming in the baseband domain to reduce the number of RF chains as compared to FD beamforming structures, e.g., the number of RF chains can be as small as the number of transmitted data streams. On the other hand, in 5G-NR systems, for example, the BS estimates the CSI from uplink sounding reference signals (SRS) emitted by the user terminals (UEs). In mm-wave systems, the CSI estimation problem is often transformed into a multi-dimensional directionof-arrival (DoA) estimation problem [9], [10], [11], thanks to the S. Gherekhloo low-rank (sparse) nature of mm-wave MIMO channels [4], where several techniques, e.g., compressed sensing [9], [10] and ESPRIT [11] can be readily employed to obtain a high CSI estimation accuracy while requiring a small number of training overhead. Differently, in sub-6 GHz-based systems, the MIMO channels often experience a high-rank nature, which makes most, if not all, mm-wave-based MIMO channel estimation methods unfeasible. To this end, classical channel estimation techniques, e.g., least-squares (LS) and minimum mean squared-error (MMSE) methods [12], [13] can be used to estimate sub-6 GHz-based MIMO channels. However, these methods were originally developed for single-antenna and small-scale MIMO systems and suffer from a severe performance degradation in difficult scenarios, e.g., with small number of training snapshots and/or a low signal-to-noise ratio (SNR). Since sub-6 GHz massive MIMO communications are, and will remain, an integral part of current and future wireless communication systems, more efficient channel estimation techniques than the classical methods are required. In this paper, we consider the channel estimation problem in sub-6 GHz uplink wideband MIMO-OFDM communication systems, where a single-user with a FD beamforming structure communicates with a BS having a HAD beamforming structure. By exploiting a hidden tensor structure in the uplink measurement matrix, we propose a novel channel estimation method called Sequential Alternating Least Squares Approximation (SALSA). Specifically, by showing that any MIMO channel matrix can be approximately decomposed into a summation of R factor matrices having a Kronecker structure, the uplink measurement matrix can be reshaped into a 3-way tensor admitting a Tucker decomposition [14]. Exploiting such a tensor representation, the MIMO channel matrix can be estimated sequentially using the classical ALS method [15]. Detailed simulation results are provided showing that the SALSA-based approach can achieve a more accurate channel estimation in difficult scenarios as compared to the classical LS-based approach. Notation: The transpose, the complex conjugate, the conjugate transpose (Hermitian), and the Kronecker product are denoted as A T , A * , A H , and ⊗, respectively. Moreover, IN is the N × N identity matrix, vec{A} forms a vector by staking the columns of A over each other, and the n-mode product of a tensor A ∈ C I 1 ×I 2 ×...,×I N with a matrix B ∈ C J×In is denoted as A ×n B. II. SYSTEM MODEL We consider an uplink single-user wideband MIMO-OFDM communication system, as depicted in Fig. 1, where a UE with NUE antennas is communicating with a BS with NBS antennas over NSC subcarriers. The UE has a FD beamforming structure while the BS has a HAD beamforming structure with NRF ≤ NBS radio-frequency (RF) chains. We assume that the NBS antennas and the NRF RF chains are divided equally 1 into NG ≥ 1 groups, where each group has NBS = N BS N G antennas andNRF = N RF N G RF chains (i.e., NBS = NG·NBS and NRF = NG ·NRF) and the RF chains in every group are connected with every antenna element in the same group. Moreover, we assume a block-fading channel model as shown in Fig. 2, where the channel coherence-time TC is divided into TBSTUE transmission time intervals (TTIs), i.e., every block has TUE snapshots. LetĀi ∈ C N BS ×N RF denote the analog combining matrix at the ith block at the BS. Then, according to our above assumptions,Āi has a block-diagonal structure given as 2 Ai = 1 N BS ·    Ā i,1 . . . 0 . . . . . . . . . 0 . . .Āi,N G     ∈ C N BS ×N RF ,(1) whereĀi,g ∈ CN BS ×N RF is the gth block-matrix with constant modulus entries, i.e., [Āi,g] [r,c] = 1, where [Āi,g] [r,c] is the (r, c)th entry ofĀi,g. The received signal by the BS in the (i, j)th TTI over the kth subcarrier, with i ∈ {1, . . . , TBS}, j ∈ {1, . . . , TUE}, k ∈ {1, . . . , NSC}, can be expressed as y k,i,j =Ā H i H k f k,j s k,j +Ā H izk,i,j ∈ C N RF ,(2) where f k,j ∈ C N UE is the (k, j)th precoding vector, s k,j ∈ C is the corresponding training symbol,zi,j ∈ C N BS is the BS additive white Gaussian noise with zero mean and variance σ 2 n , and H k ∈ C N BS ×N UE is the kth subcarrier frequency-domain MIMO channel matrix. Initially, we collect the measurement vectors {ȳ k,i,j } T UE j=1 next to each other asȲ k,i = [ȳ k,i,1 , . . . ,ȳ k,i,T UE ], which can be written as Y k,i =Ā H i H k F k +Ā H iZk,i ∈ C N RF ×T UE ,(3) where F k = [f k,1 s k,1 , . . . , f k,T UE s k,T UE ] ∈ C N UE ×T UE andZ k,i = [z k,i,1 , . . . ,z k,i,T UE ]. We assume that F k , ∀k, are designed with orthonormal rows, i.e., F k F H k = IN UE , ∀k, and TUE ≥ NUE. After applying the right-filtering to (3) we obtain Y k,i =Ȳ k,i F H =Ā H i H k + Z k,i ∈ C N RF ×N UE ,(4) where Z k,i =Ā H iZk,i F H k . Next, we collect the measurement matri- ces {Y k,i } T BS i=1 on the top of each other as Y k = Y T k,1 , . . . , Y T k,T BS T , which can be written as Y k = AH k + Z k ∈ C L×N UE ,(5) where L = TBSNRF, A = Ā 1, . . . ,ĀT BS H ∈ C L×N BS , and Z k = Z T k,1 , . . . , Z T k,T BS T . After that, we collect the measurement matrices {Y k } N SC k=1 next to each other as Y = [Y1, . . . , YN SC ] , which can be written as Y = AH + Z ∈ C L×N UE N SC ,(6) where Z = Z1, . . . , ZN SC and H = H1, . . . , HN SC ∈ C N BS ×N UE N SC is the total MIMO channel matrix. The baseline LS-based channel estimation method: Given the measurement matrix in (6), a least-squares (LS)-based method can be used to obtain an estimate of the total MIMO channel matrix aŝ HLS = [A] + Y = Ĥ 1, . . . ,ĤN SC ∈ C N BS ×N UE N SC ,(7) where [·] + denotes the Moore-Penrose pseudo-inverse. Note that, due to the left filtering, the LS-based method requires that L ≥ NBS, i.e., TBS ≥ N BS N RF to provide an accurate channel estimate. 2 Note that if N G = 1, the above analog structure coincides with the known fully-connected analog structure [5], where every RF chain is connected to every antenna element. On the other hand, if N G = N RF , the above analog structure coincides with the known partially-connected analog structure [5], where every RF chain is connected to a unique subset of antenna elements. III. THE PROPOSED SALSA METHOD To obtain a more accurate channel estimate while reducing the training overhead, we propose in this section a novel channel estimation method called SALSA, which is derived by exploiting a hidden tensor structure in the measurement matrix in (6). To show this, we first recall the following propositions from [16], [17], [18]. Proposition 1: Let X be a matrix given as X = X1 ⊗ X2 =    X1,1 . . . X1,J 1 . . . XI 1 ,1 . . . XI 1 ,J 1    ∈ C I×J ,(8) where X1 ∈ C I 1 ×J 1 , X2 ∈ C I 2 ×J 2 , I = I1I2, J = J1J2, and Xn,m = [X1] [n,m] X2 is the (n, m)th block-matrix of X. Let K ∈ C I 1 J 1 ×I 2 J 2 be a rank-one matrix given as K =              vec{X1,1} T . . . vec{XI 1 ,1} T . . . vec{X1,J 1 } T . . . vec{XI 1 ,J 1 } T              = vec{X1}vec{X2} T ,(9) with the rank-one truncated-SVD given as K = σuv H , where u ∈ C I 1 J 1 and v ∈ C I 2 J 2 are the left and right singular vectors of K, respectively, and σ is the associated singular value. Then, the optimal solution to minimize X 1 ,X 2 X − X1 ⊗ X2 2 F(10) can be obtained as X1 = reshape{ √ σu, I1, J1}(11) X2 = reshape{ √ σv * , I2, J2}.(12) Proof: Please refer to [17] for more details. Get Xr = X − r−1 r =1 X 1,r ⊗ X 2,r 5: Given Xr, get X1,r and X2,r using (11) and (12), respectively 6: end for 7: Output:X = R r=1 X1,r ⊗ X2,r ∈ C I×J Cr ⊗ Br 2 F . Here, the total MIMO channel matrix H = H 1 , . . . , H N SC ∈ C I×J is generated following the 3GPP CDL channel model [19], [20] with the main system parameters outlined in Table I. Please refer to Section IV for more details. Proposition 2: For any given I × J matrix X, it can be approximately written as a summation of R ≥ 1 factor matrices as X = R r=1 Xr = R r=1 X1,r ⊗ X2,r,(13) where Xr = X1,r ⊗ X2,r, X1,r ∈ C I 1 ×J 1 , and X2,r ∈ C I 2 ×J 2 , I = I1I2, and J = J1J2. Proof: The proof follows directly by applying Proposition 1 sequentially [18]. The corresponding Proposition is summarized in Algorithm 1. Let I = NBS and J = NUENSC. Then, from Proposition 2, the total frequency-domain MIMO channel matrix H ∈ C I×J in (6) can be approximately written as H ≈ R r=1 Cr ⊗ Br ∈ C I×J ,(14) where Br ∈ C I 1 ×J 1 , Cr ∈ C I 2 ×J 2 , I = I1I2, and J = J1J2. As shown in Fig. 3, the approximation becomes tighter as the number of channel factor matrices R increases. More importantly, we can see that in case of full rank channels, the optimal value of R, denoted in the figure by Ropt, is dependent on the division scenario of I and J, where Ropt ≈ min{I1J1, I2J2}. In other words, reducing the dimension of one of the channel factor matrices, i.e., Br ∈ C I 1 ×J 1 or Cr ∈ C I 2 ×J 2 , reduces the value of Ropt. Let L = TBSNRF. Then, by substituting (14) into (6), and assuming R is sufficiently large, we can write Y = A R r=1 Cr ⊗ Br + Z = R r=1 A(Cr ⊗ Br) + Z = R r=1 Yr + Z ∈ C L×J ,(15) where Yr = A(Cr ⊗ Br) ∈ C L×J . From (15), we note that Yr can be seen as the 1-mode unfolding of a 3-way Tucker tensor given as [14] Y r = S ×1 A ×2 B T r ×3 C T r ∈ C L×J 1 ×J 2 ,(16) where S ∈ Z I×I 1 ×I 2 is the core-tensor with the 1-mode unfolding given as [S] (1) def = II . The -mode unfolding of Y r , = {1, 2, 3}, can be expressed as [Y r ] (1) = A[S] (1) (Cr ⊗ Br) ∈ C L×J ,(17)[Y r ] (2) = B T r [S] (2) (Cr ⊗ A T ) ∈ C J 1 ×LJ 2 ,(18)[Y r ] (3) = C T r [S] (3) (Br ⊗ A T ) ∈ C J 2 ×LJ 1 .(19) From (16), the 3-way Tucker tensor form of (15) can be expressed as Y = R r=1 Y r + Z ∈ C L×J 1 ×J 2 ,(20) where Z is the 3-way tensor representation of the noise matrix Z. This latter formulation suggests that the factor matrices {Br, Cr} R r=1 can be estimated sequentially as follows. Let Y r be the tensor obtained at the rth sequential step as Y r = Y − r−1 r =1 Y r ∈ C L×J 1 ×J 2 .(21) Then, by exploiting the 2-mode and the 3-mode unfoldings, the rth factor matrices Br and Cr can be obtained using, e.g., the ALS method [15], where one factor matrix is assumed to be fixed when solving for the other. Specifically, Br and Cr can be obtained as B T r = [Y r ] (2) Ψ2 + = [Y r ] (2) Ψ H 2 [Ψ2Ψ H 2 ] −1 (22) C T r = [Y r ] (3) Ψ3 + = [Y r ] (3) Ψ H 3 [Ψ3Ψ H 3 ] −1 ,(23) where Ψ2 and Ψ3 are given as Ψ2 = [S] (2) (Cr ⊗ A T ) ∈ C I 1 ×LJ 2 (24) Ψ3 = [S] (3) (Br ⊗ A T ) ∈ C I 2 ×LJ 1 .(25) Algorithm 2 summarizes the proposed SALSA method for estimating the total MIMO channel matrix H ∈ C I×J , which is guaranteed to converge monotonically to, at least, a local optimum solution [15]. Get Y r = Y − r−1 r =1Ŷ r 6: Initialize C (0) r ∈ C I 2 ×J 2 , e.g., randomly 7: for n = 1 to Nmax-iter do GetŶ r = S ×1 A ×2B T r ×3Ĉ T r , go back to Step (5) 13: end for 14: Output:ĤSALSA = R r=1Ĉ r ⊗Br ∈ C I×J Note that, due to the right filtering, the SALSA method in Algorithm 2 requires that (C1) I1 ≤ LJ2 and (C2) I2 ≤ LJ1, i.e., TBS ≥ min I 1 N RF J 2 , I 2 N RF J 1 to provide an accurate channel estimation. Therefore, under practical settings, the SALSA method in Algorithm 2 requires less training overhead than the LS method in (7). On the other hand, assuming that the complexity of calculating the Moore-Penrose pseudo-inverse of an n×m matrix is on the order of O(min{n, m} 3 ), then the complexity of the LS method in (7) is on the order of O(min{L, J}) 3 , while for the SALSA method in Algorithm 2 the complexity is on the order of O(R ·Nmax-iter ·I 3 1 ·I 3 2 ), assuming that the (C1) and (C2) conditions are satisfied. IV. SIMULATION RESULTS We adopt the 3GPP clustered delay line (CDL) channel model described in TR 38.901 [19], where a step-by-step tutorial of it along the MATLAB scripts for channel generation is presented in [20]. Specifically, in our simulation, we first generate a time-domain channel tensor H ∈ C N BS ×N UE ×Ntaps , where Ntaps represents the number of time-domain channel taps calculated according to [20,Eqn. (64)] and using the system parameters shown in Table I. Then, we perform a NSC-point FFT operation along the third dimension for each receive-transmit antenna pair to obtain the frequency-domain channel tensor H ∈ C N BS ×N UE ×N SC , where the kth slice matrix H k = H [:,:,k] ∈ C N BS ×N UE represents the the kth subcarrier frequency-domain MIMO channel matrix. We show the simulation results in terms of the normalized mean-square-error (NMSE) that is defined as NMSE Table II. We have simulated the SALSA algorithm using all the 49 possible scenarios. In Fig. 4, we show the NMSE versus SNR results for some selected I and J division scenarios. The other scenarios are not shown, due to space limitations, but we note that their NMSE performance are inferior compared to the shown scenarios. From Fig. 4, when TBS = 12, i.e., L = TBSNRF = 48 < NBS, the analog training matrix A ∈ C L×N BS , i.e., the 1st factor matrix of the measurement tensor in (16), is left non-invertible, i.e., [A] + A = I. Therefore, the LS-based method has a very bad channel estimation accuracy NMSE. On the other hand, we can see that the best NMSE of SALSA method is achieved when I1 = 8, I2 = 8, J1 = 64, and J2 = 1, i.e., when Br ∈ C 8×64 and Cr ∈ C 8×1 , ∀r. The main reason is that by dividing I = 64 equally between I1 and I2, i.e., I1 = I2 = 8, SALSA reduces the impact of the non-invertibility of A by distributing it between the 2nd (i.e., Br) and the 3rd (i.e., Cr) factor matrices of the measurement tensor, which leads to a better channel estimation accuracy. On the other hand, by setting J1 = 64 and J2 = 1, the required number of channel factor matrices R reduces as compared to the other division scenario, as we have illustrated above in Fig. 3. = E{ H − HX 2 F }/E H 2 F }, where X ∈ {LS, SALSA}. The signal-to-noise ratio (SNR) is defined as SNR = E{ Y − Z 2 F }/E{ Z 2 F }. Differently, when TBS = 16, i.e., L = NBS, the analog training matrix A is left invertible, i.e., [A] + A = I. Therefore, the LS-based method has an accurate channel estimation accuracy. For SALSA method, on the other hand, we can see that when I1 = 8 and I2 = 8, the estimation accuracy of SALSA improves as we increase J1 and decrease J2, where the best result is obtained when we have J1 = 64 and J2 = 1, i.e., similar to the case above when TBS = 12. Nonetheless, we can see that the SALSA method can obtain a more accurate channel estimation, compered to the LS-based method, by setting I1 = 1, I2 = 64, J1 = 64, and J2 = 1, i.e., Br ∈ C 1×64 and Cr ∈ C 64×1 (or, not shown in the figure, by setting I1 = 64, I2 = 1, J1 = 1, and J2 = 64, i.e., Br ∈ C 64×1 and Cr ∈ C 1×64 ). In the both these scenarios, the channel matrix H ∈ C 64×64 in (14) is decomposed into a summation of R factor matrices Br ⊗ Cr ∈ C 64×64 , each having a rank-one, i.e., rank{Br ⊗Cr} = 1, ∀r, which leads to a better channel estimation accuracy. In Figs. 5 and 6 we show NMSE versus SNR simulation results with varying the number of channel training overhead, i.e., TBS and the number of channel factor matrices, i.e., R, respectively. From Fig. 5, we can see that the channel estimation accuracy of both methods, i.e., LS-based and SALSA improves as TBS increases. However, SALSA significantly outperforms LS-based with all TBS < 16, i.e., L < NBS scenarios, wherein the analog training matrix A ∈ C L×N BS is left non-invertible. On the other hand, we can see from Fig. 6 that the SALSA channel estimation accuracy increases with the increasing R, in the high SNR regime, while it decreases with the increasing R, in the low SNR regime. The main reason is that, in the high SNR regime, the noise impact is minimal and by increasing R, the channel estimation accuracy increases, as we have illustrated above in Fig. 3. On the other hand, in the low SNR regime, the channel measurement tensor is noise-limited and, therefore, the impact of noise increases by increasing R, i.e., after a certain R, the estimated channel factor matrices are very noisy that decreases the overall estimation accuracy. Clearly, for every SNR regime/level, there is an optimal R value, wherein the channel estimation accuracy is maximized, which we leave for a follow up future work. V. CONCLUSION In this paper, we have proposed a novel channel estimation method for MIMO-OFDM sub-6 GHz communication systems called SALSA. We have shown that an accurate channel estimation can be obtained with a small training overhead by exploiting a hidden tensor structure in the received measurement matrix, which estimates the channel matrix sequentially using an ALS-based method. Our results show that the SALSA method outperforms the conventional LS-based method, especially in the low training overhead, which makes it more appealing for practical implementations. Fig. 1 .Fig. 2 . 12The considered uplink MIMO-OFDM communication system. The channel coherence time T C division. Input: A matrix X ∈ C I×J 2: Select R, I1, J1, I2, J2 such that I = I1I2 and J = J1J2 3: for r = 1 to R do 4: Fig. 3 . 3MSE vs. the number of channel factor matrices R assuming N BS = 64, N UE = 4, and N SC = 16, where MSE = H − R r=1 Algorithm 2 2SALSA For MIMO-OFDM Channel Estimation 1: Input: Measurement matrix Y ∈ C L×J as in (6) 2: Select R ≥ 1, Nmax-iter ≥ 1, I1, I2, J1, and J2 such that I = I1I2 = NBS and J = J1J2 = NUENSC 3: Obtain the 3-way Tucker tensor Y in (20) from Y 4: for r = 1 to R do 5: Fig. 4 . 4NMSE vs. SNR for different I and J division scenarios. In all simulation scenarios, we set NBS = 64, NUE = 4, NSC = 16, TUE = NUE, NRF = 4, NG = 2, and assume a random generation of the analog decoding matrix A ∈ C T BS N RF ×N BS , where every nonzero entry is obtained as a = 1/ N BS · e jφ , where φ ∈ [0, 2π]. Initially, we show simulation results investigating the best division scenario of I and J with the constraints of I = I1I2, J = J1J2, I d ≥ 1, J d ≥ 1, and I d , J d are Natural numbers, where d ∈ {1, 2}. Recall that I = NBS and J = NUENSC. Therefore, we have I = J = 64 and the candidate numbers of I d and J d are 1, 2, 4, 8, 16, 32, and 64. Therefore, we have in total 49 different division scenarios as illustrated in Fig. 5 .Fig. 6 . 56NMSE vs. SNR with varying T BS . NMSE vs. SNR with varying R. and M. Haardt are with the Communications Research Laboratory (CRL), TU Ilmenau, Ilmenau, Germany (e-mail: {sepideh.gherekhloo, martin.haardt}@tu-ilmenau.de). K. Ardah is with Lenovo (Deutschland) GmbH (e-mail: [email protected]). TABLE I ISYSTEM PARAMETERS Parameter Value Scenario UMi Cell radius 100 m BS (UE) height 10 (1.5) m Carrier frequency fc 4 GHz Sampling frequency fs 30.72 MSamples/s No. of subcarriers N SC 16 No. of antennas at BS N BS 64 (8 × 8) No. of antennas at UE N UE 4 (2 × 2) Polarization Single TABLE II DIVISION IISCENARIOS OF I 1 , I 2 , J 1 , AND J 2Scenario No. I 1 and I 2 values J 1 and J 2 values Scenario 1 [I 1 , I 2 ] = [64, 1] [J 1 , J 2 ] = [64, 1] . . . Scenario 7 [J 1 , J 2 ] = [1, 64] . . . . . . . . . Scenario 43 [I 1 , I 2 ] = [1, 64] [J 1 , J 2 ] = [64, 1] . . . Scenario 49 [J 1 , J 2 ] = [1, 64] To simplify the exposition, we assume that N BS , N RF , and N G are selected so thatN BS andN RF are integer numbers, without loss of generality. T L Marzetta, E G Larsson, H Yang, H Q Ngo, Fundamentals of Massive MIMO. Cambridge University PressT. L. Marzetta, E. G. Larsson, H. Yang, and H. Q. Ngo, Fundamentals of Massive MIMO. Cambridge University Press, 2016. The Next Generation Wireless Access Technology. E Dahlman, S Parkvall, J Skold, Nr, Academic Press, IncUSA1st ed.E. Dahlman, S. Parkvall, and J. Skold, 5G NR: The Next Generation Wireless Access Technology, 1st ed. USA: Academic Press, Inc., 2018. Optimal design of energy-efficient multi-user MIMO systems: Is massive MIMO the answer?. E Björnson, L Sanguinetti, J Hoydis, M Debbah, IEEE Trans. Wireless Commun. 146E. Björnson, L. Sanguinetti, J. Hoydis, and M. Debbah, "Optimal design of energy-efficient multi-user MIMO systems: Is massive MIMO the answer?" IEEE Trans. Wireless Commun., vol. 14, no. 6, pp. 3059-3075, Jun. 2015. An overview of signal processing techniques for Millimeter Wave MIMO systems. R W Heath, N González-Prelcic, S Rangan, W Roh, A M Sayeed, IEEE Journal of Selected Topics in Signal Processing. 103R. W. Heath, N. González-Prelcic, S. Rangan, W. Roh, and A. M. Sayeed, "An overview of signal processing techniques for Millimeter Wave MIMO systems," IEEE Journal of Selected Topics in Signal Processing, vol. 10, no. 3, pp. 436-453, 2016. A unifying design of hybrid beamforming architectures employing phaseshifters or switches. K Ardah, G Fodor, Y C B Silva, W Cruz, F R Cavalcanti, IEEE Trans. Veh. Technol. K. Ardah, G. Fodor, Y. C. B. Silva, W. Cruz, and F. R. Cavalcanti, "A unifying design of hybrid beamforming architectures employing phase- shifters or switches," IEEE Trans. Veh. Technol., pp. 1-1, 2018. Energy-efficient hybrid analog and digital precoding for MmWave MIMO systems with large antenna arrays. X Gao, L Dai, S Han, C.-L I , R W Heath, IEEE Journal on Selected Areas in Communications. 344X. Gao, L. Dai, S. Han, C.-L. I, and R. W. Heath, "Energy-efficient hybrid analog and digital precoding for MmWave MIMO systems with large antenna arrays," IEEE Journal on Selected Areas in Communica- tions, vol. 34, no. 4, pp. 998-1009, 2016. Millimeter wave cellular networks: A MAC layer perspective. H Shokri-Ghadikolaei, C Fischione, G Fodor, P Popovski, M Zorzi, IEEE Transactions on Communications. 6310H. Shokri-Ghadikolaei, C. Fischione, G. Fodor, P. Popovski, and M. Zorzi, "Millimeter wave cellular networks: A MAC layer perspec- tive," IEEE Transactions on Communications, vol. 63, no. 10, pp. 3437- 3458, 2015. Hybrid beamforming design for downlink MU-MIMO-OFDM millimeter-wave systems. S Gherekhloo, K Ardah, M Haardt, Proc. IEEE 11th Sensor Array and Multichannel Signal Processing Workshop (SAM). IEEE 11th Sensor Array and Multichannel Signal essing Workshop (SAM)S. Gherekhloo, K. Ardah, and M. Haardt, "Hybrid beamforming design for downlink MU-MIMO-OFDM millimeter-wave systems," in Proc. IEEE 11th Sensor Array and Multichannel Signal Processing Workshop (SAM), 2020, pp. 1-5. Compressed sensing based multi-user millimeter wave systems: How many measurements are needed. A Alkhateeb, G Leus, R W Heath, Proc. IEEE International Conference on Acoustics, Speech and Signal Processing. IEEE International Conference on Acoustics, Speech and Signal essingA. Alkhateeb, G. Leus, and R. W. Heath, "Compressed sensing based multi-user millimeter wave systems: How many measurements are needed?" in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2015, pp. 2909-2913. Compressed sensing based channel estimation and open-loop training design for hybrid analog-digital massive MIMO systems. K Ardah, B Sokal, A L F De Almeida, M Haardt, Proc. 2020 IEEE International Conference on Acoustics, Speech and Signal Processing. 2020 IEEE International Conference on Acoustics, Speech and Signal essingK. Ardah, B. Sokal, A. L. F. de Almeida, and M. Haardt, "Compressed sensing based channel estimation and open-loop training design for hybrid analog-digital massive MIMO systems," in Proc. 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2020, pp. 4597-4601. Channel estimation and training design for hybrid multi-carrier mmwave massive MIMO systems: The beamspace ESPRIT approach. J Zhang, M Haardt, Proc. 25th European Signal Processing Conference. 25th European Signal essing ConferenceEUSIPCOJ. Zhang and M. Haardt, "Channel estimation and training design for hybrid multi-carrier mmwave massive MIMO systems: The beamspace ESPRIT approach," in Proc. 25th European Signal Processing Confer- ence (EUSIPCO), 2017, pp. 385-389. Optimal training design for MIMO OFDM systems in mobile wireless channels. I Barhumi, G Leus, M Moonen, IEEE Transactions on Signal Processing. 516I. Barhumi, G. Leus, and M. Moonen, "Optimal training design for MIMO OFDM systems in mobile wireless channels," IEEE Transactions on Signal Processing, vol. 51, no. 6, pp. 1615-1624, 2003. Training-based MIMO channel estimation: a study of estimator tradeoffs and optimal training signals. M Biguesh, A Gershman, IEEE Transactions on Signal Processing. 543M. Biguesh and A. Gershman, "Training-based MIMO channel estima- tion: a study of estimator tradeoffs and optimal training signals," IEEE Transactions on Signal Processing, vol. 54, no. 3, pp. 884-893, 2006. Higher-order SVD-based subspace estimation to improve the parameter estimation accuracy in multidimensional harmonic retrieval problems. M Haardt, F Roemer, G Del Galdo, IEEE Transactions on Signal Processing. 567M. Haardt, F. Roemer, and G. Del Galdo, "Higher-order SVD-based subspace estimation to improve the parameter estimation accuracy in multidimensional harmonic retrieval problems," IEEE Transactions on Signal Processing, vol. 56, no. 7, pp. 3198-3213, 2008. Tensor decompositions, alternating least squares and other tales. P Comon, X Luciani, A L F De Almeida, Journal of Chemometrics. 237-8P. Comon, X. Luciani, and A. L. F. de Almeida, "Tensor decompositions, alternating least squares and other tales," Journal of Chemometrics, vol. 23, no. 7-8, pp. 393-405, 2009. C F Van Loan, N Pitsianis, Approximation with Kronecker Products. M. S. Moonen, G. H. Golub, and B. L. R. De MoorNetherlandsSpringerC. F. Van Loan and N. Pitsianis, Approximation with Kronecker Products, M. S. Moonen, G. H. Golub, and B. L. R. De Moor, Eds. Dordrecht: Springer Netherlands, 1993. Kronecker product approximation with multiple factor matrices via the tensor product algorithm. K K Wu, SMCY Yam, SMCH Meng, SMCM Mesbahi, SMCProc. IEEE International Conference on Systems, Man, and Cybernetics. IEEE International Conference on Systems, Man, and CyberneticsK. K. Wu, Y. Yam, H. Meng, and M. Mesbahi, "Kronecker product approximation with multiple factor matrices via the tensor product algorithm," in Proc. IEEE International Conference on Systems, Man, and Cybernetics (SMC), 2016, pp. 004 277-004 282. Singular value decomposition approximation via kronecker summations for imaging applications. C Garvey, C Meng, J G Nagy, C. Garvey, C. Meng, and J. G. Nagy, "Singular value decomposition approximation via kronecker summations for imaging applications," 2018. [Online]. Available: https://arxiv.org/abs/1803.11525 Study on Channel Model for Frequencies from 0. 3GPP. 51003GPP, "Study on Channel Model for Frequencies from 0.5 to 100 . Ghz, 3rd Generation Partnership Project (3GPP), Technical Report (TR) 38.901, 2020, v16.1.0GHz," 3rd Generation Partnership Project (3GPP), Technical Report (TR) 38.901, 2020, v16.1.0. Performance analysis of multi-user MIMO schemes under realistic 3GPP 3-D channel model for 5G mmwave cellular networks. D G Riviello, F Di Stasio, R Tuninato, Electronics. 113D. G. Riviello, F. Di Stasio, and R. Tuninato, "Performance analysis of multi-user MIMO schemes under realistic 3GPP 3-D channel model for 5G mmwave cellular networks," Electronics, vol. 11, no. 3, 2022. [Online]. Available: https://www.mdpi.com/2079-9292/11/3/330
[]
[ "Fault diagnosis for PV arrays considering dust impact based on transformed graphical feature of characteristic curves and convolutional neural network with CBAM modules Author Names and Affiliations: Fault diagnosis for PV arrays considering dust impact based on transformed graphical feature of characteristic curves and convolutional neural network with CBAM modules", "Fault diagnosis for PV arrays considering dust impact based on transformed graphical feature of characteristic curves and convolutional neural network with CBAM modules Author Names and Affiliations: Fault diagnosis for PV arrays considering dust impact based on transformed graphical feature of characteristic curves and convolutional neural network with CBAM modules" ]
[ "Jiaqi Qu ", "Lu Wei [email protected] ", "Qiang Sun ", "Hamidreza Zareipour ", "Zheng Qian [email protected] ", "Jiaqi Qu ", "Zheng Qian ", "Lu Wei ", "Hamidreza Zareipour ", "Lu Wei ", "Zheng Qian ", "\nSchool of Instrumentation and Optoelectronic Engineering\nSchool of Electronics and Information Engineering\nBeihang University\nBeijingChina\n", "\nDepartment of Electrical Engineering\nQiang Sun: State Key Laboratory of Control and Simulation of Power System and Generation Equipment\nBeihang University\nBeijingChina\n", "\nThe Department of Electrical and Computer Engineering\nTsinghua University\nBeijingChina\n", "\nSchool of Electronics and Information Engineering\nUniversity of Calgary\nCalgaryCanada\n", "\nSchool of Instrumentation and Optoelectronic Engineering\nBeihang University\nBeijingChina\n", "\nBeihang University\n100191Beijing, BeijingChina., China\n" ]
[ "School of Instrumentation and Optoelectronic Engineering\nSchool of Electronics and Information Engineering\nBeihang University\nBeijingChina", "Department of Electrical Engineering\nQiang Sun: State Key Laboratory of Control and Simulation of Power System and Generation Equipment\nBeihang University\nBeijingChina", "The Department of Electrical and Computer Engineering\nTsinghua University\nBeijingChina", "School of Electronics and Information Engineering\nUniversity of Calgary\nCalgaryCanada", "School of Instrumentation and Optoelectronic Engineering\nBeihang University\nBeijingChina", "Beihang University\n100191Beijing, BeijingChina., China" ]
[]
Various faults can occur during the operation of PV arrays, and both the dust-affected operating conditions and various diode configurations make the faults more complicated. However, current methods for fault diagnosis based on I-V characteristic curves only utilize partial feature information and often rely on calibrating the field characteristic curves to standard test conditions (STC). It is difficult to apply it in practice and to accurately identify multiple complex faults with similarities in different blocking diodes configurations of PV arrays under the influence of dust. Therefore, a novel fault diagnosis method for PV arrays considering dust impact is proposed. In the preprocessing stage, the Isc-Voc normalized Gramian angular difference field (GADF) method is presented, which normalizes and transforms the resampled PV array characteristic curves from the field including I-V and P-V to obtain the transformed graphical feature matrices. Then, in the fault diagnosis stage, the model of convolutional neural network (CNN) with convolutional block attention modules (CBAM) is designed to extract fault differentiation information from the transformed graphical matrices containing full feature information and to classify faults. And different graphical feature transformation methods are compared through simulation cases, and different CNN-based classification methods are also analyzed. The results indicate that the developed method for PV arrays with different blocking diodes configurations under various operating conditions has high fault diagnosis accuracy and reliability.
null
[ "https://export.arxiv.org/pdf/2304.06493v1.pdf" ]
258,108,063
2304.06493
dadbee945cb1c76b54a2dc8f516ffad25d67444c
Fault diagnosis for PV arrays considering dust impact based on transformed graphical feature of characteristic curves and convolutional neural network with CBAM modules Author Names and Affiliations: Fault diagnosis for PV arrays considering dust impact based on transformed graphical feature of characteristic curves and convolutional neural network with CBAM modules Jiaqi Qu Lu Wei [email protected] Qiang Sun Hamidreza Zareipour Zheng Qian [email protected] Jiaqi Qu Zheng Qian Lu Wei Hamidreza Zareipour Lu Wei Zheng Qian School of Instrumentation and Optoelectronic Engineering School of Electronics and Information Engineering Beihang University BeijingChina Department of Electrical Engineering Qiang Sun: State Key Laboratory of Control and Simulation of Power System and Generation Equipment Beihang University BeijingChina The Department of Electrical and Computer Engineering Tsinghua University BeijingChina School of Electronics and Information Engineering University of Calgary CalgaryCanada School of Instrumentation and Optoelectronic Engineering Beihang University BeijingChina Beihang University 100191Beijing, BeijingChina., China Fault diagnosis for PV arrays considering dust impact based on transformed graphical feature of characteristic curves and convolutional neural network with CBAM modules Author Names and Affiliations: Fault diagnosis for PV arrays considering dust impact based on transformed graphical feature of characteristic curves and convolutional neural network with CBAM modules * Corresponding Author: Address: Xueyuan Road No.37, Haidian District,PhotovoltaicFault diagnosisDust impactGraphical feature transformationCharacteristic curves Various faults can occur during the operation of PV arrays, and both the dust-affected operating conditions and various diode configurations make the faults more complicated. However, current methods for fault diagnosis based on I-V characteristic curves only utilize partial feature information and often rely on calibrating the field characteristic curves to standard test conditions (STC). It is difficult to apply it in practice and to accurately identify multiple complex faults with similarities in different blocking diodes configurations of PV arrays under the influence of dust. Therefore, a novel fault diagnosis method for PV arrays considering dust impact is proposed. In the preprocessing stage, the Isc-Voc normalized Gramian angular difference field (GADF) method is presented, which normalizes and transforms the resampled PV array characteristic curves from the field including I-V and P-V to obtain the transformed graphical feature matrices. Then, in the fault diagnosis stage, the model of convolutional neural network (CNN) with convolutional block attention modules (CBAM) is designed to extract fault differentiation information from the transformed graphical matrices containing full feature information and to classify faults. And different graphical feature transformation methods are compared through simulation cases, and different CNN-based classification methods are also analyzed. The results indicate that the developed method for PV arrays with different blocking diodes configurations under various operating conditions has high fault diagnosis accuracy and reliability. Introduction Solar energy, with its advantages of cleanness, accessibility, and utilization, has been paid increasing attention in the fight against global warming and fossil energy shortages. The International Energy Agency (IEA) predicts that the global PV market will grow by 25% year-on-year to 197 GW of new installed capacity in 2022, with cumulative installed capacity exceeding 1,000 GW. From 2022 to 2031, the global PV gridconnected installed capacity will grow at an average annual rate of 8% [1]. However, PV systems are often subject to abnormal failures due to disturbances in the operating environment, which can result in an estimated annual energy loss of up to 18.9% [2]. PV arrays, as the core component of PV systems, are prone to multiple faults such as short-circuit, opencircuit, abnormal degradation, partial shading, etc. in complex outdoor environments [3]. Currently, the traditional DC-side protection methods, such as over-current protection (OCPDs) and ground fault protection (GFPDs), have certain difficulty in determining the type of faults [4,5]. Furthermore, due to the non-linear output characteristics of the PV array, both low mismatch faults and faults at low irradiance may cause the failures of the protection [6]. These failures can not only affect power generation but also lead to serious safety issues. In some cases, faults may even increase the risk of fire and personal danger, if not detected and corrected in time [7]. In addition, the dusty environment has a significant impact on the operation performance of PV systems [8][9][10]. Dust from PV panels can reduce the power of PV systems [11], and more importantly, the long-term dust deposition operating conditions also complicate faults, forming compound faults that are more difficult to classify [12]. Therefore, effective fault detection and diagnosis of PV arrays under the influence of dust is essential for the safety and reliability of PV systems [13]. At present, there are two main categories of techniques for detecting and diagnosing faults in PV arrays. One is the offline diagnosis methods, which include earth capacitance measurement (ECM) [14] and time domain reflection (TDR) [15,16], etc. These methods rely on specific external signal generators for testing and require the offline operation of the PV array, which interferes with normal operation and makes it hard to diagnose faults online in real-time. The other category is the online diagnostic methods that do not affect the normal operation of the PV system. According to the diagnostic indicators, these online methods are further divided into those based on the measurement of current and voltage output on the DC side of each substring of the array [4,17,18], those based on the measurement of current and voltage output on the AC side of the array [6,19,20], and those based on the I-V characteristic curves [21][22][23]. Specifically, the DC indicators-based method analyses faults through the output residuals of the current or voltage in each substring relying on a large number of sensors, which is costly and has limited universality. The AC indicators-based method does not need to install a large number of sensors and uses the residual thresholds of current and voltage output on the AC side of the array. However, the types of faults that can be distinguished between different threshold intervals are inadequate, and it is hard to diagnose multiple complex faults with complete accuracy. Since I-V characteristic curves usually contain rich information on the status of PV modules, diagnosis based on I-V curves is a hot topic [24]. Moreover, the I-V tracer currently already supports measurements for a single module or small-scale strings or arrays and has realized online measurement without changing the operational state [25,26]. In this sense, the diagnosis method based on I-V curves can be applied to all common PV installations and is easier to implement in the field. In existing studies, according to forms of curve features applied, fault diagnosis methods using I-V characteristic curves can generally be divided into: i) raw I-V curves relying on deep learning models for diagnosis and ii) extracted key features of I-V curves for diagnosis. Further, the former includes: (1) taking I-V curves data as input directly. For example, Chen et al. [27] assembled I-V curves with irradiance (G) and module temperature (T) into a 4-column matrix to classify 8 classes of PV array faults by an improved ResNet model. Gao et al. [28] designed a fusion model of convolutional neural network (CNN) and residual-gated recurrent unit (Res-GRU) to diagnose hybrid faults using a combination matrix of I-V curves, irradiance and temperature as inputs. (2) taking the residual between the measured I-V curve and the theoretical curve as input. For example, Chine et al. [29] compared the difference between the measured and the simulated PV array output power and identified faults by the attributes of differences in I-V curves. Liu et al. [21] proposed a fault diagnosis method based on stacked auto encoder (SAE) and clustering, which extracts features from the difference between the simulated and measured I-V curves to achieve classification. These methods have insufficient ability to process the full feature information contained in the original I-V curves completely, and the extraction of features mostly depends on classifiers with complex structures. In addition, they only enable the classification of a fixed degree of faults and have tiny diagnostic ability for the full fault levels of defects. The latter includes (3) identifying key features of array characteristics (e.g. V OC , I SC , V MPP , I MPP , FF, R S , and R P ) in the curves as input. For example, Fadhel et al. [30] adopt V MPP , I MPP , and P MPP as features to classify four different shading configurations. Liu et al. [31] extracted five key points from I-V curves as valid features as input to a fault diagnosis method based on variable prediction model. Besides, some similar approaches are presented in [32][33][34]. (4) calculating shape features (e.g. derivative and curvature) of the curves as input. For example, Bressan et al. [35] proposed method based on the analysis of the first and second derivative of the I-V curves, for detecting faults on series resistors and activation of bypass diodes. Ma et al. [36] analyzed the extraction of negative peaks on the derivative of the I-V curves, whereby single faults and compound faults at different levels of shading were diagnosed. However, it should be pointed out that these studies only used part information of characteristic curves. The diagnosis was completed by analyzing the current (I MPP ), voltage (V MPP ), power (P MPP ) or curve shape features at the maximum power point (MPP). In some complex scenarios such as faults considering soiling, the feature extraction process is complicated, and they may appear to have the same MPPs leading to wrong classifications. Recently, several studies have proposed additional contributions to this field. Lin et al. [37] extracted multi-scale fault features using different scales of convolution horizons and identified fixed fault parameters of faults effected by soiling, Huang et al. [38] investigated full-scale faults under the operating condition with soiling impact in PV arrays without blocking diodes, and Li et al. [39] used full characteristic information of the graphical features of I-V curves and machine learning techniques for PV arrays fault diagnosis. However, several gaps still remain. First, to the best of the authors' knowledge, there are no studies that provide a comprehensive analysis of full-scale faults with dynamic fault parameters in PV arrays with various blocking diode configurations considering the impact of soiling. Second, extracting features of array characteristics or curve shapes does not make effective use of full information contained in the I-V curves, leaving incomplete types of PV array faults to be identified. Furthermore, most current diagnostic methods using characteristic curves rely on additional measured curves of fault samples to calibrate the I-V curves to the standard test conditions, severely restricting their practical applicability. Therefore, this paper aims to develop a feature processing method using full information of characteristic curves, which does not require additional experiments to calibrate curves under different environments. Furthermore, a full-range multi-faults classification method adapted to different PV array configurations considering soiling impact is designed. It enables the complete diagnosis of complex faults for various operating conditions and configurations of PV arrays. The main contributions of the method proposed in this paper are: (1) Various fault types of PV arrays in multiple scenarios are compared and analyzed, including PV arrays with and without blocking diode configurations and compound fault types with and without the influence of soiling. It overcomes the challenge of uniqueness of array fault diagnosis methods for different objects and provides universal ability to diagnose different PV configurations with dynamic array faults. (2) The graphical feature transformation method based on Isc-Voc normalized GADF is proposed to extract common features of the same fault in different environments, and the transformed graphical features of the characteristic curves are stacked into 2-channel 2D matrices as input features for the classification model. It fills the gap of diagnosing complex faults affected by soiling using the complete characteristic curves information without calibration experiments, which greatly improves the practical application value. (3) The classification model of convolutional neural network with CBAM module is designed for fault diagnosis in multiple scenarios, which can accurately classify and identify the full range of complicated faults. It extends the performance of diagnosis methods under the influence of dust, and improves the diagnosis accuracy and robustness under multiple scenes. The rest of this paper is organized as follows: Section 2 formulates the preliminary analysis of the problem to be studied. Section 3 details the proposed fault diagnosis methodology. Section 4 presents experimental results and discussion. Finally, Section 5 concludes this work. Preliminary analysis In this section, first, the fault behaviors of PV arrays with different blocking diode configurations are analyzed. Second, the characteristic curves under normal operating condition and condition considering the influence of soiling are compared. Furthermore, the preprocessing techniques used for the current fault diagnosis are investigated, and the limitations of the existing preprocessing methods are identified. Faults analysis of PV arrays As shown in Fig. 1a, two configurations (with and without blocking diodes, denoted as Configuration 1 and Configuration 2, respectively) are designed to analyze each type of fault. In fact, during the operation of a PV array, various faults may occur, including short-circuits in substrings such as short-circuit in one or two modules, open-circuit of substrings, various degrees of shading such as shading of a single module or multiple modules, as well as series resistance degradation of array and parallel resistance degradation of array. PV arrays with different blocking diode configurations differ in the manifestation of short-circuits [6], as exemplified by the I-V characteristic curves shown in Fig. 2. Specifically, the open-circuit voltage of a normal substring (denoted V OCS ) of a PV array is proportional to the number of serial modules (denoted N SM ), and when different numbers of PV modules in a particular substring are short-circuited (denoted N LL ), then the OC voltage of the fault string is (N SM -N LL )/N SM ⋅V OCS (denoted as V OCF ). When substrings are equipped with blocking diodes, i.e. Configuration 1, the current of each string is only allowed to flow in one direction. V OCS and V MPP of the faulty string are reduced, while I MPP remains unchanged. When V OCS > V OCF , the faulty string will be disconnected from array, which is expressed in the IV/PV characteristic curve as the presence of a local minimum inflection point and the overall V OC of the array remaining stable, as shown in Fig. 2b. On the contrary, when the substrings are not equipped with blocking diodes, i.e. Configuration 2, the current from the normal string is reversed into the faulty string when V OCS > V OCF . As a result, V OC of the faulty array with short-circuits is lower than that of the normal array. It is worth noting that when V OCS < V OCF , the characteristic curves of the PV array are the same for both configurations in this interval. In addition, during outdoor operation, all modules in the PV array can be affected by various types of shading, including shading from buildings, tree shade, etc., as well as shielding from dust adhering to the surface. The essential effect of both is manifested in the reduction of irradiance intensity of the affected modules. Specifically, the former generally results in a larger reduction in irradiance typically of 20% or more, whereas dust accumulation is considered a specific form of shading that occurs across all PV modules of an array and produces a relatively small reduction in irradiance usually below 20% [40,41]. The enlarged structure of a PV module in Fig. 1a shows that every N cells are inversely connected in parallel with one bypass diode. For a single module, when the shading intensity and area are fixed, the output I-V characteristic curve of the module shows a single peak, as shown in Fig. 3a. Further, when a module in a substring containing multiple modules is shaded, the I-V curve splits into two parts [30]. When the voltage is lower than the substring voltage (denoted V S ), the shaded cells of the shaded module are bypassed by the bypass diode, and the I-V characteristic in this interval is approximately equivalent to the I-V characteristic of the remaining unshaded modules in series. When the voltage is larger than V S , the bypass diode turns off resulting in the decrease of the total output current of the string [42]. The I-V curve in this interval is mainly determined by the shaded module. Therefore, no matter what degree of shading, there is a turning peak on the I-V/P-V characteristic curve. The shadow degree can influence the inflection point of current, as shown in Fig. 3a. The current at inflection point decreases with the increase of the shading degree and the number of shading modules. As the degree of shading increases, the voltage at inflection point decreases proportionally, as shown in Fig. 3b. It also can be seen that in Fig. 2 and Fig. 3b, the characteristic curves of arrays under certain shading conditions are very similar to the short-circuited arrays configured with blocking diodes. In fact, the intensity of dust accumulation is uneven across all modules, representing a non-uniform soiling effect for the array [38,40] as shown in Fig. 1b. Therefore, under the influence of non-uniform soiling, each module of the array is equivalent to being affected by different degrees of shading, the characteristic curves of the array will also show a different number of peaks. Importantly, when other faults occur under the impact of dust, such as short-circuit and open-circuit, the characteristic curves become more complex, showing a superposition of multiple peaks and corresponding fault characteristics at the same time, as in Fig. 4b. However, most cleaning operations for dust or soiling are usually carried out on an annual basis or more frequently in areas heavily affected by dust, but other faults under the influence of non-uniform soiling can still occur during the cleaning interval [43,44]. This makes the diagnosis of complex fault types more challenging and difficult to identify accurately in time. It can be seen that various complex fault types of different array structures are rich and diverse, and have high similarities. Therefore, it is of universal significance to propose an effective method applicable to fault diagnosis in multiple scenarios. Preprocessing methods for fault diagnosis The shape of PV characteristic curves is dependent on environmental conditions such as irradiance and temperature [45]. Therefore, when using characteristic curves for PV array fault diagnosis, the influence of different environmental conditions causing different characteristics manifestations of the same fault type should be excluded to reduce confusion of features caused by factors other than the fault types [39]. At present, there are three data preprocessing methods to minimize the impact of environmental factors on the characteristic curves: (1) The input feature is a two-dimensional matrix recombined by I-V curve and ambient variables, which stems from the ability of the convolutional neural network to automatically extract features of twodimensional data. Specifically, irradiance and temperature are repeatedly supplemented into column vectors with the same points of I/V vector, and then stitched together with the I-V curves to form a final twodimensional matrix in Fig. 5, which is used as input for the deep learning network classification method. Fig. 5. The recombined GTIV matrix. This method does not directly eliminate the influence of environmental variables on the characteristic curves, but only relies on the convolutional network to extract common feature information of the same fault type under different environmental conditions. It can be inferred that its ability to deal with complicated faults diagnosis is shallow. (2) Converting key features of array characteristics in different environments to features under STC is often applied to the diagnosis method using key features identified in characteristic curves. That is, the opencircuit voltage V OC , the short-circuit current I SC , the maximum power point voltage V MPP , the maximum power point current I MPP , and the equivalent series resistance R S are converted to corresponding values under STC. The approach to obtain the feature functions is based on the traditional approximation equations [46,47], where the unknown parameters are denoted by a, b, c and d. The functions of key features can be rewritten as: OC OC,STC STC ST 1 2 3 C ln GG V V a a dT a dT GG (1) m m,STC STC ST 13 C 2 ln GG V V b b dT b dT GG (2) 3 STC ST 2 C m 1 m,STC GG I c I c dT c dT GG (3) 1 S S,STC STC C 23 ST d GG R R d dT d dT GG (4) The parameters of this non-linear static model need to be identified by searching through multiple characteristic curves. In addition, as the curves under STC still behave differently for various fault types, separate parameter identifications are required for various faults with different STC conditions, which involves additional test experiments with fault samples. Moreover, the feature-dependent diagnosis method is not effective in classifying complex fault types. (3) Correcting all points of the entire I-V characteristic curve is a preprocessing method that relies on the use of full I-V information for diagnosis. The IEC 60891 [48] defines three standard procedures for the correction of I-V curves, which are used to allow comparison of curves measured under different conditions, thus enables health monitoring of PV panels. The following are named M1, M2 and M3 respectively, as well as the correction method of M2new proposed in [49]: M1： 2 2 1 SC 2 1 1 1 G I I I T T G (5)2 1 S 2 1 2 2 1 2 1 V V R I I I T T T T  (6) M2： 2 2 1 rel 2 1 1 1 G I I T T G  (7) 2 2 1 OC1 rel 2 1 2 1 2 2 1 1 ln S G V V V T T a R I I I T T G  (8) M2new: It uses the same equation as M2 for current correction, but corrects for voltage by replacing the term " OC1 V " in (8) with " re OC1 l1 1 (25 ) V T  ".I I I I  (9) 3 1 2 1 V V V V  (10) 3 1 2 1 G G G G  (11) 3 1 2 1 T T T T (12) In fact, M1, M2, and NewM2 are correction methods based on one single curve, that requires the setting of corresponding correction factors such as  and R S . That is, after the correction parameters are determined, they can be directly corrected from any test conditions to the STC. However, it is still difficult to determine the correction factors of PV panels on site due to the rigorous experimental conditions required in the IEC 60891 procedure. In addition, research shows that due to differences in irradiance, module temperature, and severity of faults, all these methods introduce significant errors, making it difficult to perform well under all fault conditions [49]. Moreover, the distortion of curve shape in the IEC method usually leads to a relative error of 13.8% and the estimation error of fault features extracted from the correction curve also occurs frequently. It can be seen that if these features are used as defect features, it may affect the diagnosis of faults. Alternatively, M3 does not contain correction factors, but the interpolation constant  needs to be set. This method is to apply linear interpolation method on multiple I-V characteristic testing curves to obtain the I-V characteristic under the STC. Although the multiple curves-based method (M3) generally offers higher performance than the above methods based on one single curve, the conversion of a certain curve with both G and T requirements relies on multiple testing curves, and the measurement and calculation are relatively complex and inefficient, so it is not suitable for rapid field diagnosis. It is undeniable that the use of characteristic curves for PV health monitoring and fault diagnosis is a promising method. Considering the limitations of the practical application of the preprocessing methods discussed above, it will be beneficial to explore solutions based on field measurement data while reducing the dependence on the calibration process. The proposed fault diagnosis methodology In this section, firstly, the accurate modeling approach of the real PV array is presented, and the simulation method of faults in different configurations of PV arrays under different operating conditions is described. Then, a data processing method that does not depend on the calibration of the characteristic curves to STC is proposed to obtain transformed graphical feature matrices with full information of characteristic curves. Finally, the fault diagnosis model based on the convolutional neural network with CBAM module is introduced. Configuration of the simulated PV modeling To explore the complex faults of PV arrays under different operating conditions, the single-diode PV cell model proposed in [50] is used in this paper, as shown in Fig. 6. And based on this model a PV module model is built on the PSCAD/EMTDC platform with the configuration parameters from the Shell Solar SP-70 datasheet in Table 1. We compare the simulated I-V curves of this PV module model under different radiation intensities at 25°C with the manual curves supplied by manufacturer. There is a high correlation between the two, which verifies the consistency of the equivalence between the model and the actual PV module. Based on this PV module model, the PV array model in this study is further designed for a structure with three modules in series and two substrings in parallel. Among them, the substrings containing blocking diodes or not results in the two configuration types of PV arrays, whose structure is shown in Fig. 1. In order to characterize the faults under the operating environmental conditions of the real PV array as much as possible, we take one year of outdoor irradiance and temperature records measured from the PV plant as the environmental control variables for the PV array model. Table 1 Electrical characteristics of solar module Shell SP-70 at nominal condition (25°C and 1000 W/m 2 ). Parameter Value ISC (A) 4.7 VOC (V) 21.4 IMPP (A) 4.25 VMPP (V) 16.5 KV (mV/℃) -76 Ki (mA/℃) 2 NS 36 RS (Ω) 0.41 RP (Ω) 141 To fully demonstrate the generalizability of the proposed fault diagnosis method, the faults of PV arrays with two types of blocking diode configurations under different operating conditions are validated, including contamination-prone operating condition and ideal normal operating condition. The two operating conditions consist of 14 and 9 types of faults, respectively, as follows. Case1：PV arrays under operating condition with non-uniform soiling impact 1) Two types of line-to-line short-circuit (LL): one or two modules in one string are shorted (noted as LL1 and LL2 respectively). 2 Case2：PV arrays under operating condition without non-uniform soiling impact The types of faults include 1) -6) above under the ideal operating condition without the influence of non-uniform soiling. As an example, the characteristic curves for faults of the PV array with the two blocking diode configurations in Case1, including I-V and P-V are shown in Fig. 7 and Fig. 8. Importantly, what is different from [18,21,37] is that the simulations of faults in this study for Shade, ADegration, SDegration, and Soiling, as well as the severity of these faults, are not simply set as constants. This is due to the fact that these faults change dynamically with time in service, therefore the simulations should contain the full range of fault parameters. In this study, the environmental conditions of a real PV plant are used as inputs for the temperature and irradiance of the PV array model, and the fault parameter of each fault type is set randomly within its corresponding full-scale parameter range to obtain characteristic curves. Specifically, for shading, the irradiance gain is set from 20% (low shading) to 100% (full shading) for one or two modules; for soiling, a special form of shading, the irradiance gain is set randomly within 10% for all modules to simulate non-uniform soiling accumulation; for abnormal aging of ADegration and SDegration, the aging resistance values are set randomly for degradation in [20Ω, 200Ω] and [1Ω, 15Ω] respectively, adjusting for different levels of fault severity. Considering the dust-influenced operating condition, faults are simulated by superimposing other faults alongside the non-uniform soling, where the degree of soiling and dynamic fault parameters are set randomly when it comes to dynamically growing fault types. The representation of different faults with full-scale dynamic fault parameter is shown in Fig. 9, where the characteristic curves show different shapes of distortions. Graphical feature transformation method of Isc-Voc normalized GADF The graphical feature transformation method using the Gramian angular difference field (GADF) [51] enables the extraction of complete information from the resampled characteristic curves. Typically, GADF is applied to the transformation of time series signals, which mainly includes the following steps: Step x is noted as the normalized value. Step 2: Convert the rescaled sequence to the polar coordinate system, i.e. the values of the time series are treated as the cosine of the angle. The formulas for converting to polar coordinates are: arccos ,0 1, , i i i i i ii x x x X t r t N N (14) where, i t is the time stamp of point i x , N is the number of all the time points contained in the time series, and X represents the rescaled time series. Step 3: Obtain the angle difference of each pair, and then take the sine value for difference to form the GADF matrix: It is worth noting that when applied to time series, the scaling of GADF is using the full-time axis data, and thus when applied to the scaling of the I-V/P-V characteristic curves, the V-axis should be analogous to the time axis. If GADF with normal normalization following steps described above is used to convert the characteristic curves, i.e. the conversion areas are A, B, C and D respectively in Fig. 10, in essence, the current V OC and I SC of each fault state are taken as their respective normalized maximum values, which would blur the differences in V OC and I SC between the different faults. We can observe from the GADF transformation results in Fig. 11 that when GADF transformation is performed on the I-axis or P-axis with the normal normalization strategy, the same fault types under different irradiance and temperature conditions are relatively consistent in the shape of the converted graphical features, which can avoid misidentification of the same faults due to different environments. However, similarities in shapes also exist between the characteristic curves of certain different fault types, such as health, LL1, LL2, and OC in Fig. 11. In fact, these curves differ in absolute values of V OC and I SC . Therefore, it is necessary to propose a new GADF transformation method that retains consistency in the characteristics of the transformation features for the same fault in different environmental conditions, while having the ability to distinguish the similarity in the shapes of the characteristic curves for different fault types. When the external circuit is short-circuited, i.e. the load is 0, the short-circuit current I SC is: G I I k T(18) where, T is the temperature in Kelvin, G is the irradiance, and k i is the temperature coefficient of shortcircuit current. When where, the diode saturation current 0 I is expressed as [53,54]: n n 11 exp qE T II T nk T T (22) SC,n 0,n OC,n t,n exp / 1 g n 0 0,I I V nV (23) n t,n S N kT V q(24) where the band gap energy of the semiconductor is represented by E g and the nominal saturation current at the nominal temperature T n , i.e. 25°C, is represented by I 0,n via (23), the nominal thermal voltage represented by V t,n is shown in (24), I SC,n is the short-circuit current at nominal temperature, and V OC,n is the open-circuit voltage at nominal temperature. In addition, n is the ideal factor of the diode, N S is the number of series cells forming the module, k is the Boltzmann constant and q is the charge quality. Thus, it can be seen from (22) that I 0 is only related to the temperature T. It can be further concluded from (18) and (21) that the short-circuit current I SC and open-circuit voltage V OC of PV modules depend only on irradiance G and ambient temperature T, except for the constant factors with fixed values, which means that the ideal I SC and V OC are the same for different fault types under the same environmental conditions. Therefore, we propose a GADF method based on the normalization of I SC and V OC under environmental conditions, which can ensure that the transformed graphical features of the same fault types are the same in different environments, while having distinguishability between different faults with similar forms. Specifically, the main implementation process of the Isc-Voc normalized GADF method is: (1) Calculate the ideal I SC and V OC (denoted as I SC_ideal and V OC_ideal ) of the ambient conditions corresponding to each characteristic curve, expand the maximum value of V-axis to the ideal V OC , and complement the current and power of the characteristic curve by 0 over the range of the measured V OC and the ideal V OC . (2) Resample the complemented characteristic curves using the bilinear difference method to provide uniformly distributed characteristic curves with a small amount of data. Specifically, the V-axis of I-V and P-V curves is within the range of [0, V OC_ideal ] at a uniform voltage interval and the I-axis and P-axis are resampled at the same uniform interval, reducing data of each curve from the original 200 points to 50 points. (3) Convert the resampled characteristic curves to graphical features according to Isc-Voc normalized GADF. That is, the V-axis is normalized according to the range of [0, V OC_ideal ], the I-axis is normalized according to the range of [0, I SC_ideal ], and the P-axis is normalized according to the range of [0, V OC_ideal * I SC_ideal ]. (4) Then, calculate the inner product according to the differences of normalized angles, preserving the time dependence of the V-axis, and generate a GADF matrix with size of 50 2 . The I-V or P-V characteristic curve corresponding to each environmental condition is transformed into a matrix, and the I-V and P-V transformation matrices are stacked to form a 2-channel 2D matrix as the input to the fault diagnosis model. This method makes it possible to select a unified V OC under the same environmental conditions, which corresponds to the V-axis having the same time scale. In this way, all changes in characteristic curves can be reflected in the transformation matrices. When the slope of the characteristic curve changes, the diagonal regions in the matrix shrink in different directions. Further, the normalization strategy of selecting the maximum value of V OC and the maximum value of I SC in all characteristic curves, without distinguishing environmental conditions, is also compared and referred to as the global normalized GADF. More detailed discussion and analysis of the three current universal applied methods (direct I-V, RP, and GADF) as well as the proposed normalization strategies are provided in Chapter 4. Classification model of convolutional neural network with CBAM module To improve the ability of diagnosing complicated faults, we design a PV array fault diagnosis model of convolutional neural network with CBAM module, referred to as CNN-CBAM, with the transformed graphical feature matrices containing full information of the characteristic curves as the input features. The structure of the model is shown in Fig. 12, which is mainly composed of convolution modules and CBAM attention modules. The Convolution module can reduce the dimensions in time and space and lower the number of free parameters required for training [55], due to the benefits of local receptive field and weight sharing. Therefore, performance can be enhanced. Specifically, the Conv2d layer slides each filter in the input feature matrix through the local receptive field, calculates the sum of dot products on the local field, and automatically extracts the effective features from the inputs. Then, the Pooling layer divides the input area and calculates the average value of each area to complete the down sampling of the feature matrix. The CBAM module focuses on the important features and suppresses the unimportant features in the network, which can effectively improve the performance of CNN. And the CBAM is composed of the channel attention module (CAM) and the spatial attention module (SAM) [56]. Their detailed structures are shown in Fig. 13 and Fig. 14, respectively. Among them, the CAM emphasizes that the network should concentrate on useful channel features while ignoring other aspects by using the maximum pool and the average pool to compress the spatial dimension of the feature matrix. The SAM highlights that the network should focus on the local area of interest by applying the average pool and maximum pool along the channel dimension to retain the background information of the feature matrix. Specifically, first, the input feature F is multiplied by the feature matrix M C (F) generated by CAM compression along the spatial dimension to obtain F'. Then, SAM compresses F' along the channel dimension to generate the spatial feature matrix M S (F'). Finally, the optimized feature matrix F'' is obtained by multiplying M S (F') and F'. The whole process can be represented by: The CAM module processes and aggregates the input features with the maximum pool and average pool, and then inputs them into a weight-sharing multi-layer perceptron (MLP) network for summation, which is activated by sigmoid to generate the final channel attention feature M C (F). The calculation program of CAM can be expressed as: ' () C F M F F (25) '' ' ' () S F M F F (26) where, represents multiplication between elements ( )= ( ( ( )) ( ( ))) C M F MLP AvgPool F MLP MaxPool F (27) The SAM module performs maximum pool and average pool operations on the input feature F' along the channel dimension, generates a two-layer feature matrix, and cascades them together. Then, the convolution kernel with size of 7×7 is used to reduce the dimension of features, and the sigmoid is applied to generate the spatial attention feature M S (F'). The calculation program of SAM can be expressed as: ' ' ' (28) where, σ denotes the sigmoid function and f denotes the convolution operation with filter. Table 2 shows the specific structural configuration of the proposed PV array fault diagnosis model based on CNN and CBAM. The Convolution blocks with different sizes of convolutional kernels are stacked together along the depth direction to extract features at different levels. This structure can automatically extract more effective feature information directly from the original input feature matrix. In this study, two CBAM modules are respectively embedded into the convolution modules, and the classification accuracy of the faults diagnosis model is improved through the processing capability of focused feature information. ( )= ( ([ ( ); ( )])) S M F f AvgPool F MaxPool F  Results and discussion Experimental setup The data used in this study are based on the configurations described in Section 3.1. Specifically, the faults of PV arrays with two blocking diode configurations under various operating conditions are analyzed, including 14 faults under contamination-prone operating condition and 9 faults under ideal operating condition. To make the data fully reflect the real operating conditions, we take the collected annual ambient records of the actual power station as the environmental control input for each fault type to obtain the corresponding characteristic curves. The data is divided into training set and testing set, accounting for 80% and 20% respectively. And 90% of the training set is the training data and 10% is the validation data. The data volume and proportion of each data set applied to different operating conditions are shown in Table 3. Faults diagnosis evaluation indexes The widely used Precision, Recall, F1-score, and Accuracy are selected as the indicators of the effectiveness of the classification algorithm [57,58], which can be calculated as: (32) where, the number of samples that belong to the positive category and are predicted to be in the positive category are referred to as true positives (TP), the number of samples that belong to the positive category and are predicted to be in the positive category are referred to as false negatives (FN), the number of samples that belong to the negative category and are predicted to be in the positive category are referred to as false positives (FP), and the number of samples that belong to the negative category and are predicted to be in the negative category are referred to as true negatives (TN). Case 1. PV arrays under operating condition with non-uniform soiling impact This part analyzes the effectiveness of different methods to distinguish the 14 fault types of PV arrays under Case 1 operating condition with non-uniform soiling impact, including the accuracy and performance of different graphical feature transformation methods and classification algorithms for fault diagnosis. Analysis of graphical feature transformation methods First, we compare data preprocessing methods for fault diagnosis that do not rely on STC correction, as presented in Section 2.2, including methods that combine I-V curves and environmental variables to form a two-dimensional data matrix (GTIV) and methods that utilize the complete feature information of the characteristic curves. In addition, the diagnosis results of different graphical feature transformation methods such as GADF and recurrent plot (RP) using the complete information of the characteristic curves are analyzed, and three strategies of using the I-V graphical feature matrix, P-V graphical feature matrix and IV-PV stacked graphical feature matrix obtained from transformation as input are further compared. Table 4 shows the results of applying different data preprocessing methods to the characteristic curves for fault diagnosis, all of which use the multilayer-CNN classifier. It can be seen that when the graphical feature matrix transformed by single curve is used as input, i.e. the I-V or the P-V characteristic curve, the effectiveness of the information contained in the transformation matrix in distinguishing fault types is related to the feature transformation methods. Sp ecifically, the I-V graphical feature of the GADF transformation is more effective in classification than the P-V graphical feature, whereas the opposite is true for the RP transformation. In fact, the classification accuracy of the IV-PV stacked graphical feature matrix is usually higher than that of the single characteristic curve transformed matrix. Additionally, for the faults classification of PV array considering the impact of dust, the accuracies of the graphical feature matrix transformed by GADF are 98.58% and 98.48% respectively in the configurations of with and without blocking diodes, which is better than the RP transformation and GTIV matrix methods. Fig. 15 shows the classification accuracy of the optimal feature strategy for the three preprocessing methods, where the stacked IV-PV matrix is used for graphical feature transformation method. Among them, the accuracy of RP transformation method for arrays with blocking diodes is 95.12%, which is significantly higher than 90.85% without blocking diodes. This may be related to the similarity between the two types of short-circuit and the health state without blocking diode configuration when RP transformation is applied, while the GADF transformation is applicable to different PV array configurations, with high accuracy of 98.58% and 98.48% respectively. Fig. 15. Comparison of the optimal feature strategies for three preprocessing methods. The above comparative analysis confirms the effectiveness of the GADF graphical feature transformation method for fault diagnosis of PV arrays with different blocking diode configurations. Further, Table 5 compares different GADF normalization approaches. The application of GADF with good diagnostic accuracy first presented in [39] was based on the ANN method, while more types of faults and higher complexity of faults are involved in this study, therefore the classification effects of multilayer-CNN and ANN are compared. The results in Fig. 16 show that for both blocking diode configurations, the graphical feature transformation preprocessing method using Isc-Voc normalized GADF is significantly more accurate in classifying faults than either the normal normalization or global normalization strategies. In particular, when classifying with relatively simple methods such as ANN, the classification effect mainly depends on the effectiveness of the features in distinguishing faults. In the case of array with blocking diodes, for example, the classification accuracy of the Isc-Voc normalized GADF transformation is as high as 96.11%, while the highest classification accuracy of other normalization strategies is only 81.42%. This also fully demonstrates that the Isc-Voc normalization strategy can obtain features that make the GADF transformation matrix of different fault types better distinguishable compared to other normalization strategies. Moreover, the classification accuracy of the CNN is significantly improved over ANN when using the feature matrix of I-V or P-V obtained by normal normalization or global normalization strategies as input, with the diagnostic accuracy of the array without blocking diodes improved by 10.56% and 14.83% respectively. Further, when the I-V/P-V matrix or stacked IV-PV matrix processed by Isc-Voc normalized GADF is used as the input features for the CNN model, they both have higher diagnostic accuracy, which is related to the stronger learning capability of the multilayer CNN. Fig. 16. Comparison of three GADF normalization methods with different input matrices and classifiers. The loss and accuracy of training and validation in Fig. 17 show that using the IV-PV stacked feature matrix can quickly achieve higher classification accuracy, indicating that the stacked matrix has the advantage of more significant differentiation than the others. Similarly, when the features processed by the Isc-Voc normalization strategy are used as the input of the classification model, the training accuracy of the model in Fig. 18 is rapidly improved, and it takes less time to train to the highest accuracy than the features processed by other normalization strategies. In other words, the proposed Isc-Voc normalized GADF graphical feature matrix is more efficient and more accurate when applied to fault diagnosis. Analysis of convolutional neural network classification models Although the classification accuracy of multilayer CNN in the previous section is relatively considerable, the structure of this model is simple and there is still room for improvement. In this section, we further explore the improvement of classification accuracy of CNN-based models with different structures. Therefore, we design and compare the basic multilayer convolutional neural network, the multi-scale convolutional neural network presented in [37], and the proposed convolutional neural network with CBAM module, whose structures are shown in Fig. 19. Fig. 19. The structures of compared three CNN-based classification models. The results from Table 6 show that the IV-PV stacked feature matrix obtained from the Isc-Voc normalized GADF transformation has the highest classification accuracy as input to the CNN-based classification models for all three structures, regardless of the blocking diode configuration of the PV array. In addition, the multi-scale CNN performs better than the multilayer CNN when using the transformed features of all three GADF normalization strategies as input, mainly because the multi-scale CNN extracts features at different scales. When the CBAM module is embedded in the multilayer CNN, the introduction of channel attention mechanism and spatial attention mechanism enables the network to extract features of interest in the local area while focusing on the channel features, which is more selective in focus than the multi-scale CNN with different scales of convolutional kernels for feature extraction. Overall, the CNN with CBAM module has the highest classification accuracy, with the stacked IV-PV graphical feature matrix transformed by the proposed Isc-Voc normalized GADF as the input. The diagnosis accuracies applied to PV arrays with two diode configurations are 99.62% and 99.40% respectively. Fig. 20. The confusion matrices of PV arrays with blocking diodes under soiling condition. Furthermore, the preprocessing methods of Isc-Voc normalized GADF and normal normalized GADF, which are both highly accurate in using the CNN-CBAM as classifier, are compared. Fig. 20 and Fig. 21 are the confusion matrices of fault diagnosis for PV arrays with and without blocking diodes, respectively. For arrays with blocking diodes, the two normalization strategies are similar in the discrimination ability of most fault types. The specific difference is that the Isc-Voc normalized GADF performs slightly better than the normal normalized GADF in terms of OC, Adegration, and Adegation under soiling, due to the similarity of these characteristic curves under some environmental conditions and levels of faults. The former reduces the proportion of OC misdiagnosed as Adegeration by 59% (13 samples) and does not produce the 11 samples for which the latter diagnoses OC as Adegration under soiling. In addition, for arrays without blocking diodes, the characteristic curves of LL2 generated by current backflow are highly similar to those of Adegeration and OC, etc. The Isc-Voc normalized GADF has higher discrimination than the normal normalized GADF in this case, and their overall recall and F1-score both are 99.28% and 98.82, respectively. Specifically, the Isc-Voc normalized GADF can avoid 11 samples of LL2 misclassified as Adegration, 15 samples of OC misidentified as Adegration under soiling, and more 11 samples of LL1 and OC under soiling that are not correctly distinguished. Moreover, the Isc-Voc normalized GADF is 27% more accurate than the normal normalized GADF in distinguishing between Adegration under soiling and LL2 under soiling and improved the diagnostic accuracy of complex faults affected by contamination by 41.94%. Fig. 21. The confusion matrices of PV arrays without blocking diodes under soiling condition. Generally, the complexity of faults increases due to the impact of dust, there will be individual errors in the classification of all fault types. And the proposed Isc-Voc normalized GADF method has higher accuracy in differentiating complex faults and is able to converge quickly during training, and the processed features are beneficial for the rapid learning of parameters, which provide the best classification performance when applied to the proposed CNN-CBAM, as shown in Fig. 22. Fig. 22. The loss and accuracy for typical CNN-based classifiers with three GADF normalization methods. Case 2. PV arrays under operating condition without non-uniform soiling impact To further illustrate the universality of the proposed method, the performance of fault diagnosis under normal operating condition without dust impact is analyzed, which contains 9 fault states. Similarly, we analyze the influence of transformed graphical features processed by various GADF normalization methods as input on the accuracy of classification algorithms. Specifically, as shown in Table 7, for simple classifiers such as ANN, using the matrix of I-V or P-V curve converted by Isc-Voc normalized GADF as input results in a significant classification accuracy improvement of 97.57% compared to 93.26% for normal normalized GADF and 93.20% for global normalized GADF. And the accuracy of the proposed normalized GADF is up to 98.26% when the IV-PV stacked transformation matrices are used as the input features. Moreover, the CBAM-CNN exhibits general merit when the features processed by the optimal GADF strategy, i.e. stacked IV-PV transformed by Isc-Voc normalization, are applied to the CNN-based models. This is related to the fact that the types of faults to be distinguished in operation are not diverse and their complexity is general. In other words, for the faults not affected by dust, the graphical feature matrices transformed by Isc-Voc normalized GADF applied to a relatively simple structured CNN can obtain satisfactory results, but it is undeniable that the CNN embedded with CBAM module still has a slight accuracy advantage. Fig. 23 and Fig. 24 show the fault classification results of Isc-Voc normalized GADF and normal normalized GADF for PV array with the two blocking diode configurations. It can be seen that the confusion matrix of the former does not contain many sporadic misclassifications of the latter and is expressed in the overall recall and F1-score of 99.81% and 99.18%, respectively. In the fault diagnosis of PV array with blocking diode configuration, the classification error between SDegration and Non-uniform soiling is 84% less in the former than in the latter, with 8 and 50 samples respectively, and the proportion is 80.43% in the array without blocking diode. This may be due to the low fault levels of these two fault types with dynamic fault parameters, i.e. the low fault differentiation between some SDegartions with low fault levels and low degrees of dust shielding. In fact, the correct classification accuracy for both SDegration and Non-uniformed soiling, which may be misclassified, still reaches over 99% for either diode configuration of array. Overall, the proposed CBAM-CNN model based on Isc-Voc normalized GADF transformation method is able to achieve high accuracy in faults classification, which is also applicable to ideal operating condition without dust impact. Fig. 25 and Fig. 26. When the inputs are graphical features processed by Isc-Voc normalized GADF, the loss of model decreases the fastest and the accuracy rises rapidly compared to other normalization strategies, whether for ANN, CNN, or CNN-CBAM as the classifiers. It is consistent with the original intention that the graphical feature transformation method proposed can achieve high discrimination between similar faults in different environments. In particular, for arrays without blocking diode configurations, the model that classifies similar fault types using the transformed graphical features by normal normalization strategy requires a relatively long stage of learning, and the model is not as stable as models using other strategies. This is owing to the fact that the classification accuracy relies heavily on the feature learning capability of the classifiers when the discriminative ability of features is constrained. Moreover, in terms of the classification ability of models with different levels of complexity, simple models such as ANN require longer training time than CNN-based models, and the proposed Isc-Voc normalized GADF can significantly improve the diagnosis performance when applied to category of simple classifiers. For faults not affected by dust, the CNN-CBAM still has slightly better accuracy and training performance than multilayer CNN, although their differences are not significant due to the excellent strength of the CNN-based models in dealing with less complicated tasks. Discussion Compared with the most advanced research in the existing literature, our approach enables the accurate identification of dynamic fault types for arrays with multiple scenarios and configurations. Among the categories using GADF transformed graphical features as classification features, the IV-PV stacked matrix by the proposed Isc-Voc normalized GADF preprocessing method outperforms the transformed feature of single characteristic curve in [39], both in terms of classification accuracy and performance of model training. This implies that the stacked GADF graphical features can highlight the distinguishability of different fault classes in preprocessing methods without employing correction of field characteristic curves, and that the GADF based on Isc-Voc normalization has a more reliable discriminatory capability. The main reasons for this are that stacked graphical features contain richer fault information than features of single curve, and that the Isc-Voc normalized GADF overcomes the challenge of inconsistent feature characterization of the same faults in different environments. In the comparative analysis of different classifiers, the proposed CNN-CBAM has a certain accuracy advantage over other CNN-based methods, and is obviously superior to simple models such as ANN, especially under dust influenced operating condition containing multiple mixed and high-complexity dynamic faults. This is mainly because the characteristic curves jointly affected by dynamic fault parameters and changing environmental factors are more complex, and the learning ability of simple classifiers is limited. Indeed, it is worth noting that input features have a greater impact on classification performance, both in terms of accuracy and robustness, than the type of classifiers. The notion has been fully proven in terms of the significantly better diagnostic performance of Isc-Voc normalized GADF features compared to other GADF features using simple model ANN as the classifier. Furthermore, the benefits of the proposed Isc-Voc normalized GADF are still notable and effective in the condition of not considering the impact of dust, and CNN-CBAM has slight privileges over the basic CNN models due to the general complexity of faults in this scenario. These results prove that features are extremely important, that is, the original data converted into effective features can improve the discriminatory quality of the input features. It can simplify the tuning process of the classifier and improve the diagnostic performance. Moreover, the CNN structure with the introduction of the CBAM attention module, based on high-quality effective features, offers more significant advantages in dealing with the diagnosis of complex fault types. Conclusion In this paper, we propose a new fault diagnosis method for identifying and evaluating faults in PV arrays, which is able to cope with PV arrays with different blocking diode configurations under both operating conditions considering and not considering the dust impact. It is summarized that the salient aspects of our approach are as follows. To facilitate the extraction of the full information in the characteristic curves by the classifier, we transform both the I-V and P-V curves into a graphical feature matrix of GADF and stack them together. Then, for reliable and easy-to-implement practical applications, which require that features of the same faults in different environments are effectively uniform, we adopt calibration-independent field characteristic curves and normalize them by GADF using the ideal Isc and Voc under their ambient conditions. Furthermore, to overcome the challenge of classifiers for multiple dynamic faults in complex operating conditions, we employ the convolutional neural network embedded with CBAM modules to identify and classify the graphical features of complicated faults. The performance of the proposed technique is validated in terms of accuracy and stability in various scenarios. Specifically, the Isc-Voc normalized GADF transformed graphical features have higher fault discrimination than the features transformed by the other strategies, significantly simplifying the processing of the classifier and obtaining magnificent diagnostic accuracy for both conditions with or without dust effects. And it is verified that CNN-CBAM is effective for the identification of normal operating condition containing relatively simple faults or dust operating condition containing complex faults. In general, the proposed method based on GADF-transformed graphical features of characteristic curves and convolutional neural network with CBAM modules is applicable to PV arrays with or without blocking diodes and is also effective for fault diagnosis of different operating conditions, including concurrent faults affected by dust, which has economic benefits. And the proposed Isc-Voc normalized GADF transformation provides a new scheme for using full characteristic curve information with discarding STC correction and relieves the experimental dependence on correction factors, which greatly expands the universal application of fault diagnosis. Based on this research, there are still challenges that need to be addressed in future work, such as the further differentiation of different dynamic fault levels and the location of strings occurring faults while diagnosing, which will play an important role in studying the improvement of O&M efficiency and returnon-investment. Fig. 1 . 1Faults of PV arrays under different operating conditions. Fig. 2 . 2I-V curve of PV arrays with different blocking diode configurations. Fig. 3 . 3I-V curve of PV modules with different structures under the influence of different shading levels. Fig. 4 . 4I-V curves of faults under different operating conditions. Fig. 6 . 6The single-diode PV cell model. ) Open-circuit (OC): one string is open. 3) Two types of shading (Shade): 1 or 2 modules are shaded to different degrees (noted as Shade1 and Shade2, respectively). 4) Series resistance degradation of the array (Sdegration): an increase in the equivalent series resistance of array. 5) Parallel resistance degradation of the array (Adegration): a decrease in the equivalent parallel resistance of array. 6) Non-uniform soiling (Soiling): the accumulation of soiling with varying degrees on the surface of each PV module in the presence of contamination. 7) Line-to-line short-circuit under the impact of non-uniform soiling (soiling_LL): 1 or 2 modules in one string are shorted under varying soiling accumulation (denoted as soiling_LL1 and soiling_LL2, resistance degradation of the array under the impact of non-uniform soiling (soiling_Sdegartion): an increase in the equivalent series resistance of array under varying soiling accumulation. 10) Parallel resistance degradation of the array under the impact of non-uniform soiling (soiling_Adegartion): a decrease in the equivalent series resistance of array under varying soiling accumulation. Fig. 7 . 7Faults characteristic curves of PV array with blocking diodes. Fig. 8 . 8Faults characteristic curves of PV array without blocking diodes. Fig. 9 . 9I-V curves of different faults with full-scale dynamic fault parameter. I is the unit row vector [1, 1, ..., 1], i r is the value of the polar axis, and i  the value of the polar angle. Fig. 10 . 10Normal normalization area of similar faults under the same ambient condition. Through the analysis of the single-diode equivalent model of the PV module in Fig. 6, the calculation equations for the short-circuit current I SC and the open-circuit voltage V OC are obtained as follows: Fig. 11 . 11Graphical matrices transformed by GADF with different normalization strategies for similar faults under different environmental conditions (the matrices are colored for only visualization). Fig. 12 . 12The structure of proposed classification model of convolutional neural network with CBAM module. Fig. 13 . 13The structure of channel attention module (CAM). Fig. 14 . 14The structure of spatial attention module (SAM). Fig. 17 . 17The loss and accuracy for three input matrices. Fig. 18 . 18The loss and accuracy for three GADF normalization methods. Fig. 23 . 23The confusion matrix of PV arrays with blocking diodes. Fig. 24 . 24The confusion matrix of PV arrays without blocking diodes. The training and validation process of two blocking diode configurations are shown in Fig. 25 . 25The loss and accuracy of PV arrays with blocking diodes. Fig. 26 . 26The loss and accuracy of PV arrays without blocking diodes. Table 2 2Detailed configuration of the CNN-CBAM.Layer Output shape Detailed structure Input Layer (50, 50, 2) Convolution 1 (46, 46, 32) k = 3×3, filter = 8, stride = 1×1, padding = 1 k = 3×3, filter = 32, stride = 1×1, padding = 1 CBAM 1 (46, 46, 32) Convolution 2 (44, 44, 64) k = 3×3, filter = 64, stride = 1×1, padding = 1 CBAM 2 (44, 44, 64) Global Avgpool 64 Fully connected 16 Output Layer classes Table 3 3The specific information of the different dataset.PV array with blocking diodes PV array without blocking diodes 80% 20% 80% 20% Training data (90%) Validation data (10%) Testing data Training data (90%) Validation data (10%) Testing data Considering soiling impact Number of classes 14 14 14 14 14 14 Number of data 43192 4800 11998 43192 4800 11998 Without considering soiling impact Number of classes 9 9 9 9 9 9 Number of data 27766 3086 7713 27766 3086 7713 Table 4 4The fault diagnosis results of applying different data preprocessing methods.Precision Recall F1-score Accuracy Circumstance 1: With blocking diodes GADF I-V 98.40% 98.31% 98.35% 98.46% P-V 95.27% 95.16% 95.21% 95.33% IV-PV 98.53% 98.48% 98.50% 98.58% Recurrent Plot I-V 80.97% 80.03% 80.50% 81.77% P-V 86.87% 86.27% 86.57% 86.98% IV-PV 94.43% 94.23% 94.33% 95.12% Direct IV GTIV 95.86% 95.79% 95.82% 95.98% Circumstance 2: Without blocking diodes GADF I-V 98.38% 98.25% 98.31% 98.50% P-V 91.56% 91.32% 91.44% 92.06% IV-PV 98.38% 98.28% 98.33% 98.48% Recurrent Plot I-V 67.04% 65.53% 66.28% 65.94% P-V 79.32% 78.25% 78.78% 78.69% IV-PV 90.08% 89.86% 89.97% 90.85% Direct IV GTIV 95.61% 95.52% 95.56% 95.65% Table 5 5The fault diagnosis results of applying different GADF normalization methods.GADF Testing accuracy Normal normalization Global normalization Isc-Voc normalization Circumstance 1: With blocking diodes ANN IV 84.52% 86.81% 95.66% PV 80.44% 80.36% 92.97% IV-PV 91.42% 87.40% 96.11% Multilayer CNN IV 98.46% 98.25% 98.85% PV 95.33% 91.54% 97.83% IV-PV 98.58% 98.19% 99.10% Circumstance 2: Without blocking diodes ANN IV 72.30% 83.66% 94.97% PV 65.54% 75.40% 91.58% IV-PV 87.92% 82.96% 95.22% Multilayer CNN IV 98.50% 98.04% 98.42% PV 92.06% 91.23% 97.25% IV-PV 98.48% 97.79% 98.65% Table 6 6The fault diagnosis results of applying different CNN-based classification models.GADF Testing accuracy Normal normalization Global normalization Isc-Voc normalization Circumstance 1: With blocking diodes Multilayer CNN IV 98.46% 98.25% 98.85% PV 95.33% 91.54% 97.83% IV-PV 98.58% 98.19% 99.10% Multi-scale CNN IV 98.96% 98.75% 99.21% PV 96.75% 91.44% 98.81% IV-PV 99.10% 98.65% 99.31% Proposed CBAM-CNN IV 99.10% 99.02% 99.58% PV 98.35% 95.83% 99.08% IV-PV 99.42% 98.98% 99.62% Circumstance 2: Without blocking diodes Multilayer CNN IV 98.50% 98.04% 98.42% PV 92.06% 91.23% 97.25% IV-PV 98.48% 97.79% 98.65% Multi-scale CNN IV 98.60% 98.96% 98.73% PV 94.21% 92.92% 98.33% IV-PV 98.85% 98.54% 99.06% Proposed CBAM-CNN IV 98.92% 99.02% 99.32% PV 96.06% 94.10% 98.81% IV-PV 99.15% 98.77% 99.40% Table 7 7The fault diagnosis results of applying different classifiers and GADF methods.GADF Testing accuracy Normal normalization Global normalization Isc-Voc normalization Circumstance 1: With blocking diodes ANN IV 87.59% 93.20% 97.57% PV 93.26% 91.02% 97.57% IV-PV 98.09% 93.68% 98.26% Multilayer CNN IV 99.22% 98.28% 99.58% PV 98.41% 96.99% 99.38% IV-PV 99.42% 98.54% 99.58% Proposed CBAM-CNN IV 99.13% 98.61% 99.80% PV 98.83% 97.57% 99.51% IV-PV 99.18% 98.35% 99.84% Circumstance 2: Without blocking diodes ANN IV 75.28% 94.44% 97.81% PV 82.74% 91.34% 97.89% IV-PV 94.44% 94.83% 98.22% Multilayer CNN IV 99.22% 98.70% 99.25% PV 98.39% 96.95% 99.35% IV-PV 99.22% 98.61% 99.42% Proposed CBAM-CNN IV 99.19% 99.13% 99.77% PV 98.51% 97.54% 99.45% IV-PV 99.25% 98.38% 99.81% AcknowledgementThis work is supported by the National Natural Science Foundation of China (No. 61573046) and Program for Changjiang Scholars and Innovative Research Team in University (No. IRT1203). International Energy Agency-IEA. The Europa Directory of International Organizations 2022. Routledge2022. I Citaristi, Citaristi I. International Energy Agency-IEA. The Europa Directory of International Organizations 2022. Routledge2022. pp. 701-2. Line-line fault analysis and protection challenges in solar photovoltaic arrays. Y Zhao, De Palma, J-F Mosesian, J Lyons, R Lehman, B , IEEE Transactions on Industrial Electronics. 60Zhao Y, De Palma J-F, Mosesian J, Lyons R, Lehman B. Line-line fault analysis and protection challenges in solar photovoltaic arrays. IEEE Transactions on Industrial Electronics 2012;60:3784-95. Recent advances in failure diagnosis techniques based on performance data analysis for grid-connected photovoltaic systems. A Livera, M Theristis, G Makrides, G Georghiou, Renewable Energy. 133Livera A, Theristis M, Makrides G, Georghiou G E. Recent advances in failure diagnosis techniques based on performance data analysis for grid-connected photovoltaic systems. Renewable Energy 2019;133:126-43. Fault diagnosis in photovoltaic arrays using GBSSL method and proposing a fault correction system. H Momeni, N Sadoogi, M Farrokhifar, H F Gharibeh, IEEE Transactions on Industrial Informatics. 16Momeni H, Sadoogi N, Farrokhifar M, Gharibeh H F. Fault diagnosis in photovoltaic arrays using GBSSL method and proposing a fault correction system. IEEE Transactions on Industrial Informatics 2019;16:5300-8. A comparative evaluation of advanced fault detection approaches for PV systems. D S Pillai, F Blaabjerg, N Rajasekar, IEEE Journal of Photovoltaics. 9Pillai D S, Blaabjerg F, Rajasekar N. A comparative evaluation of advanced fault detection approaches for PV systems. IEEE Journal of Photovoltaics 2019;9:513-27. A fast MPPT-based anomaly detection and accurate fault diagnosis technique for PV arrays. C Li, Y Yang, K Zhang, C Zhu, H Wei, Energy Conversion and Management. 234113950Li C, Yang Y, Zhang K, Zhu C, Wei H. A fast MPPT-based anomaly detection and accurate fault diagnosis technique for PV arrays. Energy Conversion and Management 2021;234:113950. A comprehensive review on protection challenges and fault diagnosis in PV systems. D S Pillai, N Rajasekar, Renewable and Sustainable Energy Reviews. 91Pillai D S, Rajasekar N. A comprehensive review on protection challenges and fault diagnosis in PV systems. Renewable and Sustainable Energy Reviews 2018;91:18-40. Output power loss of crystalline silicon photovoltaic modules due to dust accumulation in Saharan environment. M Dida, S Boughali, D Bechki, H Bouguettaia, Renewable and Sustainable Energy Reviews. 124109787Dida M, Boughali S, Bechki D, Bouguettaia H. Output power loss of crystalline silicon photovoltaic modules due to dust accumulation in Saharan environment. Renewable and Sustainable Energy Reviews 2020;124:109787. A review of dust accumulation and cleaning methods for solar photovoltaic systems. H A Kazem, M T Chaichan, Al-Waeli A H Sopian, K , Journal of Cleaner Production. 276123187Kazem H A, Chaichan M T, Al-Waeli A H, Sopian K. A review of dust accumulation and cleaning methods for solar photovoltaic systems. Journal of Cleaner Production 2020;276:123187. Air pollution and soiling implications for solar photovoltaic power generation: A comprehensive review. Z Song, J Liu, H Yang, Applied Energy. 298117247Song Z, Liu J, Yang H. Air pollution and soiling implications for solar photovoltaic power generation: A comprehensive review. Applied Energy 2021;298:117247. Effect of dust and methods of cleaning on the performance of solar PV module for different climate regions: Comprehensive review. T Salamah, A Ramahi, K Alamara, A Juaidi, R Abdallah, M A Abdelkareem, Science of The Total Environment. 154050Salamah T, Ramahi A, Alamara K, Juaidi A, Abdallah R, Abdelkareem M A, et al. Effect of dust and methods of cleaning on the performance of solar PV module for different climate regions: Comprehensive review. Science of The Total Environment 2022;154050. Modeling of dust soiling effects on solar photovoltaic performance: A review. A Younis, Y Alhorr, Solar Energy. 220Younis A, Alhorr Y. Modeling of dust soiling effects on solar photovoltaic performance: A review. Solar Energy 2021;220:1074-88. Satellite based fault diagnosis of photovoltaic systems using recurrent neural networks. J Van Gompel, D Spina, C Develder, Applied Energy. 305117874Van Gompel J, Spina D, Develder C. Satellite based fault diagnosis of photovoltaic systems using recurrent neural networks. Applied Energy 2022;305:117874. EMC issues in high-power grid-connected photovoltaic plants. R Araneo, S Lammens, M Grossi, S Bertone, IEEE Transactions on Electromagnetic Compatibility. 51Araneo R, Lammens S, Grossi M, Bertone S. EMC issues in high-power grid-connected photovoltaic plants. IEEE Transactions on Electromagnetic Compatibility 2009;51:639-48. Experimental studies of fault location in PV module strings. T Takashima, J Yamaguchi, K Otani, T Oozeki, K Kato, M Ishida, Solar Energy Materials and Solar Cells. 93Takashima T, Yamaguchi J, Otani K, Oozeki T, Kato K, Ishida M. Experimental studies of fault location in PV module strings. Solar Energy Materials and Solar Cells 2009;93:1079-82. An irradiance-independent, robust ground-fault detection scheme for PV arrays based on spread spectrum time-domain reflectometry (SSTDR). S Roy, M K Alam, F Khan, J Johnson, J Flicker, IEEE Transactions on Power Electronics. 33Roy S, Alam M K, Khan F, Johnson J, Flicker J. An irradiance-independent, robust ground-fault detection scheme for PV arrays based on spread spectrum time-domain reflectometry (SSTDR). IEEE Transactions on Power Electronics 2017;33:7046- 57. Photovoltaic fault diagnosis via semisupervised ladder network with string voltage and current measures. S-Q Chen, Yang G-J Gao, W Guo, M-F , IEEE Journal of Photovoltaics. 11Chen S-Q, Yang G-J, Gao W, Guo M-F. Photovoltaic fault diagnosis via semisupervised ladder network with string voltage and current measures. IEEE Journal of Photovoltaics 2020;11:219-31. Random forest based intelligent fault diagnosis for PV arrays using array voltage and string currents. Z Chen, F Han, L Wu, J Yu, S Cheng, P Lin, Energy Conversion and Management. 178Chen Z, Han F, Wu L, Yu J, Cheng S, Lin P, et al. Random forest based intelligent fault diagnosis for PV arrays using array voltage and string currents. Energy Conversion and Management 2018;178:250-64. Fault diagnosis for photovoltaic array based on convolutional neural network and electrical time series graph. X Lu, P Lin, S Cheng, Y Lin, Z Chen, L Wu, Energy Conversion and Management. 196Lu X, Lin P, Cheng S, Lin Y, Chen Z, Wu L, et al. Fault diagnosis for photovoltaic array based on convolutional neural network and electrical time series graph. Energy Conversion and Management 2019;196:950-65. A new monitoring technique for fault detection and classification in PV systems based on rate of change of voltage-current trajectory. Abd El-Ghany, H A Elgebaly, A E Taha, I B M , International Journal of Electrical Power & Energy Systems. 133107248Abd el-Ghany H A, Elgebaly A E, Taha I B M. A new monitoring technique for fault detection and classification in PV systems based on rate of change of voltage-current trajectory. International Journal of Electrical Power & Energy Systems 2021;133:107248. Fault diagnosis approach for photovoltaic array based on the stacked auto-encoder and clustering with I-V curves. Y Liu, K Ding, J Zhang, Y Li, Z Yang, W Zheng, Energy Conversion and Management. 245114603Liu Y, Ding K, Zhang J, Li Y, Yang Z, Zheng W, et al. Fault diagnosis approach for photovoltaic array based on the stacked auto-encoder and clustering with I-V curves. Energy Conversion and Management 2021;245:114603. Intelligent fault identification for industrial automation system via multi-scale convolutional generative adversarial network with partially labeled samples. T Pan, J Chen, J Xie, Y Chang, Z Zhou, ISA Trans. 101Pan T, Chen J, Xie J, Chang Y, Zhou Z. Intelligent fault identification for industrial automation system via multi-scale convolutional generative adversarial network with partially labeled samples. ISA Trans 2020;101:379-89. An improved modeling method to determine the model parameters of photovoltaic (PV) modules using differential evolution (DE). K Ishaque, Z Salam, Solar Energy. 85Ishaque K, Salam Z. An improved modeling method to determine the model parameters of photovoltaic (PV) modules using differential evolution (DE). Solar Energy 2011;85:2349-59. Fault detection and diagnosis methods for photovoltaic systems: A review. A Mellit, G M Tina, S A Kalogirou, Renewable and Sustainable Energy Reviews. 91Mellit A, Tina G M, Kalogirou S A. Fault detection and diagnosis methods for photovoltaic systems: A review. Renewable and Sustainable Energy Reviews 2018;91:1-17. Slama-Belkhodja I. Fault detection and monitoring systems for photovoltaic installations: A review. A Triki-Lahiani, A B-B Abdelghani, Renewable and Sustainable Energy Reviews. 82Triki-Lahiani A, Abdelghani A B-B, Slama-Belkhodja I. Fault detection and monitoring systems for photovoltaic installations: A review. Renewable and Sustainable Energy Reviews 2018;82:2680-92. Monitoring and fault detection in photovoltaic systems based on inverter measured string IV curves. 31st European Photovoltaic Solar Energy Conference and Exhibition. WIP Wirtschaft und Infrastruktur GmbH & Co Planungs KG2015. S Spataru, D Sera, T Kerekes, R Teodorescu, Spataru S, Sera D, Kerekes T, Teodorescu R. Monitoring and fault detection in photovoltaic systems based on inverter measured string IV curves. 31st European Photovoltaic Solar Energy Conference and Exhibition. WIP Wirtschaft und Infrastruktur GmbH & Co Planungs KG2015. pp. 1667-74. Deep residual network based fault detection and diagnosis of photovoltaic arrays using current-voltage curves and ambient conditions. Z Chen, Y Chen, L Wu, S Cheng, P Lin, Energy Conversion and Management. 198111793Chen Z, Chen Y, Wu L, Cheng S, Lin P. Deep residual network based fault detection and diagnosis of photovoltaic arrays using current-voltage curves and ambient conditions. Energy Conversion and Management 2019;198:111793. A novel fault identification method for photovoltaic array via convolutional neural network and residual gated recurrent unit. W Gao, R-J Wai, IEEE Access. 8Gao W, Wai R-J. A novel fault identification method for photovoltaic array via convolutional neural network and residual gated recurrent unit. IEEE Access 2020;8:159493-510. A novel fault diagnosis technique for photovoltaic systems based on artificial neural networks. W Chine, A Mellit, V Lughi, A Malek, G Sulligoi, A Pavan, Renewable Energy. 90Chine W, Mellit A, Lughi V, Malek A, Sulligoi G, Pavan A M. A novel fault diagnosis technique for photovoltaic systems based on artificial neural networks. Renewable Energy 2016;90:501-12. PV shading fault detection and classification based on IV curve using principal component analysis: Application to isolated PV system. S Fadhel, C Delpha, D Diallo, I Bahri, A Migan, M Trabelsi, Solar Energy. 179Fadhel S, Delpha C, Diallo D, Bahri I, Migan A, Trabelsi M, et al. PV shading fault detection and classification based on IV curve using principal component analysis: Application to isolated PV system. Solar Energy 2019;179:1-10. Intelligent fault diagnosis of photovoltaic array based on variable predictive models and I-V curves. Y Liu, K Ding, J Zhang, Y Lin, Z Yang, X Chen, Solar Energy. 237Liu Y, Ding K, Zhang J, Lin Y, Yang Z, Chen X, et al. Intelligent fault diagnosis of photovoltaic array based on variable predictive models and I-V curves. Solar Energy 2022;237:340-51. An assessment of series resistance estimation techniques for different silicon based SPV modules. R Singh, M Sharma, R Rawat, C Banerjee, Renewable and Sustainable Energy Reviews. 98Singh R, Sharma M, Rawat R, Banerjee C. An assessment of series resistance estimation techniques for different silicon based SPV modules. Renewable and Sustainable Energy Reviews 2018;98:199-216. A fault diagnosis method for photovoltaic arrays based on fault parameters identification. Y Li, K Ding, J Zhang, F Chen, X Chen, J Wu, Renewable Energy. 143Li Y, Ding K, Zhang J, Chen F, Chen X, Wu J. A fault diagnosis method for photovoltaic arrays based on fault parameters identification. Renewable Energy 2019;143:52-63. Intelligent fault diagnosis of photovoltaic arrays based on optimized kernel extreme learning machine and I-V characteristics. Z Chen, L Wu, S Cheng, P Lin, Y Wu, W Lin, Applied Energy. 204Chen Z, Wu L, Cheng S, Lin P, Wu Y, Lin W. Intelligent fault diagnosis of photovoltaic arrays based on optimized kernel extreme learning machine and I-V characteristics. Applied Energy 2017;204:912-31. A shadow fault detection method based on the standard error analysis of IV curves. M Bressan, El Basri, Y Galeano, A G , Alonso C , Renewable Energy. 99Bressan M, El Basri Y, Galeano A G, Alonso C. A shadow fault detection method based on the standard error analysis of IV curves. Renewable Energy 2016;99:1181-90. Fault diagnosis of cracks in crystalline silicon photovoltaic modules through IV curve. M Ma, Z Zhang, Z Xie, P Yun, X Zhang, F Li, Microelectronics Reliability. 114113848Ma M, Zhang Z, Xie Z, Yun P, Zhang X, Li F. Fault diagnosis of cracks in crystalline silicon photovoltaic modules through IV curve. Microelectronics Reliability 2020;114:113848. Compound fault diagnosis model for Photovoltaic array using multiscale SE-ResNet. P Lin, Z Qian, X Lu, Y Lin, Y Lai, S Cheng, Sustainable Energy Technologies and Assessments. 50101785Lin P, Qian Z, Lu X, Lin Y, Lai Y, Cheng S, et al. Compound fault diagnosis model for Photovoltaic array using multi- scale SE-ResNet. Sustainable Energy Technologies and Assessments 2022;50:101785. Design of hybrid artificial bee colony algorithm and semi-supervised extreme learning machine for PV fault diagnoses by considering dust impact. J-M Huang, R-J Wai, G-J Yang, IEEE Transactions on Power Electronics. 35Huang J-M, Wai R-J, Yang G-J. Design of hybrid artificial bee colony algorithm and semi-supervised extreme learning machine for PV fault diagnoses by considering dust impact. IEEE Transactions on Power Electronics 2019;35:7086-99. Fault diagnosis of photovoltaic panels using full I-V characteristics and machine learning techniques. B Li, C Delpha, A Migan-Dubois, D Diallo, Energy Conversion and Management. 248114785Li B, Delpha C, Migan-Dubois A, Diallo D. Fault diagnosis of photovoltaic panels using full I-V characteristics and machine learning techniques. Energy Conversion and Management 2021;248:114785. Impact of soiling on IV-curves and efficiency of PV-modules. C Schill, S Brachmann, M Koehl, Solar Energy. 112Schill C, Brachmann S, Koehl M. Impact of soiling on IV-curves and efficiency of PV-modules. Solar Energy 2015;112:259-62. Dust and PV Performance in Nigeria: A review. Y N Chanchangi, A Ghosh, S Sundaram, T Mallick, Renewable and Sustainable Energy Reviews. 121109704Chanchangi Y N, Ghosh A, Sundaram S, Mallick T K. Dust and PV Performance in Nigeria: A review. Renewable and Sustainable Energy Reviews 2020;121:109704. Study of bypass diodes configuration on PV modules. S Silvestre, A Boronat, A Chouder, Applied Energy. 86Silvestre S, Boronat A, Chouder A. Study of bypass diodes configuration on PV modules. Applied Energy 2009;86:1632- 40. Detection of cleaning interventions on photovoltaic modules with machine learning. M Heinrich, S Meunier, A Same, L Queval, A Darga, L Oukhellou, Applied Energy. 263114642Heinrich M, Meunier S, Same A, Queval L, Darga A, Oukhellou L, et al. Detection of cleaning interventions on photovoltaic modules with machine learning. Applied Energy 2020;263:114642. Modeling and quantifying dust accumulation impact on PV module performance. M Al-Addous, Z Dalala, F Alawneh, C B Class, Solar Energy. 194Al-Addous M, Dalala Z, Alawneh F, Class C B. Modeling and quantifying dust accumulation impact on PV module performance. Solar Energy 2019;194:86-102. Impact of dust on the performance of solar photovoltaic (PV) systems under United Arab Emirates weather conditions. A A Hachicha, I Al-Sawafta, Z Said, Renewable Energy. 141Hachicha A A, Al-Sawafta I, Said Z. Impact of dust on the performance of solar photovoltaic (PV) systems under United Arab Emirates weather conditions. Renewable Energy 2019;141:287-97. Photovoltaic model identification using particle swarm optimization with inverse barrier constraint. J J Soon, K-S Low, IEEE Transactions on Power Electronics. 27Soon J J, Low K-S. Photovoltaic model identification using particle swarm optimization with inverse barrier constraint. IEEE Transactions on Power Electronics 2012;27:3975-83. Model-based degradation analysis of photovoltaic modules through series resistance estimation. J D Bastidas-Rodríguez, E Franco, G Petrone, C A Ramos-Paja, G Spagnuolo, IEEE Transactions on Industrial Electronics. 62Bastidas-Rodríguez J D, Franco E, Petrone G, Ramos-Paja C A, Spagnuolo G. Model-based degradation analysis of photovoltaic modules through series resistance estimation. IEEE Transactions on Industrial Electronics 2015;62:7256-65. IEC 60891, Photovoltaic Devices. Procedures for Temperature and Irradiance Corrections to Measure IV Characteristics. I Commission, International Electrotechnical Commission. Commission I E. IEC 60891, Photovoltaic Devices. Procedures for Temperature and Irradiance Corrections to Measure IV Characteristics. International Electrotechnical Commission: Geneva, Switzerland 2007. Evaluation and improvement of IEC 60891 correction methods for I-V curves of defective photovoltaic panels. B Li, A Migan-Dubois, C Delpha, D Diallo, Solar Energy. 216Li B, Migan-Dubois A, Delpha C, Diallo D. Evaluation and improvement of IEC 60891 correction methods for I-V curves of defective photovoltaic panels. Solar Energy 2021;216:225-37. Solar PV DC nanogrid dynamic modeling applying the polynomial computational method for MPPT. M F Bauomy, H Gamal, A Shaltout, Advances in Clean Energy Technologies. Elsevier2021. Bauomy M F, Gamal H, Shaltout A A. Solar PV DC nanogrid dynamic modeling applying the polynomial computational method for MPPT. Advances in Clean Energy Technologies. Elsevier2021. pp. 19-87. Imaging time-series to improve classification and imputation. Z Wang, T Oates, Twenty-Fourth International Joint Conference on Artificial Intelligence2015. Wang Z, Oates T. Imaging time-series to improve classification and imputation. Twenty-Fourth International Joint Conference on Artificial Intelligence2015. Comprehensive approach to modeling and simulation of photovoltaic arrays. M G Villalva, J R Gazoli, Ruppert Filho, E , IEEE Transactions on Power Electronics. 24Villalva M G, Gazoli J R, Ruppert Filho E. Comprehensive approach to modeling and simulation of photovoltaic arrays. IEEE Transactions on Power Electronics 2009;24:1198-208. Modeling and circuit-based simulation of photovoltaic arrays. M G Villalva, J R Gazoli, Ruppert Filho, E , Brazilian Power Electronics Conference. IEEE2009. Villalva M G, Gazoli J R, Ruppert Filho E. Modeling and circuit-based simulation of photovoltaic arrays. 2009 Brazilian Power Electronics Conference. IEEE2009. pp. 1244-54. Improvement and validation of a model for photovoltaic array performance. De Soto, W Klein, S A Beckman, W , Solar Energy. 80De Soto W, Klein S A, Beckman W A. Improvement and validation of a model for photovoltaic array performance. Solar Energy 2006;80:78-88. Review of deep learning: Concepts, CNN architectures, challenges, applications, future directions. L Alzubaidi, J Zhang, A J Humaidi, Al-Dujaili , A Duan, Y Al-Shamma, O , Journal of Big Data. 8Alzubaidi L, Zhang J, Humaidi A J, Al-Dujaili A, Duan Y, Al-Shamma O, et al. Review of deep learning: Concepts, CNN architectures, challenges, applications, future directions. Journal of Big Data 2021;8:1-74. Convolutional block attention module. S Woo, J Park, J-Y Lee, I S Kweon, Cbam, Proceedings of the European conference on computer vision (ECCV. the European conference on computer vision (ECCVWoo S, Park J, Lee J-Y, Kweon I S. Cbam: Convolutional block attention module. Proceedings of the European conference on computer vision (ECCV)2018. pp. 3-19. Credible intervals for precision and recall based on a K-fold cross-validated beta distribution. Y Wang, J Li, Neural Computation. 28Wang Y, Li J. Credible intervals for precision and recall based on a K-fold cross-validated beta distribution. Neural Computation 2016;28:1694-722. A review on evaluation metrics for data classification evaluations. M Hossin, M N Sulaiman, International journal of data mining & knowledge management process. 51Hossin M, Sulaiman M N. A review on evaluation metrics for data classification evaluations. International journal of data mining & knowledge management process 2015;5:1.
[]
[ "Denoising of discrete-time chaotic signals using echo state networks", "Denoising of discrete-time chaotic signals using echo state networks" ]
[ "André L O Duarte \nEscola Politécnica\nUniversity of São Paulo\nSão PauloBrazil\n", "Marcio Eisencraft \nEscola Politécnica\nUniversity of São Paulo\nSão PauloBrazil\n" ]
[ "Escola Politécnica\nUniversity of São Paulo\nSão PauloBrazil", "Escola Politécnica\nUniversity of São Paulo\nSão PauloBrazil" ]
[]
Noise reduction is a relevant topic when considering the application of chaotic signals in practical problems, such as communication systems or modeling biomedical signals. In this paper an echo state network (ESN) is employed to denoise a discrete-time chaotic signal corrupted by additive white Gaussian noise. The choice for applying ESNs in this context is motivated by their successful exploitation for separation and prediction of chaotic signals. The resultsshow that the processing gain of ESN is higher than that of the Wiener filter, especially when the power spectral density of the chaotic signals is white.
null
[ "https://export.arxiv.org/pdf/2304.06516v1.pdf" ]
258,108,403
2304.06516
9ff6f3cbffa670db667718228190bbe23eb64180
Denoising of discrete-time chaotic signals using echo state networks André L O Duarte Escola Politécnica University of São Paulo São PauloBrazil Marcio Eisencraft Escola Politécnica University of São Paulo São PauloBrazil Denoising of discrete-time chaotic signals using echo state networks 2010 MSC: 00-01, 99-00echo state networksdenoisingdynamical systemsmachine learningreservoir computing Noise reduction is a relevant topic when considering the application of chaotic signals in practical problems, such as communication systems or modeling biomedical signals. In this paper an echo state network (ESN) is employed to denoise a discrete-time chaotic signal corrupted by additive white Gaussian noise. The choice for applying ESNs in this context is motivated by their successful exploitation for separation and prediction of chaotic signals. The resultsshow that the processing gain of ESN is higher than that of the Wiener filter, especially when the power spectral density of the chaotic signals is white. Introduction As the number of researches on new communication schemes using chaotic signals as broadband carriers grows, see e.g. [1,2], it makes sense to look for ways to reduce the noise in chaotic signals. In fact, many different techniques have been proposed in an effort to solve 5 the denoising problem [3]. In recent years, there has been an increased interest in machine learning techniques due to their great adaptability, despite their rather simple design principles. Therefore, it is natural to investigate their performance for noise reduction. In particular, Echo State Networks (ESNs) emerged in the early 2000s in an effort to mitigate the problems faced when designing Recurrent Neural Networks (RNNs): (i) the training of RNNs is cumbersome when using classical methods based on gradient descent, such as backpropagation; (ii) the lack of convergence guarantee and (iii) the high computational effort required due to the large number of parameters to optimize [4]. 15 The distinguishing feature of ESNs is that only the weights in the output layer are optimized during training [4]. In this scenario, the network without its input and output nodes is called a reservoir. The remaining links connecting the input to the reservoir and the connections within the reservoir have weights assigned according to a random distribution [5]. These weights do not change 20 during training, minimizing the number of parameters to optimize [4]. They are widely used due to their simplicity and low computational cost, for example in many applications related to dynamical systems, such as the separation [6] and prediction [7] of chaotic signals. In [8] an investigation of ESNs for denoising was made. There, a wave signal 25 consisting of four sinusoids with random phases and time-varying envelopes is corrupted by additive white Gaussian noise (AWGN). At [9], ESNs were used to denoise electroencephalogram signals. The performance was evaluated in terms of the signal-to-noise ratio (SNR) and the root mean square error. In the present work, discrete-time chaotic signals are disturbed by AWGN. 30 The aperiodic behavior together with the sensitive dependence on initial conditions of chaotic signals [10] add to the difficulty of the denoising task. Performance is measured in terms of processing gain, providing a clearer and more meaningful understanding of the network's capabilities. The skew tent map [11] is used to obtain the chaotic signals studied in this 35 article. The power spectral density (PSD) of the orbits of this map is determined by its single parameter in a simple deterministic way [11]. The rationale for considering this map here is thus justified, because knowing the PSD in the context of noise reduction is more than welcome and useful for performing an analysis of how orbits with different PSDs affect the denoising task. Figure 1 shows a block diagram of the problem at hand. After the training period, the ESN output y(n) is expected to approximate the desired noiseless chaotic signal d(n) from the corrupted input signal u(n) = d(n) + w(n),(1) where w(n) is AWGN. It is relevant to note that in this paper we consider chaotic signals generated 45 by a discrete-time system, unlike most of the literature that generally considers off-the-shelf continuous-time dynamical systems, such as Lorenz or Rossler systems [12,13]. The task at hand here is more challenging because it does not benefit from the smoothness found in continuous-time signals. Depending on its parameter, the skew tent map generate white signals with impulsive 50 autocorrelation that are not easily distinguishable from white noise. Chaotic Signal Generator + AWGN ESN + d(n) y(n) + - w(n) u(n) The remainder of this article is divided into four sections. Section 2 gives a brief description of the chaotic signal generator considered here. Section 3 discusses the operation of the ESN. Section 4 presents and discusses the results obtained. Finally, our main conclusions are summarized in Section 5. 55 Considered Chaotic Signals Chaotic signals are bounded, aperiodic and have sensitive dependence on initial conditions (SDIC) [10]. The one-dimensional discrete-time chaotic signal d(n) considered in this paper is obtained from the skew tent map defined by [11] 60 d(n + 1) = f (d(n)) =      1−α 1+α + 2 1+α d(n), −1 < d(n) < α 1+α 1−α − 2 1−α d(n), α ≤ d(n) < 1 ,(2) with initial condition d(0) d 0 ∈ (−1, 1) and fixed parameter α ∈ (−1, 1), the x-coordinate of the peak of the tent. Figure 2(a) shows a plot of f (·) and Figure 2(b) depicts two orbits of f (·), with slightly different initial conditions, to illustrate the SDIC property. In [11] it was shown that the PSD of the orbits of f (·) is given by α = −0.9 α = −0.7 α = 0 α = 0.7 α = 0.9 α d f (d) n d(n) ω/π P (ω) 10 −2P (ω) = 1 − α 2 3(1 + α 2 − 2α cos ω) .(3) As shown in Figure 2(c), P (ω) is white for α = 0. Also, the higher the value of |α|, the more power is concentrated at high (α > 0) or low (α < 0) frequencies. In other words, the further away from α = 0, the more the spectral characteristics of the generated chaotic signal will differ from a white noise signal. This direct relationship between the α parameter and the PSD is what led us to 70 choose this map. It allows us to analyze, in Section 4, the influence of the PSD of the chaotic signal on the performance of the ESN in the noise reduction task. through which information is exchanged [4,5]. Echo State Networks reservoir input layer output layer In the following, we review the main aspects of each layer of the considered ESN. u(n) ∈ R Nu y(n) d(n) ∈ R Nd r(n) ∈ R N Reservoir The reservoir consists of N nodes connected to each other. The weight of each link is randomly determined and are the entries of the internal matrix W N ×N . They are obtained as follows: first, the entries of an auxiliary matrix 100 W aux N ×N are drawn from a uniform distribution in [−1, 1]. Then its spectral radius ρ is determined. Finally, we compute W = λ (W aux /ρ), where λ is a parameter. Since W aux /ρ has unit spectral radius, it follows that W has spectral radius λ. This parameter is selected here using the algorithm described in Section 4. 105 Each node k, 1 ≤ k ≤ N , has an internal state r k ∈ R, which forms the internal state vector r(n) ∈ R N . At each time n + 1, the internal state vector is updated according to the leaky-integrator model equation r(n + 1) = (1 − a) r(n) + a tanh W in 1 u(n) T + W r(n) ,(4) where the leakage parameter is a ∈ [0, 1] and the initial condition is r(− + 1) = 0. Output layer 115 The goal of the ESN is to have an output that approximates a desired signal d(n) ∈ R N d . Let D N d Numerical simulations The desired one-dimensional chaotic signal d(n) is generated by (2) Selecting the ESN parameters To select the ESN parameters a, λ, p and q, we have generated u(n) with an arbitrary d 0 ∈ [−1, 1], α = 0.1 and SNR in = 2.0 dB. This procedure is repeated, obtaining a opt i , λ opt i , p opt i , q opt i , i = 2, 3, . . .. The search continues until the selected values for each parameter do not change. 150 The evolution of the selected parameters along the procedure is shown in Figure 6 shows examples of the estimated signals using ESN and a Wiener Filter (WF) [14] for SNR in = 2.0 dB and skew tent map parameter α = 0.9. We denote the ESN estimated signal by d(n) and the WF estimated signal by Using the optimal parameters obtained in Section 4.1, the ESN was trained 165 and tested for different values of α ∈ (−1, 1). Figure 7 presents the obtained results in terms of mean processing gain after five training/testing scenarios. As a benchmark, a WF with 10 taps was employed to perform the same denoising task. The error bars indicate the standard deviation calculated from the five repetitions. of α. Furthermore, as expected, the performance of both techniques degrades as α approaches zero. Note that for α = 0, the PSD of the chaotic signal is white and the processing gain for the WF is zero. The processing gain is maximum for α = 0.95, where it reaches 7.17 ± 0.05 dB. 175 In papers such as [12, 13] processing gains ranging from 6.0 dB to 25.0 dB have been obtained using other techniques but always considering continuoustime chaotic systems. As mentioned in Section 1, we consider the task of denoising discrete-time chaotic signals to be much more difficult. Conclusions 180 In this paper, we have analyzed the use of an ESN for noise reduction in a chaotic signal. Unlike other works in the literature, we considered signals generated by discrete-time maps, whose lack of smoothness makes the task more challenging. As chaotic signal generator, we considered the skew tent map be- cause the dependence of its PSD on its single parameter is known, which allows 185 an analysis of the influence of spectral whiteness on the denoising performance. We have shown that by selecting and tuning the network parameters, the ESN outperforms a WF, the optimal linear filtering technique, in all cases, especially when the PSD of the chaotic signal is white, resulting in zero gain for the WF. For increasing values of |α|, the PSD of the chaotic signals becomes 190 narrowband and the obtained processing gain is higher. As a future investigation, we intend to evaluate how the proposed noise reduction technique can be used to improve the performance of chaos-based communication systems in noisy channels. Figure 1 : 1Problem formulation: noise reduction of a chaotic signal using ESN. Figure 2 2: a) The skew tent map (2); b) two orbits for α = 0.7 with initial conditions d 0 = 0 (red line) and d 0 = 10 −6 (blue line) showing SDIC; c) PSD for different values of α. Figure 3 3shows a schematic of an ESN. Its purpose is to use an input signal u(n) to approximate a target signal d(n) after a training period. It consists of 75 (i) an input layer, (ii) the so-called reservoir and (iii) an output layer. Each of these parts is composed of information processing nodes connected by links, Figure 3 : 3ESN architecture, indicating the signals u(n), y(n), d(n) and the internal state vector r(n). role of the input layer is to preprocess the input signal in order to control the amount of nonlinearity of the ESN [5].Given an input signal u(n) ∈ R Nu , the input layer computesW in [ 1 u(n) ] T ,where W in is N × N u + 1. So there are N u + 1 entry nodes, one for each 85 dimension of u(n) and an extra one for the bias. As it is shown in the followingsubsection, W in [ 1 u(n) ]T is part of the argument of a tanh(·) function, which is present in the state vector update equation(4). Since tanh(·) has approximately linear behavior for arguments close to zero, but nonlinear for larger arguments, it follows that W in effectively determines the amount of nonlinearity behind90 the ESN operation. The matrix W in is called the input matrix and its entries are samples from uniform distributions. The first column entries are drawn from a uniform distribution in [−p, p], while the remaining entries follow a uniform distribution in [−q, q]. Both p and q are chosen here according to the algorithm described in 95 Section 4. 110 After a transient period of time steps the training period begins. During L ∈ N training time steps, the corresponding state vectors are collected in the trajectory matrix T N ×L , defined by T = [ r(1) r(2) ... r(L) ]. which is passed to the output layer. ×L be the matrix of samples of the desired signal available for training, i.e., D = [ d(1) d(2) ... d(L) ]. The optimized weights of the connections from the reservoir to the output are given by W out = DT + , where T + is the Moore-Penrose pseudoinverse of T . After training, the output signal of the 120 ESN is calculated as the linear combination y(n) = W out r(n).At this point, our description of the ESN is almost complete. Only four parameters remain to be adjusted: (i) the leakage parameter a, (ii) the spectral radius λ of W and the scalars (iii) p and (iv) q that define the intervals of the uniform distributions used to obtain W in . Our methodology for choosing them 125 is described along with the numerical results in Section 4. for a given α and initial condition d 0 . The corrupted input u(n) of the ESN is then (1), where the AWGN w(n) power is determined by a chosen SNR in .Figure 4 shows examples of d(n), w(n) and u(n) for d 0 = 0, α = 0.7 and SNR in = 2.0 dB.Our goal is to train the ESN to reduce the noise component in u(n). This means that after sufficient training, the output signal y(n) of the network should mimic the desired chaotic signal d(n), using only u(n) as input.The ESN was designed with N = 500 nodes, a transient of = 200 samples,135 and L = 25000 samples for training. For the estimation of the output SNR (SNR out ), 10 6 samples were considered. 140Figure 4 : 0 40As a first step, we consider λ = 0.05, p = 0 and q = 0.5. Then, keepingthem fixed, we have varied a from 0 to 1 in 0.05 steps. In each case, the ESN Examples of training signals: a) noiseless chaotic signal d(n) for α = 0.7 and d 0 = 0; b) AWGN for SNR in = 2.0 dB; c) input signal u(n) (blue line) for the ESN; d(n) is also shown for comparison. was trained and tested. Thus, the value a = a opt 1 that maximizes G[dB] SNR out [dB] − SNR in [dB] is determined. Then, analogously, we get λ opt 1 , where λ varies between 0.05 and 1 in 0.05 steps, using a = a opt 1 , p = 0 and q = , in that order, by first varying p between 0 and 10 in 0.5 steps and q between 0.5 and 10 in 0.5 steps, keeping all other parameters at their current selected values. Figure 5 . 5It took 12 iterations for all parameters to stop changing. The final selected parameters are a = 0.8, λ = 0.75, p = 1.5 and q = 1.0 which are used in our simulations. Figure 5 : 5Selection of the ESN parameters. Parameters at each iteration: a) a: leakage parameter ; b) λ: spectral radius of W ; c) p and d) q: parameters of the uniform distributions used to determine the entries of W in ; e) processing gain G[dB] at each iteration.d W (n). In this case, SNR out = 7.7 dB for the ESN and SNR out = 5.1 dB for the WF, showing that the ESN outperforms the WF in this scenario.160As can be seen from (3) andFigure 2(c), the PSD of the chaotic signal becomes whiter as |α| approaches zero. Thus, in terms of autocorrelation it becomes similar to the corrupting noise, denoising it becomes more difficult and one can expect the processing gain to decrease with |α|. Figure 6 : 6Noise reduction for α = 0.9 and SNR in = 2.0 dB. Black lines represent the desired signal d(n). a) Input signal u(n); b) WF estimated signal d W (n); c) ESN estimated signal d(n). Figure 7 : 7Processing gain in dB as a function of the skew tent map parameter α using an ESN and a WF. M Eisencraft, R R F Attux, Chaotic Signals in Digital Communications. R. SuyamaTaylor & Francis IncM. Eisencraft, R. R. F. Attux, R. Suyama (Eds.), Chaotic Signals in Dig- ital Communications (Electrical Engineering & Applied Signal Processing Series), Taylor & Francis Inc, 2013. Chaos for communication. M S Baptista, 10.1007/s11071-021-06644-4Nonlinear Dynamics. 1052M. S. Baptista, Chaos for communication, Nonlinear Dynamics 105 (2) (2021) 1821-1841. doi:10.1007/s11071-021-06644-4. Advanced Digital Signal Processing and Noise Reduction. S V Vaseghi, Wiley4th EditionS. V. Vaseghi, Advanced Digital Signal Processing and Noise Reduction, 4th Edition, Wiley, 2009. The "echo state" approach to analysing and training recurrent neural networks. H Jaeger, German National Research Center for Information TechnologyTech. Rep. 148H. Jaeger, The "echo state" approach to analysing and training recurrent neural networks, Tech. Rep. 148, German National Research Center for Information Technology (2001). A practical guide to applying echo state networks. M Lukoševičius, 10.1007/978-3-642-35289-8_36Lecture Notes in Computer Science. Berlin HeidelbergSpringerM. Lukoševičius, A practical guide to applying echo state networks, in: Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2012, pp. 659-686. doi:10.1007/978-3-642-35289-8_36. Separation of chaotic signals by reservoir computing, Chaos: An Interdisciplinary Journal of Non-215 linear. S Krishnagopal, M Girvan, E Ott, B R Hunt, 10.1063/1.5132766Science. 30223123S. Krishnagopal, M. Girvan, E. Ott, B. R. Hunt, Separation of chaotic sig- nals by reservoir computing, Chaos: An Interdisciplinary Journal of Non- 215 linear Science 30 (2) (2020) 023123. doi:10.1063/1.5132766. Hybrid regularized echo state network for multivariate chaotic time series prediction. M Xu, M Han, T Qiu, H Lin, 10.1109/tcyb.2018.2825253IEEE Transactions on Cybernetics. 496M. Xu, M. Han, T. Qiu, H. Lin, Hybrid regularized echo state network for multivariate chaotic time series prediction, IEEE Transactions on Cyber- netics 49 (6) (2019) 2305-2315. doi:10.1109/tcyb.2018.2825253. Clustered and deep echo state 220 networks for signal noise reduction. L De Oliveira Junior, F Stelzer, L Zhao, 10.1007/s10994-022-06135-6Machine Learning. 1118L. de Oliveira Junior, F. Stelzer, L. Zhao, Clustered and deep echo state 220 networks for signal noise reduction, Machine Learning 111 (8) (2022) 2885- 2904. doi:10.1007/s10994-022-06135-6. EEG denoising through a wide and deep echo state network optimized by UPSO algorithm. W Sun, Y Su, X Wu, X Wu, Y Zhang, 10.1016/j.asoc.2021.107149Applied Soft Computing. 105107149W. Sun, Y. Su, X. Wu, X. Wu, Y. Zhang, EEG denoising through a wide and deep echo state network optimized by UPSO algorithm, Applied Soft Computing 105 (2021) 107149. doi:10.1016/j.asoc.2021.107149. 225 Chaos: An Introduction to Dynamical Systems. K T Alligood, T Sauer, SpringerK. T. Alligood, T. Sauer, Chaos: An Introduction to Dynamical Systems, Springer, 1997. Spectral properties of chaotic signals generated by the skew tent map. M Eisencraft, D Kato, L Monteiro, 10.1016/j.sigpro.2009.06.018Signal Processing. 901M. Eisencraft, D. Kato, L. Monteiro, Spectral properties of chaotic signals generated by the skew tent map, Signal Processing 90 (1) (2010) 385-390. doi:10.1016/j.sigpro.2009.06.018. Noise smoothing for nonlinear time series using wavelet soft threshold. M Han, Y Liu, J Xi, W Guo, 10.1109/lsp.2006.881518IEEE Signal Processing Letters. 141M. Han, Y. Liu, J. Xi, W. Guo, Noise smoothing for nonlinear time series using wavelet soft threshold, IEEE Signal Processing Letters 14 (1) (2007) 62-65. doi:10.1109/lsp.2006.881518. Chaotic signal denoising based on simplified convolutional denoising auto-encoder. S Lou, J Deng, S Lyu, 10.1016/j.chaos.2022.112333Chaos, Solitons and Fractals. 161112333S. Lou, J. Deng, S. Lyu, Chaotic signal denoising based on simplified con- volutional denoising auto-encoder, Chaos, Solitons and Fractals 161 (2022) 235 112333. doi:10.1016/j.chaos.2022.112333. Adaptive filter theory. S S Haykin, Prentice HallS. S. Haykin, Adaptive filter theory, Prentice Hall, 2002.
[]
[ "Interactron: Embodied Adaptive Object Detection", "Interactron: Embodied Adaptive Object Detection" ]
[ "Klemen Kotar \nInstitute for AI\n\n", "PRIOR @Roozbeh Mottaghi \nInstitute for AI\n\n", "Allen \nInstitute for AI\n\n" ]
[ "Institute for AI\n", "Institute for AI\n", "Institute for AI\n" ]
[]
Over the years various methods have been proposed for the problem of object detection. Recently, we have witnessed great strides in this domain owing to the emergence of powerful deep neural networks. However, there are typically two main assumptions common among these approaches. First, the model is trained on a fixed training set and is evaluated on a pre-recorded test set. Second, the model is kept frozen after the training phase, so no further updates are performed after the training is finished. These two assumptions limit the applicability of these methods to real-world settings. In this paper, we propose Interactron, a method for adaptive object detection in an interactive setting, where the goal is to perform object detection in images observed by an embodied agent navigating in different environments. Our idea is to continue training during inference and adapt the model at test time without any explicit supervision via interacting with the environment. Our adaptive object detection model provides a 7.2 point improvement in AP (and 12.7 points in AP 50 ) over DETR[5], a recent, high-performance object detector. Moreover, we show that our object detection model adapts to environments with completely different appearance characteristics, and performs well in them. The code is available at: https://github.com/allenai/interactron.
10.1109/cvpr52688.2022.01444
[ "https://export.arxiv.org/pdf/2202.00660v3.pdf" ]
246,442,335
2202.00660
e23cb74c60a46a7bfc5b3b89178f38f00648af44
Interactron: Embodied Adaptive Object Detection Klemen Kotar Institute for AI PRIOR @Roozbeh Mottaghi Institute for AI Allen Institute for AI Interactron: Embodied Adaptive Object Detection Over the years various methods have been proposed for the problem of object detection. Recently, we have witnessed great strides in this domain owing to the emergence of powerful deep neural networks. However, there are typically two main assumptions common among these approaches. First, the model is trained on a fixed training set and is evaluated on a pre-recorded test set. Second, the model is kept frozen after the training phase, so no further updates are performed after the training is finished. These two assumptions limit the applicability of these methods to real-world settings. In this paper, we propose Interactron, a method for adaptive object detection in an interactive setting, where the goal is to perform object detection in images observed by an embodied agent navigating in different environments. Our idea is to continue training during inference and adapt the model at test time without any explicit supervision via interacting with the environment. Our adaptive object detection model provides a 7.2 point improvement in AP (and 12.7 points in AP 50 ) over DETR[5], a recent, high-performance object detector. Moreover, we show that our object detection model adapts to environments with completely different appearance characteristics, and performs well in them. The code is available at: https://github.com/allenai/interactron. Introduction Object detection has been a central problem in computer vision since the inception of the field. There has been an extensive literature over the past decades proposing various methods ranging from constellation [14,15,18], regionbased [22,50,51], and hierarchical [25,45,61] models to the more recent powerful CNN [19,20,43] and Transformer [5,7,62] based models to tackle this problem. Typically, there are two main assumptions in these works: (1) There is a fixed training set and a test set. (2) The model is frozen after the training stage (i.e., it cannot be updated) and is evaluated on the pre-defined test set. These assumptions pose certain limitations for object detection in real world applications. First, in many applica- Figure 1. We introduce INTERACTRON a novel method for object detection. The idea is to adapt the detection model in an interactive setting during inference without any explicit supervision. The top row shows a standard detector that is kept frozen during inference. The bottom row shows our model that is updated during inference without any supervision by using future observations. tions (e.g., autonomous driving or home assistant robots), the model continuously receives new observations from the environment. The new observations might help the model correct its belief. For example, a partially occluded object might not be detected confidently in the current frame, but there might be a better (unoccluded) view of that object in later observations. The model should use this signal to improve its confidence in similar situations in the future. Second, freezing the weights after training inhibits further improvement and adaptation of the model. We believe there are strong self-supervised signals in the inference phase that an embodied agent can leverage via interacting with its environment to adapt the model. There has been work to adapt object detector in an unsupervised way (e.g., [11,47,52,55]). However, they assume a pre-recorded set of observations during inference. The idea of the proposed method is to continue training during inference while interacting with an environment. Our hypothesis is that interacting with the environment en-ables the embodied agent to capture better observations during inference leading to better adaptation and higher performance. In stark contrast to common object detection works, there is no distinct boundary between training and inference phases, and the model learns to take actions and adapt without any explicit supervision during inference. More specifically, there is an agent that interacts within indoor environments and relies on an object detector trained fully supervised to recognize objects. Our goal is to improve the object detection performance by adapting the model during inference while the agent interacts with the environment according to a learned policy ( Figure 1). During training, the agent learns a loss function using the supervised data, i.e. it learns to mimic the gradients produced during training using the labeled data. During inference, there is no supervision available for object detection. However, the model can generate gradients for the input images. Therefore, the model can be updated at inference time using the generated gradients. Basically, the model is updated without any explicit supervision at test time. We evaluate our adaptive object detection model, referred to as Interactron 1 , using the AI2-THOR [27] framework which includes 125 different object categories that appear in 120 indoor scenes. The task is to detect objects in all frames that the agent observes while navigating in a scene. Our experiments show that by learning to adapt, the DETR [5] model, which is a recent, high-performance object detection model, improves by 7.2 points in mAP. In addition to this strong result, we show that our adaptive model trained on AI2-THOR achieves near par results with a model trained with full supervision for the Habitat [44] framework, which includes scenes with completely different appearance characteristics. In summary, we propose an embodied adaptive object detection approach, where the model is updated during both training and inference. This approach is in contrast to traditional object detection, where the network is frozen after training. The model learns to adapt during inference via interaction with the environment and without any explicit supervision. We show our model significantly outperforms strong non-adaptive baselines and generalizes well to environments with different appearance distributions. Related Work Object detection. Various methods have been proposed to tackle the problem of object detection. Part-based and region-based models [14,16,54] were among the highperforming methods before the emergence of CNN based approaches. CNN based detectors [19,20,34,42,43] and the recently proposed Transformer based approaches [5,7,62] have achieved remarkable performance on detection bench- 1 Inspired from Detectron [21], the popular object detection framework. marks. One of the main assumptions of these works is that the model is kept frozen after training i.e. the weights of the model cannot change at test time. In contrast, in this paper, the model is updated in a self-supervised way to improve the detection performance. There have also been various works in segmentation and detection that adapt the network [11,47,49,52,55]. However, they have access to only a fixed set of images and there is no mechanism to interact with an environment. Embodied adaptive methods. Our detection model falls in the category of models that adapt at test time. We describe a few examples of these approaches for embodied tasks. [37] propose a method to adapt online to novel terrains, crippled body parts, and highly-dynamic environments. [32] propose an algorithm to enable visual odometry networks to continuously adapt to new environments. [31] propose a domain adaptation method for visual navigation so methods trained in simulation better generalize to real environments. However, it is different from the adaptive methods in the sense that it has access to some target-domain images during training. [56] propose a meta-learning based approach to adapt to new test environments for the task of navigation towards objects. In [30], there is a meta-learner that learns a set of transferable navigation skills. The agent can then quickly adapt to combine these skills when the navigationspecific reward is provided. [53] learn to adapt to new camera configurations using a few examples at test time for the task of vision and language navigation. [29] meta-learn a set of tasks (environment configurations) to better generalize to new tasks using a few samples. In contrast to these approaches, we focus on improving object detection. More importantly, unlike these approaches (except [56]), we propose to learn a loss function instead of relying on a predefined loss function. Our method shares similarities with continual learning [48] approaches. However, most continual learning works focus on passive non-embodied scenarios (e.g., [4,41,46]). A recent work by [33] proposes a continual learning method for a navigation scenario. Continual learning works typically focus on learning without forgetting while our objective is to adapt efficiently to test scenarios without any supervision. Active vision. Active vision [3] typically involves an agent that moves in an environment to have a better perception for the defined task or to perform the task more efficiently. The active vision literature addresses various types of tasks such as 3D reconstruction [9,10,28], object recognition [1,24,26], 3D pose estimation [8,57,60], and 3D scene modeling [2]. The main difference of our approach with these works is that our model is updated on-the-fly based on a loss that is learned self-supervised (as opposed to the typical manually defined measures of uncertainty). [58] is closer to our work and infers a policy to better recognize objects. Again, it differs from our approach since they freeze their model after training and it is based on full supervision. [38] propose to actively select a view in a scene and request annotation for that view. In contrast, our approach is adaptive and does not request annotations. [13] use pseudo-labels for self-supervised training of object detection. In contrast, we learn a policy and more importantly, continue training during inference. Embodied self-supervision. There are various works that learn self-supervised representations via embodied interactions [12,35,36,39,40]. Our goal is different in that we learn a loss function in a self-supervised fashion to change the weights on an object detector in a new environment to adapt to that environment. [6] addresses self-supervised learning for object detection by relying on a pose sensor, depth images, and 3D consistency. In contrast, we use only RGB images and do not rely on a perfect pose assumption. Embodied Adaptive Learning In this section, we introduce our approach to applying embodied, adaptive learning during inference to the object detection task. The main idea is that we do not freeze the weights of the model after training and instead let the model adapt during inference without any explicit supervision while an embodied agent explores the environment. Task Definition We first introduce a new flavor of the object detection task, suited for interactive environments (such as AI2-THOR [27] or Habitat [44]). The task consists of predicting a bounding box and category label for every object in the egocentric RGB frame of an embodied agent. Formally we are given a scene S ∈ S, and a position p and asked to predict every object o ∈ O S,p , the set of all objects visible in f S,p (the egocentric RGB frame at position p in scene S). The agent is also allowed to take n actions from the action set A according to some policy P and record the n additional RGB frames that it observed. We call the sequence of the n frames observed by our agent F. We then use some model M, which takes F as input, to predict a bounding box and class label for every object in O S,p (for a certain vocabulary of object categories). Note that we perform the detection only on the initial frame. Otherwise, the agent will be encouraged to "cheat" by simply moving to an area with few easily detectable objects. For every position p in every scene S, there are many possible sequences of frames F as there are many trajectories that the agent can explore. We call these sequences of frames rollouts, and we define the set of all rollouts for a given scene S and position p as R S,p . Finally, since there are many scenes and positions, each of which can be the starting point for many rollouts, we can define the set R all of all the possible rollouts for all the possible positions in all of our scenes, such that F ∈ R S,p ⊂ R all . In summary, each instance of an interactive object detection task T contains a scene S and a starting position p and is drawn from some distribution d(T , S) of all task instances given a set of scenes. Standard Approaches The most trivial approach to solving this problem is by using an off-the-shelf detector M exist and simply performing object detection on the initial frame f S,p . Here our policy P no−op is simply to not take any actions at all. We can boost the performance by pre-training the object detector on data from the same domain as our interactive environment. A more powerful approach will use a random policy to move the agent and collect several frames around the starting position p. Then a multi-frame model M mf can be trained to perform object detection on the initial frame using all of the frames as input. A model trained on such sequences can learn to leverage the multiple perspectives of the objects collected by the agent as it moves around to improve object detection. Adaptive Learning Intuitively, training an object detector in one particular local area of an environment (be it a room, building, or scene) increases the performance of the object detector on other nearby frames in that local area, since these environments (and in fact the natural world) are continuous. We confirm this intuition by empirical results, so we proceed to formulate this task as a meta learning problem where each instance of the interactive object detection task T represents a new task to fit to. At training time this abstraction works well as we can treat each frame in F, and their corresponding ground truth labels, as a task example and apply a version of the MAML algorithm [17]. We can then produce an object detector M θ meta , parameterized by θ. We train this model by doing a forward pass with all the frames in F, then computing the backward pass by using the ground truth labels and an object detection loss L det . We then take a gradient step and update our parameters such that θ = θ − α∇ θ L det (θ, F). We then optimize the model by minimizing the detection loss L det . We repeat this process on many tasks from d(T , S train ), where S train is a set of training scenes. At test time, however, this approach is infeasible, as we are not given labels for any of the frames. We can overcome this by adding another loss, one that is not based on the labels but rather just the frames in F. This loss can be hand designed, or it can be learned. Taking inspirations from [56,59], we learn the loss function. In our case, we use a learned loss produced by a model called the adaptive loss or L φ ada parameterized by φ that takes as input all of the frames in F as well as the predictions M θ meta (F) to pro-duce a gradient used for dynamic adaptation. There is no explicit objective for the learned loss. Instead, we simply encourage that minimizing this loss improves the detection ability of our model. Thus the learning objective for this model is m θ,φ in F∈R all L det (θ − α∇ θ L φ ada (θ, F), F)(1) As mentioned above L det is not available at test time, so the parameters of L φ ada are frozen and only M θ meta is trained according to L φ ada . This method allows us to dynamically adapt our object detector to its local environment, using the information contained in the frames in F, which were obtained by a random policy P rand . Interactive Adaptive Learning In standard adaptive and meta learning applications, we generally operate under the assumption that the distribution of data samples for each task is fixed and can not be influenced by us. In the interactive setting this is not true, as the samples we use for adaptation are collected by our agent. Formally, at each time step our agent takes an action a according to some policy P 2 , which takes as input all of the previous frames it has seen. Following the intuition that not all samples provide the same quality and quantity of information, we can learn a policy P int , which is a neural network parameterized by ρ and optimize it to guide the agent along a sequence of frames F that will allow M meta to easily adapt to the new task. In order to learn P int , we must first find a way to assign a value to each rollout F obtained by our actions. We do this by measuring the similarity of the gradients of θ produced by the detection loss L det (computed based on the labeled data) and those produced by the learned loss L ada . More specifically, we measure the 1 distance between the gradients produced by L det using the ground truth labels of the first frame and the gradients produced by L ada using the sequence of frames our agent collected. This way we are encouraging the agent to collect frames that will help the learned loss emulate the supervision provided by the ground truth labels. We call this value the Initial Frame Gradient Alignment IF GA and define it for any sequence F as follows: IF GA(F) = |∇ θ L φ ada (θ, F) − ∇ θ L det (θ, [f S,p ])| (2) This allows us to extract another useful training signal from our learned loss. Note that we only compute the IF GA of complete sequences of length n + 1 (the initial frame plus the n frames collected by the agent) as our learned loss estimator takes n+1 frames as input to compute 2 We denote learned policies by P and pre-defined ones with P . Algorithm 1 Training (d(T , S train ), θ, φ, ρ, α, β 1 , β 2 , β 3 , n) 1: while not converge do 2: for mini-batch of tasks τ i ∈ d(T , S train ) do 3: θ i ← θ 4: t ← 0 5: F i ← [f Si,pi ] 6: while t < n do 7: Sample action a from P ρ int (F i ) 8: Take action a and collected frame f 9: F i ← F i + [f ] 10: t ← t + 1 11: θ i ← θ i − α∇ θi L φ ada (θ i , F i ) 12: θ ← θ − β 1 i ∇ θ L det (θ i , f Si,pi ) 13: φ ← φ − β 2 i ∇ φ L det (θ i , F i ) 14: ρ ← ρ − β 3 i ∇ ρ L pol (θ i , P ρ int (F i )) the adaptive gradient. We can then define a full exploitation policy P exp which given any incomplete sequence of frames F inc where len(F inc ) < n explores every possible completion of the sequence and outputs the action that leads toward a complete sequence of frames with the lowest IF GA. This ideal policy is computed during training by exploring every possible trajectory the agent can take from the starting frame f S,p , but at test time this is not possible. Instead, we use a neural network P ρ int parameterized by ρ as our policy and train it to clone the behavior of P exp using the following loss function: L pol (P ρ int , F) = −P exp (F) T log P ρ int (F)(3) For a task where our agent is permitted to take n steps in the environment we define the adaptive gradient step learning rate as α, and the detector, learned loss, and policy learning rates as β 1 , β 2 and β 3 , respectively. Then we write down our interactive adaptive training algorithm as in Algorithm 1. The inference procedure for adaptive interactive learning is the same as the one for adaptive learning described in Section 3.3, with the exception that frames are not rolled out randomly by P rand , but are rather obtained by following P int . With this method, we are not only exploiting the ability of our model to fit to a local area of the scene, but also the fact that frames that would make good training examples for our model to be fit, also tend to contain useful information. Models While the methods described in Section 3 are fundamentally model agnostic, certain architectures naturally fit this approach. In this section, we describe the specifics of the models studied in this paper. Figure 2 shows an overview of these models. Figure 2. The architecture of the four main models presented in Section 4. A single frame baseline, which is just an off-the-shelf object detector, a multi-frame baseline which includes an inter-frame fusion Transformer and two INTERACTRON models (w/ and w/o a learned policy), which use a Transformer to learn a self-supervised loss function. INTERACTRON Model Our pipeline consists of two models: the Detector which performs the task of object detection and is adapted to the local environment at test time and the Supervisor which consists of the learned loss L ada and learned policy P int and is frozen at test time. The Detector can be any off-the-shelf object detection model, but we use DEtection TRansformer (DETR) [5] for our experiments. It is architecturally simple and yet very powerful. It utilizes a ResNet backbone which produces image features and a Transformer model which attends to all of the features to produce object detection embeddings. Each object detection embedding is then passed through an MLP to extract bounding box coordinates and a predicted object class. Although we use Transformers elsewhere in our model, using a Transformer-based object detector is not a requirement of this architecture. (FasterRCNN [43] results have also been provided in the Appendix). The Supervisor is a Transformer model that functions as both the policy and the learned loss. Image features and object detection embeddings produced by the detector, collectively called detection tokens, are passed into the Transformer. The transformer outputs of these tokens are passed through an MLP then reduced to a scalar to compute the adaptive gradient. In addition to these, a learnable policy token is passed into the Transformer for each action the agent needs to take, and its output is used to compute the policy. Learned Loss Training is performed by passing the outputs of the Detection Tokens through an MLP and taking the 2 norm of all the features to obtain a scalar which is the adaptive learning objective. The 2 norm is a trick (also in [59]) to combine the sequence of vectors produced by the Supervisor transformer into a single scalar loss. As the learning objective in Eq. 1 illustrates, the parameters of the Supervisor φ are optimized such that the gradients produced by the Supervisor ∇θL φ ada result in a reduction of the loss of the Detector. During training, the ground truth loss is computed from the object annotations, and the predictions made by the adapted Detector. The gradients used to adapt the Detector parameters are produced by the Supervisor. We can backpropagate through the adaptive gradients to update the Supervisor parameters, such that it produces better adaptive gradients (meta-training). At test time there is no object annotations, but the Supervisor has now been optimized to produce good gradients using the other frames and detections in the sequence. Policy Training is performed by treating the problem as a sequence prediction task. We learn n different embeddings to produce n different Policy Tokens. The Transformer outputs of the Policy Tokens are passed through an MLP to produce the action probability distribution. All possible trajectories of length n are explored during training and the policy is optimized to select the actions which lead to a complete rollout F with the lowest IF GA. When n is small it is possible to roll out every trajectory, but as n gets larger stochastic exploration or reinforcement learning methods are preferable, we leave this to future work. The Detection Tokens are fed into the Transformer one frame at a time, followed by a Policy Token, the output of which is used to predict the next action the agent should take. The model has access to all of the previous frames when deciding which action to take next. The adaptive gradient is only computed once all n steps have been taken. Figure 4 in the Appendix provides more details about our model. Ablation Models The INTERACTRON-Rand Model is essentially the same as the interactive adaptive learning model described above, except that it does not feed policy tokens to the Transformer, and instead utilizes a random policy P rand . The Multi-Frame Baseline is architecturally the same as the Adaptive Learning Model, but instead of using the Transformer as a learned loss function L ada , it uses it as a fusion layer, combining the detector output of all the frames in the sequence to produce detections for the first frame. This architecture corresponds to the model M mf described in Section 3.2. The Single Frame Baseline is simply an off-the-self detector (in our case DETR) that has been pre-trained on some data including images from our interactive environment and corresponds to M exist . Experiments We perform an extensive set of experiments to show the advantages of our INTERACTRON model. 3 We compare our method with a variety of baselines: a non-adaptive baseline, a non-adaptive baseline that aggregates information across multiple frames, and an adaptive baseline that explores a scene randomly. We also perform an experiment on an environment different from the one used for training and evaluate how well our model adapts. We also perform a set of ablation experiments to better analyze the proposed model. Implementation details. We conduct the bulk of our experiments in the AI2-iTHOR [27] interactive environment, as it offers many similarly sized scenes, fast rendering, and the ability to perform domain randomization. For our main task, we consider 5 frames (the initial frame plus 4 frames collected by the agent interacting with the environment). We use the action set {MoveForward, MoveBackward, Ro-tateLeft, RotateRight} and 30 degree rotation angles. We perform our evaluations with 300x300 images. Performing the object detections at higher resolutions could offer a significant performance improvement, but it also comes with significantly increased computational complexity so we leave exploring different resolutions for future work. For our main experiment, we train INTERACTRON models using the train set d(T , S train ) and test it on d(T , S test ). 3 The results in this section are different from those in the CVPR 2022 version. The trends are still the same and we observe a huge gap between the performance of the proposed approach and the baselines. The discrepancy is due to a bug in our evaluation code that selected the best checkpoint on the test set. We train our model for 2,000 epochs on the training set, using SGD to perform the meta training step and the Adam optimizer to train the learned loss and policy models. For training details see Appendix E. We ensure that all possible trajectories are explored during the entire training run. The single frame baseline is just the pre-trained object detector, while the multi-frame baseline is trained on d(T , S train ) for 1,000 epochs using the Adam optimizer. We use the standard COCO metrics for results. Dataset. We collect two datasets, d(T , S train ) and d(T , S test ), where S train consists of AI2-iTHOR [27] training and validations scenes (100 scenes in total) and S test consists of AI2-iTHOR test scenes (20 scenes). The datasets consist of scene id S, agent starting positions p as well as the labels for every object visible from the starting position O S,p . The starting positions (which consist of positional and rotational coordinates) are randomly sampled from all available positions for a given scene. After each sampling, the locations of all the objects in a given scene are randomized. We uniformly draw a total of 1000 pairs S, p from the training and validation scenes and 100 pairs from the test scenes. Note that the agent can explore different trajectories depending on the policy. So only the initial frames in the test set are fixed. We also collected a pre-training dataset of 10K frames from the training scenes with object detection annotations to train the base object detector. We employ the same sampling methods as above. We create a new set of detection classes which is the union of all the object categories from LVIS [23] and AI2-iTHOR [27]. This results in a total of 1,235 object categories and a "background" category. For our AI2-iTHOR evaluations, we only use the 125 iTHOR object classes, ignoring the others. Pre-training the model. We pre-train the model, using the DETR [5] codebase on a dataset of 124K LVIS images and 10K images from the AI2-iTHOR pre-training dataset. We use the standard 500 epoch training schedule. Table 1 shows that our method outperforms the baselines by a significant margin. We compare our method with a non-adaptive single-frame baseline DETR [5], a nonadaptive multi-frame baseline, referred to as "Multi-frame", an adaptive baseline with a random policy, referred to as "INTERACTRON-Rand". Our random policy INTERACTRON uses the same data and neural network architecture as the multi-frame baseline, yet it outperforms it, showcasing the merits of adaptive training even when our agent is not following an optimal policy. Our full INTERACTRON model further widens the performance gap by selecting good frames for computing the learned loss. The overall improvement between the offthe-shelf detector (DETR [5]) that the single frame baseline represents and our model is 7.2 AP (and 12.7 AP 50 ). INTERACTRON Results Transfer Results One of the key goals of this work is adapting our model to novel environments. In order to evaluate the crossenvironment adaptability performance of INTERACTRON models, we evaluate them on the Habitat [44] environment that has completely different appearance characteristics (natural images vs synthetic AI2-iTHOR images). We train our model in the AI2-iTHOR environment and perform adaptive inference in the Habitat environment. We test our model on a dataset of Habitat task instances, which are generated similarly to our AI2-iTHOR task instances (see Appendix C). For this set of tasks we only used the intersection of the iTHOR categories which we trained our model on, and the Habitat categories (22 object categories in total). Table 2 shows INTERACTRON models trained on AI2-iTHOR images are able to perform almost on par with a detector pre-trained with full supervision in Habitat. This shows that our method allows a model to fit to a new environment without access to training data or labels from that environment as well as a model that does have access to the labeled data in that environment. It is interesting to note that for small objects, INTERACTRON outperforms the baseline approaches (refer to AP S column). This indicates that our method may in fact leverage the signal in the later frames to update its belief about small or partially visible objects which are typically hard to detect. Ablations No Training at test. We profile the performance contribution of our learned loss L ada by measuring the performance of our model on the test set without it. We roll out the trajectory according to P int , then simply omit applying the gradient update according to L ada . Table 3 shows the meta trained model, without the train-at-test gradient update performs significantly worse than our full model, showcas- Table 5. Ablation -Repetition of the same frame. ing the contribution of training at test time. In fact, without the test time adaptation, our model performs worse than the off-the-shelf detector baseline. Varying number of frames. Since we have shown that gathering information from 5 frames is more useful than just relying on a single frame, it naturally follows to inquire if adding even more frames helps improve our model even further. To test this, we train INTERACTRON models with rollouts of length 7 and 9, respectively. We slow down the training schedule and increase the number of epochs when training these, as there are significantly more possible trajectories to explore with these sequence lengths (see Appendix E for details). Table 4 shows the results. We find that any improvement gained by adding more frames is within the training noise of our model. Multiple copies of the same frame. We explore training a model which sees five repetitions of the first frame instead of five unique frames. Training a policy with this model is meaningless, so we use just the INTERACTRON-Rand model. Table 5 show the results. The model trained to look at just the first frame performs better than the offthe-shelf DETR model, but worse than the model looking at 5 different frames. The improvement can be attributed to the extra learning capacity of the supervisor Transformer model. This verifies that our adaptive approach in fact benefits from extracting information from all of the frames in our trajectory to adapt the model to the current environment. Variance analysis. We repeat the AI2-iTHOR training to compute the performance variance between runs. We find that the standard deviation between the AP 50 of four different training runs to be 0.01. DETR Interactron Step 2 Step 3 Step 3 Step 4 Figure 3. Qualitative results. in the AI2-iTHOR and Habitat environments. The first column displays the results produced by DETR [5], the 2nd column displays the results of INTERACTRON, and the subsequent columns depict the interactive steps that the model took. Note that for the result shown in the third row, INTERACTRON has never seen Habitat images during training. The second row displays both a success and a failure case for our model. DETR only sees the arm of the sofa and classifies it as an armchair, while by looking from other angles, INTERACTRON discovers it is a sofa. It is also able to correctly detect the statue. On the other hand, DETR can detect the newspaper, which our model misses, and INTER-ACTRON falsely labels the fireplace as a laptop, an object it sees later in the rollout, and incorrectly associates with the position of the fireplace in the first image. Qualitative Results The third row shows an example of our transfer results in the Habitat environment. The agent starts positioned close to the sofa, so it is difficult to identify. DETR does not detect it while INTERACTRON takes a few steps backwards and is then able to correctly label it as a sofa. Note that IN-TERACTRON has never seen Habitat images during training. Discussion Constant Adaptation. One of the benefits of using an adaptive model versus one that simply views more frames at a time is that after the adaptation we are left with a model that only needs one frame to work well in the local area. Sub-policy. If a certain complex task (such as navigation or object manipulation) requires a high confidence detection of the objects visible from a certain position, the interactive policy we propose can be used as a sub-policy. Limitations One of the main limitations of this approach is that it works only for a specific set of object categories (125 categories in our experiments), and it is not capable of learning about new categories. Another limitation is that the proposed policy training approach only works with short trajectories, which can be explored with reasonable space and time complexity. Further work can address alternative policy training approaches for tasks that require longer rollouts. Conclusion We introduce INTERACTRON an adaptive object detection model that adapts to its test environment without explicit supervision. The model gathers information about the scene via a learned policy deployed on an embodied agent that navigates in the environment. We show our approach substantially improves a state-of-the-art object detector (7.2 point improvement in object detection AP). Moreover, we showcase the strengths of our approach in adapting the model to new environments, performing near on par with a model trained fully supervised for that environment. We use the DETR [5] model with a ResNet50 backbone for our detector. We freeze the backbone and Transformer encoder and only meta-train the decoder portion of DETR. We use the loss function based on the Hungarian matching algorithm described in the DETR paper as our ground truth detection loss L det . We set a maximum of 50 detections per image frame. For our supervisor we use a 4 layer, 8 head Transformer based on the GPT architecture with an internal dimension of 512. Our supervisor also consists of two separate embedding layers for the Image Features and Object Detection Features respectively. What we refer to as "Object Detection Features" are described as the output embeddings of the query tokens in the DETR paper. A positional embedding is learned for each token in the sequence. The supervisor also contains two decoders that consists of three consecutive linear layers each, with a hidden dimension of 512, separated by the GeLU non-linearity. One of the decoders consumes the outputs of the detection tokens and is used to compute the learned loss, while the other is used to decode the output of the Policy Tokens and produce a policy. The learned loss is computed by taking the 2 norm of the outputs of all the Detection Tokens passed through the decoder. Figure 4 illustrates a diagram of the INTERACTRON pipeline with tensor dimensions. C. Data Collection Details C.1. iTHOR Data Collection Details For all of the iTHOR datasets the objects are labeled as one of the 1,235 classes which comprise the union of all the LVIS object categories and all the iTHOR object categories. The iTHOR Pre-Training Dataset is collected from the 100 scenes in the train and val splits (scenes 0-25, 200-225, 300-325, 400-425). We uniformly sample frames from these scenes to collect a total of 10,000 frames and their an-notations. A random shuffle of the placement of objects in the scene is performed before each sample is collected. If a frame has less than 3 objects visible in it, it is rejected and a new sample is drawn. The annotations are stored in LVIS format. The data is added to the LVIS dataset to collectively form the iTHOR+LVIS pre-training dataset. The iTHOR Training Set is collected from the 100 scenes in the train and val splits. We uniformly sample starting positions from these scenes to collect a total of 1,000 starting location frames. A random shuffle of the placement of objects in the scene is performed before each sample is collected. If a frame has less than 3 objects visible in it, it is rejected and a new sample is drawn. In addition to this, every frame in every possible trajectory the agent could take from any sampled starting location is also collected. This implementation detail allows us to pre-cache every possible trajectory the agent could take to improve training efficiency. The iTHOR Test Set is collected from the 20 scenes in the test split (scenes 25-30, 225-230, 325-330, 425-430). We uniformly sample starting positions from these scenes to collect a total of 100 starting location frames. If a frame has less than 3 objects visible in it, it is rejected and a new sample is drawn. In addition to this, every frame in every possible trajectory the agent could take from any sampled starting location is also collected. This implementation detail allows us to pre-cache every possible trajectory the agent could take to improve training efficiency. C.2. Habitat Data Collection Details For all of the Habitat datasets the objects are labeled as one of the 1,255 classes which comprise the union of all the LVIS object categories, all the iTHOR object categories and all the Habitat object categories. The Habitat Pre-Training Dataset is collected from the 56 scenes in the MP3D train split. We uniformly sample frames from these scenes to collect a total of 10,000 frames and their annotations. Each scene has an equal number of samples, regardless of its size. If a frame has less than 3 objects visible in it, it is rejected and a new sample is drawn. The annotations are stored in LVIS format. The data is added to the iTHOR+LVIS pre-training dataset to collectively form the Habitat+iTHOR+LVIS dataset. The Habitat Test Dataset is collected from the 11 scenes in the MP3D val split. We uniformly sample starting positions from these scenes to collect a total of 100 starting location frames. Each scene has an equal number of samples, regardless of its size. If a frame has less than 3 objects visible in it, it is rejected and a new sample is drawn. In addition to this, every frame in every possible trajectory the agent could take from any sampled starting location is also collected. This implementation detail allows us to precache every possible trajectory the agent could take to im- prove training efficiency. D. DETR Pre-Training Details We pre-train the DETR object detector using the official DETR codebase, modified to accept the format of our dataset. We use the same training parameters as proposed in the original DETR paper to train one detector on the iTHOR+LVIS pre-training dataset and another on the Habi-tat+iTHOR+LVIS dataset. We scale all of the pre-training images such that one side has a dimension of 300. E. Training Details E.1. Interactron Training The INTERACTRON model is trained for 2,000 epochs with the Adam optimizer, using an initial learning rate of 1e−4 and a linearly annealing schedule. Both the supervisor model producing the learned loss and policy and the detector itself are trained using these parameters. To stabilize the training we record the weights after each of the last 500 epochs and average them to produce the final model. The training takes approximately 120 hours using a single Nvidia RTX 3090. During training we do not follow the policy of the model, but rather explore every possible trajectory the agent could take, from a given starting location. We then record the IF GA value for each possible trajectory and update it whenever we revisit the same trajectory. We optimize our policy to always pursue the trajectory with the lowest recorded IF GA. In slight contradiction to common nomenclature, one epoch does not represent a pass through every single frame in our dataset, as the agent can explore multiple trajectories from each starting location. For our 5 frame (4 step) experiments, for example, we can only guarantee that every possible frame in the dataset has been explored after 256 epochs of training. E.2. Interactron 7 Frame Training For training the 7 frame model (where the agent takes 6 steps) we utilize a training regime of 5,000 epochs using the same initial learning rate and annealing schedule. This way we ensure that every possible trajectory the agent could take from each initial frame is explored multiple times. E.3. Interactron 9 Frame Training For training the 9 frame model (where the agent takes 8 steps) we utilize a training regime of 9,000 epochs using the same initial learning rate and annealing schedule. This way we ensure that every possible trajectory the agent could take from each initial frame is explored at least once. Figure 3 3shows a selection of rollouts produced by IN-TERACTRON. The bounding boxes displayed represent detections with a confidence score greater than 0.5. The first row showcases an example of situations in which INTERACTRON excels. The cup, which is placed at the edge of the view of the agent is not detected by DETR, but INTERACTRON adjusts the position of the agent to get a better view of the cup. Similarly, other drawers are also detected by INTERACTRON. Figure 4 . 4The detailed architecture of the INTERACTRON model. The first column describes each layer, while the second column provides the dimensions of the intermediate tensors where applicable. Table 1 . 1Object detection results. We compare our method INTERACTRON with a set of baseline approaches. The standard COCO detection metrics have been reported. Our method provides a massive gain compared to the single-frame DETR[5] baseline.AP AP50 AP75 APS APM APL Adaptive Policy DETR [5] 0.256 0.448 0.207 0.112 0.368 0.605 No Move Multi-frame 0.288 0.517 0.299 0.132 0.452 0.697 Random INTERACTRON-Rand (ours) 0.313 0.551 0.320 0.167 0.477 0.710 Random INTERACTRON (ours) 0.328 0.575 0.331 0.174 0.480 0.733 Learned Policy Dataset AP AP50 AP S AP M AP LDETR [5] LVIS + AI2-iTHOR 0.17 0.20 0.08 0.17 0.30 DETR [5] LVIS + AI2-iTHOR + Habitat 0.24 0.30 0.13 0.22 0.39 INTERACTRON-Rand LVIS + AI2-iTHOR 0.22 0.26 0.14 0.19 0.32 INTERACTRON LVIS + AI2-iTHOR 0.22 0.27 0.16 0.20 0.32 Table 2. Transfer results. We train our model on the AI2- iTHOR [27] framework and perform adaptive inference in the Habitat [44] environment. Transformer Output MLPsAdaptive Loss and Model PolicyResNet 50 ResNet 50 ResNet 50 ResNet 50 ℒ ada ia ℒ ada ia ℒ ada ia ℒ ada ia ℒ ada DETR DETR DETR DETR DETR ResNet 50 RGB Frames ResNet Backbone Image Features DETR Detector Object Detection Features Learned Policy Embedding (Model Parameter) Interaction Supervisor 3x300x300 1024x19x19 1024x50 1024x1 MLP MLP MLP MLP MLP MLP MLP MLP MLP Acknowledgements. We thank Aaron Walsman for the helpful discussions and Winson Han for his help with the figures.AppendixA. Pseudo LabelsWe investigate the viability of using pseudo-labels instead of a learned loss using the Multi-frame model and a simple pseudo-label method. We take the pre-trained Multiframe model, and during evaluation apply a gradient update using the detections as labels, weighed by the detection confidence. This provides a small performance improvement of about 2 points with an AP 50 of 0.538. While our learned loss still outperforms this model, this ablation shows that a pseudo-label approach could also be viable in our problem setting. A dataset for developing and benchmarking active vision. Phil Ammirato, Patrick Poirson, Eunbyung Park, Jana Košecká, Alexander C Berg, ICRA. Phil Ammirato, Patrick Poirson, Eunbyung Park, Jana Košecká, and Alexander C Berg. A dataset for developing and benchmarking active vision. In ICRA, 2017. 2 Data acquisition and view planning for 3-d modeling tasks. S Paul, Blaer, Peter K Allen, IROS. Paul S Blaer and Peter K Allen. Data acquisition and view planning for 3-d modeling tasks. In IROS, 2007. 2 Active vision. Andrew Blake, Alan Yuille, MIT pressAndrew Blake and Alan Yuille. Active vision. MIT press, 1993. 2 Incremental robot learning of new objects with fixed update time. Raffaello Camoriano, Giulia Pasquale, Carlo Ciliberto, Lorenzo Natale, Lorenzo Rosasco, Giorgio Metta, ICRA. Raffaello Camoriano, Giulia Pasquale, Carlo Ciliberto, Lorenzo Natale, Lorenzo Rosasco, and Giorgio Metta. In- cremental robot learning of new objects with fixed update time. In ICRA, 2017. 2 End-toend object detection with transformers. Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, Sergey Zagoruyko, ECCV. 11Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to- end object detection with transformers. In ECCV, 2020. 1, 2, 5, 6, 7, 8, 11 SEAL: selfsupervised embodied active learning using exploration and 3d consistency. Devendra Singh Chaplot, Murtaza Dalal, Saurabh Gupta, Jitendra Malik, Ruslan Salakhutdinov, NeurIPS. Devendra Singh Chaplot, Murtaza Dalal, Saurabh Gupta, Jitendra Malik, and Ruslan Salakhutdinov. SEAL: self- supervised embodied active learning using exploration and 3d consistency. In NeurIPS, 2021. 3 Ting Chen, Saurabh Saxena, Lala Li, J David, Geoffrey Fleet, Hinton, Pix2seq: A language modeling framework for object detection. arXiv. 1Ting Chen, Saurabh Saxena, Lala Li, David J Fleet, and Ge- offrey Hinton. Pix2seq: A language modeling framework for object detection. arXiv, 2021. 1, 2 Sotiris Malassiotis, and Tae-Kyun Kim. Recovering 6d object pose and predicting next-best-view in the crowd. Andreas Doumanoglou, Rigas Kouskouridas, CVPR. Andreas Doumanoglou, Rigas Kouskouridas, Sotiris Malas- siotis, and Tae-Kyun Kim. Recovering 6d object pose and predicting next-best-view in the crowd. In CVPR, 2016. 2 Next best view planning for active model improvement. Enrique Dunn, Jan-Michael Frahm, BMVC. Enrique Dunn and Jan-Michael Frahm. Next best view plan- ning for active model improvement. In BMVC, 2009. 2 Developing visual sensing strategies through next best view planning. Enrique Dunn, Jur Van Den, Jan-Michael Berg, Frahm, IROS. Enrique Dunn, Jur Van Den Berg, and Jan-Michael Frahm. Developing visual sensing strategies through next best view planning. In IROS, 2009. 2 One-shot unsupervised cross-domain detection. Francesco Cappio Antonio D&apos;innocente, Silvia Borlino, Barbara Bucci, Tatiana Caputo, Tommasi, ECCV. 1Antonio D'Innocente, Francesco Cappio Borlino, Silvia Bucci, Barbara Caputo, and Tatiana Tommasi. One-shot un- supervised cross-domain detection. In ECCV, 2020. 1, 2 Selfsupervised transfer learning for instance segmentation through physical interaction. Andreas Eitel, Nico Hauff, Wolfram Burgard, IROS. Andreas Eitel, Nico Hauff, and Wolfram Burgard. Self- supervised transfer learning for instance segmentation through physical interaction. In IROS, 2019. 3 Move to see better: Towards selfsupervised amodal object detection. Zhaoyuan Fang, Ayush Jain, Gabriel Sarch, Adam W Harley, Katerina Fragkiadaki, BMVC. Zhaoyuan Fang, Ayush Jain, Gabriel Sarch, Adam W. Harley, and Katerina Fragkiadaki. Move to see better: Towards self- supervised amodal object detection. In BMVC, 2021. 3 Object detection with discriminatively trained part-based models. F Pedro, Ross B Felzenszwalb, David Girshick, Deva Mcallester, Ramanan, TPAMI. 12Pedro F Felzenszwalb, Ross B Girshick, David McAllester, and Deva Ramanan. Object detection with discriminatively trained part-based models. TPAMI, 2009. 1, 2 Efficient matching of pictorial structures. F Pedro, Felzenszwalb, P Daniel, Huttenlocher, CVPR. Pedro F Felzenszwalb and Daniel P Huttenlocher. Efficient matching of pictorial structures. In CVPR, 2000. 1 Alan Yuille, and Raquel Urtasun. Bottom-up segmentation for top-down detection. Sanja Fidler, Roozbeh Mottaghi, CVPR. Sanja Fidler, Roozbeh Mottaghi, Alan Yuille, and Raquel Ur- tasun. Bottom-up segmentation for top-down detection. In CVPR, 2013. 2 Modelagnostic meta-learning for fast adaptation of deep networks. Chelsea Finn, Pieter Abbeel, Sergey Levine, ICML. Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model- agnostic meta-learning for fast adaptation of deep networks. In ICML, 2017. 3 The representation and matching of pictorial structures. A Martin, Robert A Fischler, Elschlager, IEEE Transactions on Computers. 1Martin A Fischler and Robert A Elschlager. The representa- tion and matching of pictorial structures. IEEE Transactions on Computers, 1973. 1 Fast r-cnn. Ross Girshick, ICCV. 1Ross Girshick. Fast r-cnn. In ICCV, 2015. 1, 2 Rich feature hierarchies for accurate object detection and semantic segmentation. Ross Girshick, Jeff Donahue, Trevor Darrell, Jitendra Malik, CVPR. 1Ross Girshick, Jeff Donahue, Trevor Darrell, and Jitendra Malik. Rich feature hierarchies for accurate object detection and semantic segmentation. In CVPR, 2014. 1, 2 Piotr Dollár, and Kaiming He. Ross Girshick, Ilija Radosavovic, Georgia Gkioxari, Ross Girshick, Ilija Radosavovic, Georgia Gkioxari, Piotr Dollár, and Kaiming He. Detectron. https://github. com/facebookresearch/detectron, 2018. 2 Recognition using regions. Chunhui Gu, Joseph J Lim, Pablo Arbeláez, Jitendra Malik, CVPR. Chunhui Gu, Joseph J. Lim, Pablo Arbeláez, and Jitendra Malik. Recognition using regions. In CVPR, 2009. 1 Lvis: A dataset for large vocabulary instance segmentation. Agrim Gupta, Piotr Dollar, Ross Girshick, CVPR. Agrim Gupta, Piotr Dollar, and Ross Girshick. Lvis: A dataset for large vocabulary instance segmentation. In CVPR, 2019. 6 Look-ahead before you leap: end-to-end active recognition by forecasting the effect of motion. Dinesh Jayaraman, Kristen Grauman, ECCV. Dinesh Jayaraman and Kristen Grauman. Look-ahead before you leap: end-to-end active recognition by forecasting the effect of motion. In ECCV, 2016. 2 Context and hierarchy in a probabilistic image model. Ya Jin, Stuart Geman, CVPR. Ya Jin and Stuart Geman. Context and hierarchy in a proba- bilistic image model. In CVPR, 2006. 1 Pairwise decomposition of image sequences for active multiview recognition. Edward Johns, Stefan Leutenegger, Andrew J Davison, CVPR. Edward Johns, Stefan Leutenegger, and Andrew J Davison. Pairwise decomposition of image sequences for active multi- view recognition. In CVPR, 2016. 2 Ai2-thor: An interactive 3d environment for visual ai. Eric Kolve, Roozbeh Mottaghi, Winson Han, Eli Vanderbilt, Luca Weihs, Alvaro Herrasti, Daniel Gordon, Yuke Zhu, Abhinav Gupta, Ali Farhadi, 67Eric Kolve, Roozbeh Mottaghi, Winson Han, Eli VanderBilt, Luca Weihs, Alvaro Herrasti, Daniel Gordon, Yuke Zhu, Ab- hinav Gupta, and Ali Farhadi. Ai2-thor: An interactive 3d environment for visual ai. arXiv, 2017. 2, 3, 6, 7 Autonomous generation of complete 3d object models using next best view manipulation planning. Michael Krainin, Brian Curless, Dieter Fox, ICRA. Michael Krainin, Brian Curless, and Dieter Fox. Au- tonomous generation of complete 3d object models using next best view manipulation planning. In ICRA, 2011. 2 Kiran Lekkala, Sami Abu-El-Haija, Laurent Itti, Meta adaptation using importance weighted demonstrations. arXiv. Kiran Lekkala, Sami Abu-El-Haija, and Laurent Itti. Meta adaptation using importance weighted demonstrations. arXiv, 2019. 2 Unsupervised reinforcement learning of transferable meta-skills for embodied navigation. Juncheng Li, Xin Wang, Siliang Tang, Haizhou Shi, Fei Wu, Yueting Zhuang, William Yang Wang, CVPR. Juncheng Li, Xin Wang, Siliang Tang, Haizhou Shi, Fei Wu, Yueting Zhuang, and William Yang Wang. Unsupervised re- inforcement learning of transferable meta-skills for embod- ied navigation. In CVPR, 2020. 2 Unsupervised domain adaptation for visual navigation. Shangda Li, Devendra Singh Chaplot, Yao-Hung Hubert Tsai, Yue Wu, Louis-Philippe Morency, Ruslan Salakhutdinov, arxiv. 2Shangda Li, Devendra Singh Chaplot, Yao-Hung Hu- bert Tsai, Yue Wu, Louis-Philippe Morency, and Ruslan Salakhutdinov. Unsupervised domain adaptation for visual navigation. arxiv, 2020. 2 Self-supervised deep visual odometry with online adaptation. Shunkai Li, Xin Wang, Yingdian Cao, Fei Xue, Zike Yan, Hongbin Zha, CVPR. 2020Shunkai Li, Xin Wang, Yingdian Cao, Fei Xue, Zike Yan, and Hongbin Zha. Self-supervised deep visual odometry with online adaptation. In CVPR, 2020. 2 A lifelong learning approach to mobile robot navigation. Bo Liu, Xuesu Xiao, Peter Stone, IEEE Robotics and Automation Letters. 2Bo Liu, Xuesu Xiao, and Peter Stone. A lifelong learning approach to mobile robot navigation. IEEE Robotics and Automation Letters, 2021. 2 Ssd: Single shot multibox detector. Wei Liu, Dragomir Anguelov, Dumitru Erhan, Christian Szegedy, Scott Reed, Cheng-Yang Fu, Alexander C Berg, ECCV. Wei Liu, Dragomir Anguelov, Dumitru Erhan, Christian Szegedy, Scott Reed, Cheng-Yang Fu, and Alexander C Berg. Ssd: Single shot multibox detector. In ECCV, 2016. 2 Learning about objects by learning to interact with them. Martin Lohmann, Jordi Salvador, Aniruddha Kembhavi, Roozbeh Mottaghi, NeurIPS. 2020Martin Lohmann, Jordi Salvador, Aniruddha Kembhavi, and Roozbeh Mottaghi. Learning about objects by learning to interact with them. In NeurIPS, 2020. 3 A self-supervised learning system for object detection using physics simulation and multi-view pose estimation. Chaitanya Mitash, Kostas E Bekris, Abdeslam Boularias, IROS. Chaitanya Mitash, Kostas E Bekris, and Abdeslam Boular- ias. A self-supervised learning system for object detection using physics simulation and multi-view pose estimation. In IROS, 2017. 3 Learning to adapt in dynamic, real-world environments through meta-reinforcement learning. Anusha Nagabandi, Ignasi Clavera, Simin Liu, S Ronald, Pieter Fearing, Sergey Abbeel, Chelsea Levine, Finn, ICLR. Anusha Nagabandi, Ignasi Clavera, Simin Liu, Ronald S Fearing, Pieter Abbeel, Sergey Levine, and Chelsea Finn. Learning to adapt in dynamic, real-world environments through meta-reinforcement learning. In ICLR, 2019. 2 Embodied visual active learning for semantic segmentation. David Nilsson, Aleksis Pirinen, Erik Gärtner, Cristian Sminchisescu, AAAI. David Nilsson, Aleksis Pirinen, Erik Gärtner, and Cristian Sminchisescu. Embodied visual active learning for semantic segmentation. In AAAI, 2021. 3 Sören Pirk, Mohi Khansari, Yunfei Bai, Corey Lynch, Pierre Sermanet, Online object representations with contrastive learning. arXivSören Pirk, Mohi Khansari, Yunfei Bai, Corey Lynch, and Pierre Sermanet. Online object representations with con- trastive learning. arXiv, 2019. 3 Selfsupervisory signals for object discovery and detection. Etienne Pot, Alexander Toshev, Jana Kosecka, arXivEtienne Pot, Alexander Toshev, and Jana Kosecka. Self- supervisory signals for object discovery and detection. arXiv, 2018. 3 icarl: Incremental classifier and representation learning. Alexander Sylvestre-Alvise Rebuffi, Georg Kolesnikov, Christoph H Sperl, Lampert, CVPR. Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, Georg Sperl, and Christoph H Lampert. icarl: Incremental classifier and representation learning. In CVPR, 2017. 2 You only look once: Unified, real-time object detection. Joseph Redmon, Santosh Divvala, Ross Girshick, Ali Farhadi, CVPR. Joseph Redmon, Santosh Divvala, Ross Girshick, and Ali Farhadi. You only look once: Unified, real-time object de- tection. In CVPR, 2016. 2 Faster r-cnn: Towards real-time object detection with region proposal networks. Kaiming Shaoqing Ren, Ross He, Jian Girshick, Sun, NeurIPS. 15Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In NeurIPS, 2015. 1, 2, 5 Habitat: A platform for embodied ai research. Manolis Savva, Abhishek Kadian, Oleksandr Maksymets, Yili Zhao, Erik Wijmans, Bhavana Jain, Julian Straub, Jia Liu, Vladlen Koltun, Jitendra Malik, ICCV. 27Manolis Savva, Abhishek Kadian, Oleksandr Maksymets, Yili Zhao, Erik Wijmans, Bhavana Jain, Julian Straub, Jia Liu, Vladlen Koltun, Jitendra Malik, et al. Habitat: A plat- form for embodied ai research. In ICCV, 2019. 2, 3, 7 Discriminative structure learning of hierarchical representations for object detection. Paul Schnitzspan, Mario Fritz, Stefan Roth, Bernt Schiele, CVPR. Paul Schnitzspan, Mario Fritz, Stefan Roth, and Bernt Schiele. Discriminative structure learning of hierarchical representations for object detection. In CVPR, 2009. 1 Incremental learning of object detectors without catastrophic forgetting. Konstantin Shmelkov, Cordelia Schmid, Karteek Alahari, ICCV. Konstantin Shmelkov, Cordelia Schmid, and Karteek Ala- hari. Incremental learning of object detectors without catas- trophic forgetting. In ICCV, 2017. 2 Adapting object detectors with conditional domain normalization. Peng Su, Kun Wang, Xingyu Zeng, Shixiang Tang, Dapeng Chen, Di Qiu, Xiaogang Wang, ECCV. 1Peng Su, Kun Wang, Xingyu Zeng, Shixiang Tang, Dapeng Chen, Di Qiu, and Xiaogang Wang. Adapting object detec- tors with conditional domain normalization. In ECCV, 2020. 1, 2 Lifelong robot learning. Robotics and autonomous systems. Sebastian Thrun, M Tom, Mitchell, Sebastian Thrun and Tom M Mitchell. Lifelong robot learn- ing. Robotics and autonomous systems, 1995. 2 Learning to adapt structured output space for semantic segmentation. Yi-Hsuan Tsai, Wei-Chih Hung, Samuel Schulter, Kihyuk Sohn, Ming-Hsuan Yang, Manmohan Chandraker, CVPR. Yi-Hsuan Tsai, Wei-Chih Hung, Samuel Schulter, Ki- hyuk Sohn, Ming-Hsuan Yang, and Manmohan Chandraker. Learning to adapt structured output space for semantic seg- mentation. In CVPR, 2018. 2 Segmentation as selective search for object recognition. E A Koen, Van De Sande, R R Jasper, Theo Uijlings, Arnold W M Gevers, Smeulders, ICCV. Koen E. A. van de Sande, Jasper R. R. Uijlings, Theo Gev- ers, and Arnold W. M. Smeulders. Segmentation as selective search for object recognition. In ICCV, 2011. 1 Efficient region search for object detection. Sudheendra Vijayanarasimhan, Kristen Grauman, CVPR. Sudheendra Vijayanarasimhan and Kristen Grauman. Effi- cient region search for object detection. In CVPR, 2011. 1 Mega-cda: Memory guided attention for category-aware unsupervised domain adaptive object detection. V S Vibashan, Vikram Gupta, Poojan Oza, A Vishwanath, Sindagi, Patel, CVPR. 1Vibashan VS, Vikram Gupta, Poojan Oza, Vishwanath A Sindagi, and Vishal M Patel. Mega-cda: Memory guided attention for category-aware unsupervised domain adaptive object detection. In CVPR, 2021. 1, 2 Visual perception generalization for vision-and-language navigation via meta-learning. arXiv. Ting Wang, Zongkai Wu, Donglin Wang, Ting Wang, Zongkai Wu, and Donglin Wang. Visual percep- tion generalization for vision-and-language navigation via meta-learning. arXiv, 2020. 2 Regionlets for generic object detection. Xiaoyu Wang, Ming Yang, Shenghuo Zhu, Yuanqing Lin, ICCV. Xiaoyu Wang, Ming Yang, Shenghuo Zhu, and Yuanqing Lin. Regionlets for generic object detection. In ICCV, 2013. 2 Domain-specific suppression for adaptive object detection. Yu Wang, Rui Zhang, Shuo Zhang, Miao Li, Yangyang Xia, Xishan Zhang, Shaoli Liu, CVPR. 1Yu Wang, Rui Zhang, Shuo Zhang, Miao Li, YangYang Xia, XiShan Zhang, and ShaoLi Liu. Domain-specific suppres- sion for adaptive object detection. In CVPR, 2021. 1, 2 Learning to learn how to learn: Self-adaptive visual navigation using meta-learning. Mitchell Wortsman, Kiana Ehsani, Mohammad Rastegari, Ali Farhadi, Roozbeh Mottaghi, CVPR. 23Mitchell Wortsman, Kiana Ehsani, Mohammad Rastegari, Ali Farhadi, and Roozbeh Mottaghi. Learning to learn how to learn: Self-adaptive visual navigation using meta-learning. In CVPR, 2019. 2, 3 3d shapenets: A deep representation for volumetric shapes. Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang Zhang, Xiaoou Tang, Jianxiong Xiao, CVPR. Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Lin- guang Zhang, Xiaoou Tang, and Jianxiong Xiao. 3d shapenets: A deep representation for volumetric shapes. In CVPR, 2015. 2 Embodied amodal recognition: Learning to move to perceive objects. Jianwei Yang, Zhile Ren, Mingze Xu, Xinlei Chen, J David, Devi Crandall, Dhruv Parikh, Batra, ICCV. Jianwei Yang, Zhile Ren, Mingze Xu, Xinlei Chen, David J Crandall, Devi Parikh, and Dhruv Batra. Embodied amodal recognition: Learning to move to perceive objects. In ICCV, 2019. 2 One-shot imitation from observing humans via domain-adaptive meta-learning. Tianhe Yu, Chelsea Finn, Annie Xie, Sudeep Dasari, Tianhao Zhang, P Abbeel, Sergey Levine, In RSS. 35Tianhe Yu, Chelsea Finn, Annie Xie, Sudeep Dasari, Tianhao Zhang, P. Abbeel, and Sergey Levine. One-shot imitation from observing humans via domain-adaptive meta-learning. In RSS, 2018. 3, 5 Robotic pick-and-place of novel objects in clutter with multi-affordance grasping and cross-domain image matching. Andy Zeng, Shuran Song, Kuan-Ting Yu, Elliott Donlon, Maria Francois R Hogan, Daolin Bauza, Orion Ma, Melody Taylor, Eudald Liu, Romo, ICRA. Andy Zeng, Shuran Song, Kuan-Ting Yu, Elliott Donlon, Francois R Hogan, Maria Bauza, Daolin Ma, Orion Taylor, Melody Liu, Eudald Romo, et al. Robotic pick-and-place of novel objects in clutter with multi-affordance grasping and cross-domain image matching. In ICRA, 2018. 2 Unsupervised structure learning: Hierarchical recursive composition, suspicious coincidence and competitive exclusion. Long Leo Zhu, Chenxi Lin, Haoda Huang, Yuanhao Chen, Alan Yuille, ECCV. Long Leo Zhu, Chenxi Lin, Haoda Huang, Yuanhao Chen, and Alan Yuille. Unsupervised structure learning: Hierarchi- cal recursive composition, suspicious coincidence and com- petitive exclusion. In ECCV, 2008. 1 Deformable detr: Deformable transformers for end-to-end object detection. Xizhou Zhu, Weijie Su, Lewei Lu, Bin Li, Xiaogang Wang, Jifeng Dai, 1arXivXizhou Zhu, Weijie Su, Lewei Lu, Bin Li, Xiaogang Wang, and Jifeng Dai. Deformable detr: Deformable transformers for end-to-end object detection. arXiv, 2020. 1, 2
[ "https://github.com/allenai/interactron." ]
[ "Building Intelligence in the Mechanical Domain -Harvesting the Reservoir Computing Power in Origami to Achieve Information Perception Tasks", "Building Intelligence in the Mechanical Domain -Harvesting the Reservoir Computing Power in Origami to Achieve Information Perception Tasks" ]
[ "Jun Wang [email protected] \nDepartment of Mechanical Engineering\nVirginia Tech 181 Durham Hall, 1145 Perry Street24061BlacksburgVAUSA\n", "Suyi Li \nDepartment of Mechanical Engineering\nVirginia Tech 181 Durham Hall, 1145 Perry Street24061BlacksburgVAUSA\n" ]
[ "Department of Mechanical Engineering\nVirginia Tech 181 Durham Hall, 1145 Perry Street24061BlacksburgVAUSA", "Department of Mechanical Engineering\nVirginia Tech 181 Durham Hall, 1145 Perry Street24061BlacksburgVAUSA" ]
[]
In this paper, we experimentally examine the cognitive capability of a simple, paper-based Miura-ori -using the physical reservoir computing framework -to achieve different information perception tasks. The body dynamics of Miura-ori (aka. its vertices displacements), which is excited by a simple harmonic base excitation, can be exploited as the reservoir computing resource. By recording these dynamics with a high-resolution camera and image processing program and then using linear regression for training, we show that the origami reservoir has sufficient computing capacity to estimate the weight and position of a payload. It can also recognize the input frequency and magnitude patterns. Furthermore, multitasking is achievable by simultaneously applying two targeted functions to the same reservoir state matrix. Therefore, we demonstrate that Miura-ori can assess the dynamic interactions between its body and ambient environment to extract meaningful information -an intelligent behavior in the mechanical domain. Given that Miura-ori has been widely used to construct deployable structures, lightweight materials, and compliant robots, enabling such information perception tasks can add a new dimension to the functionality of such a versatile structure.
10.48550/arxiv.2302.05517
[ "https://export.arxiv.org/pdf/2302.05517v1.pdf" ]
256,827,210
2302.05517
332d1e9bcfd30c4648865b09ab3a632710f7c2e1
Building Intelligence in the Mechanical Domain -Harvesting the Reservoir Computing Power in Origami to Achieve Information Perception Tasks Jun Wang [email protected] Department of Mechanical Engineering Virginia Tech 181 Durham Hall, 1145 Perry Street24061BlacksburgVAUSA Suyi Li Department of Mechanical Engineering Virginia Tech 181 Durham Hall, 1145 Perry Street24061BlacksburgVAUSA Building Intelligence in the Mechanical Domain -Harvesting the Reservoir Computing Power in Origami to Achieve Information Perception Tasks *Correspondent email address:OrigamiEmbodies IntelligenceReservoir ComputingInformation Perception In this paper, we experimentally examine the cognitive capability of a simple, paper-based Miura-ori -using the physical reservoir computing framework -to achieve different information perception tasks. The body dynamics of Miura-ori (aka. its vertices displacements), which is excited by a simple harmonic base excitation, can be exploited as the reservoir computing resource. By recording these dynamics with a high-resolution camera and image processing program and then using linear regression for training, we show that the origami reservoir has sufficient computing capacity to estimate the weight and position of a payload. It can also recognize the input frequency and magnitude patterns. Furthermore, multitasking is achievable by simultaneously applying two targeted functions to the same reservoir state matrix. Therefore, we demonstrate that Miura-ori can assess the dynamic interactions between its body and ambient environment to extract meaningful information -an intelligent behavior in the mechanical domain. Given that Miura-ori has been widely used to construct deployable structures, lightweight materials, and compliant robots, enabling such information perception tasks can add a new dimension to the functionality of such a versatile structure. Introduction We are witnessing an ever-increasing demand for the next generation of multi-functional structures and material systems that can behave intelligently according to their mission needs and dynamic working conditions. Ideally, these intelligent systems should observe their environment, extract critical information from sensory inputs, learn from past experiences, decide on the action plan, and execute control commands -in a highly integrated and distributed setup and in real time. Such intelligence is traditionally implemented in the digital domain with the help of, for example, onboard computers. However, there has been an increasing interest in offloading and distributing some of the intelligence into the physical domain without a centralized digital computer or even without any electronics [1][2][3]. In this regard, animals have offered us many inspirations [4,5] as they can outsource many information processing and locomotion control tasks from their brain to the body -by exploiting its physical morphology, mechanics, and neuro-muscular structure. For example, the octopus' hydrostat-muscular arm has an elaborated muscular layout with a distributed sensory-neural network so that it can accomplish many complex locomotions and manipulations without the direct involvement of its brain [6]. Inspired by these lessons from nature, researchers developed many new strategies for achieving (or mimicking) intelligent behaviors in the mechanical domain. For example, one can use the multi-stability embedded in soft materials to sequence locomotion gaits [7], replace digital circuitry with fluidic ones for logic operation in entirely soft substrates [8], and exploit wave phenomena to achieve computation directly in metamaterial systems [9] . Overall, achieving intelligence using mechanical components (we refer to as "mechano-intelligence" hereafter [10]) could bring significant advantages such as lower power consumption, less analog-digital conversion, higher overall speed, and better survivability in harsh environments. Nonetheless, intelligence is a broad concept consisting of many interconnected facets, including (and not limited to) information perception, decision-making, learning with memory, and command execution. The studies mentioned above are limited to a particular intelligence task and mostly focused on command execution. We still need a versatile foundation for achieving multiple intelligent tasks of different natures. To this end, this study examines the potential of physical reservoir computing (PRC) as such a foundation. Reservoir computing is a branch within the discipline of recurrent artificial neural networks. In a (c) Figure 1: The overall concept of physical reservoir computing and its implementation in this study: (a) The conceptual setup of a generic reservoir computer, where only the connection weights between the reservoir's nodal response and the output layer are trained. (b) A well-studied example of a physical reservoir computer that uses a random network of mass and nonlinear spring as the computing kernel [11]. (c) Overall setup for the origami-based reservoir in this study. Here, a very simple, paper-folded origami is the physical kernel, the dynamic vibrations of its vertices are the reservoir states, and the target intelligent behaviors are information perception tasks. In detail, the folded Miura-ori is attached to the shaker receiving vibration signal. Then displacements for 28 vertices (attached with green cards) are recorded, acting as reservoir states. A group of linear readout weights could be obtained by training this set of vectors with linear regression and setting the target output as perception information (e.g. payload weight, payload position, input patterns). This set of readout weight then could be used to predict different information received by Miura-ori. reservoir computer, the interconnection weights inside the neural network's kernel remain fixed, and only output weights are trained to reach the targeted output ( Figure 1a) [12]. Such a fixed neural net kernel is called the "reservoir." Reservoir computing has been widely used in time-series prediction tasks like robotic motion planning [13] and text recognition [14]. More importantly, since the reservoir does not change during training, one can use a physical body as the reservoir and harvest its nonlinear and high-dimensional dynamic responses as the computational resource [11] (essentially, the physical body itself becomes the neural network). For example, a recurrently connected mass-spring-damper system could act as a physical reservoir and achieve different machine learning tasks like emulation, pattern generation, and even text recognition (Figure 1b). Following up on this idea, researchers used many other physical systems to achieve reservoir computing, including soft robots [15][16][17], tensegrity structure [18,19], and pneumatically driven apparatus [20]. A key advantage of physical reservoir computing is its simplicity and flexibility to achieve different tasks -so it can become the foundation for multi-faceted mechano-intelligence. In the physical reservoir, only a group of linear readout weights are needed for its state vectors (e.g., spring lengths in Figure 1b) to achieve the targeted output, and one can train these readout weights with simple linear regressions. This setup means the nonlinear dynamics of the physical body are most responsible for complex computation. Moreover, one can apply separate sets of readout weights to the same reservoir to achieve different machine-learning tasks concurrently. Finally, PRC is also a powerful method for edge computing devices [21] since it can store memory and process information simultaneously without time delay. Therefore, this study examines the use of physical reservoir computing as the foundation to achieve mul-tiple and complex intelligent tasks. More specifically, we use a very simple, paper-folded Miura-ori as the physical kernel ( Figure 1c) and show that it has sufficient reservoir computing capacity by projecting input signals (e.g., forces from the environment) to a high-dimensional state space (aka. nonlinear vibrations in its body). As a result, the Miura-ori can analyze the dynamic interactions between its body and the surrounding environment to extract valuable knowledge (aka. information perception) -a vital but sparsely studied component of intelligence in the mechanical domain. We use extensive experimentation to show that an origami reservoir can "observe" and "analyze" its mechanical vibration responses and achieve three information perception tasks. 1) The origami reservoir can accurately estimate the weight of payloads applied at its top edge, 2) it can classify the position of these payloads, and 3) it can recognize different input signal patterns (frequency and amplitude). Moreover, we experimentally show that the origami reservoir can simultaneously achieve any combination of the two tasks (aka. multi-tasking). One can achieve these tasks simply by extracting origami's displacement fields from videos and training the corresponding readout weights with linear regressions. In what follows, we first explain the experimental setup, including the physical platform and computing framework for origami reservoir computing. Next, we detail the results regarding the information perception tasks. We further study the influence of reduced computing dimension on intelligence performance. Finally, we end this paper with a summary and discussion. Physical Reservoir Computing Setup Physical Kernel In all experiments, we use a traditionally folded Miura-ori sheet made from 1.3mm thin print paper as the computation kernel ( Figure 2a). Miura-ori is a classical origami pattern that has been used as the backbone of many deployable structures [22], multi-functional materials [23], and compliant robotics [24] . It is a periodic tessellation of unit cells consisting of four identical quadrilateral facets connected by mountain and valley creases. We first perforate the crease pattern into the print paper on a mechanical plotter machine (GraphicTech FCX4000-60) and manually fold the Miura-ori until the dihedral "folding" angle between facets is between 45 • to 60 • . Then we place the folded Miura-ori on a large-stroke exciter (APS 113) and fully fix origami to the exciter table on its left bottom corner. This way, the origami can receive base excitation u(t) generated via a National Instrument DAQ system with a power amplifier ( Figure 1c). During the tests, we attach small green markers to Miura-ori's vertices-points where the crease lines meet-and record their displacements using a video camera at 25 frames per second with 720p resolution. The size of the origami reservoir (or the number of green markers) is, therefore, 4 × 7. Then, by post-processing the video footage in MATLAB, we obtain all vertical displacements of the 28 vertices (or nodes), labeled as s 1 , . . . , s n shown in Figure 1(c). These data are used as the reservoir state vectors for information perception tasks. With linear regression, readout weights (w 0 , w 1 , . . . , w n ) can be obtained to get desired output. Computing Framework Since the origami body is the fixed reservoir kernel, training is reduced to finding a set of static linear readout weight W out = [w 0 , w 1 , . . . , w n ] via simple linear regression: W out = [I S] + y = Φ + y,(1) where matrix S is compiled by assembling reservoir state s j at every training time steps, I is a column of ones for calculating bias term. [.] + refers to the Moore-Penrose pseudo-inverse to accommodate nonsquare matrices. y represents reference signals in each time step, and it becomes a matrix Y when more than one reference is provided for multi-tasking. y is the only variable that needs to be adjusted manually in training for different tasks. With obtained readout weights, the reservoir output (shown in Figure 1c) is: y(t) = w 0 + N i=1 w i s i .(2) The performance is evaluated based on the rooted mean squared error (RMSE): RM SE = 1 M M j=1 y j − ∧ y j 2 ,(3) where M is the number of the video frame in evaluation, y j is the reservoir output in the j th frame, and ∧ y j is the target value in the j th frame. Dynamic Responses The input signal u(t) in the information perception tasks is a sinusoidal function: u(t) = A sin(2πf t),(4) where A is excitation amplitude, f is the input frequency, and t is the elapsed time in seconds. We conducted multiple experiments, each with a unique combination of payload weight, payload position, and input frequency. The payload, made of small magnets, weighs from 3 to 18 grams (16 different levels). For reference, the origami weighs 6 grams. Its position varies from the upper left corner of the origami reservoir (labeled as "a") to the upper right corner (labeled as "h"). The input frequency is set between 1 and 6 Hz (7 different levels). Therefore, 896 groups of reservoir state vectors (16 × 8 × 7) are collected to comprehensively investigate the best training method and corresponding intelligent task performance. In every test, vibratory input lasts for 15 seconds. The first and last 5 seconds of data (125 frames) are discarded to eliminate transient responses, and the remaining displacement data are used for reservoir training and testing. Figure 2(d) shows four typical time responses of the 28 vertices (aka. state vectors). In these tests, the payload is located at the top left corner of origami; the payload weights and input frequencies are <3g, 1Hz>, <17g, 1Hz>, <3g, 6Hz>, and <17g, 6Hz>. Comparing these data shows that the origami vertices' displacements are close to the excitation signal u(t) when the input frequency is low at 1 Hz and the payload weight is small at 3 grams. In this testing condition, the Miura-ori behaves like a rigid body with minimal folding deformation. However, when the Miura-ori carries a much heavier payload at 17 grams (about three times its weight), its vertices displacements deviate from the input signal and show a much more significant nonlinearity. Such nonlinearity originates from the non-uniform deformation from facet bending [25], and it is required for successful reservoir computing [11]. It is worth noting that we test payloads up to 18 grams. Any heavier weights could generate chaotic responses and even buckles the Miura-ori. On the other hand, when the input frequency increases to 6Hz, the nonlinearity is not captured clearly in the recorded data even if the payload is heavy and the non-uniform dynamic deformation still exists. Two factors contribute to the significantly weaker nonlinearity at higher input frequencies. First, the output amplitude from the shaker naturally decreases as the frequency increases due to its power limit. Secondly, the data sampling rate -the camera's frame per second (FPS) setting -is 25, meaning that the camera can only capture about 4 data points per cycle with a high input frequency of 6 Hz. Such a relatively low sampling rate might lead to the loss of some critical information. Therefore, even though a relatively high-frequency actuation might be beneficial to achieve high computing power in the physical reservoir, we are still constrained by sensors' sensitivity and operating frequency. Information Processing by Origami Reservoir Task 1: Payload Weight Estimation In this section, we investigate whether the origami reservoir can directly predict the weight of its payload by assessing its influence on the time response. To train the origami to achieve this task, we need to assemble a state vector matrix by combining the reservoir state responses from two different "training" payloads. That is S = [S m 1 ,pa ; S mn,pa ], where S m 1 ,pa is 5 seconds of state vector matrix from placing the 3-gram payload on position "a" (upper left corner of the origami), and S mn,pa is another 5 seconds of state vector matrix from the other payload. The corresponding target output is a piece-wise step function whose value equals the training payload weight in that: y 1 (t) = m 1 = 3 (0 < t < 5) m n (5 ≤ t < 10) .(5) The readout weight W payload is then determined based on Eq. (1), and all other state vectors (not used for training) are used for evaluating the prediction performance. The final predicted payload is then the average y j over 125 frames of test data. Here, to rigorously assess the best training approach, we choose the second training mass m n between 4 and 18 grams, and then survey the corresponding payload weight estimation performance (the input frequency is fixed at 3 Hz). The matrix plot in Figure 3(a) summarizes the results. Every row in this matrix represents the results based on a combination of two training masses, highlighted by the orange dashed lines. For example, in the third to last row, we select the 3-gram payload as the first training mass and the 16-gram one as the second training mass to obtain the readout weights. Then, this set of readout weights is applied to the reservoir state vectors from other payloads to estimate their weight. Correspondingly, the rest of this row is the estimation result. Figure 3(b) shows the reservoir outputs from this 3 and 16-gram payload training. The reservoir output clearly shows separability (i.e., a small difference in the input would result in a significant change in the output so that different inputs give different outputs). This is a prerequisite for a successful reservoir. In the matrix plot, we mark the block with yellow if the estimation error is less than 30%, which is considered a successful estimation, and blue otherwise. By carefully surveying the results from all selections of training weights, one can see that the successful cases concentrate in the lower triangle of the matrix, indicating that the origami reservoir can better predict unknown payload weights if they are within the range of two training payloads. The prediction becomes more reliable when the second training payload is heavier, which is likely the benefit of the more nonlinear and high-dimensional responses discussed earlier. In addition, an unsymmetrical setup (e.g., payload at the corner of origami) gives a better prediction than a symmetric one (e.g., payload at the center of origami). The influence of input frequency on prediction accuracy is also essential to be understood. To this end, we fix the training payloads to be 3 and 16 grams at the upper left corner of origami -the favored training setup -and increase the input frequency from 1 to 6 Hz (1, 1.5, 2, 3.5, 4, 5.5, and 6Hz). For each tested input frequency, we obtain a separate set of readout weights from the two training payloads and apply it to predict all other payload weights (from 4 to 15 grams). predictions under different input frequencies show good consistency with the ground truth. However, some discrepancies still exist. The origami reservoir underestimates when the true payload weight is less than two times its weight, especially when the input frequency is low in the 1-3.5 Hz range. In contrast, the reservoir overestimates the payload weight when the payload is heavier than 12 grams at higher input frequencies. This error might be caused by information loss due to the sampling rate, as we discussed before. Overall, the average rooted mean square error is 0.364g for all tested input frequencies. Task 2: Payload Position Classification For this task, we investigate whether the origami reservoir can predict the position of different payloads on its body. We use a similar setup as the payload weight estimation task to obtain the reservoir state vector matrices. For training, we assemble state vectors from the same payload but in different positions "a" (upper left corner) and "h" (upper right corner, Figure 4a). That is S = [S m i ,pa ; S m i ,ph ], and each state vector matrix component is 5 seconds long. The target output is again a piece-wise constant step function: y 2 (t) = −1 (0 < t < 5) 1 (5 ≤ t < 10) .(6) Based on this training setup, we obtain a unique set of readout weights W position for each payload and then apply it to other state vectors from this payload located at positions "b" to "g." Since the targeted output is -1 for position "a" and 1 for position "g," the averaged reservoir outputs fall into a range of [-1, 1] when the payload is at these intermediate positions. Therefore, the payload is classified as on the origami's left side when the averaged reservoir output is less than 0 and on the right side otherwise. Figure 4(b) summarizes the results of the position classification task from three different input frequencies and ten different payloads (ranging from 8 to 17 grams). We find that the origami reservoir could not provide reliable results when the payload's weight is roughly the same as or less than the origami (6 grams). This is likely because Miura-ori deforms relatively uniformly with small external force, so it cannot distinguish different payload positions. We collect all successful cases with payload positions ranging from "b" to "g" in Figure 4(b). The overall error rate -corresponding to 1, 3, and 5 Hz input frequencies -is 17/60 (22%), 10/60 (17%), and 8/60 (13%), respectively. Therefore, increasing input frequency gives higher accuracy. Figure 4(c) shows Task 3: Input Pattern Recognition example outputs from the reservoir under three frequencies, all carrying a 16-gram payload. These outputs clearly show the characteristic separation property (aka. different payload positions generate significantly different reservoir outputs) even though the predicted payload position is sometimes inaccurate. Overall, the origami reservoir can classify the payload position more accurately when these payloads are heavier (around three times the origami's weight) and located far away from the origami's center. Since a relatively low input frequency is desirable for payload weight estimation but a higher frequency works better for payload position classification, an input frequency of around 2 to 3 Hz gives a balanced performance between these two tasks. Task 3: Input Pattern Recognition Input signal recognition and classification is an essential component of intelligent behavior, and it is the prerequisite for more complex learning, decision-making, and control tasks. To this end, we investigate whether the origami reservoir can learn and memorize different input frequency and amplitude patterns from the underlying shaker. If origami has such a capability, it should recognize unknown combinations of these input parameters with sets of pre-learned readout weights. First, we train the origami reservoir to identify input frequencies. We set up three different input frequency patterns -4 Hz as pattern A, 2 Hz as pattern B, and 6 Hz as pattern C -while the amplitude is constant at "level 2" in the shaker controller. For the training, we place the 6-gram payload at position "a" to induce sufficiently nonlinear dynamic responses. Then we excite the origami by these three patterns in a sequence, each for 5 seconds, and collect the corresponding reservoir state vectors S = [S A ; S B ; S C ]. The target output y 3 (t) is set as the magnitude of input frequencies as a function of time (orange dashed line in Figure 5a). Next, in a separate training, we train the same origami reservoir to identify different input amplitudesusing the input magnitude of level 2 as pattern I, level 1 as pattern II, and level 4 as pattern III -and collect the corresponding reservoir state matrix S = [S I ; S II ; S III ]). Note that the input frequency is constant at 4Hz. The target output is set up similarly (black dashed line in Figure 5a). By completing these two sets of training, we obtain two separate sets of readout weights W amp and W f req . Figures 5(a) summarize the training results of frequency and amplitude pattern identification, respectively. One can see that the origami reservoir can be successfully trained to output the correct frequencies and magnitudes. For comparison, we calculate another set of outputs without the origami reservoir by applying the readout weights only to the bottom four origami vertices' displacement (whose time responses are closest to the shake input). The corresponding results are shown by blue lines, which exhibit no input identification capability. This comparison proves the necessity of an origami reservoir with its nonlinear dynamics to accomplish computing and machine learning. Once the training is complete, we test if the origami reservoir can recognize an unknown input signal with a random combination of the three frequencies and magnitudes. For example, Figure 5(c) shows two input patterns produced with a random sequence of three frequencies and amplitudes, each lasting for a random duration. Once we apply the two sets of pre-trained readout weights W amp and W f req to the origami's vertice's displacements from these random excitations (a few examples shown in Figure 5d), we can obtain the reservoir's prediction on the corresponding input parameters (Figure 5e, f). To eliminate the influences of high-frequency fluctuation in the data, we sample and average the reservoir output every 0.2 s as the final result (solid red line in Figure 5e,f), which is close to the true input frequencies and magnitudes. Multi-Tasking Multi-tasking in physical reservoirs has been achieved with emulation (i.e., emulating several nonlinear filters with the same state vectors) [26], but it has yet to be proved with more complex intelligent behaviors. Here, we use two case studies to explore multi-tasking in the origami reservoir and examine whether it can extra two types of information from the same state vectors. In the first case study, we task the origami reservoir to estimate the payload weight and position simultaneously. In the second case, we train the origami to recognize payload weight and input frequency simultaneously. Such multi-tasking requires more state vectors for training. For the first case study, four groups of state vectors are assembled for training: including the payload of 8 or 17 grams located at positions "a" or "h." Therefore, the reservoir state vector matrix for training is S = [S 8g,pa ; S 17g,pa ; S 8g,ph ; S 17g,ph ], each component consisting of 5 seconds of data. Two target outputs are defined according to the real value of the payload weight and position (Figure 6), and each target output will give a set of readout weights. Training for payload weight and position simultaneously generates near-perfect fits to the targeted outputs (Figure 6a). Then we apply the two sets of readout weights to other reservoir state vectors from random payload weight and position combinations, such as <16g, position b>, <9g, position b>, <9g, position g>, <11g, position c>). The results show that the origami reservoir can correctly classify the payload position and, at the same time, accurately estimate its weight with less than 10% errors. Estimating an unknown payload's weight is more straightforward than classifying its position. One can add more position data to training. However, adding more training data under this setting of position target function do not promise to increase the accuracy rate of classification. For the second case study of concurrent payload weight and input frequency recognition, training requires six groups of state vectors, including 3 and 15-gram payloads, as well as 4, 2, and 6 Hz input fre-quencies ( Figure 6c). Therefore, S = [S 15g,4Hz ; S 15g,2Hz ; S 15g,6Hz ; S 3g,4Hz ; S 3g,2Hz ; S 3g,6Hz ]), each components containing 5 seconds of data. The two target outputs are set up similarly as shown in Figure 6(b). The training results for input frequency recognition show some fluctuations, but those for payload weight estimation are consistent. Similarly, we obtain two sets of readout weights from the training and then apply them to state vectors from four randomly selected input settings: <6g, 2Hz>, <6g, 6Hz>, <9g, 4Hz>, and <9g, 6Hz>. Surprisingly, even though the training result does not clearly separate the 2Hz and 4Hz input frequencies, the origami reservoir still precisely predicts the testing frequency under a different payload. In multi-perception task, the influence of different observing elements on the performance of each perception task should be noted. In general, we observe that the payload weight strongly influences the multitasking performance. For example, for the input frequency and magnitude recognition task, the origami reservoir's prediction is more accurate when the payload is near the same weight as the origami itself (Figure 7a). Taking a closer look at the reservoir's output when the payload is 3, 6, and 9 grams, one can find it difficult to distinguish the reservoir's output between lower input frequencies (2 and 4Hz) and lower input amplitude (level 1 and 2 in shaker controller, as shown in data circled with the dashed lines in the first plot of Figure 7b,c). On the other hand, if the payload becomes heavier at 9 grams, more Training results for simultaneous payload weight and input frequency recognition. In this case, we place the payload of 15g on position "a" and excited the origami using 4, 2, and 6Hz input frequencies, each for 5 seconds in sequence. Then we repeat the process with the 3-gram payload. (b, bottom) Two groups of reservoir outputs under four prediction cases when <magnitude, frequency> is chosen as <6g, 2Hz>, <6g, 6Hz>, <9g, 4Hz>, and <9g, 6Hz>. Reduced Reservoir Dimensionality chaotic data occur (data circled with dashed-dotted lines in the third plot). Reduced Reservoir Dimensionality In the case studies above, we collected all 28 vertices displacements to construct the state vector matrices (S). Such design can, however, lead to a complicated mechatronic setup if one wants to use embedded sensors to measure these reservoir states. Therefore, it is vital to uncover the influence of reducing reservoir dimension on information perception performance by reducing the number of vertices displacement used in these tasks. We analyze the rooted mean square error of payload weight estimation and frequency/amplitude recognition tasks based on randomly selected 4 (14% of total vertices), 8 (28%), 12 (42%), 16 (57%), 20 (71%), 24 (86%), and 28 (100%) vertices displacements. Figure 8(a) shows that the error of payload weight estimation rapidly decreases when we increase the dimension of computing data. About 30% of vertices displacement is sufficient to obtain an accurate prediction. This observation is consistent with the authors' previous simulation results [27] [ref] . Figure 8(b) details some examples of the reservoir output with a 15-gram payload. The output from only 14% of computing nodes deviates significantly from the actual payload weight. But when we use 28% of the vertices (or nodal) displacement, the averaged reservoir output quickly shifts to the true value, although there is a significant fluctuation in the output data. As we use more vertices displacements, such fluctuation decreases. The situation is different for the input parameter recognition tasks. The errors in input frequency and amplitude recognition decrease more gradually with increasing computing dimension. As a result, roughly 50% of the vertex displacement is necessary for accurate predictions (Figure 8c and 8e, respectively). When closely comparing the reservoir outputs with 14%, 57%, 71%, and 100% of the vertices displacements, one can see that the origami reservoir cannot filter undesired signals and recognize the input parameters precisely with less than 60% of nodal displacements. Therefore, input pattern recognition demands more computing power than payload weight estimation. Summary and Discussion Via extensive experimentation, we demonstrate that origami structures harbor sufficient physical reservoir computing capacity to perform intelligent information perception tasks like input recognition, payload weight estimation, payload position classification, and a combination of two tasks. In all these tasks, the computation or machine learning occurs in the physical vibrations of the origami, and only a simple linear regression is required for training. We also obtained insights into how to set up the origami reservoir for better intelligence performance. Here we discuss some important observations and conclusions. First, we need two payloads -one relatively light compared to the origami and the other heavy -to train the readout weights for payload weight estimation. The accuracy of payload weight estimation is high when the weight of other payloads falls between the two training values, and the input frequency is low (1 -3.5 Hz). Second, precisely estimating the payload position is challenging. Therefore, we use the origami reservoir as a classifier to predict whether the payload is on the left or right half. This classification task is more likely to succeed when the payload is much heavier than the origami, and the input frequency is relatively high (3 -6Hz). Under these conditions, origami can exhibit significantly nonuniform deformation to separate the response from different payload positions. Therefore, a moderate input frequency (3Hz) is the ideal choice to estimate the payload magnitude and position simultaneously (aka. multi-tasking). Finally, the origami reservoir accomplished input pattern recognition tasks, a prerequisite embedded controlling and behavior switch. Besides these three information perception tasks, we prove that multi-tasking is achievable by applying two target functions to one group of state vector matrices (e.g., estimating payload weight and position; or predicting payload weight and input frequency). However, the increased training burden will naturally sacrifice accuracy, especially when some tasks negatively impact the performance of others (e.g., in Figure 7). Thus, it is crucial to consider the trade-off between multi-tasking and desired accuracy. Despite these promising results, several challenges arise during the experiment effort. The most significant is the robustness and repeatability. The origami exhibits plastic deformation during repeated vibration, so the resting positions of vertices inevitably drift slightly over time. The origami and the payload could receive minor disturbance by accident, which will also influence the results of nonlinear projection. Therefore, ensuring an identical experimental setup between training and testing is vital. The other challenging phenomenon is the fluctuations in reservoir outputs, especially in the input recognition tasks. One possible explanation is that we use the vertices' displacement as state vectors, which shows less nonlinearity than other state variables, such as creases angle. In conclusion, a successful setup of an origami reservoir for information perception depends on robust training of readout weights, consistent design of the physical platform, careful choice of reservoir state vectors, and a sufficiently large dimension for computing. Overall, the results of this study are stepping stones for building more sophisticated and practical intelligent tasks for constructing advanced materials and robots with mechano-intelligence. Figure 2 : 2Setting up the origami reservoir computing. (a) The crease geometry of a Miura-ori shows the tessellation of mountain folds (dashed line) and valley folds (dash-dotted line). (b) 3D illustration of the folding process. Note that the folding angle is the dihedral angle between two quadrilateral facets. (c) Paper-based Miura-ori on plotter cutter and after folding. (d) Vertices (or nodal) displacements recorded by the camera. The payload is 3 and 17 grams, and the input frequency is 1 or 6 Hz. Different color lines represent the different vertices in Miura-ori. Figure 3 : 3Estimating payload weight with different input excitation frequencies. (a) Summary of the origami's reservoirs predictions on different payload weights based on different training setups. Each row in this matrix corresponds to a unique selection of the two training masses (highlighted by orange dashed lines). For example, the second row shows the results from training with the 3 and 4-gram payloads, and the last row is from training with the 3 and 18-gram payloads. Each column in this matrix shows the origami's prediction corresponding to a different payload, so the value inside each block is the predicted payload weight, and the background color is set as yellow if the error is less than 30%. (b) The reservoir's output corresponding to payloads from 4 to 15 grams, while readout weight is trained with the 3 and 16-gram payloads. (c) Prediction results under different excitation frequencies. The solid orange line is the ground truth. (d) The root-mean-square error of prediction with different frequencies. Figure 3 (Figure 4 : 34c) summarizes all the reservoir predictions, andFigure 3(d) shows the corresponding root mean square errors (RMSE). Results of payload position classification task under different payload and input frequencies (1Hz, 3Hz, 5Hz). (a) The definition of positions "a" to "h" on the origami body. (b) Color block diagram summarizing the origami reservoir's prediction on payload position. The payload is classified to be on the left half of origami if the output is less than 0 (blue color) and, otherwise, the right half (red color). Note that the first and last columns are training results, while columns 2 to 7 are the classification results. Each row represents the results from a different payload weighing 8 to 17 grams. (c) Example reservoir outputs when the payload is 16 grams, and the base excitation input frequencies are 1, 3, and 5 Hz, respectively. Figure 5 : 5Identification of input patterns with different frequencies and amplitudes: (a) Training for input pattern recognition regarding frequency (up) and magnitude (below). The dashed lines are target functions for training. They are constant piecewise functions whose value equals the input frequency magnitude (4, 2, and 6Hz) or input magnitude (level 2, 1, 4 in the shaker controller), respectively. The solid red line is the training results with the reservoir, while the blue line is the training output without involving the origami reservoir. (b) Two input patterns for testing, with random combinations of three different frequency or amplitude patterns, respectively. (c) Example data showing the displacements of 8 vertices under the testing input frequency pattern shown in (b). (d) The results of the input pattern recognition task for frequency (up) and amplitude (below), respectively. Dashed lines are the true input patterns, and solid red lines are the origami reservoir's predictions. Figure 6 : 6Multi-tasking with origami reservoir. (a, top) Training results for simultaneous payload weight and payload position estimation. In this case, we place an 8-gram training payload and then a 17-gram one on position "a" of the origami reservoir, each for 5 seconds in sequence. Then, they are put on position "h." (a, bottom) The two concurrent outputs of origami reservoir under four testing conditions, when <magnitude, position> is chosen as <16g, pb>, <9g, pg>, <9g, pb>, <11g, p3>. (b, top) Figure 7 : 7Influence of payload weight on the accuracy of input patterns recognition. (a) Relationship between RMSE of pattern recognition and the payload weight. (b, c) Reservoir outputs for input frequency and magnitude recognition with the 3g, 6g, and 9g payload, respectively. The true input parameters are the same as shown inFigure 5(b). Figure 8 : 8Influence of reservoir dimensionality (aka., number of vertices displacement) on information perception tasks performance. (a) The RSME of the weight estimation task for 4, 6, 8, 10, 12, 14, and 15-gram payloads using different amounts of vertices displacements. The payload is located in position "a," with a 4Hz input frequency. (b) The reservoir output corresponding to the 15-gram payload and different computing dimensions. (c, e) The RSME for input frequency and amplitude pattern recognition based on various percentages of vertices displacements, respectively. (d, f) The reservoir outputs for the predicted frequency and amplitude patterns, respectively, when 8, 16, 24, and 28 nodal displacements are used. The true input parameters are the same as shown inFigure 5(b). AcknowledgementsThe authors acknowledge the support from the National Science Foundation (CMMI-1933124). . M Sitti, Extreme Mechanics Letters. 46101340M. Sitti, Extreme Mechanics Letters 2021, 46 101340. . F Iida, F Giardina, Robotics, and Autonomous Systems. 6Annual Review of ControlF. Iida, F. Giardina, Annual Review of Control, Robotics, and Autonomous Systems 2022, 6. R Pfeifer, F Iida, Embodied Artificial Intelligence: International Seminar. Dagstuhl Castle, GermanySpringerR. Pfeifer, F. Iida, In Embodied Artificial Intelligence: International Seminar, Dagstuhl Castle, Ger- many, July 7-11, 2003. Revised Papers. Springer, 2004 1-26. . G Mengaldo, F Renda, S L Brunton, M Bächer, M Calisti, C Duriez, G S Chirikjian, C Laschi, Nature Reviews Physics. 2022G. Mengaldo, F. Renda, S. L. Brunton, M. Bächer, M. Calisti, C. Duriez, G. S. Chirikjian, C. Laschi, Nature Reviews Physics 2022, 1-16. . C Laschi, B Mazzolai, IEEE Robotics & Automation Magazine. 23107C. Laschi, B. Mazzolai, IEEE Robotics & Automation Magazine 2016, 23, 3 107. K Nakajima, H Hauser, R Kang, E Guglielmino, D G Caldwell, R Pfeifer, Frontiers in computational neuroscience 2013. 791K. Nakajima, H. Hauser, R. Kang, E. Guglielmino, D. G. Caldwell, R. Pfeifer, Frontiers in compu- tational neuroscience 2013, 7 91. . P Bhovad, J Kaufmann, S Li, Extreme Mechanics Letters. 32100552P. Bhovad, J. Kaufmann, S. Li, Extreme Mechanics Letters 2019, 32 100552. E G Hevia, L De La Rochefoucauld, R J Wood, 2022 International Conference on Robotics and Automation (ICRA). IEEEE. G. Hevia, L. De La Rochefoucauld, R. J. Wood, In 2022 International Conference on Robotics and Automation (ICRA). IEEE, 2022 7138-7144. . A Silva, F Monticone, G Castaldi, V Galdi, A Alù, N Engheta, Science. 6167160A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, N. Engheta, Science 2014, 343, 6167 160. . M A Mcevoy, N Correll, Science. 62281261689M. A. McEvoy, N. Correll, Science 2015, 347, 6228 1261689. . H Hauser, A J Ijspeert, R M Füchslin, R Pfeifer, W Maass, Biological cybernetics. 5355H. Hauser, A. J. Ijspeert, R. M. Füchslin, R. Pfeifer, W. Maass, Biological cybernetics 2011, 105, 5 355. B Schrauwen, D Verstraeten, J Van Campenhout, Proceedings of the 15th european symposium on artificial neural networks. p. the 15th european symposium on artificial neural networks. pB. Schrauwen, D. Verstraeten, J. Van Campenhout, In Proceedings of the 15th european symposium on artificial neural networks. p. 471-482 2007. 2007 471-482. Advanced Technologies for Enhanced Quality of Life. F Wyffels, B Schrauwen, IEEEF. Wyffels, B. Schrauwen, In 2009 Advanced Technologies for Enhanced Quality of Life. IEEE, 2009 118-122. . F Araujo, M Riou, J Torrejon, S Tsunegi, D Querlioz, K Yakushiji, A Fukushima, H Kubota, S Yuasa, M D Stiles, Scientific. 20201F. Abreu Araujo, M. Riou, J. Torrejon, S. Tsunegi, D. Querlioz, K. Yakushiji, A. Fukushima, H. Kubota, S. Yuasa, M. D. Stiles, et al., Scientific reports 2020, 10, 1 1. T Li, K Nakajima, M Cianchetti, C Laschi, R Pfeifer, 2012 IEEE International Conference on Robotics and Automation. IEEET. Li, K. Nakajima, M. Cianchetti, C. Laschi, R. Pfeifer, In 2012 IEEE International Conference on Robotics and Automation. IEEE, 2012 4918-4924. K Nakajima, Brain Evolution by Design. SpringerK. Nakajima, In Brain Evolution by Design, 403-414. Springer, 2017. Y Horii, K Inoue, S Nishikawa, K Nakajima, R Niiyama, Y Kuniyoshi, ALIFE 2021: The 2021 Conference on Artificial Life. MIT Press2021Y. Horii, K. Inoue, S. Nishikawa, K. Nakajima, R. Niiyama, Y. Kuniyoshi, In ALIFE 2021: The 2021 Conference on Artificial Life. MIT Press, 2021 . . K Caluwaerts, M D&apos;haene, D Verstraeten, B Schrauwen, Artificial life. 1935K. Caluwaerts, M. D'Haene, D. Verstraeten, B. Schrauwen, Artificial life 2013, 19, 1 35. . K Fujita, S Yonekura, S Nishikawa, R Niiyama, Y Kuniyoshi, Journal of the Institute of Industrial Applications. 692K. Fujita, S. Yonekura, S. Nishikawa, R. Niiyama, Y. Kuniyoshi, Journal of the Institute of Indus- trial Applications Engineers 2018, 6, 2 92. . T Kawase, T Miyazaki, T Kanno, K Tadano, Y Nakajima, K Kawashima, Sensors and Materials. 82803T. Kawase, T. Miyazaki, T. Kanno, K. Tadano, Y. Nakajima, K. Kawashima, Sensors and Materials 2021, 33, 8 2803. T Yamane, H Numata, J B Héroux, N Kanazawa, S Takeda, G Tanaka, R Nakane, A Hirose, D Nakano, International Conference on Neural Information Processing. SpringerT. Yamane, H. Numata, J. B. Héroux, N. Kanazawa, S. Takeda, G. Tanaka, R. Nakane, A. Hirose, D. Nakano, In International Conference on Neural Information Processing. Springer, 2018 635-643. . A A Deleo, J O&apos;neil, H Yasuda, M Salviato, J Yang, Composites Science and Technology. 191108060A. A. Deleo, J. O'Neil, H. Yasuda, M. Salviato, J. Yang, Composites Science and Technology 2020, 191 108060. . S J Wu, H Yuk, J Wu, C S Nabzdyk, X Zhao, Advanced Materials. 33S. J. Wu, H. Yuk, J. Wu, C. S. Nabzdyk, X. Zhao, Advanced Materials 2021, 33, 11 2007667. . D Rus, M T Tolley, Nature Reviews Materials. 3101D. Rus, M. T. Tolley, Nature Reviews Materials 2018, 3, 6 101. . S Li, H Fang, S Sadeghi, P Bhovad, K.-W Wang, Advanced materials. 51805282S. Li, H. Fang, S. Sadeghi, P. Bhovad, K.-W. Wang, Advanced materials 2019, 31, 5 1805282. . P Bhovad, S Li, Scientific Reports. 11P. Bhovad, S. Li, Scientific Reports 2021, 11, 1 1. In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. J Wang, S Li, American Society of Mechanical Engineers. 86281J. Wang, S. Li, In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, volume 86281. American Society of Mechanical Engineers, 2022 V007T07A064.
[]
[ "A Stellar Population Synthesis Model for the Study of Ultraviolet Star Counts of the Galaxy", "A Stellar Population Synthesis Model for the Study of Ultraviolet Star Counts of the Galaxy" ]
[ "Ananta C Pradhan \nTata Institute of Fundamental Research\nHomi Bhabha Road400005MumbaiIndia\n", "D K Ojha \nTata Institute of Fundamental Research\nHomi Bhabha Road400005MumbaiIndia\n", "A C Robin \nInstitut Utinam\nUMR 6213\nCNRS\nOSU THETA\nUniversité de Franche-Comté\n41bis avenue de l'Observatoire25000BesançonFrance\n", "S K Ghosh \nTata Institute of Fundamental Research\nHomi Bhabha Road400005MumbaiIndia\n\nNational Centre for Radio Astrophysics\nTata Institute of Fundamental Research\n411007PuneIndia\n", "John J Vickers \nAstronomisches Rechen-Institut, Zentrum für Astronomie\nUniversität Heidelberg\nMönchhofstr, 12-1469120HeidelbergGermany\n" ]
[ "Tata Institute of Fundamental Research\nHomi Bhabha Road400005MumbaiIndia", "Tata Institute of Fundamental Research\nHomi Bhabha Road400005MumbaiIndia", "Institut Utinam\nUMR 6213\nCNRS\nOSU THETA\nUniversité de Franche-Comté\n41bis avenue de l'Observatoire25000BesançonFrance", "Tata Institute of Fundamental Research\nHomi Bhabha Road400005MumbaiIndia", "National Centre for Radio Astrophysics\nTata Institute of Fundamental Research\n411007PuneIndia", "Astronomisches Rechen-Institut, Zentrum für Astronomie\nUniversität Heidelberg\nMönchhofstr, 12-1469120HeidelbergGermany" ]
[]
Context. Galaxy Evolution Explorer (GALEX), the first all sky imaging ultraviolet (UV) satellite, has imaged a large part of the sky providing an excellent opportunity for studying UV star counts. Combining photometry from the different wavelengths in the infrared (from Wide-field Infrared Survey (WISE) and Two Micron All Sky Survey (2MASS)) to UV allows us to extract a real star catalog from the GALEX source catalog. Aims. The aim of our study is to investigate in detail the observed UV star counts obtained by GALEX vis-a-vis the model simulated catalogs produced by the Besançon model of stellar population synthesis in various Galactic directions, and to explore the potential for studying the structure of our Galaxy from images in multiple near-UV (NUV) and far-UV (FUV) filters of the forthcoming Ultraviolet Imaging Telescope (UVIT) to be flown onboard ASTROSAT. Methods. We have upgraded the Besançon model of stellar population synthesis to include the UV bands of GALEX and UVIT. Depending on the availability of contiguous GALEX, Sloan Digital Sky Survey (SDSS), WISE and 2MASS overlapping regions, we have chosen a set of nineteen GALEX fields which spread over a range of Galactic directions. We selected a sample of objects from the GALEX database using the CASjobs interface and then cross-matched them with the WISE+2MASS and SDSS catalogs. UV stars in the GALEX catalog are identified by choosing a suitable infrared (IR) colour, J − W1 (W1 is a WISE band at 3.4 µm), which corresponds to a temperature range from 1650 K to 65000 K. The IR colour cut method, which is used for the first time for separation of stars, is discussed in comparison with the GALEX+SDSS star counts method. Results. We present the results of the UV star counts analysis carried out using the data from GALEX. We find that the Besançon model simulations represent the observed star counts of both the GALEX All-sky Imaging Survey (AIS) and Medium Imaging Survey (MIS) well within the error bars in various Galactic directions. Based on the model analysis, we separated out white dwarfs (WDs) of the disc and blue horizontal branch stars (BHBs) of the halo from the observed sample by selecting a suitable FUV − NUV colour.Conclusions. The Besançon model is now ready for further comparisons in the UV domain and will be used for prospective studies for the UVIT instrument to be flown onboard ASTROSAT.
10.1051/0004-6361/201321757
[ "https://arxiv.org/pdf/1403.2561v1.pdf" ]
54,665,839
1403.2561
210e187cc28dcc9c76f1e62c9a068a6c47c23739
A Stellar Population Synthesis Model for the Study of Ultraviolet Star Counts of the Galaxy 11 Mar 2014 May 9, 2014 Ananta C Pradhan Tata Institute of Fundamental Research Homi Bhabha Road400005MumbaiIndia D K Ojha Tata Institute of Fundamental Research Homi Bhabha Road400005MumbaiIndia A C Robin Institut Utinam UMR 6213 CNRS OSU THETA Université de Franche-Comté 41bis avenue de l'Observatoire25000BesançonFrance S K Ghosh Tata Institute of Fundamental Research Homi Bhabha Road400005MumbaiIndia National Centre for Radio Astrophysics Tata Institute of Fundamental Research 411007PuneIndia John J Vickers Astronomisches Rechen-Institut, Zentrum für Astronomie Universität Heidelberg Mönchhofstr, 12-1469120HeidelbergGermany A Stellar Population Synthesis Model for the Study of Ultraviolet Star Counts of the Galaxy 11 Mar 2014 May 9, 2014Astronomy & Astrophysics manuscript no. ms˙astro-ph c ESO 2014Stars: general -Ultraviolet: stars -Galaxy: disc -Galaxy: stellar content -Galaxy: halo Context. Galaxy Evolution Explorer (GALEX), the first all sky imaging ultraviolet (UV) satellite, has imaged a large part of the sky providing an excellent opportunity for studying UV star counts. Combining photometry from the different wavelengths in the infrared (from Wide-field Infrared Survey (WISE) and Two Micron All Sky Survey (2MASS)) to UV allows us to extract a real star catalog from the GALEX source catalog. Aims. The aim of our study is to investigate in detail the observed UV star counts obtained by GALEX vis-a-vis the model simulated catalogs produced by the Besançon model of stellar population synthesis in various Galactic directions, and to explore the potential for studying the structure of our Galaxy from images in multiple near-UV (NUV) and far-UV (FUV) filters of the forthcoming Ultraviolet Imaging Telescope (UVIT) to be flown onboard ASTROSAT. Methods. We have upgraded the Besançon model of stellar population synthesis to include the UV bands of GALEX and UVIT. Depending on the availability of contiguous GALEX, Sloan Digital Sky Survey (SDSS), WISE and 2MASS overlapping regions, we have chosen a set of nineteen GALEX fields which spread over a range of Galactic directions. We selected a sample of objects from the GALEX database using the CASjobs interface and then cross-matched them with the WISE+2MASS and SDSS catalogs. UV stars in the GALEX catalog are identified by choosing a suitable infrared (IR) colour, J − W1 (W1 is a WISE band at 3.4 µm), which corresponds to a temperature range from 1650 K to 65000 K. The IR colour cut method, which is used for the first time for separation of stars, is discussed in comparison with the GALEX+SDSS star counts method. Results. We present the results of the UV star counts analysis carried out using the data from GALEX. We find that the Besançon model simulations represent the observed star counts of both the GALEX All-sky Imaging Survey (AIS) and Medium Imaging Survey (MIS) well within the error bars in various Galactic directions. Based on the model analysis, we separated out white dwarfs (WDs) of the disc and blue horizontal branch stars (BHBs) of the halo from the observed sample by selecting a suitable FUV − NUV colour.Conclusions. The Besançon model is now ready for further comparisons in the UV domain and will be used for prospective studies for the UVIT instrument to be flown onboard ASTROSAT. Introduction The Milky Way is the best studied Galaxy in the universe; its structure and evolution have been studied by a variety of techniques. In the early 20 th century, Kapteyn (1922) first studied the geometrical structure of the Galaxy using the star counts method whereby he counted stars on the photographic plates in selected areas of the sky. Since then the star counts method has been used as one of the preferred methods to constrain the structural parameters of the Galaxy effectively. Several reviews (Bahcall 1986;Freeman 1987;Gilmore et al. 1988;Majewski 1993;Helmi 2008;Ivezić et al. 2012) have discussed the connection of star counts to the Galactic structure. The advent of instruments with better resolution and greater sensitivity have enabled us to obtain photometric observations covering large parts of the sky in several wavelength bands. The population synthesis models of the Milky Way are well supported by these observations in predicting the different structural parameters of the Galaxy, such as stellar densi-ties, scale length, scale height, etc. Among the models built to understand the Galactic structure by star counting method, one can cite: Bahcall & Soneira (1980), Gilmore & Reid (1983), Robin & Crézé (1986), Robin et al. (2003), Girardi et al. (2005) and Jurić et al. (2008). However, the above Galaxy models are predominantly based on the visible and IR photometric surveys. Very few attempts (Brosch 1991;Cohen et al. 1994) had been made to study the star counts in UV prior to GALEX due to a lack of availability of UV photometric surveys. The advent of GALEX, which provided a wide sky coverage in UV, now allows new analysis of the UV sky (Xu et al. 2005;Bianchi et al. 2011aBianchi et al. ,b, 2013. An attempt has also been made to predict the star counts in the X-ray band (Guillout et al. 1996) by extending the Besançon model of stellar population synthesis (Robin & Crézé 1986) to the ROSAT PSPC energy bands. Indeed, the UV surveys, among others, could help in tracing the spiral structures which mainly contain very young stars. The UV surveys also help in constraining the shape of the ini-tial mass function (IMF) towards the high-mass star end as well as elucidating the recent star formation history. Moreover, they also trace very blue populations such as WDs and BHBs deep in the halo population, which in turn trace the streams and relics of ancient accretion in the Milky Way halo. GALEX has covered a large part of the sky which provides an opportunity to explore and characterize these hot sources in the FUV (1344 -1786 Å, λ eff = 1538.6 Å) and NUV (1771 -2831 Å, λ eff = 2315.7 Å) wavebands with better resolution and greater sensitivity than the previous surveys. A vivid description of the source selection, FUV and NUV magnitude error cuts and the statistical analysis of the GALEX catalog is provided by Bianchi et al. (2007), Bianchi (2009) and Bianchi et al. (2011aBianchi et al. ( , 2013. Detection of WDs and BHBs is one of the main achievements of GALEX as these sources are elusive in the other wavelength bands of the electromagnetic spectrum due to their high temperature. WDs and BHBs are integral to the study of stellar evolution and structure of the Milky Way as they belong to different stellar populations of the Galaxy. We have upgraded the Besançon model of stellar population synthesis to include the UV bands of GALEX and the upcoming UVIT 1 (which will be flown onboard ASTROSAT) to predict star counts in different parts of the sky (Todmal et al. 2010). UVIT will image the sky in the FUV (1300 -1800 Å) and NUV (2000 -3000 Å) channels, each having five filters, at a high resolution of 1.8 ′′ (Postma et al. 2011;Kumar et al. 2012a,b). Better positional accuracy of UVIT as compared to GALEX will enable more reliable cross correlation with other catalogs which will be of great utility in inferring the Galactic structure using the star counts technique. The transmission curves (effective area versus wavelength) for the FUV and NUV bands of GALEX together with each of the five FUV (left panel) and NUV (right panel) filters of the upcoming UVIT/ASTROSAT are shown in Figure 1. We have included the effective area curves of both the GALEX and all the UVIT/ASTROSAT bands in the model to simulate the UV star counts in these bands. Apart from the GALEX bands, we will discuss the model simulated star counts of the BaF2 (FUV: 1370 -1750 Å, λ eff = 1504 Å) and NUVB4 (NUV: 2505 -2780 Å, λ eff = 2612 Å) bands of UVIT/ASTROSAT. The expected sensitivity limits (5σ) in AB magnitude system in the UVIT BaF2 (FUV) and NUVB4 (NUV) wavebands, for an exposure time of 200 seconds, are 20.0 and 21.2 magnitudes, respectively (ASTROSAT Handbook 2013; private communication). It is worth mentioning here that throughout the paper we have used AB system for the GALEX, UVIT and SDSS data sets, whereas the 2MASS and WISE data sets are in the Johnson system (see Section 2). We give details of the observations and selection of UV stars in Section 2. We describe about the Besançon model Galaxy model in Section 3 and discuss the comparison of the GALEX+WISE+2MASS and GALEX+SDSS star counts in Section 4. We present the comparison of the model with the observations in Section 5, and discuss the distribution of the model star counts in Section 6. We mention the identification of WDs and BHBs using FUV − NUV colour in Section 7. Finally, we summarize our conclusions in Section 8. 1 http://www.iiap.res.in/Uvit Observations and cross-correlation of GALEX sources 2.1. GALEX data GALEX was an orbiting space telescope launched in April, 2003, which was terminated in mid-February, 2012. The satellite and on-orbit performance are described in Martin et al. (2005) and Morrissey et al. (2005Morrissey et al. ( , 2007. It observed the sky in two UV bands, FUV and NUV, simultaneously, with a spatial resolution of 4.2 ′′ and 5.3 ′′ , respectively. The field of view is 1.25 • in diameter and the images are sampled with 1.5 ′′ pixels. The typical AB magnitude limits (5σ depth) met by AIS for an exposure time of 100 seconds and MIS for an exposure time of 1500 seconds are 19.9/20.8 (FUV/NUV) and 22.6/22.7 (FUV/NUV), respectively (Morrissey et al. 2007). The AIS has the largest sky coverage when compared to the other GALEX surveys. GALEX has observed a large part of the sky (∼75%), excepting the Galactic plane and some regions of the Magellanic Clouds due to safety concerns of the detectors. In this paper, we have used the GALEX GR6 data which is available in Multi-mission Archive at Space Telescope Science Institute (MAST 2 ). Selection of GALEX fields We have selected nineteen GALEX fields for which both the detectors of GALEX were turned on. We retained sources which had a reliable NUV detection, however, FUV detections are available for ∼3.5% and ∼6.8% of the NUV detections in the selected AIS and MIS fields, respectively. The rest of the NUV sources do not have a FUV detection because their FUV fluxes are too low to be detected. We include only regions within a radius of 0.5 • from the center of the tiles to eliminate edge artifacts and bad sources along the edge as well as to avoid overlapping areas and duplication of the sources. The coverage areas of the observed fields are calculated by summing up the areas of all the tiles in a field. The fields are selected in the footprints of the GALEX, SDSS, WISE and 2MASS surveys. The various fields are as follows: -Four GALEX tiles were chosen at the southern Galactic latitudes: two each in AIS and MIS. -Eight fields with large area coverage of the sky were chosen in several northern Galactic directions. The fields include: two regions towards the Galactic center (GC) (one each in AIS and MIS), two regions towards the Galactic anticenter (GAC) (one each in AIS and MIS), and one region each in AIS towards the Galactic antirotation (GAR), Galactic low latitude (GLL), Galactic high latitude (GHL), and Galactic pole (GP) directions. -Seven fields in AIS were chosen at 10 • intervals of b around l ∼ 50 • in order to study the latitude variation of UV star counts. The center coordinates, survey types, area coverages, location in the Galaxy, number of GALEX tiles and the range of exposure times of NUV and FUV observations of each of the fields are given in Table 1. WISE+2MASS data The AIS and MIS of GALEX overlap with the 2MASS and WISE footprints. 2MASS (Skrutskie et al. 2006) has observed the entire sky in the J (1.24 µm), H (1.66 µm) and K s (2.16 µm) near-IR (NIR) bands with angular resolutions of 2.9 ′′ , 2.8 ′′ , and 2.9 ′′ respectively; while WISE (Wright et al. 2010) has mapped the sky in the W1 (3.4 µm), W2 (4.6 µm), W3 (12 µm), and W4 (22 µm) mid-IR bands, with angular resolutions of 6.1 ′′ , 6.4 ′′ , 6.5 ′′ , and 12.0 ′′ respectively. The 5σ point source sensitivities of the four WISE bands are better than 0.08, 0.11, 1 and 6 mJy (equivalent to 16.6, 15.6, 11.3, and 8.0 Vega magnitude) in unconfused regions on the ecliptic (Wright et al. 2010). The existing WISE+2MASS cross-matched catalog available at Infrared Science Archive (IRSA 3 ) has been used for convenience. This catalog has been produced using a 3.0 ′′ matching radius, which was found to be adequate considering the positional accuracy and resolution. We have made use of the Virtual Astronomical Observatory (VAO 4 ) for cross-matching GALEX sources with WISE+2MASS sources. GALEX sources were uploaded into the VAO, seeking their WISE and 2MASS counterparts using a match radius of 3.0 ′′ . We found most of the real matched sources within this radius, with a very small fraction (< 1%) having multiple matches which were removed from the final catalog. We also estimated the possible contamination by spurious matches (random coincidences) for the matched sources following the method of Bianchi et al. (2011a). For this purpose we used a match radius of 6.0 ′′ , which is equivalent to the resolution of WISE, to find the GALEX counterparts of WISE sources. The spurious matches were found to be ∼10% of the total matched sources, and 75% of these spurious matched sources lie beyond a distance of 3.0 ′′ . SDSS data So far, SDSS has mapped over 35% of the full sky in five optical photometric bands (u, g, r, i, z) covering the wavelength range from 3000 to 11000 Å (Aihara et al. 2011). GALEX GR6 has been cross-matched against SDSS DR7 and the provided cross-matched table is xSDSSDR7. Several works (Seibert et al. 2005;Budavári et al. 2009;Bianchi et al. 2007Bianchi et al. , 2011a have explained the cross-matching of the GALEX catalog with SDSS, astrophysical source classifications and related statistical analyses. We uploaded the GalexIDs of the objects into the GALEX CASjobs 5 SQL (Structured Query Language) interface to determine their SDSS counterparts in a search radius of 3.0 ′′ . We have eliminated the multiple matches (< 1%) from the GALEX+SDSS final catalog. The estimated spurious matches in case of GALEX+SDSS are found to be ∼7% within 3.0 ′′ radius. The SDSS star/galaxy classifications have been adopted while performing the match in order to separate out point sources from the source list. SDSS classified point sources (GALEX+SDSS) include both stars and quasi stellar objects (QSOs), out of which we selected QSOs using the SDSS colour cuts from Richards et al. (2002) and removed them from the GALEX+SDSS point sources and termed the clean sample as 'GALEX+SDSS stars'. 2.5. Selection of stars from the GALEX catalog by IR colour cut method Figure 2 shows J − W1 versus NUV colour-magnitude diagram (CMD) of all GALEX sources that are cross-matched with WISE+2MASS sources for the regions in the directions of the GC and the GAC, each covering 69.9 deg 2 of the sky. QSO candidates are selected using the SDSS colour cuts from Richards et al. (2002) and are represented by blue crossed symbols in the plot. We clearly see two groups of sources in the figure well separated by J − W1 colour. The stars verified by their SDSS classification as point sources in a cross-matched sample are identified to be bluer than J − W1 < 1.2 and the extra-galactic objects (e.g. galaxies, QSOs, etc.) are redder, with J−W1 > 1.2. Since the contamination by SDSS-identified QSOs is estimated to be negligible in the J − W1 < 1.2 star counts (< 0.1% of the whole sample), we have used the J − W1 colour cut procedure for all the fields to separate the stars from the extragalactic objects. Henceforth in the paper, we refer to GALEX and WISE+2MASS cross-matched sources with J − W1 < 1.2 (GALEX+WISE+2MASS) as 'UV-IR stars'. Figure 3 shows the distribution of UV-IR stars as a function of the GALEX UV magnitudes for AIS and MIS. Stars with NUV and FUV magnitude errors less than 0.5, 0.4, 0.3, 0.2 and 0.15 are displayed with magenta, green, red, blue and cyan colour lines respectively, whereas the black line represents the stars without any magnitude error cut. The typical 5σ magnitude limits of the NUV and FUV bands for AIS and MIS (see Section 2.1) are shown by vertical dashed lines. As seen from the histograms, a progressive stringent error cut eliminates the fainter stars. The completeness limits need to be established according to a given magnitude error. If we consider all stars without accounting for errors, the star counts go deeper but their values are not reliable due to the uncertainty on the magnitude measurement. This is particularly true for the FUV filter where some spurious detections can occur. Finally, we retained stars with magnitude error less than 0.2 in both bands as this error cut gives magnitude limits almost similar to the typical 5σ limits of the GALEX bands for AIS and MIS which are provided by Morrissey et al. (2007). We have applied magnitude error cuts (similar to the one shown in Figure 3 for UV-IR stars) in the original GALEX source catalog that includes all the GALEX detections and also in the matched GALEX source catalog obtained after crossmatching with the WISE+2MASS catalog. We find a loss of GALEX sources in the matched catalog when compared with the original GALEX source catalog at a specific magnitude error cut. The completeness limits for the original GALEX sources for NUV and FUV magnitude error cuts of 0.2 are 20.5/21.0 magnitude (FUV/NUV) in AIS and 22.5/22.5 magnitude (FUV/NUV) in MIS. The completeness limits at the same magnitude error cuts for the matched catalog become 20.0/20.5 magnitude (FUV/NUV) in AIS and 22.5/22.0 magnitude (FUV/NUV) in MIS and these limits are the same for the UV-IR stars too. For a specific magnitude error cut, the FUV and NUV completeness limits of the observed sources which are cross-matched to the surveys at longer wavelengths become brighter than the completeness limits of the unmatched GALEX source catalog due to the loss of faint sources in the former. Photometric error cuts and completeness limits In order to examine which objects are affected by the limits of the combined surveys (GALEX+WISE+2MASS), we split the stars into two NUV − W1 colour intervals: hot (NUV − W1 < 5) and cool (NUV −W1 > 5) stars. We checked the completeness limit of the NUV band (AIS) for these two colour ranges. For hot stars, we found that the completeness limit of GALEX NUV (AIS) is reduced by 0.5 magnitude (i.e., the effective magnitude limit gets brighter). The GALEX completeness limit (AIS) of hot stars is therefore limited by the depth of WISE, and similarly by the depth of 2MASS. For cool stars, the NUV (AIS) completeness limit is the same in the GALEX catalog alone and in the combined catalog with the near-IR surveys. Besanç on Galaxy model The Besançon model is a population synthesis model based on a scenario of Galactic evolution and constrained by dynamics. In the model, five populations are taken into account: thin disc, thick disc, stellar halo, bar, and bulge (Robin et al. 2012). The previous versions of the model are extensively described in Robin et al. (2003). We use the newest version of the model (version of April 2013; Robin et al. 2012) which has been upgraded to include the FUV and NUV passbands of GALEX and the upcoming UVIT/ASTROSAT, by applying their filter transmission curves to produce UV star counts in various Galactic directions. The model uses a set of evolutionary tracks, a star formation rate and an IMF as defined in Haywood et al. (1997), to generate different stellar populations. The colours are computed from the Basel Stellar Library (BaSeL3.1) model atmospheres (Westera et al. 2002). In this new version of the model, DA and DB type WDs are included using the evolutionary tracks and atmosphere models from Holberg & Bergeron (2006). The luminosity functions are obtained assuming an initial-to-finalmass ratio (m I = 9.1743m f − 3.6147) from Kalirai (2008). The distribution in age is computed assuming a lifetime on the main sequence (MS) from Eggleton et al. (1989) and a lifetime on the giant branch of 15% of the time on the MS. The repartition in DA (WD with hydrogen rich atmosphere) and DB (WD with helium rich atmosphere) is computed assuming that at T e f f > 40000 K they are all DA, and at T e f f < 20000 K, 50% are DB with a linear variation between 20000 K and 50000 K. The final luminosity function is normalised to fit Harris et al. (2006). Similarly, the BHBs are incorporated in the model by taking the evolutionary tracks from BaSTI (A Bag of Stellar Tracks and Isochrones) models (Pietrinferni et al. 2004). Ultimately, the model produces UV star counts by Monte Carlo simulations using a set of stellar atmospheric models, observational photometric errors and extinction. The model incorporates an extinction (A V ) assuming an ellipsoidal distribution of diffuse absorbing matter, which follows an Einasto extinction law and is depicted by an adjustable normalization (extinction gradient) of 0.7 mag/kpc in the V band. We produced the model simulations towards various Galactic directions assuming the default extinction gradient. However, the default value of diffuse extinction (0.7 mag/kpc in the V band), which may not be appropriate at low latitudes, can be adjusted by adding a few absorbing clouds with a given adhoc distance and extinction from the Schlegel et al. (1998) maps. This has been illustrated in Section 5.1. The ratios between UV band to visual extinction are taken to be 2.67 and 2.64 for the FUV and NUV band of GALEX, respectively, following the extinction law of Cardelli et al. (1989). Stars in the simulated GALEX catalog have a UV colour, FUV and NUV magnitudes, a temperature range from 1650 K to 65000 K, log g from -1 to 9, all luminosity classes and a range of metallicities. In the simulations done for comparison with the GALEX observed star counts, the simulated stars are mostly MS stars (∼77%) with a small contribution from giants and subgiants (∼17%). The WDs are ∼6% of the sample and reside at the bluer end of FUV − NUV colour (see Sections 7). Figure 4 shows the distribution of the GALEX AIS star counts (for field 5 in Table 1) as a function of NUV magnitude for the model simulation (solid line), GALEX+SDSS stars (dashed line), UV-IR stars (GALEX+WISE+2MASS: dasheddotted line) and GALEX+WISE star counts with no 2MASS detection (dotted line). The error bars shown in the model star counts are due to Poisson noise. The NUV 5σ detection limit (NUV magnitude = 20.8; Morrissey et al. 2007) and the completeness limit (∼ 20.5 magnitude; see Section 2.5) for AIS are demarcated by the solid and dashed vertical lines, respectively. Stars with NUV magnitude up to the completeness limit are well detected by the GALEX, SDSS, WISE and 2MASS surveys; a good agreement in star counts among the cross-matched surveys and the model simulations can clearly be seen in Figure 4. Comparison of the GALEX+WISE+2MASS and GALEX+SDSS stars It is also evident from Figure 4 that the GALEX+SDSS stars are slightly more than the UV-IR stars in the NUV band at the fainter magnitudes. This discrepancy could be caused by 2MASS: since the time gap between the WISE and 2MASS surveys is ∼12 years, high proper motion stars may have moved outside the cross-matching radii. However, stars with proper motions high enough to move by 3.0 ′′ in ∼12 years are very rare in a survey of a few square degrees. Another possibility is that the 2MASS J band, which has a 10σ point source sensitivity limit of about 15.8 magnitude, does not penetrate deeply enough to provide counterparts for all WISE detections. Though GALEX+SDSS has a smaller sky area coverage and a fainter limit compared to GALEX+WISE+2MASS, both the selections yield a close match of the star counts at the brighter end. It is also possible that the GALEX+SDSS stars are still contaminated by faint galaxies and quasars. So, we preferred to use the star counts determined by the J − W1 colour cut (UV-IR stars) rather than the GALEX+SDSS stars. Data and model comparison We modelled the stellar density distribution of the Milky Way in UV using the Besançon model of stellar population synthesis (as described in Section 3) for different regions of the sky. Four simulated catalogs for each of the fields chosen for our study were produced in order to reduce the statistical noise. Appropriate photometric errors were applied in the model to produce realistic simulations and the error information was assumed from the observed data which is a polynomial function of the magnitude. We can simulate the catalogs using the 'small field' option which assumes that the density does not vary across the field, or using the 'larger field' option with a given step in longitude and latitude, to account for the fields where the density can vary. We have used 'small field' option for the small fields (e.g., area < 15 deg 2 ) by providing the center l and b coordinates of the fields along with their coverage area. For the larger fields (e.g., area > 15 deg 2 ), we have used the 'larger field' option where we provide the range of l and b coordinates and a step size (e.g., from 1.0 • to 2.5 • for small to large fields, respectively) to cover the field. However, the gradients in the fields (Table 1) are small enough that considering either the center of the field or the range of l/b does not make any difference in the predicted star counts. Comparison of observed UV star counts with the model in various fields Initially, simulations were performed for four GALEX tiles (fields 1 -4 in Table 1) Table 1). Similarly, Figures 6c and 6d represent the comparison of star counts for MIS in the directions of the GC and the GAC, covering an area of 22.77 deg 2 and 18.85 deg 2 , respectively (fields 7 -8 in Table 1). These fields are chosen at the northern intermediate Galactic latitude of the Galaxy. The error bars shown in the model-predicted star counts are due to Poisson noise. The maximum estimated asymmetric error in the observed counts is ∼2% -10% depending on the NUV magnitude bins (i.e., error increases towards the fainter magnitude bins), which is not shown in the plots. The model shows a good agreement with the observation (UV-IR stars and GALEX+SDSS stars) down to an NUV magnitude of ∼20.5 for AIS and 22.0 for MIS (see Figures 5 and 6). We have also produced the model-predicted star counts for one of the passbands (NUVB4: 2505 -2780 Å, λ eff = 2612 Å) of upcoming UVIT/ASTROSAT which is shown by a dasheddotted line. Star counts are enhanced in the UVIT NUVB4 band compared to the GALEX NUV band because NUVB4 covers a smaller wavelength range and its effective wavelength is longer than the effective wavelength of the NUV band. Most of the stars have flux peaks at longer wavelengths, such that NUV −NUV B4 is positive. Since the magnitudes are normalized to AB system, the integral of the filter does not matter while computing magnitudes, though narrower filters will demand longer exposure times to get the required magnitude. The model-predicted star counts for the regions at the GHL, the GAR and the GP (solid line: fields 9 -12 in Table 1) match well the UV-IR stars (solid circles) and the GALEX+SDSS stars (open circles) except the region at the GLL (see Figures 7a, 7b, 7c and 7d). As seen in Figure 7d, the model simulated NUV star counts (solid line) produced using the standard diffuse extinction do not match observation beyond NUV magnitude fainter than 18.5. This mismatch could be due to the default extinction gradient being used in the model not being sufficient at the GLL. We took the line of sight extinction (A V = 0.1 magnitude) for the GLL from the Schlegel et al. (1998) maps and then corrected the extinction by adding a cloud of A V = 0.1 magnitude at a distance of 1 Kpc (Section 3). The model-predicted star counts after correcting the extinction (dashed line) show a good agreement with the UV-IR stars. In Figure 8, we have shown the distribution in FUV magnitudes of the UV-IR stars (solid circles) and model-simulated (solid histograms) star counts for AIS and MIS (fields 5 -8 in Table 1) towards the GC and the GAC. Despite the poor statistics, the model fit well the observations up to the completeness limit of the data sets (see Section 2.5). We have also produced the model-predicted star counts for one of the FUV passbands (BaF2: 1370 -1750 Å, λ eff = 1504 Å) of the forthcoming UVIT/ASTROSAT which is shown by a dashed-dotted line in Figure 8. Since the BaF2 passband range is close to the GALEX FUV passband, the UVIT model simulated FUV star counts match the GALEX observed FUV star counts reasonably well. Overall the Besançon model of stellar population synthesis upgraded to the UV passbands simulates star counts which are consistent with the observed GALEX star counts and can be used efficiently for the study of Galactic structure parameters. Latitude variation in star counts In order to study the latitude variation of UV star counts, we have chosen GALEX fields at 10 • Galactic latitude intervals for l ∼ 50 • . We determined NUV star counts per square degree in each field separately for the GALEX+SDSS stars and the UV-IR stars. As shown in Figure 9, the solid circles represent the UV-IR stars while the open circles show the GALEX+SDSS stars. The solid line represents the model generated star counts. The model errors due to Poisson noise are shown in the plot while the asymmetric errors on the UV-IR star counts which arise due to the propagation of photometric errors are not shown. The stellar density decreases from lower to higher Galactic latitudes in case of both observed and model star counts. The UV-IR star counts with NUV magnitude brighter than 20.5 magnitude match model simulations at the intermediate and high Galactic latitudes. However, a slight deviation of model simulated counts from observed counts is seen at low Galactic latitudes. This could be due to the default extinction gradient used in the model which might be inappropriate at low latitudes because some clouds can be present as discussed above for Figure 7(d). Comparison with the TRILEGAL model The predictions from the TRILEGAL model (Girardi et al. 2005), which is another stellar population code, have been compared with UV star counts by Bianchi et al. (2011a). It was found that the TRILEGAL-predicted NUV star counts which show an overall good match to observations at brighter magnitudes are better at the northern high latitudes and the southern low latitudes. We produced NUV star counts using the 3 alternative IMFs that the TRILEGAL website 6 proposes. However, we see in Figure 10 that the Besançon model produces a better fit to real star counts than TRILEGAL does in the GHL field close to the pole as well as in the GAR field at the intermediate latitudes. Here we use a WD modeling similar to TRILEGAL, with small differences. The initial-to-final mass relation from Kalirai (2008) is used in the Besançon model while TRILEGAL alternatively uses Marigo & Girardi (2007) or Weidemann (2000), the latter giving a better fit to the GALEX data (see Figure 9 in Bianchi et al. 2011a). We also use different atmosphere models (Holberg & Bergeron 2006), while TRILEGAL uses either Koester (2008) or TLUSTY models (Hubeny & Lanz 1995). Bianchi et al. (2011a) pointed out that the difference between these two models is not larger than 0.05 magnitude in FUV − NUV colour for most of the WDs. Finally, TRILEGAL does not consider DB WDs because it includes only WDs hotter than 18000 K, while we have taken them into account. However, the difference between TRILEGAL and the Besançon model predictions is mainly due to the more detailed account for the settling of the disc with age in the Besançon Galaxy model (the dynamical constraint which is used, forces the sub-components of the thin disc to follow a tied age/vertical scale height relation in agreement with the observed age/velocity dispersion relation). Distribution of the stars We find that the model reproduces the observed UV star counts as selected from the GALEX data. The star counts are dominated by MS stars, WDs and BHBs. The vertical distribution of different stellar populations depends on their structural parameters. In Figure 11a, we show the contribution of the thin disc (dotted line), thick disc (dashed line), halo (dashed-dotted line) and sum of the three populations (solid line) predicted by the model for an AIS field towards the GC at the intermediate Galactic latitude. The relatively bright stars are dominated by the thin disc at NUV magnitudes brighter than 18.5 whereas the thick disc and halo stars become significant at NUV magnitudes fainter than ∼18.5 and ∼19.5 respectively. This is very similar to the comparison made by Bianchi et al. (2011a) for hot star candidates. Considering the stars with NUV magnitudes brighter than 20.5, we found that the thick disc stars are the most dominant population and ∼54% -∼60% of the total population (depending on the Galactic direction). We have shown the vertical distribution of the model simulated stars in Figure 11b. It is evident that the thin disc star counts (dotted line) dominate up to a distance of 1.5 kpc over the Galactic plane whereas the thick disc star counts (dashed line) dominate at distances between 1.5 and 4.0 kpc beyond which the halo stars (dashed-dotted line) dominate the total stellar population. A similar trend has been observed by both Du et al. (2003) for BATC (Beijing-Arizona-Taiwan-Connecticut) multicolour photometric survey star counts and Phleps et al. (2000) for CADIS (Calar Alto Deep Imaging Survey) deep star counts for regions at intermediate Galactic latitudes. Blue hot stars FUV − NUV colour is an important indicator of the spectral type of the stars. Particularly, UV colour can be used to identify hot BHBs and WDs (Kinman et al. 2007;Bianchi et al. 2011a), which emit most of their light in UV because of their high temperatures. The BHBs are comparatively more luminous in UV than the other population II stars. Similarly, the WDs which are the end product of the stellar evolution of the intermediate and low mass stars, provide important information about the Galactic disc star formation history. Comparing the observed FUV − NUV colour of stars with the model, we were able to separate out the halo BHBs and disc WDs from the whole sample of stars. Figure 12 shows the comparison of GALEX FUV − NUV colours for the UV-IR stars (solid circles) and model simulated star counts (solid-lined histogram) for the AIS fields towards the GC and the GAC. We have considered stars with NUV magnitude < 20.5 and FUV magnitude < 20.0 for the GALEX AIS survey. The FUV − NUV colours of WD (dotted line) and BHB (long-dashed line) populations are also shown along with the UVIT FUV − NUV (BaF2 -NUVB4) colour (dashed line) in the plot. Looking at the FUV − NUV model predictions, the sources can be classified into two groups, the one with FUV − NUV > 2.5 are the red cool stars and the other with FUV − NUV < 2.5 are blue hot stars. The blue stars exhibit a bimodal distribution indicating the existence of two populations; the peak at FUV − NUV ∼ -0.5 are the hot WDs of the disc and the peak at FUV − NUV ∼ 2.0 are BHBs of the Galactic halo. In the Besançon model, the temperature range of WDs is from 10000 K to 27000 K and that of BHBs is from 5000 K to 20000 K. Hotter stars with temperature greater than 27000 K are rare to be found in significant numbers in the data considered here. The colour distributions in Figure 12 towards both the GC and the GAC show some differences between the model and observations. Specially, we notice that the very blue peak at FUV − NUV < 0, due to hot WDs, is too high in the model. Moreover, there is a lack of stars in the GC field at 0 < FUV − NUV < 1.5. In the colour range where the BHBs dominate, the number of predicted stars is well in agreement with the observations in both fields, indicating that the halo BHB density is well simulated. There is a dearth of model-simulated stars in the colour range, 2 < FUV − NUV < 3.5, which is not understood yet and will be investigated in a further study. At FUV − NUV > 4, the model lacks stars but more towards the GAC than towards the GC. This colour domain is mostly dominated by the thick disc MS stars. We guess that it is due to the scale length which will be investigated in a forthcoming paper. Both, photometry and spectroscopy can be used to identify WDs and BHBs. Several large area sky surveys such as 2MASS, SDSS and GALEX have been used to distinguish them by appropriate colour selections and it is worth mentioning a few of the works. Kleinman et al. (2013) produced the latest catalog of spectroscopically confirmed DA and DB type WDs from SDSS Data Release 7. Using the data from GALEX FUV and NUV imaging, Bianchi et al. (2011b) presented a catalog of hot star candidates, particularly WDs. Similarly, the first selection of BHBs from SDSS colours was made by Yanny et al. (2000) and then followed by many others (Sirko et al. 2004;Bell et al. 2010;Deason et al. 2011;Vickers et al. 2012). We have identified WD and BHB candidates using suitable GALEX FUV − NUV colours. It was found from the model FUV − NUV colour (Figure 11) that BHB and WD star candidates occupy the colour range, 1.5 < FUV − NUV < 2.5 and FUV − NUV < 0.5, respectively. In the mentioned colour range, we obtain a clean sample of WD candidates, whereas in the sample of BHB candidates, a contamination of non-BHB candidates, such as WDs and MS stars, constitute about 7%. These colour ranges have been used for the separation of WD and BHB candidates from other populations in the observed sample. In order to substantiate our identification of the WD and BHB star candidates using GALEX FUV − NUV colour, we compared them with their known 2MASS colours. E(B − V) values for the stars were measured from Schlegel et al. (1998) and converted to NUV, J and H extinction using Cardelli et al. (1989) Figure 13a shows the (J −K) o versus (NUV − J) o colour-colour diagram for the BHB candidates. The sources at different latitude intervals are represented by different symbols. The dashed parallelogram encloses the area used by Kinman et al. (2007) which contains 66% of the BHB candidates selected on the basis of FUV − NUV colour. Similarly, Figure 13b shows the H − K versus J − H colour-colour diagram for the WD candidates. The dashed rectangle encloses the area in the colour-colour diagram chosen from Hoard et al. (2007) that contains a majority of the WD candidates of our sample. The location of our selected WD and BHB star candidates in the re-spective 2MASS colour window indicates that the FUV − NUV colour can also be used as a potential tool in identifying WD and BHB candidates. This is a preliminary investigation and we will use this in our future work of an all sky study of these sources using the GALEX data. Conclusions The Besançon model of stellar population synthesis has been previously checked at many different wavelengths from visible (U band) to mid-IR (12 µm). The model produces accurate star counts up to magnitude ∼ 22 in the visible or 18 in the K band. However, the stars that dominate the counts in the UV were not previously checked vis-a-vis model predictions. The availability of the GALEX data gives opportunity to check model predictions for high temperature, blue stars, specially BHBs from the halo and WDs from the disc. We have shown that the model performs very well for these types of stars as it does for other types. The model provides a good check that the population synthesis scheme gives predictions which are consistent with each other at all wavelengths. To do so, we make use of Holberg & Bergeron (2006) models which provide good stellar atmospheres and cooling tracks for WDs. However, the ratio between DA and DB type WDs has to be investigated more deeply. We have generally considered a simple dust distribution while limiting the comparisons to |b| > 20 • . In future, we will compare the model at lower latitudes, in particular for the sake of analysis of the spiral structure, assuming the 3D extinction map from Marshall et al. (2006). We also compared predictions in the UV bands from the TRILEGAL model with our model and found that the predictions of the Besançon model are in better agreement with the observation than the TRILEGAL model as shown in Figure 10. However, in the faintest NUV magnitude bins TRILEGAL seems to be better in the GAR field. It will be something to look at carefully in the future using the all sky observations of GALEX and WISE, and we aim to present a detailed comparison between observations and the model. We plan to complete the analysis by comparing model predictions with a variety of models of WDs, varying the tracks and investigating whether it could be possible to constrain the star formation history of the disc from the WDs distribution. Moreover, an analysis of the thick disc WD luminosity function could also be interesting for constraining the formation history of this old population, but it would require complementary kinematical data. We have seen that BHBs are a major component of GALEX stars. An analysis of this component could lead to constraints on the shape of the halo, once the contamination by extra-galactic objects is eliminated. The final model can be safely used to predict star counts of various types in the UV wavelengths at the level of a few percent in many Galactic directions; the model produces star counts that match well down to FUV ∼ 20.0, NUV ∼ 20.5 for AIS, and FUV ∼ 22.5, NUV ∼ 22.0 for MIS. However, for the hot WDs, there is a mismatch of UV colours between the model and observation. A more detailed study is planned to explain the discrepancies by changing the WD luminosity function and the scale lengths alternatively. A study is also going on to better constrain the thick disc shape from large surveys in the visible and near-IR (Robin et al., in prep). We plan to further investigate the UV star counts with this revised model and the GALEX survey in the near future. The Besançon model is also developed to predict star counts in the UV passbands of the forthcoming UVIT telescope to be flown onboard ASTROSAT. We compared the model-predicted star counts at two of the UVIT filters with that of the GALEX observed star counts because of the similar wavelength coverage of both the instruments. The UVIT-predicted star counts are sensitively different from the GALEX observed star counts due to the differences in effective wavelengths. UVIT star counts will be very useful to separate out different stellar populations since they have several UV colours and better angular resolution compared to GALEX, which in turn will help us to estimate the structural parameters of the Galaxy with better precision. Table 1). The dotted line represents the GALEX+WISE sources with no 2MASS counterparts. The star counts are binned in 0.5 magnitude interval in NUV magnitude. The error bars in the model-simulated star counts are due to Poisson noise. The NUV 5σ detection limit (NUV magnitude = 20.8) is shown by a solid vertical line. The UV-IR star counts show a turn over at NUV magnitude ∼ 20.5 (demarcated by a dashed vertical line). Table 1). The error bars shown in the model star counts are due to Poisson noise, while the asymmetric errors in the observed star counts are not shown in the plot. Table 1). The dashed-dotted line represents FUV star counts for the BaF2 band of UVIT/ASTROSAT (1350 -1750 Å, λ eff = 1504 Å). The FUV magnitudes are binned in intervals of 0.5. The model error bars shown in the plots are due to Poisson noise. Table 1). The UV colours shown are for the stars with FUV < 20.0 magnitude, NUV < 20.5 magnitude and the photometric errors < 0.2 (in both FUV and NUV bands). FUV − NUV colours of the model-predicted WDs and BHBs are shown by a dotted line and a long-dashed line, respectively. The UVIT (BaF2 − NUV B4) colour coverage is indicated by a dashed line. Table 1. The parallelogram encloses the area occupied by the BHB samples of Kinman et al. (2007). The sources at various latitudes are represented by different symbols. b) H − K versus J − H colour-colour diagram for the WD candidates of the combined AIS fields in Table 1. The dashed line rectangle encloses the area used from Hoard et al. (2007). A reddening vector (the arrow) of A V = 1 magnitude and the mean error bars of the colours are displayed in both the diagrams. , each covering an area of 0.785 deg 2 at the southern intermediate Galactic latitudes. We binned the model and the UV-IR stars in 0.5 magnitude intervals in the NUV band, for respective tiles of AIS and MIS, in the directions of the GC and the GAC. As shown inFigure5, we found that the model star counts (solid line) match well the UV-IR stars (solid circles) as well as the GALEX+SDSS stars (open circles) up to the completeness limits of AIS (20.5 magnitude) and MIS (22.0 magnitude) for the regions at the southern intermediate Galactic latitudes.In order to check the universal validity of the model, we produced simulated catalogs in various directions of the Galaxy covering a large area of the sky.Figures 6a and 6bshow the comparison of the model-predicted NUV star counts (solid line) with the UV-IR stars (solid circles) as well as with the GALEX+SDSS stars (open circles) for AIS in the directions of the GC and the GAC, each covering 69.9 deg 2 area of the sky (fields 5 -6 in extinction law : A(NUV) = 8.90E(B−V), A(J) = 0.874E(B−V), and A(H) = 0.589E(B−V). Fig. 1 : 1Effective area versus wavelength plots for the GALEX FUV and NUV bands are shown in relation to the FUV and NUV filters of UVIT/ASTROSAT for imaging mode. The left panel shows the five FUV filters of UVIT/ASTROSAT in different colours along with the GALEX FUV filter (dashed line) and the right panel shows the same for the NUV filters. Fig. 2 : 2The diagrams show J (2MASS) − W1 (WISE) versus NUV CMD for the GALEX and WISE+2MASS cross-matched sources for the AIS fields towards the GC and the GAC. The matched sources are clearly separated in two groups indicating isolation of stars (J − W1 < 1.2) from the extra-galactic sources (J − W1> 1.2). The vertical dashed line shows the limit that we choose for selecting the point sources (J − W1 < 1.2). QSOs are shown by blue crossed symbols (see the text). Fig. 3 : 3Distribution of the UV-IR stars as a function of the GALEX NUV and FUV magnitudes for AIS and MIS. The stellar sources obtained after applying various magnitude error cuts are shown by different line styles in colours. The vertical dashed lines represent the respective 5σ detection limits of the GALEX bands for typical exposure times as mentioned in Section 2. Fig. 4 : 4Distribution of the UV-IR stars (GALEX+WISE+2MASS : dashed-dotted line), GALEX+SDSS stars (dashed line) and model-simulated star counts (solid line) for the AIS field towards the GC covering 69.9 deg 2 of the sky (field 5 in Fig. 5 : 5Comparison of the UV-IR stars (solid circles) with model-predictions (solid line) as a function of NUV magnitudes for the regions at the southern intermediate Galactic latitudes. The open circles represent the GALEX+SDSS stars. The plots are for the fields towards the GC and the GAC for individual GALEX AIS and MIS tiles, each covering an area of 0.785 deg 2 (fields 1-4 in Fig. 6 :Fig. 7 : 67Comparison of the model-predicted star counts (solid line) with the UV-IR stars (solid circles) as well as with the GALEX+SDSS stars (open circles) for the GALEX fields at the northern intermediate Galactic latitudes. The Galactic coordinate ranges, survey types and area coverages are mentioned in each panel. The dashed-dotted line shows the model-simulated star counts for the NUVB4 band of UVIT/ASTROSAT (2505 -2780 Å, λ eff = 2612 Å). The error bars shown in the model counts are due to Poisson noise. Comparison of model-predictions (solid line) with the UV-IR stars (solid circles) and GALEX+SDSS stars (open circles) as a function of the GALEX NUV magnitude for the regions at the GHL (7a), the GP(7b), the GAR (7c) and the GLL (7d). In plot 7d, the solid line represents the model NUV star counts produced assuming the standard diffuse extinction (as in other fields) whereas the dashed line is the same after correcting the extinction using the value from theSchlegel et al. (1998) maps (see Section 5). The model error bars shown in the plots are due to Poisson noise. Fig. 8 : 8Distribution in FUV magnitudes of the UV-IR stars (solid circles) and model-predicted (solid line) star counts for the AIS and MIS fields towards the GC and the GAC (fields 5 -8 in Fig. 9 : 9The latitude variation of NUV star counts (per deg 2 ) for both the observation and model simulation at l ∼ 50 • . The UV-IR stars, GALEX+SDSS stars and model-predicted star counts are represented by solid circles, open circles and solid line, respectively. The UV-IR stars and the GALEX+SDSS stars shown in the plot are for NUV magnitude errors < 0.2. The error bars displayed in the model star counts are due to Poisson noise. The asymmetric errors in the UV-IR star counts which arise due to the propagation of photometric errors are not shown. Fig. 10 : 10Comparison of NUV star counts predicted by the Besançon (solid line) and TRILEGAL (dashed-dotted line) models of stellar population synthesis for the fields towards the GHL and the GAR. The solid circles represent the UV-IR stars. The Besançon model error bars shown in the plots are due to Poisson noise. Fig. 11 : 11The left panel represents the distribution of various stellar populations produced by the Besançon model of stellar population synthesis in the direction of GC. Different lines are explained in the legend. Similarly, the right panel represents the vertical distribution of the stellar populations in the same direction as the left panel. Fig. 12 : 12Comparison of FUV − NUV colour between the model-predicted star counts (solid line) and the UV-IR stars (solid circles) for the AIS fields towards the GC and the GAC (fields 5 -6 in 13: a) (J − K) 0 versus (NUV − J) 0 colour-colour diagram for the BHB candidates of the combined AIS fields given in Table 1 : 1Details of the GALEX fields. The areas of different fields are chosen depending on the availability of GALEX, WISE+2MASS and SDSS overlapping regions. http://galex.stsci.edu/GR6 http://irsa.ipac.caltech.edu/Missions/wise.html 4 http://vao-web.ipac.caltech.edu/applications/VAOSCC 5 http://galex.stsci.edu/casjobs http://stev.oapd.inaf.it/cgi-bin/trilegal Acknowledgements. The authors thank the anonymous referee for useful comments and suggestions that improved the content of the paper.GALEX (Galaxy Evolution Explorer) is a NASA small explorer launched in 2003 April. We gratefully acknowledge NASA's support for construction, operation, and science analysis for the GALEX mission, developed in cooperation with the Centre National d'Etudes Spatiales of France and the Korean Ministry of Science and Technology. This work has made use of the data products from the Wide-field Infrared Survey Explorer (WISE), Two Micron All Sky Survey (2MASS) and Sloan Digital Sky Survey (SDSS). We also thank the UVIT/ASTROSAT team for providing the UVIT filter response curves.Simulations were executed on computers from the Utinam Institute of the Université de Franche-Comté, supported by the Région de Franche-Comté and Institut des Sciences de l'Univers (INSU). We acknowledge the support of the French "Agence Nationale de la Recherche" under contract ANR-2010-BLAN-0508-01OTP. Many thanks to Bernard Debray who is responsible for providing the web interface for the Besançon Galaxy model. . H Aihara, C Prieto, D An, ApJS. 19329Aihara, H., Allende Prieto, C., An, D., et al. 2011, ApJS, 193, 29 . J N Bahcall, R M Soneira, ApJ. 23817Bahcall, J. N., & Soneira, R. M. 1980, ApJ, 238, 17 . J N Bahcall, ARAA. 24577Bahcall, J. N. 1986, ARAA, 24, 577 . E Bell, X Xue, H. -W Rix, AJ. 1401850Bell, E., Xue, X., & Rix, H. -W. 2010, AJ, 140, 1850 . L Bianchi, L Rodriguez-Merino, M Viton, ApJS. 173659Bianchi, L., Rodriguez-Merino, L., Viton, M., et al. 2007, ApJS, 173, 659 . L Bianchi, Ap&SS. 32011Bianchi, L. 2009, Ap&SS, 320, 11 . L Bianchi, B Efremova, J Herald, MNRAS. 4112770Bianchi, L., Efremova, B., & Herald, J., et al. 2011a, MNRAS, 411, 2770 . L Bianchi, J Herald, B Efremova, Ap&SS. 335161Bianchi, L., Herald, J., & Efremova, B., et al. 2011b, Ap&SS, 335, 161 . L Bianchi, A Conti, B Shiao, arXiv:1312.3281J.Adv. Space Res. (in PressBianchi, L., Conti, A., & Shiao, B. 2013, J.Adv. Space Res., (in Press), arXiv:1312.3281 . N Brosch, MNRAS. 250780Brosch, N. 1991, MNRAS, 250, 780 . T Budavári, S Heinin, A S Szalay, ApJ. 6941281Budavári, T., Heinin, S., & Szalay, A. S., et al. 2009, ApJ, 694, 1281 . J A Cardelli, G C Clayton, J S Mathis, ApJ. 345245Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245 . M Cohen, T P Saseen, S Bowyer, ApJ. 427848Cohen, M., Saseen, T. P., & Bowyer, S. 1994, ApJ, 427, 848 . A Deason, V Belokurov, N Evans, MNRAS. 4162903Deason, A., Belokurov, V., & Evans, N., et al. 2011, MNRAS, 416, 2903 . C Du, X Zhou, J Ma, A&A. 407541Du, C., Zhou, X., & Ma, J., et al. 2003, A&A, 407, 541 . P P Eggleton, M J Fitchett, C A Tout, ApJ. 347998Eggleton, P. P., Fitchett, M. J., & Tout, C. A., et al. 1989, ApJ, 347, 998 . K C Freeman, ARA&A. 25603Freeman, K. C. 1987, ARA&A, 25, 603 . G Gilmore, N Reid, MNRAS. 2021025Gilmore, G., & Reid, N. 1983, MNRAS, 202, 1025 . G Gilmore, R F G Wyse, K Kuijken, ARA&A. 27555Gilmore, G., Wyse, R. F. G., & Kuijken, K. 1988, ARA&A, 27, 555 . L Girardi, M A T Groenewegen, E Htziminaoglou, L Da Costa, A&A. 436895Girardi, L., Groenewegen, M. A. T., Htziminaoglou, E., & da Costa, L. 2005, A&A, 436, 895 . P Guillout, M Haywood, C Motch, A C Robin, A&A. 31689Guillout, P., Haywood, M., Motch, C., & Robin, A. C. 1996, A&A, 316, 89 . M Haywood, A C Robin, M Crézé, A&A. 320440Haywood, M., Robin, A. C., & Crézé, M. 1997, A&A, 320, 440 . H C Harris, J A Munn, Kilic, AJ. 131571Harris, H. C., Munn, J. A., & Kilic, Mukremin 2006, AJ, 131, 571 . Amina Helmi, A&ARv. 15145Helmi, Amina 2008, A&ARv., 15, 145 . D W Hoard, S Wachter, L K Sturch, AJ. 13426Hoard, D. W., Wachter, S., and Sturch, L. K. 2007, AJ, 134, 26 . J B Holberg, P Bergeron, AJ. 1321221Holberg, J. B., & Bergeron, P. 2006, AJ, 132, 1221 . I Hubeny, T Lanz, ApJ. 439875Hubeny, I., & Lanz, T., 1995, ApJ, 439, 875 . Ž Ivezić, T C Beers, M Jurić, ARA&A. 50251Ivezić,Ž., Beers, T. C., & Jurić, M. 2012, ARA&A, 50, 251 . M Jurić, Ž Ivezić, A Brooks, ApJ. 673864Jurić, M., Ivezić,Ž., & Brooks, A., et al. 2008, ApJ, 673, 864 . J S Kalirai, B M S Hansen, D D Kelson, ApJ. 676594Kalirai, J. S., Hansen, B. M. S., & Kelson, D. D. 2008, ApJ, 676, 594 . J C Kapteyn, ApJ. 55302Kapteyn, J. C. 1922, ApJ, 55, 302 . T D Kinman, S Salim, L Clewey, ApJ. 662111Kinman, T. D., Salim, S., & Clewey, L. 2007, ApJ, 662, L111 . S J Kleinman, S O Kepler, D Koester, ApJS. 2045Kleinman, S. J., Kepler, S. O., & Koester, D. 2013, ApJS, 204, 5 . D Koester, Mem. Soc. Astron. Ital. 75282Koester, D. 2008, Mem. Soc. Astron. Ital., 75, 282 Ultraviolet to Gamma Ray. A Kumar, S K Ghosh, J Hutchings, Proc. SPIE 8443, 84434R, on "Space Telescopes and Instrumentation. SPIE 8443, 84434R, on "Space Telescopes and InstrumentationA Model for the UV Star Counts of the GalaxyKumar, A., Ghosh, S. K., & Hutchings, J., et al. 2012a, Proc. SPIE 8443, 84434R, on "Space Telescopes and Instrumentation 2012: Ultraviolet to Gamma Ray" Pradhan et al.: A Model for the UV Star Counts of the Galaxy A Kumar, S K Ghosh, P U Kamath, Proc. SPIE 8443, 84431N, on "Space Telescopes and Instrumentation 2012: Ultraviolet to Gamma Ray" Majewski. SPIE 8443, 84431N, on "Space Telescopes and Instrumentation 2012: Ultraviolet to Gamma Ray" Majewski31575Kumar, A., Ghosh, S. K., & Kamath, P. U., et al. 2012b, Proc. SPIE 8443, 84431N, on "Space Telescopes and Instrumentation 2012: Ultraviolet to Gamma Ray" Majewski, S. R. 1993, ARA&A, 31, 575 . D J Marshall, A C Robin, C Reylé, A&A. 453635Marshall, D. J., Robin, A. C., & Reylé, C., et al. 2006, A&A, 453, 635 . P Marigo, L Girardi, A&A. 469239Marigo, P., & Girardi L. 2007, A&A, 469, 239 . D C Martin, J Fanson, D Schiminovich, ApJ. 6191Martin, D. C., Fanson, J., & Schiminovich, D., et al. 2005, ApJ, 619, L1 . P Morrissey, D Schiminivoch, T A Barlow, ApJ. 6197Morrissey, P., Schiminivoch, D., & Barlow, T. A., et al. 2005, ApJ, 619, L7 . P Morrissey, T Conrow, T A Barlow, ApJS. 173682Morrissey, P., Conrow, T., & Barlow, T. A., et al. 2007, ApJS, 173, 682 . S Phleps, K Meisenheimer, B Fuchs, C Wolf, A&A. 356108Phleps, S., Meisenheimer, K., Fuchs, B., & Wolf, C. 2000, A&A, 356, 108 . H Ohlendorf, T Preibisch, B Gaczkowski, A&A. 55214Ohlendorf, H., Preibisch, T., & Gaczkowski, B. 2013, A&A, 552, 14 . A Pietrinferni, S Cassisi, M Salaris, F Castelli, ApJ. 612168Pietrinferni, A., Cassisi, S., Salaris, M., & Castelli, F. 2004, ApJ, 612, 168 . J Postma, J B Hutchings, D Leahy, PASP. 123833Postma, J., Hutchings, J. B., & Leahy, D. 2011, PASP, 123, 833 . G T Richards, X Fan, H J Newberg, AJ. 1232945Richards, G. T., Fan, X., & Newberg, H. J., et al. 2002, AJ, 123, 2945 . A C Robin, M Crézé, A&A. 15771Robin, A. C., & Crézé, M. 1986, A&A, 157, 71 . A C Robin, C Reylé, S Derriére, S Picaud, A&A. 409523Robin, A. C., Reylé, C., Derriére, S., & Picaud, S. 2003, A&A, 409, 523 . A C Robin, D J Marshall, M Schultheis, C Reylé, A&A. 538106Robin, A. C., Marshall, D. J., Schultheis, M., & Reylé, C. 2012, A&A, 538, 106 . F P Santos, A Roman-Lopes, G A P Franco, ApJ. 751138Santos, F. P., Roman-Lopes, A., & Franco, G. A. P. 2012, ApJ, 751, 138 . D J Schlegel, D P Finkbeiner, M Davis, ApJ. 500525Schlegel, D.J., Finkbeiner, D.P., & Davis, M. 1998, ApJ, 500, 525 . M Seibert, T Budavári, J Rhee, ApJ. 23Seibert, M., Budavári, T., & Rhee, J., et al. 2005, ApJ, 619, L23 . E Sirko, J Goodman, G R Knapp, AJ. 1342236Sirko, E., Goodman, J., & Knapp, G. R., et al. 2004, AJ, 134, 2236 . M F Skrutskie, R M Cutri, R Stiening, AJ. 1311163Skrutskie, M. F., Cutri, R. M., & Stiening, R., et al. 2006, AJ, 131, 1163 S S Todmal, S K Ghosh, D K Ojha, A C Robin, ASI Conference Series, Interstellar Matter and Star Formation : A Multiwavelength Perspective. D. K. Ojha Vickers, J. J., Grebel, E. K., & Huxor, A. P. 2012186Todmal, S.S., Ghosh, S.K., Ojha, D. K., & Robin, A. C. 2010, ASI Conference Series, Interstellar Matter and Star Formation : A Multiwavelength Perspective, Vol. 1, pp. 245, ed. by D. K. Ojha Vickers, J. J., Grebel, E. K., & Huxor, A. P. 2012, AJ, 143, 86 . V Wademann, A&A. 363647Wademann, V. 2000, A&A, 363, 647 . P Westera, T Lejeune, R Buser, A&A. 381524Westera, P., Lejeune, T., & Buser, R., et al. 2002, A&A, 381, 524 . E L Wright, P R M Eisenhardt, A K Mainzer, AJ. 1401868Wright, E. L., Eisenhardt, P. R. M., & Mainzer, A. K., et al. 2010, AJ, 140, 1868 . C K Xu, J Donas, S Arnouts, ApJ. 61911Xu, C. K., Donas, J., & Arnouts, S., et al. 2005, ApJ, 619, L11 . B Yanny, H J Newberg, S Kent, ApJS. 540825Yanny, B., Newberg, H. J., & Kent, S., et al. 2000, ApJS, 540, 825
[]
[ "Paracompositionality, MWEs and Argument Substitution", "Paracompositionality, MWEs and Argument Substitution" ]
[ "Cem Bozşahin [email protected]@gmail.com \nCognitive Science Department\nInformatics Institute Middle East Technical University (ODTÜ)\nAnkara Turkey\n", "Arzu Burcu Güven \nCognitive Science Department\nInformatics Institute Middle East Technical University (ODTÜ)\nAnkara Turkey\n" ]
[ "Cognitive Science Department\nInformatics Institute Middle East Technical University (ODTÜ)\nAnkara Turkey", "Cognitive Science Department\nInformatics Institute Middle East Technical University (ODTÜ)\nAnkara Turkey" ]
[]
Multi-word expressions, verb-particle constructions, idiomatically combining phrases, and phrasal idioms have something in common: not all of their elements contribute to the argument structure of the predicate implicated by the expression.Radically lexicalized theories of grammar that avoid string-, term-, logical form-, and tree-writing, and categorial grammars that avoid wrap operation, make predictions about the categories involved in verb-particles and phrasal idioms. They may require singleton types, which can only substitute for one value, not just for one kind of value. These types are asymmetric: they can be arguments only. They also narrowly constrain the kind of semantic value that can correspond to such syntactic categories. Idiomatically combining phrases do not subcategorize for singleton types, and they exploit another locally computable and compositional property of a correspondence, that every syntactic expression can project its head word. Such MWEs can be seen as empirically realized categorial possibilities rather than lacuna in a theory of lexicalizable syntactic categories.
10.1007/978-3-662-57784-4_2
[ "https://arxiv.org/pdf/1805.08438v1.pdf" ]
46,893,523
1805.08438
257cf7f418e133412ad8fe6c35455e2cf1acb765
Paracompositionality, MWEs and Argument Substitution 22 May 2018 Cem Bozşahin [email protected]@gmail.com Cognitive Science Department Informatics Institute Middle East Technical University (ODTÜ) Ankara Turkey Arzu Burcu Güven Cognitive Science Department Informatics Institute Middle East Technical University (ODTÜ) Ankara Turkey Paracompositionality, MWEs and Argument Substitution 22 May 2018SyntaxsemanticsCCGmulti-word expressionidiomverb-particlelexical insertiontype theory Multi-word expressions, verb-particle constructions, idiomatically combining phrases, and phrasal idioms have something in common: not all of their elements contribute to the argument structure of the predicate implicated by the expression.Radically lexicalized theories of grammar that avoid string-, term-, logical form-, and tree-writing, and categorial grammars that avoid wrap operation, make predictions about the categories involved in verb-particles and phrasal idioms. They may require singleton types, which can only substitute for one value, not just for one kind of value. These types are asymmetric: they can be arguments only. They also narrowly constrain the kind of semantic value that can correspond to such syntactic categories. Idiomatically combining phrases do not subcategorize for singleton types, and they exploit another locally computable and compositional property of a correspondence, that every syntactic expression can project its head word. Such MWEs can be seen as empirically realized categorial possibilities rather than lacuna in a theory of lexicalizable syntactic categories. Introduction A type is a set of values. When we write a syntactic type, say NP, we mean a set of expressions (values) which can substitute for that type. This type serves to distinguish some expressions from for example the set of expressions that can substitute for a VP type. The distinction is crucial for solving the correspondence problem in syntax-semantics. For this purpose we talk about semantic types, for example e for things and t for propositions. The concepts that can substitute for semantic types are not expressions in the sense that syntactic expressions are, because they are not observable, but they leverage a theory to hypothesize about the kind of semantics that these types stand for. These two species of types are then put in a correspondence in a theory of syntaxsemantics connection. The understanding is that if one substitutes a certain expression for a syntactic type, then its corresponding semantic type substitutes for a certain kind of semantic value. We know less about the semantic values; but, at the level of the correspondence problem, this is not very critical. It is however crucial to make the distinctions and propagate them in a parsing mechanism rather than solving all typeinterpretation problems in one go. We need a theory which provides explicit vocabulary and mechanism for the correspondence, to be more specific about the equal relevance of substitution for subexpressions which purportedly do not contribute to the meaning of the expression. In the categorial grammar parlance, for which we will use Combinatory Categorial Grammar [30,31], hereafter CCG, we can exemplify the correspondence as follows, where we use the "result-first argument-next" notation: (1) a. hits := (S\NP 3s )/NP: λ xλ y.hit ′ xy b. hit := VP inf /NP: λ xλ y.hit ′ xy Some syntactic types are further narrowed down by features, such as NP 3s above for third person singular NP, which are, in CCG, not re-entrant. We argue in the paper that in a radically lexicalized theory which adheres to transparency of derivations by type substitution (rather than lexical insertion), such as CCG, there are built-in degrees of freedom to support Multi-word Expressions (MWEs) and idioms without complicating the mechanism. Paracompositionality is key to projection of their properties in a derivation. It is the idea that, in addition to the compositionality of the lexical correspondence, which is compositional partly because it relies on non-vacuous abstractions, type substitution by (i) what we call singleton types and (ii) what is called head-dependencies in the NLP literature is also compositional because it spells non-vacuous abstraction as part of the correspondence, but as something related to the contingency of the predicate, rather than the argument structure of the predicate. In a radically lexicalized grammar both sources are available in a lexical item. These types are paracompositional also in the sense that whether we have an idiom reading or compositional one is already decided by the category of the head in the derivational process. The term contingency is used here in the sense of Moens and Steedman [24] where it relates to extension of happenings. In the case of events (culminations, points, processes and culminating processes), which have definite extension, it is an event modality of space, time and manner; and, in the case of states where extension is indefinite (e.g. understand) it is some property of the state. From now on when we use the term 'contingency' we mean something related to extension of the predicate, rather than who does what to whom in the predicate. MWEs are expressions involving more than one word in which the properties of the expression are not determined by the composition of the properties of the constituent words, which would be the case for phrases. There is a tendency to treat them as single lexical units [10,33]; but, as we shall see, CCG does not require the single unit to be the phonological representation to the left of ':=' in the format of (1). This property of CCG naturally extends to coverage of verb-particle constructions e.g. look the word up as discontiguous MWEs headed by a lexical item. Phrasal idioms and idiomatically combining phrases are classes identified by Nunberg, Sag and Wasow [25] to account for systematic variation in syntactic productivity of idioms. Typewise they will relate to singleton types (phrasal idioms) and head-word subcategorization (idiomatically combining phrases) in our formulation. As a preview of the article, we can think of the meaning distinctions as ranging from "beans" i.e. the nounphrase beans itself as a category (this is what we call the singleton type); to NP beans as the category of an NP headed by the word beans, which has wider range of substitution; and, to the polyvalent NP with the widest substitution for that type. This much is categorial grammar with type substitution. CCG as an empirical theory adds to this the claim that there is an asymmetry in the range of substitutions: the singleton types can be arguments only, and arguments of arguments and results, but never the result. We shall see that this has implications for the linguist's choice of handling syntactic productivity in a grammar. Some implications follow: Because of paracompositionality, all expressions requiring a singleton type would involve the semantic type of a predicate, and all idiomatically combining phrases requiring a different interpretation than the compositional one would have the same consequence independent of their syntactic productivity. In short, every idiom must contain a predicate (but not necessarily a verb). We cover these implications in the article. Substitution in a Derivation In (1a), the '/NP' can be substituted for by certain kinds of expressions, for example John, me, the ball, a stone in the corner, etc. Its corresponding semantic counterpart in the logical form (LF), written after the colon, has the placeholder x which can be typed as e, to be suitably substituted for by a semantic value described above. The '\NP 3s ' can be substituted by narrower expressions, for example eliminating I, you. Because this is an indirect correspondence, its semantic counterpart y can have the same type e. The tacit assumption of indirectness is sometimes made explicit, for example in Bach's [2] rule-by-rule hypothesis: The derivational process operates with syntactic types only, and when it applies the semantics of the rule, its semantics works only with LF objects. Quoting from Bach: "Neither type of rule has access to the representations of the other type except at the point where a translation rule corresponding to a given syntactic rule is applied." The "syntactic rule" in a lexicalized grammar such as CCG is the combinatory syntactic type of a lexical correspondence. The" translation rule" is the lexically-specified logical form, LF, as in (1). The derivational process reveals partially derived types, for example S\NP 3s : λ y.hit ′ s ′ y for (1a), if function application substitutes say a stone for '/NP', with some semantic value s ′ . The semantic type of such derived categories is concomitantly functional, e.g. e → t for this syntactic type. John hits is e → t too, with category S/NP: λ x.hit ′ x john ′ . We can see the relevance of derived types to substitutability in a closer look at (1b). If function application substitutes for the '/NP' in the example, the derived category would be VP inf : λ y.hit ′ s ′ y in this case. This is also an e → t type semantically. However, its syntax is narrower so that we can account for the expressions in (2). 1 (2) John persuaded Mary to/* hit/*hits the target. The derivational process works as below, with VP to-inf distinct from VP inf . NP ((S\NP)/VP to-inf )/NP NP VP to-inf /VP inf VP inf /NP NP : j ′ : λ xλ pλ y.persuade ′ (px)xy : m ′ : λ p.p : λ xλ y.hit ′ xy : h ′ > (S\NP)/VP to-inf : λ pλ y.persuade ′ (pm ′ )m ′ y > VP inf : λ y.hit ′ h ′ y > VP to-inf : λ y.hit ′ h ′ y > S\NP : λ y.persuade ′ (hit ′ h ′ m ′ )m ′ y < S : persuade ′ (hit ′ h ′ m ′ )m ′ j ′ Here function application is shown in forward form (>) and backward form (<). Derivation proceeds from top to bottom in display, as standard in CCG; i.e., bottom-up as far as parsing is concerned, and one at a time. For brevity alternative derivations using function composition are not shown; their implications for constituency are discussed in Steedman references. We also eschew the slash modalities of Baldridge and Kruijff [3] to avoid digression, which can further restrict the combination possibilities of syntactic types. They are mentioned later when they are relevant to discussion. The LF contains a structured form, viz. the predicate-argument structure, which is written in linear notation for simplicity; for example hit ′ xy is same as ((hit ′ x)y); i.e., it is left-associative. In preparation for final discussion of substitution ( §6) in relation to the wrapping operation, we can redraw this derivation by showing the substituting expressions as we proceed, which we do in Figure 1. MWEs present a challenge for substitution in such correspondences. In Schuler and Joshi's [29]:25 words: "In the pick .. up example, there is no coherent meaning for Up such that pick X Up = Pick( X , Up)." They go on to show how tree-write in the form of TAG transformations, rather than string-rewrite of CFG transformations such as [28], can deliver different meanings of such expressions after a fully compositional tree is established for 'pick', '..' and 'up'. In such systems, post-processing and reanalysis of a categorial surface derivation are possible, both for TAG and HPSG, 2 therefore these transformations are possible, indeed useful, to simplify large-scale grammar development. as it does in English, such as in Dyirbal's control construction, where the controlled absolutive argument can be the patient NP of the transitive clause or syntactic subject of an intransitive clause, but not the ergative NP of the transitive clause. It seems to require VP: λ x.pred ′ x where x's role in the controlled clause pred ′ is determined by verbal morphology of the controlled clause; see [23] for the phenomenon. Assuming a VP cross-linguistically makes narrower predictions about control. We handle this problem elsewhere. 2 TAG transformations take a phrase structure tree and decompose it to elementary structures to deliver an LF. [21] is a different TAG way to incorporate meaning postulates of [26]. HPSG uses phrasal post-classification to the same effect. For example [4,28] For radically lexicalized grammars such as CCG where such options are not available, three paths to maintaining compositionality in the presence of "non-compositional" and/or idiomatic parts seem to be available: (4) a. letting the logical form change the compositional meaning, b. introducing surface wrap, c. reassessing the substitutability of argument types, to the extent that (i) they can be narrowed by head-dependencies, and (ii) the semantic contribution of some parts of the correspondence to the predicate-argument structure can be ignored in a principled way, and locally. The problem is exacerbated by phrasal idioms which seem to have partially active syntax in some non-compositional parts, for example kick the (proverbial/old) bucket, but note ♯the bucket that John kicked, ♯kick the great bucket in the sky, and *the breeze was shot. (♯ is used to indicate unavailability of idiomatic reading. The last two examples and judgments are from [28].) However, there are also phrasal idioms which are syntactically quite active, e.g. the beans that John spilled, and spilling the musical/artistic/juicy beans. (hit ′ h ′ m ′ )m ′ j ′ Option (4a) does not always necessitate post-processing of MWEs in CCG, but, as we shall see later in (23), it does not guarantee locality of derivations either. One way to realize it is the following: (5) kicked := (S\NP)/NP : λ xλ y.if head(x) = bucket ′ then die ′ y else kick ′ xy This approach to phrasal idioms which is similar to meaning postulates for the same task such as [26] would then have to make sure that the head meaning bucket ′ has some predefined cluster of modifiers such as proverbial or old, but not much else, for example ♯kick the bucket that overflowed. It would also have to overextend itself to avoid the idiomatic reading in ♯the bucket that you kicked. As an alternative, the type NP bucket below is inspired by trainable stochastic CFGs which can distinguish argument PPs from adjunct PPs by encoding head dependencies for CFG rules, for example VP put → V put NP PP on : (We shall fix the unaccounted vacuous abstraction in it later in the paper.) (6) kicked := (S\NP)/NP bucket : λ xλ y.die ′ y It might appear to be LF-motivated just like (5) above; but, it is actually a case of (4c/i). NP bucket , meaning NP headed by bucket, can be made distinct from NP buckets because different surface expressions can be substituted for them. (6) overgenerates for the examples given above, but it might be the right degree of freedom to exploit in the syntax-semantics correspondence of idiomatically combining MWEs such as NP beans for spill the beans. In the remainder of the paper, we show that option (4c/ii) has been implicit in CCG theory all along but never used, in the form of syntactic types for which only one value can substitute ( §3). We call them singleton types. This way of lexical categorization and subcategorization predicts very limited syntax, but not as metalinguistic marking that [28] proposed for kick the proverbial/old bucket. It is due to having to enumerate different senses and contingencies of phrasal idioms (e.g. proverb bucket for senses above, also covering e.g. when I face the proverbial bucket), and pick up for MWEs. In §4 we show that idiomatically combining phrases have principled distinctions from singleton types. Head-word subcategorization such as (6) is the more promising option for them, which radically lexicalized grammars can handle without extension. There are also idioms which require analysis combining both options such as those with semantic reflexives where the referent is not part of the idiom, e.g. I twiddle my/*his thumbs. §5 covers these cases. These findings reveal some aspects of type substitution and its projection when the expressions are not fully compositional at the level of the predicate-argument structure. As such they may have implications beyond CCG. Finally we show that adopting option (4b) to analyze for example pick · · · up as pick up (· · ·) wrap overgenerates in the combinatory version of wrap ( §6), and complicates the grammar with a domino-effect in the surface version of wrap; therefore, it would do more damage than good if adopted for (discontiguous) MWEs and phrasal idioms. CCG can continue to avoid all forms of wrap in the presence of all kinds of MWEs and phrasal idioms. Singleton Types A brief preview of the proposal for (4c/ii) is as follows. A singleton syntactic type selfrepresents because it can substitute for one value only. We designate such types with strings, such as "up" or "the bucket"; for example: (7) a. picked := (S\NP)/"up"/NP: λ yλ xλ z.cause ′ (init ′ (hold x ′ yz))z b. kicked := (S\NP)/"the bucket": λ xλ y.die x ′ y (Init ′ is a function that yields a culminating state in the sense of [24].) We call categories in (7) 'paracompositional' to highlight the fact that, although their LF correspondence is intact so that the derivational process is transparent, they might have seemingly vacuous abstraction from the perspective of the predicate-argument structure, symbolized by the placeholders x above. 3 However, one can make a case that this abstraction, corresponding respectively to singleton categories "up" and "the bucket", might have a role inside the LF constants shown in primes, as contingencies. We write them for example as die ′ x y (as ceremonial death, reported death, etc.), rather than die ′ y. These LF 'constants' are convenient generalizations in CCG standing in for a plethora of features anyway, so it seems natural to think of them as having their own abstraction. (The semantic types corresponding to these contingencies are then α → t for some α.) It will be seen in §3.2 that the examples in (7) differ in their sense from picked up the book and kicked the blue bucket, therefore a separate grammar entry is empirically justified. The sense distinction is reflected explicitly in the LF, as we shall see later. Both possibilities for substitution, for the syntactic type and for its placeholder in the LF, are principally restricted by CCG. Singletons also engender a way for such entries to be morphologically more transparent, for example by being susceptible to inflection, e.g. picking, by providing a segmental alternative to contiguous but MWE pick up · · ·, which would need a morphological pointer for inflection, as noted by [28,33] for their analyses. Nunberg, Sag and Wasow's [25] dichotomy between phrasal idioms and idiomatically combining items also vanishes, because of the singleton types and head-word subcategorized argument types. The distinction between syntactically pseudo-active kick the bucket and more active spill the beans naturally follows from whether the idiomatic part has a role in the predicate-argument structure, which we capture by systematically choosing between option (4c/i) and (4c/ii) per lexical correspondence. Parsing and Correspondence with Singleton Types The crucial property of a category in a lexical correspondence such as α := A/"s" with singleton s, is that the string "s" as a category does have its own correspondence. This cannot be a literal match without categorial processing of the surface string, with s to the right of α. It is a compositional derivational process arising from (a) below, to lead to (b). The lexically specifiable difference from a polyvalent category such as NP, VP is that the item α subcategorizes for the string s, hence treat it as a category, rather than subcategorize for the category of s, viz. B in the example. To obtain B, the derivational process works as usual for s, independent of the item α. We shall see in (9) that rules of function application need no amendment for this interpretation. (8b) is lexically determined by α. In other words, the surface string s is derived by the derivational process as well. It is just that the item α carrying the singleton type as an argument decides what to do with its semantics, which we indicated schematically above as modal contribution to contingency of p, as p x of α. This is not post-processing of a category in a radically lexicalized grammar, in which all and only head functors decide what to do with the semantics of their arguments. It means that, whether an argument type is polyvalent or singleton, there has to be an LF placeholder for it, otherwise the derivational process, which is completely driven by syntactic types in CCG, cannot proceed. It can be seen in the basic primitive of CCG, viz. function application: (9) X/Y: f Y: a → X: f a (>) Y: a X\Y: f → X: f a (<) The LF of the functor, f , has to be a lambda abstraction, to be able to take any Y and yield f a. This is true of singleton '/Y' and '\Y' too. We can clearly see the role of substitution rather than insertion in projection of types. The rule (>) above is in fact realized as below (similarly for others): (10) α := X/Y: f β := Y: a → αβ := X: f a (>) There is no sense in which we can insert something into α and β as they form αβ because these are surface expressions. The singleton types present an asymmetry in argument-result (or domain-range) specification. Functors such as A/B and A\B have domain B and range A, and, apart from trivial identities where A and B are the same singleton, the interpretation where the range itself (A) is a singleton is problematic. Since A|B is a function into A for some slash '|', if it is not a trivial case of singleton identity, say "up"/"up", it is difficult to see how A can be singleton. Although there are no formal reasons to avoid singleton results, and results of results, we conjecture that singletons are arguments, and arguments of results and arguments, because there seems to be no nontrivial function of a singleton-result with grammatical significance. A related argument can be made about a singleton's potential to be the overall syntactic category of a lexical item. The notion of extending the phonological range of an item such as (a) below coincides naturally with "words with spaces" idea (e.g. ad hoc, by and large, every which way), which is common in NLP of MWEs, but (b) is also an option. (11) a. every which way := (S\NP)\(S\NP): λ pλ x.omni ′ px b. every which way := "every which way": omniway ′ Notice that (b) is different than having scored := (S\NP)/"every which way" for lexically specified verbal adjunction in the manner of [13], which, given (8), must either use entries similar to (11), or derive every which way syntactically, and choose to trump its category because it wants a narrower LF due to singleton subcategorization. However we think that both options may be redundant, because of the following. In CCG the head functor decides the semantics of its entry even if it subcategorizes for a singleton category. Therefore the entries in (a-b) above which we use in (a-b) below may be redundant if the words in "words with spaces" are part of the grammar, and if they can combine in any way, say as in (c) below for some A, B, C: S\NP There would be no post-processing or reanalysis in these cases; they would be multiple analyses because of redundancy. The transparency of derivation requires that in configurations like (8b) the constituents of the rule applying can themselves be derived. The rules that allow CCG to rise above function application in projection, composition and substitution also maintain the transparency of the syntactic process, by being oblivious to the nature of argument types in these rules: 4 (13) X/Y: f Y/Z: g → X/Z: λ x. f (gx) (> B) X/Y/Z: f Y/Z: g → X/Z: λ x. f x(gx) (> S) If the result categories are not singletons, as we argued, then the rules above never face a case where Y is a singleton. This means that, since singletons are arguments, meaning they bear a slash, say '|A' for some slash '|' in {\, /}, the slash is inherently application-only, equivalently '| ⋆ A' in [3] terminology. 5 This is corroborated by examples like below where there is no idiomatic reading: (We show the derivation for the hypothetical case where singletons would be allowed 4 We show only one directional variant of each rule for brevity. The same idea applies to all variants; see Steedman references for a standard set of rules, and [5] for review of proposals for combinatory extensions. Bozşahin [5]: §10 shows that all projection rules of CCG can be packed into one monad to enable monadic computation with just one rule of projection. This is possible because CCG is radically lexicalized in the sense that combinatory rules cannot project anything which is not in the lexicon. What appears to be rule choice when presented as (9/13) becomes dependency passing within monad with one rule of combination. 5 The way this is implemented in many CCG systems including ours is for example to constrain the slashes as follows: X/ ⋆ Y: f Y: a → X: f a (>) X/ ⋄ Y: f Y/ ⋄ Z: g → X/ ⋄ Z: λ x. f (gx) (> B) It is easier to describe slash-modal control from the perspective of syntactic types of expressions accessing these rules. '⋆-rules' are accessible by all categories, '⋄' and '×' are compatible only with themselves, and with the most permissive slash. to compose. Typing the singleton as '/ ⋆ "the bucket"' eliminates the derivation. The slashes in the paper are harmonic '\ ⋄ ' or '/ ⋄ ' unless stated otherwise.) (14) For a singleton, its functor (and there must be one, since they can only be arguments) decides lexically whether there is a predicate-argument structural role for the placeholder in the LF, as we see in the distinction of spill the beans, where secret ′ is an argument of divulge ′ , versus kick the bucket, where bucket ′ or anything related to it is not an argument of die ′ . Therefore, for CCG, MWEs and phrasal idioms are not exceptions that need nontransparent derivation, apart from lexical specification as something special. They are consequences of the nature of categories and radical lexicalization. Also because of the properties described in this section, a string as a category cannot be empty, which would violate CCG's principle of adjacency and principle of transparency (see Steedman references). No rule in (9) or (13) can apply if one of the categories is empty. Therefore the surface string itself for the singleton (s in example (8)) cannot be empty either. Having explored the possibilities for the singleton types in combinatory categories, we look at their use. Verb-particles and Phrasal Idioms with Singleton Types In verb-particle constructions, the differences in the syntax-semantics correspondence force the following lexical distinctions. We now write the categories in more detail than in the preview. c. picked := (S\NP)/NP : λ xλ y.pick ′ xy ∧ choose ′ xy The features above are all finite-state computable, just like morphological ones, as phonological weight (∓heavy) and lexical content (∓lexc) in an expression substituting for a category. All CCG category features can be interpreted this way, because combinators do all the syntactic work. The reason for having two different grammar entries (a-b) for pick up follows from the fact that they are not equally substitutable, for example as an answer to What did you do? (15b) leads to achievement, and (15a) to culmination. Both cases also differ from (c), which provides wider substitution for NP, and with a different meaning. We treat (a-c) distinctions surface-compositionally, which are transparently projected without wrap: (16) I picked the book up NP 1s (S\NP)/"up"/NP -heavy NP ((S\NP)\(S\NP))/NP : i ′ : λ yλ xλ z.cause ′ (init ′ (hold ′ x yz))z : def ′ book ′ : λ xλ pλ y.up ′ (py)x > (S\NP)/"up" : λ xλ z.cause ′ (init ′ (hold ′ x (def ′ book ′ )z))z > S\NP : λ z.cause ′ (init ′ (hold ′ λ xλ pλ y.up ′ (py)x (def ′ book ′ )z))z < S : cause ′ (init ′ (hold ′ λ xλ pλ y.up ′ (py)x (def ′ book ′ )i ′ ))i ′ where hold ′ at the end of the derivation can interpret its event modality (contingency) compositionally, since it is a closed lambda term. Notice that the word up knows nothing about the verb-particle construction. Its category is for a PP head, say PP up , as a predicate modifier. It is the verb that delivers the distinct meaning. Its subcategorization is for a singleton, which eschews the syntactic category of the word up but not its phonology and semantics, as described in (8b). (15b) can be assumed to arise from the syntactic category VP/NP +lexc /"up" by finite inflection. CCG has options here, to accommodate morphology without having to have a "morphological insertion point" in a contiguous but MWE entry pick up := VP/NP +lexc , to avoid ?pick upped. 6 This is made possible by singleton types. Examples (15a-b) use a degree of freedom which is relevant to phrasal idioms. The singleton syntactic type "up" corresponding to the LF placeholder x maintains the compositionality of the correspondence; but, it may have no contribution to the predicateargument structure at all in some cases, which would make it paracompositional, because its semantic type is a closed lambda term as far as predicate-argument structure is concerned. Notice that in (8b), s ′ is not in the predicate-argument structure of p; it is a contingency of p. Consider the following examples in this regard, where x for bucket ′ as an event modality might mean 'ceremonial death', 'reported death', etc.: NP Given the polyvalent argument category of the relative pronoun, we can see that relativization out of phrasal idioms would not be possible even if we allowed composition of singleton types, therefore the syntactic productivity of idiomatically combining phrases arises from their use of head-dependencies rather than singletons, as we shall soon see in derivations similar to (b), in (26). We note that carrying the head-word in a polyvalent category to have the same effect, for example kick := (S\NP)/ ⋆ NP bucket , overgenerates the idiom reading, because the bucket that John thought overflowed can substitute for NP bucket . The direct approach to categories that we see in radically lexicalized grammars, whether they are polyvalently substitutable or not, contrasts with systems of rewrite and/or record keeping in which post-processing is possible. For example there is no reanalysis or post-processing mechanism needed to eliminate the idiomatic reading below: (19) ♯Mary dragged and John kicked the bucket. S/(S\NP) (S\NP)/NP S/(S\NP) (S\NP)/ ⋆ "the bucket" > B *> B We can then follow [32] in assuming that passive is a polyvalent lexical process headed by the passive morpheme, mapping for example VP inf /NP to VP pass , which eliminates passivization *the breeze was shot from the entry: (20) shoot :=VP inf /"the breeze": λ xλ y.smalltalk ′ x one ′ y Idioms such as at any rate, beside the point further demonstrate that all idioms needing restricted types must contain a predicative element in the domain of locality of their head because we are required by paracompositionality to record the special reading and contingency, for example as extension of discursive clarification (a) and comparison (b): (21) a. at := (S/S)/"any rate" : λ xλ s.more ′ exactly ′ s x b. at := (S\S)/"any rate" : λ xλ s.contrastwith ′ x s Head-word Subcategorization and Idioms The difference between idiomatically combining phrases and phrasal idioms such as kicking the bucket is clear: The syntactically active ones are active because the idiomatic part has a role in the predicate-argument structure. 'Secret' is an argument of 'divulge', whereas 'bucket' is not an argument of 'die'. For example, spill the beans seems to require categorization such as (a) below in the manner of (6), rather than (b) fashioned from (5) or singleton-subcategorizing (c). Cf. also the non-idiomatic spill in (d). Tense morphology renders finite versions of VP inf below as S\NP, eg. spilled:=(S\NP)/NP beans for (a). (22) a. spill := VP inf /NP beans : λ xλ y.divulge ′ x secret ′ y b. spill := VP inf /NP : λ xλ y.if head(x) = beans ′ then divulge ′ x secret ′ y else spill ′ xy c. spill := (VP inf /"beans")/PredP: λ pλ xλ y.divulge ′ px secret ′ y d. spill := VP inf /NP: λ xλ y.spill ′ xy PredP is a predicative phrase type, which includes the quantifier phrase. The syntactic type of the idiomatic argument in (a) encodes the head-dependency from surface structure. It avoids the idiomatic reading in to spill the bean, which (b) may not. (b)-style solutions would depend on LF objects, which may not always reflect surface forms in full. In fact (b) requires post-processing to eliminate the idiom reading in the following example: λ xλ y.cook ′ xy : def ′ beans ′ > B > B S/NP S/NP : λ x.if head(x) = · · · : λ x.cook ′ xm ′ & S/NP : λ z.and ′ (if head(z) = · · ·)(cook ′ z m ′ ) > S/NP: and ′ (if head(def ′ beans ′ ) = beans ′ then divulge ′ secret ′ you ′ · · ·)(cook ′ (def ′ beans ′ ) m ′ ) This is still the case if we treat the construction as multi-headed, as [15]:238 do, by also assuming the beans := NP beans : secret ′ , and changing the LF choice condition of spill to 'if head(x)=secret ′ then divulge ′ xy else spill ′ xy'. Cook ′ does not refer to this entry. The process of marking head-word dependencies requires statistical learning, as the category such as NP beans in (22a) implies. It has been known in TAG systems with supertags since [6] that disambiguating such categories is feasible with training. The earliest approach to such marking in CCG is [8,9] as far as we know, where probabilistic CCGs are similarly trained. Later work such as [1] shows further progress in disambiguation of head-dependencies. NP beans is a polyvalent type, not a singleton. Therefore we get the following accounted for by (22a) (some of the examples are from [33]): (24) a. spill /several/the musical/the artistic/mountains of/loads of/ beans b. spill the beans no one cares about Head-marking of an argument category by the idiom's head is required because of examples such as below, where an idiomatic reading is eliminated despite relatively free syntax because the coordinands would not be like-typed: (25) When the head of the construction does not require identical types as does the conjunction above, head-projection works with simple term match; cf. the one for kicking the bucket in (18a) (h is for head-word feature): (26) the beans that you spilled NP h /N h N beans (N h \N h )/(S/NP) S/(S\NP 2s ) (S\NP)/NP beans > B S/NP beans > N h \N h < N beans > NP beans The example also shows that argument types of idiomatically combining phrases must be composable; therefore; (22c) is inadequate. 8 Idioms Requiring a Combined Approach There seems to be cases where a combination of singletons and head-marked polyvalent subcategorization is needed. The give creeps construction, which is sometimes considered not an idiom because of its compositionality [19], is paracompositional in our sense, and idiomatically combining in [25] terminology, because although creeps seems to be an event modality of revulse ′ rather than its argument, fear ′ is an argument. A simple head-marking approach such as '/NP creeps ' would overgenerate in cases such as ♯give me some creeps, but we have give me the absolute/shivering/full-on creeps. Notice also that the construction and related items resist dative shift (judgments are from [20]; '*' seems to be equivalent to '♯' in our terms): (27) a. The Count gave me the creeps./ *The Count gave the creeps to me. b. His boss gave Max the boot./ *His boss gave the boot to Max. Richards [27] observes that (a) below can be the unaccusative of give; and, (b) is widely attested in the web (but recall ♯give me some creeps). (28) a. Mary got the creeps. b. give some creeps c. give := VP/N creeps /"the"/NP : λ xλ yλ zλ w.cause ′ (init ′ (revulse ′ z fear ′ y x))w Assuming that dative shift is polyvalent, following [32], in the form of lexical mapping from VP/NP/NP to VP/PP to /NP, we can eliminate it for the type in (c), which we think captures the insight of Richards, and permits adjunction within an N, e.g. mountains of creeps. Another class of idioms forces a combined approach as well. Semantic reflexives in I twiddled my thumbs/ate my words/racked my brain/lose my mind are not morphological reflexives and they are inherently possessive, for example: (29) twiddled := (S\NP agr )/"thumbs"/NP -lexc,+poss,agr : λ xλ yλ z.pass ′ y time ′ (self ′ z) z ∧ inalien ′ (xyz) The LF captures the properties that the subject idles on his own time, the lexical possessive in the LF of x which is presumably lexically poss ′ is inalienable and belongs to the subject. This is a reflexive in the sense that it must be bound in its local domain determined by pass. ′ The referent (z) is available in one domain of locality in a radically lexicalized grammar because the head of the idiom does not require a VP in phrasestructure sense but a clause. Agreement is locally available too; by insisting on same agreement features. The head-dependency is that the argument does not contain lexical material, leaving out examples such as John twiddled John's thumbs as an idiom. No Wrap We have seen that options (4c/i) and (4c/ii) are not mutually exclusive. We also suggested that singleton type is a forced move to avoid loss of meaning composition. One consequence of this is the treatment of verb-particles without wrap, which are not related to idioms although they are MWEs. We now consider option (4b) in more detail from this perspective, which at first sight seems to be just as lexical as the two alternatives we have considered so far. The projection principle of CCG, which says that lexical specification of directionality and order of combination cannot be overridden during derivations, eliminates (30) from projection because it has the second-combining argument (Y) of a function applying before its first-combining argument (Z), an operation of the general class that has been proposed in other categorial approaches under the name of "wrap." (30) (X/Y)/Z: f Y: a → X/Z: λ z. f za (*) Wrap of the kind in (30) has a combinatory equivalent, namely Curry's combinator C (see [11]). CCG's adjacency principle eliminates this combinator on empirical grounds, rather than formal, as a freely operating rule. Adding (30) to CCG's projection has the effect of treating VSO and VOS as both grammatical, which is not the case for Welsh, and to carry the same meaning, which is not the case for Tagalog although both VSO and VOS are fine. These properties must be part of a lexicalized grammar rather than syntactic projection. The version of wrap which [2,12,16] employ is different, which was eliminated from consideration so far because it is non-combinatory; and, it violates adjacency of functors and arguments. That wrap is the following: (31) s 1 s 2 X/ W Y: f Y: a wrap first(s 1 ) s 2 rest(s 1 ) := X: f a where f irst() function gives the first element in a list of surface expressions for Bach [2], or first word for Dowty [12]; and, rest() returns the rest of the expression. The wrapping slash '/ W ' of Jacobson [16] does the infixation of s 2 . Semantically, it is function application. Syntactically, no combinator can do what this rule does to its input expressions, which is to rip apart one surface expression (s 1 ) and insert into it. It differs from C, which wraps one independent expression in two independent expressions. The appeal of surface wrap to MWEs was to be able to write a category for pick · · · up as for example pick := (S\NP)/ W NP/P up : λ xλ yλ z.pick x ′ yz; cf. (16). Syntactic wraps such as above, whether combinatory or non-combinatory, have domino effects on dependency and constituency, unlike 'lexical wrap', where a lexical entry specifies its correspondence; for example, for the strictly VSO Welsh verb gwelodd := (S/NP)/NP 3s : λ xλ y.saw ′ yx; note the LF. An example of global complications in grammar caused by wrap can be seen below, where dashed boxes denote wrapped-in material; cf. Figure 1 (31). Given these categories which involve wrap, there is one interpretation for (b), where the adverb can only modify persuade. With the unwrapped version of persuade in (c), two interpretations are possible: one modifies the VP complement of persuade, and the other, persuade John, both of which are required for adequacy. Conclusion One point of departure of CCG from other categorial grammars and from tree-rewrite systems is that (i) we can complicate the basic vocabulary of the theory, but (ii) not its basic mechanism such as introducing wrap, if a better explanation can be achieved. The first point has been made by Chomsky repeatedly since [7]:68. Singleton types could be viewed as one way of doing that, much like S\NP vs. VP distinction. We have argued that it is actually not a complication at all in CCG's case, because the possibility has been available, in the notion of type as a set of values, which can be a singleton set. CCG differs from Chomskyan notion of category substitution by eliminating move, empty categories and lexical insertion altogether, which means that all computation is local, type-driven, and there is no action-at-a-distance, to address the second point. The expressions substituting for these types are then locally available in the course of a derivation. This seems critical for MWEs. The possibility of a singleton value is built-in to any type. The asymmetry of CCG's singletons' categorization, that they can be arguments, and arguments of arguments and results, and, their inherent applicative nature, deliver MWEs and phrasal idioms as natural consequences rather than stipulation or a "pain in the neck for NLP." Syntactically active idioms are not singleton-typed because they have relevance to predicateargument structure; and, their narrower syntax, compared to free syntax, seems to necessitate head-marking of some argument categories, which is known to be probabilistically learnable. Some implications of our analyses are that all idioms can be made compositional at the level of a lexical correspondence without losing semantic distinctions, and without meaning postulates or reanalysis. Categorial post-processing of MWEs and phrasal idioms, and multi-stage processing of them in the lexicon, as done by [10,33], may be unnecessary if we assume type substitution to be potentially having one value, and surface head-marking to be an option for polyvalent argument types. One conjecture is that any idiom in any language has to involve a predicate implicated by some predicative element in the expression to keep the meaning assembly paracompositional. The analyses in the article can be replicated by running the CCG tool at github.com/bozsahin/ccglab. The particular fragment in the chapter is at github.com/bozsahin/ccglab-grammars/cb-ag-fg2018-grammar. Fig. 1 . 1Substitution of syntactic expressions for syntactic types. Boxes show segments combined. We display some one-at-a-time derivations on the same line to save space. "s": λ x.p x B: s ′ > αs := A: p s ′ Same idea applies to backward application, for α := A\"s" and the sequence sα. (( NP (S\NP)/"every which way" (S\NP)\(S\NP) S\NP)/"every which way" "every which way" S\NP)/"every which way" A/B B/C C S /"the bucket" (X\ ⋆ X)/ ⋆ X S/(S\NP) (S\NP)/VP inf VP inf /VP inf VP inf /"the bucket" types, one-to-one correspondence of syntactic types and placeholder types is meant to capture the thematic structure in CCG, for example for the door opened versus someone opened the door, by having two different (albeit related) correspondences for open. ( 15) a. picked := (S\NP)/"up"/NP -heavy : λ yλ xλ z.cause ′ (init ′ (hold x ′ yz))z b. picked := (S\NP)/NP +lexc /"up": λ xλ yλ z.pick x ′ yz ( 17) a. kicked := (S\NP)/ ⋆ "the bucket": λ xλ y.die ′ x y b. kicked := (S\NP)/NP: λ xλ y.kick ′ xy They anticipate very limited syntax in the semantically paracompositional part in the idiom reading (a) because of having to enumerate them (kick the old/proverbial bucket vs kick the bucket that John thought overflowed).7 These assumptions cannot give rise to the idiom reading in the bucket that you kicked, with no further stipulation than singleton categories in a lexical entry (cf. a-b; '*' on the right of a derivation means it is not possible):(18) a. ♯the bucket that you kicked (N\N)/(S/NP) S/(S\NP) (S\NP)/ ⋆ "the bucket" NP/N N (N\N)/(S/NP) S/(S\NP) (S\NP)/NP λ p.p you ′ : λ xλ y.if · · · : λ pλ qλ z.and ′ (pz)(qz) : λ p.p m ′ :♯You spilled and Mary cooked the beans S/(S\NP) (S\NP)/NP (X\ ⋆ X)/ ⋆ X S/(S\NP) (S\NP)/NP NP beans : VP inf / W NP/VP inf VPinf NP persuade to do the dishes := VP inf / W NP wrap persuade John to do the dishes := VP inf VP inf / W NP/VP inf VP inf NP persuade to do the dishes := VP inf / W NP wrap persuade John to do the dishes := VP inf persuade John to do the dishes easily := VP inf c. persuade John to do the dishes easily VP inf /VP inf /NP NP VP inf VP inf /VP inf Derivation (a) is Bach's use of non-combinatory wrap rule in. (32) a. persuade to do the dishes John > b. persuade to do the dishes John easily > < > This is equivalent to saying that in CCG the type VP is not always an abbrevation for S\NP, which might be the case in other brands of categorial grammars. The English facts above could be taken care of by featural distinctions such as S inf , S to-inf , S fin in S\NP, rather than also positing a VP. But in ergative languages the '\NP' does not always coincide with the same LF role The diversity of approaches in the volume for idioms[14] is testimony to the practice that the idioms are decisive factors in polishing our theories linguistically, psychologically and computationally. van der Linden[22], which is another categorial approach to idioms, allows vacuous abstractions, i.e. define semantics without mention of x in the LF of (7b). Apart from our empirical claim that they have a place in LF because they relate to contingency, vacuous abstractions seem to open ways to resource insensitivity which is unheard of in natural language; for example, the K combinator with its vacuous abstraction λ xλ y.x can delete things from LF. We have yet to find a word or morpheme that does this; see[5]:81 for some speculation.[22]'s treatment of phrasal idioms such as kick the bucket assumes partial involvement of the head verb kick for the semantics of the idiom, whereas in our conception it is fully responsible for the idiom with the aid of singleton types. The fact that this form is also attested in child and adult language suggests that these entries may be bonafide lexical options.7 It is tempting to try NP proverbial bucket : proverb ′ death ′ for kick the proverbial bucket which is a head-subcategorizing category; but, we would have to overextend ourselves to eliminate the idiom reading in kick the proverbial bucket that overflowed if we have to. In this sense we suggest that phrasal idioms are best treated with singleton types. One way to put it altogether is to use a feature such as ∓special in addition to h, which ordinary verbs negatively specify, heads of idiomatic combination positively specify, and heads of syntactic constructions eg. coordinators and relative markers (under)specify as they see fit. The value '+special' need not be further broken down for singletons because they are selfrepresenting, and, presumably, featureless. For example phonological weight is intrinsically captured in "the beans"; also, lexical content. Broad-coverage CCG semantic parsing with AMR. Y Artzi, K Lee, L Zettlemoyer, Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing. the 2015 Conference on Empirical Methods in Natural Language ProcessingLisbon, PortugalArtzi, Y., Lee, K., Zettlemoyer, L.: Broad-coverage CCG semantic parsing with AMR. In: Proceedings of the 2015 Conference on Empirical Methods in Natural Language Process- ing. pp. 1699-1710. Lisbon, Portugal (2015) An extension of Classical Transformational grammar. E Bach, Problems in Linguistic Metatheory: Proceedings of the 1976 Conference at Michigan State University. Lansing, MIMichigan State UniversityBach, E.: An extension of Classical Transformational grammar. In: Problems in Linguistic Metatheory: Proceedings of the 1976 Conference at Michigan State University. pp. 183- 224. Michigan State University, Lansing, MI (1976) Multi-modal Combinatory Categorial Grammar. J Baldridge, G J Kruijff, Proceedings of 11th Annual Meeting of the European Association for Computational Linguistics. 11th Annual Meeting of the European Association for Computational LinguisticsBudapestBaldridge, J., Kruijff, G.J.: Multi-modal Combinatory Categorial Grammar. In: Proceedings of 11th Annual Meeting of the European Association for Computational Linguistics. pp. 211-218. Budapest (2003) Feeling our way to an analysis of english possessed idioms. F Bond, J Q Ho, D Flickinger, Proceedings of the 22nd International Conference on Head-Driven Phrase Structure Grammar. Müller, S.the 22nd International Conference on Head-Driven Phrase Structure GrammarStanford, CACSLI PublicationsBond, F., Ho, J.Q., Flickinger, D.: Feeling our way to an analysis of english possessed idioms. In: Müller, S. (ed.) Proceedings of the 22nd International Conference on Head- Driven Phrase Structure Grammar. pp. 61-74. CSLI Publications, Stanford, CA (2015) C Bozşahin, Combinatory Linguistics. BerlinDe Gruyter MoutonBozşahin, C.: Combinatory Linguistics. De Gruyter Mouton, Berlin (2012) New models for improving supertag disambiguation. J Chen, S Bangalore, K Vijay-Shanker, Proceedings of the ninth conference on European chapter of the Association for Computational Linguistics. the ninth conference on European chapter of the Association for Computational LinguisticsAssociation for Computational LinguisticsChen, J., Bangalore, S., Vijay-Shanker, K.: New models for improving supertag disam- biguation. In: Proceedings of the ninth conference on European chapter of the Associa- tion for Computational Linguistics. pp. 188-195. Association for Computational Linguis- tics (1999) Some empirical issues in the theory of transformational grammar. N Chomsky, Goals of Linguistic Theory. Peters, S.Englewood Cliffs, NJPrentice-HallChomsky, N.: Some empirical issues in the theory of transformational grammar. In: Peters, S. (ed.) Goals of Linguistic Theory. Prentice-Hall, Englewood Cliffs, NJ (1972) Wide-coverage efficient statistical parsing with CCG and log-linear models. S Clark, J R Curran, Computational Linguistics. 334Clark, S., Curran, J.R.: Wide-coverage efficient statistical parsing with CCG and log-linear models. Computational Linguistics 33(4), 493-552 (2007) Building deep dependency structures with a wide-coverage CCG parser. S Clark, J Hockenmaier, M Steedman, Proceedings of the 40th Annual Meeting on Association for Computational Linguistics. the 40th Annual Meeting on Association for Computational LinguisticsClark, S., Hockenmaier, J., Steedman, M.: Building deep dependency structures with a wide-coverage CCG parser. In: Proceedings of the 40th Annual Meeting on Association for Computational Linguistics. pp. 327-334 (2002) Representing idioms. A Copestake, HPSG Conference. CopenhagenCopestake, A.: Representing idioms. In: HPSG Conference. Copenhagen (1994) . H B Curry, R Feys, Combinatory Logic. North-HollandAmsterdamCurry, H.B., Feys, R.: Combinatory Logic. North-Holland, Amsterdam (1958) Dative movement and Thomason's extensions of Montague Grammar. D Dowty, Linguistics, Philosophy, and Montague Grammar. Davis, S., Mithun, M.AustinUniversity of Texas PressDowty, D.: Dative movement and Thomason's extensions of Montague Grammar. In: Davis, S., Mithun, M. (eds.) Linguistics, Philosophy, and Montague Grammar, pp. 153-222. Uni- versity of Texas Press, Austin (1979) The dual analysis of adjuncts/complements in categorial grammar. D Dowty, in [18Dowty, D.: The dual analysis of adjuncts/complements in categorial grammar. pp. 33-66 (2003), in [18] M Everaert, E J Van Der Linden, R Schreuder, Idioms: structural and psychological perspectives. Schenk, A.New JerseyEveraert, M., Van der Linden, E.J., Schreuder, R., Schenk, A. (eds.): Idioms: structural and psychological perspectives. Lawrence Erlbaum, New Jersey (1995) Generalized Phrase Structure Grammar. G Gazdar, E Klein, G Pullum, I Sag, Harvard University PressGazdar, G., Klein, E., Pullum, G., Sag, I.: Generalized Phrase Structure Grammar. Harvard University Press (1985) Flexible categorial grammars: Questions and prospects. P Jacobson, Formal Grammar. Levine, R.OxfordOxford University PressJacobson, P.: Flexible categorial grammars: Questions and prospects. In: Levine, R. (ed.) Formal Grammar, pp. 129-167. Oxford University Press, Oxford (1992) A lexical theory of phrasal idioms (ms), available at: www1.icsi. P Kay, I A Sag, D Flickinger, Kay, P., Sag, I.A., Flickinger, D.: A lexical theory of phrasal idioms (ms), available at: www1.icsi.berkeley.edu/∼kay/idiom-pdflatex.11-13-15.pdf Modifying adjuncts. E Lang, C Maienborn, C Fabricius-Hansen, Walter de GruyterLang, E., Maienborn, C., Fabricius-Hansen, C.: Modifying adjuncts. Walter de Gruyter (2003) On the double object construction. R Larson, Linguistic Inquiry. 19Larson, R.: On the double object construction. Linguistic Inquiry 19, 335-392 (1988) On "dative idioms" in English. R Larson, Workshop on Syntax-semantics. Fuji Women's University. Larson, R.: On "dative idioms" in English. In: Workshop on Syntax-semantics. Fuji Women's University (2012) Same syntax, different semantics: A compositional approach to idiomaticity in multi-word expressions. T Lichte, L Kallmeyer, Empirical Issues in Syntax and Semantics. Piñón, C.ParisCSSP11Lichte, T., Kallmeyer, L.: Same syntax, different semantics: A compositional approach to idiomaticity in multi-word expressions. In: Piñón, C. (ed.) Empirical Issues in Syntax and Semantics, vol. 11, pp. 111-140. CSSP, Paris (2016) Incremental processing and the hierarchical lexicon. E J Van Der Linden, Computational linguistics. 182van der Linden, E.J.: Incremental processing and the hierarchical lexicon. Computational linguistics 18(2), 219-238 (1992) Ergativity: Argument Structure and Grammatical Relations. CSLI, Stanford. C D Manning, CAManning, C.D.: Ergativity: Argument Structure and Grammatical Relations. CSLI, Stan- ford, CA (1996) Temporal ontology and temporal reference. M Moens, M Steedman, The Language of Time: A Reader. Inderjeet Mani, James Pustejovsky, and Robert GaizauskasOxford University Press14Moens, M., Steedman, M.: Temporal ontology and temporal reference. Computational Linguistics 14, 15-28 (1988), reprinted in Inderjeet Mani, James Pustejovsky, and Robert Gaizauskas (eds.) The Language of Time: A Reader. Oxford University Press, 93-114. . G Nunberg, I A Sag, T Wasow, Idioms. Language. 703Nunberg, G., Sag, I.A., Wasow, T.: Idioms. Language 70(3), 491-538 (1994) The recognition and interpretation of idioms. S G Pulman, Idioms: Processing, structure, and interpretation. Cacciari, C., Tabossi, P.Lawrence Erlbaum, Hillsdale, NJPulman, S.G.: The recognition and interpretation of idioms. In: Cacciari, C., Tabossi, P. (eds.) Idioms: Processing, structure, and interpretation, pp. 249-270. Lawrence Erlbaum, Hillsdale, NJ (1993) An idiomatic argument for lexical decomposition. N Richards, Linguistic inquiry. 321Richards, N.: An idiomatic argument for lexical decomposition. Linguistic inquiry 32(1), 183-192 (2001) Multiword expressions: A pain in the neck for NLP. I A Sag, T Baldwin, F Bond, A Copestake, D Flickinger, International Conference on Intelligent Text Processing and Computational Linguistics. Mexico CitySag, I.A., Baldwin, T., Bond, F., Copestake, A., Flickinger, D.: Multiword expressions: A pain in the neck for NLP. In: International Conference on Intelligent Text Processing and Computational Linguistics. pp. 1-15. Mexico City (2002) Tree-rewriting models of multi-word expressions. W Schuler, A Joshi, Proceedings of the ACL Workshop on Multiword Expressions: from Parsing and Generation to the Real World. the ACL Workshop on Multiword Expressions: from Parsing and Generation to the Real WorldPortland, ORSchuler, W., Joshi, A.: Tree-rewriting models of multi-word expressions. In: Proceedings of the ACL Workshop on Multiword Expressions: from Parsing and Generation to the Real World. pp. 25-30. ACL, Portland, OR (2011) M Steedman, The Syntactic Process. Cambridge, MAMIT PressSteedman, M.: The Syntactic Process. MIT Press, Cambridge, MA (2000) Taking Scope: The Natural Semantics of Quantifiers. M Steedman, MIT PressCambridge, MASteedman, M.: Taking Scope: The Natural Semantics of Quantifiers. MIT Press, Cambridge, MA (2012) Combinatory Categorial Grammar. M Steedman, J Baldridge, Non-transformational syntax. Borsley, R., Börjars, K.OxfordBlackwellSteedman, M., Baldridge, J.: Combinatory Categorial Grammar. In: Borsley, R., Börjars, K. (eds.) Non-transformational syntax, pp. 181-224. Blackwell, Oxford (2011) Lexical encoding of MWEs. A Villavicencio, A Copestake, B Waldron, F Lambeau, Proceedings of the Workshop on Multiword Expressions: Integrating Processing. the Workshop on Multiword Expressions: Integrating ProcessingAssociation for Computational LinguisticsVillavicencio, A., Copestake, A., Waldron, B., Lambeau, F.: Lexical encoding of MWEs. In: Proceedings of the Workshop on Multiword Expressions: Integrating Processing. pp. 80-87. Association for Computational Linguistics (2004)
[]
[ "The Nucleon Anapole Moment and Parity-Violating ep Scattering", "The Nucleon Anapole Moment and Parity-Violating ep Scattering" ]
[ "Shi-Lin Zhu \nDepartment of Physics\nUniversity of Connecticut\n06269StorrsCTUSA\n", "S J Puglia \nDepartment of Physics\nUniversity of Connecticut\n06269StorrsCTUSA\n", "B R Holstein \nDepartment of Physics and Astronomy\nUniversityof Massachusetts\n01003AmherstMAUSA\n", "M J Ramsey-Musolf \nDepartment of Physics\nUniversity of Connecticut\n06269StorrsCTUSA\n\nTheory Group\nThomas Jefferson National Laboratory\n23606Newport NewsVAUSA\n" ]
[ "Department of Physics\nUniversity of Connecticut\n06269StorrsCTUSA", "Department of Physics\nUniversity of Connecticut\n06269StorrsCTUSA", "Department of Physics and Astronomy\nUniversityof Massachusetts\n01003AmherstMAUSA", "Department of Physics\nUniversity of Connecticut\n06269StorrsCTUSA", "Theory Group\nThomas Jefferson National Laboratory\n23606Newport NewsVAUSA" ]
[]
Parity-violating (PV) interactions among quarks in the nucleon induce a PV γN N coupling, or anapole moment (AM). We compute electroweak gauge-independent contributions to the AM through O(1/Λ 2 χ ) in chiral perturbation theory. We estimate short-distance PV effects using resonance saturation. The AM contributions to PV electron-proton scattering slightly enhance the axial vector radiative corrections, R p A , over the scale implied by the Standard Model when weak quark-quark interactions are neglected. We estimate the theoretical uncertainty associated with the AM contributions to R p A to be large, and discuss the implications for the interpretation PV of ep scattering.
10.1103/physrevd.62.033008
[ "https://export.arxiv.org/pdf/hep-ph/0002252v3.pdf" ]
2,228,377
hep-ph/0002252
3fde814f3d0f2ef90e3722efd2ce93b2bcd27dbb
The Nucleon Anapole Moment and Parity-Violating ep Scattering arXiv:hep-ph/0002252v3 10 Apr 2000 Shi-Lin Zhu Department of Physics University of Connecticut 06269StorrsCTUSA S J Puglia Department of Physics University of Connecticut 06269StorrsCTUSA B R Holstein Department of Physics and Astronomy Universityof Massachusetts 01003AmherstMAUSA M J Ramsey-Musolf Department of Physics University of Connecticut 06269StorrsCTUSA Theory Group Thomas Jefferson National Laboratory 23606Newport NewsVAUSA The Nucleon Anapole Moment and Parity-Violating ep Scattering arXiv:hep-ph/0002252v3 10 Apr 2000Indices: 2130+y1340Ks1388+e1130Er Parity-violating (PV) interactions among quarks in the nucleon induce a PV γN N coupling, or anapole moment (AM). We compute electroweak gauge-independent contributions to the AM through O(1/Λ 2 χ ) in chiral perturbation theory. We estimate short-distance PV effects using resonance saturation. The AM contributions to PV electron-proton scattering slightly enhance the axial vector radiative corrections, R p A , over the scale implied by the Standard Model when weak quark-quark interactions are neglected. We estimate the theoretical uncertainty associated with the AM contributions to R p A to be large, and discuss the implications for the interpretation PV of ep scattering. I. INTRODUCTION The SAMPLE collaboration at MIT-Bates has recently reported a value for the strange-quark magnetic form factor measured using backward angle parity-violating (PV) electron-proton scattering [1]: G (s) M (Q 2 = 0.1 GeV 2 /c 2 ) = 0.61 ± 0.27 ± 0.19 ,(1) where the first error is experimental and the second is theoretical. The dominant contribution to the theoretical error is uncertainty associated with radiative corrections to the axial vector term in the backward angle left-right asymmetry A LR [2]: A LR ∝ Q P W + Q N W G n M G p M + Q (0) W G (s) M G p M − (1 − 4sin 2 θ W ) 1 + 1/τ G p A G p M ,(2) where Q P W and Q N W are the proton and neutron weak charges, respectively, Q W is the SU(3)-singlet weak charge * , θ W is the weak mixing angle, and τ = Q 2 /4M 2 n . The axial form factor is normalized at the photon point as G p A (0) = −g A [1 + R p A ](3) where g A = 1.267 ± 0.004 [3] is the nucleon's axial charge as measured in neutron β-decay and R p A denotes process-dependent electroweak radiative corrections to the V (e) × A(p) scattering amplitude. The radiative correction R p A is the subject of the present study. It was first analyzed in Ref. [5] and found to be large, negative in sign, and plagued by considerable theoretical uncertainty. Generally, R p A contains two classes of contributions. The first involve electroweak radiative corrections to the elementary V (e)×A(q) amplitudes, where q is any one of the quarks in the proton. These radiative corrections, referred to henceforth as "onequark" radiative corrections, are calculable in the Standard Model. They contain little theoretical uncertainty apart from the gentle variation with Higgs mass and long-distance QCD effects involving light-quark loops in the Z − γ mixing tensor. The one-quark contributions can be large, due to the absence from loops of the small (1 − 4sin 2 θ W ) factor appearing at tree level (see Eq. (2) ) and the presence of large logarithms of the type ln(m q /M Z ). A second class of radiative corrections, which we refer to as "many-quark" corrections, involve weak interactions among quarks in the proton. In this paper, we focus on those many-quark corrections which generate an axial vector coupling of the photon to the proton (see Figure 1). This axial vector ppγ interaction, also known as the anapole moment (AM), has the form L AM = e Λ 2 χN (a s + a v τ 3 )γ µ γ 5 N∂ ν F νµ .(4) (Here, we have elected to normalize the interaction to the scale of chiral symmetry breaking, Λ χ = 4πF π .) These many-quark anapole contributions to R p A , which are independent of the electroweak gauge parameter [7], were first studied in Ref. [4,5] and found in Ref. [5] to carry significant theoretical uncertainty. The scale of this uncertainty was estimated in Ref. [5], and this value was used to obtain the theoretical error in Eq. (1). (Note that the central value for G (s) M given in Eq. (1) is obtained from the experimental asymmetry using the calculation of Ref. [5]). In order to better constrain the error in G (s) M associated with R p A , the SAMPLE collaboration performed a second backward angle PV measurement using quasielastic (QE) scattering from the deuteron. The asymmetry A LR (QE) is significantly less sensitive to G (s) M than is A LR (ep), but retains a strong dependence on R T =1 A , the isovector part of R p A . The calculation of Ref. [5] found the uncertainty in R p A to be dominated by this isovector component-R T =1 A ≈ −0.34 ± 0.20-and the goal of the deuterium measurement was, therefore, to constrain the size of this largest term. A preliminary deuterium result was reported at the recent Bates25 Symposium at MIT, and suggests that R T =1 A has the same negative sign as computed in Ref. [5] but has considerably larger magnitude, possibly of order unity [8]. Combining this result with the previous A LR (ep) measurement would yield a nearly vanishing value for G (s) M , rather than the large and positive value quoted in Eq. (1). The prospective SAMPLE result for R T =1 A is remarkable, indicating that a higher-order electroweak radiative correction is of the same magnitude as, and cancels against, the treelevel amplitude! The occurance of such enhanced electroweak radiative corrections is rare. Nevertheless, there does exist at least one other instance in which higher-order electroweak processes can dominate the axial vector hadronic response, namely, the nuclear anapole moment. The anapole moment of a heavy nucleus grows as A 2/3 (see, e.g. Refs. [6,7,9] and references therein). Because of the scaling with mass number, the nuclear AM contribution to a V (e) × A(nucleus) amplitude can be considerably larger than the corresponding treelevel Z 0 -exchange amplitude, and this A 2/3 enhancement is consistent with the size of the Cesium AM recently determined by the Boulder group using atomic parity-violation [10]. The reason behind the enhancement of R T =1 A for the few-nucleon system, however, is not understood. The goal of the present paper is to investigate whether there exist c onventional, hadronic physics effects which can explain the enhancement apparently implied by the SAMPLE deuterium measurement. In order to address this question, we revisit the analysis of Ref. [5]. Following Ref. [11], we re-cast that analysis into the framework of heavy baryon chiral perturbation theory (HBChPT) [12,13]. We carry out a complete calculation of R T =1 A and R T =0 A to order 1/Λ 2 χ , including loop diagrams not considered in Refs. [5,11]. We also extend those analyses to include decuplet as well as octet intermediate states, magnetic insertions, and SU(3) chiral symmetry. As in Ref. [5], we estimate the chiral counterterms at O(1/Λ 2 χ ) using vector meson saturation. However, we go beyond that previous analysis and determine the sign of this vector meson contribution phenomenologically. We find that decuplet intermediate states and magnetic insertions do not contribute up to the chiral order at which we truncate. Also, the effect of SU(3) symmetry, in the guise of kaon loops, is generally smaller than the pion loops considered previously. In the end, we express our results in terms of effective PV hadronic couplings. Some of these couplings may be determined from nuclear and hadronic PV experiments or detailed calculations (for reviews, see Refs. [14,15]), while others are presently unconstrained by measurement. Guided by phenomenology and the dimensional analysis of Ref. [11], we estimate the range of possible values for the new couplings. We suspect that our estimates are overly generous. Nevertheless, we find thateven under liberal assumptions -the AM contributions to R T =1 A appear unable to enhance the one-quark corrections to the level apparently observed by the SAMPLE collaboration and, in our conclusions, we speculate on possible additional sources of enhancement not considered here. The remainder of the paper is organized as follows. In Section 2, we relate the anapole couplings a s,v to the radiative corrections, R T =0,1 A , and in Section 3, we outline our formalism for computing these couplings in HBChPT. A reader already familiar with this formalism may wish to skip to Section 4, where we compute the chiral loop contributions to the nucleon anapole moment through O(1/Λ 2 χ ). We also include the leading 1/m N terms in the heavy baryon expansion, which generate contributions of O(1/Λ χ m N ). Section 5 contains the vector meson estimate of the chiral counterterms and the determination of the sign, while Section 6 gives our numerical estimate of the AM contributions to R T =0,1 A . We briefly discuss the phenomenology of hadronic and nuclear PV and what that phenomenology may imply about the scale of the unknown low-energy constants. Section 7 summarizes our conclusions. The Appendices give a detailed discussion of (A) our formalism, (B) the full set of hadronic PV Lagrangians allowed under SU(3) symmetry, and (C) graphs, nominally present at O(1/Λ 2 χ ) but whose contributions vanish. II. ANAPOLE CONTRIBUTIONS TO R A The electron-nucleon parity violating amplitude is generated by the diagrams in Figure 2. At tree level this amplitude reads iM P V = iM P V AV + iM P V V A ,(5) where iM P V AV = i G µ 2 √ 2 l λ5 < N|J λ |N >(6) and iM P V V A = i G µ 2 √ 2 l λ < N|J λ5 |N > = −i 1 − 4sin 2 θ W 2 √ 2 g A G µē γ λ eN τ 3 γ λ γ 5 N .(7) at tree-level in the Standard Model ( Figure 2a). Here, J λ (J λ5 ) and l λ (l λ5 ) denote the vector (axial vector) weak neutral currents of the quarks and electron, respectively [2]. The anapole moment interaction of Eq. (4) generates additional contributions to M P V V A when a photon is exchanged between the nucleon and the electron (Figure 2b). The corresponding amplitude is iM P V AM = i (4πα) Λ 2 χē γ λ eN(a s + a v τ 3 )γ λ γ 5 N .(8) Note that unlike iM P V V A , iM P V AM contains no (1 − 4sin 2 θ W ) suppression. Consequently, the relative importance of the anapole interaction is enhanced by 1/(1 − 4sin 2 θ W ) ∼ 10. This enhancement may be seen explicitly by converting Eqs. (6) and (8) into R T =0,1 A : R T =0 A | anapole = − 8 √ 2πα G µ Λ 2 χ 1 1 − 4sin 2 θ W a s g A (9) R T =1 A | anapole = − 8 √ 2πα G µ Λ 2 χ 1 1 − 4sin 2 θ W a v g A(10) The constants a s,v contain contributions from loops generated by the Lagrangians given in Section 3 and from counterterms in the tree-level effective Lagrangian of Eq. (4): a s,v = a L s,v + a CT s,v .(11) In HBChPT, only the parts of the loop amplitudes non-analytic in quark masses can be unambigously indentified with a L s,v . The remaining analytic terms are included in a CT s,v . In what follows, we compute explicityly the various loop contributions up through O(1/Λ 2 χ ), while in principle, a CT s,v should be determined from experiment. In Section 5, however, we discuss a model estimate for a CT s,v . Before proceeding with details of the calculation, it is useful to take note of the scales present in Eqs. (9). The constants a s,v are generally proportional to a product of strong and weak meson-baryon couplings. The former are generally of order unity, while the size of weak, PV couplings can be expressed in terms of g π = 3.8 × 10 −8 , the scale of charged current contributions [16]. One then expects the AM contributions to the axial radiative corrections to be of order R T =0,1 A ∼ − 8 √ 2πα G µ Λ 2 χ 1 1 − 4sin 2 θ W g π g A ≈ −0.01 .(12) In some cases, the PV hadronic couplings may be an order of magnitude larger than g π . Alternatively, chiral singularities arising from loops may also enhance the AM effects over the scale in Eq. (12). Thus, as we show below, the net effect of the AM is anticipated to be a 10-20 % contribution to R T =0,1 A . III. NOTATIONS AND CONVENTIONS Since much of the formalism for HBChPT is standard, we relegate a detailed summary of our conventions to Appendix A. However, some discussion of the effective Lagrangians used in computing chiral loop contributions to a s,v is necessary here. Specifically, we require the parity-conserving (PC) and parity-violating (PV) Lagrangians involving pseudoscalar meson, spin-1/2 and spin-3/2 baryon, and photon fields. For the moment, we restrict ourselves to SU(2) flavor symmetry and generalize to SU(3) later. The relativistic PC Lagrangian for π, N, ∆, and γ interactions needed here is L P C = F 2 π 4 T rD µ ΣD µ Σ † +N (iD µ γ µ − m N )N + g AN A µ γ µ γ 5 N + e Λ χN (c s + c v τ 3 )σ µν F + µν N −T µ i [(iD ij α γ α − m ∆ δ ij )g µν − 1 4 γ µ γ λ (iD ij α γ α − m ∆ δ ij )γ λ γ ν + g 1 2 g µν A ij α γ α γ 5 + g 2 2 (γ µ A ij ν + A ij µ γ ν )γ 5 + g 3 2 γ µ A ij α γ α γ 5 γ ν ]T ν j +g πN ∆ [T µ i (g µν + z 0 γ µ γ ν )ω ν i N +Nω ν † i (g µν + z 0 γ ν γ µ )T µ i ] −ie c ∆ q i Λ χT µ i F + µν T ν i + [ ie Λ χT µ 3 (d s + d v τ 3 )γ ν γ 5 F + µν N + h.c.](13) where D µ is a chiral and electromagnetic (EM) covariant derivative, Σ = exp(i τ · π/F π ) is the conventional non-linear representation of the pseudoscalar field, N is a nucleon isodoublet field, T i µ is the ∆ field in the isospurion formalism, F µν is the photon field strength tensor, and A µ is the axial field involving the pseudoscalars A µ = − D µ π F π + O(π 3 )(14) with D µ being the EM covariant derivative. Explicit expressions for the fields and the transformation properties can be found in Appendix A. The constants c s , c v determined in terms of the nucleon isoscalar and isovector magnetic moments, c ∆ is the ∆ magnetic moment, d s , d v are the nucleon and delta transition magnetic moments, and z 0 is the off-shell parameter which is not relevant in the present work [17]. Our convention for γ 5 is that of Bjorken and Drell [18]. In order to obtain proper chiral counting for the nucleon, we employ the conventional heavy baryon expansion of L P C , and in order to cosistently include the ∆ we follow the small scale expansion proposed in [17]. In this approach energy-momenta and the delta and nucleon mass difference δ are both treated as O(ǫ) in chiral power counting. The leading order vertices in this framework can be obtained via P + ΓP + where Γ is the original vertex in the relativistic Lagrangian and P ± = 1 ± v 2 .(15) are projection operators for the large, small components of the Dirac wavefunction respectively. Likewise, the O(1/m N ) corrections are generally propotional to P + ΓP − /m N . In previous work the parity conserving πN∆γ interaction Lagrangians have been obtained to O(1/m 2 N ) [17]. We collect some of the relevant terms below: L P C v =N [iv · D + 2g A S · A]N − iT µ i [iv · D ij − δ ij δ + g 1 S · A ij ]T j µ +g πN ∆ [T µ i ω i µ N +N ω i † µ T µ i ] + 1 2m NN (v · D) 2 − D 2 + [S µ , S ν ][D µ , D ν ] −ig A (S · Dv · A + v · AS · D) N + · · ·(16) where S µ is the Pauli-Lubanski spin operator and δ ≡ m ∆ − m N . The PV analog of Eq. (13) can be constructed using the chiral fields X a L,R defined in Appendix A and the spacetime transformation properties of the various fields in Eq. (13). We find it convenient to follow the convention in Ref. [11] and separate the PV Lagrangian into its various isospin components. The hadronic weak interaction has the form H W = G µ √ 2 J λ J λ † + h.c. ,(17) where J λ denotes either a charged or neutral weak current built out of quarks. In the Standard Model, the strangeness conserving charged currents are pure isovector, whereas the neutral currents contain both isovector and isoscalar components. Consequently, H W contains ∆T = 0, 1, 2 pieces and these channels must all be accounted for in any realistic hadronic effective theory. Again for simplicity, we restrict our attention first to the light quark SU(2) sector. (A general SU(3) PV meson-baryon Lagrangian is given in the Appendix and is considerably more complex.) We quote the relativistic Lagrangians, but employ the heavy baryon projections, as described above, in computing loops. It is straightforward to obtain the corresponding heavy baryon Lagrangians from those listed below, so we do not list the PV heavy baryon terms below. For the πN sector we have L πN ∆T =0 = h 0 VN A µ γ µ N (18) L πN ∆T =1 = h 1 V 2N γ µ NT r(A µ X 3 + ) − h 1 A 2N γ µ γ 5 NT r(A µ X 3 − ) (19) − h π 2 √ 2 F πN X 3 − N L πN ∆T =2 = h 2 V I abN [X a R A µ X b R + X a L A µ X b L ]γ µ N (20) − h 2 A 2 I abN [X a R A µ X b R − X a L A µ X b L ]γ µ γ 5 N . The above Lagrangian was first given by Kaplan and Savage (KS) [11]. However, the coefficients used in our work are slightly different from those of Ref. [11] since our definition of A µ differs by an overall phase (see Appendix A). Moreover, the coefficient of the second term in the original PV ∆T = 2 NNππ Lagrangian in Eq. (2.18) was misprinted in the work of KS, and should be 2h 2 A in their notation instead of h 2 V as given in Eq. (2.18) of [11]. The term proportional to h π contains no derivatives and, at leading-order in 1/F π , yields the PV NNπ Yukawa coupling traditionally used in meson-exchange models for the PV NN interaction [16,19]. The PV γ-decay of 18 F can be used to constrain the value of h π in a nuclear model-independent way as discussed in Ref. [19], resulting in h π = (0.7±2.2)g π [15]. Future PV experiments are planned using light nuclei to confirm the 18 F result. The coupling h π has also received considerable theoretical attention [16,29,20,21] and is particularly interesting since it receives no charged current contributions at leading order. Unlike the PV Yukawa interaction, the vector and axial vector terms in Eqs. (18)(19)(20) contain derivative interactions. The terms containing h 1 A , h 2 A start off with NNππ interactions, while all the other terms start off as NNπ. Such derivative interactions have not been included in conventional analyses of nuclear and hadronic PV experiments. Consequently, the experimental constraints on the low-energy constants h i V , h i A are unknown. The authors of Ref. [11] used simple dimensional arguments and factorization limits to estimate their values, and we present additional phenomenological considerations in Section 6 below. We emphasize, however, that the present lack of knowledge of these couplings introduces additional uncertainties into R T =0,1 A . In addition to purely hadronic PV interactions, one may also write down PV EM interactions involving baryons and mesons † . The anapole interaction of Eq. (4) represents one such interaction, arising at O(1/Λ 2 χ ) and involving no π's. There also exist terms at O(1/Λ χ ) which include at least one π: L γN P V = c 1 Λ χN σ µν [F + µν , X 3 − ] + N + c 2 Λ χN σ µν F − µν N + c 3 Λ χN σ µν [F − µν , X 3 + ] + N .(21) The corresponding PV Lagrangians involving a N → ∆ transition are somewhat more complicated. The analogues of Eqs. (18)(19)(20) are L π∆N ∆I=0 = f 1 ǫ abcN iγ 5 [X a L A µ X b L + X a R A µ X b R ]T µ c +g 1N [A µ , X a − ] + T µ a + g 2N [A µ , X a − ] + T µ a + h.c.(22)L π∆N ∆I=1 = f 2 ǫ ab3N iγ 5 [A µ , X a + ] + T µ b + f 3 ǫ ab3N iγ 5 [A µ , X a + ] − T µ b +g 3N [(X a L A µ X 3 L − X 3 L A µ X a L ) − (X a R A µ X 3 R − X 3 R A µ X a R )]T µ a +g 4 {N [3X 3 L A µ (X 1 L T 1 µ + X 2 L T 2 µ ) + 3(X 1 L A µ X 3 L T 1 µ + X 2 L A µ X 3 L T 2 µ ) † Note that the hadronic derivative interactions of Eqs. (18)(19)(20) also contain γ fields as required by gauge-invariance −2(X 1 L A µ X 1 L + X 2 L A µ X 2 L − 2X 3 L A µ X 3 L )T 3 µ ] − (L ↔ R)} + h.c. (23) L π∆N ∆I=2 = f 4 ǫ abd I cdN iγ 5 [X a L A µ X b L + X a R A µ X b R ]T µ c +f 5 ǫ ab3N iγ 5 [X a L A µ X 3 L + X 3 L A µ X a L + (L ↔ R)]T µ b +g 5 I abN [A µ , X a − ] + T µ b + g 6 I abN [A µ , X a − ] + T µ b + h.c. ,(24) where the terms containing f i and g i start off with single and two pion vertices, respectively. Finally, we consider PV γ∆N interactions: L γ∆N P V = ie d 1 Λ χT µ 3 γ ν F + µν N + ie d 2 Λ χT µ 3 γ ν [F + µν , X 3 + ] + N (25) +ie d 3 Λ χT µ 3 γ ν [F + µν , X 3 + ] − N + ie d 4 Λ χT µ 3 γ ν γ 5 F − µν N +ie d 5 Λ χT µ 3 γ ν γ 5 [F + µν , X 3 − ] + N + ie d 6 Λ χT µ 3 γ ν γ 5 [F − µν , X 3 + ] + N +ie d 7 Λ χT µ 3 γ ν [F − µν , X 3 − ] + N + ie d 8 Λ χT µ 3 γ ν [F − µν , X 3 − ] − N + h.c..(26) The PV γ∆N vertices d 1−3 , d 4−6 and d 7−8 are associated at leading order in 1/F π with zero, one and two pion vertices, respectively. All the vertices in (18) -(25) are O(p) or O(1/Λ χ ) except h π , which is Yukawa interaction and of O(p 0 ). As we discuss in Appendix C, we do not require PV interactions involving two ∆ fields. IV. CHIRAL LOOPS The contributions to a s,v arising from the Lagrangians of Eqs . (18)(19)(20) are shown in Figure 3. We regulate the associated integrals using dimensional regularization (DR) and absorb the divergent-1/(d−4)-terms into the counterterms, a CT s,v . The leading contributions arise from the PV Yukawa coupling h π contained in the loops of 3a-f. To O(1/Λ 2 χ ), the diagrams 3e,f containing a photon insertion on a nucleon line do not contribute. The reason is readily apparent from examination of the integral associated with the amplitude of Figure 3e: iM 3e = ie N h π v · ε √ 2g A F π d D k (2π) D i(S · k) v · k i v · (q + k) i k 2 − m 2 π + iǫ = −ie N h π v · ε 2 √ 2g A F π S µ ∞ 0 sds 1 0 du d D k (2π) D k µ [k 2 + sv · k + usv · q + m 2 π ] 3 ,(27) where q µ is the photon momentum, ε is the photon polarization vector, s has the dimensions of mass, and we have Wick rotated to Euclidean momenta in the second line. From this form it is clear that iM 3e ∝ S · v = 0. The sum of the non-vanishing diagrams Figure 3a-d yields a gauge invariant leading order result, which is purely isoscalar: a L s (Y 1) = − √ 2 24 eg A h π Λ χ m π .(28) As the PV Yukawa interaction is of order O(p 0 ), we need to consider higher order corrections involving this interaction, which arise from the 1/m N expansion of the nucleon propagator and various vertices. Since P + · 1 · P − = 0, there is no 1/m N correction to the PV Yukawa vertex. From the 1/m NN N terms in Eq. (13) we have a L s (Y 2) = 7 √ 2 48π eg A h π Λ χ m N ln( µ m π ) 2 ,(29) where µ is the subtraction scale introduced by DR. Finally, the 1/m N correction to the strong πNN vertex, contained in the term ∝ g A in Eq. (13), yields a L s (Y 3) = − √ 2 48π eg A h π Λ χ m N ln( µ m π ) 2 .(30) These terms are also isoscalar, and the results in Eqs. (28)(29)(30) are fully contained in the previous analyses of Refs. [5,7,11]. For the interactions in Eqs. (18)(19)(20) containing h i V , the eight diagrams Figure 3a-h must be considered. Their contribution is purely isovector- a L v (V ) = 1 6 eg A (h 0 V + 4 3 h 2 V ) ln( µ m π ) 2 .(31) -and was not included in previous analyses. The contribution generated from the two-pion PV axial vertices in Eqs. (19)(20) comes only from the loop Figure 3i and contains both isovector and isoscalar components: a L s (A) + a L v (A)τ 3 = − 1 3 e(h 1 A + h 2 A τ 3 ) ln( µ m π ) 2 .(32) a result first computed in Ref. [11]. In principle, a variety of additional contributions will arise at O(1/Λ 2 χ ). For example, insertion of the nucleon magnetic moments (i.e. the terms in Eq. (13) containing c s,v ) into the loops Figure 3e,f-resulting in the loops of Figure 5a,b-would in principle generate terms of O(1/Λ 2 χ ) when the PV Yukawa interaction is considered. As shown in Appendix C, however, such contributions vanish at this order. Similarly, the entire set of ∆ intermediate state contributions shown in Figure 4, as well as those generated by L γN P V and L γ∆N P V in Figure 6, vanish up to O(1/Λ 2 χ ). The reasons for the vanishing of these various possible contributions is discussed in Appendix C. Thus, the complete set of SU(2) loop contributions up to O(1/Λ 2 χ ) are given in Eqs. (28)(29)(30)(31)(32). Because m c − m s >> m s − m u,d and Λ χ >> m s , it may be appropriate to treat the lightest strange and non-strange hadrons on a similar footing and extend the foregoing discussion to SU(3) chiral symmetry. A similar philosophy has been adopted by several authors in studying the axial charges and magnetic moments of the lightest baryons [12,[22][23][24][25]. In what follows, we consider the possibility that kaon loop contributions, introduced by the consideration of SU(3) symmetry, may further enhance the anapole contribution to R A . Before proceeding along these lines, however, one must raise an important caveat. When kaon loop corrections are included in a HBChPT analysis, higher order chiral corrections may go as powers of m K /Λ χ ∼ 0.5. Consequently, the convergence of the SU(3) chiral expansion remains a subject of debate [26]. Fortunately, no such factors appear in the present analysis through O(1/Λ 2 χ ) so that at this order, we find that kaon loop effects in R A are generally tiny compared to those involving pion loops. Whether or not higher-order terms (e.g., those of O(1/Λ 2 χ × m K /Λ χ ) contribute as strongly as those considered here remains a separate, open question. To set our notation, we give the leading strong-interaction SU(3) Lagrangian. Since the K 0 and η are neutral, loops containing these mesons do not contribute to the AM through O(1/Λ 2 χ ) and we do not include their strong couplings below. For the proton the possible intermediate states are Σ 0 K + , ΛK + while for the neutron only Σ − K + can appear. The necessary vertices derive from L = 2g AN S · AN + 2g N ΛK [(N S · K)Λ +Λ(S · K † N)] (33) +2g N ΣK [S · K †Σ N +N ΣS · K] , where g N ΛK = −[(1 + 2α)/ √ 6]g A , g N ΣK = (1 − 2α)g A with g A = D + F , α = F/(D + F )L 1π Yukawa = −ih π (pnπ + −npπ − ) − ih pΣ 0 K (pΣ 0 K + −Σ 0 pK − ) −ih nΣ − K (nΣ − K + −Σ − nK − ) − ih pΛK (pΛK + −ΛpK − ) + · · · .(34) In terms of the SU(3) couplings listed in Appendix B, the h BBM have the form h π = −2 √ 2(h 1 + h 2 ) h pΣ 0 K = −[h 1 − h 2 + √ 3(h 3 − h 4 )] h nΣ − K = √ 2h pΣ 0 K h pΛK = h 1 √ 3 + √ 3h 2 + h 3 + 3h 4 .(35) Similarly, we write for the vector PV interaction L 1π V = − h pnπ + V √ 2F πp γ µ nD µ π + − h pΣ 0 K + V √ 2F πp γ µ Σ 0 D µ K + − h nΣ − K + V √ 2F πn γ µ Σ − D µ K + − h pΛK + V √ 2F πp γ µ ΛD µ K + + h.c. + · · · ,(36) and for the axial PV two pion and kaon interactions L 2π A = i h pπ A F 2 πp γ µ γ 5 p(π + D µ π − − π − D µ π + ) + i h pK A F 2 πp γ µ γ 5 p(K + D µ K − − K − D µ K + ) +i h nπ A F 2 πn γ µ γ 5 n(π + D µ π − − π − D µ π + ) + i h nK A F 2 πn γ µ γ 5 n(K + D µ K − − K − D µ K + ) + · · · .(37) Expressions for these PV vector and axial coupling constants in terms of SU(3) constants appear in Appendix B. For illustrative purposes, it is useful to express the nucleon-pion couplings in terms of the h i V,A of Eqs. (18)(19)(20) for the SU(2) sector: h pnπ + V = h 0 V + 4 3 h 2 V h pπ A = h 1 A + h 2 A h nπ A = h 1 A − h 2 A .(38) The leading order contributions to a s,v arise only from the loops in Figure 3 where a photon couples to a charged meson. The charged kaon loop contributions to the a s,v can be obtained from the corresponding formulae for the π-loop terms by making simple replacements of couplings and masses. For example, for the PV Yukawa interactions, these replacements are: (a) for the proton case, m π → m K , h π → h pΣ 0 K , g A → g N Σ 0 K + = g N ΣK / √ 2 for Σ 0 K + intermediate states and h π → h pΛK , g A → g N ΛK for ΛK + ; (b) for the neutron case, h π → h nΣ − K , g A → g N Σ − K + = g N ΣK for Σ − K + intermediate state for the neutron case. Similar replacements hold for the vector PV coupling contributions. For the axial PV two-pion contribution we need only make the replacement h pπ A → h pK A , m π → m K . Upon making these substitutions, we obtain the complete heavy baryon loop contribution to O(1/Λ 2 χ ) in SU (3): a L s = √ 2 24 g A h π [− Λ χ m π + 3 π Λ χ m N ln( µ m π ) 2 ] − √ 3 144 (1 + 2α)g A h pΛK [− Λ χ m K + 3 π Λ χ m N ln( µ m K ) 2 ] + √ 2 32 (1 − 2α)g A h nΣ − K [− Λ χ m K + 3 π Λ χ m N ln( µ m K ) 2 ] − 1 6 (h pπ A + h nπ A ) ln( µ m π ) 2 − 1 6 (h pK A + h nK A ) ln( µ m K ) 2 + 1 12 (1 − 2α)g A (h nΣ − K + V + h pΣ 0 K + V √ 2 ) ln( µ m K ) 2 − √ 6 72 (1 + 2α)g A h pΛK + V ln( µ m K ) 2 (39) a L v = − √ 2 96 (1 − 2α)g A h nΣ − K [− Λ χ m K + 3 π Λ χ m N ln( µ m K ) 2 ] − √ 3 144 (1 + 2α)g A h pΛK [− Λ χ m K + 3 π Λ χ m N ln( µ m K ) 2 ] − 1 6 (h pπ A − h nπ A ) ln( µ m π ) 2 − 1 6 (h pK A − h nK A ) ln( µ m K ) 2 − 1 6 g A h pnπ + V ln( µ m π ) 2 + 1 12 (1 − 2α)g A (−h nΣ − K + V + h pΣ 0 K + V √ 2 ) ln( µ m K ) 2 − √ 6 72 (1 + 2α)g A h pΛK + V ln( µ m K ) 2 .(40) V. LOW-ENERGY CONSTANTS AND VECTOR MESONS A pure ChPT treatment of the anapole contributions to R A would use a measurment of the axial term in A LR (ep) and A LR (QE), together with the non-analytic, long-distance loop contributions, a L s,v , to determine the low-energy constants, a CT s,v . In the present case, however, we wish to determine whether there exist reasonable hadronic mechanisms which can enhance the low-energy constants to the level suggested by the SAMPLE results. Thus, we attempt to estimate a CT s,v theoretically. Because they are governed in part by the short-distance (r > 1/Λ χ ) strong interaction, a CT s,v are difficult to compute from first principles in QCD. Nevertheless, experience with ChPT in the pseudscalar meson sector and with the phenomenology of nucleon EM form factors suggests a reasonable model approach. It is well known, for example, that in the O(p 4 ) chiral Lagrangian describing pseudoscalar interactions, the low-energy constants are well-described by the exchange of heavy mesons [27]. In particular, the charge radius of the pion receives roughly a 7% long-distance loop contribution, while the remaining 93% is saturated by t-channel exchange of the ρ 0 . Similarly, in the baryon sector, dispersion relation analyses of the isovector and isoscalar nucleon electromagnetic form factors indicate important contributions from the lightest vector mesons [28]. Thus, it seems reasonable to assume that t-channel exchange of vector mesons also plays an important role in the short-distance physics associated with the anapole moment. With these observations in mind, we estimate the coefficients a CT s,v in the approximation that they are saturated by t-channel exchange of the lightest vector mesons, as shown in Figure 7. Here parity-violation enters through the vector meson-nucleon interaction vertices. We also use a similar picture for the electromagnetic nucleon form factors to determine the overall phase of a CT s,v in the vector meson dominance approximation. To that end we require the PC and PV vector meson-nucleon Lagrangians [16]: L P C ρN N = g ρN NN [γ µ + κ ρ iσ µν q ν 2m N ]τ · ρ µ N (41) L P C ωN N = g ωN NN [γ µ + κ ω iσ µν q ν 2m N ]ω µ N (42) L P C φN N = g φN NN [γ µ + κ φ iσ µν q ν 2m N ]φ µ N(43) and L P V ρN N =Nγ µ γ 5 ρ 0 µ [h 1 ρ + (h 0 ρ + h 2 ρ √ 6 )τ 3 ]N (44) L P V ωN N =Nγ µ γ 5 ω µ [h 0 ω + h 1 ω τ 3 ]N (45) L P V φN N =Nγ µ γ 5 φ µ [h 0 φ + h 1 φ τ 3 ]N .(46) (Note that we have adopted a different convention for γ 5 than used in Ref. [16].) The coupling constants h i ρ,ω,φ were estimated in Refs. [16,29] and have also been constrained by a variety of hadronic and nuclear parity-violating experiments (for a review, see Ref. [19]). For the V − γ transition amplitude, we use L V γ = e 2f V F µν V µν ,(47) where e is the charge unit, f V is the γ-V conversion constant (V = ρ 0 , ω, φ), and V µν is the corresponding vector meson field tensor. (This gauge-invariant Lagrangian ensures that the diagrams of Figure 7 do not contribute to the charge of the nucleon.) The amplitude of Figure 7 then becomes a CT s (V MD) = h 1 ρ f ρ ( Λ χ m ρ ) 2 + h 0 ω f ω ( Λ χ m ω ) 2 + h 0 φ f φ ( Λ χ m φ ) 2 ,(48)a CT v (V MD) = h 0 ρ + h 2 ρ / √ 6 f ρ ( Λ χ m ρ ) 2 + h 1 ω f ω ( Λ χ m ω ) 2 + h 1 φ f φ ( Λ χ m φ ) 2 .(49) The parity violating rho-pole contribution was first derived in [5,7]. However, the relative sign between h i ρN and f ρ is undetermined from the diagram of Figure 7 alone. Nevertheless, we can fix the overall phase using two phenomenological inputs. Parity violating experiments in the p-p system constrain the sign of the combination g ρN h i ρN [30,31,19]. In particular, the scale of the longitudinal analyzing power, A L , is set by the combination of constants A L ∝ g ρN N (2 + κ V )[h 0 ρ + h 1 ρ + h 2 ρ / √ 6] + g ωN N (2 + κ S )[h 0 ω + h 1 ω ] ,(50) where the constant of proportionality is positive, κ V = 3.7 and κ S = −0.12. Using the standard values for the strong V NN couplings, one finds that A L has roughly the same sensitivity to each of the h i V appearing in Eq. (50) (modulo the 1/ √ 6 coefficient of h 2 ρ ). From the 45 MeV experiment performed at SIN [32], for example, one obtains the approximate constraint [15] h 0 ρ + h 1 ρ + h 2 ρ / √ 6 + h 0 ω + h 1 ω ∼ −28 ± 4 ,(51) where the h i V have are expressed in units of g π and where a positive sign has been assumed for g V N N . Given this constraint, it is very unlikely that the product (h 0 ρ +h 2 ρ / √ 6)g ρN N > 0 unless the corresponding products involving h 1 ρ and h 0,1 ω in Eq. (50) obtain anomalously large, negative values. In fact, a fit to hadronic and nuclear PV observables in Ref. [19] strongly favors a phase difference between the strong and weak V NN couplings. Experimentally, one also knows the isovector nucleon charge radius r 2 T =1 EXP = 6 dF 1 (q 2 ) dq 2 | q 2 =0 > 0 ,(52) where < p ′ |j T =1 µ (0)|p >= eū(p ′ )[F 1 (q 2 ) + iσ µν q ν 2m N F 2 (q 2 )]u(p) .(53) One may reasonably approximate the ρ 0 contribution to r 2 T =1 using VMD [28]. The calculation is the same as above but with the weak hadronic coupling replaced by the strong coupling. The result is F ρ 0 1 (q 2 ) = g ρN N f ρ q 2 q 2 − m 2 ρ . (54) Then we have dF V M D 1 (q 2 ) dq 2 | q 2 =0 = − g ρN N f ρ m 2 ρ .(55) Comparing Eqs. (52) and (55), and noting that the ρ 0 generates a positive contribution to r 2 T =1 [28], we arrive at g ρN N /f ρ < 0. Combining this result with g ρN N h i ρ < 0 as favored by the pp experiments [30,31,19] we obtain the relative sign between h i ρ and f ρ : h i ρ /f ρ > 0. Accordingly we determine the relative signs for PV ω, φ-nucleon coupling constants. VI. THE SCALE OF R A Expressions for the anapole contributions to R T =0 A and R T =1 A in terms of the a s,v appear in Eq. (9). We may now use these expressions, along with the results in Eqs. (39)(40) and (48-49), to obtain a numerical estimate for the R T =0,1 A | anapole . To do so, we use the global fit value for the weak mixing angle in the on-shell scheme, sin 2 θ W = 0.2230 [3], g A = 1.267 ± 0.004 [3], f ρ = 5.26 [33], f ω = 17, f φ = 13 [34], α = F/(D + F ) = 0.36, µ = Λ χ . We express all the PV coupling constants in units of g π = 3.8 × 10 −8 as is traditionally done [29,16]. We obtain R T =0 A | anapole = 10 −2 {0.17h π + h 1 A − 0.0036h nΣ − K −0.033(h nΣ − K + V + h pΣ 0 K + V √ 2 ) + 0.2(h pK A + h nK A ) − 0.006h pΛK +0.088h pΛK + V − 0.26|h 1 ρ | − 0.08|h 0 ω | − 0.05|h 0 φ |} (56) R T =1 A | anapole = 10 −2 {h 2 A − 0.6(h 0 V + 4 3 h 2 V ) − 0.0012h nΣ − K − 0.033(−h nΣ − K + V + h pΣ 0 K + V √ 2 ) + 0.2(h pK A − h nK A ) − 0.006h pΛK + 0.088h pΛK + V −0.26(|h 0 ρ | + |h 2 ρ | √ 6 ) − 0.087|h 1 ω | + 0.05|h 1 φ |} ,(57) where we have set the phase of the vector meson contributions as discussed above, and used the relations in Eq. (38). The expressions in Eqs. (56-57) illustrate the sensitivity of the radiative corrections to the various PV hadronic couplings. As expected on general grounds, the overall scale of R T =0,1 A is at about the one percent level [see Eq. (12)]. In terms of the conventional PV couplings, R T =0 A is most sensitive to h π and h 1 ρ , while R T =1 A is most strongly influenced by h 0 ρ + h 2 ρ / √ 6. The corrections also display strong dependences on the couplings h i V,A not included in the standard analysis of nuclear and hadronic PV. In particular, the couplings h 2 A and h 0 V + 4h 2 V /3 are weighted heavily in R T =1 A . In general, the sensitivity to the PV NY K couplings is considerably weaker than the sensitivity to the NNπ and NNρ couplings. In order to make an estimate of R T =0,1 A , we require inputs for the PV couplings. To that end, we use the "best values" for h π , h i ρ , and h i ω given in Ref. [29]. These values are consistent with the fit of Ref. [19]. For the h i φ we use the "best values" of Ref. [16]. The analyses given in Refs. [16,19,29], together with experimental input, also allow for the standard couplings to take on a range of values. For example, the ranges for the h i ω given in Refs. [16,29] correspond to − 33 ≤ h 0 ω + h 1 ω ≤ 13 .(58) In order to maintain consistency with the experimental constraint of Eq. (51), one then requires 0 ≤ h 0 ρ + h 1 ρ + h 2 ρ / √ 6 ≤ −45 .(59) We adopt this range even though it is smaller than the range given in Ref. [29]. Indeed, allowing the h i ρ to assume the full ranges given in Ref. [29] would require the h i ω to vary outside their corresponding theoretical "reasonable ranges" if the constraint of Eq. (51) is to be satisfied. Since one expects |h 1 ρ | << |h 0,2 ρ | [16,29], we have a reasonable range of values for the important isoscalar ρ contribution in Eq. (57), and the rather broad range of values allowed for the h i ρ contributes significantly to our estimated uncertainty in R T =1 A . For h i ω,φ , we use the ranges of Refs. [16,29] ‡ . In contrast to the situation with the h 0 ρ contribution, however, the variation in the h i ω,φ over their "reasonable ranges" has negligible impact on our estimated theoretical uncertainty. Estimating values for the Yukawa couplings h N Y K and for the h V,A is more problematic-to date, no calculation on the level of Ref. [16] has been performed for such couplings. Estimates for h V,A , based on dimensional and factorization arguments, were given in Ref. [11] and generally yielded values for h V,A in the non-strange sector on the order of g π . For our central values, then, we take h i V,A = g π , resulting in roughly 1% contributions from the PV vector and axial vector interactions. Without performing a detailed calculation as in Ref. [16], one might also attempt to determine reasonable ranges for these parameters by looking to phenomenology. To that end, the authors of Ref. [11] considered analogies between the axial vector PV operators of Eq. (19)(20) and contact operators needed to explain the size of ∆I = 1/2 hyperon P-wave decay amplitudes. From this analogy, these authors conclude that |h i A | ∼ 10g π may be reasonable. However, whether such large ranges are consistent with nuclear PV data remains to be determined. In the absence of such an analysis, which goes beyond the scope of the present work, we adopt the range −10g π ≤ h i A ≤ 10g π suggested in Ref. [11]. The corresponding uncertainties in the R T =0,1 A are roughly ±10%. The implications of phenomenology for the h i V are even less clear than for the h i A . However, we note that large values h i V ∼ ±10g π do not appear to be ruled out by hadronic and nuclear PV data. At tree-level, for example, the vector terms in L πN ∆T =0,1,2 do not contribute to the PV NN interaction through the one π-exchange amplitudes of Figure 8a. It is straightforward to show that the corresponding amplitude vanishes for on-shell nucleons § . Thus, at this level, purely hadronic PV processes are insensitive to the h i V and provide no constraints on these couplings. In PV electromagnetic processes, however, the h i V do contribute through PV two-body currents, such as those shown in Figure 8b. Nevertheless, one expects the impact of PV two-body currents to be considerably weaker than that of the PV NN potential. The PV γ-decay of 18 F, for example, is dominated by the mixing of a nearly-degenerate pair of (J π , T ) = (0 − , 0) and (0 + , 1) states. The small energy denominator associated with this parity-mixing enhances the relative importance of the PV NN potential by roughly two orders of magnitude over the generic situation with typical nuclear level spacings. By contrast, the PV two-body currents do not participate in parity-mixing and receive no such enhancements. A similar situation holds for PV electromagnetic processes in other nuclei of interest. Hence, we expect the PV γ-decays of light nuclei to be relatively insensitive to the h i V , even if the latter are on the order of 10g π . Consequently, we rather generously take −10g π ≤ h 0 V + 4h 2 V /3 ≤ 10g π , yielding a ‡ Allowing the h i ω to assume positive values would require a sign change on the correspnding terms in Eqs. (56,57). § The on-shell approximation is generally used in deriving the PV NN potential from Feynman diagrams. ±7% contribution to the uncertainty in R T =1 A . Allowing similarly large ranges for the PV NY K couplings has a negligible impact on the uncertainty in the R T =0,1 A . With these input values for the PV couplings, we arrive at the anapole contributions to R T =0,1 A shown in Table I. The latter must be added to the one-quark Standard Model contributions, also shown in Table I. We compute the one-quark corrections using the on-shell parameters given in Refs. [3,35]. We emphasize that the quoted values for the R T =0,1 A are renormalization scheme-dependent. The relative size of the isovector one-quark corrections are smaller, for example, in the MS scheme, where one has R T =1 A (SM) = −0.18 and R T =0 A (SM) = 0.07. The corresponding tree-level amplitude, however, is also smaller by a factor of ∼ 1.44 than the on-shell tree-level amplitude. A reader working in the MS scheme should, therefore, take care to adjust the tree-level amplitude and SM radiative corrections appropriately from the on-shell values used here. Moreover, the anapole contributions to the R (T ) A will be a factor of 1.44 larger in the MS scheme since the tree-level amplitude is correspondingly smaller * * . Adding the one-quark and anapole contributions yields a large, negative value for R T =1 A . This result contains considerable theoretical uncertainty, mostly due to our liberal assignment of reasonable ranges to the h V,A . Even with this generous theoretical uncertainty, however, R T =1 A is still roughly a factor of two away from the apparent SAMPLE result. Compared with the one-quark SM contribution, the many-quark anapole contribution is relatively small -though it does push the total in the right direction. The isoscalar correction, R T =0 A , is considerably smaller in magnitude than R T =1 A yet retains a sizeable theoretical uncertainty. * * Note that R T =0 A gives the ratio of the isoscalar, axial vector amplitude to the tree-level isovector, axial vector amplitude. The sign of R T =0 A as defined here is opposite that of Ref. [2]. TABLES Source R T =1 A R T =0 VII. CONCLUSIONS In view of the preliminary SAMPLE result for PV quasielastic electron scattering from 2 H, we have up-dated our previous calculation of the axial vector radiative corrections R T =0,1 A . Using the framework of HBChPT, we have computed all many-quark anapole contributions through O(1/Λ 2 χ ). We include new one-loop contributions involving the PV vector couplings, h i V and estimate the scale of the analytic, low-energy constants using resonance saturation. We fix the sign of the latter using the phenomenology of PV pp scattering and of nucleon EM form factors. We also show that large classes of loops involving decuplet intermediate states, magnetic insertions, and PV EM insertions vanish through O(1/Λ 2 χ ). Finally, we extend the previous analyses to include SU(3) symmetry, and determine that the impact of kaon loops is generally negligible. In the end, we find that R T =1 A -though large and negative-is still a factor of two or so away from the suggestion that R T =1 A ∼ −1 from the SAMPLE experiment. Even allowing for considerable theoretical uncertainty -dominated by the PV couplings h i V,A -there remains a sizable gap between our result and the preliminary experimental value. There exist a number of possible additional contributions to R T =0,1 A not considered here which may ultimately account for the apparent experimental result. The most obvious include higher-order chiral corrections. This appears, however, to be an unlikely source of large contributions. On general grounds, we expect the size of the O(1/Λ 3 χ ) contributions to be suppressed by m/Λ χ relative to those considered here, where m denotes a pseudoscalar mass. For kaon loops, this suppression factor is only ∼ 1/2; however, at O(1/Λ 2 χ ) kaon loops generate at most a few percent contribution to R T =0,1 A . The suppression factor for the next order pionic contributions is closer to 0.1. Hence, it would be surprising if the next order in the chiral expansion could close the factor of two gap with experiment. More promising sources of sizeable contributions include Z − γ box graph contributions, where the full tower of hadronic intermediate states is included, as well as paritymixing in the deuteron wavefunction. At a more speculative level, one might also consider contributions from physics beyond the Standard Model. For example, the presence of an additional, relatively light neutral gauge boson might modify the SM V (e) × A(q) amplitudes and contribute to R T =1 A . A popular class of Z ′ models are generated by E 6 symmetry [36]. The contribution of an extra, neutral weak E 6 gauge boson Z ′ is given by R T =1 A (new) = 4 5 1 1 − 4sin 2 θ W sin 2 φ G ′ φ G µ ,(60) where φ is a mixing angle which governs the structure of an additional U(1) group in E 6 theories [36] and G ′ φ is the Fermi constant associated with the new U(1) group [37]. Note that this contribution has the wrong sign to account for the large negative value of R T =1 A . Alternatively, one might consider new tree-level interactions generated by supersymmetric extensions of the SM. Such interactions arise when R-parity, or equivalently, B −L, is not conserved (B and L denote baryon and lepton number, respectively). The contribution from R-parity violating SUSY interactions is given by [38,37,39] R T =1 A (new) = 1 1 − 4x ∆ ′ 11k (d k R ) − ∆ ′ 1j1 (q j L ) − ∆ 12k (ẽ k R )(1 − 4x + 4λ x ) ,(61) where x = sin 2 θ W , λ x = x(1 − x) 1 − 2x 1 1 − ∆r ∼ 0.3 ,(62) ∆r is a radiative correction, and where ∆ ijk (f ) = 1 4 √ 2 |λ ijk | 2 G µ M 2 f ,(63) withf denoting the superpartner of fermion f and i, j, k labeling fermion generations. The terms having a prime are semileptonic whereas the un-primed terms are purely leptonic. In principle, the correction in Eq. (61) could generate a negative contribution to R T =1 A . However, the various other electroweak data constrain the terms appearing in this expression. For example, relations between G µ and other SM parameters require − 0.0023 ≤ ∆ 12k (ẽ k R ) ≤ 0.0028 ,(64) at 90 % C.L., so that the first term in Eq. (61) cannot provide the large negative contribution needed to explain the SAMPLE result. Similarly, assuming only the semileptonic R-parity violating interactions modify the weak charge of nuclei, the recent determination of the cesium weak charge by the Boulder group [40,10] implies that 0.0026 ≤ 2.6∆ ′ 11k (d k R ) − 2.9∆ ′ 1j1 (q j L ) ≤ 0.015 ,(65) at 95 % C.L. (for m H = 300 GeV). Thus, it appears unlikely that the second term in Eq. (61) could enhance R T =1 A by a factor of two. In short, two of the most popular new physics scenarios having implications for lowenergy phenomenology appear unlikely to enhance R T =1 A significantly. Thus, if more conventional hadronic and nuclear processes cannot account for the SAMPLE result, one may be forced to consider more exotic alternatives. APPENDIX A: FORMALISM In this section we first review the general parity and CP conserving Lagrangians including N, π, ∆, γ in the relativistic form. We follow standard conventions and introduce Σ = ξ 2 , ξ = e iπ Fπ , π = 1 2 π a τ a(A1) with F π = 92.4 MeV being the pion decay constant. The chiral vector and axial vector currents are given by A µ = − i 2 (ξD µ ξ † − ξ † D µ ξ) = − D µ π F π + O(π 3 ) V µ = 1 2 (ξD µ ξ † + ξ † D µ ξ) .(A2) and we require also the gauge and chiral covariant derivativs D µ π = ∂ µ π − ieA µ [Q, π] D µ = D µ + V µ ,(A3)with Q = 2 3 0 0 − 1 3 (A4) and A µ being the photon field. The chiral field strength tensors are F ± µν = 1 2 (∂ µ A ν − ∂ ν A µ )(ξQ ′ ξ † ± ξ † Q ′ ξ) (A5) with Q ′ = 1 0 0 0 (A6) acting in the space of baryon isodoublets. For the moment, we restrict our attention to SU(2) flavor space and consider just π, N, and ∆ degrees of freedom. We represent the nucleon as a two component isodoublet field, while for the ∆, we use the isospurion formalism, treating the ∆ field T i µ (x) as a vector spinor in both spin and isospin space [17] with the constraint τ i T i µ (x) = 0. The components of this field are T 3 µ = − 2 3 ∆ + ∆ 0 µ , T + µ = ∆ ++ ∆ + / √ 3 µ , T − µ = − ∆ 0 / √ 3 ∆ − µ . (A7) The field T i µ also satisfies the constraints for the ordinary Schwinger-Rarita spin-3 2 field, γ µ T i µ = 0 and p µ T i µ = 0 . (A8) We eventually convert to the heavy baryon expansion, in which case the latter constraint becomes v µ T i µ = 0 with v µ the heavy baryon velocity. It is useful to review the spacetime and chiral transformation properties of the various fields. Under a chiral transformation, ξ → LξU † = UξR † A µ → UA µ U † D µ → UD µ U † ,(A9) and N → UN , T µ → UT µ , Σ → LΣR † etc.(A10) In the SU(2) sector parity violating effects are conveniently described by introducing the operators [11]: X a L = ξ † τ a ξ , X a R = ξτ a ξ † , X a ± = X a L ±X a R .(A11) which transform as X a L,R → UX a L,R U † ,(A12) with the index a rotating like a vector of SU(2) L and SU(2) R respectively. The P and CP transformation properties of these fields are shown in Table 2. µν TABLE II. Parity (P) and CP transformation properties of chiral fields. Here, T denotes the transpose, C is the charge conjugation matrix (C = iγ 2 γ 0 in the Dirac representation) and δ(i) = 1, i = 1, 3 and δ(2) = −1. Field P CP A µ −A µ −A T µ N γ 0 N γ 0 CN T T µ −γ 0 T µ −δ(a)γ 0 CT T a µ X a L X a R δ(a)X T a L X a R X a L δ(a)X T a R F ± µν ±F ± µν −F ±T Finally, we note that in the Lagrangians of Section III, one has the following definitions: D ij µ = δ ij D µ − 2iǫ ijk V k µ ω i µ = T r[τ i A µ ] A ij µ = ξ ik 3/2 A µ ξ kj 3/2 ,(A13) where ξ ij 3/2 = 2 3 δ ij − i 3 ǫ ijk τ k is the isospin 3/2 projection operator. APPENDIX B: THE SU (3) PARITY VIOLATING AND CP CONSERVING LAGRANGIAN In this Appendix we list the parity violating and CP conserving SU(3) Lagrangian for the pseudosclar meson octet and baryon octet. We are interested in the diagonal case of the parity violating electron nucleon scattering. Hence, we include only those interaction terms that ensure strangeness and charge conservation at each vertex. In the following we use ξ = e iπ Fπ , π = 1 2 π a λ a , X a L = ξ † λ a ξ, X a R = ξλ a ξ † , X a ± = X a L ±X a R , [A, B] ± = AB ± BA. We classify the parity violating Lagrangian according to isospin violation ∆T = 0, 1, 2, which arises from the operators of X a L , X a R , their combinations and products. The ∆T = 2 piece comes from the operators I ab {X a L OX b L ± (L ↔ R)} with O = N,N , A µ and I ab = 1 3    1 0 0 0 1 0 0 0 −2    ,(B1) where a, b = 1, 2, 3. Several operators contribute to the ∆T = 1 part, like X 3 ± , f 3ab {X a L OX b L ± (L ↔ R)}, d 3ab {X a L OX b L ± (L ↔ R)} where f abc , d abc are the antisymmetric and symmetric structure constants of SU(3) algebra. With the requirement that the final Lagrangian be hermitian, parity-violating and CP-conserving, the operator with f 3ab vanishs. For the ∆T = 0 part relevant operators are 1, X 8 ± , f 8ab {X a L OX b L ±(L ↔ R)}, d 8ab {X a L OX b L ± (L ↔ R)}, δ ab {X a L OX b L ± (L ↔ R)}. For the same reason the f 8ab structure does not contribute. Note the matrix identity λ a λ b λ a = 4(C 2 (3) − 1 2 C 2 (8))λ b , where C 2 (3), C 2 (8) are the Casimir invariants of the basic and adjoint representations of SU(3) group respectively. Hence, the operator containing δ ab is identical to the unit operator. Based on these considerations, we obtain L PV ∆T =0 = h 3 F π T rN[X 8 − , N] + + h 4 F π T rN[X 8 − , N] − + v 1 T rNγ µ [A µ , N] + +v 2 T rNγ µ [A µ , N] − + v 7 2 T rNγ µ A µ NX 8 + + v 8 2 T rNγ µ X 8 + NA µ + v 9 2 T rNγ µ [X 8 + , A µ ] + N + v 10 2 T rNγ µ N[X 8 + , A µ ] + + a 5 T rNγ µ γ 5 A µ NX 8 − +a 6 T rNγ µ γ 5 X 8 − NA µ + a 7 T rNγ µ γ 5 [X 8 − , A µ ] + N + a 8 T rNγ µ γ 5 N[X 8 − , A µ ] + + √ 3v 11 d 8ab T r{Nγ µ NX a L A µ X b L + (L ↔ R)} + √ 3v 12 d 8ab T r{Nγ µ X a L A µ X b L N + (L ↔ R)} + √ 3v 13 d 8ab T r{Nγ µ X a L N[X b L , A µ ] + + (L ↔ R)} + √ 3v 14 d 8ab T r{Nγ µ [X a L , A µ ] + NX b L + (L ↔ R)} + √ 3a 9 d 8ab T r{Nγ µ γ 5 NX a L A µ X b L − (L ↔ R)} + √ 3a 10 d 8ab T r{Nγ µ γ 5 X a L A µ X b L N − (L ↔ R)} + √ 3a 11 d 8ab T r{Nγ µ γ 5 X a L N[X b L , A µ ] + − (L ↔ R)} + √ 3a 12 d 8ab T r{Nγ µ γ 5 [X a L , A µ ] + NX b L − (L ↔ R)} ,(B2)L PV ∆T =1 = h 1 F π T rN[X 3 − , N] + + h 2 F π T rN[X 3 − , N] − + v 3 2 T rNγ µ A µ NX 3 + + v 4 2 T rNγ µ X 3 + NA µ + v 5 2 T rNγ µ [X 3 + , A µ ] + N + v 6 2 T rNγ µ N[X 3 + , A µ ] + +a 1 T rNγ µ γ 5 A µ NX 3 − + a 2 T rNγ µ γ 5 X 3 − NA µ + a 3 T rNγ µ γ 5 [X 3 − , A µ ] + N +a 4 T rNγ µ γ 5 N[X 3 − , A µ ] + + v 15 d 3ab T r{Nγ µ NX a L A µ X b L + (L ↔ R)} +v 16 d 3ab T r{Nγ µ X a L A µ X b L N + (L ↔ R)} +v 17 d 3ab T r{Nγ µ X a L N[X b L , A µ ] + + (L ↔ R)} +v 18 d 3ab T r{Nγ µ [X a L , A µ ] + NX b L + (L ↔ R)} +a 13 d 3ab T r{Nγ µ γ 5 NX a L A µ X b L − (L ↔ R)} +a 14 d 3ab T r{Nγ µ γ 5 X a L A µ X b L N − (L ↔ R)} +a 15 d 3ab T r{Nγ µ γ 5 X a L N[X b L , A µ ] + − (L ↔ R)} +a 16 d 3ab T r{Nγ µ γ 5 [X a L , A µ ] + NX b L − (L ↔ R)} ,(B3)L PV ∆T =2 = v 19 2 I ab T r{Nγ µ NX a L A µ X b L + (L ↔ R)} + v 20 2 I ab T r{Nγ µ X a L A µ X b L N + (L ↔ R)} + v 21 2 I ab T r{Nγ µ X a L N[X b L , A µ ] + + (L ↔ R)} + v 22 2 I ab T r{Nγ µ [X a L , A µ ] + NX b L + (L ↔ R)} + a 17 2 I ab T r{Nγ µ γ 5 NX a L A µ X b L − (L ↔ R)} + a 18 2 I ab T r{Nγ µ γ 5 X a L A µ X b L N − (L ↔ R)} + a 19 2 I ab T r{Nγ µ γ 5 X a L N[X b L , A µ ] + − (L ↔ R)} + a 20 2 I ab T r{Nγ µ γ 5 [X a L , A µ ] + NX b L − (L ↔ R)} .(B4) These Lagrangians contain 4 Yukawa couplings, 20 axial vector couplings and 22 vector couplings, all of which should be fixed from the experimental data or from model calculations. In reality, however, we have only limited information which constrains a few of them. It is useful to expand the above Lagrangians to the order involving the minimum number of Goldstone bosons and to collect those vertices needed in the calculation of R A : L 1π Y ukawa = 2 √ 2i(h 1 + h 2 )(pnπ + −npπ − ) +i[h 1 − h 2 + √ 3(h 3 − h 4 )](pΣ 0 K + −Σ 0 pK − ) + √ 2i[h 1 − h 2 + √ 3(h 3 − h 4 )](nΣ − K + −Σ − nK − ) −i[ h 1 √ 3 + √ 3h 2 + h 3 + 3h 4 ](pΛK + −ΛpK − ) + · · · . (B5) L 1π V = − h pnπ + V F πp γ µ nD µ π + − h pΣ 0 K + V F πp γ µ Σ 0 D µ K + − h nΣ − K + V F πn γ µ Σ − D µ K + − h pΛK + V F πp γ µ ΛD µ K + + h.c. + · · · ,(B6) where h pnπ + V = v 1 +v 2 √ 2 + 4 √ 2 violating vertices in Eqs. (22)- (24) arise first at two-loop order and contribute to the nucleon anapole moment at the order of O(1/Λ 3 χ ). Although the PV N∆π interactions nominally contribute at lower order, in this case such contributions vanish up to O(1/Λ 2 χ ). The reason is as follows. Each of the parity violating and CP conserving single pion vertices has the same Lorentz structure-iγ 5 . In the heavy baryon expansion, the relevant vertices are obtained by the substitution P + iγ 5 P + , which vanishes. The leading nonzero contribution arises at first order in the 1/m N expansion. Consequently, its contribution to the nucleon anapole moment appears only at O(1/Λ 2 χ m N ), and since in this work we truncate at O(1/Λ 2 χ ), the PV π∆N vertices do not contribute. Magnetic moment insertions The nucleon has a large isovector magnetic moment. We thus consider associated possible PV chiral loop corrections which lead to a nucleon anapole moment. The relevant diagrams are shown in Figure iM 5a + iM 5b = iǫ µναβ ε µ q ν v α √ 2g A eµ N h πN m N F π [S β , S σ ] + d D k (2π) D k σ v · k 1 v · (q + k) 1 k 2 − m 2 π + iǫ ,(C1) where e µ is the photon polarization vector and µ N is the nucleon magnetic moment. The denominator of the integrand in (C1) is nearly the same as for M 3e . The numerator contains a single factor of S · k. Hence, Figures 5a and 5b vanish for the same reason as does M 3e . For the nucleon delta transition magnetic moment insertion we have iM 5c + iM 5d = − 2 √ 3 g πN ∆ eµ ∆N h πN m N F π (q σ ǫ ν − ǫ σ q ν )[P µν 3/2 S σ + S σ P νµ 3/2 ] × d D k (2π) D k µ v · k 1 v · (q + k) 1 k 2 − m 2 π + iǫ ,(C2) where µ ∆N is the nucleon delta transition magnetic moment and P µν 3/2 = g µν −v µ v ν + 4 3 S µ S ν is the spin 3 2 projection operator in the heavy baryon chiral perturbation framework. Since the integrand is the same as in M 3e the integral is proportional to v µ . Moreover, v µ P µν 3/2 = 0, so that M 5c + M 5d = 0. Finally, the ∆ magnetic insertions of Figures 5e-h require the PV N∆π vertex, which starts off at O(1/m N F π ). Thus, the latter do not contribute up to O(1/Λ 2 χ ). In short, none of the magnetic insertions contribute at the order to which we work in this analysis. and D, F are the usual SU(3) symmetric and antisymmetric coupling constants. The general pesudoscalar octet and baryon octet PV Lagrangians are given in the Appendix B. They contain four independent PV Yukawa couplings, 20 axial vector couplings (h A -type), and 22 vector couplings (h V -type). For simplicity, we combine the SU(3) couplings into combinations specific to various hadrons-e.g. the leading PV Yukawa interactions are . One-quark Standard Model (SM) and many-quark anapole contributions to V (A) × A(N ) radiative corrections. Values are computed in the on-shell scheme using sin 2 θ W = 0.2230 . 5 . 5At O(1/Λ 2 χ ) there are only four relevant diagrams Figures 5a-d. Since the magnetic moment is of O(1/Λ χ ) and the strong pion baryon vertex is of O(1/F π ), the remaining PV vertex must be a Yukawa coupling if the loop is to contribute at O(1/Λ 2 χ ) or lower. For the nucleon magnetic moment insertion we have, for example, Figure Captions FIG 1 . 1Axial vector γNN coupling, generated by PV hadronic interactions Figure 2 . 2Feynman diagrams for polarized electron nucleon scattering.Figure 2agives tree-level Z 0 -exchange amplitude,while FIG. 2bgives the anapole moment contribution.The dark circle indicates an axial vector coupling. Figure 3 . 3Meson-nucleon intermediate state contributions to the nucleon anapole moment. The shaded circle denotes the PV vertex. The solid, dashed and curly lines correspond to the nucleon, pion and photon respectively. For the SU(3) case the intermediate states can also be hyperons and kaons. Figure 4 . 4The contribution to the nucleon anapole moment from PV π∆N vertices. The double line is the ∆ intermediate state. Figure 5 . 5Anapole moment contributions generated by insertions of the baryon magnetic moment operator, denoted by the cross, and the PV hadronic couplings, denoted by the shaded circle. Figure 6 . 6PV electromagnetic insertions, denoted by the overlapping cross and shaded circle. Figure 7 . 7Vector meson contribution to the anapole moment. Shaded circle indicates PV hadronic coupling. Figure 8 . 8Contributions to (a) PV NN interaction and (b) PV two-body current generated by the vector terms in Eqs.(18)(19)(20). . PV electromagnetic insertions ACKNOWLEDGEMENTWe wish to thank D. Beck, E.J. Beise, and R. McKeown for useful discussions. This work was supported in part under U.S. Department of Energy contract #DE-AC05-84ER40150, the National Science Foundation and a National Science Foundation Young Investigator Award.. Finally, the PV γ∆∆ vertices contain one π. Since the ∆ can only appear as an intermediate state, this vertex contributes at two-loop order and is of higher-order in chiral counting than we consider here (the corresponding diagrams are not shown). Thus, to O(1/Λ 2 χ ), the PV electromagnetic insertions do not contribute. . D T Spayde, the SAMPLE CollaborationPhys. Rev. Lett. 841106D.T. Spayde et al., the SAMPLE Collaboration, Phys. Rev. Lett. 84, 1106 (2000). . M J Musolf, Phys. Reports. 2391M.J. Musolf et al., Phys. Reports 239, 1 (1994). . E Caso, Particle Data GroupEuro. Phys. J. C. 31E. Caso et al., Particle Data Group, Euro. Phys. J. C 3, 1 (1998). Parity Nonconservation in Atomic Phenomena. I B Khriplovich, Gordon and Breach. I. B. Khriplovich, Parity Nonconservation in Atomic Phenomena, Gordon and Breach, London, 1991. . M J Musolf, B R Holstein, Phys. Lett. B. 242461M. J. Musolf and B. R. Holstein, Phys. Lett. B 242, 461 (1990). . V V Flambaum, I B Khriplovich, O P Suschkov, Phys. Lett. B. 146367V.V. Flambaum, I.B. Khriplovich and O.P. Suschkov, Phys. Lett. B 146, 367 (1984). . M J Musolf, B R Holstein, Phys. Rev. D. 432956M. J. Musolf and B. R. Holstein, Phys. Rev. D 43, 2956 (1991). E J H Beise ; ; D, R D Beck, Mckeown, talk given at Bates25 Symposium. Cambridge, MAprivate communiationE.J. Beise, talk given at Bates25 Symposium, Nov. 3-5, 1999, Cambridge, MA; D.H. Beck and R.D. McKeown, private communiation. . W C Haxton, E M Henley, M J Musolf, Phys. Rev. Lett. 63949W. C. Haxton, E. M. Henley and M. J. Musolf, Phys. Rev. Lett. 63, 949 (1989). . C S Wood, Science. 2751759C. S. Wood et al., Science 275, 1759 (1997). . D B Kaplan, M J Savage, Nucl. Phys. A. 556653D. B. Kaplan and M. J. Savage, Nucl. Phys. A 556, 653 (1993); . Erratum-Ibid, A. 570833Erratum-ibid. A 570, 833 (1994); . Erratum-Ibid, A. 580679Erratum-ibid. A 580, 679 (1994). . E Jenkins, A V Manohar, Phys. Lett. B. 255558E. Jenkins and A. V. Manohar, Phys. Lett. B 255, 558 (1991); . B. 259353B 259, 353 (1991). . - G Ulf, Meissner, Int. J. Mod. Phys. E. 1561Ulf-G. Meissner, Int. J. Mod. Phys. E 1, 561 (1992). Weak Interactions in Nuclei. B R Holstein, World ScientificB.R. Holstein, Weak Interactions in Nuclei, World Scientific, 1989. Holstein in Symmetries and Fundamental. W Haeberli, B R , Nuclei, W.C. Haxton and E.M. HenleyWorld Scientific17SingaporeW. Haeberli and B.R. Holstein in Symmetries and Fundamental Interactions in Nu- clei, W.C. Haxton and E.M. Henley, Eds., World Scientific, Singapore, 1995, p. 17. . B Desplanques, J F Donoghue, B R Holstein, Ann. Phys. 124449B. Desplanques, J. F. Donoghue and B. R. Holstein, Ann. Phys. 124, 449 (1980). . T R Hemmert, B R Holstein, J Kambor, J. Phys. G. 241831T. R. Hemmert, B. R. Holstein and J. Kambor, J. Phys. G 24, 1831 (1998). B J Bjorken, S Drell, Relativistic Quantum Fields. New YorkMcGraw-HillB. J. Bjorken and S. Drell, Relativistic Quantum Fields, McGraw-Hill, New York, 1965. . E G Adelberger, W C Haxton, Ann. Rev. Nucl. Part. Sci. 35501E. G. Adelberger and W. C. Haxton, Ann. Rev. Nucl. Part. Sci. 35, 501 (1985). . N Kaiser, Ulf.-G Meissner, Nucl. Phys. A. 499759ibid. AN. Kaiser and Ulf.-G. Meissner, Nucl. Phys. A 499, 699 (1989); ibid. A 510, 759 (1990); . .-G Ulf, H Meissner, Weigel, Phys. Lett. B. 4471Ulf.-G. Meissner and H. Weigel, Phys. Lett. B 447, 1 (1999). . E M Henley, W.-Y P Hwang, L S Kisslinger, Phys. Lett. B. 36721E. M. Henley, W.-Y. P. Hwang and L. S. Kisslinger, Phys. Lett. B 367, 21 (1996); . Erratum-Ibid, B. 440449Erratum-ibid. B 440, 449 (1998). . E Jenkins, Phys. Lett. B. 302482E. Jenkins et al., Phys. Lett. B 302, 482 (1993); . B. 388866B 388, 866 (1996) [E]. . - G Ulf, S Meissner, Steininger, Nucl. Phys. B. 499349Ulf-G. Meissner and S. Steininger, Nucl. Phys. B 499, 349 (1997). . L Durand, P Ha, Phys. Rev. D. 5813010L. Durand and P. Ha, Phys. Rev. D 58, 013010 (1998). . S Puglia, M J Ramsey-Musolf, hep-ph/9911542S. Puglia and M. J. Ramsey-Musolf, hep-ph/9911542. . .-G Ulf, S Meissner, Steininger, Nucl. Phys. B. 499349Ulf.-G. Meissner and S. Steininger, Nucl. Phys. B 499, 349 (1997); . .-G Ulf, Meissner, hep-ph/9709402Ulf.-G. Meissner, hep-ph/9709402. . G Ecker, J Gasser, A Pich, E De Rafael, Nucl. Phys. B. 321311G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B 321, 311 (1989). . G Höhler, Nucl. Phys. 114505G. Höhler et al., Nucl. Phys. B114, 505 (1976). . G B Feldman, Phys. Rev. C. 43863G. B. Feldman et al., Phys. Rev. C 43, 863 (1991). . R Balzer, Phys. Rev. C. 301409R. Balzer et al., Phys. Rev. C 30, 1409 (1984). . B Desplanques, Nucl. Phys. A. 335147B. Desplanques, Nucl. Phys. A 335, 147 (1980). . S Kistyrn, Phys. Rev. Lett. 581616S. Kistyrn et al., Phys. Rev. Lett. 58, 1616 (1987). Currents and mesons, the. J J Sakurai, University of Chicago PressJ. J. Sakurai, Currents and mesons, the University of Chicago Press, 1969. . H.-W Hammer, M J Ramsey-Musolf, Phys. Rev. C. 6045205H.-W. Hammer and M.J. Ramsey-Musolf, Phys. Rev. C 60, 45205 (1999). The Particle Data Group. Phys. Rev. D. 541The Particle Data Group, Phys. Rev. D 54, 1 (1996). . D London, J L Rosner, Phys. Rev. D. 341530D. London and J.L. Rosner, Phys. Rev. D 34, 1530 (1986). . M J Ramsey-Musolf, Phys. Rev. C. 6015501M.J. Ramsey-Musolf, Phys. Rev. C 60, 015501 (1999). . V Barger, G F Guidice, T Han, Phys. Rev. D. 402987V. Barger, G.F. Guidice, T. Han, Phys. Rev. D 40, 2987 (1989). . M J Ramsey-Musolf, hep-ph/0004062M.J. Ramsey-Musolf, hep-ph/0004062. For recent analyses of the theoretical implications of these measurements, see for example. S C Bennett, C E Wieman, ; , R Casalbuoni, S Curtis, D Dominici, R Gatto, S Riemann, hep-ph/0001215Phys. Rev. Lett. 822484S.C. Bennett and C.E. Wieman, Phys. Rev. Lett. 82, 2484 (1999). For recent analyses of the theoretical implications of these measurements, see for example, R. Casalbuoni, S. De Curtis, D. Dominici, R. Gatto, S. Riemann, hep-ph/0001215.
[]
[ "Fair and Efficient Allocations of Chores under Bivalued Preferences *", "Fair and Efficient Allocations of Chores under Bivalued Preferences *" ]
[ "Jugal Garg ", "Aniket Murhekar [email protected] ", "John Qin [email protected] " ]
[]
[]
We study the problem of fair and efficient allocation of a set of indivisible chores to agents with additive cost functions. We consider the popular fairness notion of envy-freeness up to one good (EF1) with the efficiency notion of Pareto-optimality (PO). While it is known that an EF1+PO allocation exists and can be computed in pseudo-polynomial time in the case of goods, the same problem is open for chores.Our first result is a strongly polynomial-time algorithm for computing an EF1+PO allocation for bivalued instances, where agents have (at most) two disutility values for the chores. To the best of our knowledge, this is the first non-trivial class of indivisible chores to admit an EF1+PO allocation and an efficient algorithm for its computation.We also study the problem of computing an envy-free (EF) and PO allocation for the case of divisible chores. While the existence of an EF+PO allocation is known via competitive equilibrium with equal incomes, its efficient computation is open. Our second result shows that for bivalued instances, an EF+PO allocation can be computed in strongly polynomial-time.
10.1609/aaai.v36i5.20436
[ "https://arxiv.org/pdf/2110.09601v1.pdf" ]
239,024,873
2110.09601
509c56f46aae8b5f03cf5dc0740a0da5250c4623
Fair and Efficient Allocations of Chores under Bivalued Preferences * 18 Oct 2021 Jugal Garg Aniket Murhekar [email protected] John Qin [email protected] Fair and Efficient Allocations of Chores under Bivalued Preferences * 18 Oct 2021 We study the problem of fair and efficient allocation of a set of indivisible chores to agents with additive cost functions. We consider the popular fairness notion of envy-freeness up to one good (EF1) with the efficiency notion of Pareto-optimality (PO). While it is known that an EF1+PO allocation exists and can be computed in pseudo-polynomial time in the case of goods, the same problem is open for chores.Our first result is a strongly polynomial-time algorithm for computing an EF1+PO allocation for bivalued instances, where agents have (at most) two disutility values for the chores. To the best of our knowledge, this is the first non-trivial class of indivisible chores to admit an EF1+PO allocation and an efficient algorithm for its computation.We also study the problem of computing an envy-free (EF) and PO allocation for the case of divisible chores. While the existence of an EF+PO allocation is known via competitive equilibrium with equal incomes, its efficient computation is open. Our second result shows that for bivalued instances, an EF+PO allocation can be computed in strongly polynomial-time. Introduction The problem of fair division is concerned with allocating items to agents in a fair and efficient manner. Formally introduced by Steinhaus [29], fair division is an active area of research studied across fields like computer science and economics. Most work has focused on the fair division of goods: items which provide non-negative value (or utility) to the agents to whom they are allocated. However, several practical scenarios involve chores (or bads). Chores are items which impose a cost (or disutility) to the agent to whom they are allocated. For instance, household chores such as cleaning and cooking often need to be fairly distributed among members of the household. Likewise, teachers have to divide teaching load, stakeholders have to divide liabilities upon dissolution of a firm, etc. These examples highlight the importance of allocating chores in a fair and efficient manner. Agencies responsible for designing such allocations must take into account the differences in preferences of agents in order for the allocation to be acceptable to all those involved. Arguably, the most popular notion of fairness is envy-freeness (EF) [16,30], which requires that every agent weakly prefers the bundle of items allocated to them over the bundle of any other agent. When items are divisible, i.e., can be shared among agents, EF allocations are known to exist. However, in the case of indivisible items, EF allocations need not exist. For instance, while dividing one chore between two agents, the agent who is assigned the chore will envy the other. Next, we study the problem of computing an EF+PO allocation of divisible chores. For goods, it is known that an EF+PO allocation always exists [30] and is in fact polynomial-time computable via the Eisenberg-Gale convex program [27]. This is done by computing the competitive equilibrium with equal incomes (CEEI). Here, the idea is to provide each agent with the same amount of fictitious money, and then find prices and an allocation of items such that all items are completely bought and each agent buys her most preferred bundle subject to her budget constraint. This is an example of a market where demand (of agents) equals supply (of items), and is known as the Fisher market. For goods, there are polynomial-time algorithms for computing the competitive equilibrium (CE) [15,28,32]. For chores, the problem is harder: Bogomolnaia et al. [10] showed that the CE rule can be non-convex, multi-valued and disconnected. Algorithms with exponential run-times are known for computing CE for chores [11,20,14,19], but designing a polynomial-time algorithm is an open problem. Working towards this goal, our second result shows: Result 2. For bivalued instances with n agents and m divisible chores, an EF+fPO allocation can be computed in poly(n, m)-time. Further related work Barman et al. [5] showed that for n agents and m goods, an EF1+PO allocation can be computed in time poly(n, m, V ), where V is the maximum utility value. Their algorithm first perturbs the values to a desirable form, and then computes an EF1+fPO allocation for the perturbed instance, which for a small-enough perturbation is EF1+PO for the original instance. Their approach is via integral market equilibria, which guarantees fPO at every step, and the concept of price-envyfreeness up to one good (pEF1) which is a strengthening of EF1. Using similar tools, Garg and Murhekar [26] showed that an EF1+fPO allocation can be computed in poly(n, m, V )-time. They also showed that an EF1+fPO allocation can be computed in poly(n, m)-time for k-ary instances (agents have at most k values for the goods) where k is a constant. It may seem a natural idea to try and use these approaches for chores, however they do not extend easily. While our algorithm also uses integral market equilibria to obtain the fPO property and pEF1 for chores to argue about EF1, our algorithm and its analysis is much more involved and significantly different from previous works. Bivalued preferences are a well-studied class in literature. The following results are for the goods setting. Aziz et al. [3] showed PO is efficiently verifiable for bivalued instances and coNP-hard for 3valued instances; Aziz [2], and Vazirani and Yannakakis [31] studied the Hylland-Zeckhauser scheme for probabilistic assignment of goods in bivalued instances; and Bogomolnaia and Moulin [9] studied matching problems with bivalued (dichotomous) preferences. More generally, instances with few values have also been studied: Barman et al. [6] showed that EF1+PO allocations can be computed for binary valuations; Babaioff et al. [4] studied truthful mechanisms for dichotomous valuations; Golovin [23] presented approximation algorithms and hardness results for computing max-min fair allocations in 3-valued instances;Bliem et al. [8] studied fixed-parameter tractability for computing EF+PO allocations with parameter n + z, where z is the number of values; and Garg et al. [22] studied leximin assignments of papers ranked by reviewers on a small scale, in particular they present an efficient algorithm for 2 ranks, i.e., "high or low interest" and show NP-hardness for 3 ranks. Such instances have also been studied in resource allocation contexts, including makespan minimization with 2 or 3 job sizes [33,13]. The fairness notion of equitability requires that each agent get the same amount of utility or disutility. Similar to EF1 and EFX, equitability up to one (resp. any) item (EQ1 (resp. EQX)) are relaxations of equitability. Using approaches inspired by [5], pseudo-polynomial time algorithms for computing EQ1+PO allocations were developed for both goods [17] and chores [18]. For bivalued goods, an EQX+PO allocation is polynomial time computable [21]. Preliminaries Problem instance. A fair division instance (of chores) is a tuple (N, M, C), where N = [n] is a set of n ∈ N agents, M = [m] is a set of m ∈ N indivisible chores, and C = {c 1 , . . . , c n } is a set of cost or disutility functions, one for each agent i ∈ N . Each cost function c i : M → R ≥0 is specified by m numbers c ij ∈ R ≥0 , one for each chore j ∈ M , which denotes the cost agent i has for performing (receiving) chore j. We assume that the cost functions are additive, that is, for every agent i ∈ N , and for S ⊆ M , c i (S) = j∈S c ij . For notational ease, we write c(S \ j) instead of c(S \ {j}). We call a fair division instance (N, M, C) a bivalued instance if there exist a, b ∈ R ≥0 , with a ≥ b, such that for all i ∈ N and j ∈ M , c ij ∈ {a, b}. That is, the cost of any chore to any agent is one of at most two given numbers. By scaling the costs, we can assume w.l.o.g. for bivalued instances that all costs c ij ∈ {1, k}, where k = a/b ≥ 1. Such a scaling is w.l.o.g., since our fairness and efficiency properties are scale-invariant. Given such an instance, we partition the set of chores into sets of low-cost chores M low and high-cost chores M high : • M low = {j ∈ M : ∃i ∈ N s.t. c ij = 1}, and • M high = {j ∈ M : ∀i ∈ N, c ij = k}. Further, we assume that both a, b > 0, since if b = 0 then re-scaling the values transforms the instance into the simpler binary case, for which efficient algorithms are known (Footnote 1). We can additionally assume for bivalued instances that for every agent i, there is at least one chore j s.t. c ij = 1. This is w.l.o.g., since if c ij = k for all j ∈ M , then we can re-scale costs to set c ij = 1 for all j ∈ M . Allocation. An allocation x of chores to agents is an n-partition (x 1 , . . . , x n ) of the chores, where agent i is allotted x i ⊆ M and gets a total cost of c i (x i ). A fractional allocation x ∈ [0, 1] n×m is a fractional assignment such that for each chore j ∈ M , i∈N x ij = 1. Here, x ij ∈ [0, 1] denotes the fraction of chore j allotted to agent i. Fairness notions. An allocation x is said to be envy-free up to one chore (EF1) if for all i, h ∈ N , there exists a chore j ∈ x i such that c i (x i \ j) ≤ c i (x h ). We say that an agent i EF1-envies an agent h if for all j ∈ x i , c i (x i \ j) > c i (x h ), i.e., the EF1 condition between i and h is violated. A (fractional) allocation x is said to be envy-free if for all i, h ∈ N , c i (x i ) ≤ c i (x h ). We say that an agent i envies an agent h if c i (x i ) > c i (x h ), i.e., the EF condition between i and h is violated. Pareto-optimality. An allocation y dominates an allocation x if c i (y i ) ≤ c i (x i ), ∀i and there exists h s.t. c h (y h ) < c h (x h ). An allocation is said to be Pareto optimal (PO) if no allocation dominates it. Further, an allocation is said to be fractionally PO (fPO) if no fractional allocation dominates it. Thus, a fPO allocation is PO, but not vice-versa. Fisher markets. A Fisher market or a market instance is a tuple (N, M, C, e), where the first three terms are interpreted as before, and e = {e 1 , . . . , e n } is the set of agents' minimum payments, where e i ≥ 0, for each i ∈ N . In this model, chores can be allocated fractionally. Given a payment vector, also called a price 2 vector, p = (p 1 , . . . , p m ), each chore j pays p j per unit of chore. Agents perform chores in exchange for payment. Given chore payments, each agent i aims to obtain the set of chores that minimizes her total cost subject to her payment constraint, i.e., receiving a total payment of at least e i . Given a (fractional) allocation x with a price vector p, the spending 3 of an agent i under (x, p) is given by p(x i ) = j∈M p j x ij . We define the bang-per-buck ratio α ij of chore j for an agent i as α ij = c ij /p j , and the minimum bang-per-buck (mBB) ratio as α i = min j α ij . We define mBB i = {j ∈ M : c ij /p j = α i }, called the mBB-set, to be the set of chores that give mBB to agent i at prices p. We say (x, p) is 'on mBB' if for all agents i and chores j, x ij > 0 ⇒ j ∈ mBB i . For integral x, this means that x i ⊆ mBB i for all i ∈ N . A market equilibrium or market outcome is a (fractional) allocation x of the chores to the agents and set of prices p of the chores satisfying the following properties: • the market clears, i.e., all chores are fully allocated. Thus, for all j, i∈N x ij = 1, • each agent receives their minimum payment, for all i ∈ N , p(x i ) = j∈M x ij p j = e i , and, • agents only receive chores that give them minimum bang-per-buck, i.e., (x, p) is on mBB. Given a market outcome (x, p) with x integral, we say it is price envy-free up to one chore (pEF1) if for all i, h ∈ N there is a chore j ∈ x i such that p(x i \ j) ≤ p(x h ). We say that an agent i pEF1-envies an agent h, if for all j ∈ x i , p(x i \ j) > p(x h ), i.e., the pEF1 condition between i and h is violated. For integral market outcomes on mBB, the pEF1 condition implies the EF1 condition. Lemma 1. Let (x, p) be an integral market outcome on mBB. If (x, p) is pEF1 then x is EF1 and fPO. Proof. We first show that (x, p) forms a market equilibrium for the Fisher market instance (N, M, C, e), where for every i ∈ N , e i = p(x i ). It is easy to see that the market clears and each agent receives their minimum payment. Further x is on mBB as per our assumption. Now the fact that x is fPO follows from the First Welfare Theorem [25], which shows that for any market equilibrium (x, p), the allocation x is fPO. Since (x, p) is pEF1, for all pairs of agents i, h ∈ N , there is some chore j ∈ x i s.t. p(x i \ j) ≤ p(x h ). Since (x, p) is on mBB, x i ⊆ mBB i . Let α i be the mBB-ratio of i at the prices p. By the definition of mBB, c i (x i \ j) = α i p(x i \ j), and c i (x h ) ≥ α i p(x h ). Combining these implies x is EF1. Our Algorithm 2 begins with and maintains an integral market outcome (x, p) on mBB 4 , and modifies x and p appropriately to eventually arrive at an outcome (x, p) on mBB where the pEF1 condition is satisfied. Lemma 1 then ensures that x is EF1+fPO. We now define least spenders as agents with minimum spending, and big spenders as agents with maximum spending after the removal of their highest-priced chore. Definition 1 (Least and big spenders). An agent ℓ ∈ argmin i∈N p(x i ) is referred to as a least spender (LS). An agent b ∈ argmax i∈N min j∈x i p(x i \ j) is referred to as a big spender (BS). We break ties arbitrarily to decide a unique LS and BS. Together with Lemma 1, the following lemma shows that in order to obtain an EF1 allocation, it is sufficient to focus on the pEF1-envy the big spender has towards the least spender. Lemma 2. Let (x, p) be an integral market outcome on mBB. If x is not EF1, then the big spender b pEF1-envies the least spender ℓ. Proof. If x is not EF1, then Lemma 1 implies that x is not pEF1. Hence there is a pair of agents i, h s.t. for every chore j ∈ x i , p(x i \ j) > p(x h ). By the definition of big spender, we know p(x b \ j ′ ) ≥ p(x i \ j), for every j ′ ∈ x b . By the definition of least spender, p(x i ) ≥ p(x ℓ ). Putting these together we get p(x b \ j ′ ) > p(x ℓ ) for every j ′ ∈ x b , implying that b pEF1-envies ℓ. Given a market outcome (x, p) on mBB, we define the mBB graph to be a bipartite graph G = (N, M, E) where for an agent i and chore j, (i, j) ∈ E iff j ∈ mBB i . Further, an edge (i, j) is called an allocation edge if j ∈ x i , otherwise it is called an mBB edge. For agents i 0 , . . . , i ℓ and chores j 1 , . . . , j ℓ , a path P = (i 0 , j 1 , i 1 , j 2 , . . . , j ℓ , i ℓ ) in the mBB graph, where for all 1 ≤ ℓ ′ ≤ ℓ, j ℓ ′ ∈ x i ℓ ′ −1 ∩ mBB i ℓ ′ , is called a special path. We define the level λ(h; i 0 ) of an agent h w.r.t. i 0 to be half the length of the shortest special path from i 0 to h, and to be n if no such path exists. A path P = (i 0 , j 1 , i 1 , j 2 , . . . , j ℓ , i ℓ ) is an alternating path if it is special, and if λ(i 0 ; i 0 ) < λ(i 1 ; i 0 ) < · · · < λ(i ℓ ; i 0 ), i.e. the path visits agents in increasing order of their level w.r.t. i 0 . Further, the edges in an alternating path alternate between allocation edges and mBB edges. Typically, we consider alternating paths starting from a big spender agent. Definition 2 (Component C i of a big spender i). For a big spender i, define C ℓ i to be the set of all chores and agents which lie on alternating paths of length ℓ starting from i. Call C i = ℓ C ℓ i the component of i, i.e., the set of all chores and agents reachable from i through alternating paths. EF1+fPO allocation of indivisible chores In this section, we present our main result: Theorem 1. Given a bivalued fair division instance (N, M, C) of indivisible chores with all c ij ∈ {a, b} for some a, b ∈ R + , an EF1+fPO allocation can be computed in strongly polynomial-time. We prove Theorem 1 by showing that our Algorithm 2 computes an EF1+fPO allocation in polynomial-time. To ensure smooth presentation, we defer some proofs to Appendix A. Obtaining Initial Groups Recall that we can scale the costs so that they are in {1, k}, for some k > 1. The first step of Algorithm 2 is to obtain a partition of the set N of agents into groups N 1 , . . . , N R with desirable properties. For this, we use Algorithm 1 (called MakeInitGroups). Algorithm 1 starts with a cost-minimizing market outcome (x, p) where each chore j is assigned to an agent who has minimum cost for j. This ensures the allocation is fPO. The chore prices are set as follows. Each low-cost chore j is assigned to an agent i s.t. c ij = 1. If an agent values all chores at k, then we can re-scale all values to 1. Each low-cost chore is priced at 1, and each high-cost chore is priced at k. This pricing ensures that the mBB ratio of every agent is 1. The algorithm then eliminates pEF1-envy from the component of the big spender b by identifying an agent i in C b that is pEF1-envied by b, and transferring a single additional chore j ℓ to i from an agent h ℓ−1 who lies along a shortest alternating path from b to i (Lines 7 & 8). Note that the identity of the big spender may change after transferring j ℓ if j ℓ belonged to b, so we must check who the big spender is after each transfer (Line 10). Once the component of the current big spender b is pEF1, the same process is applied to the next big spender outside the previously made components. Repeated application of this process creates disjoint partial components H 1 , . . . , H R of agent sets N 1 , . . . , N R , where R ≤ n, all of which are pEF1. We refer to N 1 , . . . , N R as agent groups, and H 1 , . . . , H R as initial (partial) components. Note also that the spending (up to the removal of the biggest chore) 1: (x, p) ← initial cost minimizing integral market allocation, where p j = c ij for j ∈ x i . 2: R ← 1, N ′ ← N 3: while N ′ = ∅ do 4: b ← argmax i∈N ′ min j∈x i p(x i \ j) ⊲ Big Spender 5: C b ← Component of b ⊲ See Definition 2 6: while ∃ agent i ∈ C b s.t. ∀j ∈ x b , p(x b \ j) > p(x i ) 7: Let (b, j 1 , h 1 , j 2 , . . . , h ℓ−1 , j ℓ , i) be the shortest alternating path from b to i 8: x h ℓ−1 ← x h ℓ−1 \ {j ℓ } ⊲ Chore transfer 9: x i ← x i ∪ {j ℓ } 10: b ← argmax i∈N ′ min j∈x i p(x i \ j) 11: H R ← C b ∩ (N ′ ∪ x N ′ ) ⊲ Partial component 12: N R ← H R ∩ N ⊲ Agent group 13: N ′ ← N ′ \ N R , R ← R + 1 14: return (x, p, {N r } r∈[R] ) of the big spender h r of H r is weakly decreasing with r. We now record several properties of the output of Algorithm 1 . Lemma 3. Algorithm 1 returns in poly(n, m)-time a market outcome (x, p) with agents partitioned into groups N 1 , . . . , N R , with the following properties: (i) For all low-cost chores j ∈ M , p j = 1, and for all high-cost chores j ∈ M , p j = k. (ii) The mBB ratio α i of every agent i is 1. (iii) Let H r be the collection of agents N r and chores allocated to them in (x, p). Then each H r is a partial component of some agent. That is, for each r ∈ [R], there is an agent h r ∈ H r s.t. H r comprises of all agents and chores not in r ′ <r H r ′ reachable through alternating paths from h r . Further, h r is the big spender among agents not in r ′ <r H r ′ : h r ∈ argmax i / ∈( r ′ <r H r ′ ) min j∈x i p(x i \ j) (iv) The spending (up to removal of the largest chore) f (r) of the big spender in H r weakly decreases with r. Here f (r) = max i∈Hr min j∈x i p(x i \ j). (v) Each group is pEF1, i.e., an agent does not pEF1-envy other agents in the same group. (vi) For every agent i ∈ H r and chore j ∈ H r ′ with r ′ < r, c ij = k. (vii) All high-cost chores belong to H R . Proof. Properties (i) and (ii) follow from the construction of the initial allocation in Line 1. Property (iii) follows from the construction of the set H r in Line 11. The agent h r is the last agent b chosen in Line 10. Property (iv) then follows from the last result of (iii). To see why (v) holds, we examine the loop in Lines 6-10. This loop terminates only once the big spender b in the component does not pEF1-envy any other agent in the component. Since every agent spends (up to the removal of one chore) less than the big spender, it must be that every agent does not pEF1-envy any other agent in the component. Thus, the component is pEF1. Next, suppose for some agent i ∈ H r and some chore j ∈ H r ′ for r ′ < r, c ij = 1. Then j is a low-cost chore and must be priced at 1. Hence, α ij = c ij /p j = 1 = α i , implying that j ∈ mBB i . However this means that i would have been added to H r ′ , since there is an alternating path from h r ′ to i via j ∈ H r ′ . This is a contradiction, thus showing (vi). Next, we show (vii). Suppose a high-cost chore j belongs to H r for r < R. We know j is priced at k. Then for an agent i ∈ H R , α ij = c ij /p j = 1 = α i , implying that j ∈ mBB i . However this means i belongs to H r since there is an alternating path from h r to i via j, which is a contradiction. Finally, we argue that in making a group starting with the a new big spender in N ′ outside of previously established pEF1 groups, we do not disturb the previously established groups. Let N r be the current group being made, and let N r ′ , where r ′ < r, be a previously established group. Let b and b ′ be the big spenders in N r and N r ′ , respectively. We show that no agent in N r ′ gains or loses a chore. Note that there are no high-cost chores in N r ′ , as high-cost chores are mBB for all agents so a group containing a high-cost chore must be the last group N R . Since all chores are priced 1 and N r ′ is pEF1, every agent in N r ′ has total spending either p( x b ′ ) or p(x b ′ ) − 1. We also have that p(x b ) ≤ p(x b ′ ) by construction. Thus, it cannot be that b pEF1-envies any agent in N r ′ , so agents in N r ′ will not receive any chores. Agents in N r ′ also will not lose any chores, since they can only transfer chores to another agent in N r ′ . However, we have already shown that agents in N r ′ cannot receive chores since they will not be pEF1-envied by b, so these transfers are impossible. Thus, all previous groups remain undisturbed while establishing a later group. Finally, we argue that: Overview of Algorithm 2 Our main algorithm (Algorithm 2) begins by calling Algorithm 1, which returns a market outcome (x, p) and a set of agent groups {N r } r∈[R] (with associated partial components {H r } r∈[R] ) satisfying properties in Lemma 3. In the subsequent discussion, we refer to (x, p) as the initial allocation. Also in the subsequent discussion, all mentions of an agent receiving or losing chores are relative to this initial allocation. The following is an important invariant of Algorithm 2 (after the initial allocation is constructed). Lemma 5. The spending of the least spender does not decrease in the run of Algorithm 2. We say that a group N r is above (resp. below) group N s if r < s (resp. r > s). Lemma 3 shows that each group N r is initially pEF1. Hence if the initial allocation (x, p) is not pEF1, then the big spender b and the least spender ℓ must be in different components. Since b ∈ H 1 , it must be the case that ℓ ∈ H s for some s > 1. Since we want to obtain an fPO allocation, we can only transfer along mBB edges. Hence we raise the prices of all chores in H 1 . We show that doing so creates an mBB edge from all agents i / ∈ H 1 to all chores j ∈ H 1 (Lemma 9 below). In particular, there is an mBB edge from ℓ to a chore assigned to b. Hence we transfer a chore directly from b to ℓ, thus reducing the pEF1-envy of b. This may change the identity of the big and least spenders. If the allocation is not yet pEF1, we must continue this process. At an arbitrary step in the run of the algorithm, let b and ℓ be the big and least spenders. If the allocation is not pEF1, then b pEF1-envies ℓ (Lemma 2). We consider cases based on the relative positions of b and ℓ. First we argue that b and ℓ cannot lie in the same group, by showing that: Algorithm 2 Computing an EF1+fPO allocation Input: Fair division instance (N, M, C) with c ij ∈ {1, k} Output: An integral allocation x 1: (x, p, {N r } r∈[R] ) ← MakeInitGroups(N, M, V) 2: U ← [R] ⊲ Unraised groups 3: x 0 ← x ⊲ Copy of initial allocation, used in Line 22 4: b ← argmax i∈N min j∈x i p(x i \ j) ⊲ Big Spender 5: ℓ ← argmin i∈N p(x i ) ⊲ Least Spender 6: while (x, p) is not pEF1 and ℓ ∈ U do 7: b ← argmax i∈N min j∈x i p(x i \ j) 8: ℓ ← argmin i∈N p(x i ) 9: Let (r, s) s.t. b ∈ N r , ℓ ∈ N s ⊲ r < s 10: if r ∈ U then 11: Raise prices of chores owned by agents in N r by a factor of k 12: U ← U \ {r} 13: else 14: Transfer a chore from b to ℓ along an mBB edge 15: while (x, p) is not pEF1 do 16: b ← argmax i∈N min j∈x i p(x i \ j) 17: ℓ ← argmin i∈N p(x i ) 18: Let (r, s) s.t. b ∈ N r , ℓ ∈ N s 19: if s > r then 20: Transfer a chore from b to ℓ along an mBB edge 21: else if s < r then 22: ∃ i ∈ N r ′ s.t. r ′ ∈ U and ∃ j ∈ x i s.t. j ∈ x 0 ℓ 23: Transfer j from i to ℓ 24: Transfer a chore from b to i 25: return (x, p) Lemma 6. Throughout the run of Algorithm 2, each group N r remains pEF1. Hence b and ℓ must lie in different groups. Once again, since we want to transfer chores away from b to reduce the pEF1-envy, and we want to obtain an fPO allocation, we only transfer chores along mBB edges. Doing so may require that we raise the prices of all chores belonging to certain agents in order to create new mBB edges to facilitate chore transfer. In our algorithm, all agents in a group undergo price-rise together. We call a group N r a raised group if its agents have undergone price-rise, else it is called an unraised group. The set U (Line 2) records the set of unraised components. We will use the terms time-step or iteration interchangeably to denote either a chore transfer or a price-rise step. We say 'at time-step t', to refer to the state of the algorithm just before the event at t happens. We denote by (x t , p t ) the allocation and price vector at time-step t. Let T be the first time-step that the current LS enters a raised group. Note that such an event may or may not happen. Our algorithm performs instructions in Lines 6-14 before T , and Lines 15-24 after T , as we describe below. Algorithm prior to T (Lines 6-14) We first record some properties of the algorithm prior to T . These observations follow directly from the algorithm. Lemma 7. Prior to T , the following hold: 1. Any transfer of chores only takes place directly from the big spender b to the least spender ℓ. Thus, an agent receives a chore only if she is a least spender, and an agent loses a chore only if she is a big spender. 2. An agent ceases to be a least spender only if she receives a chore. An agent ceases to be a big spender only if she loses a chore. 3. A group undergoes price-rise at t only if the group contains the big spender at t. We now show that: Lemma 8. If at any point in the run of Algorithm 2 prior to T , the big spender lies in a group which contains a former least spender, then the allocation is pEF1. This allows us to show that: Lemma 9. Prior to T , the following hold: (i) Let r be the number of price-rise steps until time-step t, where t < T . Then the raised groups are exactly N 1 , . . . , N r . Furthermore they underwent price-rise exactly once and in that order. (ii) For any chore j allocated to an agent in a raised group N r and any agent i in an unraised group N r ′ , where r ′ > r, j ∈ mBB i . (iii) For each r ′ ∈ [r], at the time of price-rise of N r ′ , no agent in N r ′ has either received or lost a chore since the initial allocation. Proof. We prove (i), (ii) and (iii) by induction. For r = 0, they are trivially true since there are no raised groups. Assume that at some time-step t, (i) groups N 1 , . . . , N r have undergone price-rise, once and in that order, for some r ≥ 1, and (ii) and (iii) hold. Note that our algorithm only raises the prices of chores owned by a group if the group contains the big spender at the time (Lemma 7). If the current BS is in a raised group, then the induction hypothesis ensures that there is an mBB edge from the LS (who is in an unraised group prior to T ) to chores owned by the BS. The algorithm therefore performs a direct chore transfers and no price-rise is necessary. If eventually the BS enters an unraised group, then a price-rise step is potentially necessary. Suppose b ∈ N r+1 is an agent who has received a chore prior to time-step t. If this happens then b must have been a former LS by Lemma 7. Then Lemma 8 shows that the allocation must already be pEF1. Hence we assume b has not received a new chore since the initial allocation in Line 1. Furthermore since b pEF1-envies the LS ℓ, it must be the case that ℓ ∈ N s where s ≥ r + 2. Lemma 3 shows that there is no mBB edge from ℓ to chores owned by b, hence a direct chore transfer is not possible and it is necessary for N r+1 to undergo price-rise. This shows (i). Now if an agent i ∈ N r+1 had previously received a chore, then i is a former LS at t. Then b ∈ N r+1 is in a group containing a former LS i, hence Lemma 8 shows that the allocation is pEF1. Similarly Lemma 7 shows that no agent in N r+1 can have lost a chore. This is because only BS agents lose chores. Prior to t, no agent of N r+1 can be the BS. Hence no agent in N r+1 has received or lost a chore since the initial allocation at the time N r+1 undergoes price-rise, thus showing (iii). The algorithm next raises the prices of all chores owned by N r+1 by a factor of k, and N r+1 becomes a raised group. Consider an agent i ∈ N r ′ for r ′ ≥ r + 2 and a chore allocated to an agent in N r+1 . Since the mBB ratio only changes upon a price-rise, the mBB ratio of i is 1 since N r ′ does not undergo a price-rise before N r+1 . Observe that since i / ∈ N r+1 , there is no alternating path from agents in N r+1 to i. Hence j / ∈ mBB i before the price-rise. Thus c ij /p t j > 1, showing c ij = k and p t j = 1. After the price-rise, we have that p t+1 j = k, and α i = c ij /p t+1 j . Thus, j ∈ mBB i after the price-rise, which shows (ii). To summarize the behavior of the Algorithm prior to T , we have argued in the above proof that if the allocation is not pEF1, we can always (i) transfer a chore directly from b to ℓ, or (ii) perform a price-rise on the group of b and then transfer a chore from b to ℓ. Further, we argue that the algorithm makes progress towards getting a pEF1 allocation. Lemma 10. Algorithm 2 performs at most poly(n, m) steps prior to T . Proof. Prior to T , the LS always remains in an unraised group. Chores are transferred away from agents who become big spenders in raised groups. Once an agent undergoes price-rise, she cannot gain any additional chores, since doing so would mean she is the LS in a raised group, which cannot happen prior to T . When the BS is in an unraised group, the group undergoes a price-rise. Thus, effectively, either agents in raised components only lose chores, or the BS 'climbs-down' in the group list N 1 , . . . , N R , while the LS remains below the BS. Since there are R ≤ n groups, and at most m chores allocated to raised groups, after poly(n, m) steps either of two events happen: (i) the LS and BS both belong to the same group, or (ii) the LS enters a raised group. In the former case, the allocation is pEF1 due to Lemma 6, and the algorithm terminates in poly(n, m) steps. We discuss the latter case in the next section. Thus, there are at most poly(n, m) steps prior to T . Algorithm after T (Lines 15-24) We now describe the algorithm after T , i.e., once the LS enters a raised group (Lines 15-24). We show that subsequent to T , as long as the allocation is not pEF1, we can either (i) transfer a chore directly from b to ℓ, or (ii) transfer chores via an alternating path containing 3 agents. We do not perform any price-rises subsequent to T . From Lemma 9, we know that at T , groups N 1 , . . . , N r have undergone price-rise, for some r ∈ [R]. Let N <r = r ′ <r N r ′ , and N >r = r ′ >r N r ′ . The allocation at T need not be pEF1, but we argue that it is already very close to being pEF1. Specifically, we show: Lemma 11. At T , agents in N <r are pEF1 towards others. Lemma 12. At T , agents in N >r are pEF1 towards others. The above two lemmas imply that if the BS is not in N r , then the allocation is pEF1. Let us assume that the allocation is not pEF1 at T . Let b, the BS at T be in N r , and ℓ, the LS at T be in N <r , since the LS is in a raised group at T . Suppose ℓ has never lost a chore. Let ℓ ∈ N r ′ , where r ′ < r, and t ′ be the time when N r ′ underwent price-rise. Let b ′ the BS at t ′ . Since the spending of the BS (up to removal of one chore) just after price-rises does not increase, we have: p T (x T b \ j) ≤ p t ′ (x t ′ b ′ \ j ′ ) ≤ p t ′ (x t ′ ℓ ) = p T (x T ℓ ), for some chores j ∈ x T b , j ′ ∈ x t ′ b ′ . The intermediate transition follows from the property that N r ′ is pEF1. This shows that the allocation is pEF1. On the other hand, suppose ℓ has lost at least one chore j prior to T . At T , j must be assigned to some unraised agent i (Lemma 9). Further, there is a chore j ′ ∈ x T b s.t. j ′ ∈ mBB i . Thus, b x − → j ′ mBB − −− → i x − → j mBB − −− → ℓ is an alternating path. The algorithm now transfers chores along this alternating path. Note that as long as ℓ does not own a chore that she initially owned, such a path is available, and such a transfer is possible. If not, then it is as if ℓ has never lost a chore, and in that case the previous argument shows that the allocation must be pEF1. Once again, doing such transfers may change the identities of the BS and LS, and we must continue the algorithm. We show that: Lemma 13. After T , the following are invariant: (i) Agents in N <r do not pEF1-envy any other agent. (ii) Agents in N >r do not pEF1-envy any other agent. (iii) Each group is pEF1. Just after T , the BS is in N r and the LS is in N <r . After a chore transfer, the identity of the LS or BS can change. If the BS enters either N <r or N >r , then using Lemma 13 the allocation would be pEF1. While the BS is in N r : (i) if the LS is in N r , the allocation would be pEF1, (ii) if the LS is in N >r , then we can transfer from BS to LS directly along an mBB edge (which exists due to Lemma 9), (iii) if the LS is in N <r , then we can transfer from the BS to LS via an alternating path with three agents as described above. Finally we argue termination in polynomial-time: Lemma 14. Algorithm 2 performs at most poly(n, m) steps after T and terminates with a pEF1 allocation. Proof. Call the difference between the spending (up to the removal of the biggest chore) of the big spender and the spending of the least spender as the spending gap. If the allocation is not pEF1, the spending gap is positive. After T , there are no price-rises, hence the spending gap weakly decreases. As long as the allocation is not pEF1, the BS must be in N r . Based on whether the LS is in N <r or N >r we can perform chore transfers which weakly decrease the spending gap. While the allocation is not pEF1, such a transfer is always possible. Further, each such transfer reduces the number of chores owned by agents in N r , and such agents do not receive any chores again. Hence there can only be poly(n, m) steps after T eventually terminating in a pEF1 allocation. Summarizing our EF1+fPO algorithm We summarize Algorithm 2. 1. Algorithm 2 first calls Algorithm 1 to partition agents into groups N 1 , . . . , N R with properties as in Lemma 3 to obtain an initial allocation. Lemma 4 shows this takes poly(n, m) steps. 2. When the current allocation (x, p) is not pEF1, the BS b pEF1-envies the LS ℓ. While there is an mBB edge from ℓ to a chore owned by b, we transfer a chore directly from b to ℓ. If not, in order to transfer along mBB edges, we may have to raise the prices of chores belonging to the group of b, creating raised groups. 3. Let T be the first time-step when the LS enters a raised group. Prior to T , while the allocation is not pEF1, the algorithm either performs a direct chore transfer from the BS to the LS, or performs price-rise on the group of b. Lemma 9 shows that the groups are raised exactly once and in order of N 1 , . . . , N R . Lemma 10 shows the algorithm runs for poly(n, m) steps before T . 4. Once the LS enters a raised group at T , there are no more price-rise steps. The algorithm then performs chore transfers from the BS to LS via alternating paths with at most 3 agents. Lemma 13 and Lemma 14 show that the algorithm performs at most poly(n, m) steps after T and terminates with a pEF1 allocation. 5. Finally, we note that allocation is always fPO, since (i) Algorithm 1 returns a market outcome which is fPO, and (ii) any transfer of chores happens along mBB edges. Examples We illustrate two examples of an execution of Algorithm 2 for a fair division instance, one in which the algorithm terminates before a time T where the LS enters a raised component, and one in which the algorithm terminates after such a time T . Algorithm terminates before T Consider the instance captured by the table of disutilities below. Chores j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 j 10 j 11 j 12 j 13 Agents a 1 1 1 1 1 1 k k k k k k k k a 2 k k k k k 1 1 1 1 k k k k a 3 k k k k k k k k k 1 k k k a 4 k k k k k k k k k k 1 k k a 5 k k k k k k k k k k k 1 k a 6 k k k k k k k k k k k k 1 Algorithm 2 begins by calling Algorithm 1 which returns the following initial allocation x. • x a 1 = {j 1 , j 2 , j 3 , j 4 , j 5 } • x a 2 = {j 6 , j 7 , j 8 , j 9 } • x a 3 = {j 10 } • x a 4 = {j 11 } • x a 5 = {j 12 } • x a 6 = {j 13 } Suppose k = 5. Note that here each agent lies in its own group. Since x is not pEF1, Algorithm 2 proceeds to raises the prices of the chores of a 1 by a factor of k. Then, a chore is transferred from a 1 to each of a 3 , a 4 , a 5 , and a 6 . This gives us the following pEF1 (and thus EF1) allocation x ′ . • x ′ a 1 = {j 5 } • x ′ a 2 = {j 6 , j 7 , j 8 , j 9 } • x ′ a 3 = {j 1 , j 10 } • x ′ a 4 = {j 2 , j 11 } • x ′ a 5 = {j 3 , j 12 } • x ′ a 6 = {j 4 , j 13 } Algorithm terminates after T We now consider a slight alteration of the previous instance, with an additional agent a 7 and additional chore j 14 . The table of disutilities is given below. Chores j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 j 10 j 11 j 12 j 13 j 14 Agents a 1 1 1 1 1 1 k k k k k k k k k a 2 k k k k k 1 1 1 1 k k k k k a 3 k k k k k k k k k 1 k k k k a 4 k k k k k k k k k k 1 k k k a 5 k k k k k k k k k k k 1 k k a 6 k k k k k k k k k k k k 1 k a 7 k k k k k k k k k k k k k 1 Again, Algorithm 2 first calls Algorithm 1 which returns the following initial allocation y. • y a 1 = {j 1 , j 2 , j 3 , j 4 , j 5 } • y a 2 = {j 6 , j 7 , j 8 , j 9 } • y a 3 = {j 10 } • y a 4 = {j 11 } • y a 5 = {j 12 } • y a 6 = {j 13 } • y a 7 = {j 14 } Again let k = 5. As before, Algorithm 2 raises the prices of the chores of a 1 by a factor of k and transfers a chore from a 1 to each of a 3 , a 4 , a 5 , and a 6 . Now, however, the allocation is not pEF1. The BS, a 2 , still pEF1-envies the LS a 7 . Thus, the prices of a 2 's chores are raised, and a chore is transferred from a 2 to a 7 . We now have the following allocation y ′ . • y ′ a 1 = {j 5 } • y ′ a 2 = {j 7 , j 8 , j 9 } • y ′ a 3 = {j 1 , j 10 } • y ′ a 4 = {j 2 , j 11 } • y ′ a 5 = {j 3 , j 12 } • y ′ a 6 = {j 4 , j 13 } • y ′ a 7 = {j 6 , j 14 } We see that y ′ is not pEF1 as a 2 pEF1-envies a 1 , and in fact a 1 is the LS. This marks the first time T in which the LS has entered a raised group. Then, to achieve pEF1, j 1 is returned to a 1 from a 3 , and j 7 is transferred from a 2 to a 3 as a replacement. This gives the pEF1 allocation y ′′ . • y ′′ a 1 = {j 1 , j 5 } • y ′′ a 2 = {j 8 , j 9 } • y ′′ a 3 = {j 7 , j 10 } • y ′′ a 4 = {j 2 , j 11 } • y ′′ a 5 = {j 3 , j 12 } • y ′′ a 6 = {j 4 , j 13 } • y ′′ a 7 = {j 6 , j 14 } EF+PO allocation of divisible chores In this section, we prove our second result: Theorem 2. Given a bivalued fair division instance (N, M, V ) of divisible chores with all c ij ∈ {a, b} for some a, b ∈ R + , an EF+fPO allocation can be computed in strongly polynomial-time. We prove Theorem 2 by showing that Algorithm 4 computes an EF+fPO allocation in strongly polynomial-time. First note that we can scale the costs of the chores so that they are in {1, k}. Describing the Approach Since we are interested in an EF+fPO allocation, we try to construct a (fractional) market outcome (x, p) which is price-envy-free (pEF) as opposed to pEF1. A market outcome (x, p) is said to be pEF if for all agents i, h ∈ N , p(x i ) ≥ p(x h ), i.e., all agents have equal spending. Similar to Lemma 1, we can show that if (x, p) is a pEF market outcome, then x is EF. Thus, our aim is to obtain a market outcome in which all agents have the same spending. As in Algorithm 2, we begin with an initial market outcome (x, p) on mBB, with certain desirable properties. Specifically, we partition the agents into groups N 1 , . . . , N R so that each group spends the same amount, each group is a partial component of some agent, and the peragent spending of the group N r decreases with r. Now, if the allocation is not pEF, it must be the case that any agent in the pool of biggest spenders B (here B = {b : b ∈ argmax i∈N p(x i )}) envies any agent in the pool of least spenders L (here L = {ℓ : ℓ ∈ argmin i∈N p(x i )}), i.e., p(x b ) > p(x ℓ ) for any b ∈ B and ℓ ∈ L. The natural idea following Algorithm 2 is to reduce the pEF-envy by draining chores uniformly from B to L along mBB edges. Further, if the pool B has chores which are not on mBB for L, then we raise the prices of certain chores in B by a factor of k, which creates the required mBB edges following Lemma 9. Initially, B equals N 1 , and L equals N R . At a certain point in the run of the algorithm, B equals i≤r N i and L equals i≥r ′ N r ′ , where r ≤ r ′ . When eventually the pools of biggest and least spenders coincide, the algorithm terminates with a pEF allocation. This approach is similar to Algorithm 2 and terminates in polynomial time. However, we observe a stronger property of this approach, which we use to design Algorithm 4. We claim that effectively, this approach is equivalent to raising the prices of all chores in the first r * groups, and then draining chores carefully, eventually resulting in a pEF allocation. Therefore, we can simply iterate over r = 1 to R − 1 ≤ n as a guess for a right value of r * . We now describe the parts of the algorithm and also prove the above argument. Obtaining Initial Groups Algorithm 3 obtains a partition of the set N of agents into groups N 1 , . . . , N R similar to Algorithm 1, with the strengthened condition that the partial components are price envy-free (pEF) rather than pEF1. The price vector p is set the same way as in Algorithm 1. Then all low-cost chores are priced at 1 and high-cost chores are priced at k. To allocate these chores, however, we use a balanced flow formulation [15]. This gives us an allocation with the key property that if agent i spends more than agent j, then there is no alternating path from i to j. Then, each b selected in Line 4 of Algorithm 3 is a biggest spender in its partial component, but also cannot spend more than any agent in its partial component. Thus, it must be that the partial component (i.e., group) is pEF. Algorithm 3 runs in polynomial time as the balanced flow allocation is achieved by at most m max-flow computations (see [15]) and the remaining structure is analogous to Algorithm 1. j = c ij for j ∈ x i . 2: R ← 1, N ′ ← N 3: while N ′ = ∅ do 4: b ← argmax i∈N ′ p(x i ) ⊲ Biggest Spender 5: C b ← Component of b ⊲ See Definition 2 6: H R ← C b ∩ (N ′ ∪ x N ′ ) ⊲ Partial component 7: N R ← H R ∩ N ⊲ Agent group 8: N ′ ← N ′ \ N R , R ← R + 1 9: return (x, p, {N r } r∈[R] ) Discussion on Algorithm 4 Algorithm 4 proceeds similarly to Algorithm 2 with two key differences: (1) Multiple groups are raised simultaneously rather than one at a time. (2) Transfer are made between a pool of biggest spenders and a pool of least spenders. That is, we transfer (fractions of) chores from all biggest spenders to all least spenders. We show that there always exists some r * , 1 ≤ r * ≤ R − 1, such that simultaneously raising the first r * groups allows us to reach a pEF allocation by performing chore transfers along mBB edges from biggest spenders to least spenders. Recalling that the partial components associated with each group N i are pEF, let s(N i ) denote the per agent spending of N i . By construction, we have that s(N i ) ≥ s(N j ) for i ≤ j. Suppose we do an initial raise of the first r groups. Let β ∈ [n] be such that the pool of biggest spenders B is the union of raised groups β i=1 N i . Similarly, let λ ∈ [n] be such that the pool of least spenders L is the union of unraised groups R i=λ N i . Algorithm 4 then proceeds to transfer chores out of agents in B at a uniform rate ρ B and into agents in L at a uniform rate ρ L , such that |B|ρ B = |L|ρ L . Then, s(B) will fall towards s(N β+1 ) while s(L) rises towards s(N λ−1 ). When we transfer enough chores so that s(B) reaches s(N β+1 ) or s(L) reaches s(N λ−1 ), whichever occurs first, we add the agents in that group to the respective pool. Note that in this process both B and L are only growing. Once an agent enters B or L, she never leaves. In addition, we have that s(B) is always decreasing while s(L) is always increasing, i.e., the spending gap s(B) − s(L) is weakly decreasing. When the spending gap is zero, i.e. s(B) = s(L), the allocation is pEF. However, to guarantee that transfers occur along mBB edges, we must have that all agents in B are in raised groups, and all agents in L are in unraised groups. Thus, we give the two conditions under which the algorithm cannot obtain a pEF allocation after raising the first r groups: -Condition 1. Raised groups have too much total spending. The spending of agents in L rises to spending level of N r before the spending of agents in B falls to the level of N r . That is, a raised group enters L. -Condition 2. Raised groups have too little total spending. The spending of agents in B falls to the level of N r+1 before the spending of agents in L rises to the level of N r+1 . That is, an unraised group enters B. for agents ℓ ∈ L do 20: Algorithm 4 Computing an EF+fPO allocation x b ← x b \ 1 |L| C 21: x ℓ ← x ℓ ∪ 1 |L| C 22: if (x, p) is pEF then 23: return (x, p) β ← β + 1 If neither of these conditions hold, then B always contains only raised groups, and L always holds only unraised groups. It is clear then that we can perform transfers until the spending gap between B and L closes entirely and we have a pEF allocation. Specifically, this is possible when the algorithm reaches a point where B contains all the raised groups and L contains all the unraised groups (see Line 12). We show that is guaranteed for some r * , 1 ≤ r * ≤ R − 1. Suppose we raise only the single group N 1 . It cannot be that Condition 1 holds here, as L rising to the level of N 1 implies that the allocation is pEF. It must be that we have Condition 2, or we would be done as the allocation will be pEF and the algorithm would halt. Now suppose inductively that we have raised the first r groups and found that the allocation is not pEF and Condition 1 is impossible for the first r groups, i.e., Condition 2 holds. Then, it must also be that Condition 1 is impossible if we raise the first r + 1 groups if the allocation is still not pEF. Let σ = s(N r+1 ) be the per-agent spending of N r+1 just before N r+1 undergoes price-rise. Since raising r groups results in Condition 2, we know that s(B) falls to σ before s(L) rises to σ. Surely then, in raising r + 1 groups, s(B) falls to σ · k before s(L) rises to σ · k, as σ · k > σ. This shows Condition 1 cannot hold after raising the first r + 1 groups, given that the allocation is not yet pEF. Finally, suppose that we have raised the first R − 1 groups, i.e., all but the last group, and the allocation is not pEF. Then, Condition 2 cannot hold, as s(B) falling to the level of s(N R ) implies that the allocation is pEF. Thus, if the allocation is not pEF, Condition 1 must hold, but this is impossible by our inductive argument. Hence it must be that for some r * , 1 ≤ r * ≤ R − 1, a pEF allocation was obtained. Following this logic, Algorithm 4 checks each possible r until r * is found and a pEF allocation is obtained. Further, since all chore transfers are along mBB, the allocation remains fPO throughout. Thus the algorithm computes a pEF+fPO allocation. For each r ∈ [R], the algorithm performs at most O(n 2 ) transfers before adding an agent group to either B or L. A group cannot be added to both B and L, nor can it leave B or L after being added. Thus, since R ≤ n, the total number of transfers performed to check if some r is correct is O(n 3 ), so checking all possible r takes O(n 4 ) transfers. Hence, Algorithm 4 terminates in strongly polynomial-time, proving Theorem 2. Pseudocode Description We describe the details of the pseudocode of Algorithm 4. As described in Section 4.1, Algorithm 4 iteratively searches for a right value r * of the number of groups to raise. This corresponds to the for loop starting in Line 1. Line 2 calls Algorithm 3 to obtain initial groups as described in Section 4.2. Lines 4-5 initialize the pool of biggest spenders B to N 1 and set the index of the lowest group in B, β, to 1. Lines 6-7 initialize the pool of least spenders L to N R and set the index of the lowest group in L, λ, to R. The loop starting in Line 8 is active while the pool of biggest spenders doesn't contain an unraised component (β ≤ r) and the pool of least spenders doesn't contain a raised component (λ ≥ r + 1). We then want to decide how much spending (chores) to drain from B to L. As described in Section 4.3, either the per-agent spending of B can climb down to the per-agent spending of N β+1 , or the per-agent spending of L can climb up to the per-agent spending of N λ−1 . Line 9 computes the total amount of spending d B to be transferred in the former case and Line 10 computes the total amount of spending d L to be transferred in the latter case. The loop starting in Line 11 considers agents b ∈ B one by one and computes an appropriately priced bundle of chores that should be taken away from them. If B contains all the raised groups and L contains all the unraised groups, i.e., if β = r and λ = r + 1 (Line 12), then we are a step away from obtaining a pEF allocation. We compute the average spending of all agents, q, in Line 13. This is the amount to which we want to bring every agent's spending. Hence, for each agent b in consideration, we compute a subset C ⊆ x b with value s(B) − q (Line 14), and distribute it evenly to all agents in L (Lines 20-21). Here, 1 |L| C denotes a (fractional) bundle of chores C ′ where C ′ ij = 1 |L| C ij for agent i and chore j (revisit Section 2 for the definition of a fractional allocation). Otherwise, if d B ≥ d L , then the per-agent spending of L reaches the per-agent spending of N λ−1 before the per-agent spending of B reaches the per-agent spending of N β+1 . Hence each agent ℓ ∈ L must obtain an additional spending of d L |L| from agents in B. We do this by computing a subset C of x b of value d L |B| from every b ∈ B (Line 16) and giving a 1 |L| -sized fraction of C to ℓ (Line 20-21). Thus, ℓ obtains a spending 1 |L| · d L |B| from every agent in B, which totals to d L |L| , the required amount. Now, we update the pool of least spenders to include N λ−1 , and update λ (Lines [25][26]. Analogously, when d B < d L , the per-agent spending of B reaches the per-agent spending of N β+1 before the per-agent spending of L reaches the per-agent spending of N λ−1 . Hence, each agent b ∈ B loses an amount equalling d B |B| . For each b, we compute a set C with spending d B |B| (Line 18), which is then transferred equally among all agents in L (Line 20-21). We also update the pool of biggest spenders to include N β+1 , and update β (Lines [28][29]. We terminate if the allocation is pEF (Lines 22-23). If not, then we continue our search for a right value r * by incrementing r (Line 1). The arguments of Section 4.3 show that for some 1 ≤ r ≤ R − 1, Algorithm 4 will terminate with a pEF+fPO allocation. Discussion In this paper, we presented a strongly polynomial-time algorithm for computing an EF1+fPO allocation of indivisible chores to agents with bivalued preferences, constituting the first non-trivial result for the EF1+PO problem for chores. Our algorithm is novel and relies on several involved arguments. Given that the general case is a challenging open problem, we believe extending our algorithm and its analysis to the class of k-ary chores is an interesting and natural next step. Another interesting question is whether we can compute an EFX allocation in this setting. We also presented a strongly polynomial-time for computing an EF+fPO allocation of divisible bivalued chores. Computing an EF+fPO allocation of divisible k-ary chores in polynomial-time is also a compelling direction for future work. A Missing Proofs from Section 3 Proof. Consider the time needed to make the component of the big spender pEF1. We show that this takes time poly(n, m). Note that when the identity of the big spender b does not change, a transfer of chore j strictly increases the level of j. Thus, while the big spender b is unchanging, chore j can be transferred at most n times. Thus, the total number of chore transfers with the same big spender is at most mn. We now bound the total number of times the identity of the big spender may change. Let x t be the allocation at iteration t, and p be the price vector. Furthermore, let j (i,t) denote the highest-priced chore of agent i at time t. We claim that if an agent b ceases to be the BS at iteration t and subsequently becomes the BS again at t ′ , then her spending (up to removal of one chore) strictly goes down, i.e., p( t) ). Note that b must lose some chore at time t to cease being the BS. Then, since the spending of b up to one chore at time t ′ is weakly greater than at time t, b must also have gained a chore between t and t ′ . Let the most recent time b gained a chore be t ′′ ∈ (t, t ′ ), and let b ′ (resp. b ′′ ) be the BS at time t ′ (resp. t ′′ ). Note that the spending of the big spender up to one chore weakly decreases with each iteration, so p( t) ). Since b receives a chore j at time t ′′ , b must be envied by b ′′ at time t ′′ (Line 6). This gives us p( t) ), thus proving the claim. Thus, if an agent i ceases to be the BS at time t and later becomes the BS again at t ′ , the spending of i is strictly smaller at t ′ than at t. It follows that i has strictly smaller disutility at t ′ since the prices have not changed. In any allocation x, if s i (resp. t i ) is the number of chores in x i with cost b (resp. a) by i, the disutility of i is u i = s i + t i k. Since 0 ≤ s i , t i ≤ m, the number of different disutility values i can have in any allocation is at most O(m 2 ). Thus, for any agent i, the number of times her disutility decreases is at most O(m 2 ). Thus, an agent can become the big spender only O(m 2 ) times. Hence, the number of identity changes of the BS is at most O(m 2 n). x t ′ b \ j (b,t ′ ) ) < p(x t b \ j (b,x t ′ b ′ \ j (b ′ ,t ′ ) ) ≤ p(x t ′′ b ′′ \ j (b ′′ ,t ′′ ) ) ≤ p(x t b \ j (b,x t ′′ b ) < p(x t ′′ b ′′ \j (b ′′ ,t ′′ ) ). As j is the last chore gained by b, we have p(x t ′ b \ j (b,t ′ ) ) ≤ p(x t ′ b \ j) ≤ p(x t ′′ b ) < p(x t ′′ b ′′ \ j (b ′′ ,t ′′ ) ) ≤ p(x t b \ j (b, Thus, the time needed to make the component of the big spender pEF1 is O(m 3 n 2 ). Since we need construct at most n pEF1 components, the total time needed is O(m 3 n 3 ). In conclusion, Algorithm 1 terminates in poly(n, m) time. Lemma 5. The spending of the least spender does not decrease in the run of Algorithm 2. Proof. The spending of the LS cannot decrease during a price-rise step. Suppose t is a chore transfer step and s is the spending of the LS at t. If t ≤ T , transfers are from the BS b to the LS ℓ. Further, this happens only if b pEF1-envies ℓ. Thus, b cannot spend less than s after the chore transfer due to the pEF1 condition. If t > T , chore transfers are along alternating paths b → j ′ → i → j → ℓ, where j ′ ∈ x t b ∩ mBB i and j ∈ x t i ∩ mBB ℓ , and b pEF1-envies ℓ. As discussed in Section 3.4, the spending of i does not change during this chore transfer, hence i cannot spend less than s after the transfer. As before, b cannot spend less than s after the transfer due to the pEF1 condition. Thus, the spending of the least spender does not decrease in the run of Algorithm 2. Lemma 6. Throughout the run of Algorithm 2, each group N r remains pEF1. Proof. We prove this lemma in two parts: before T , and after T , the latter is proved in Lemma 13 (iii). For the run of the algorithm prior to T , we proceed by induction. As the base case, we know from Lemma 3 that the groups {N r } r∈ [R] are each pEF1 at the end of MakeInitGroups. As the inductive hypothesis, assume that each group is pEF1 at time-step t. After the event at t, a group can cease to be pEF1 potentially because of a (i) chore transfer, or (ii) price-rise. 1. Chore transfer step. From the algorithm it is clear that any chore transfer prior to T happens from the big spender b to ℓ. Let N r be a group. We consider four cases: • Both b and ℓ lie outside N r , then the bundles of agents in N r are unaffected, and the allocation continues to remain pEF1 after the transfer. • Both b and ℓ lie in N r . However, if this happens then the algorithm terminates with a pEF1 allocation, since N r is pEF1 at time-step t by assumption. • b lies in N r and ℓ does not. Let i = b be an agent in N r and (x, p) be the allocation at t. Since N r is pEF1, we have p(x i ) ≥ p(x b \ j) for some j ∈ x b . Since N r is pEF1, b does not pEF1-envy i at t. Since b loses a chore at t, b does not pEF1-envy i after the transfer. Further, since b is the big spender in H r , p( x i \ j ′ ) ≤ p(x b \ j) for some j ′ ∈ x i . Thus, p(x ′ i \ j ′ ) ≤ p(x ′ b ) , where x ′ is the allocation after the transfer. Hence, i does not pEF1-envy b after the transfer. Thus, N r remains pEF1 after the transfer. • ℓ lies in N r and b does not. Let i = ℓ be an agent in N r and (x, p) be the allocation at t. Since N r is pEF1, we have p(x i \ j) ≤ p(x ℓ ) for some j ∈ x i . Since ℓ gains a chore j ′ at time t, the allocation x ′ after the event at t satisfies p(x ′ ℓ \ j ′ ) = p(x ℓ ) ≤ p(x i ) = p(x ′ i ). Hence ℓ does not pEF1-envy i after the transfer. Further, since ℓ gains an additional chore and N r is pEF1 at t, i will not pEF1-envy ℓ after the transfer. Thus, N r remains pEF1 after the transfer. 2. Price-rise step. Suppose N r undergoes a price-rise at t. At t, N r is pEF1 by assumption. Since the prices of all chores in N r are raised by the same factor, it continues to be pEF1 after the price-rise. In both cases, we see that N r continues to remain pEF1 after the event at t. We conclude by induction that each group remains pEF1 throughout the run of the algorithm. We denote by (x t , p t ) the allocation and price vector at time-step t. Lemma 8. If at any point in the run of Algorithm 2 prior to T , the big spender lies in a group which contains a former least spender, then the allocation is pEF1. Proof. Let t be the first time-step when the big spender (BS) b lies in a group N r which also contains an agent i, who was a least spender (LS) most recently at time-step t ′ ≤ t. If t ′ = t, then N r contains both the LS and the BS, in which case the allocation must be pEF1 due to Lemma 6. Thus, t ′ < t. Suppose N r underwent a price-rise at a time-step t ′′ between t ′ and t. Then the BS h at t ′′ must be in N r . If that happens, then at t ′′ , the big spender lies in a group which also contains a former LS i. Since t ′′ < t, this contradicts the definition of t. Thus, N r does not undergo a price-rise between t ′ and t. We next show that: Claim 1. b cannot be a former least spender, unless the allocation is already pEF1. Proof. For sake of contradiction, assume b was an LS most-recently at time-step t 0 < t. From the properties of the algorithm (Lemma 7), b ceased to be an LS at t 0 because she received a chore j. Subsequent to this, b cannot receive another chore, since t 0 is the most recent time-step when b was an LS, and only an LS can receive a chore (Lemma 7). Thus we must have x t b ⊆ x t 0 b ∪ {j}. Further, b ∈ H r does not undergo price-rise between t 0 and t, otherwise it would contradict the definition of t. Hence we obtain: p t (x t b \ j) ≤ p t 0 (x t 0 b ). Let ℓ be the LS at t. From Lemma 5, we know p t (x t ℓ ) ≥ p t 0 (x t 0 b ), since b was the LS at t 0 . Putting it together, we get p t (x t ℓ ) ≥ p t (x t b \ j), i.e., (x, p) is pEF1 at t, since b is the BS at t. With Claim 1 in hand, we can now prove Lemma 8. Since b is not a former LS, b has not received a chore. Also, b has not undergone a price-rise. Hence: p t (x t b \ j) ≤ p t ′ (x t ′ b \ j ′ ) ≤ p t ′ (x t ′ i ), for some j ∈ x t b and j ′ ∈ x t ′ b , since H r is pEF1 at t ′ by Lemma 6. Finally, by Lemma 5, we get p t ′ (x t ′ i ) ≤ p t (x t ℓ ), where ℓ is the LS at t. Putting these together, we get: p t (x t ℓ ) ≥ p t (x t b \ j), showing that (x, p) is pEF1 at t. Lemma 11. At T , agents in N <r are pEF1 towards others. Proof. Let t ≤ T be the time-step at which N r was raised, i.e., the most recent price-rise. Let h be the big spender at t. We know that h ∈ H r . Let i be an arbitrary agent in N <r . Since i is in a raised group N r ′ , where r ′ < r, i cannot have gained a chore before T . Let t ′ < t be the time at which N r ′ was raised. Then: • if i gained a chore before t ′ , then N r ′ cannot have been undisturbed at t ′ . • if i gained a chore at time t ′′ ∈ (t ′ , T ), then i had to have been the LS at t ′′ (Lemma 7). Hence at t ′′ < T , the LS was in a raised group. This contradicts the definition of T . Hence i can only lose chores. Just after t ′ , all chores of i are priced at k. Hence the spending of i after t ′ is always a multiple of k. Further, until t, h is undisturbed, and contains only chores of price 1. Let c = p t (x t h ) be the number of (price 1) chores initially assigned to h. Note that N r ′ must contain an agent who initially has at least c price 1 chores, since N r ′ undergoes price-rise before N r . Since each group is pEF1, i must have at least c − 1 price 1 chores. Hence just after t ′ , we have p t ′ (x t ′ i ) ≥ k(c − 1). After t ′ , the spending of i potentially drops because of transfer of chores of price k. At t, since h is the BS, we know: p t (x t i ) − k ≤ p t (x t h ) − 1. Since the spending of i is a multiple of k, we have: p t (x t i ) ≤ k + k · p t (x t h ) − 1 k = k + k · c − 1 k Just before the time before t that i lost her last chore, since h is not the BS, the spending of i has to be at least k + (c − 1). Hence the spending of i at t is gk where: • if k|(c − 1), then g ∈ {(c − 1)/k, 1 + (c − 1)/k} • otherwise, g = 1 + ⌊(c − 1)/k⌋ Since i is an arbitrary agent in N <r , we see that every agent in N <r spends almost the same amount (up to k). Hence there is no pEF1-envy among agents of N <r at t. From t to T , agents in N <r only lose chores, and the gap between the spendings of any two agents cannot ever exceed k. Hence at T , agents in N <r do not pEF1-envy each other. Since the LS is in N <r at T , we conclude that agents in N <r do not pEF1-envy any other agent. We see from the previous analysis that p t (x t i ) ≥ c − 1, where c = p t (x t h ). Lemma 12. At T , agents in N >r are pEF1 towards others. Proof. Let t ≤ T be the time-step at which N r was raised, i.e., the most recent price-rise. Let i be an arbitrary agent in N >r . Since i is in an unraised group, i cannot have undergone a price-rise or ever lost a chore. Suppose t ′ is the last time i gained a chore j. Then at t ′ , i is a LS. Thus, x t ′ i ∪ {j} = x T i , and p T (x T i \ j) = p t ′ (x t ′ i ). Since the spending of the LS weakly increases, p t ′ (x t ′ i ) ≤ p T (x T ℓ ), where ℓ is the LS at T . Hence we obtain: p T (x T i \ j) ≤ p T (x T ℓ ), showing that i does not pEF1-envy the LS. Suppose i did not ever gain a chore. Then, the bundle of i has been undisturbed from the start. Since h is the BS at t, we have p T (x T i \ j ′ ) = p t (x t i \ j ′ ) ≤ c − 1, for some j ′ ∈ x T i . We know that ℓ spends at least c − 1 at t. We know ℓ does not gain a chore. We show that it is not possible for ℓ to lose a chore after t, unless the allocation is already pEF1. Suppose ℓ loses a chore j for the last time at t ′ ∈ (t, T ). At t ′ , ℓ was the BS. Since there are no price-rises after t, the spending of the BS up to the removal of one chore weakly decreases. Hence p T (x T b \ j ′ ) ≤ p t ′ (x t ′ ℓ \ j), where b is the BS at T . Since ℓ doesn't lose a chore after t ′ , p t ′ (x t ′ ℓ \ j) = p T (x T ℓ ). Thus the allocation is already pEF1. If ℓ did not lose any chore after t, then p T (x T ℓ ) = p t (x t ℓ ) ≥ c − 1. Since p T (x T i \ j ′ ) ≤ c − 1, for some j ′ ∈ x T i if i has not gained a chore, we see that in this case too i does not pEF1-envy the LS at T . Since i was an arbitrary agent in N >r , we conclude that no agent in N >r pEF1-envies any other agent. Lemma 13. After T , the following are invariant: (i) Agents in N <r do not pEF1-envy any other agent. (ii) Agents in N >r do not pEF1-envy any other agent. (iii) Each group is pEF1. Proof. Recall that Lemmas 11 and 12 shows that (i) and (ii) hold just after T , and Lemma 6 shows (iii) holds just after T . With this as our base case, we proceed by induction. Assume (i)-(iii) hold at some time step t > T . The event at t can either be: • A direct chore transfer from b ∈ N r to ℓ ∈ N >r . Such a transfer does not newly cause any agent to pEF1-envy ℓ since ℓ gains a chore, nor any agent to pEF1-envy b since b is the BS. Also, b does not newly pEF1-envy any agent since she loses a chore, and neither does ℓ newly pEF1-envy any agent since she is the LS. • A chore transfer of the form b → j ′ → i → j → ℓ, where i ∈ N >r , j ′ ∈ x t b , j ∈ x t i , and j is a chore that ℓ lost prior to T . Hence p t j = p t j ′ = k. Hence, p t+1 (x t+1 i ) = p t (x t i ). Thus, this step effectively amounts to transferring a chore from b to ℓ. As argued in the previous paragraph, this does not cause any agent to newly start pEF1-envying another agent. Hence, by induction, (i)-(iii) hold after t as well. Algorithm 1 1MakeInitGroups Input: Fair division instance (N, M, C) with c ij ∈ {1, k} Output: Integral alloc. x, prices p, agent groups {N r } r∈[R] Lemma 4 . 4Algorithm 1 terminates in time poly(n, m). Algorithm 3 3MakeInitGroupsDiv Input: Fair division instance (N, M, C) with c ij ∈ {1, k} Output: Fractional alloc. x, prices p, agent groups {N r } r∈[R]1: (x, p) ← initial cost minimizing balanced flow fractional market allocation, where p CC Input: Fair division instance (N, M, C) with c ij ∈ {1, k} Output: A fractional allocation x 1: for r ∈ [R − 1] do 2: (x, p, {N r } r∈[R] ) ← MakeInitGroupsDiv(N, M, V) 3:Raise prices of chores of agents in{N i } i∈[r] ← subset of x b s.t. p(C) = s(B) − q 15: else if d B ≥ d L then 16: C ← subset of x b s.t. p(C) ← subset of x b s.t. p(C) = d B |B| 19: Lemma 4 . 4Algorithm 1 terminates in time poly(n, m). In fact, we can assume the two values are positive, since one of them being zero implies the setting is binary, in which case computing an EF1+PO allocation is trivial by first assigning chores to agents which have 0 cost for them, and then allocating almost equal number of chores of cost 1 to everyone. We refer to payments as prices for sake of similarity with the Fisher market model in the goods case.3 This is actually the earning of agent i, but we refer to earning as spending for sake of similarity with the Fisher market model in the goods case. Note that although our algorithm only maintains the allocation x and prices p, the associated Fisher market instance is always implicitly present by setting ei = p(xi) as in the proof of Lemma 1 Aris Filos-Ratsikas, Alexandros Hollender, and Alexandros A. Voudouris. Maximum Nash welfare and other stories about EFX. Georgios Amanatidis, Georgios Birmpas, Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, (IJCAI). the Twenty-Ninth International Joint Conference on Artificial Intelligence, (IJCAI)Georgios Amanatidis, Georgios Birmpas, Aris Filos-Ratsikas, Alexandros Hollender, and Alexandros A. Voudouris. Maximum Nash welfare and other stories about EFX. In Proceed- ings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, (IJCAI), pages 24-30, 2020. The Hylland-Zeckhauser rule under bi-valued utilities. CoRR, abs. Haris Aziz, Haris Aziz. The Hylland-Zeckhauser rule under bi-valued utilities. CoRR, abs/2006.15747, 2020. Efficient reallocation under additive and responsive preferences. Haris Aziz, Péter Biró, Jérôme Lang, Julien Lesca, Jérôme Monnot, Theoretical Computer Science. 790Haris Aziz, Péter Biró, Jérôme Lang, Julien Lesca, and Jérôme Monnot. Efficient reallocation under additive and responsive preferences. Theoretical Computer Science, 790:1 -15, 2019. Fair and truthful mechanisms for dichotomous valuations. Moshe Babaioff, Tomer Ezra, Uriel Feige, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence35Moshe Babaioff, Tomer Ezra, and Uriel Feige. Fair and truthful mechanisms for dichotomous valuations. Proceedings of the AAAI Conference on Artificial Intelligence, 35(6):5119-5126, May 2021. URL: https://ojs.aaai.org/index.php/AAAI/article/view/16647. Finding fair and efficient allocations. Siddharth Barman, Rohit Sanath Kumar Krishnamurthy, Vaish, Proceedings of the 19th ACM Conference on Economics and Computation (EC). the 19th ACM Conference on Economics and Computation (EC)Siddharth Barman, Sanath Kumar Krishnamurthy, and Rohit Vaish. Finding fair and efficient allocations. In Proceedings of the 19th ACM Conference on Economics and Computation (EC), pages 557-574, 2018. Greedy algorithms for maximizing Nash social welfare. Siddharth Barman, Rohit Sanath Kumar Krishnamurthy, Vaish, Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems (AAMAS). the 17th International Conference on Autonomous Agents and MultiAgent Systems (AAMAS)Siddharth Barman, Sanath Kumar Krishnamurthy, and Rohit Vaish. Greedy algorithms for maximizing Nash social welfare. In Proceedings of the 17th International Conference on Au- tonomous Agents and MultiAgent Systems (AAMAS), page 7-13, 2018. On approximate envy-freeness for indivisible chores and mixed resources. Umang Bhaskar, A R Sricharan, Rohit Vaish, arXiv:2012.06788CoRR. Umang Bhaskar, A. R. Sricharan, and Rohit Vaish. On approximate envy-freeness for indivisible chores and mixed resources. CoRR, abs/2012.06788, 2020. URL: https://arxiv.org/abs/2012.06788, arXiv:2012.06788. Complexity of efficient and envyfree resource allocation: Few agents, resources, or utility levels. Bernhard Bliem, Robert Bredereck, Rolf Niedermeier, Proceedings of the 25th International Joint Conference on Artificial Intelligence (IJCAI). the 25th International Joint Conference on Artificial Intelligence (IJCAI)Bernhard Bliem, Robert Bredereck, and Rolf Niedermeier. Complexity of efficient and envy- free resource allocation: Few agents, resources, or utility levels. In Proceedings of the 25th International Joint Conference on Artificial Intelligence (IJCAI), page 102-108, 2016. Random matching under dichotomous preferences. Anna Bogomolnaia, Herve Moulin, Econometrica. 721Anna Bogomolnaia and Herve Moulin. Random matching under dichotomous preferences. Econometrica, 72(1):257-279, 2004. Competitive division of a mixed manna. Anna Bogomolnaia, Hervé Moulin, Fedor Sandomirskiy, Elena Yanovskaya, Econometrica. 856Anna Bogomolnaia, Hervé Moulin, Fedor Sandomirskiy, and Elena Yanovskaya. Com- petitive division of a mixed manna. Econometrica, 85(6):1847-1871, 2017. URL: http://www.jstor.org/stable/44955184. Algorithms for competitive division of chores. CoRR. Simina Brânzei, Fedor Sandomirskiy, arXiv:1907.01766Simina Brânzei and Fedor Sandomirskiy. Algorithms for competitive division of chores. CoRR, abs/1907.01766, 2019. URL: http://arxiv.org/abs/1907.01766, arXiv:1907.01766. The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Eric Budish, Journal of Political Economy. 1196Eric Budish. The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy, 119(6):1061 -1103, 2011. On (1, e)-restricted assignment makespan minimization. Deeparnab Chakrabarty, Sanjeev Khanna, Shi Li, Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Deeparnab Chakrabarty, Sanjeev Khanna, and Shi Li. On (1, e)-restricted assignment makespan minimization. In Proceedings of the 26th Annual ACM-SIAM Symposium on Dis- crete Algorithms (SODA), page 1087-1101, 2015. Competitive allocation of a mixed manna. Jugal Bhaskar Ray Chaudhury, Peter Garg, Ruta Mcglaughlin, Mehta, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference. the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual ConferenceBhaskar Ray Chaudhury, Jugal Garg, Peter McGlaughlin, and Ruta Mehta. Competitive allocation of a mixed manna. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 -13, 2021, pages 1405-1424, 2021. Market equilibrium via a primal-dual algorithm for a convex program. R Nikhil, Christos H Devanur, Amin Papadimitriou, Vijay V Saberi, Vazirani, J. ACM. 555Nikhil R. Devanur, Christos H. Papadimitriou, Amin Saberi, and Vijay V. Vazirani. Market equilibrium via a primal-dual algorithm for a convex program. J. ACM, 55(5), 2008. Resource allocation and the public sector. D K Foley, Yale Economic Essays. 71D.K. Foley. Resource allocation and the public sector. Yale Economic Essays, 7(1):45-98, 1967. Equitable allocations of indivisible goods. Rupert Freeman, Sujoy Sikdar, Rohit Vaish, Lirong Xia, Proceedings of the 28th International Joint Conference on Artificial Intelligence (IJCAI). the 28th International Joint Conference on Artificial Intelligence (IJCAI)Rupert Freeman, Sujoy Sikdar, Rohit Vaish, and Lirong Xia. Equitable allocations of indivis- ible goods. In Proceedings of the 28th International Joint Conference on Artificial Intelligence (IJCAI), pages 280-286, 2019. Equitable allocations of indivisible chores. Rupert Freeman, Sujoy Sikdar, Rohit Vaish, Lirong Xia, Proceedings of the 19th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS '20. the 19th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS '20Rupert Freeman, Sujoy Sikdar, Rohit Vaish, and Lirong Xia. Equitable allocations of indivis- ible chores. In Proceedings of the 19th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS '20, page 384-392, 2020. When dividing mixed manna is easier than dividing goods: Competitive equilibria with a constant number of chores. Jugal Garg, Martin Hoefer, Peter Mcglaughlin, Marco Schmalhofer, Algorithmic Game Theory -14th International Symposium, SAGT 2021. Aarhus, Denmark12885Jugal Garg, Martin Hoefer, Peter McGlaughlin, and Marco Schmalhofer. When dividing mixed manna is easier than dividing goods: Competitive equilibria with a constant number of chores. In Algorithmic Game Theory -14th International Symposium, SAGT 2021, Aarhus, Denmark, September 21-24, 2021, Proceedings, volume 12885, pages 329-344, 2021. Computing competitive equilibria with mixed manna. Jugal Garg, Peter Mcglaughlin, Proceedings of the 19th International Conference on Autonomous Agents and Multiagent Systems, AAMAS '20. the 19th International Conference on Autonomous Agents and Multiagent Systems, AAMAS '20Auckland, New ZealandInternational Foundation for Autonomous Agents and Multiagent SystemsJugal Garg and Peter McGlaughlin. Computing competitive equilibria with mixed manna. In Proceedings of the 19th International Conference on Autonomous Agents and Multiagent Systems, AAMAS '20, Auckland, New Zealand, May 9-13, 2020, pages 420-428. International Foundation for Autonomous Agents and Multiagent Systems, 2020. Computing fair and efficient allocations with few utility values. Jugal Garg, Aniket Murhekar, Proc. 14th Symp. Algorithmic Game Theory (SAGT). 14th Symp. Algorithmic Game Theory (SAGT)2021Jugal Garg and Aniket Murhekar. Computing fair and efficient allocations with few utility values. In In Proc. 14th Symp. Algorithmic Game Theory (SAGT), 2021. Assigning papers to referees. Naveen Garg, Telikepalli Kavitha, Amit Kumar, Kurt Mehlhorn, Julián Mestre, 58Algorithmica, v.x, x-xNaveen Garg, Telikepalli Kavitha, Amit Kumar, Kurt Mehlhorn, and Julián Mestre. Assigning papers to referees. Algorithmica, v.x, x-x (2009), 58, 01 2009. Max-min fair allocation of indivisible goods. D Golovin, CMU-CS-05-144Technical ReportD. Golovin. Max-min fair allocation of indivisible goods. Technical Report, CMU-CS-05-144, 2005. On approximately fair allocations of indivisible goods. R J Lipton, E Markakis, E Mossel, A Saberi, Proceedings of the 5th ACM Conference on Electronic Commerce (EC). the 5th ACM Conference on Electronic Commerce (EC)R. J. Lipton, E. Markakis, E. Mossel, and A. Saberi. On approximately fair allocations of indivisible goods. In Proceedings of the 5th ACM Conference on Electronic Commerce (EC), pages 125-131, 2004. A Mas-Colell, M D Whinston, J R Green, Microeconomic Theory. Oxford University PressA. Mas-Colell, M.D. Whinston, and J.R. Green. Microeconomic Theory. Oxford University Press, 1995. On fair and efficient allocations of indivisible goods. Aniket Murhekar, Jugal Garg, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence35Aniket Murhekar and Jugal Garg. On fair and efficient allocations of indivisible goods. Pro- ceedings of the AAAI Conference on Artificial Intelligence, 35(6):5595-5602, May 2021. Algorithmic Game Theory. N Nisan, T Roughgarden, E Tardos, V V Vazirani, Cambridge University PressN. Nisan, T. Roughgarden, E. Tardos, and V.V. Vazirani. Algorithmic Game Theory. Cam- bridge University Press, 2007. Improved algorithms for computing Fisher's market clearing prices: Computing fisher's market clearing prices. James B Orlin, James B. Orlin. Improved algorithms for computing Fisher's market clearing prices: Comput- ing fisher's market clearing prices. 2010. Sur la division pragmatique. H Steinhaus, Econometrica. 171H. Steinhaus. Sur la division pragmatique. Econometrica, 17(1):315-319, 1949. Equity, envy, and efficiency. Hal R Varian, Journal of Economic Theory. 91Hal R Varian. Equity, envy, and efficiency. Journal of Economic Theory, 9(1):63 -91, 1974. Computational complexity of the Hylland-Zeckhauser scheme for one-sided matching markets. V Vijay, Mihalis Vazirani, Yannakakis, Proceedings of the 12th Innovations in Theoretical Computer Science Conference. the 12th Innovations in Theoretical Computer Science Conference2021Vijay V. Vazirani and Mihalis Yannakakis. Computational complexity of the Hylland- Zeckhauser scheme for one-sided matching markets. In Proceedings of the 12th Innovations in Theoretical Computer Science Conference, (ITCS), 2021. A strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives. A László, Végh, SIAM J. Comput. 455László A. Végh. A strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives. SIAM J. Comput., 45(5):1729-1761, 2016. A polynomial-time approximation scheme for maximizing the minimum machine completion time. Gerhard J Woeginger, Operations Research Letters. 204Gerhard J. Woeginger. A polynomial-time approximation scheme for maximizing the minimum machine completion time. Operations Research Letters, 20(4):149 -154, 1997.
[]
[ "Unsupervised Domain Adaptive Fundus Image Segmentation with Few Labeled Source Data", "Unsupervised Domain Adaptive Fundus Image Segmentation with Few Labeled Source Data" ]
[ "Qianbi Yu \nSchool of Computer Science\nUniversity of Sydney Sydney\nAustralia\n", "Dongnan Liu [email protected] \nSchool of Computer Science\nUniversity of Sydney Sydney\nAustralia\n", "Chaoyi Zhang \nSchool of Computer Science\nUniversity of Sydney Sydney\nAustralia\n", "Xinwen Zhang \nSchool of Computer Science\nUniversity of Sydney Sydney\nAustralia\n", "Weidong Cai \nSchool of Computer Science\nUniversity of Sydney Sydney\nAustralia\n" ]
[ "School of Computer Science\nUniversity of Sydney Sydney\nAustralia", "School of Computer Science\nUniversity of Sydney Sydney\nAustralia", "School of Computer Science\nUniversity of Sydney Sydney\nAustralia", "School of Computer Science\nUniversity of Sydney Sydney\nAustralia", "School of Computer Science\nUniversity of Sydney Sydney\nAustralia" ]
[]
Deep learning-based segmentation methods have been widely employed for automatic glaucoma diagnosis and prognosis. In practice, fundus images obtained by different fundus cameras vary significantly in terms of illumination and intensity. Although recent unsupervised domain adaptation (UDA) methods enhance the models' generalization ability on the unlabeled target fundus datasets, they always require sufficient labeled data from the source domain, bringing auxiliary data acquisition and annotation costs. To further facilitate the data efficiency of the cross-domain segmentation methods on the fundus images, we explore UDA optic disc and cup segmentation problems using few labeled source data in this work. We first design a Searching-based Multi-style Invariant Mechanism to diversify the source data style as well as increase the data amount. Next, a prototype consistency mechanism on the foreground objects is proposed to facilitate the feature alignment for each kind of tissue under different image styles. Moreover, a cross-style self-supervised learning stage is further designed to improve the segmentation performance on the target images. Our method has outperformed several state-of-theart UDA segmentation methods under the UDA fundus segmentation with few labeled source data.
10.48550/arxiv.2210.04379
[ "https://export.arxiv.org/pdf/2210.04379v1.pdf" ]
252,781,157
2210.04379
de3bc68d712eb59c2773af1275a869c6a1210e55
Unsupervised Domain Adaptive Fundus Image Segmentation with Few Labeled Source Data Qianbi Yu School of Computer Science University of Sydney Sydney Australia Dongnan Liu [email protected] School of Computer Science University of Sydney Sydney Australia Chaoyi Zhang School of Computer Science University of Sydney Sydney Australia Xinwen Zhang School of Computer Science University of Sydney Sydney Australia Weidong Cai School of Computer Science University of Sydney Sydney Australia Unsupervised Domain Adaptive Fundus Image Segmentation with Few Labeled Source Data YU ET AL.: UNSUPERVISED DOMAIN ADAPTATION WITH FEW SHOT DATA 1 Deep learning-based segmentation methods have been widely employed for automatic glaucoma diagnosis and prognosis. In practice, fundus images obtained by different fundus cameras vary significantly in terms of illumination and intensity. Although recent unsupervised domain adaptation (UDA) methods enhance the models' generalization ability on the unlabeled target fundus datasets, they always require sufficient labeled data from the source domain, bringing auxiliary data acquisition and annotation costs. To further facilitate the data efficiency of the cross-domain segmentation methods on the fundus images, we explore UDA optic disc and cup segmentation problems using few labeled source data in this work. We first design a Searching-based Multi-style Invariant Mechanism to diversify the source data style as well as increase the data amount. Next, a prototype consistency mechanism on the foreground objects is proposed to facilitate the feature alignment for each kind of tissue under different image styles. Moreover, a cross-style self-supervised learning stage is further designed to improve the segmentation performance on the target images. Our method has outperformed several state-of-theart UDA segmentation methods under the UDA fundus segmentation with few labeled source data. Introduction Glaucoma is a chronic eye condition that causes progressive damage to the optic nerve and eventually leads to blindness if left untreated [1]. In clinical practice, accurate examination of the head of the optic nerve i.e. cup-to-disc ratio is crucial for early detection and treatment of glaucoma diagnosis [27]. Recently, deep learning-based models have been widely used for Figure 1: Overview of the proposed framework for cross-domain optic disc and cup segmentation with few labeled data. The first stage model is trained using target and synthesis images with additional discriminative and consistency loss. The customized consistency loss utilizes the mask pooling technique and computes the cosine similarity of the class-prototype vectors. automatic optic cup and disc segmentation in fundus images [7,31] and achieved appealing performance. Nevertheless, these methods will suffer from performance drop when validated on new datasets with unseen distributions due to the domain shift issue [9,29]. To this end, unsupervised domain adaptation (UDA) methods have been proposed to enhance the models' generalization ability, by transferring the knowledge from the labeled source data to the unlabeled target data [11,15,18,22,23,25]. Recently, several methods have been proposed to further facilitate the data-efficiency of the UDA medical image segmentation [4,19,44]. [19] proposed to train the UDA model with few target images, and [4] explored a source-free UDA setting without sharing the raw source data with the target domain. However, they still require the model to be optimized with sufficient labeled source images, which might be challenging to fulfill in practical applications. Acquiring the pixel-level annotations for the optic discs and cups is time-consuming and error-prone for automatic fundus segmentation datasets [35]. On the other side, directly training the UDA models with insufficient labeled source data can easily cause over-fitting problems and limit the segmentation performance. To further reduce the amount of data required and maintain high-performance cross-domain segmentation, we explore the UDA fundus image segmentation problem with few labeled source data. Although [44] also explores a similar problem as ours via a teacher-student framework, they ignore the category relationship of the foreground tissues. Given the extremely insufficient supervision (e.g., given no more than 10% of the labeled training data), the model would suffer from misalignment due to the lack of semantic-level information learning. In this work, a novel framework is proposed to transfer the knowledge from the extremely limited labeled source data to the unlabeled target data for cross-domain optic disc and cup segmentation. First, we propose to diversify the domain knowledge by a Searchingbased Multi-Style Invariant (SMSI) mechanism, enriching the image distributions by creat-ing transformed styles based on the synthesized images through searching strategies. Secondly, considering the similarity of the foreground content within the same category under various styles, a new Class-Prototype Consistency (CPC) mechanism is also introduced. Moreover, we design a Cross-Style Self-supervised Learning (CSSL) strategy with pseudo labels to further boost the overall segmentation of the unlabeled target images. Our proposed method is validated on two cross-domain optic disc and cup segmentation experiments with limited labeled source data available. By outperforming other state-of-the-art UDA segmentation methods, our proposed framework is demonstrated to be more effective and can conquer the domain shift under low resource situations, which is, therefore, more practical and important in real-world applications. Related Work Unsupervised domain adaptation Performing pixel-level domain mapping using image-toimage translation is a typical solution to reduce the domain gap at the appearance level [14,46]. In addition, feature-level adaptation can also alleviate the cross-domain discrepancy by inducing domain-invariant features learning [10,22,33,34,42]. Various methods have involved generative adversarial training, but non-GAN-based techniques have also achieved competitive results, especially those via frequency space learning [13,39,41]. These methods mostly conduct frequency alignment or frequency modification to achieve image stylization. By introducing little extra computations to the framework, the frequency learning-based methods can achieve style transformation in a more efficient manner than the GAN-based ones. However, current frequency space methods heavily rely on non-learnable parameter selection, such as parameter β in [39] and parameter p in [13]. To avoid massive experiments for selecting the appropriate parameters for the improved synthesized images, we design a SMSI module based on AutoML techniques, which have been widely investigated for efficient medical image analysis [30,37,38]. Methodology Searching-based Multi-Style Invariant Mechanism (SMSI) To alleviate the domain gap at the appearance level as well as enlarge the data-scarce source domain, we propose a Searching-based Multi-Style Invariant Mechanism (SMSI) for the source domain based on Fourier transform [39]. Specifically, each channel of an input image x is firstly transformed into the frequency space F(x) via: F(x) = ∑ h,w x(h, w)e − j2π( h H m+ w W n) , where j 2 = −1. Next, this frequency signal can be further decomposed into an amplitude spectrum F A and a phase spectrum F P , which respectively represent the low-level (e.g., appearance) and high-level (e.g., content) characteristics of each image [16,39]. By obtaining the amplitude F A s and phase F P s from each source image x s , its corresponding synthesis image in the target-like style can be generated via: X s→t = F −1 [(M β F A t + (1 − M β )F A s , F P s ],(1) where the F −1 is the inverse Fourier transform. To ensure that each synthesis image contains comprehensive appearance-level information under the target distributions, we propose to replace the F A s with the average amplitude spectrum from all the target images, denote as Figure 2: K-folds automatic Fourier transform searching algorithm. Source dataset X s and target dataset X t are split in to k-folds, Fourier transform with different proportion parameter β applied to the source dataset, segmentation network trained with target dataset is used to evaluate the transformed data X s→t . The parameters selected from each fold are appended to the final Fourier transform policy list F * . F A t . M β is used to control the proportion of the target amplitude during synthesis and controlled by a parameter β ∈ (0, 1) , defined as M β = I (h,w)∈[−β H:β H,−βW :βW ] . As indicated in previous works [39], different M β choices can induce distinct domain adaption performance. However, it is cost-intensive to conduct massive experiments for selecting the appropriate parameters for each specific application scenario. To tackle this issue, we propose an efficient searching strategy to find the optimal parameters for the synthesis images which can achieve better cross-domain segmentation performance. Specifically, the search space is first defined as F(X s ; β ). Given the above search space, the search processes are formulated as: (i) Train a plain segmentation model with original source data X s . (ii) Initialized by the model from step (i), several models are further optimized on K groups of synthesis images according to Equation 1. In each group, the β for M β is randomly initialized within (0, 1). (iii) Let each model in step (ii) learn the ideal β by searching controllers and appending to the final policy set following [5,20]. As the Fourier transformation is only changing the style of each image, instead of its content, the segmentation ground truth of the synthesis images is similar to the original one [39]. Therefore, the objective function of the policy search is designed to maximize the validation dice on transformed data X s→t with original source label: F * = argmax D(θ G |F(X s )),(2) where θ G is the parameter of the segmentation network used to optimize L seg and D is the validation dice. The search controller can be implemented efficiently using the Treestructured Parzen Estimators algorithm in [2]. Figure 2 indicates the detailed process of the SMSI mechanism. After optimal parameters are determined, each source image can generate n × k synthesis image by varying the n number of β parameters that control the likelihood of s → t and taking k average amplitude values from different parts of the target images. This procedure generally expands the source domain dataset and provides a useful regularization technique to increase the diversity of the dataset. Although training the models with the synthesis images can alleviate the domain gap at the appearance levels, it can still incur domain shifts at the feature level [12,23]. As such, we introduce feature invariant induction learning based on the searching-based multi-style synthesis process. Specifically, additional adversarial domain discriminators are utilized to generate domain-invariant features for the synthesis images and target images on top of the traditional supervised loss. Denote that the source dataset is X s ⊂ R H×W ×3 with ground truth C-class segmentation maps Y s ⊂ (1,C) H×W , synthesis source dataset is X s→t with the same ground truth maps. The target dataset is X t with no ground truth label. A discriminator θ D is trained adversarially to distinguish between the synthesis source set and target set with discrimination loss L D . Simultaneously, the segmentation network is trained to fool the discriminator as: min θ D 1 |X s→t | ∑ x s→t L D (I x s→t , 1) + 1 |X t | ∑ x t L D (I x t , 0) min θ G 1 |X t | ∑ x t L D (I x t , 1)(3) where I x s→t and I x t are the weighted self-information maps following ADVENT [34]. Summarily, there are two discriminators D 1 and D 2 implemented to distinguish (i) X s→t and X t (ii) X s and X t as indicated in the top part of Figure 1. Class-Prototype Consistency Mechanism (CPC) The synthesis images produced from the SMSI mechanism and their corresponding source images should have the same image content but in different styles. Motivated by previous works that the class-aware features under the same category should maintain the same across different domains [45], we propose a class-prototype consistency mechanism for the synthesis images. The class prototype is created by using the high-level feature maps from the model's encoder and the ground truth source masks. The source masks are first resized and converted into binary masks for each class. Then, they are multiplied by the feature maps extracted from synthesis images and source images respectively, generating class-relevant masked feature maps. Taking a global average pooling further converts the feature maps into class prototype vectors. Global average pooling has the ability to sum out the spatial data and enforce the correspondences between feature maps and classes. Denote the mask of class c as M c and f s , f s→t as the feature maps, the class prototypes of c class are defined as: p c s = 1 N h×w ∑ c M c f s p c s→t = 1 N h×w ∑ c M c f s→t(4) where h and w are the height and width of the feature maps. This masked pooling technique enables the network to focus on the target content of images instead of intensity and illumination variation. To narrow the gap between the features under the same class in different synthesis domains, we propose to enlarge the similarity between them. The overall CPC Mechanism is illustrated in the bottom part of Figure 1, where binary masks for optic cups and discs are used. Specifically, the overall consistency loss function can be defined as: L con = ∑ c=(0,1) 1 − p c s · p c s→t max(||p c s || 2 · ||p c s→t || 2 , ε) ,(5) where ε is the small value to avoid division by zero, p c s and p c s→t are the prototypes of the class c for the features from the FFT synthesized images and source images. The overall optimization function for the segmentation network with SMSI and CPC mechanism is defined as: Figure 3: Demonstration of the pseudo label self-supervised learning process. Specifically, a second stage model θ 2 G is initialized by θ 1 G and trained over the target data X t and the FFT target data X t→s with pseudo labels. L total = L seg (X s ,Y s ) + L seg (X s→t ,Y s ) + λ (L D 1 + L D 2 ) + L con .(6) Cross-Style Self-supervised Learning (CSSL) When dealing with model adaptation towards the target domain, a consummate resource is the target ground truth masks, which are not available in UDA settings. As compensation, highly-confident pseudo labels can be created for unlabeled target images by using prediction probabilities P v from the trained model θ 1 G on the v-th pixel. The pseudo labels can be defined asŷ v t = I[P v ≥ γ], where I is the indicator function and γ ∈ (0, 1) is the probability threshold to determine the binary mask. However, solely training the model with the target pseudo labels brings noise to the optimization process due to the gap between the pseudo and real labels. To stabilize the training process, we propose a cross-style self-supervised learning strategy, to jointly re-train the model using the target images X t and the Fourier transformed target images with source-like styles X t→s and their pseudo labelsŷ v t , andŷ v t→s , respectively. The source-like synthesized images are obtained following the process in Section 2.1. Since the segmentation learning for the model in the first stage is based on the annotated source data, the pseudo labels for the source-like synthesis images contain less noise and therefore can be a complement to the target supervised loss. Figure 3 demonstrates these two supervised segmentation losses. In general, the model learns combined distribution and gets further improvement using self-supervised training in the second stage. Experiments Datasets and Implementation Details The experiments aim to segment the cup and disc components in the multi-center fundus images, which are obtained from various patients using different eye examination equipment. The source dataset REFUGE [28] contains 400 annotated images. There are two target datasets, RIM ONE-r3 [8] contains 99 training images and 60 testing images, and Drishti-GS [32] contains 50 training images and 51 testing images. All datasets used are publicly available. In our experiments with few labeled source data, only 10 random source images are accessible during model training. For the image synthesis process in the first stage, n = 3 and k = 5 are selected for each source image, with a total of 150 FFT source images generated. In the second stage, n = 3 and k = 1 are selected for each target image, 150 and 297 FFT target images are generated respectively. The experiment results under other selections for the source images as well as more detailed experimental and implementation settings are available in the supplementary material. The network used in our experiments is a MobileNetv2 with a DeepLabv3+ backbone based on the structure in [35]. A semantic segmentation module called Atrous Spatial Pyramid Pooling (ASPP) in DeepLab re-samples a given feature layer at various rates before convolution. The overall model size is 7.62M and the inference time is 30.57s for one image. In the first stage, the segmentation model is trained with Adam optimizer under a 1e-3 learning rate, the discriminators have been trained with SGD optimizer with a 2.5e-5 learning rate, 8 batch size, and 200 training epochs. The weighting factor λ in Equation 6 is set as 0.5. In the second stage, the segmentation model is trained with Adam optimizer with a 2e-3 learning rate, 8 batch size, and 20 training epochs. The probability threshold γ = 0.75 is used to generate the pseudo labels. Segmentation results are evaluated by the Dice coefficient and Average Surface Distance (ASD). The framework is implemented on Pytorch 1.7.1 using a NVIDIA RTX3090 GPU. Comparison Experiments The proposed method is compared to state-of-the-art (SOTA) unsupervised domain adaptation methods, as well as two recent UDA approaches particularly for few labeled source images. In supplementary material, extra SOTA fundus image segmentation methods [6,21,24,36,43] are also compared. CyCADA [12] translates the source images into the target style using cycle-consistent adversarial networks and trains the adversarial network with the translated images. AdvEnt [34] brings in entropy loss and adversarial loss respectively to address the domain shift problem. FDA [39] adopts frequency swap method for image stylization and evaluate the segmentation model with multi-band transfer. PixMatch [26] develops a new component to ensure that the model's predictions on a target image and a perturbed version of the same image are pixel-wise consistent. LTIR [17] learns texture invariant features from different domains using Style-Wrap to change the images' appearance. Consider from another perspective, the two recently-developed methods MT [44] and PCS [40] have a similar experimental setting, they both focus on domain adaptation with few source data. MT follows the mean teacher paradigm and adopts dual teacher models to provide both semantic and structural knowledge to the student model, whereas PCS performs in-domain and cross-domain learning using prototypes from feature memory banks. Some other latest fundus image segmentation baselines are also evaluated. BEAL [35] suggests boundary prediction and entropy-driven during adversarial training and achieves excellent results for cross-domain prediction. DPL [4] is a novel proposal for source-free domain adaptation in the field of fundus image segmentation, with a pseudo-label denoising technique. It utilizes a pre-trained source model to generate pseudo-labels. For comparison, we follow the same experimental settings in these segmentation models, i.e., only 10 randomly selected source images will be accessible throughout the experiment, even for the pre-trained model. Quantitative analysis. As indicated in Table 1, the segmentation performance of all other comparison UDA approaches is at the same level. This indicates their adaptation abilities are limited due to the lack of sufficient supervision learning. For the MT [44] and PCS [40] which were originally designed for UDA with few labeled source data, we notice that their performance is suboptimal. For MT, the lack of consideration of the cross-domain category information makes the model learn insufficient semantic-level knowledge given the extremely limited labeled source data for segmentation supervision learning, which further incurs inferior performance on the target testing data. Although PCS proposes a class-aware UDA framework, it was particularly designed for UDA classification under the small domain gap. When validated on the UDA fundus image segmentation with a large domain bias, its segmentation results are limited by ignoring the appearance-level domain bias and the particular designs for segmentation. On the other hand, our method can tackle the aforementioned challenges by the SMSI for appearance-level adaption, CPC for cross-domain category-aware information processing, and the CSSL for further performance gain without auxiliary annotations. Overall, our method has outperformed others, achieving 6.22% Dice, 10.71 ASD pixel for RIM-ONE-r3 and 4.44% Dice, 5.92 ASD pixel for Drishti-Gs. We have also conducted a two-tailed paired t-test on the comparison studies, and given the p-value smaller than 0.01, our improvements are statistically significant. Qualitative analysis. As presented in Figure 4, the segmentation results of some experiments show that focusing on content rather than appearance enables the network to better distinguish target objects from irrelevant backgrounds. The segmentation predictions from several comparison methods are significantly distracted by the background noise. Additionally, regardless of domain differences, the network faces difficulties when attempting to determine the spatial prior knowledge of the optic disc and optic cup. Our predictions alleviate these issues, have a much clear boundary between the cup and disc, and exhibit much fewer background segmentation error. Ablation Studies Ablation studies are conducted to evaluate the effectiveness of our proposed modules. In Table 2, the source-only and target-only experiments provide lower and upper bounds of this setting. The source-only experiment trains the segmentation network using only source images and directly adapts to the target domain. By contrast, a target-only experiment trains the network using annotated target data under a supervised learning setting. The first implemented module is standard adversarial training with additional discriminator loss. The improvement under two metrics suggests that the concept of adversarial training can be drawn on this task. Then SMSI further boosts the performance by diversifying the source styles and inducing the domain invariant feature generation, providing a large quantity of labeled data for few-shot learning. In addition, a novel class prototype consistency loss allows the network to particularly align the features at the category level. Both Dice and ASD metrics indicate that the proposed method significantly increases the adaptation ability of the segmentation network with limited labeled data. Settings Dice On top of these modules at the first stage, our proposed Cross-Style Self-supervised Learning (CSSL) module brings an improvement of about 14% in dice value and 18 pixels in ASD over the non-adaptation model. We also conduct ablation experiments by conducting the self-supervised learning only on the target images, which introduces less performance gain than the CSSL. This further demonstrates the claim in Section 3.3 that our CSSL can alleviate the noises from the pseudo labels and lead to a better self-supervised segmentation performance. By jointly conducting our proposed strategies, the segmentation performance of the source-only model can be improved to a level similar to that of the fully supervised model. In addition, we also explore the models' effectiveness under different thresholds γ for the pseudo label learning stage introduced in Section 3.3. As shown in Figure 5, the best segmentation performance under the Dice and ASD metrics is obtained under both settings when the threshold γ is 0.75. Conclusion In this work, we propose a novel framework for domain adaptive optic disc and cup segmentation given only a few labeled source data. To alleviate the domain bias issue under the data-scarce setting, the SMSI, CCP, and CSSL modules are designed. In comparison to alternative domain adaptation strategies and even fully supervised networks, the model has been trained to reach competitive outcomes. In this work, we notice the principle bias between the two domains results from the different image styles due to the device variation, and there are no severe distinctions between the morphological structures for the foreground objects in the two domains. As such, future studies are suggested on the cross-domain segmentation problems with larger distinctions in the labeling space. Table 3: The selection of the optimal β parameters for Fourier transform. Referring to the searching-based multi-style invariant mechanism (SMSI), the transformation of REFUGE to the other two target domains is required in the first stage as X s→t , transformation of two target domains to REFUGE is required in the second stage as X t→s . Table 3 for each dataset, three optimal β values are selected. Table 4: Experimental results of our proposed method in terms of Dice and ASD metrics on RIM-ONE-r3 and Drishti-GS target datasets with randomly selected 10, 20, and 30 shots source data. Supplementary Material Methods Cup Dice Disc Dice Average Dice RIM-ONE-r3 CADA [24] 64.04 76.64 70.34 TAU [43] 54.20 78.60 66.40 ECSD-Net [21] 80 Table 5: Comparing our method to other fundus segmentation methods. the results of other methods are obtained by using 400 fully labeled source domain images, whereas our method only uses 40 source images. The training and testing images in the target domain used by all comparison methods are the same. The results for other methods are directly referenced from the articles. *Our method can achieve comparable and even better UDA segmentation performance only using 10% labeled source data, which indicates the effectiveness of our method, as well as maintaining the data efficiency. Table 6: Experimental results of different domain adaptation approaches in terms of Dice and ASD metrics on RIM-ONE-r3 and Drishti-GS target datasets with 40 (10%) labeled source data. Models' training time and models' sizes are also shown. Figure 4 : 4Segmentation results from some of the comparison experiments. Figure 5 : 5UDA Segmentation performance under different selections of the threshold γ for the pseudo label learning stage. RIM and GS indicate our experiment settings of using the RIM-ONE-r3 and Drishti-GS as the target domain, respectively. Figure 6 : 6Exhibition of synthesis images based on SMSI. Referring to Figure 7 : 7Exhibition of synthesis images based on original FDA and SMSI. For original FDA, β = 0.01/0.05/0.09 values are selected. Table 1 : 1Experimental results for comparison of different domain adaptation approaches in terms of Dice and ASD metrics on RIM-ONE-r3 and Drishti-GS target datasets. The numbers shown in the table are average values, taken from three sets of experiments under different groups of randomly selected 10 labeled source images. The numbers in parenthesis are the SD values. Table 2 : 2Ablation results with implemented domain adaptation modules in terms of Dice and ASD metrics on RIM-ONE-r3 and Drishti-GS target datasets. MethodsDice Metric [%] ASD Metric [pixel] Training Time [s/iter]Model Size [M] Cup Disc Average Cup Disc Average RIM-ONE-r3 CyCADA 66.61 76.99 71.80 47.35 41.62 44.48 12.14 31.04 ADVENT 67.99 80.67 74.33 42.04 33.43 37.74 12.19 42.61 PixMatch 70.50 75.20 72.85 16.33 35.90 26.12 13.26 42.61 LTIR 69.28 79.82 74.55 15.52 27.10 21.31 11.04 28.91 MT 70.04 82.66 76.35 13.23 20.54 16.88 8.23 31.04 PCS 65.71 78.00 71.86 18.04 26.09 22.06 14.13 59.34 Ours 83.47 87.85 85.66 7.33 11.33 8.64 79.91 7.62 Drishti-GS CyCADA 81.83 91.54 86.68 12.55 12.32 12.43 7.71 31.04 ADVENT 81.82 92.32 87.07 12.43 10.59 15.25 7.33 42.61 PixMatch 75.31 93.13 84.22 16.91 8.34 12.63 8.71 42.61 LTIR 76.72 94.17 85.44 15.82 7.20 11.51 5.94 42.61 MT 75.33 91.62 83.48 16.53 9.79 13.16 8.29 31.04 PCS 78.67 89.63 84.15 17.09 13.64 15.36 8.26 59.34 Ours 86.68 96.17 91.43 8.85 4.35 6.60 86.43 7.62 © 2022. The copyright of this document resides with its authors. It may be distributed unchanged freely in print or electronic forms. Optic disc and optic cup segmentation methodologies for glaucoma image detection: a survey. Ahmed Almazroa, Ritambhar Burman, Kaamran Raahemifar, Vasudevan Lakshminarayanan, Journal of ophthalmology. Ahmed Almazroa, Ritambhar Burman, Kaamran Raahemifar, and Vasudevan Lakshmi- narayanan. Optic disc and optic cup segmentation methodologies for glaucoma image detection: a survey. Journal of ophthalmology, 2015. Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures. James Bergstra, Daniel Yamins, David Cox, Proceedings of the 30th International Conference on Machine Learning. the 30th International Conference on Machine LearningJames Bergstra, Daniel Yamins, and David Cox. Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures. In Proceedings of the 30th International Conference on Machine Learning, 2013. Source-free domain adaptive fundus image segmentation with denoised pseudo-labeling. Cheng Chen, Quande Liu, Yueming Jin, Qi Dou, Pheng-Ann Heng, International Conference on Medical Image Computing and Computer-Assisted Intervention. Cheng Chen, Quande Liu, Yueming Jin, Qi Dou, and Pheng-Ann Heng. Source-free domain adaptive fundus image segmentation with denoised pseudo-labeling. In Inter- national Conference on Medical Image Computing and Computer-Assisted Interven- tion, 2021. Source-free domain adaptive fundus image segmentation with denoised pseudo-labeling. Cheng Chen, Quande Liu, Yueming Jin, Qi Dou, Pheng-Ann Heng, International Conference on Medical Image Computing and Computer-Assisted Intervention. Cheng Chen, Quande Liu, Yueming Jin, Qi Dou, and Pheng-Ann Heng. Source-free domain adaptive fundus image segmentation with denoised pseudo-labeling. In Inter- national Conference on Medical Image Computing and Computer-Assisted Interven- tion, 2021. Autoaugment: Learning augmentation policies from data. Barret Ekin Dogus Cubuk, Quoc V Le, Conference on Computer Vision and Pattern Recognition. Dandelion Mane, Vijay Vasudevan,Ekin Dogus Cubuk, Barret Zoph, Dandelion Mane, Vijay Vasudevan, and Quoc V. Le. Autoaugment: Learning augmentation policies from data. In Conference on Computer Vision and Pattern Recognition, 2019. Unsupervised domain adaptive fundus image segmentation with category-level regularization. Wei Feng, Lin Wang, Lie Ju, Xin Zhao, Xin Wang, Xiaoyu Shi, Zongyuan Ge, International Conference on Medical Image Computing and Computer-Assisted Intervention. Wei Feng, Lin Wang, Lie Ju, Xin Zhao, Xin Wang, Xiaoyu Shi, and Zongyuan Ge. Unsupervised domain adaptive fundus image segmentation with category-level regu- larization. In International Conference on Medical Image Computing and Computer- Assisted Intervention, 2022. Joint optic disc and cup segmentation based on multi-label deep network and polar transformation. Huazhu Fu, Jun Cheng, Yanwu Xu, Damon Wing Kee Wong, Jiang Liu, Xiaochun Cao, IEEE transactions on medical imaging. 377Huazhu Fu, Jun Cheng, Yanwu Xu, Damon Wing Kee Wong, Jiang Liu, and Xiaochun Cao. Joint optic disc and cup segmentation based on multi-label deep network and polar transformation. IEEE transactions on medical imaging, 37(7):1597-1605, 2018. Interactive tool and database for optic disc and cup segmentation of stereo and monocular retinal fundus images. Francisco Fumero, Jose Sigut, Silvia Alayón, Marta González-Hernández, Marta González De La, Rosa , International Conference in Central Europe on Computer Graphics. Francisco Fumero, Jose Sigut, Silvia Alayón, Marta González-Hernández, and Marta González de la Rosa. Interactive tool and database for optic disc and cup segmentation of stereo and monocular retinal fundus images. International Conference in Central Europe on Computer Graphics, 2015. Unsupervised domain adaptation by backpropagation. Yaroslav Ganin, Victor Lempitsky, International conference on machine learning. Yaroslav Ganin and Victor Lempitsky. Unsupervised domain adaptation by backprop- agation. In International conference on machine learning, 2015. Domain-adversarial training of neural networks. The journal of machine learning research. Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, Victor Lempitsky, 17Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. Domain-adversarial train- ing of neural networks. The journal of machine learning research, 17(1):2096-2030, 2016. A novel domain adaptation framework for medical image segmentation. Amir Gholami, Shashank Subramanian, Varun Shenoy, Naveen Himthani, Xiangyu Yue, Sicheng Zhao, Peter Jin, George Biros, Kurt Keutzer, In International MICCAI Brainlesion Workshop. Amir Gholami, Shashank Subramanian, Varun Shenoy, Naveen Himthani, Xiangyu Yue, Sicheng Zhao, Peter Jin, George Biros, and Kurt Keutzer. A novel domain adapta- tion framework for medical image segmentation. In International MICCAI Brainlesion Workshop, 2018. Cycada: Cycle-consistent adversarial domain adaptation. Judy Hoffman, Eric Tzeng, Taesung Park, Jun-Yan Zhu, Phillip Isola, Kate Saenko, Alexei Efros, Trevor Darrell, International conference on machine learning. Judy Hoffman, Eric Tzeng, Taesung Park, Jun-Yan Zhu, Phillip Isola, Kate Saenko, Alexei Efros, and Trevor Darrell. Cycada: Cycle-consistent adversarial domain adap- tation. In International conference on machine learning, pages 1989-1998, 2018. Fsdr: Frequency space domain randomization for domain generalization. Jiaxing Huang, Dayan Guan, Aoran Xiao, Shijian Lu, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionJiaxing Huang, Dayan Guan, Aoran Xiao, and Shijian Lu. Fsdr: Frequency space domain randomization for domain generalization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6891-6902, 2021. Image-to-image translation with conditional adversarial networks. Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, Alexei A Efros, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionPhillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros. Image-to-image trans- lation with conditional adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1125-1134, 2017. Domain adaptation for biomedical image segmentation using adversarial training. Mehran Javanmardi, Tolga Tasdizen, IEEE 15th International Symposium on Biomedical Imaging (ISBI 2018). Mehran Javanmardi and Tolga Tasdizen. Domain adaptation for biomedical image segmentation using adversarial training. In 2018 IEEE 15th International Symposium on Biomedical Imaging (ISBI 2018), pages 554-558, 2018. Harmofl: Harmonizing local and global drifts in federated learning on heterogeneous medical images. Meirui Jiang, Zirui Wang, Qi Dou, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence36Meirui Jiang, Zirui Wang, and Qi Dou. Harmofl: Harmonizing local and global drifts in federated learning on heterogeneous medical images. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pages 1087-1095, 2022. Learning texture invariant representation for domain adaptation of semantic segmentation. Myeongjin Kim, Hyeran Byun, Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. the IEEE/CVF conference on computer vision and pattern recognitionMyeongjin Kim and Hyeran Byun. Learning texture invariant representation for do- main adaptation of semantic segmentation. In Proceedings of the IEEE/CVF confer- ence on computer vision and pattern recognition, pages 12975-12984, 2020. Domain adaptive nuclei instance segmentation and classification via category-aware feature alignment and pseudo-labelling. Canran Li, Dongnan Liu, Haoran Li, Zheng Zhang, Guangming Lu, Xiaojun Chang, Weidong Cai, International Conference on Medical Image Computing and Computer-Assisted Intervention. Canran Li, Dongnan Liu, Haoran Li, Zheng Zhang, Guangming Lu, Xiaojun Chang, and Weidong Cai. Domain adaptive nuclei instance segmentation and classification via category-aware feature alignment and pseudo-labelling. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 715-724, 2022. Few-shot domain adaptation with polymorphic transformers. Shaohua Li, Xiuchao Sui, Jie Fu, Huazhu Fu, Xiangde Luo, Yangqin Feng, Xinxing Xu, Yong Liu, Daniel Ting, Rick Siow Mong Goh, International Conference on Medical Image Computing and Computer-Assisted Intervention. Shaohua Li, Xiuchao Sui, Jie Fu, Huazhu Fu, Xiangde Luo, Yangqin Feng, Xinxing Xu, Yong Liu, Daniel Ting, and Rick Siow Mong Goh. Few-shot domain adaptation with polymorphic transformers. In International Conference on Medical Image Computing and Computer-Assisted Intervention, 2021. Fast autoaugment. Sungbin Lim, Ildoo Kim, Taesup Kim, Chiheon Kim, Sungwoong Kim, Advances in Neural Information Processing Systems (NeurIPS). Sungbin Lim, Ildoo Kim, Taesup Kim, Chiheon Kim, and Sungwoong Kim. Fast au- toaugment. In Advances in Neural Information Processing Systems (NeurIPS), 2019. Ecsd-net: A joint optic disc and cup segmentation and glaucoma classification network based on unsupervised domain adaptation. Bingyan Liu, Daru Pan, Zhenbin Shuai, Hui Song, Computer Methods and Programs in Biomedicine. 213106530Bingyan Liu, Daru Pan, Zhenbin Shuai, and Hui Song. Ecsd-net: A joint optic disc and cup segmentation and glaucoma classification network based on unsupervised domain adaptation. Computer Methods and Programs in Biomedicine, 213:106530, 2022. Unsupervised instance segmentation in microscopy images via panoptic domain adaptation and task re-weighting. Dongnan Liu, Donghao Zhang, Yang Song, Fan Zhang, O&apos; Lauren, Heng Donnell, Mei Huang, Weidong Chen, Cai, Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. the IEEE/CVF conference on computer vision and pattern recognitionDongnan Liu, Donghao Zhang, Yang Song, Fan Zhang, Lauren O'Donnell, Heng Huang, Mei Chen, and Weidong Cai. Unsupervised instance segmentation in mi- croscopy images via panoptic domain adaptation and task re-weighting. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 4243- 4252, 2020. Pdam: A panoptic-level feature alignment framework for unsupervised domain adaptive instance segmentation in microscopy images. Dongnan Liu, Donghao Zhang, Yang Song, Fan Zhang, O&apos; Lauren, Heng Donnell, Mei Huang, Weidong Chen, Cai, IEEE Transactions on Medical Imaging. 401Dongnan Liu, Donghao Zhang, Yang Song, Fan Zhang, Lauren O'Donnell, Heng Huang, Mei Chen, and Weidong Cai. Pdam: A panoptic-level feature alignment frame- work for unsupervised domain adaptive instance segmentation in microscopy images. IEEE Transactions on Medical Imaging, 40(1):154-165, 2020. Cada: Multi-scale collaborative adversarial domain adaptation for unsupervised optic disc and cup segmentation. Peng Liu, Charlie T Tran, Bin Kong, Ruogu Fang, Neurocomputing. 469Peng Liu, Charlie T Tran, Bin Kong, and Ruogu Fang. Cada: Multi-scale collabora- tive adversarial domain adaptation for unsupervised optic disc and cup segmentation. Neurocomputing, 469:209-220, 2022. Semi-supervised learning with generative adversarial networks for chest x-ray classification with ability of data domain adaptation. Ali Madani, Mehdi Moradi, Alexandros Karargyris, Tanveer Syeda-Mahmood, IEEE 15th International symposium on biomedical imaging. Ali Madani, Mehdi Moradi, Alexandros Karargyris, and Tanveer Syeda-Mahmood. Semi-supervised learning with generative adversarial networks for chest x-ray classifi- cation with ability of data domain adaptation. In IEEE 15th International symposium on biomedical imaging, pages 1038-1042, 2018. Pixmatch: Unsupervised domain adaptation via pixelwise consistency training. Luke Melas, -Kyriazi , Arjun K Manrai, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionLuke Melas-Kyriazi and Arjun K Manrai. Pixmatch: Unsupervised domain adaptation via pixelwise consistency training. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12435-12445, 2021. Bi-gcn: boundary-aware input-dependent graph convolution network for biomedical image segmentation. Yanda Meng, Hongrun Zhang, Dongxu Gao, Yitian Zhao, Xiaoyun Yang, Xuesheng Qian, Xiaowei Huang, Yalin Zheng, British Machine Vision Conference. Yanda Meng, Hongrun Zhang, Dongxu Gao, Yitian Zhao, Xiaoyun Yang, Xuesheng Qian, Xiaowei Huang, and Yalin Zheng. Bi-gcn: boundary-aware input-dependent graph convolution network for biomedical image segmentation. British Machine Vision Conference, 2021. Refuge challenge: A unified framework for evaluating automated methods for glaucoma assessment from fundus photographs. José Ignacio Orlando, Huazhu Fu, João Barbosa Breda, Karel Van Keer, R Deepti, Andrés Bathula, Ruogu Diaz-Pinto, Pheng-Ann Fang, Jeyoung Heng, Joonho Kim, Lee, Medical image analysis. 59101570José Ignacio Orlando, Huazhu Fu, João Barbosa Breda, Karel van Keer, Deepti R Bathula, Andrés Diaz-Pinto, Ruogu Fang, Pheng-Ann Heng, Jeyoung Kim, JoonHo Lee, et al. Refuge challenge: A unified framework for evaluating automated meth- ods for glaucoma assessment from fundus photographs. Medical image analysis, 59: 101570, 2020. Visual domain adaptation: A survey of recent advances. M Vishal, Raghuraman Patel, Ruonan Gopalan, Rama Li, Chellappa, IEEE signal processing magazine. 323Vishal M Patel, Raghuraman Gopalan, Ruonan Li, and Rama Chellappa. Visual domain adaptation: A survey of recent advances. IEEE signal processing magazine, 32(3):53- 69, 2015. Hypersegnas: Bridging one-shot neural architecture search with 3d medical image segmentation using hypernet. Cheng Peng, Andriy Myronenko, Ali Hatamizadeh, Vishwesh Nath, Md Mahfuzur Rahman Siddiquee, Yufan He, Daguang Xu, Rama Chellappa, Dong Yang, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionCheng Peng, Andriy Myronenko, Ali Hatamizadeh, Vishwesh Nath, Md Mah- fuzur Rahman Siddiquee, Yufan He, Daguang Xu, Rama Chellappa, and Dong Yang. Hypersegnas: Bridging one-shot neural architecture search with 3d medical image seg- mentation using hypernet. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 20741-20751, 2022. Optic disc and cup segmentation methods for glaucoma detection with modification of u-net convolutional neural network. Artem Sevastopolsky, Pattern Recognition and Image Analysis. 273Artem Sevastopolsky. Optic disc and cup segmentation methods for glaucoma detec- tion with modification of u-net convolutional neural network. Pattern Recognition and Image Analysis, 27(3):618-624, 2017. Drishti-gs: Retinal image dataset for optic nerve head (onh) segmentation. Jayanthi Sivaswamy, Datt Sr Krishnadas, Madhulika Joshi, A Ujjwaft Syed Jain, Tabish, IEEE 11th international symposium on biomedical imaging (ISBI). Jayanthi Sivaswamy, SR Krishnadas, Gopal Datt Joshi, Madhulika Jain, and A Ujjwaft Syed Tabish. Drishti-gs: Retinal image dataset for optic nerve head (onh) segmentation. In 2014 IEEE 11th international symposium on biomedical imaging (ISBI), pages 53-56, 2014. Adversarial discriminative domain adaptation. Eric Tzeng, Judy Hoffman, Kate Saenko, Trevor Darrell, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionEric Tzeng, Judy Hoffman, Kate Saenko, and Trevor Darrell. Adversarial discrimina- tive domain adaptation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7167-7176, 2017. Advent: Adversarial entropy minimization for domain adaptation in semantic segmentation. Tuan-Hung Vu, Himalaya Jain, Maxime Bucher, Matthieu Cord, Patrick Pérez, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionTuan-Hung Vu, Himalaya Jain, Maxime Bucher, Matthieu Cord, and Patrick Pérez. Advent: Adversarial entropy minimization for domain adaptation in semantic segmen- tation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2517-2526, 2019. Boundary and entropy-driven adversarial learning for fundus image segmentation. Shujun Wang, Lequan Yu, Kang Li, Xin Yang, Chi-Wing Fu, Pheng-Ann Heng, International Conference on Medical Image Computing and Computer-Assisted Intervention. Shujun Wang, Lequan Yu, Kang Li, Xin Yang, Chi-Wing Fu, and Pheng-Ann Heng. Boundary and entropy-driven adversarial learning for fundus image segmentation. In International Conference on Medical Image Computing and Computer-Assisted Inter- vention, pages 102-110, 2019. Patchbased output space adversarial learning for joint optic disc and cup segmentation. Shujun Wang, Lequan Yu, Xin Yang, Chi-Wing Fu, Pheng-Ann Heng, IEEE transactions on medical imaging. 3811Shujun Wang, Lequan Yu, Xin Yang, Chi-Wing Fu, and Pheng-Ann Heng. Patch- based output space adversarial learning for joint optic disc and cup segmentation. IEEE transactions on medical imaging, 38(11):2485-2495, 2019. Bix-nas: Searching efficient bi-directional architecture for medical image segmentation. Xinyi Wang, Tiange Xiang, Chaoyi Zhang, Yang Song, Dongnan Liu, Heng Huang, Weidong Cai, International Conference on Medical Image Computing and Computer-Assisted Intervention. Xinyi Wang, Tiange Xiang, Chaoyi Zhang, Yang Song, Dongnan Liu, Heng Huang, and Weidong Cai. Bix-nas: Searching efficient bi-directional architecture for medical image segmentation. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 229-238, 2021. Towards bi-directional skip connections in encoder-decoder architectures and beyond. Tiange Xiang, Chaoyi Zhang, Xinyi Wang, Yang Song, Dongnan Liu, Heng Huang, Weidong Cai, Medical Image Analysis. 78102420Tiange Xiang, Chaoyi Zhang, Xinyi Wang, Yang Song, Dongnan Liu, Heng Huang, and Weidong Cai. Towards bi-directional skip connections in encoder-decoder architectures and beyond. Medical Image Analysis, 78:102420, 2022. Fda: Fourier domain adaptation for semantic segmentation. Yanchao Yang, Stefano Soatto, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionYanchao Yang and Stefano Soatto. Fda: Fourier domain adaptation for semantic seg- mentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pat- tern Recognition, pages 4085-4095, 2020. Prototypical cross-domain selfsupervised learning for few-shot unsupervised domain adaptation. Xiangyu Yue, Zangwei Zheng, Shanghang Zhang, Yang Gao, Trevor Darrell, Kurt Keutzer, Alberto Sangiovanni Vincentelli, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionXiangyu Yue, Zangwei Zheng, Shanghang Zhang, Yang Gao, Trevor Darrell, Kurt Keutzer, and Alberto Sangiovanni Vincentelli. Prototypical cross-domain self- supervised learning for few-shot unsupervised domain adaptation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 13834- 13844, 2021. Spectral unsupervised domain adaptation for visual recognition. Jingyi Zhang, Jiaxing Huang, Zichen Tian, Shijian Lu, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionJingyi Zhang, Jiaxing Huang, Zichen Tian, and Shijian Lu. Spectral unsupervised domain adaptation for visual recognition. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9829-9840, 2022. Deep least squares alignment for unsupervised domain adaptation. Youshan Zhang, Brian D Davison, British Machine Vision Conference. Youshan Zhang and Brian D Davison. Deep least squares alignment for unsupervised domain adaptation. British Machine Vision Conference, 2021. Tau: Transferable attention u-net for optic disc and cup segmentation. Knowledge-Based Systems. Yuhao Zhang, Xiangrui Cai, Ying Zhang, Hong Kang, Xin Ji, Xiaojie Yuan, 213106668Yuhao Zhang, Xiangrui Cai, Ying Zhang, Hong Kang, Xin Ji, and Xiaojie Yuan. Tau: Transferable attention u-net for optic disc and cup segmentation. Knowledge-Based Systems, 213:106668, 2021. Mt-uda: Towards unsupervised cross-modality medical image segmentation with limited source labels. Ziyuan Zhao, Kaixin Xu, Shumeng Li, Zeng Zeng, Cuntai Guan, International Conference on Medical Image Computing and Computer-Assisted Intervention. Ziyuan Zhao, Kaixin Xu, Shumeng Li, Zeng Zeng, and Cuntai Guan. Mt-uda: Towards unsupervised cross-modality medical image segmentation with limited source labels. In International Conference on Medical Image Computing and Computer-Assisted In- tervention, pages 293-303, 2021. Cross-domain object detection through coarse-to-fine feature adaptation. Yangtao Zheng, Di Huang, Songtao Liu, Yunhong Wang, Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. the IEEE/CVF conference on computer vision and pattern recognitionYangtao Zheng, Di Huang, Songtao Liu, and Yunhong Wang. Cross-domain object detection through coarse-to-fine feature adaptation. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 13766-13775, 2020. Unpaired image-toimage translation using cycle-consistent adversarial networks. Jun-Yan Zhu, Taesung Park, Phillip Isola, Alexei A Efros, Proceedings of the IEEE international conference on computer vision. the IEEE international conference on computer visionJun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to- image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pages 2223-2232, 2017.
[]